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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 HEARTH REMELTING by DAVID WILLIAM TRIPP A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE 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 this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial- gain shall not be allowed without my written permission. Department of M&T7*Z&> hjK>D ffT&tZ,, &t$> (sJ&ZtL/' The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6G/81) A b s t r a c t 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 used i n r e f i n i n g of s u p e r a l l o y s , t i t a n i u m a l l o y s and t h e r e c y c l i n g of these m a t e r i a l s . The removal of i m p u r i t i e s and exhogenous p a r t i c l e s d u r i n g the h e a r t h m e l t i n g o p e r a t i o n depends p r i m a r i l y on the time a t temperature r e l a t i o n s h i p d e v e l o p e d w i t h i n a p o o l of molten m e t a l . In the p a s t h e a r t h m e l t e r s have a c t e d l a r g e l y on e m p i r i c a l e v i d e n c e t o s p e c i f y such parameters as melt r a t e s , power l e v e l s and s k u l l s i z e s . 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 model which c o u l d be used t o p r e d i c t c e r t a i n parameters (such as p o o l volume or a l l o y element e v a p o r a t i o n r a t e s ) when g i v e n s k u l l geometry, power i n p u t and melt r a t e . A t h r e e d i m e n s i o n a l s t e a d y s t a t e heat t r a n s f e r model of bo t h the s k u l l and water c o o l e d copper mould d u r i n g e l e c t r o n beam h e a r t h r e m e l t i n g has been d e v e l o p e d . The model has been used t o i n v e s t i g a t e the e f f e c t s of s u r f a c e t e m p e r a t u r e , l i q u i d m o t i o n , power i n p u t , s k u l l geometry, presence of the h e a r t h mould and melt r a t e on parameters such as p o o l volume d u r i n g s k u l l m e l t i n g . In g e n e r a l the c h o i c e of any c o m b i n a t i o n of o p e r a t i n g parameters depends on a b a l a n c e between the r e f i n i n g c a p a c i t i y of the p r o c e s s ( i . e . l i q u i d volume) and the l o s s of a l l o y elements by e v a p o r a t i o n . In the case of i i m e l t i n g pure m a t e r i a l s (e.g. CP t i t a n i u m ) the b a l a n c e i s between r e f i n i n g c a p a c i t y and e f f i c i e n t energy use. I t was found t h a t f o r c e d c o n v e c t i o n i s s i g n i f i c a n t l y more e f f e c t i v e i n i n c r e a s i n g the volume of the l i q u i d p o o l than any o t h e r s i n g l e parameter. I n c r e a s i n g the power i n p u t t o the s k u l l , i n c r e a s i n g the s k u l l w i d t h and removing the water c o o l e d copper mould from around the s k u l l a l s o i n c r e a s e the p o o l volume. The e v a p o r a t i o n r a t e s of a l l o y elements w i t h i n the s k u l l were most e f f e c t e d by changes i n the power d i s t r i b u t i o n and the degree of l i q u i d m o t i o n . T a b l e of C o n t e n t s A b s t r a c t i i Ta b l e of C o n t e n t s i v L i s t of T a b l e s i x L i s t of F i g u r e s x L i s t of Symbols x v i Acknowledgements x v i i i 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 Systems 2 1.1.1 Beam G e n e r a t i o n and C o n t r o l 2 1.1.2 Vacuum Systems 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 and 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 Advantages 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 of EBCHR 11 1.2.2.3.1 R e c y c l i n g of T i t a n i u m S c r a p 11 1.2.2.3.2 Removal of Oxide I n c l u s i o n s from S u p e r a l l o y s 17 1.2.2.3.3 In-Spec T i t a n i u m E l e c t r o d e s 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 1.3. The Thermal Regime D u r i n g EBCHR 19 2.0 M o d e l l i n g Heat Flow i n the E l e c t r o n Beam H e a r t h 22 2.1. The Heat Flow E q u a t i o n 22 i v V 2.2. S k u l l Boundary C o n d i t i o n s 23 2.2.1 Top Boundary C o n d i t i o n 23 2.2.2 S i d e Boundary C o n d i t i o n s 26 2.2.3 Bottom Boundary C o n d i t i o n 27 2.2.4 C e n t e r l i n e Boundary C o n d i t i o n 27 2.3. H e a r t h Boundary C o n d i t i o n s 28 2.3.1 E x t e r i o r S u r f a c e Boundary C o n d i t i o n 28 2.3.2 I n t e r i o r S u r f a c e 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 Boundary C o n d i t i o n 29 2.3.4 Water Channel Boundary 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 Model 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 Model 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 Boundary C o n d i t i o n 34 3.2.1.2 S i d e Boundary C o n d i t i o n 37 3.2.1.3 Bottom Boundary C o n d i t i o n 39 3.2.1.4 Thermal C o n d u c t i v i t y 39 3.2.2 H e a r t h Boundary 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 Water Channel Boundary C o n d i t i o n s 46 3.2.2.4 Thermal C o n d u c t i v i t y 49 3.3. B l o c k Model 49 3.4. N u m e r i c a l Technique 49 v i 3.4.1 Non L i n e a r i t i e s 50 3.4.1.1 Thermal 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 Model R e s u l t s 55 4.1. I n t r o d u c t i o n 55 4.2. Temperature D i s t r i b u t i o n 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 Movement 66 4.3. Power D i s t r i b u t i o n 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 Rate Adjustments 80 4.3.2 R e s u l t s 88 4.3.2.1 E f f e c t of Power D e n s i t y 88 4.3.2.2 I n f l u e n c e of L i q u i d M o t i o n 94 4.3.3 The Heat B a l a n c e 94 4.4. Beam Spot Temperature 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 Design and O p e r a t i o n 116 5.1. F a c t o r s A f f e c t i n g H e a r t h Design and O p e r a t i o n ...116 5.2. H e a r t h Design 116 5.2.1 Geometry 117 5.2.2 The E f f e c t of the H e a r t h Mould 127 5.3. Furnace O p e r a t i o n s 132 5.3.1 E f f e c t of M e l t Rate 133 5.3.2 Power D i s t r i b u t i o n 136 v i i 5.4. Summary 150 6.0 Summary and Recommendations f o r F u t u r e Work 152 6.1. Summary 152 6.2. Recommendations 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 154 6.2.2 Power D i s t r i b u t i o n and Power L e v e l 154 6.2.3 E v a p o r a t i o n Rates 155 6.3. C o n c l u d i n g Remarks 155 L i s t of R e f e r e n c e s 156 APPENDIX 1 Beam Spot Model 159 L i s t of T a b l e s T a b l e 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 C o n c e n t r a t i o n s i n T i t a n i u m A l l o y s 16 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 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 Rates f o r V a r i o u s Temperature D i s t r i b u t i o n s 65 4.3 Summary of P o o l Data and 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 Runs 73 4.4 Summary of Heat B a l a n c e C a l c u l a t i o n s , P o o l Data and E v a p o r a t i o n Rates f o r Power D i s t r i b u t i o n Model. ..99 4.5 Thermal E f f i c i e n c y of E l e c t r o n Beam M e l t i n g Under V a r i o u s C o n d i t i o n s 105 5.1 R e s u l t s of C a l c u l a t i o n s on the EB S k u l l as a F u n c t i o n of T h i c k n e s s 121 5.2 R e s u l t s of C a l c u l a t i o n s on the EB S k u l l as a F u n c t i o n of Width 125 5.3 E v a p o r a t i o n R a t e s , P o o l Data and Heat Bal 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 and Without a H e a r t h Mould 131 5.4 Power L e v e l s Used as a F u n c t i o n of M e l t Rate 135 5.5 P o o l D a t a , E v a p o r a t i o n Rates and Heat Balance 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 v i i i IX 5.6 P o o l Data, Heat B a l a n c e C a l c u l a t i o n s and E v a p o r a t i o n R ates f o r Power D i s t r i b u t i o n Runs 149 F i g u r e L i s t of F i g u r e s Page 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 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 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 Phase Diagram 14 1.6 T i t a n i u m - Carbon Phase Diagram 15 2.1 Heat Flow Schematic of the EBCHR P r o c e s s 25 3.1 Geometry of the S k u l l and H e a r t h Used by V i k i n g M e t a l l u r g i c a l 35 3.2 Thermal C o n d u c t i v i t y of T i t a n i u m 6A1-4V as a F u n c t i o n of Temperature 41 3.3 Thermal C o n d u c t i v i t y of T i t a n i u m 6A1-4V as 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 M o t i o n 43 3.4 X-Ray Photograph of a Johnson S k u l l Showing Large 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 Temperature = 150 °C 57 x 2 Contour Map Showing the B o u n d a r i e s of the L i q u i d P o o l , C o n t o u r i n g Temperature = 1625 °C, Superheat = 1 1 0 °C, LTCMF = 1 3 Contour Map Showing the B o u n d a r i e s of the S o l i d S k u l l , C o n t o u r i n g Temperature = 1595 °C, Superheat = 1 1 0 °C, LTCMF = 1 4 P o o l P r o f i l e s a t 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, Superheat = 110 °C, LTCMF = 1 5 X-ray Photograph of an A. Johnson S k u l l Showing the L o c a t i o n of the S o l i d u s 6 Contour Map Showing the B o u n d a r i e s of the L i q u i d P o o l , C o n t o u r i n g Temperature = 1625 °C, Superheat = 150 °C, LTCMF = 1 7 Contour Map Showing the B o u n d a r i e s of the S o l i d S k u l l , C o n t o u r i n g Temperature = 1595 °C, Superheat = 150 °C, LTCMF = 1 8 P o o l P r o f i l e s a t 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, Superheat = 150 °C, LTCMF = 1 9 Contour Map Showing the B o u n d a r i e s of the L i q u i d P o o l , C o n t o u r i n g Temperature = 1625 °C, Superheat = 200 °C, LTCMF = 1 10 Contour Map Showing the B o u n d a r i e s of the S o l i d S k u l l , C o n t o u r i n g Temperature = 1595 °C, Superheat = 200 °C, LTCMF = 1 11 P o o l P r o f i l e s a t 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, Superheat = 200 °C, LTCMF = 1 12 P o o l P r o f i l e s a t 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, Superheat = 110 °C, LTCMF = 1 4.13 P o o l P r o f i l e s a t 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, Superheat = 110 °C, LTCMF = 2 76 4.14 P o o l P r o f i l e s a t 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, Superheat = 110 °C, LTCMF = 5 77 4.15 Schematic Diagram of I n t e r a c t i o n s Between an Beam and 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 Energy Loss Mechanisms 79 4.16 Number of B a c k s c a t t e r e d E l e c t r o n s as a F u n c t i o n of Atomic Number of the I r r a d i a t e d M a t e r i a l f o r a Normal Beam, A c c e l e r a t i n g V o l t a g e = 10 KV. 2 81 4.17 Number of B a c k s c a t t e r e d E l e c t r o n s as a F u n c t i o n of An g l e of I n c i d e n c e of the Beam, A c c e l e r a t i n g V o l t a g e = 10 KV. 2 82 4.18 Energy D i s t r i b u t i o n of B a c k s c a t t e r e d E l e c t r o n s as a F u n c t i o n of Atomic Number f o r a Normal Beam, A c c e l e r a t i n g V o l t a g e = 10 KV. 2 83 4.19 Power Losses 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 as a F u n c t i o n of Atomic Number f o r a Normal Beam, A c c e l e r a t i n g V o l t a g e = 10 KV. 2 84 4.20 Schematic Heat Flow Diagram of the E l e c t r o n Beam S k u l l 86 4.21 A T y p i c a l Power D i s t r i b u t i o n Used By V i k i n g M e t a l l u r g i c a l . .89 4.22 Contour Map Showing the B o u n d a r i e s of the L i q u i d P o o l , C o n t o u r i n g Temperature = 1625 °C, T o t a l Power = 33 KW, LTCMF = 1 90 4.23 Contour Map Showing the B o u n d a r i e s of the S o l i d S k u l l , C o n t o u r i n g Temperature = 1595 °C, T o t a l Power = 33 KW, LTCMF = 1 91 x i i i 4.24 P o o l P r o f i l e s a t 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 92 4.25 S u r f a c e Temperature D i s t r i b u t i o n , 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 4.26 Contour Map Showing the B o u n d a r i e s of the L i q u i d P o o l , C o n t o u r i n g Temperature = 1625 °C, T o t a l Power = 36 KW, LTCMF = 1 95 4.27 Contour Map Showing the B o u n d a r i e s of the S o l i d S k u l l , C o n t o u r i n g Temperature = 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 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 97 4.29 S u r f a c e Temperature D i s t r i b u t i o n , 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 4.30 P o o l P r o f i l e s a t 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 = 2 100 4.31 S u r f a c e Temperature D i s t r i b u t i o n , 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 = 2 101 4.32 P o o l P r o f i l e s a t 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 = 5 102 4.33 S u r f a c e Temperature D i s t r i b u t i o n , 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 = 5 103 4.34 Unsteady S t a t e Response of the Temperature Under the Beam Spot as a F u n c t i o n of Power Input 107 x i v 4.35 Unsteady S t a t e Response of the Temperature Under the Beam Spot 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 Photomacrograph of the B l o c k Used D u r i n g the Experiment a t V i k i n g Showing the 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 : 0.375X , Beam Power : 80 KW 111 4.37 Photomacrograph of t h e B l o c k Used D u r i n g the Experiment a t Johnson Showing the 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 : 0.580X 112 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 of the 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 the B l o c k Model f o r the 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 Used t o E v a l u a t e the E f f e c t of S k u l l T h i c k n e s s 119 5.2 The V a r i a t i o n of P o o l Volume w i t h S k u l l T h i c k n e s s 120 5.3 V a r i a t i o n of Average Heat F l u x w i t h S k u l l T h i c k n e s s 122 5.4 V a r i a t i o n of P o o l Volume w i t h S k u l l Width 124 5.5 V a r i a t i o n of P o o l Parameters w i t h D i m e n s i o n l e s s S k u l l Width 1 26 5.6 Power D i s t r i b u t i o n Used t o E v a l u a t e the E f f e c t s of the H e a r t h Mould 128 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 C o o l e d Copper H e a r t h 129 XV 5.8 P o o l P r o f i l e s f o r EB S k u l l Without a Water C o o l e d Copper H e a r t h 130 5.9 Shape of the Power D i s t r i b u t i o n Used t o E v a l u a t e the E f f e c t of M e l t Rate. 134 5.10 Thermal E f f i c i e n c y as a F u n c t i o n of M e l t Rate 138 5.11 P o o l Volume and E v a p o r a t i o n Rate of A l as a F u n c t i o n of M e l t Rate 139 5.12 Base Case 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 of the E f f e c t of Power D i s t r i b u t i o n on the Thermal Regime 141 5.13 Power D i s t r i b u t i o n 1. 3.15 KW C i r c l e i n A d d i t i o n t o the Base Case 142 5.14 Power D i s t r i b u t i o n 2. Base Case Power D i s t r i b u t i o n 1/2 as Wide, Same T o t a l Power 143 5.15 Power D i s t r i b u t i o n 3. R e c t a n g u l a r Shape of Same T o t a l Area as the Base Case 144 5.16 P o o l P r o f i l e s f o r the Base Case 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 D e f i n i t i o n Tg - Temperature of t h e S k u l l T u - Temperature of t h e H e a r t h hi T A - Ambient Temperature T w - Temperature of t h e C o o l i n g Water kg - Thermal C o n d u c t i v i t y of the S k u l l k H - Thermal C o n d u c t i v i t y of the H e a r t h eg - E m i s s i v i t y of the S k u l l e H - E m i s s i v i t y of the H e a r t h a - Bandwidth of the G a u s s i a n D i s t r i b u t i o n or the S t e f a n Boltzman Constant nS/H ~ S k u l l / H e a r t h Heat T r a n s f e r C o e f f i c i e n t h w - Copper/Water Heat T r a n s f e r C o e f f i c i e n t N R e - Reynolds Number N„ - P r a n d t l Number Pr Q. - Heat F l u x V"w - Water V e l o c i t y 5 - Mesh S i z e A - Area I - B a c k s c a t t e r e d C u r r e n t I, - Beam C u r r e n t x v i x v i i a - Angle of Beam I n c i d e n c e E - Energy e - Fundamental Charge of the E l e c t r o n _ Beam A c c e l e r a t i n g V o l t a g e P r - B a c k s c a t t e r i n g Power P - Beam Power Acknowledgements I would l i k e t o acknowledge the support and encouragement p r o v i d e d by the boys i n the o f f i c e , s p e c i f i c a l l y Bob, M i k e , Steve and B a r r y as w e l l as a l l the o t h e r p e o p l e i n the department who c o n t r i b u t e d t o t h i s t h e s i s i n any way, no m a t t e r how s m a l l . My s i n c e r e s t thanks t o Dr. A l e c M i t c h e l l f o r h i s guidance and i n s i g h t s i n t o the s p e c i a l m e l t i n g i n d u s t r y . My thanks a l s o t o A. Johnson M e t a l s , V i k i n g M e t a l l u r g i c a l Co. and G e n e r a l E l e c t r i c Co. f o r t h e i r support d u r i n g the work. Thanks a l s o go out t o E l Dorado Resources f o r t h e i r f i n a n c i a l s u p p o r t . Tom N i c h o l of the computing c e n t e r was a l s o 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 problems d u r i n g the work. F i n a l l y , I would l i k e t o thank my f a m i l y f o r t h e i r i n t e r e s t , h e l p , support and sometimes c o n s t a n t b a d g e r i n g which a i d e d i n the c o m p l e t i o n of t h i s work. x v i i i CHAPTER 1 I n t r o d u c t i o n In r e c e n t y e a r s i t has become apparent t h a t the t r a d i t i o n a l p r o c e s s r o u t e s f o r the p r o d u c t i o n of super c l e a n m a t e r i a l s and a l l o y s f o r the a e r o s p a c e i n d u s t r y a r e i n a d e q u a t e . Vacuum i n d u c t i o n m e l t i n g (VIM) f o l l o w e d by e l e c t r o s l a g r e m e l t i n g (ESR) or vacuum a r c r e m e l t i n g (VAR) does not produce m a t e r i a l w i t h i n c l u s i o n s i z e s and d i s t r i b u t i o n s t h a t meet the s p e c i f i c a t i o n of the engine m a n u f a c t u r e r s . As w e l l , vacuum a r c p r o c e s s i n g does not remove e i t h e r the h i g h d e n s i t y i n c l u s i o n s or the h a r d - a l p h a d e f e c t s from t i t a n i u m a l l o y s . In a d d i t i o n , buy t o f l y r a t i o s i n the aerospace i n d u s t r y a r e e x t r e m e l y h i g h - on the o r d e r of 10:1 or g r e a t e r - e s p e c i a l l y i n the p r o d u c t i o n of c r i t i c a l components f o r j e t e n g i n e s . T h i s means t h a t t h e r e i s a s i g n i f i c a n t amount of both s u p e r a l l o y and t i t a n i u m s c r a p a v a i l a b l e t o be 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 of b oth m a t e r i a l s , exogenous p a r t i c l e s i n t r o d u c e d i n t o the s c r a p d u r i n g the m a n u f a c t u r i n g p r o c e s s e s may f i n d t h e i r way i n t o the f i n i s h e d p r o d u c t . For example, h i g h d e n s i t y i n c l u s i o n s (HDI's) - u s u a l l y t o o l b i t c h i p s - i n t i t a n i u m w i l l s u r v i v e even a t r i p l e vacuum a r c r e m e l t i n g t r e a t m e n t . 1 2 The aerospace i n d u s t r y and more s p e c i f i c a l l y t he a i r c r a f t engine m a n u f a c t u r e r s have been i n t e r e s t e d i n a p r o c e s s o f f e r i n g an a l t e r n a t i v e t o the t r a d i t i o n a l super c l e a n p r o c e s s i n g r o u t e s . I n g e n e r a l , i t has been found t h a t the c o l d h e a r t h r e m e l t i n g p r o c e s s e s and more s p e c i f i c a l l y 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 p r o c e s s (EBCHR) has a l l o w e d f o r the p r o d u c t i o n of super c l e a n s u p e r a l l o y m a t e r i a l s and f o r the r e c y c l i n g of t i t a n i u m s c r a p t o h i g h q u a l i t y f o r g i n g s t o c k . 1.1. E l e c t r o n Beam Systems E l e c t r o n beam m e l t i n g systems r e q u i r e two i n t e g r a t e d systems t o o p e r a t e . These a r e the beam g e n e r a t i o n and c o n t r o l systems and the vacuum system. Both of t h e s e systems a l s o have e x t e n s i v e computer c o n t r o l i n o r d e r t o guide the beam and i n s u r e t h a t the v a r i o u s v a l v e s open and c l o s e i n t h e c o r r e c t sequence. 1.1.1 Beam G e n e r a t i o n and C o n t r o l The g e n e r a t i o n and c o n t r o l of e l e c t r o n beams a r e based on the p h y s i c s of h i g h energy p a r t i c l e s and e l e c t r o n 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 of e l e c t r o n guns ar e manufactured r a n g i n g i n power from the m i c r o w a t t to the megawatt l e v e l s , the fundamentals remain the same. E l e c t r o n s a r e g e n e r a t e d by h e a t i n g an e l e c t r o n e m i t t e r ( u s u a l l y t u n g s t e n f o r m e l t i n g a p p l i c a t i o n s ) e i t h e r d i r e c t l y or i n d i r e c t l y . A h i g h e l e c t r i c a l p o t e n t i a l (20 - 50 3 KV f o r m e l t i n g ) i s then a p p l i e d between the hot e m i t t e r cathode and 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 or anode. T h i s causes e l e c t r o n s t o be e m i t t e d from the cathode and a c c e l e r a t e d t o h i g h energy l e v e l s i n an e l e c t r o s t a t i c f i e l d . The c h o i c e of h e a t i n g method, cathode 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 depends on the a p p l i c a t i o n , the o p e r a t i n g parameters of the gun, c o n d i t i o n s i n the m e l t chamber and d e s i r e d cathode l i f e 1 . In a d d i t i o n t o the two r e q u i r e d e l e c t r o d e s (anode and cathode) a beam g e n e r a t i o n system may c o n t a i n e l e c t r o d e s f o r f o c u s s i n g and s h a p i n g . E l e c t r o d e s e l e c t i o n and d e s i g n d e termine the main beam parameters such as c u r r e n t d e n s i t y , a p e r t u r e and l o c a t i o n of the f o c a l s p o t . Once g e n e r a t e d and f o c u s s e d the beam can be 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 l e n s e s . These c o n t r o l schemes are based on the p r i n c i p l e s of e l e c t r o n o p t i c s and p r o v i d e e x c e l l e n t c o n t r o l of the e l e c t r o n beam. At the p r e s e n t t i m e , a x i a l guns a r e p r e f e r r e d f o r m e l t i n g o p e r a t i o n s . F u r t h e r d i s c u s s i o n of r i n g , l i n e and 2-4 t r a n s v e r s e guns can be found i n the l i t e r a t u r e . 1.1.2 Vacuum Systems E l e c t r o n beam systems f o r m e l t i n g r e q u i r e r a t h e r - 4 e x t e n s i v e vacuum systems. A p r e s s u r e on the o r d e r of 10 Pa (10 ^ t o r r ) i n the gun chamber i s r e q u i r e d t o a l l o w the beam 4 g e n e r a t i o n system t o o p e r a t e . W i t h d i f f e r e n t i a l pumping, c l o s e l y d e s i g n e d passages between the gun and the melt chamber and 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 a t -1 5 p r e s s u r e s as h i g h as 13 Pa (10 t o r r ) i n the melt chamber. Chamber p r e s s u r e s a r e more l i k e l y t o be around 0.13 Pa (10 t o r r ) f o r c o n v e n t i o n a l m e t a l p r o c e s s i n g . The presence of t h i s r e a s o n a b l y good vacuum a l l o w s f o r both advantages and d i s a d v a n t a g e s i n e l e c t r o n beam r e m e l t i n g . Because of the vacuum l e v e l 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 gases from the f e e d s t o c k m a t e r i a l . In a d d i t i o n the vacuum p r o v i d e s f o r the removal of v o l a t i l e i m p u r i t i e s such as magnesium and manganese from a l l o y t i t a n i u m . On the o t h e r hand, the vacuum environment w i l l a l s o cause the e v a p o r a t i o n of d e s i r e d a l l o y i n g elements such as aluminum from t i t a n i u m 6% aluminum 4% vanadium and chromium from s u p e r a l l o y s . 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 E l e c t r o n beam r e m e l t i n g on an i n d u s t r i a l s c a l e has 5 been c o m m e r c i a l l y v i a b l e s i n c e the mid 1950's. S i n c e t h a t time e l e c t r o n beams have p r o v i d e d the power s o u r c e f o r m e l t i n g and r e f i n i n g r e f r a c t o r y m e t a l s f o r use i n aerospace components, f o r the r e c y c l e of r e a c t i v e m e t a l s c r a p and f o r the p r o d u c t i o n of super c l e a n s u p e r a l l o y m a t e r i a l . 1.2.1 D r i p and P o o l M e l t i n g E l e c t r o n beam d r i p m e l t i n g i s used p r i m a r i l y f o r the m e l t i n g and r e f i n i n g of the r e f r a c t o r y m e t a l s (Nb, V, T a ) . In t h i s p r o c e s s bar f e e d s t o c k i s i n t r o d u c e d t o the m e l t i n g chamber i n e i t h e r a h o r i z o n t a l or v e r t i c a l f a s h i o n (see f i g u r e 1.1). One or more e l e c t r o n beams then impinge on the s u r f a c e of the bar and drops of molten m a t e r i a l f a l l i n t o a water c o o l e d copper c r u c i b l e where the m a t e r i a l i s s o l i d i f i e d i n a c o n t r o l l e d way t o produce an i n g o t . V e r t i c a l f e e d i n g i s the 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 r e g i o n of shadow i n the c r u c i b l e . I n v e r t i c a l d r i p m e l t i n g the e l e c t r o d e i s u s u a l l y r o t a t e d i n o r d e r t o f a c i l i t a t e even m e l t i n g . P o o l m e l t i n g i s v i r t u a l l y i d e n t i c a l t o d r i p m e l t i n g e xcept t h a t the f e e d s t o c k i s u s u a l l y p a r t i c u l a t e or s m a l l p i e c e s ( i . e . sponge, s c r a p ) as shown i f f i g u r e 1.2. In t h i s c a s e , m a t e r i a l i s f e d i n t o the l i q u i d p o o l i n the i n g o t c r u c i b l e where i t m e l t s u s i n g the heat p r e s e n t w i t h i n the c r u c i b l e . P o o l m e l t i n g i s g e n e r a l l y f o l l o w e d by d r i p r e m e l t i n g the r e s u l t i n g i n g o t one or more t i m e s . 6 REMELT STICK ELECTON GUN CRUCIBLE F i g u r e 1.1 E l e c t r o n Beam D r i p M e l t i n g . 7 ELECTRON GUNS SCRAP HOPPER INGOT CRUCIBLE F i g u r e 1.2 E l e c t r o n Beam P o o l M e l t i n g . 8 1.2.2 H e a r t h M e l t i n g 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 one of the p r o c e s s e s which i s a b l e t o guarantee the absence of some d e f e c t s from aerospace q u a l i t y m a t e r i a l s . I t i s c u r r e n t l y b e i n g used as a complimentary s t e p t o one or two vacuum a r c r e m e l t s i n t i t a n i u m p r o c e s s i n g and i s a l s o used f o r t h e removal of o x i d e 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 g e n e r a l l e v e l , EBCHR i n v o l v e s r e m e l t i n g f e e d s t o c k a t one end of a water c o o l e d copper t r o u g h , m a i n t a i n i n g the m e t a l i n the l i q u i d s t a t e over the l e n g t h of the t r o u g h and a l l o w i n g i t t o f l o w i n t o an i n g o t mould a t the o t h e r end (see f i g u r e 1.3). 1.2.2.1 P r o c e s s Advantages The p r i m a r y m e t a l l u r g i c a l advantages of e l e c t r o n beam h e a r t h r e m e l t i n g a r e t w o f o l d . F i r s t the presence of a good vacuum a l l o w s v a r i o u s v o l a t i l e i m p u r i t i e s (Mn, Mg i n T i ) t o e v a p o r a t e thus r e f i n i n g the m a t e r i a l . S e c o n d l y , the presence of a q u a n t i t y of l i q u i d m e t a l a l l o w s f o r g r a v i m e t r i c s e p a r a t i o n of h i g h d e n s i t y and low d e n s i t y i n c l u s i o n s from the b u l k as w e l l as p r o v i d i n g a r e a c t i o n volume f o r the d i s s o l u t i o n of i n t e r - m e t a l l i c p a r t i c l e s . I n a d d i t i o n the n a t u r e and f l e x i b i l i t y of the e l e c t r o n beam power sour c e p r o v i d e s a number of p r o c e s s a d vantages. Because the beam can be c o n t r o l l e d w i t h g r e a t a c c u r a c y , i t i s p o s s i b l e t o c a s t a wide v a r i e t y of shapes 9 ELECTRON GUNS SCRAP FEEDER INGOT CRUCIBLE F i g u r e 1.3 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 . 10 30 ( i . e . s q u a r e s , s l a b s , rounds, h o l l o w i n g o t s , e t c . ) . The d i s c o n n e c t i o n of the power s o u r c e from the f e e d m a t e r i a l (heat can be a p p l i e d w i t h o u t m e l t i n g a d d i t i o n a l f e e d s t o c k ) reduces the amount of m a c r o s e g r e g a t i o n due t o melt r a t e i n s t a b i l i t i e s found i n VAR i n g o t s ^ improves the s u r f a c e q u a l i t y of the f i n i s h e d i n g o t and reduces t h e amount of p r o d u c t l o s t due t o the f o r m a t i o n of s h r i n k a g e c a v i t i e s d u r i n g f i n a l s o l i d i f i c a t i o n . The f e e d i n g systems commonly used i n h e a r t h r e m e l t i n g a l s o g i v e a g r e a t degree of f l e x i b i l i t y w i t h r e s p e c t t o i n p u t f e e d s t o c k , c h e m i s t r i e s and me l t p r a c t i c e . 1.2.2.2 P r o c e s s D i s a d v a n t a g e s The b i g g e s t d i s a d v a n t a g e of 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 t h a t a l t h o u g h v o l a t i l e c o n t a m i n e n t s a r e removed from the i n p u t f e e d s t o c k d u r i n g m e l t i n g , d e s i r e d a l l o y i n g 7 elements a l s o e v a p o r a t e . C h e m i s t r y c o n t r o l when r e m e l t i n g a l l o y m a t e r i a l s becomes ve r y d i f f i c u l t , e s p e c i a l l y when the a l l o y i n g element cannot be added t o the charge i n s u f f i c i e n t q u a n t i t i e s t o make up the d e f i c i t . A n other problem a s s o c i a t e d w i t h a l l o y r e m e l t i n g i s the l a c k of c h e m i c a l homogeneity t h a t r e s u l t s from poor m i x i n g w i t h i n the l i q u i d m e t a l . Even though e l e c t r o n beam t e c h n o l o g y i s w e l l u n d e r s t o o d , i t i s a l s o v e r y complex r e q u i r i n g a broad 11 knowledge of p h y s i c s , vacuum systems and e l e c t r o n i c s . T h e r e f o r e a v e r y h i g h l e v e l of o p e r a t o r t r a i n i n g i s r e q u i r e d . In a d d i t i o n , the t r a n s f o r m e r s , computer c o n t r o l s and vacuum systems a r e c o s t l y making the p r o c e s s v e r y c a p i t a l i n t e n s i v e . 1.2.2.3 A p p l i c a t i o n s of EBCHR 1.2.2.3.1 R e c y c l i n g of T i t a n i u m S c r a p R e c y c l i n g of t i t a n i u m s c r a p i s d e s i r a b l e f o r two re a s o n s . F i r s t , v i r g i n t i t a n i u m sponge i s r e l a t i v e l y e x p e n s i v e and second a r e a s o n a b l y l a r g e amount of t i t a n i u m s c r a p ( p a r t i c u l a r l y machine t u r n i n g s ) i s a v a i l a b l e . There a r e two s i g n i f i c a n t problems a s s o c i a t e d w i t h the t i t a n i u m s c r a p r e c y c l e . High d e n s i t y i n c l u s i o n s (HDI's) - u s u a l l y t u n g s t e n c a r b i d e t o o l b i t s - a r e o f t e n p r e s e n t i n machine t u r n i n g s and so u r c e s f o r type I , ha r d a l p h a d e f e c t s a r e a l s o common. I t i s p o s s i b l e t o remove h i g h d e n s i t y i n c l u s i o n s from s c r a p w i t h out making use of an e l e c t r o n beam h e a r t h 8 f u r n a c e . In a d d i t i o n t o c l e a n i n g the s c r a p of grease and o i l , i t i s passed over a magnetic s e p a r a t o r removing the bu l k of the HDI's. In non-EB i n s t a l l a t i o n s the s c r a p i s then x - r a y e d and v i s u a l l y i n s p e c t e d and any h i g h d e n s i t y p a r t i c l e s a r e removed. T h i s p r o c e s s i s e x p e n s i v e but does work. When u s i n g e l e c t r o n beam r e m e l t i n g , the x - r a y and 12 v i s u a l i n s p e c t i o n s t e p s a r e e l i m i n a t e d . Any h i g h d e n s i t y i n c l u s i o n s which e n t e r the h e a r t h f u r n a c e w i l l s i n k t o the bottom of the l i q u i d p o o l , where they become t r a p p e d by the s o l i d s k u l l and d i s s o l v e over a p e r i o d of t i m e . T h i s has been e x p e r i m e n t a l l y v e r i f i e d u s i n g p l a n t t r i a l s by E n t r e k i n g a t A. Johnson M e t a l s . Hard a l p h a (or type I ) d e f e c t s i n t i t a n i u m a r i s e from p a r t i c l e s of s t r o n g l y a - s t a b i l i z e d p a r t i c l e s e n t e r i n g the l i q u i d and r e m a i n i n g through t o the f i n i s h e d p r o d u c t . The s t r o n g e s t a - s t a b i l i z e r s a r e the i n t e r s t i t i a l elements (C, 0, N). S m a l l c o n c e n t r a t i o n s of t h e s e elements s t a b i l i z e t he a l p h a phase v e r y s t r o n g l y and a l s o i n c r e a s e the m e l t i n g or l i q u i d u s t e m p e r a t u r e s u b s t a n t i a l l y (see f i g u r e s 1.4 t o 1.6). T h e r e f o r e b u r n t sponge and flame c u t s c r a p a r e prime s o u r c e s of ha r d a l p h a d e f e c t s . The p r e s e n c e of the i n t e r s t i t i a l elements i n t i t a n i u m i s not n e c e s s a r i l y d e t r i m e n t a l t o the m e c h a n i c a l p r o p e r t i e s . Oxygen up t o a l e v e l of 2000 p.p.m. i s c o n s i d e r e d an a l l o y i n g element and n i t r o g e n around the 100 p.p.m. l e v e l w i l l exceed a l l a l l o y s p e c i f i c a t i o n s 1 ^ (see t a b l e 1.1). L o c a l c o n c e n t r a t i o n s of these elements r e s u l t i n g from hard a l p h a p r o d u c i n g p a r t i c l e s not b e i n g c o m p l e t e l y d i s s o l v e d may be d e t r i m e n t a l t o the f a t i g u e p r o p e r t i e s of the m a t e r i a l . U n f o r t u n a t e l y the presence of thes e c o n c e n t r a t i o n s of n i t r o g e n and oxygen i s i m p o s s i b l e t o d e t e c t by n o n - d e s t r u c t i v e means. U s u a l l y the hard a l p h a 13 F i g u r e 1.4 T i t a n i u m - N i t r o g e n Phase Diagram. °c Ti 5 10 15 20 Weight Percent Oxygen F i g u r e 1.5 T i t a n i u m - Oxygen Phase Diagram. 15 °C 3100 2700 2300 1900 1500 1100 I / s --* y ' / 31 / 1 / 380- \ 6.5% \ v-L / / / 1 1 / ] / / / / 1 1 \ \ X : V" 1650° \ 6 I y I L. Ti U 8 12 16 20 Weight Percent Carbon F i g u r e 1.6 T i t a n i u m - Carbon Phase Diagram. 16 A l l o y N (max) Commercial P u r i t y 300 (CP) T i - 5 A l - 2 . 5 S n 500 T i - 5 A l - 2 . 5 S n ELI 700 Ti-6A1-4V 500 Ti-6A1-4V ELI 500 T i - 1 0 V - 2 F e - 3 A l 500 Ti-3Al-8V-6Cr-4Mo-4Zr 300 (Beta-C) Ti-5Al-2Sn-2Zr-4Mo-4Cr 400 ( T i t a n i u m - 1 7 ) I m p u r i t y L i m i t s p.p.m. 0 C H (max) (max) (max) 2000 1000 150 2000 800 200 1200 800 125 2000 1000 125 1300 800 125 1600 500 150 1200 500 200 1300 500 125 T a b l e 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 C o n c e n t r a t i o n s i n T i t a n i u m A l l o y s . 17 d e f e c t forms a c o n t i n u o u s g r a i n boundary w i t h the s u r r o u n d i n g m a t r i x which i s i n v i s i b l e t o u l t r a s o n i c s . M e t a l l o g r a p h i c t e c h n i q u e s can be used t o d e t e c t the p r e s e n c e of a - s t a b i l i z a t i o n but t h i s method i s o n l y u s e f u l i f the d e f e c t i s p r e s e n t on the s u r f a c e of the p a r t b e i n g examined. The i n c r e a s e i n l i q u i d u s temperature which accompanies the i n t e r s t i t i a l e l e ments means t h a t e x t r e m e l y h i g h t e m p e r a t u r e s or l o n g t i m e s a t h i g h e r t e m p e r a t u r e s a r e r e q u i r e d t o melt or d i s s o l v e the p a r t i c l e s r e s p o n s i b l e f o r the 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 s u p p l y i n g e i t h e r c o n d i t i o n . EBCHR, on the o t h e r hand, can manage both depending on the d e s i r e d m e l t i n g c o n d i t i o n s . The presence of a l i q u i d volume a t a r e a s o n a b l y h i g h temperature d u r i n g 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 n s u r e s t h a t most, i f not a l l , of the hard a l p h a d e f e c t s w i l l be removed. F u r t h e r work, b o t h i n the l a b o r a t o r y and on an i n d u s t r i a l s c a l e , must be done i n o r d e r t o s p e c i f y the times and t e m p e r a t u r e s t h a t a r e r e q u i r e d to guarantee the complete removal of hard a l p h a d e f e c t s . 1.2.2.3.2 Removal of Oxide I n c l u s i o n s from S u p e r a l l o y s The removal of oxygen and o x i d e i n c l u s i o n s from s u p e r a l l o y s has been demonstrated b o t h on a s m a l l s c a l e 1 1 11 12 and on an i n d u s t r i a l s c a l e . ' The removal of t h e o x i d e i n c l u s i o n s from the f i n i s h e d p r o d u c t i s a c c o m p l i s h e d u s i n g one or more water c o o l e d copper dams or making e f f e c t i v e use 18 of beam sweep p a t t e r n s . I t has been shown t h a t s i g n i f i c a n t improvements i n t o t a l oxygen c o n t e n t , maximum p a r t i c l e s i z e and m e c h a n i c a l p r o p e r t i e s can be a c h i e v e d . 1.2.2.3.3 In-Spec T i t a n i u m E l e c t r o d e s C u r r e n t p r a c t i c e f o r r e c y c l i n g of t i t a n i u m a l l o y 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 by EB m e l t i n g t o produce an e l e c t r o d e d e p l e t e d i n a l l o y e l ements. A l l o y a d d i t i o n s a re then made add i n g master a l l o y s t o the i n g o t and vacuum a r c r e m e l t i n g t o g i v e the prop e r a l l o y c h e m i s t r y , h o m o g e n i z a t i o n and s t r u c t u r e . O b v i o u s l y t h e r e a r e economic advantages t o removing the r e m e l t s t e p u s i n g the vacuum a r c f u r n a c e . The a l l o y l o s s e s d u r i n g e l e c t r o n beam h e a r t h r e m e l t i n g a re u n a v o i d a b l e . The a l l o y c h e m i s t r y can be a c h i e v e d by a d d i n g a d d i t i o n a l q u a n t i t i e s of the a l l o y i n g element t o the f e e d s t o c k h o p i n g t o a c h i e v e the d e s i r e d c h e m i c a l c o m p o s i t i o n s i n the i n g o t . To d a t e t h i s approach 1 3 has met w i t h l i m i t e d s u c c e s s but the a l l o y i s not u s u a l l y 1 4 homogenous. Takagi and M i t c h e l l have suggested u s i n g x - r a y a n a l y s i s as an o n - l i n e q u a n t i t a t i v e t o o l f o r c h e m i c a l c o m p o s i t i o n c o n t r o l . Work i s p r o c e e d i n g on the development of t h i s t e c h n i q u e . 1.2.2.4 O p e r a t i n g H e a r t h Furnaces In N o r t h America t h e r e a r e c u r r e n t l y a number of o p e r a t i n g h e a r t h r e m e l t i n g f u r n a c e s . The two b i g g e s t 19 i n s t a l l a t i o n s a re a t V i k i n g M e t a l l u r g i c a l i n V e r d i , Nevada and A. Johnson M e t a l s i n Morgantown, P e n n s y l v a n n i a . At V i k i n g , t h e r e are two f u r n a c e s , a 1.2 MW i n s t a l l a t i o n and a 2.4 MW i n s t a l l a t i o n which uses 2 1.2 MW Von Ardenne guns. Both t h e s e f u r n a c e s a r e used p r i m a r i l y f o r r . e c y c l i n g t i t a n i u m s c r a p . A. Johnson o p e r a t e s a s i n g l e 2.4 MW f u r n a c e which has 4 600 KW Von Ardenne guns. E l e c t r o M e t a l s (a d i v i s i o n of DeGussa E l e c t r o n i c s ) a l s o o p e r a t e s a s m a l l h e a r t h r e m e l t i n g f u r n a c e i n V a l l e j o , C a l i f o r n i a . T h i s f a c i l i t y i s used f o r r e m e l t i n g s u p e r a l l o y m a t e r i a l s f o r foundry a p p l i c a t i o n s . There are a d d i t i o n a l h e a r t h r e m e l t i n g f u r n a c e s o p e r a t i n g throughout the w o r l d . The l a r g e s t s i n g l e h e a r t h f u r n a c e i s i n the S o v i e t Union and i s r e p o r t e d t o have a r a t e d m e l t i n g power of 9 MW. The h e a r t h and s k u l l d e s i g n i n t h e v a r i o u s i n s t a l l a t i o n s a re by no means s t a n d a r d . For i n s t a n c e , i n the 2.4 MW V i k i n g 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 square a r e a w i t h a narrow t r o u g h e x i t i n g from one s i d e . The l a r g e a r e a i s used as the melt end and the narrow t r o u g h a c t s as a pour l i p . At A. Johnson, the h e a r t h i s a r e c t a n g u l a r t r o u g h 20 w i t h a s e m i - c i r c u l a r c r o s s s e c t i o n . T h i s v a r i a t i o n i n f u r n a c e 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 w i t h r e s p e c t t o melt r a t e s , melt p r a c t i c e and may i n f l u e n c e the a b i l i t y of a p a r t i c u l a r f u r n a c e t o melt e f f i c i e n t l y . 1.3. The Thermal Regime D u r i n g EBCHR The s u c c e s s of the e l e c t r o n beam h e a r t h r e m e l t i n g p r o c e s s i s l a r g e l y dependent on c o n t r o l l i n g the temp e r a t u r e f i e l d s w i t h i n the s k u l l . For example, i n o r d e r t o make mass t r a n s f e r c a l c u l a t i o n s p e r t a i n i n g t o the d i s s o l u t i o n of type I d e f e c t s i n t i t a n i u m , i t i s n e c e s s a r y t o have adequate d a t a on p o o l volume, r e s i d e n c e time and te m p e r a t u r e . To d a t e , t h e r e i s l i t t l e or no a v a i l a b l e d a t a on the t h e r m a l f i e l d g e n e r a t e d d u r i n g EBCHR. In a d d i t i o n t o f i n d i n g ways t o make the EBCHR p r o c e s s more e f f i c i e n t , i t would a l s o be b e n e f i c i a l t o have some method of d e s i g n i n g h e a r t h s t o g i v e b e t t e r r e s u l t s . C u r r e n t h e a r t h d e s i g n seems t o be a h i t or miss a f f a i r . A l t h o u g h 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 moulds e f f e c t i v e l y (some f o r many y e a r s ) t h e r e i s no guarantee t h a t these d e s i g n s a r e the most e f f i c i e n t . 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 i s e x t r e m e l y d i f f i c u l t t o experiment on o p e r a t i n g h e a r t h f u r n a c e s . T h i s i s p r i m a r i l y because of the e l e v a t e d t e m p e r a t u r e s , the vacuum t h a t i s p r e s e n t and the super c l e a n n a t u r e of the m a t e r i a l s b e i n g m e l t e d . As p r e v i o u s l y 21 mentioned, e l e c t r o n beam m e l t i n g i n g e n e r a l i s c a p i t a l i n t e n s i v e making p i l o t p l a n t (and t o some e x t e n t l a b o r a t o r y s c a l e ) c o n s t r u c t i o n u n r e a l i s t i c . T h e r e f o r e the o n l y way i n which t o stu d y the heat f l o w c h a r a c t e r i s t i c s of 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 by c o n s t r u c t i n g a m a t h e m a t i c a l model. 22 CHAPTER 2 M o d e l l i n g Heat Flow i n the E l e c t r o n Beam H e a r t h 2.1. The Heat Flow E q u a t i o n Because of the c o n t r o l of the e l e c t r o n beam i n a p p l y i n g power t o the s k u l l i n an EBCHR f u r n a c e , the geo m e t r i e s t h a t can and are b e i n g used a r e v a r i e d . In a d d i t i o n , energy must be d e l i v e r e d i n an non - u n i f o r m manner over the s u r f a c e of the s k u l l . These p o i n t s must be taken 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 a m a t h e m a t i c a l model f o r the EBCHR p r o c e s s . In i t s most g e n e r a l form the heat f l o w e q u a t i o n i s as shown i n e q u a t i o n 2.1. T h i s e q u a t i o n d e s c r i b e s unsteady s t a t e heat f l o w i n t h r e e d i m e n s i o n s . 3 3T 3 3T 3 3T 3T — ( k — ) + — ( k — ) + — ( k — ) = pC ( — ) . 3x 3x 3y 3y 3z 3z P 3t ...(2.1) D u r i n g the 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 i n p u t power i s c o n t r o l l e d by programming the beam t o move over the s k u l l i n some s o r t of p a t t e r n . T h i s r e s u l t s i n a r e g i o n of v e r y h i g h t e m p e r a t u r e a t the beam sp o t (up t o 2cm i n d i a m e t e r ) which moves about w i t h the beam. T h i s would i n d i c a t e t h a t the EBCHR p r o c e s s p r e s e n t s an unsteady s t a t e 22 23 heat t r a n s f e r problem. The problem can be s i m p l i f i e d by i n t e g r a t i n g the power i n p u t over time t o g i v e a power d i s t r i b u t i o n . When the power d i s t r i b u t i o n i s then imposed on the s k u l l as boundary c o n d i t i o n , the unsteady s t a t e n a t u r e of the problem i s removed. As p r e v i o u s l y i n d i c a t e d the o p e r a t i n g EB h e a r t h s a t A. Johnson 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 not s i m p l e r e c t a n g u l a r boxes. V i k i n g M e t a l l u r g i c a l uses a T-shaped h e a r t h s e c t i o n i n which the melt b a s i n forms the c r o s s and t h e runout t r o u g h and pour l i p form the s t i c k . A. Johnson use a r e c t a n g u l a r shaped t r o u g h w i t h a s e m i - c i r c u l a r bottom. In b o t h th e s e g e o m e t r i e s , i t would be i n a c c u r a t e t o n e g l e c t any of the t h r e e heat f l o w d i r e c t i o n s as they a l l c o n t r i b u t e t o the th e r m a l regime. T h e r e f o r e we can w r i t e the heat f l o w e q u a t i o n s f o r the h e a r t h and s k u l l as : 9 9 T S 9 9 T S 9 9 T S — (k_ ) + — ( k _ ) + — ( k . ) = 0. 9x b 9 x 9y ^ y 9z b 9 z ...(2.2) 9 — (k 9x H 9 x 9 T H 9 9 T H ) + — (k T 9T 9y H 9 y ) + — ( k H 9z H 9 z ) = 0 ...(2.3) 24 2.2. S k u l l Boundary C o n d i t i o n s 2.2.1 Top Boundary C o n d i t i o n A heat f l o w schematic f o r the EBCHR p r o c e s s i s shown i n F i g u r e 2.1. At the t o p s u r f a c e of the s k u l l heat i s b e i n g t r a n s f e r r e d by two p r o c e s s : energy i s d e l i v e r e d by the i m p i n g i n g e l e c t r o n beam and heat i s r a d i a t e d t h r o u g h a vacuum t o the s u r r o u n d i n g f u r n a c e . U s i n g the r a d i a t i o n boundary c o n d i t i o n the t o p boundary c o n d i t i o n i s g i v e n by 9T S - k c = P_(x,y) - a e c ( T ^ - T?) where z=0. b 3 z b ...(2.4) The boundary c o n d i t i o n g i v e n by e q u a t i o n 2.4 i s comp l e t e . 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 . I n s t e a d of g i v i n g a power d i s t r i b u t i o n on the t o p s u r f a c e , a tem p e r a t u r e d i s t r i b u t i o n i s s p e c i f i e d . T h i s i s a v a l i d approach s i n c e any temperature d i s t r i b u t i o n under a g i v e n s e t of c o n d i t i o n s can be a t t r i b u t e d t o o n l y one power d i s t r i b u t i o n . More f o r m a l l y 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 s u r f / \ M * \ ^,^---*^r out <>EB 1 1 k s ^ ^nould ^bottom Mnould * ^m^Ti-V QEB - f(x.y) z > z c z < z F i g u r e 2.1 Heat Flow Schematic of the EBCHR P r o c e s s . 26 An added b e n e f i t of u s i n g the power d i s t r i b u t i o n approach i s t h a t the model then becomes independent of power sour c e i n t e r a c t i o n s w i t h the s k u l l . In f a c t i t makes the model independent of power sou r c e a l l o w i n g i t s use w i t h the o t h e r p o t e n t i a l h e a r t h r e m e l t i n g power s o u r c e , plasma a r c h e a t i n g p r o v i d e d the t h e r m a l c o n d i t i o n s i n plasma a r c r e m e l t i n g a r e a d e q u a t e l y known. 2.2.2 S i d e Boundary C o n d i t i o n s At the e x t e r i o r s u r f a c e s of the s k u l l heat i s t r a n s f e r r e d from the s k u l l i n t o the s u r r o u n d i n g water c o o l e d copper h e a r t h . Depending on the s t a r t u p p r o c e d u r e , method of s k u l l c o n s t r u c t i o n and the power d i s t r i b u t i o n imposed on the s k u l l , the h e a r t h and s k u l l may form a s o l i d / s o l i d c o n t a c t i n the r e g i o n of the s k u l l s u r f a c e . In t h i s a r e a the boundary c o n d i t i o n i s w r i t t e n as : 9 T S 3 T S R s 3 x k s 3 y h S / H ( T S T H ) ...(2.6) at the 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 the x and y d i r e c t i o n s . Below t h i s r e g i o n of s o l i d / s o l i d c o n t a c t a vacuum 27 gap w i l l form. S i n c e the o n l y way f o r heat t o t r a n s f e r a c r o s s a vacuum gap i s by r a d i a t i o n , the boundary c o n d i t i o n i n t h i s r e g i o n i s : b b 4 4 " k Q = ~ k c = - o ( e c T c - e u T u ) . b g x S 9 y H H ...(2.7) I t i s a l s o p o s s i b l e t h a t some degree of p o i n t c o n t a c t may d e v e l o p i n the vacuum gap r e g i o n due t o the s c r a p m a t e r i a l s used i n c o n s t r u c t i n g the s k u l l . I f the degree t o which t h i s p o i n t c o n t a c t o c c u r s i s l a r g e some average between the two modes of heat t r a n s f e r may be r e q u i r e d . 2.2.3 Bottom Boundary C o n d i t i o n The bottom boundary c o n d i t i o n i s s i m i l a r t o t h e s i d e boundary c o n d i t i o n except t h a t t h e r e i s not l i k e l y t o be a r e g i o n of s o l i d / s o l i d c o n t a c t . I t i s q u i t e p o s s i b l e t h a t a l a r g e number of p o i n t c o n t a c t s w i l l d e v e l o p on the bottom n e c e s s i t a t i n g the use of the average mentioned above. The o n l y o t h e r mode of heat t r a n s f e r a c r o s s t h e gap i s r a d i a t i o n g i v i n g the boundary c o n d i t i o n as : 3T_ b 4 4 - k Q = -a(e„T„ - e T ) S 3 z ...(2.8) 28 2.2.4 C e n t e r l i n e Boundary C o n d i t i o n I f we assume the t h e r m a l f i e l d i s s y m m e t r i c a l about the c e n t e r l i n e we p e r m i t the c e n t e r l i n e t o d e f i n e the f i n a l s u r f a c e . There w i l l be no heat f l o w a c r o s s t h i s s u r f a c e g i v i n g : 9 T S - k c = 0 where y=0 S 3 y ...(2.9) 2.3. H e a r t h Boundary C o n d i t i o n s 2.3.1 E x t e r i o r S u r f a c e Boundary C o n d i t i o n The e x t e r i o r s u r f a c e s of the water c o o o l e d copper h e a r t h a r e those exposed t o the f u r n a c e w a l l and not t o the s k u l l . S i n c e a r e a s o n a b l y good vacuum i s p r e s e n t , the o n l y p o s s i b l e mode of heat f l o w i s r a d i a t i o n , t h u s the boundary c o n d i t i o n i s 9 T H 3 T H 9 T H , 4 4, , x -k„ = - k u = -k„ = -ae H(T„ - T*) ...(2.10) Hg x H 9 y H g z H H A a t the a p p r o p r i a t e l o c a t i o n s . 29 2.3.2 I n t e r i o r S u r f a c e Boundary C o n d i t i o n s The i n t e r i o r s u r f a c e s of the h e a r t h a r e tho s e 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 i n p u t t o the h e a r t h must be e q u a l t o the heat out of the s k u l l . T h e r e f o r e the heat f l u x e s a r e i d e n t i c a l i n magnitude t o tho s e s t a t e d i n s e c t i o n s 2.2.2 and 2.2.3 but o p p o s i t e i n s i g n . Thus the f o l l o w i n g e q u a t i o n s h o l d a t the a p p r o p r i a t e l o c a t i o n s . _k !!u . _k !!5 (T - T ) H 9 x Hg y S/H H S ...(2.11) 9 T H 9 T H 9 T H • 4 4^ " kI-U = " kH^ = ~kR~T~ = " a ( e H T H " e q T q ) H 9 x H 9 y H 9z H H s s ...(2.12) 2.3.3 C e n t e r l i n e Boundary C o n d i t i o n A g a i n u s i n g the assumption of l o n g i t u d i n a l symmetry, the s u r f a c e d e f i n e d by the c e n t e r l i n e becomes the f i n a l s u r f a c e and due t o the s y m m e t r i c a l n a t u r e of the t h e r m a l f i e l d t he boundary c o n d i t i o n i s : 9 T H -k = 0 where y=0 H 9 Y ...(2.13) 30 2.3.4 Water Channel Boundary 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 i n the problem i s t h a t of the h e a r t h / w a t e r i n t e r f a c e . At the s e s u r f a c e s heat f l o w s from the copper h e a r t h t o t h e c o o l i n g w a t e r . T h i s boundary c o n d i t i o n i s be s t r e p r e s e n t e d by : 3T H 3T H 3T H ""HIT = " V = " kH^ = "VTH " V ...,2.,4) a t the a p p r o p r i a t e l o c a t i o n s . The heat t r a n s f e r c o e f f i c i e n t h w i s d e t e r m i n e d based on the n a t u r e of the i n t e r f a c e w i t h r e s p e c t t o b o i l i n g and the v e l o c i t y of the c o o l i n g w a t e r . 2.4. B l o c k Model In a d d i t i o n t o the EBCHR p r o c e s s model, a model has been c o n s t r u c t e d t o d e s c r i b e the t h e r m a l regime i n a b l o c k of m a t e r i a l f r e e l y r a d i a t i n g t o the f u r n a c e w a l l s . T h i s model p r e s e n t s one of t h e l i m i t i n g c a s e s i n the c o l d h e a r t h p r o c e s s . In t h i s case i t i s not n e c e s s a r y t o i n f e r 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 the boundary c o n d i t i o n s a t a l l but the top s u r f a c e can be w r i t t e n as : 9 T B 9 T B 9 T B 4 4 - k R = - k R = -k = -ae (T? - T l ) B 9 x B 9 y B 9z B B A ...(2.15) a t the a p p r o p r i a t e l o c a t i o n s . 31 At the t o p s u r f a c e , the boundary c o n d i t i o n s t a t e d i n s e c t i o n 2.2.1 w i t h the power d i s t r i b u t i o n w i l l a p p l y . 9 T R - k n = P-,(x,y) - a e n ( T n - T, ) where z = 0 B 9 z B B B A ...(2.16) 2.5. Summary In summary, the model i n v o l v e s the stea d y s t a t e t h r e e d i m e n s i o n a l heat f l o w e q u a t i o n w i t h boundary c o n d i t i o n s . The boundary c o n d i t i o n s a r e e x p r e s s e d as g e n e r a l l y as p o s s i b l e t o make the model a p p l i c a b l e t o a wide v a r i e t y of g e o m e t r i e s , m e l t i n g m a t e r i a l s and heat f l o w c o n d i t i o n s . T h i s r e q u i r e s the s p e c i f i c a t i o n of heat t r a n s f e r c o e f f i c i e n t s f o r the boundary c o n d i t i o n s r a t h e r than the heat f l u x e s . A l l of the computer runs were made u s i n g the f o l l o w i n g e q u a t i o n s and boundary c o n d i t i o n s . 9 9T 9 9T 9 9T 9T — ( k — ) + — ( k — ) + — ( k — ) = pC — 9x 9x 9y 9y 9z 9z P9t ...(2.1) 9 9 T S 9 9 T S 9 9 T S — ( k c ) + — ( k _ ) + — ( k _ ) = 0 9x b 9 x 9y b 9 y 9z b 9 z ...(2.2) 9 9 T H 9 9 T H 9 9 T H — (k„ ) + — (k„ ) + — ( k H ) = 0 9x H 9 x 9y H 9 y 9z H 9 z ...(2.3) 9T C S 4 4 - k c = P c ( x , y ) - a e c ( T c - T.) where z=0 S 9 z S ...(2.4) 32 T Q = T(x,y) where z=0 9 T s 9 T s o s9x = -k s9y = " h S / H ( T S - T ) 9 T S s9x 9 T S = -k s 9 y • - * ( e s T s " " e H T H } 9 T S -k s 9 z " eH^H ^  9 T S b9y = 0 where y=0 9 T H " kH H9x 9 T H = -k H 9 y 9 T H = _ k H H 9z = - « H < T * ' TA> 9 T H " kH H3x 9 T H " k H 9 y " _ h S / H ( T H 9 T H " kH H 9 x 9T H  = " kH H 9 y 9T H " k« 9z " - a U H T H - e T4) 9T H " kH H 9 y = 0 where y=0 9 T H k H 9 x 9 T H = " kH H 9 y 9 T H " k» 9z = " hW ( TH " V 9 T B " k R B9x 9 T B = " k R B9y 9 T B = " k R B 9z " - ^ B ( T B - T2> 9 T B ~ k R B 9 z = P B ( x , y ) " - B ( T B " T4) where A z = 0 ...(2.5) ...(2.6) ...(2.7) ...(2.8) ...(2.9) ..(2.10) (2.11) ..(2.12) ...(2.13) ...(2.14) ...(2.15) . .(2.16) 33 CHAPTER 3 Model F o r m u l a t i o n 3 . 1 . I n t r o d u c t i o n By f a r the most c r i t i c a l p o r t i o n of any m o d e l l i n g work i s the s p e c i f i c a t i o n of boundary c o n d i t i o n s . In the case of C o l d H e a r t h R e m e l t i n g t h i s s p e c i f i c a t i o n can be r a t h e r d i f f i c u l t due t o the p r o p r i e t a r y n a t u r e of the i n d u s t r y and the w e l l u n d e r s t o o d r e l u c t a n c e of o p e r a t i n g i n d u s t r i a l f i r m s t o a l l o w e x p e r i m e n t a t i o n on t h e i r f u r n a c e s . I t i s a l s o q u i t e e x p e n s i v e t o b u i l d l a b o r a t o r y s c a l e h e a r t h f u r n a c e s due t o the l a r g e c a p i t a l c o s t s t h a t would be i n c u r r e d . T h e r e f o r e the f i e l d s of i n g o t r e m e l t i n g and c o n t i n u o u s c a s t i n g p r o v i d e c r i t i c a l i n s i g h t i n t o the d e t e r m i n a t i o n of some of the n e c e s s a r y p a r a m e t e r s . 3.2. C o l d H e a r t h Model The h e a r t h model i s d i f f e r e n t from the b l o c k model o n l y i n t h a t the t i t a n i u m s k u l l i s surrounded by a water c o o l e d copper h e a r t h . Thus much of the d i s c u s s i o n which f o l l o w s a l s o a p p l i e s t o the b l o c k model d i s c u s s e d l a t e r . As mentioned above a number of d i f f e r e n t g e o m e t r i e s a r e p o s s i b l e f o r the EBCHR p r o c e s s . Of the s e the ex a g g e r a t e d T-shape c u r r e n t l y b e i n g used by V i k i n g 33 34 M e t a l l u r g i c a l was chosen f o r s t u d y . A diagram of the assumed geometry i s shown i n F i g u r e 3.1. The l a r g e r p o r t i o n ( t h e c r o s s on the T) i s the melt b a s i n t o which t h e feed s t o c k i s added. The molten t i t a n i u m (or o t h e r m a t e r i a l ) then f l o w s down the run out t r o u g h ( the s t i c k on the T) and i n t o the i n g o t mould a t the pour l i p . The h e a r t h geometry used d u r i n g the i n i t i a l s t a g e s of t h i s work i s a s c a l e d down v e r s i o n of the V i k i n g h e a r t h ( a p p r o x i m a t e l y 7.5 t i m e s s m a l l e r ) . A l t h o u g h the w a l l s of the h e a r t h i n the V i k i n g i n s t a l l a t i o n a r e t a p e r e d t o p r o v i d e f o r easy removal of the s k u l l and s m a l l e r gaps a l l o w i n g f o r enhanced heat f l o w , i t was assumed i n the study t h a t the h e a r t h w a l l s a re v e r t i c a l and t h a t t h i s c o n f i g u r a t i o n has no a p p r e c i a b l e e f f e c t on the heat t r a n s f e r c o n d i t i o n s i n the h e a r t h . The water c h a n n e l s used by V i k i n g a r e p l a c e d i n a r a t h e r complex arrangment. T h i s arrangement would be d i f f i c u l t t o d e s c r i b e m a t h e m a t i c a l l y . For t h i s reason an a l t e r n a t e s e t of c o o l i n g water c h a n n e l s has been used which s h o u l d be a r e a s o n a b l e a p p r o x i m a t i o n of the c o n d i t i o n s found i n i n d u s t r i a l p r a c t i c e . 35 520 MELT BASIN RUN THROUGH TROUGH ft 190 POUR LIP T 280 ft HEARTH v T " 1gO +*— 350 SKULL 200 *_ •1200 1900 230 350 ALL DIMENSIONS IN MILLIMETERS NOT TO SCALE F i g u r e 3 . 1 Geometry of the S k u l l and H e a r t h Used by V i k i n g M e t a l l u r g i c a l . 4 36 3.2.1 S k u l l Boundary C o n d i t i o n s 3.2.1.1 Top Boundary C o n d i t i o n As p r e v i o u s l y d i s c u s s e d the t o p boundary c o n d i t i o n can be w r i t t e n as : 9T_ 5 4 4 - k c = P c ( x , y ) - a e c ( T c - T.) where z=0 b 3 z b ...(3.1) The two a d j u s t a b l e parameters i n t h i s e q u a t i o n a r e 1) the e m i s s i v i t y of the s k u l l e g and 2) the power d i s t r i b u t i o n P s ( x , y ) . The e m i s s i v i t y of t i t a n i u m i n a vacuum has been 1 5 r e p o r t e d by the Defence M a t e r i a l s I n f o r m a t i o n C e n t e r t o be = 0.4 between 1 200 °C and the a p p r o ximate m e l t i n g p o i n t of 1600 °C. A l t h o u g h i t would be more c o r r e c t t o v a r y the e m i s s i v i t y of t i t a n i u m w i t h t e m p e r a t u r e i t was f e l t t h a t the s t a t e d v a l u e of 0.4 was r e a s o n a b l y good over the t emperature range e x p e c t e d . S i n c e the model i s s t e a d y s t a t e , the power d i s t r i b u t i o n s u p p l i e d as the boundary c o n d i t i o n must a l s o be 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 power d i s t r i b u t i o n 37 i s a n y t h i n g but s t e a d y s t a t e . W i t h a p o i n t power s o u r c e moving a t h i g h speed a c r o s s the s u r f a c e of the s k u l l , the d e l i v e r y of power t o the s k u l l s u r f a c e v a r i e s g r e a t l y w i t h t i m e . The a c t u a l power d i s t r i b u t i o n was i n t e g r a t e d over time t o produce a time averaged power d i s t r i b u t i o n . The d i s t r i b u t i o n s used i n the c o u r s e of the work were p r o v i d e d 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 Boundary C o n d i t i o n In both c o n t i n u o u s c a s t i n g and the i n g o t r e m e l t i n g p r o c e s s e s , a r e g i o n of c o n t a c t between the water c o o l e d copper mould and a s o l i d i f y i n g s h e l l of the c a s t m a t e r i a l e x i s t s near the s u r f a c e of the l i q u i d m e t a l . As the s o l i d i f i c a t i o n p r o g r e s s e s t h i s s h e l l s h r i n k s away from the copper due t o t h e r m a l c o n t r a c t i o n l e a v i n g an a i r or vacuum gap depending on the p r o c e s s . In the EBCHR p r o c e s s e s , the s k u l l i s c o n s t r u c t e d by m e l t i n g s c r a p i n t o the h e a r t h . Due t o the n a t u r e of the s c r a p g e n e r a l l y m e l t e d i n these p r o c e s s (machine t u r n i n g s and t u b u l a r s e c t i o n s ) the c o n t a c t i s p r o b a b l y not c o n t i n u o u s but more l i k e l y t o be a s e r i e s of d i s c r e t e p o i n t c o n t a c t s . V i s u a l e x a m i n a t i o n s of a number of e l e c t r o n beam s k u l l s p r o v i d e d by V i k i n g M e t a l l u r g i c a l i n d i c a t e t h a t t h i s was the c a s e . There appeared t o be a r e g i o n of s o l i d / s o l i d or p o i n t c o n t a c t t h a t extended a p p r o x i m a t e l y 3 t o 4 cm below the s u r f a c e of the s k u l l . T h i s 38 r e g i o n was f o l l o w e d by an a r e a of s p a r s e p o i n t c o n t a c t between th e h e a r t h and s k u l l i n d i c a t i n g t h e e x i s t e n c e of a vacuum gap. The 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 w r i t t e n as : 9 T S 9 T S -k = -k = -h / (T - T ) S 9 x S 9 y S/H S H ...(3.2) 3T C 3T C o b , 4 4 . -k = -k = - o ( e TZ - e T*) S 9 x s 3 y H H ...(3.3) The heat t r a n s f e r c o e f f i c i e n t hg/j_j c a n be e v a l u a t e d by e x a m i n i n g the c o n t a c t r e g i o n i n more d e t a i l . B a l l a n t y n e 1 ^ g i v e s the heat t r a n s f e r c o e f f i c i e n t f o r t h e VAR p r o c e s s as _ 2 -1 — 1 0.02 c a l cm sec °C which r e p r e s e n t s a t h e r m a l 2 — 1 r e s i s t a n c e of 50 °C cm sec c a l . Of t h i s r e s i s t a n c e ' 2 -1 1 °C cm sec c a l b e l o n g s t o the copper used as the i n g o t 2 — 1 mould and 1-2 °C cm sec c a l belongs t o the mould/water i n t e r f a c e . S i n c e t h e s e two r e s i s t a n c e s a r e not t o be i n c l u d e d i n the s k u l l / h e a r t h heat t r a n s f e r c o e f f i c i e n t , a 2 — 1 t h e r m a l r e s i s t a n c e of about 48 °C cm sec c a l s h o u l d be e x p e c t e d f o r the r e s i s t a n c e i n q u e s t i o n . However, the c o n t a c t between the s k u l l and the h e a r t h i s not p e r f e c t . The p r e s e n c e of these i m p e r f e c t i o n s w i l l t e n d t o r a i s e the t h e r m a l r e s i s t a n c e . I f the v a l u e i s a r b i t r a i l y i n c r e a s e d by 2 -1 30%, a r e s i s t a n c e of 60 °C cm sec c a l i s o b t a i n e d which c o r r e s p o n d s t o a heat t r a n s f e r c o e f f i c i e n t of — 2 — 1 -1 0.016 c a l cm sec °C . T h i s v a l u e f a l l s i n the range 39 -2 0„-1 of 0.011 - 0.03 c a l cm sec C found by Fenech and 1 7 Rohsenow and a l s o f a l l s i n the range used by E i s e n and •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 and 19 -o - 1 - 1 G e i g e r used 0.015 c a l cm sec °C i n work they d i d i n m o d e l l i n g the ESR p r o c e s s . 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 from K r i e t h who quotes a v a l u e of 0.1. S i n c e the two s u r f a c e s a r e e s s e n t i a l l y p a r a l l e l p l a t e s i n 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 view f a c t o r s a r e r e q u i r e d . 3.2.1.3 Bottom Boundary C o n d i t i o n At the bottom of the s k u l l the boundary c o n d i t i o n can be w r i t t e n as : s 4 4, - k q = - a ( e Q T „ - e T ) S 3 z S S H H ...(3.4) T h i s i s e x a c t l y the same c o n d i t i o n as t h a t of the vacuum gap on the s i d e s . The f o r m a t i o n of p o i n t c o n t a c t s on the bottom s u r f a c e i s a l s o a p o s s i b i l i l t y . The s k u l l s were examined f o r t h i s p o s s i b i l i t y and, i n g e n e r a l , the bottom s u r f a c e s were r e a s o n a b l y f r e e from e x t r u s i o n s which r e s u l t i n p o i n t c o n t a c t s . Areas around the edges of the s k u l l d i d e x t r u d e beyond the l e v e l of the b u l k but the c o n t r i b u t i o n of t h e s e r e g i o n s t o the o v e r a l l heat t r a n s f e r s u r f a c e a t the bottom 40 of the s k u l l was deemed n e g l i g i b l e . 3.2.1.4 Thermal C o n d u c t i v i t y The t h e r m a l c o n d u c t i v i t y of T i - 6 A l - 4 V as a f u n c t i o n of temperature has been r e p o r t e d by the Defence 1 5 M a t e r i a l s I n f o r m a t i o n Center and i s shown i n f i g u r e 3.2. R e s e a r c h e r s i n the i n g o t r e m e l t i n g and c o n t i n u o u s c a s t i n g f i e l d s have found t h a t the l i q u i d m e t a l motion t h a t r e s u l t s from n a t u r a l or f o r c e d c o n v e c t i o n i s b e s t s i m u l a t e d by s e t t i n g the t h e r m a l c o n d u c t i v i t y of the l i q u i d t o some f a c t o r m u l t i p l i e d by the t h e r m a l c o n d u c t i v i t y a t the s o l i d u s t e m p e r a t u r e . B a l l a n t y n e 1 ^ found t h a t the m u l t i p l y i n g f a c t o r s h o u l d be 2-3 f o r ESR m e l t s and sometimes as h i g h as 10 f o r VAR m e l t s u s i n g h i g h melt r a t e s . These v a l u e s a r e i n good agreement w i t h o b s e r v a t i o n s by o t h e r workers i n t h i s 1 ft 0 1 0 ? 0 Q f i e l d . As w e l l , H a r r i s o n 3 found t h a t n a t u r a l c o n v e c t i o n i n l i q u i d t i n i n c r e a s e s the t h e r m a l c o n d u c t i v i t y by a f a c t o r which i s not g r e a t e r than 7-10. S i n c e an e l e c t r o n beam has a v e r y low momentum, l i t t l e or no s t i r r i n g s h o u l d r e s u l t from i n t e r a c t i o n s of the l i q u i d m e t a l and the power s o u r c e . The l i q u i d m e t a l i s a l s o i n a t h e r m a l s t a b l e c o n f i g u r a t i o n and t h e r e f o r e t h e r e s h o u l d be a r e l a t i v e l y s m a l l amount of l i q u i d motion when compared t o the ESR p r o c e s s 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 t h a t the m u l t i p l y i n g f a c t o r i n an e l e c t r o n beam f u r n a c e w i t h no enhancement of the f l u i d f l o w s h o u l d be on the o r d e r of 1-2. 0.5-1 £ 0.4 J 0.1-1 1 1 r-1000 1300 1600 1900 Temperature °C F i g u r e 3.2 Thermal C o n d u c t i v i t y of T i t a n i u m 6A1-4V as a F u n c t i o n of Temperature. 42 I n i t i a l l y the i n c r e a s e d t h e r m a l c o n d u c t i v i t y was s p e c i f i e d u s i n g a s t e p f u n c t i o n . T h i s p r o c e d u r e produced an i n s t a b i l i t y i n the s o l u t i o n which caused the n u m e r i c a l s o l u t i o n t o o s c i l l a t e . I f the t h e r m a l c o n d u c t i v i t y was not r a i s e d i n a s t e p - w i s e manner ( i . e . at t h e s o l i d u s t e m p e r a t u r e ) but was i n c r e a s e d over a t emperature range ( i . e . s o l i d u s t o l i q u i d u s ) then the n u m e r i c a l i n s t a b i l i t y d i s a p p e a r e d and one s o l u t i o n was o b t a i n e d . The r e s u l t i n g t h e r m a l c o n d u c t i v i t y ( w i t h enhancement f o r l i q u i d motion) as a f u n c t i o n of temperature i s shown i n f i g u r e 3.3. I t s h o u l d be noted t h a t when no enhanced l i q u i d motion was used, the n u m e r i c a l s o l u t i o n a l w a y s converged. D u r i n g the r e s e a r c h programme some of the s k u l l s r e c e i v e d from A. Johnson and V i k i n g M e t a l l u r g i c a l were c u t t hrough the c r o s s s e c t i o n s . These s k u l l s r e v e a l e d t h a t the r e g i o n of the s k u l l near the bottom was u s u a l l y f i l l e d w i t h v o i d s and t h e r e f o r e had a h i g h p o r o s i t y (see f i g u r e 3.4). These v o i d s c o u l d have a s i g n i f i c a n t e f f e c t on the t h e r m a l c o n d u c t i v i t y of the s k u l l i n t h i s a r e a . S i n c e t h e r e i s no adequate way t o q u a n t i f y t h i s e f f e c t w i t h o u t engaging i n an e x t e n s i v e program of s k u l l s e c t i o n i n g or x - r a y e v a l u a t i o n , F i g u r e 3.3 Thermal C o n d u c t i v i t y of T i t a n i u m 6A1-4V as 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 M o t i o n . F i g u r e 3.4 X-Ray Photograph of a Johnson S k u l l Showing Large F r a c t i o n of V o i d s . 45 i t 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 Boundary C o n d i t i o n s The boundary c o n d i t i o n s i n the h e a r t h can be d i v i d e d i n t o 3 p a r t s : 1) t h e i n t e r i o r , 2) the e x t e r i o r and 3) the water c h a n n e l s . 3.2.2.1 I n t e r i o r Boundary C o n d i t i o n The boundary c o n d i t i o n s on the i n t e r i o r s u r f a c e s have a l r e a d y been d e s c r i b e d and c h a r a c t e r i z e d i n the s e c t i o n s on the s k u l l boundary c o n d i t i o n s . I t i s o n l y n e c e s s a r y t o i n s u r e t h a t the boundary c o n d i t i o n s on the h e a r t h a r e c o n s i s t e n t w i t h those of the s k u l l . 3.2.2.2 E x t e r i o r Boundary C o n d i t i o n s The e x t e r i o r s u r f a c e s of the h e a r t h a r e exposed t o the w a l l s of the f u r n a c e and a r e t h e r e f o r e a l l o w e d t o f r e e l y r a d i a t e h e a t . There a r e some assumptions t h a t can be made r e g a r d i n g the magnitude of t h i s heat f l u x , however. F i r s t l y , i f the water c o o l i n g i s s u f f i c i e n t , the t e m p e r a t u r e s a t the h e a r t h s u r f a c e s h o u l d be r e l a t i v e l y low and the heat f l o w between t h e s e s u r f a c e s and the f u r n a c e w a l l n e g l i g i b l e i n comparison t o the heat f l o w i n t o the c o o l i n g water. S e c o n d l y , the temperature g r a d i e n t s i n the r e g i o n between the water c h a n n e l and the e x t e r i o r s u r f a c e of the h e a r t h a re ex 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 v e r y l i t t l e heat w i l l f l o w i n t o the water from the s i d e o p p o s i t e the s k u l l . For 46 t h i s reason i t i s unnecessary t o i n c l u d e t h e m a t e r i a l between the c o o l i n g c h a n n e l s and the e x t e r i o r s u r f a c e s . T h e r e f o r e the e x t e r i o r boundary c o n d i t i o n s become : H 3 x H 3 y H 9z ...(3.5) The o n l y e x c e p t i o n t o t h i s statement i s a t the t o p s u r f a c e . Here the tempe r a t u r e s are e x p e c t e d t o be h i g h enough t o ge n e r a t e s i g n i f i c a n t heat f l o w from the h e a r t h t o the f u r n a c e w a l l s . T h e r e f o r e a r a d i a t i o n boundary c o n d i t i o n e x i s t s : H 9 z H H A ...(3.6) 3.2.2.3 Water Channel Boundary C o n d i t i o n s In o p e r a t i n g i n d u s t r i a l h e a r t h s the water c h a n n e l s ar e c o n s t r u c t e d by e i t h e r d r i l l i n g h o l e s i n the p r o p e r l o c a t i o n s i n a copper b l o c k or c a s t i n g copper around t u b i n g 23 a r r a n g e d i n the d e s i r e d f a s h i o n . Both of th e s e c o n s t r u c t i o n methods produce c y l i n d r i c a l water c h a n n e l s which a r e d i f f i c u l t t o handle i n a r e c t a n g u l a r c o o r d i n a t e f i n i t e d i f f e r e n c e system. T h e r e f o r e the model uses c o o l i n g c h a n n e l s of square c r o s s - s e c t i o n . P r o v i d e d the square c h a n n e l s have the same wett e d p e r i m e t e r as the c y l i n d r i c a l c h a n n e l s they r e p l a c e , the d i f f e r e n c e i n the heat t r a n s f e r c h a r a c t e r i s t i c s between the two are n e g l i g i b l e . 47 The network of c o o l i n g c h a n n e l s employed a t V i k i n g M e t a l l u r g i c a l i s complex and c o n s e q u e n t l y e x t r e m e l y d i f f i c u l t t o model a c c u r a t e l y . F or t h i s reason an a r b i t r a r y arrangement of c o o l i n g water c h a n n e l s was chosen f o r the h e a r t h model. The c o o l i n g water c h a n n e l s used i n the model are spaced e v e n l y over the whole of the h e a r t h mould. The boundary c o n d i t i o n f o r the c o o l i n g c h a n n e l s has 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 as : 9 T H 3 T H 9 T H " KH A X = " K H 9 Y = " KH-^ = VTH " V . . . ( 3 . 7 ) Thus the s p e c i f i c a t i o n of h w s p e c i f i e s t he boundary c o n d i t i o n . Two e f f e c t s govern t he magnitude of the heat t r a n s f e r c o e f f i c i e n t h w > These a r e : 1) the n a t u r e of the i n t e r f a c e w i t h r e s p e c t t o b o i l i n g and 2) the water v e l o c i t y a t the i n t e r f a c e . C o l d c r u c i b l e and c o l d mould p r o c e s s e s g e n e r a l l y o p e r a t e i n e i t h e r the n o n - b o i l i n g or n u c l e a t e b o i l i n g regime. I n the case of n u c l e a t e b o i l i n g , s u r f a c e e x c e s s temperature - d e f i n e d as the d i f f e r e n c e between the s u r f a c e temperature and the s a t u r a t i o n t e m p e r a t u r e - i s i n the range of 5 - 100 °C. T h e r e f o r e , i n o r d e r t o a c h i e v e n u c l e a t e b o i l i n g s u r f a c e t e m p e r a t u r e s must be i n the range of 105 200 °C at the water/copper i n t e r f a c e . In the case of the c o l d h e a r t h p r o c e s s e s the te m p e r a t u r e s e x p e c t e d a t the i n t e r f a c e are much l e s s i n d i c a t i n g t h a t n o n - b o i l i n g heat 48 t r a n s f e r i s the o p e r a t i v e mode. In n o n - b o i l i n g heat t r a n s f e r under f o r c e d c o n v e c t i o n , the S e i d e r - T a t e r e l a t i o n s h i p produces good r e s u l t s f o r the v a l u e of t h e heat t r a n s f e r c o e f f i c i e n t . The r e l a t i o n s h i p i s : u - n no-i K I 0 - 8 M 1 / 3 / B \ ° - 1 4 Nu " ° ' 0 2 3 Re N P r ( ~ > ,, .. where N_, > 1 0000 and 0.7 « N„ Re 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 by Samarasekera and Brimacombe, J o s h i , and B a l l a n t y n e . A p p l y i n g the r e l a t i o n s h i p u s i n g water a t 20 °C and 1.25 cm square c o o l i n g c h a n n e l s the e q u a t i o n f o r the heat t r a n s f e r c o e f f i c i e n t becomes : h„ = 0.01257 ( V r 7 ) 0 , 8  W W ...(3.9) where V i s i n cm sec 1. U s i n g a t y p i c a l v e l o c i t y of 200 cm sec 1 f o r t h e water v e l o c i t y g i v e s — 2 — 1 h w = 0.871 w a t t s cm °C . I f a t h e r m a l r e s i s t a n c e type of c a l c u l a t i o n i s then used t o det e r m i n e the r e l a t i v e magnitude of t h i s t h e r m a l r e s i s t a n c e , i t i s found t h a t the h e a r t h / w a t e r i n t e r f a c e a c c o u n t s f o r l e s s than 1% of the t o t a l heat t r a n s f e r r e s i s t a n c e . The b u l k of the heat t r a n s f e r r e s i s t a n c e i s a c c o u n t e d f o r by the vacuum gap 49 (---94%) w i t h the o t h e r heat t r a n s f e r r e s i s t a n c e s a c c o u n t i n g f o r the r e m a i n i n g 6%. From t h i s r e s u l t i t would appear t h a t the v a l u e s o b t a i n e d u s i n g e q u a t i o n 3.9 a r e s u f f i c i e n t l y a c c u r a t e . 3.2.2.4 Thermal C o n d u c t i v i t y Over the temperature range e x p e c t e d i n the water c o o l e d copper h e a r t h , the t h e r m a l c o n d u c t i v i t y of copper i s e s s e n t i a l l y c o n s t a n t a t 3.9 w a t t s cm 1 °C 1 . 2 ^ T h i s a l l o w s the o v e r a l l heat f l o w e q u a t i o n i n the h e a r t h t o be w r i t t e n as : 2 2 2 3 Z T H 3 Z T H 9^T H + + = 0 3 x 2 3 y 2 3 z 2 ...(3.10) 3.3. B l o c k Model The b l o c k model which i s used under c o n d i t i o n s of e l e c t r o n beam power i n p u t i s e s s e n t i a l l y the s k u l l model w i t h o u t the p r o v i s i o n f o r s o l i d / s o l i d or p o i n t c o n t a c t . That i s t o say the b l o c k i s f r e e t o r a d i a t e t o the f u r n a c e w a l l s 50 i n a l l d i r e c t i o n s . 3.4, N u m e r i c a l Technique A f t e r e xamining the m a t h e m a t i c a l l y s t a t e d model i t i s r e a s o n a b l y c l e a r t h a t an a n a l y t i c a l s o l u t i o n would be d i f f i c u l t i f not i m p o s s i b l e . Thus a n u m e r i c a l s o l u t i o n t e c h n i q u e was chosen. As w i t h o t h e r n u m e r i c a l problems i t i s p o s s i b l e t o choose e i t h e r t h e f i n i t e d i f f e r e n c e or f i n i t e element method f o r the s o l u t i o n t e c h n i q u e . Both of th e s e methods i n v o l v e the s o l u t i o n of a system of s i m u l t a n e o u s e q u a t i o n s . S i n c e no r e d u c t i o n of the system i s p o s s i b l e u s i n g the f i n i t e d i f f e r e n c e t e c h n i q u e i n the stea d y s t a t e c a s e , the o n l y advantage t h i s method has over the f i n i t e element approach i s i t s s i m p l i c i t y i n 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 . U s i n g the f i n i t e d i f f e r e n c e approach a l a r g e s p a r s e m a t r i x r e p r e s e n t i n g the system of l i n e a r e q u a t i o n s i s g e n e r a t e d . In o r d e r t o s o l v e f o r the temperature f i e l d , t h i s system of e q u a t i o n s must be reduced t o the s o l u t i o n v e c t o r . A f t e r t e s t i n g a v a r i e t y of the t e c h n i q u e s a v a i l a b l e , i t was found t h a t s u c c e s s i v e symmetric o v e r r e l a x a t i o n w i t h 2 6 c o n j u g a t e g r a d i e n t a c c e l e r a t i o n was the b e s t i n terms of 51 b o t h speed and c o s t . 3.4.1 Non L i n e a r i t i e s U n f o r t u n a t e l y u s i n g the f i n i t e d i f f e r e n c e t e c h n i q u e r e q u i r e s the system of e q u a t i o n s t o be l i n e a r . In the c u r r e n t problem two n o n - l i n e a r i t i e s e x i s t . These a r e : 1) v a r i a t i o n of t h e r m a l c o n d u c t i v i t y w i t h temperature and 2) the f o u r t h power of t emperature i n the r a d i a t i o n boundary c o n d i t i o n . 3.4.1.1 Thermal C o n d u c t i v i t y The f i n i t e d i f f e r e n c e t e c h n i q u e i s c o n d u s i v e t o the s o l v i n g of the t h e r m a l c o n d u c t i v i t y problem. The t h e r m a l c o n d u c t i v i t y a t each node i s e v a l u a t e d on the b a s i s of the t e mperature at the p r e v i o u s i t e r a t i o n (or on the b a s i s of some i n i t i a l g u e s s ) . The system of e q u a t i o n s i s then g e n e r a t e d and s o l v e d and the p r o c e s s r e p e a t e d . One o t h e r c o n d i t i o n of v a r y i n g t h e r m a l c o n d u c t i v i t y w i t h t emperature i s t h a t c a r e must be taken i n e v a l u a t i n g the t h e r m a l c o n d u c t i v i t y a t a node. I t i s p o s s i b l e t o produce l a r g e e r r o r s i f the t h e r m a l c o n d u c t i v i t y used i s not an average v a l u e . For example, c o n s i d e r two a d j a c e n t nodes at t e m p e r a t u r e s T 1 and T 2 r e s p e c t i v e l y . These two nodes have t h e r m a l c o n d u c t i v i t i e s k, and k 0 a s s o c i a t e d 52 w i t h them. G i v e n t h a t T 2 i s g r e a t e r than T 1 then the heat e n t e r i n g node 1 i s Q ( T 2 " V (-). = k A 1 1 A ...(3.12) and the heat l e a v i n g node two i s Q ( T 2 " V (-)_ = k, A A ...(3.13) I f k 1 * k 2 (as i s the case here) then (Q/A) 1 * (Q/A) 2 which i s p h y s i c a l l y i m p o s s i b l e . To s o l v e the problem an average t h e r m a l c o n d u c t i v i t y k A V i s d e f i n e d as ( k 1 + k 2 } ^ = = * ...(3.14) then (-) = - ( - ) , = k —2 L_ A 1 A A ...(3.15) 5 3 3 . 4 . 1 . 2 R a d i a t i o n The second n o n - l i n e a r i t y which d e v e l o p s i s t h a t a s s o c i a t e d w i t h the f o u r t h power of temperature i n the r a d i a t i o n e q u a t i o n : " " 8 < T S " T J ' . . . ( 3 . , 6 ) The f i n i t e d i f f e r e n c e method i s not as amenable t o s o l v i n g t h i s n o n - l i n e a r i t y . In t h i s case the boundary c o n d i t i o n i s l i n e a r i z e d by d e f i n i n g a pseudo-heat t r a n s f e r c o e f f i c i e n t : h R = or hR " a e C ( T 4 - TJ ) S S A ( T - T . ) . . . ( 3 . 1 7 ) S A a ( e S T S - eH TH> ( T g - T H ) . . . ( 3 . 1 7 ) U s i n g t h i s method a l s o r e q u i r e s an i t e r a t i v e t y pe of s o l u t i o n , i n which the pseudo-heat t r a n s f e r c o e f f i c i e n t i s e v a l u a t e d based on the p r e v i o u s i t e r a t i o n 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 I t e r a t i v e S o l u t i o n Both n o n - l i n e a r i t i e s i n the model r e q u i r e t h a t the s o l u t i o n be o b t a i n e d i t e r a t i v e l y . To a c c o m p l i s h t h i s s o l u t i o n , an i n i t i a l guess must be p r o v i d e d t o the system. 54 T h i s i n i t i a l guess i s then used t o generate a s o l u t i o n . To a l l o w f o r q u i c k e r convergence t o the f i n a l s o l u t i o n , a s h o o t i n g t e c h n i q u e which combines the c u r r e n t s o l u t i o n and one or more p r e v i o u s s o l u t i o n s i s used. At some p o i n t i n the i t e r a t i v e p r o c e s s an e n d p o i n t must be reached. I t was p o s s i b l e t o d e f i n e t h e e n d p o i n t i n a number of d i f f e r e n t ways. The most o b v i o u s way was t o d e f i n e the d i f f e r e n c e i n t e m p e r a t u r e between i t e r a t i o n s and s t o p when 1) the average of t h e s e d i f f e r e n c e s i s s m a l l or 2) the maximum ( a b s o l u t e v a l u e ) of these d i f f e r e n c e s i s s m a l l . I t was a l s o p o s s i b l e t o c a l c u l a t e the heat l o s t from the s k u l l and compare t h i s t o t h e i n p u t power and then t o cease i t e r a t i o n s when the d i f f e r e n c e between t h e s e two v a l u e s d i f f e r s by a s u f f i c i e n t l y s m a l l v a l u e over a p e r i o d of i t e r a t i o n s . In g e n e r a l i t was found t h a t u s i n g a maximum p e r m i s s i b l e d i f f e r e n c e of about 1 °C produced a p e r c e n t a g e e r r o r i n the power c a l c u l a t i o n of l e s s than 5% and an average d i f f e r e n c e of l e s s than 0.01 °C. Only the maximum d i f f e r e n c e and p e r c e n t a g e e r r o r convergence c r i t e r i a have been used t o produce r e s u l t s . 55 CHAPTER 4 Model R e s u l t s 4.1. I n t r o d u c t i o n The EBCHR model can be run u s i n g e i t h e r a temperature or power d i s t r i b u t i o n as the t o p s u r f a c e boundary c o n d i t i o n . A l t h o u g h f o r any s e t of c o n d i t i o n s a p a r t i c u l a r power d i s t r i b u t i o n w i l l produce a s p e c i f i c t e mperature d i s t r i b u t i o n ( i n o t h e r words, i n o r d e r t o m a i n t a i n a temperature d i s t r i b u t i o n a s p e c i f i c power d i s t r i b u t i o n i s r e q u i r e d ) , t h e l i n k between t h e two i s not c l e a r . I t i s advantageous t o examine the e f f e c t s of some v a r i a b l e s u s i n g a temperature d i s t r i b u t i o n where i t i s p o s s i b l e t o i g n o r e the m e l t i n g power and l i q u i d m e t a l superheat a t the pour l i p . Even though no comprehensive program t o v e r i f y the model was c a r r i e d o u t , c e r t a i n o b s e r v a t i o n s made d i r e c t l y on EBCHR s k u l l s and of m e l t i n g o p e r a t i o n s i n the i n d u s t r i a l f u r n a c e s have a l l o w e d a s m a l l degree of c o n f i d e n c e 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 the model. 4.2. Temperature D i s t r i b u t i o n Model A. Johnson m e t a l s have conducted measurements of the s u r f a c e temperature i n t h e i r h e a r t h f u r n a c e u s i n g 55 56 2 6 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 melt b a s i n as h i g h as 80 - 140 °C (150 - 250 °F) above the l i q u i d u s t e mperature of the m e l t s t o c k . Top temperature d i s t r i b u t i o n s were c o n s t r u c t e d t o r e f l e c t t h e s e measurements. To keep the temp e r a t u r e d i s t r i b u t i o n as s i m p l e as p o s s i b l e , t e m p e r a t u r e s were v a r i e d l i n e a r l y from some lower v a l u e a t t h e p e r i m e t e r of t h e s k u l l t o a maximum of v a l u e of the l i q u i d u s t e m p e r a t u r e p l u s some su p e r h e a t a t a p l a t e a u a t a f i x e d d i s t a n c e form the s k u l l p e r i m e t e r . An example of t h i s t y pe of temp e r a t u r e d i s t r i b u t i o n i s shown i n f i g u r e 4.1. The model was run u s i n g t i t a n i u m 6 weight % aluminum 4 weight % vanadium (Ti6A14V or 6-4) as the melt s t o c k . U s i n g a l i q u i d t h e r m a l c o n d u c t i v i t y m u l t i p y i n g f a c t o r (LTCMF) of 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 a 110 °C s u p e r h e a t , the b o u n d a r i e s of the l i q u i d p o o l were g e n e r a t e d . These b o u n d a r i e s a r e shown i n f i g u r e 4.2 th r o u g h f i g u r e 4.4. F i g u r e 4.2 and 4.3 a r e c o n t o u r maps of the l i q u i d u s / s o l i d u s i s o t h e r m s and f i g u r e 4.4 shows the p o o l p r o f i l e a t the c e n t e r l i n e of the s k u l l . U s i n g f i g u r e s 4.2 and 4.3 i t can be seen t h a t the l i q u i d p o o l extends over more t h a t h a l f of t h e s u r f a c e a r e a of the s k u l l and t o a depth of a p p r o x i m a t e l y 1 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 l e n g t h and t h e r e f o r e l i q u i d i s p r e s e n t t o a maximum depth of 1.75 cm. 57 o r - U~> — o o I I 1 I I I I I 1 I I I T V s"ti 's-6 s'z. s*s 9'£ s't eo-Sd3l3NI1N33 F i g u r e 4.1 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 i i r~ S "M 9*6 i — i — i — i i—r~ o in — o ro C M — o — C M — o C D — o -- o in CO — • L U 1 1 1 - u 1 c n o CD in o cn i i r F i g u r e 4.2 Contour Map Showing t h e B o u n d a r i e s of the L i q u i d P o o l , C o n t o u r i n g Temperature = 1625 Superheat = 1 1 0 °C, LTCMF = 1. (C, 59 i — i — r 9 ' l l 9/6 i i i m i—r S'£ 9'9 S"E 9'1 m ' o . c n ' o . r-' o LU ' C D o o ' LT) O S"0-F i g u r e 4.3 Contour Map Showing the B o u n d a r i e s of t h e S o l i d S k u l l , C o n t o u r i n g Temperature = 1595 °C, Superheat = 1 1 0 °C, LTCMF = 1. 60 i—i—i—r i i I i i i n i i i — r . O'SZMO/QtSIO'SZW 0'6 0'L 0'£ OT 0"I O'l dU31 • JdHS ScJ313WIlN33 F i g u r e 4.4 P o o l P r o f i l e s a t 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, Superheat = 110 °C, LTCMF = 1. 61 T h i s p o o l depth i s r e a s o n a b l y c l o s e t o t h a t o b s e r v e d d u r i n g f u r n a c e o p e r a t i o n . Both A. Johnson and V i k i n g i n d i c a t e t h a t the melt p o o l i s from 2.5 - 5 cm (1 - 2 i n ) i n d e p t h . The x-ray e x a m i n a t i o n of a s i m i l a r s k u l l (see f i g u r e 4.5) a l s o shows the p o o l depth t o be s h a l l o w . The p o o l depth i s marked by the b r i g h t spot on the photograph. T h i s b r i g h t spot i s a h i g h d e n s i t y t u n g s t e n c a r b i d e p a r t i c l e which was d e l i b e r a t e l y p l a c e d i n the f e e d s t o c k t h a t produced t h i s s k u l l . S i n c e t u n g s t e n c a r b i d e i s s i g n i f i c a n t l y denser than t i t a n i u m , the l o c a t i o n of the WC p a r t i c l e , marks the l i q u i d / s o l i d i n t e r f a c e a t t h i s p a r t i c u l a r l o c a t i o n . I t i s d i f f i c u l t t o make any d e f i n i t e c o n c l u s i o n s about the v a l i d i t y of the model based on t h i s e v i d e n c e a l o n e due t o the r a t h e r a b s t r a c t n a t u r e of the temperature d i s t r i b u t i o n imposed on the s k u l l and the d i f f e r e n c e i n the g e o m e t r i e s of the A. Johnson s k u l l and the m o d e l l e d one. I t i s i n f o r m a t i v e t o c o n s t r u c t a heat b a l a n c e on the s k u l l . R e s u l t s of t h i s c a l c u l a t i o n a r e shown i n t a b l e 4.1. The power r e q u i r e d t o produce t h e c a l c u l a t e d t e m perature f i e l d i s on the o r d e r of 19 KW. Of t h i s 19 KW, 6.7 KW or <*35% i s r a d i a t e d back t o the f u r n a c e from the s u r f a c e of t h e s k u l l . The b a l a n c e ( or 12.4 KW) i s removed from the s k u l l t h r ough the h e a r t h by the c o o l i n g w a t e r . A l s o of i n t e r e s t i s the c a l c u l a t i o n of the e v a p o r a t i o n r a t e of a l l o y elements from the l i q u i d m e t a l . .7 U s i n g d a t a p r o v i d e d by Tagaki and assuming t h a t e v a p o r a t i o n 62 F i g u r e 4.5 X-ray Photograph of an A. Johnson S k u l l Showing the L o c a t i o n of the S o l i d u s . 63 Superheat S u r f a c e H e a r t h T o t a l Temperature L o s s e s Heat Power T S H °C KW (%) KW (%) KW (%) 110 6.7 ( 3 4 . 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 Balance 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 r a t e l i m i t i n g s t e p , the e v a p o r a t i o n r a t e of aluminum, vanadium and 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 e q u a t i o n : p°7 C A *x 'x X m = x p/27rRMxT ...(4.1) where, p° (Pa) i s the vapour p r e s s u r e of the pure element a t the temperature T (°K), 7 i s the a c t i v i t y c o e f f i c i e n t of component x a t the t e m p e r a t u r e T, M i s the m o l e c u l a r weight of component x, p i s the d e n s i t y i n moles/cm , 3 C x i s the c o n c e n t r a t i o n of x i n (g/cm ) and A i s the a r e a . Making the g r o s s assumption t h a t the c o n c e n t r a t i o n C x i s c o n s t a n t over the e n t i r e l i q u i d p o o l , the 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 w i l l be g i v e n by e q u a t i o n 4.1. The t o t a l mass f l u x from the s k u l l can then be c a l c u l a t e d by i n t e g r a t i n g e q u a t i o n 4.1 over the a r e a of the l i q u i d p o o l . A p p l y i n g t h i s e q u a t i o n t o the temperature d i s t r i b u t i o n used t o g e n e r a t e f i g u r e s 4.2 t h r o u g h 4.4 r e s u l t s i n 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 of S u r f a c e Temperature I n c r e a s i n g the s u r f a c e superheat temperature t o 150 °C has the e f f e c t of i n c r e a s i n g both the s u r f a c e a r e a 65 Superheat E v a p o r a t i o n Rates Temperature kg/hr T S H °C T i A l v 110 0.056 1.40 0.00 150 0.096 2.19 0.00 200 0.158 3.03 0.00 T a b l e 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 Rates f o r V a r i o u s Temperature D i s t r i b u t i o n s . 66 and the depth of the l i q u i d p o o l (as shown i n f i g u r e s 4.6 t o 4.8). The p o o l depth i n c r e a s e s t o a p p r o x i m a t e l y 1.5 cm w i t h the l i q u i d / s o l i d gap r e m a i n i n g a t a p p r o x i m a t e l y 0.75 cm. A 200 °C superheat temperature produces a l i q u i d p o o l of =2 cm depth w i t h the same 0.75 cm l i q u i d / s o l i d gap (see f i g u r e s 4.9 t o 4.11). These r e s u l t s of the heat b a l a n c e c a l c u l a t i o n s as w e l l as the e v a p o r a t i o n r a t e s f o r the t h r e e superheat t e m p e r a t u r e s a re summarized i n t a b l e 4.3. As the s u r f a c e s u p e r h e a t t e m p e r a t u r e r i s e s from 110 °C t o 200 °C the depth and volume of the l i q u i d p o o l a l s o i n c r e a s e . These i n c r e a s e s a r e o f f s e t however by i n c r e a s e s i n the amount of heat r a d i a t e d t o the f u r n a c e and the maximum e v a p o r a t i o n r a t e . On the two extremes, a t 110 °C superheat the p o o l volume i s a p p r o x i m a t e l y 66 cm, the heat l o s t t o the f u r n a c e i s 6.7 KW and the aluminum e v a p o r a t i o n r a t e i s 1.4 kg/hr. On the o t h e r hand, a t 200 °C 3 superheat t h e r e i s a p o o l volume of 135 cm , 7.4 KW of r a d i a t e d heat and an aluminum e v a p o r a t i o n r a t e of 3.03 kg/hr r e p r e s e n t i n g i n c r e a s e s of 105%, 10% and 116% r e s p e c t i v e l y . I t i s apparent from t h i s d a t a t h a t some i n c r e a s e i n p o o l volume can be ex p e c t e d when the temperature a t the s u r f a c e 67 i i i i i i i I i i i i r i £' II S"6 S'Z. S'S S'E S" I • S'O-Sd313WIiN33 F i g u r e 4.6 Contour Map Showing the B o u n d a r i e s of the L i q u i d P o o l , C o n t o u r i n g Temperature = 1625 °C, Superheat = 1 5 0 °C, LTCMF = 1. 68 F i g u r e 4.7 Contour Map Showing the B o u n d a r i e s of the S o l i d S k u l l , C o n t o u r i n g Temperature = 1595 °C, Superheat = 1 5 0 °C, LTCMF = 1. o F i g u r e 4.8 P o o l P r o f i l e s a t 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, Superheat = 150 °C, LTCMF = 1. 70 " i— i i — i — r 9"I1 S"B i i i— L 9"9 9'£ 9' in C M n o r \ j o . e n o . r -' o . in • L U - <rnL L U o r-o in o m 9 '0 -F i g u r e 4.9 Contour Map Showing the B o u n d a r i e s of the L i q u i d P o o l , C o n t o u r i n g Temperature = 1625 Superheat = 200 °C, LTCMF = 1. 'C, 71 - m . i—i—i i — i— i — i — i i i r S ' l l S'6 S'S £"£ £ ' I Sd313HIlN33 . in C M m C M o C M O . C D ' O . r-' o . in • a • U J U J C D o r -a m CD m S" o-F i g u r e 4.10 Contour Map Showing the B o u n d a r i e s of the S o l i d S k u l l , C o n t o u r i n g Temperature = 1595 °C, Superheat = 200 °C, LTCMF = 1. F i g u r e 4.11 P o o l P r o f i l e s a t 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, Superheat = 200 °C, LTCMF = 1. 73 Superheat Temperature °C LTCMF S u r f a c e L o s s e s KW H e a r t h Heat KW T o t a l Power KW P o o l Depth cm P o o l Volume cm 3 110 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 4.25 171 T a b l e 4.3 Summary of P o o l Data and 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 Runs. 74 of the s k u l l i s i n c r e a s e d . T h i s i n c r e a s e i s o b t a i n e d a t no s m a l l expense i n terms of e v a p o r a t i o n r a t e s of a l l o y i n g e l e m e n t s . 4.2.2 E f f e c t of L i q u i d Movement The e f f e c t s of l i q u i d movement can be o b t a i n e d by a r t i f i c i a l l y i n c r e a s i n g the t h e r m a l c o n d u c t i v i t y of the l i q u i d m e t a l by some f a c t o r . As p r e v i o u s l y d i s c u s s e d v e r y s m a l l v e l o c i t i e s can have a major e f f e c t on the t h e r m a l c o n d u c t i v i t y of the l i q u i d m e t a l . The r e s u l t s of v a r y i n g the l i q u i d t h e r m a l c o n d u c t i v i t y f a c t o r LTCMF from 1 t o 5 a r e shown i n f i g u r e s 4.12 t o 4.14 f o r a 150 °C s u p e r h e a t . The r e s u l t s of the heat b a l a n c e c a l c u l a t i o n s a r e a l s o t a b u l a t e d i n t a b l e 4.3. I t i s q u i t e o b v i o u s from the d a t a c o m p i l e d i n t a b l e 4.3 t h a t s m a l l l i q u i d v e l o c i t i e s a l s o have a s i g n i f i c a n t e f f e c t on the e x t e n t and volume of the l i q u i d p o o l . T h i s e f f e c t i s much g r e a t e r than the e f f e c t of i n c r e a s i n g s u r f a c e temperature w i t h o u t the i n c r e a s e d l o s s e s of heat by r a d i a t i o n and a l l o y element e v a p o r a t i o n . The i n c r e a s e i n l i q u i d movement a l s o causes an i n c r e a s e i n the t o t a l power n e c e s s a r y t o m a i n t a i n the tem p e r a t u r e d i s t r i b u t i o n . T h i s i n c r e a s e i n t o t a l power i s a l s o apparent when the s u r f a c e temperature i n i n c r e a s e d and F i g u r e 4.12 P o o l P r o f i l e s a t 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, Superheat = 110 °C, LTCMF = 1. i F i g u r e 4.13 P o o l P r o f i l e s a t 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, Superheat = 110 °C, LTCMF = 2 . F i g u r e 4.14 P o o l P r o f i l e s a t 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, Superheat = 110 °C, LTCMF = 5. 78 i s of the same o r d e r of magnitude. 4.3. Power D i s t r i b u t i o n Model 4.3.1 The Power D i s t r i b u t i o n O b t a i n i n g a power d i s t r i b u t i o n f o r use i n the power d i s t r i b u t i o n model i s somewhat more complex than o b t a i n i n g a temperature d i s t r i b u t i o n . T y p i c a l l y programmed beam t r a v e r s e s c o n t a i n i n f o r m a t i o n as t o the power l e v e l s on the gun or guns and i n f o r m a t i o n about the l o c a t i o n and d u r a t i o n of a p a r t i c u l a r p a t t e r n . In o r d e r t o f i t the c o n s t r a i n t of the s t e a d y s t a t e heat f l o w e q u a t i o n t h e power d i s t r i b u t i o n must be time averaged and magnitude a d j u s t m e n t s f o r m e l t i n g power and beam l o s s e s must be made. 4.3.1.1 Beam Losses Energy l o s s e s i n e l e c t r o n beams r e s u l t from v a r i o u s beam i n t e r a c t i o n s as i n d i c a t e d i n f i g u r e 4.15. The energy l o s s e s due t o x - r a y g e n e r a t i o n , t h e r m i o n i c emmisions and secondary e l e c t r o n s amount t o l e s s than 0.5% of the t o t a l i n c i d e n t beam energy. B a c k s c a t t e r e d e l e c t r o n s can account f o r 10 - 40 % of the i n c i d e n t energy depending on 2 the m a t e r i a l b e i n g 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 . The number of b a c k s c a t t e r e d e l e c t r o n s i s r e l a t e d o n l y t o the atomic number of the i r r a d i a t e d m a t e r i a l , a c c e l e r a t i n g v o l t a g e and the a n g l e of i n c i d e n c e of the 79 Electron Beam 090000 X-Rays © Backscattered Electrons © Secondary Electrons © Thermionic Emission F i g u r e 4.15 Schematic Diagram of I n t e r a c t i o n s Between an Beam and 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 Energy Loss Mechanisms. 80 i m p i n g i n g beam as shown i n f i g u r e s 4.16 and 4.17. The energy d i s t r i b u t i o n of the s e b a c k s c a t t e r e d e l e c t r o n s can a l s o be d e t e r m i n e d e x p e r i m e n t a l l y and thes e r e s u l t s a r e shown i n f i g u r e 4.18. Doing an a p p r o p r i a t e i n t e g r a t i o n of the energy d i s t r i b u t i o n r e s u l t s i n the 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 as shown i n f i g u r e 4.19. The a p p l i c a t i o n of the s e e x p e r i m e n t a l o b s e r v a t i o n s t o the power d i s t r i b u t i o n i n e l e c t r o n beam r e m e l t i n g i s not s t r a i g h t f o r w a r d . A l t h o u g h the b a c k s c a t t e r i n g of e l e c t r o n s i s independent of the p h y s i c a l s t a t e of the i r r a d i a t e d m a t e r i a l , the s u r f a c e of the s k u l l i s r a r e l y p l a n e . I t has been o b s e r v e d d u r i n g m e l t i n g uranium t h a t the l i q u i d s u r f a c e 27 deforms, f o r m i n g a d e p r e s s i o n a t the beam s p o t . I t would seem u n l i k e l y t h a t the p r e v i o u s r e l a t i o n s h i p s would a p p l y t o the m e l t i n g s i t u a t i o n . In the absence of t h e proper i n f o r m a t i o n the r e s u l t s f o r a p l a n e s u r f a c e have been used. In m e l t i n g t i t a n i u m a normal beam i n c i d e n t on a p l a n e s u r f a c e can expect t o l o s e 20% of i t s energy t o b a c k s c a t t e r e d e l e c t r o n s . In h e a r t h r e m e l t i n g , i t i s r e a s o n a b l e t o assume t h a t an a d d i t i o n a l 10% of t h e i n c i d e n t beam energy w i l l be l o s t because the beam i s not always normal t o the s k u l l s u r f a c e . T h e r e f o r e i n h e a r t h m e l t i n g t i t a n i u m , beam l o s s e s w i l l be on the o r d e r of 30%. In m e l t i n g h e a v i e r m a t e r i a l s such as n i o b i u m , t a n t a l u m and the s u p e r a l l o y s beam l o s s e s up t o 50% would not be out of the q u e s t i o n . 81 F i g u r e 4.16 Number of B a c k s c a t t e r e d E l e c t r o n s as a F u n c t i o n of Atomic Number of the I r r a d i a t e d M a t e r i a l f o r a Normal Beam, A c c e l e r a t i n g V o l t a g e = 10 KV. 2 82 F i g u r e 4.17 Number of B a c k s c a t t e r e d E l e c t r o n s as a F u n c t i o n of Angle of I n c i d e n c e of the Beam, A c c e l e r a t i n g V o l t a g e = 10 KV. 2 83 F i g u r e 4.18 Energy D i s t r i b u t i o n of B a c k s c a t t e r e d E l e c t r o n s as a F u n c t i o n of Atomic Number f o r a Normal Beam, A c c e l e r a t i n g V o l t a g e = 10 KV. 2 84 F i g u r e 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 as a F u n c t i o n of Atomic Number f o r a Normal Beam, A c c e l e r a t i n g V o l t a g e = 10 KV. 2 85 4.3.1.2 M e l t Rate A d j u s t m e n t s In h e a r t h m e l t i n g the power r e q u i r e d t o melt the m a t e r i a l i s i m p o r t a n t . I t i s advantageous i n the steady s t a t e model t o remove t h i s heat i n p u t from the 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 by examining the o v e r a l l heat b a l a n c e on the s k u l l (see f i g u r e 4.20). The o v e r a l l heat b a l a n c e i s g i v e n by : Q I N + Q E B Q L Q EB + Q I N = QOUT + QSURF + QHRTH* ( 4 2 ) S i n c e Q I N and Q Q U T a r e a s s o c i a t e d w i t h the heat c o n t e n t of the m a t e r i a l b e i n g m e l t e d the d e f i n i t i o n of the b a s e l i n e f o r thes e terms a l l o w s them t o be c a l c u l a t e d . U s i n g room temperature as the b a s e l i n e g i v e s the f o l l o w i n g : Q I N = 0 I N ...(4.3) and • • • QOUT = QMELT + QSH ( 4 4 ) where Q M E L T i s the m e l t i n g power and Q g H i s the power 86 QEB Q, Q Q'« EB Q IN SURF Q OUT V ^HRTH F i g u r e 4.20 Schematic Heat Flow Diagram of t h e E l e c t r o n Beam S k u l l . 87 r e q u i r e d t o superheat the m a t e r i a l . T h i s a l l o w s the heat b a l a n c e t o be w r i t t e n as : • • • • • • • Q E B ~ Q L = Q EB = QMELT + QSH + QSURF + QHRTH* ,_ v For t i t a n i u m Q M E L T * S ° » 4 3 kWhr/kg and Q G H i s 0 . 1 9 2 Whr/kg°C. U s i n g the r e s u l t s o b t a i n e d from the temperature model, Q g H amounts t o a p p r o x i m a t e l y 2% of the t o t a l heat b a l a n c e and s i n c e i t i s tempe r a t u r e dependent r e s u l t i n g i n i n c r e a s e d n u m e r i c a l i n s t a b i l i t y i t has been i g n o r e d . T h i s approach w i l l g i v e the model a tendency t o p r e d i c t s l i g h t l y h i g h e r t e m p e r a t u r e s a t the pour l i p of the s k u l l . F i n a l l y the heat b a l a n c e can be w r i t t e n : Q E B Q L Q EB QMELT + QSURF + QHRTH or Q E B Q L QMELT Q EB QSURF + QHRTH* . . . ( 4 . 6 ) . . . ( 4 . 7 ) For a power d i s t r i b u t i o n i t i s i n a c c u r a t e t o s i m p l y remove the m e l t i n g power from the o v e r a l l power i n p u t as the m e l t i n g t a k e s p l a c e l o c a l l y a t the melt b a s i n . T h e r e f o r e the m e l t i n g power d i s t r i b u t i o n must be s u b t r a c t e d from the a p p l i e d power d i s t r i b u t i o n . 88 The power d i s t r i b u t i o n s used d u r i n g t h e c o u r s e of the work have been d e r i v e d from those t y p i c a l l y used by V i k i n g u s i n g the pr o c e d u r e d e s c r i b e d above. A t y p i c a l power d i s t r i b u t i o n 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 r u n n i n g the model u s i n g the power d i s t r i b u t i o n shown i n f i g u r e 4.21 and no enhanced heat t r a n s f e r due t o l i q u i d motion a r e shown i n f i g u r e s 4.22 t o 4.25. The r e g u l a r p o o l p r o f i l e produced and the p o o l depth i n d i c a t e d ( f i g u r e 4.24) a r e i n agreement w i t h t h o s e r e s u l t s produced e a r l i e r u s i n g the temperature d i s t r i b u t i o n boundary c o n d i t i o n and c o n s e q u e n t l y w i t h o b s e r v a t i o n s made i n o p e r a t i n g h e a r t h f u r n a c e s . The s u r f a c e t e m p e r a t u r e s c a l c u l a t e d do not agree w i t h those e x p e r i m e n t a l l y d e t e r m i n e d 27 by A. Johnson. T h i s d i s c r e p a n c y i n s u r f a c e t emperature may be due t o the s c a l e down i n s i z e of both the power d i s t r i b u t i o n and the s i z e of the h e a r t h . As w e l l d i f f e r e n c e s i n the o p e r a t i n g p r a c t i c e s of V i k i n g and A. Johnson may cause some d i s c r e p a n c i e s i n the r e s u l t i n g t h e r m a l f i e l d s . 4.3.2.1 E f f e c t of Power D e n s i t y The e f f e c t of an i n c r e a s e i n power d e n s i t y was de t e r m i n e d by i n c r e a s i n g the t o t a l power i n p u t t o the h e a r t h by 10%. As e x p e c t e d , the power i n c r e a s e r e s u l t s i n an i n c r e a s e i n the p o o l d e p t h , p o o l volume, peak s u r f a c e t e mperature i n the p o o l (see f i g u r e s 4.26 t o 4.29) as w e l l C to 3; > n- »-3 H-"rJ c 03 O 0) n no Q) o rt cr c rr O 3 G u> (D Oi 3 POWER DISTRIBUTION q 10 o 0 2 O z Q a: O O o o w. >-q I P O W E R S T A G E S T A G E 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 - 2 . 0 — r — 3.0 8.0 X C O O R D 13.0 I N A T E ( C M ) 18.0 23.0 28. 90 T—I ^~ CD CD CM O F i g u r e 4.22 Contour Map Showing the B o u n d a r i e s of the L i q u i d P o o l , C o n t o u r i n g Temperature = 1625 °C, T o t a l Power = 33 KW, LTCMF = 1. 91 o o s" 11 s :6 g/z. S'S S ' E SM S ' O -F i g u r e 4.23 Contour Map Showing the B o u n d a r i e s of the S o l i d S k u l l , C o n t o u r i n g Temperature = 1595 °C, T o t a l Power = 33 KW, LTCMF = 1. F i g u r e 4.24 P o o l P r o f i l e s a t 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 " l l 9'6 S'L. 9"9 9'E 9'I 9"0-r wni ' nyoon 1 F i g u r e 4.25 S u r f a c e Temperature D i s t r i b u t i o n , 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. 94 as the r a d i a t i o n and l o s s of a l l o y elements due t o e v a p o r a t i o n (see t a b l e 4.4). 4.3.2.2 I n f l u e n c e of L i q u i d M o t i o n The r e s u l t s o b t a i n e d from i n c r e a s i n g the l i q u i d t h e r m a l c o n d u c t i v i t y f a c t o r u s i n g the power d i s t r i b u t i o n i n d i c a t e d i n f i g u r e 4.21 a r e shown i n f i g u r e s 4.30 t h r o u g h 4.33 and summarized i n t a b l e 4.4. In g e n e r a l , enhancing heat f l o w i n the l i q u i d u s i n g f o r c e d c o n v e c t i o n produces i n c r e a s e s i n the p o o l volume and p o o l depth as w e l l as f l a t t e n i n g out the temp e r a t u r e g r a d i e n t s . As a r e s u l t of the l i q u i d m o t i o n , s u r f a c e t e m p e r a t u r e s a r e reduced b r i n g i n g about a c o r r e s p o n d i n g drop i n the heat l o s t t o the f u r n a c e environment and a l l o y element e v a p o r a t i o n . 4.3.3 The Heat Ba l a n c e As shown p r e v i o u s l y the heat b a l a n c e on the EB s k u l l i s • • • *> • • Q E B Q L QMELT Q EB QSURF + QHRTH* ,, ...(4.7) U s i n g t h i s r e l a t i o n s h i p and the r e s u l t s of the power 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 t o c a l c u l a t e the t h e r m a l 95 S' I F i g u r e 4.26 Contour Map Showing the B o u n d a r i e s of the L i q u i d P o o l , C o n t o u r i n g Temperature = 1625 °C, T o t a l Power = 36 KW, LTCMF = 1. 96 o CD i i i i i i i i i i i i r S ' l l S'6 S'£ S'S S'E S ' l S 'O-(N3) 'QdOOG A F i g u r e 4.27 Contour Map Showing the B o u n d a r i e s of the S o l i d S k u l l , C o n t o u r i n g Temperature = 1595 °C, T o t a l Power = 36 KW, LTCMF = 1. F i g u r e 4.28 P o o l P r o f i l e s a t 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 S u r f a c e Temperature D i s t r i b u t i o n , 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. 99 Beam Power KW LTCMF S u r f a c e Heat L o s s KW H e a r t h Power KW Peak S u r f a c e Temp. °C P o o l P o o l DepthVolume cm cm 3 E v a p o r a t i o n Rates kg/hr T i A l 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 10.1 1865 42.5 175 0.0062 0.1096 T a b l e 4.4 Summary of Heat B a l a n c e C a l c u l a t i o n s , P o o l Data and E v a p o r a t i o n Rates f o r Power D i s t r i b u t i o n Model. F i g u r e 4.30 P o o l P r o f i l e s a t 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 F i g u r e 4.31 S u r f a c e Temperature D i s t r i b u t i o n , 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. 102 0"SZ910"S£910"SZSl "dW31 'jdns m C X I c n c \ j o ' CM ' o . c n ' o , r -o o in o on o i i i i i i i — i — i — i — r 0'6 O'L 0"S 0'£ 0"l 0"l-(N3) 'QdOOD Z F i g u r e 4.32 P o o l P r o f i l e s a t 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 = 5. 1 03 o i i i i i i i i i i — i — i — r , 9 ' l l 9'6 9"Z. 9'9 9' £ 9*1 9 '0 -(N3) 'Gd003 k F i g u r e 4.33 S u r f a c e Temperature D i s t r i b u t i o n , 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 04 e f f i c i e n c y of the 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 can be d e f i n e d as : QMELT + QHRTH T?„ = " Q ...(4.8) EB As summarized i n t a b l e 4.5 the t h e r m a l e f f i c i e n c y of the h e a r t h f u r n a c e i s on the o r d e r of 50%. The e f f i c i e n c y d e c r e a s e s w i t h i n c r e a s i n g power l e v e l and i n c r e a s e s w i t h i n c r e a s i n g l i q u i d movement. These v a r i a t i o n s i n t h e r m a l e f f i c i e n c y can be a t t r i b u t e d t o the v a r i a t i o n s i n s u r f a c e t e mperature r e s u l t i n g from i n c r e a s e d i n p u t power or l i q u i d m o t i o n . 4.4. Beam Spot Temperature S i n c e the power d i s t r i b u t i o n s used as i n p u t f o r the model have been time a v e r a g e d , the te m p e r a t u r e f i e l d s t h a t a r e c a l c u l a t e d a re a l s o t ime averaged. The use of a v e r y i n t e n s e moving p o i n t s o u r c e s h o u l d r e s u l t i n some l a r g e t r a n s i e n t s i n the s k u l l , t he l a r g e s t a t the s u r f a c e over the a r e a t o which the power i s a p p l i e d . T h i s r a i s e s some q u e s t i o n s as t o the degree t o which the c a l c u l a t e d t e m p e r a t u r e s would agree w i t h a c t u a l t e m p e r a t u r e s w i t h i n the h e a r t h . In o r d e r to examine the temperature t r a n s i e n t s produced underneath the beam s p o t , an unsteady s t a t e heat 1 05 LTCMF M e l t i n g S u r f a c e H e a r t h Beam T o t a l Thermal Power R a d i a t i o n Power L o s s e s I n p u t E f f i c i e n c y MLT SURF QHRTH Q L Q E B V KW KW KW KW KW % 1 8.5 6.7 8.5 10.2 33.9 50.1 1 8.5 7.7 8.9 10.8 35.9 48.5 2 8.5 5.9 9.2 10.2 33.9 52.2 5 8.5 5.1 10.0 10.2 33.9 54.6 T a b l e 4.5 Thermal E f f i c i e n c y of E l e c t r o n Beam M e l t i n g Under V a r i o u s C o n d i t i o n s . 106 t r a n s f e r model has been d e v e l o p e d (see Appendix 1 ) . I n t u i t i v e l y t he time taken t o r e a c h a s p e c i f i c t emperature s h o u l d d e c r e a s e as the power l e v e l i n c r e a s e s . The model p r e d i c t s t h i s as shown i n f i g u r e 4.34. In f i g u r e 4.35 power has been a p p l i e d t o the melt s u r f a c e a t the c e n t e r l i n e f o r a p e r i o d of 20 m i l l i s e c o n d s and then the melt has been a l l o w e d t o c o o l . T h i s i s a worst case s c e n a r i o i n t h e EB h e a r t h as the beam r a r e l y remains i n any l o c a t i o n f o r t h i s l e n g t h of t i m e . As the f i g u r e shows the melt a t the c e n t e r l i n e c o o l s down t o the b u l k t e m p e r a t u r e i n l e s s than 2 seconds. T h i s i s a l o n g time r e l a t i v e t o the c y c l e t i m e s used i n a i n d u s t r i a l EB h e a r t h f u r n a c e ( V i k i n g uses c y c l e t i m e s of 14 seconds t y p i c a l l y ) . I f the c o o l down time i s l o n g compared t o the c y c l e time then the temp e r a t u r e f l u c t u a t i o n s caused by the moving power source w i l l not be s i g n i f i c a n t . I n t h i s c a s e the time averaged temperature w i l l be o n l y s l i g h t l y l e s s t han the peak temperature a t t a i n e d under the beam s p o t . O b s e r v a t i o n s of the molten p o o l i n the Johnson s k u l l s u p p o r t t h i s c o n c l u s i o n . In the Johnson h e a r t h f u r n a c e two of the f o u r guns d e s c r i b e e l l i p t i c a l p a t t e r n s over the s k u l l i n between the melt end and the pour l i p . To the naked eye o n l y v e r y s l i g h t changes i n b r i g h t n e s s i n t h i s r e g i o n a r e o b s e r v e d . In o t h e r a r e a s where the beam p a t t e r n i s c i r c u l a r and remains s t a t i o n a r y f o r l o n g p e r i o d s of t i m e , t h e r e i s an e a s i l y d i s c e r n a b l e b r i g h t spot and l a r g e 107 F i g u r e 4.34 Unsteady S t a t e Response of the Temperature Under the Beam Spot as a F u n c t i o n of Power I n p u t . 108 BEAM SPOT TEMPERATURE o TIME (SEC) F i g u r e 4.35 Unsteady S t a t e Response of the Temperature Under the Beam Spot A f t e r 20 m i l l i s e c o n d s a t 200 KW. 109 v a r i a t i o n s i n b r i g h t n e s s . 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 Do t o the c o m p l e x i t y and s i z e of the h e a r t h f u r n a c e a t V i k i n g , v e r i f i c a t i o n e x p e r i m e n t s on the V i k i n g s k u l l were not co n d u c t e d . I n s t e a d , e x p e r i m e n t s were conducted on s m a l l e r b l o c k s of m a t e r i a l a t bo t h Johnson and V i k i n g . In the V i k i n g experiment a 28 cm x 48 cm x 15 cm (11" x 19" x 6") b l o c k of T i 6 A l 4 V was used. The b l o c k was p l a c e d i n s i d e a 60 cm (24") d i a m e t e r c r u c i b l e mould and a hand f u l l of n a i l s , f o r marking the l i q u i d p o o l , were p l a c e d on the upper s u r f a c e . The b l o c k was then p l a c e d i n s i d e the f u r n a c e chamber. An 80 KW beam d e s c r i b i n g a 10 cm (4") di a m e t e r c i r c l e was then c e n t e r e d on the upper f a c e and a l l o w e d t o remain t h e r e . A f t e r a p p r o x i m a t e l y 40 minutes the e l e c t r o n beam was shut o f f and the b l o c k was a l l o w e d t o c o o l . In the experiment c o n d u c t e d a t Johnson, much the same pr o c e d u r e was c a r r i e d o u t . A c y l i n d r i c a l b l o c k of T i 6 A l 4 V and a number of i r o n washers were p l a c e d i n a c r u c i b l e mould. The b l o c k was then heated by a c e n t e r e d 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 p a t t e r n f o r a p e r i o d of time l o n g enough f o r the b l o c k t o a r r i v e a t s t e a d y s t a t e . The beam was then e x t i n g u i s h e d and the b l o c k a l l o w e d t o c o o l . 1 10 In each c a s e , the r e s u l t i n g b l o c k s of m a t e r i a l were s e c t i o n e d and e t c h e d t o show the e x t e n t of the l i q u i d p o o l . The photomacrographs showing the b o u n d a r i e s of the l i q u i d p o o l s a r e shown i n f i g u r e s 4.36 and 4.37 Due t o the c y l i n d r i c a l geometry employed a t Johnson, m o d e l l i n g the experiment c o u l d not be done. The V i k i n g t r i a l d i d use r e c t a n g u l a r geometry and the r e s u l t i n g p o o l p r o f i l e c o u l d be compared t o a computer run of the b l o c k model. An i n i t i a l run u s i n g a beam l o s s f a c t o r of 30% and assuming t h a t the beam was v e r y t i g h t , produced s u r f a c e t e m p e r a t u r e s on the o r d e r of 3000 °C and h i g h e r and a p o o l boundary (see f i g u r e 4.38) t h a t bears l i t t l e or no resemblence t o the one shown i n f i g u r e 4.36. F u r t h e r runs made by s p r e a d i n g out the i n p u t power over a l a r g e r a r e a and 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 produced the b e s t f i t t o the a c t u a l p o o l p r o f i l e as shown i n f i g u r e 4.39. A l t h o u g h the model can approach the p o o l p r o f i l e s a c t u a l l y o b t a i n e d i n the experiment the f i t i s u n s a t i s f a c t o r y . The r e s u l t s a r e a l s o poor c o n s i d e r i n g the power d i s t r i b u t i o n i n p u t i n t o the model and the power a c t u a l l y used i n the e x p e r i m e n t . C l e a r l y t h e r e a r e o t h e r f a c t o r s c o n t r i b u t i n g t o the p o o l p r o f i l e . One e x p l a n a t i o n f o r the d i s c r e p a n c y between the p r e d i c t e d and a c t u a l p o o l p r o f i l e s i s a d i f f e r e n c e i n the F i g u r e 4.36 Photomacrograph of t h e B l o c k Used D u r i n g the Experiment a t V i k i n g Showing the 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 : 0.375X , Beam Power : 80 KW. F i g u r e 4.37 Photomacrograph of the B l o c k Used D u r i n g the Exp e r i m e n t a t Johnson Showing the 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. 580X. to F i g u r e 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 of the 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 . 1 1 4 © 'o -R _ r— " a •UJ i i i i i i i — i — i i i — i — i — i — i — i — i — i — i — i — i — r 0 081 O K I 009 O'Sl OEI a'II 06 OL OS OE 0! 01-( , o i x ) dW3i j a n s iwJ) - d a o o o z F i g u r e 4.39 P o o l P r o f i l e s of the B l o c k Model f o r the 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 and a c t u a l f l u i d f l o w regimes. For example, the v e r y h i g h t e m p e r a t u r e s under the beam spot and i n the c e n t e r of a c i r c u l a r beam p a t t e r n c o u l d cause i n c r e a s e d f l u i d f l o w i n the x - y p l a n e w i t h l i t t l e or no i n c r e a s e d f l o w i n t h e z d i r e c t i o n . The i m p o s i t i o n of t h i s f l u i d f l o w regime would cause heat t o f l o w more r e a d i l y i n the x - y p l a n e c a u s i n g the p o o l t o s p r e a d . T h i s would a l s o r e s u l t i n a s h a l l o w e r p o o l . I t has been found by r e s e a r c h e r s m o d e l l i n g the 31 32 w e l d i n g p r o c e s s ' t h a t s u r f a c e t e n s i o n e f f e c t s can c o n t r i b u t e s i g n i f i c a n t l y t o the f l u i d f l o w regime d u r i n g w e l d i n g . These i n v e s t i g a t i o n s i n t o the f l u i d regime i n the weld p o o l i n d i c a t e t h a t the f l u i d v e l o c i t y due t o s u r f a c e t e n s i o n e f f e c t s may be as h i g h as t w i c e t h o s e caused by n a t u r a l c o n v e c t i o n . O b v i o u s l y these i n c r e a s e d f l u i d v e l o c i t i e s on the s u r f a c e of the l i q u i d p o o l may account f o r the wider p o o l observed d u r i n g the e x p e r i m e n t a l work. 116 CHAPTER 5 H e a r t h Design and 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 The d e s i g n and o p e r a t i o n of an e l e c t r o n beam h e a r t h f u r n a c e i s a f f e c t e d by a number of c o n s i d e r a t i o n s . These i n c l u d e the e v a p o r a t i o n of a l l o y i n g e l ements, the f l u i d f l o w and t h e r m a l regimes t h a t d e v e l o p and the a b i l i t y of the h e a r t h t o p r o v i d e adequate r e a c t i o n volume f o r the d i s s o l u t i o n or s e p a r t i o n of i n t e r and n o n - m e t a l l i c compounds. The importance of any of t h e s e c o n s i d e r a t i o n s i s p r i m a r i l y dependent on the end use of the i n g o t and the s t a r t i n g m a t e r i a l . For i n s t a n c e , e v a p o r a t i o n r a t e s a r e of l i t t l e or no co n c e r n when m e l t i n g CP t i t a n i u m but a r e of paramount importance when m e l t i n g any a l l o y m a t e r i a l . The heat t r a n s f e r model can be used t o e v a l u a t e the e f f e c t s of the m i x i n g parameters ( i n a q u a l i t a t i v e way) on the t h e r m a l regime, the e f f e c t of t h e r m a l c o n d i t i o n s a t 1 16 1 17 the s u r f a c e of the s k u l l and the i n f l u e n c e of geometry. In a d d i t i o n the 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 me l t r a t e and power d i s t r i b u t i o n on the t h e r m a l regimes and e v a p o r a t i o n r a t e s can be examined. 5.2. H e a r t h Design D e s i g n of an e l e c t r o n beam h e a r t h i s c u r r e n t l y a t r i a l and e r r o r e x e r c i s e . A l t h o u g h i t i s p o s s i b l e t o e s t i m a t e c e r t a i n parameters r e q u i r e d (such as h e a r t h l e n g t h ) from e x p e r i e n c e , i t i s d i f f i c u l t t o a s s e s s some of the o t h e r parameters such as e v a p o r a t i o n r a t e s . U s i n g a m a t h e m a t i c a l 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 t o be e v a l u a t e d i n terms of the t h e r m a l regime d e v e l o p e d d u r i n g o p e r a t i o n . The mass 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 r a t e parameters can then be o b t a i n e d from the t h e r m a l p r o f i l e . U s i n g the model i t i s r e l a t i v e l y easy t o e v a l u a t e the e f f e c t s of geometry and t h e r m a l boundary c o n d i t i o n s on the p o o l volume, heat b a l a n c e and e v a p o r a t i o n r a t e s . 5.2.1 Geometry Geometry t a k e s i n t o account both the s i z e and shape of the h e a r t h . The e f f e c t of changing the l e n g t h of the h e a r t h i s d i f f i c u l t t o det e r m i n e because d i f f e r e n t power d i s t r i b u t i o n s would be r e q u i r e d f o r the d i f f e r e n t h e a r t h s i z e s i n o r d e r t o m a i n t a i n l i q u i d m e t a l from the mel t b a s i n t o t he pour l i p . 118 The i n f l u e n c e of s k u l l t h i c k n e s s on the t h e r m a l p r o f i l e i s e a s i l y d e t e r m i n e d s i n c e the power d i s t r i b u t i o n i s independent of the s k u l l t h i c k n e s s . The e f f e c t of a l t e r i n g s k u l l t h i c k n e s s was de t e r m i n e d u s i n g a r e c t a n g u l a r s k u l l 40 cm x 10 cm. The power d i s t r i b u t i o n shown i n f i g u r e 5.1 was used and the t h i c k n e s s v a r i e d from 3 cm t o 11 cm. The v a r i a t i o n of p o o l volume w i t h s k u l l t h i c k n e s s i s shown i n f i g u r e 5.2 and i s summarized i n t a b l e 5.1 As i s r e a d i l y seen, the p o o l volume i n u n a f f e c t e d by changes i n s k u l l t h i c k n e s s . As w e l l the maximum p o o l depth has l i t t l e v a r i a t i o n w i t h s k u l l t h i c k n e s s . T h i s would i n d i c a t e t h a t f o r a g i v e n power d i s t r i b u t i o n , any s k u l l t h i c k n e s s l a r g e r t h a t the p r e d i c t e d p o o l depth would be adequate. From t h i s r e s u l t , t h e r e would appear t o be l i t t l e or no advantage t o u s i n g a t h i c k e r s k u l l . A t h i c k s k u l l r e p r e s e n t s a l a r g e r , b u l k i e r p i e c e of m a t e r i a l r e q u i r i n g b i g g e r and b e t t e r h a n d l i n g equipment and a b i g g e r e n c l o s u r e . The advantage t o u s i n g a t h i c k e r s k u l l becomes apparent when the heat f l u x t o the water c o o l e d copper h e a r t h i s examined. The average heat f l u x t h r o u g h any p a r t i c u l a r s u r f a c e d e c r e a s e s d r a m a t i c a l l y w i t h i n c r e a s i n g s k u l l t h i c k n e s s (see f i g u r e 5.3). T h e r e f o r e u s i n g a t h i c k s k u l l a l l o w s f o r the use of a tube and c a s t c o n s t r u c t i o n i n the water c o o l e d h e a r t h . T h i s t ype of c o n s t r u c t i o n i n unable t o handle h i g h heat f l u x e s due t o the poor t h e r m a l c o n t a c t between the c a s t \\9 o. «0 1 20 Pool Volume vs. Skull Thickness 90 65 -f 1 1 1 1 ! 2 4 6 3 10 12 Thickness (cm) F i g u r e 5.2 The V a r i a t i o n of P o o l Volume w i t h S k u l l T h i c k n e s s . 121 S k u l l P o o l M e l t T h i c k n e s s Volume E f f i c i e n c y (cm) (cm 3) (%) 3.0 89.4 56.2 5.0 81.6 56.2 7.0 77.1 56.1 9.0 71.3 56.1 11.0 68.1 56.1 E v a p o r a t i o n T o t a l R a t es Average Heat F l u x ( k g / h r ) (w cm" 2) T i A l 0.0185 0.2099 20.65 0.0161 0.1894 16.26 0.0151 0.1797 13.42 0.0139 0.1687 11.42 0.0132 0.1623 9.94 T a b l e 5.1 R e s u l t s of C a l c u l a t i o n s on the EB S k u l l as a F u n c t i o n of T h i c k n e s s . d n> cn CO H < n t--7T PJ CD t-"-in o cn 3 > < 0) 0) 0) »-• c rt cn Average HeaL F l u x Legend A m e l t b a s i n x l e n g t h e d g e • p o u r l i p  G 3 b o t t o m ffi o v e r a l l T h i c k n e s s (cm) 123 23 . copper and the tube network. I t i s a l s o s i g n i f i c a n t l y c heaper than the u s u a l method of d r i l l i n g the c o o l l i n g c h a n n e l s i n a copper b l o c k . A l t e r i n g the w i d t h of the s k u l l w i t h o u t changing the power d i s t r i b u t i o n has the o p p o s i t e e f f e c t on the p o o l volume (see f i g u r e 5.4). The b e n e f i t s of the i n c r e a s e i n p o o l volume are o f f s e t somewhat by the i n c r e a s e d s u r f a c e t e m p e r a t u r e s t h a t r e s u l t i n h i g h e r s u r f a c e heat l o s s e s . The i n c r e a s e d s u r f a c e t emperature r e p r e s e n t s a l o s s i n t h e r m a l " e f f i c i e n c y " but no l o s s i n the m e l t i n g e f f i c i e n c y (or the s p e c i f i c m e l t i n g e n e r g y ) . The e v a p o r a t i o n r a t e s a l s o i n c r e a s e due t o a c o m b i n a t i o n of g r e a t e r s u r f a c e a r e a and i n c r e a s e d 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 w i d t h of the s k u l l a l s o r e s u l t s i n a d e c r e a s e of the average heat f l u x t o the mould. The e f f e c t s of s k u l l w i d t h are r e a d i l y apparent when the p o o l parameters a r e p l o t t e d a g a i n s t a d i m e n s i o n l e s s s k u l l w i d t h ( d e f i n e d as the r a t i o of the w i d t h of the power d i s t r i b u t i o n and the a c t u a l s k u l l w i d t h ) . The v a r i o u s p o o l p a rameters ( i . e . p o o l volume, e v a p o r a t i o n r a t e of aluminum and heat f l u x ) a l l v a r y l i n e a r l y w i t h t h i s d i m e n s i o n l e s s parameter as shown i n f i g u r e 5.5. From the p o i n t of view of h e a r t h mould 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 of a t h i c k , wide s k u l l s h o u l d be used. O b v i o u s l y the c h o i c e of s k u l l d i m e n s i o n s w i l l a l s o depend on t h e s i z e of containment v e s s e l , the • q^PTM l i n n s M 4 T R T sumiOA Tjood 3 0 u o i ^ e i j e A t * 9 S J H B T J Pool Volume (cm**3) o CO c a 3 r o C5 • o c co o _ L _ o o r o c c r o Mass Flux Al ( K g / h r ) r o o o CO o > 13 O O 13 O o ^ CO C O r o Average Heat Flux (W/cm +*2) cn 1 25 S k u l l P o o l M e l t W i dth Volume E f f i c i e n c y (cm) (cm 3) (%) 5.0 77.13 49.9 7.0 98.38 49.8 10.0 103.38 49.8 12.0 106.38 49.7 15.0 108.38 49.6 E v a p o r a t i o n T o t a l Rates Average Heat F l u x ( k g / h r ) (W cm" 2) T i A l 0.0151 0.1797 13.42 0.0193 0.2193 10.64 0.0204 0.2292 9.33 0.0212 0.2358 8.33 0.0218 0.2409 7.17 T a b l e 5.2 R e s u l t s of C a l c u l a t i o n s on the EB S k u l l as a F u n c t i o n of Width 127 equipment r e q u i r e d t o remove the s k u l l from the h e a r t h mould and the t o l e r a n c e f o r the e v a p o r a t i o n of a l l o y e l e m e n t s . S i n c e the cha n g i n g of both the t h i c k n e s s and w i d t h of the h e a r t h o n l y a minor e f f e c t on p o o l volume, i t i s r e a s o n a b l e t o c o n c l u d e t h a t the q u a n t i t y of l i q u i d m e t al i s c o n t r o l l e d by the l e n g t h of the h e a r t h and the power d i s t r i b u t i o n a p p l i e d t o the s k u l l . 5.2.2 The E f f e c t of the H e a r t h Mould U s i n g a h e a r t h mould around the s k u l l r e p r e s e n t s one l i m i t i n g case f o r heat t r a n s f e r d u r i n g EBCHR. The o t h e r extreme i s t o a l l o w a l l s u r f a c e s of the s k u l l t o r a d i a t e f r e e l y t o the containment v e s s e l . A l t h o u g h the h e a r t h mould f u n c t i o n s as p r o t e c t i o n a g a i n s t breakout and o v e r f l o w of l i q u i d m e t a l , as a pour l i p a t the c r u c i b l e end of the s k u l l and as s u p p o r t f o r unmelted f e e d s t o c k i t a l s o i n c r e a s e s the heat f l o w from the s k u l l . The computer model was run u s i n g a s k u l l and s i m i l a r t o the V i k i n g geometry w i t h the power d i s t r i b u t i o n used e a r l i e r (see f i g u r e 5.5). For compa r i s o n , the temperature p r o f i l e s were o b t a i n e d f o r a s k u l l of the same geometry and u s i n g the same power d i s t r i b u t i o n but w i t h o u t the water c o o l e d copper mould. The r e s u l t s of t h e s e runs a r e shown i n f i g u r e s 5.7 and 5.8 and summarized i n t a b l e 5.3. I t i s f a i r l y o b v i o u s t h a t removing the water c o o l e d copper h e a r t h from the around the s k u l l i n c r e a s e s the id C n cn POWER DISTRIBUTION r t rt> CD 0) n r t rr 3 o c I— 1 a o rt) in rt r| cr c r t o 3 G ui rt) a r t O n < 0> M c 0> rt (D p d -O UJ o a: O O a P O W E R S T A G E CE S T A G E 1 eZl S T A G E 2 S T A G E 3 S T A G E 4 K 2 S T A G E 5 r t rt> n rt) o r t q IN -2.0 3.0 8.0 X C O O R D 13.0 I N A T E ( C M ) 18.0 23.0 28.0 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 Co o l e d Copper H e a r t h . 1 30 o F i g u r e 5.8 P o o l P r o f i l e s f o r EB S k u l l Without a Water C o o l e d Copper H e a r t h . 131 Mould Peak P o o l Thermal E v a p o r a t i o n S u r f a c e Volume E f f i c i e n c y R a t e s Temperature (°C) (cm 3) (%) ( k g / h r ) T i A l YES 2175 113.5 51.5 0.0754 0.5313 NO ~ 2275 252.9 43.5 0.3287 1.5373 T a b l e 5.3 E v a p o r a t i o n R a t e s , P o o l Data and Heat Balance C a l c u l a t i o n s f o r an EB S k u l l With and Without a H e a r t h Mould. 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 i n c r e a s e i n p o o l volume i s accompanied by a t h r e e f o l d i n c r e a s e i n the maximum e v a p o r a t i o n r a t e of aluminum from 6/4. The t h e r m a l e f f i c i e n c y of the f u r n a c e a l s o d e c r e a s e s but t h i s i s due t o an i n c r e a s e i n s u r f a c e heat l o s s . The m e l t i n g e f f i c i e n c y of a h e a r t h f r e e s k u l l i s not a p p r e c i a b l y d i f f e r e n t than the copper s u r r o u n d e d s k u l l . One of the p e r i l s of o p e r a t i n g an e l e c t r o n beam m e l t i n g f u r n a c e w i t h o u t u s i n g a water c o o l e d copper h e a r t h i s i n d i c a t e d i n f i g u r e 5.8. From the diagram of p o o l p r o f i l e s i t i s apparent t h a t l i q u i d m e t a l w i l l f l o w from the pour l i p i n t o the i n g o t c r u c i b l e and a l s o from the melt b a s i n . I t would be easy t o remedy t h i s s i t u a t i o n by c h o o s i n g a d i f f e r e n t power d i s t r i b u t i o n (or i n c r e a s i n g t h e s i z e of the s k u l l ) . However, the 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 l i k e l y t oo g r e a t t o g a i n wide acceptance i n commercial o p e r a t i o n s . 5.3. Furnace O p e r a t i o n s From the t h e r m a l s t a n d p o i n t , the s i n g l e b i g g e s t f a c t o r i n d e t e r m i n i n g the parameters of the mo l t e n p o o l d u r i n g EB h e a r t h r e m e l t i n g i s the power l e v e l and d i s t r i b u t i o n . The improper s e l e c t i o n of a power d i s t r i b u t i o n w i l l 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 t o f a i l r e g a r d l e s s of the c a r e and a t t e n t i o n p a i d t o o t h e r p e r t i n e n t a r e a s (such as h e a r t h d e s i g n ) . 133 U s i n g the proc e d u r e d e s c r i b e d e a r l i e r i n t h i s work (see s e c t i o n 4.3.1) i t i s p o s s i b l e t o a r r i v e a t power d i s t r i b u t i o n t h a t can be used i n the model s t a r t i n g w i t h an a p p l i e d power d i s t r i b u t i o n and making a d j u s t m e n t s f o r b o t h beam l o s s e s and melt r a t e . T h e r e f o r e the i n f l u e n c e of melt r a t e on the t h e r m a l regime can be i n v e s t i g a t e d . I t i s a l s o common p r a c t i c e d u r i n g a m e l t i n g campaign t o have some degree of o p e r a t o r c o n t r o l over the power d i s t r i b u t i o n . T h i s c o n t r o l i s a t t a i n e d e i t h e r by a l l o w i n g the o p e r a t o r t o p h y s i c a l l y c o n t r o l one or more of the guns, or g i v i n g the o p e r a t o r a c c e s s t o the programmable c o n t r o l l e r which c o n t r o l s the power d i s t r i b u t i o n . The e f f e c t s of t h e s e p r a c t i c e s can be examined u s i n g the t h e r m a l model d e v e l o p e d h e r e . An a d d i t i o n a l a p p l i c a t i o n of the model i s the d e s i g n of a power d i s t r i b u t i o n t h a t w i l l produce a g i v e n t e mperature d i s t r i b u t i o n . A l t h o u g h p o s s i b l e t h i s p r o c e d u r e has not been demonstrated i n t h i s work due t o the c o n s i d e r a b l e expense i n v o l v e d . 5.3.1 E f f e c t of M e l t Rate The e f f e c t of a l t e r i n g the melt r a t e on the t h e r m a l regime was e v a l u a t e d u s i n g a s e r i e s of power d i s t r i b u t i o n s of the form shown i f f i g u r e 5.9. The power l e v e l s and melt r a t e s a s s o c i a t e d w i t h them a r e summarized i n t a b l e 5.4. M e l t Rate (kg/h r ) Stage 1 Power L e v e l (W cm" 2) Stage 2 Stage 3 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 T a b l e 5.4 Power L e v e l s Used as a F u n c t i o n of M e l t Rate. 136 The r e s u l t s of the c a l c u l a t i o n s are o u t l i n e d i n t a b l e 5.5. For t h i s p a r t i c u l a r c o n f i g u r a t i o n of s k u l l geometry and power l e v e l , melt r a t e s l e s s t h a t 40 kg/hr produce p o o l depths g r e a t e r than the s k u l l t h i c k n e s s and melt r a t e s g r e a t e r than 90 kg/hr produce l i q u i d p o o l s of i n s u f f i c i e n t volume. Not s u p r i s i n g l y , b oth the t h e r m a l e f f i c i e n c y and the m e l t i n g e f f i c i e n c y r i s e as the melt r a t e i s i n c r e a s e d (see f i g u r e 5.10). T h i s i n c r e a s e i n t h e s e two parameters i s matched by a d e c r e a s e i n both the p o o l volume and the maximum e v a p o r a t i o n r a t e of the a l l o y i n g elements (see f i g u r e 5.11). T h i s i n d i c a t e s t h a t the h i g h e r the m e l t i n g r a t e the b e t t e r the m e t a l t h r o u g h p u t , the more e f f i c i e n t the m e l t i n g p r o c e s s and b e t t e r p r o d u c t i s a c h i e v e d . T h i s i s t r u e i f the s o l e aim of the m e l t i n g s t a g e i s t o c o n s o l i d a t e CP s c r a p or produce a p r i m a r y e l e c t r o d e from m a t e r i a l which does not c o n t a i n exhogenous p a r t i c l e s ( i . e . WC or T i N ) . I f t h e s e d e f e c t s a r e p r e s e n t i n the f e e d s t o c k then the p o o l volume (a measure of the r e s i d e n c e time of the m a t e r i a l i n the l i q u i d s t a t e ) s h o u l d a l s o be l a r g e enough t o a l l o w f o r the removal of the o f f e n d i n g s p e c i e s . Thus some b a l a n c e between throughput and p a r t i c l e s e p a r t i o n and d i s s o l u t i o n r a t e s must be a t t a i n e d . 5.3.2 Power D i s t r i b u t i o n W i th the s i n g l e e x c e p t i o n of l i q u i d m o t i o n , the power d i s t r i b u t i o n has the g r e a t e s t e f f e c t on the t h e r m a l 1 37 M e l t P o o l S p e c i f i c E v a p o r a t i o n Rate Volume Energy R a t e s ( k g / h r ) (cm 3) (KWH/kg) (k g / h r ) T i A l 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 Ta b l e 5.5 P o o l Data, E v a p o r a t i o n Rates and Heat Bal 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 R a t e . 138 60 T h e r m a l Efficiency As A Funct ion of Melt Rate •2.75 O C D • —4 O C£3 50 A 40H 30 H Specific Energy Thermal Efficiency Melt Efficiency •2.25 h 1.75 C D c O o C D 1 2 5 # 20 30 •0.75 50 70 90 Melt Rate (Kg/hr) n o F i g u r e 5.10 Thermal E f f i c i e n c y as a F u n c t i o n of M e l t R a t e . 139 Pool Parameters As A Function of Melt Rate 40 50 60 70 80 90 100 Melt Rate (Kg/hr) F i g u r e 5.11 P o o l Volume and E v a p o r a t i o n Rate of A l as a F u n c t i o n of M e l t R a t e . 140 regime 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 of i n c r e a s i n g the t o t a l power s u p p l i e d t o the s k u l l has been p r e v i o u s l y shown (see s e c . 4.3.2.1). I t was found t h a t i n c r e a s i n g the power l e v e l r e s u l t s i n s i g n i f i c a n t i n c r e a s e s i n p o o l volume, s u r f a c e heat l o s s e s and e v a p o r a t i o n r a t e s . W i t h e l e c t r o n beam power i n p u t i t i s a l s o p o s s i b l e t o change the shape and s i z e of the power i n p u t a r e a . These changes i n the power d i s t r i b u t i o n can a l s o have a l a r g e e f f e c t on the t h e r m a l regime i n the EB s k u l l . The power d i s t r i b u t i o n chosen f o r the base case i s shown i n f i g u r e 5.12. I t i s the same power d i s t r i b u t i o n used f o r a melt r a t e of 80 kg/hr used i n the p r e v i o u s s e c t i o n . The f i r s t a l t e r a t i o n t o t h i s power d i s t r i b u t i o n i s the a d d i t i o n of a c i r c l e of r a d i u s 5 cm r e p r e s e n t i n g a t o t a l power of 3.15 KW t o the base power d i s t r i b u t i o n (see f i g u r e 5.13). The e f f e c t of a l t e r i n g the s i z e of the power d i s t r i b u t i o n was demonstrated by s h r i n k i n g the base case power d i s t r i b u t i o n i n the w i d t h d i r e c t i o n (see f i g u r e 5.14) and m a i n t a i n i n g the same t o t a l power. The f i n a l case was o b t a i n e d by c o r r e c t i n g the e l l i p s e shape used i n the base case t o a r e c t a n g u l e of i d e n t i c a l a r e a . The r e s u l t i n g power d i s t r i b u t i o n i s shown i n f i g u r e 5.15. The model was then run u s i n g each of these power d i s t r i b u t i o n s . The r e s u l t s of the runs a r e shown i n f i g u r e s 5.16 t o 5.19 and summarized i n t a b l e 5.6. \ 0 id C cn > T3 Co O Q. C (-•• (0 rt- it (-•• o a in rr rt 0 ^  rt CT V C (D rt W O 01 3 01 ro — o 0) CO 01 • ro — • cn o n i — • n > POWER DISTRIBUTION U J (—' az o 5 <° a c e o o o o PONER LEVEL 48.0 H/CN**2 W 201.0 W/CM**2 B l 335.0 W/CM*x2 67.0 H/CM*M2 • 268.0 W/CM**2 Z Z 67.0 W/CM**2 •D 134.0 W/CMx*2 ES3 153.0 W/CM*x2 ' 1— -5.0 10.0 25.0 40.0 COORDINATE 55.0 I 70.0 85.0 [CM) 4^ ro 143 CO c at cn no o o o in CD X o r t ar ID W 01 in o 0) in ft) i — i r . 0 3 . 0 9 . 0 1 5 . 0 2 1 . 0 - 1 1 1 1 I n I I I 2 7 0 3 3 . 0 3 9 . 0 4 5 . 0 51 X C O O R D I N A T E ( C M ) i — i r i — r . 0 5 7 . 0 6 3 . 0 6 9 . 0 7 5 0 81 4^  cn CTi 147 3 i—i—r i—IT 03 .m . c n CO .cn CO in i n ' _ C J . in T L U I— o C E or—• O . r-' CM " o ' rg " a . in o ' cn a ' on C J .CO I S -2LI S ' U S ^ l - 016 0'6 0"C-( t 0 I X ) ' d W 3 1 JcjfWD) "QcJOOG Z F i g u r e 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 a F i g u r e 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 1 49 Power P o o l Thermal E v a p o r a t i o n D i s t r i b u t i o n Volume E f f i c i e n c y Rates (cm 3) (%) (kg/hr) T i A l 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 Tab l e 5.6 P o o l Data, Heat Bal a n c e C a l c u l a t i o n s and E v a p o r a t i o n Rates f o r Power D i s t r i b u t i o n Runs. 150 The a d d i t i o n of the e x t r a c i r c l e of power produces an i n c r e a s e i n p o o l volume of ^30%. Accompanying t h i s i n c r e a s e i n p o o l volume i s an i n c r e a s e i n the 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 of aluminum of over 100%. S h r i n k i n g the power d i s t r i b u t i o n i n the w i d t h d i r e c t i o n i s a l s o d i s a d v a n t a g e o u s i n terms of i n c r e a s i n g the e v a p o r a t i o n r a t e . In t h i s case an i n c r e a s e i n p o o l volume of 43% i s accompanied by a f o u r f o l d i n c r e a s e i n the e v a p o r a t i o n r a t e of aluminum. F i n a l l y the change i n shape from e l l i p t i c a l t o a r e c t a n g u l a r power d i s t r i b u t i o n p r o v i d e d l i t t l e or no v a r i a t i o n i n both the p o o l volume or e v a p o r a t i o n r a t e s . From t h i s d a t a a s i g n i f i c a n t c o n c l u s i o n can be made. For the purpose of m a x i m i z i n g p o o l volume w h i l e m i n i m i z i n g e v a p o r a t i o n r a t e s , the i n p u t power s h o u l d be spread u n i f o r m l y over as l a r g e a s u r f a c e a r e a as p o s s i b l e . The p r a c t i c e of c o n c e n t r a t i n g power i n any one p a r t i c u l a r l o c a t i o n s h o u l d not be used as t h i s i n c r a a s e s e v a p o r a t i o n l o s s e s w i t h o u t a p p r e c i a b l y i n c r e a s i n g p o o l volume. F i n a l l y the use of 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 seems t o produce s l i g h t l y more u n i f o r m temperature d i s t r i b u t i o n s w i t h i n the melt a l t h o u g h e l l i p t i c a l or c i r c u l a r power d i s t r i b u t i o n s produce s l i g h t l y h i g h e r p o o l volumes. 5.4. Summary Four a s p e c t s of h e a r t h o p e r a t i o n and d e s i g n have been examined as t o t h e i r e f f e c t s on the p o o l parameters and the t h e r m a l regime. The c o n c l u s i o n s a r r i v e d a t a r e : 151 p o o l volume i s d e t e r m i n e d p r i m a r i l y by h e a r t h ( s k u l l ) l e n g t h and power d i s t r i b u t i o n and i s l a r g e l y independent of s k u l l depth or w i d t h , p o o l volume i s i n c r e a s e d (or melt r a t e s can be i n c r e a s e d ) . b y removing the water c o o l e d copper h e a r t h from around the s k u l l - t h i s p r a c t i c e i n c r e a s e s b o t h the e v a p o r a t i o n r a t e s and the r i s k s , More e f f i c i e n t use of the i n p u t power i s made when m e t a l throughput r a t e s a r e i n c r e a s e d - t h i s e f f i c i e n t use of power must be weighed a g a i n s t r e f i n i n g and the most e f f i c i e n t c o m b i n a t i o n of h i g h p o o l volume w i t h low e v a p o r a t i o n r a t e i s o b t a i n e d by s p r e a d i n g the i n p u t power over as l a r g e an a r e a as p o s s i b l e . 152 CHAPTER 6 Summary and Recommendations f o r F u t u r e Work 6.1. Summary In t h i s work a t h r e e d i m e n s i o n a l steady s t a t e heat t r a n s f e r model of the e l e c t r o n beam h e a r t h has been de v e l o p e d . The p r i n c i p a l o b j e c t i v e of the work was t o use the model t o p r e d i c t c e r t a i n parameters (such as p o o l volume) which a r e i m p o r t a n t i n d e t e r m i n i n g the most e f f i c i e n t method of o p e r a t i o n of the e l e c t r o n beam h e a r t h f u r n a c e . D u r i n g the work i t was shown t h a t the two most c r u c i a l parameters f o r d e t e r m i n i n g the th e r m a l regime i n the h e a r t h a r e the f l u i d f l o w and the power d i s t r i b u t i o n / l e v e l a p p l i e d t o the s k u l l . These two f a c t o r s e s s e n t i a l l y d etermine the s i z e , shape and temperature d i s t r i b u t i o n of 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 as t o t h e i r e f f e c t s on the l i q u i d p o o l . These i n c l u d e d g e o m e t r i c a l f a c t o r s ( d e p t h , w i d t h ) , the presence of the h e a r t h mould and the i n f l u e n c e of melt r a t e . These o t h e r f a c t o r s a l s o a f f e c t the t h e r m a l regime but t o a f a r l e s s e r e x t e n t than the p r i m a r y c o n s i d e r a t i o n s of f l u i d f l o w and power i n p u t . The 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 of aluminum from a t i t a n i u m a l l o y m e l t was a l s o c a l c u l a t e d . In 1 52 153 g e n e r a l i t was found t h a t any c o m b i n a t i o n of i n p u t parameters which produced an i n c r e a s e d l i q u i d p o o l volume (and t h e r e f o r e i n c r e a s e d r e f i n i n g c a p a c i t y , d i s s o l u t i o n time and s e t t l i n g t i m e s ) a l s o i n c r e a s e d the s u r f a c e e v a p o r a t i o n r a t e . I n c r e a s i n g the f l u i d v e l o c i t y was the o n l y e x c e p t i o n t o t h i s g e n e r a l i z a t i o n . I n c r e a s e d l i q u i d movement s i g n i f i c a n t l y i n c r e a s e d p o o l volume w h i l e r e d u c i n g the s u r f a c e e v a p o r a t i o n r a t e of a l l o y i n g e l e m e n t s . The melt r a t e employed d u r i n g e l e c t r o n beam m e l t i n g was r e f l e c t e d i n the power d i s t r i b u t i o n a p p l i e d t o the s k u l l . An i n c r e a s e d melt r a t e r e s u l t e d i n l e s s energy b e i n g d e l i v e r e d t o the s k u l l . T h e r e f o r e an i n c r e a s e i n melt r a t e r e s u l t s i n a decrease i n the r e f i n i n g c a p a b i l i t y , a dec r e a s e i n the e v a p o r a t i o n r a t e of a l l o y elements and a more e f f i c i e n t use of the i n p u t energy f o r m e l t i n g . The melt r a t e s used i n commercial o p e r a t i o n s must a r r i v e a t some ba l a n c e of r e f i n i n g , power usage and e v a p o r a t i o n r a t e s . S k u l l geometry ( i . e . d e p t h , w i d t h ) has a s m a l l e f f e c t on p o o l volume and e v a p o r a t i o n r a t e s . S k u l l d i m e n s i o n s do have a s i g n i f i c a n t e f f e c t on the average heat f l u x out of t h e s k u l l . As the d e p t h and w i d t h d i m e n s i o n s a r e i n c r e a s e d , the average heat f l u x t h r o u g h the s k u l l s u r f a c e 154 d e c r e a s e s . T h i s a l l o w s f o r c o n s t r u c t i o n of h e a r t h mould by the l e s s c o s t l y tube and c a s t method i f the s k u l l were l a r g e enough. 6.2. Recommendations f o r F u t u r e Work 6.2.1 V e r i f i c a t i o n At the p r e s e n t t ime the model i s e s s e n t i a l l y u n v e r i f i e d . A l t h o u g h i t p r e d i c t s t r e n d s o b s e r v e d d u r i n g o p e r a t i o n - such as p o o l depths i n the range of 2 - 4 cm and e v a p o r a t i o n r a t e s i n c r e a s i n g w i t h i n c r e a s i n g power i n p u t the degree t o which the t h e r m a l regime i s p r e d i c t e d i s unknown. Only an e x t e n s i v e program of m o d e l l i n g s m a l l e r s k u l l s and t h e i r accompanying moulds f o l l o w e d by c o n d u c t i n g e x p e r i m e n t s on s i m i l a r equipment can v e r i f y the model c o m p l e t e l y . 6.2.2 Power D i s t r i b u t i o n and Power L e v e l •The exact d e t a i l s of the d e l i v e r y of power t o the s u r f a c e of a l i q u i d m e t a l by an e l e c t r o n beam a r e not c o m p l e t e l y known. For i n s t a n c e , the shape of t h e l i q u i d s u r f a c e , the programmed a n g l e of the e l e c t r o n beam and the m a t e r i a l b e i n g i r r a d i a t e d a l l c o n t r i b u t e t o the power l o s t due t o e l e c t r o n b a c k s c a t t e r i n g . F u r t h e r work i n d e t e r m i n i n g t h i s type of i n f o r m a t i o n i s r e q u i r e d . I t would a l s o be h e l p f u l t o have a c c e s s t o good d e s c r i p t i o n s of power d i s t r i b u t i o n s used i n i n d u s t r i a l 155 h e a r t h f u r n a c e s . A l t h o u g h an o p e r a t o r can always t e l l e x a c t l y where the beam i s a c t i n g on the s u r f a c e of the s k u l l , t h i s l o c a t i o n does not always c o r r e s p o n d t o the programmed l o c a t i o n . The i n c r e a s e i n the r e l i a b i l i t y of t h i s type of d a t a would r e q u i r e s i g n i f i c a n t f i n a n c i a l s u p port from the p r i v a t e s e c t o r and i s not l i k e l y t o be f o r t h c o m i n g . 6.2.3 E v a p o r a t i o n Rates F u r t h e r work on the m o d e l l i n g of e v a p o r a t i o n r a t e s d u r i n g e l e c t r o n beam r e m e l t i n g i s r e q u i r e d . The model used i n t h i s work i s a t be s t p r i m i t i v e , a t t e m p t i n g o n l y t o g i v e some i n d i c a t i o n of the e x p e c t e d e v a p o r a t i o n r a t e s . I n v e s t i g a t i o n i n t o the unsteady s t a t e n a t u r e of the t h e r m a l regime and the f l u i d f l o w regime s h o u l d accompany any program exa m i n i n g e v a p o r a t i o n r a t e s . Both of these f a c t o r s can g r e a t l y e f f e c t the amount of e v a p o r a t i o n . 6.3. C o n c l u d i n g Remarks The model d e v e l o p e d i n t h i s work r e p r e s e n t s a f i r s t a t t e mpt a t q u a n t i f y i n g the t h e r m a l and mass f l o w regimes d u r i n g e l e c t r o n beam h e a r t h r e m e l t i n g . A g r e a t d e a l of work remains t o be done t o a l l o w the p r o c e s s t o o p e r a t e more e f f i c i e n t l y w i t h r e s p e c t t o energy usage, e v a p o r a t i o n r a t e s and r e f i n i n g c h a r a c t e r i s t i c s . L i s t of R e f e r e n c e s 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 Reviews, 3J_, No. 2, 1986, 77-89. 2. S. S c h i l l e r , U. H e i s i g , S. Panzer : " E l e c t r o n Beam  Technology", 1982, New York, W i l e y . 3. H.R. Smith : " E l e c t r o n Beam P r o c e s s i n g " , 1972, Dayton, Ohio, U n i v e r s a l Technology 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 and Ion Beam Technology", 1974, P r i n c e t o n , NJ, The E l e c t r o c h e m i c a l S o c i e t y , 482 - 517. 5. H. Ranke, V. Bauer, J . Heimel : " E l e c t r o n Beam R e m e l t i n g and R e f i n i n g - S t a t e of the A r t 1986",, P r o c . Conf., 1987, B a k i s h M a t e r i a l s C o r p o r a t i o n , Englewood, NJ., 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 onference on T i t a n i u m P r o d u c t s and A p p l i c a t i o n s " , P r o c . Conf., 1987, T i t a n i u m Development A s s o c . , Dayton, Ohio, 1011 - 1019. 7. K. Takagi : M a s t e r ' s T h e s i s , U n i v e r s i t y of B r i t i s h C o lumbia, 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 Conference on T i t a n i u m  P r o d u c t s and A p p l i c a t i o n s " , P r o c . Conf., 1987, T i t a n i u m Development A s s o c . , Dayton, Ohio, 879 -883. 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 and R e f i n i n g - S t a t e of the A r t 1985 P t . I " , P r o c . Conf., 1985, B a k i s h M a t e r i a l s C o r p o r a t i o n , Englewood, NJ., 40 - 47. 10. ASM M e t a l s Handbook, V o l 3, American S o c i e t y f o r M e t a l s , M e t a l s Park, Ohio, 357. 156 157 11. E.E. Brown, 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 and  R e f i n i n g - S t a t e of the A r t 1985 P t . I I " , P r o c . Conf., 1985, B a k i s h M a t e r i a l s C o r p o r a t i o n , Englewood, NJ., 103 - 117. 12. C. d'A Hunt, 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 and R e f i n i n g - S t a t e of t h e A r t 1985 P t . I " , P r o c . Conf., 1985, B a k i s h M a t e r i a l s C o r p o r a t i o n , Englewood, NJ., 58 - 70. 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 and A p p l i c a t i o n s " , P r o c . Conf., 1987, T i t a n i u m Development A s s o c . , Dayton, Ohio, 904 - 917. 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 and  R e f i n i n g - S t a t e of the 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 , Englewood, NJ., 88 - 99. 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 I n f o r m a t i o n C e n t e r , B a t t e l l e M e m o r i a l 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 of B r i t i s h C o l u m b i a , 1978. 17. H. Fenech, W.M. Rohsenow : T r a n s . ASME, J . of Heat T r a n s . 85, 1963, 15. 18. W.B. E i s e n , A. Campagna : Met. T r a n s . , J_, 1 970, 849. 19. L.F. C a r v a j a l , G.E. G e i g e r : Met. T r a n s . , 2, 1971, 2087. 20. F. K r i e t h , W.Z. B l a c k : " B a s i c Heat T r a n s f e r " , 1980, New York, Harper and Row. 21. M. Choudhary, J . S z e k e l y : Met. T r a n s . , _1_1_B, 1980, 439 -453. 22. 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 of 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 and R e f i n i n g - S t a t e of the A r t 1986", P r o c . Conf., 1987, B a k i s h M a t e r i a l s C o r p o r a t i o n , Englewood, NJ., 45 - 52. 24. I.V. Samarasekera, J.K. Brimacombe : Can. Met. Q u a r t e r l y , JJ8, 1979, 251 - 266. 25. L. Hageman, D. Young : " A p p l i e d I t e r a t i v e Methods", Academic P r e s s , New York, 1981. 26. C.H. E n t r e k i n : U n p u b l i s h e d work. 27. H.S. K h e s h g i , P.M. Gresho : " E l e c t r o n Beam R e m e l t i n g and  R e f i n i n g - S t a t e of the A r t 1986", P r o c . Conf., 1987, B a k i s h M a t e r i a l s C o r p o r a t i o n , Englewood, NJ., 68 - 97. 28. R. Pasko : U n p u b l i s h e d 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 of 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 onference on  T i t a n i u m P r o d u c t s and A p p l i c a t i o n s " , P r o c . Conf., 1987, T i t a n i u m Development A s s o c . , Dayton, Ohio, 939 - 947. 31. H e i p l e and Roper : Weld J . , 6_1_, 1982, pp. 97, 32. Oreper e t . a l . : Weld J . , 62, 1983, pp. 307. APPENDIX 1 Beam Spot Model I n t r o d u c t i o n The power d i s t r i b u t i o n s used i n the h e a r t h model a r e time averaged due t o the r e s t r i c t i o n s of the s t e a d y s t a t e heat f l o w e q u a t i o n . T h i s i m p l i e s t h a t a l l of t h e t e m p e r a t u r e s r e p o r t e d a r e time averaged as w e l l . In a c t u a l o p e r a t i o n , t emperature v a r i a t i o n s a t the s u r f a c e o c c u r due t o the moving p o i n t power s o u r c e . These v a r i a t i o n s , i f s i g n i f i c a n t l y l a r g e , may be i m p o r t a n t i n d e t e r m i n i n g heat l o s s e s due t o r a d i a t i o n and a l l o y element l o s s e s due t o e v a p o r a t i o n . In o r d e r t o e v a l u a t e the magnitude of the temperature v a r i a t i o n s and d e t e r m i n e the r e l a t i o n s h i p between them and power d e n s i t y , beam sweep speed and o t h e r time dependent v a r i a b l e s , a s i m p l e unsteady s t a t e heat f l o w 16 28 model p a t t e r n e d a f t e r B a l l a n t y n e and Pasko has been d e v e l o p e d . The model i s c a p a b l e of p r e d i c t i n g t e m p e r a t u r e s 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 s e c t i o n . F o r m u l a t i o n S i n c e the i n c l u s i o n of time i n a heat f l o w problem r e p r e s e n t s the i n c l u s i o n of an e x t r a d i m e n s i o n , i t i s advantageous t o choose a c o o r d i n a t e system which a l l o w s the 159 160 e x c l u s i o n of one of t h e heat f l o w d i r e c t i o n s . The o n l y system which p r o v i d e s t h i s c o n s i d e r i n g the c o n s t r a i n t s on the p h y s i c a l system i s c y l i n d r i c a l c o o r d i n a t e s w i t h a x i a l symmetry (see f i g u r e A1.1). Making the a x i a l symmetry assu m p t i o n the unsteady s t a t e heat f l o w e q u a t i o n i s 3 3T k3T 3 3T 3T — ( k — ) + + — ( k — ) = pC — 3x 3r r 3r 3z 3z P3t ...(A1.1) In an attempt t o i s o l a t e the a f f e c t s of the beam from a l l o t h e r e f f e c t s , the boundary c o n d i t i o n s were made as s i m p l e as p o s s i b l e . T h e r e f o r e the p e r i m e t e r and bottom s u r f a c e s were c o n s i d e r e d t o be a d i a b a t i c or : 3T k q — = 0 where r = r S 9r m a x ...(A1.2) 3T • k c — = 0 where z = z „ S 9 z m a x ...(A1.3) The assumption of a x i a l symmetry i m p l i e s t h a t 3T k c — = 0 where r = 0 S 9r ...(A1.4) The t o p boundary c o n d i t i o n r e f l e c t s the t h r e e heat t r a n s f e r p r o c e s s e s t a k i n g p l a c e a t the s u r f a c e . At the top s u r f a c e energy i s b e i n g d e l i v e r e d by the e l e c t r o n beam, heat i s b e i n g r a d i a t e d t o the f u r n a c e atmosphere and m a t e r i a l i s 161 162 e v a p o r a t i n g r e q u i r i n g the l a t e n t heat of v a p o r i z a t i o n . The power and r a d i a t i o n boundary c o n d i t i o n s a r e e q u i v a l e n t t o t h a t used i n the h e a r t h f u r n a c e model ( e q u a t i o n 2.4) w i t h t h e e x c e p t i o n t h a t a time averaged power d i s t r i b u t i o n i s not r e q u i r e d . 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 l o a d on the 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 the l a t e n t heat of e v a p o r a t i o n by the e v a p o r a t i o n r a t e of the m a t e r i a l b e i n g m e l t e d . The r a t e of e v a p o r a t i o n i s g i v e n by the Langmuir e q u a t i o n ( e q u a t i o n 4.1). T h i s e q u a t i o n i s v a l i d p r o v i d i n g the r a t e l i m i t i n g s t e p f o r e v a p o r a t i o n i s the t r a n s f o r m a t i o n from l i q u i d t o gas and not d i f f u s i o n i n e i t h e r the l i q u i d or the gas phases. U s i n g a pure m a t e r i a l under a good vacuum w i t h i n f i n i t e pumping speed i n s u r e s t h i s . The t o p boundary c o n d i t i o n can then be w r i t t e n as : AH p°C 3T 4 4 -k,,— = P 0 ( r ) - o e ^ i T Z - T*) -S 9 z S S S A p/2irRMT ...(A1.5) Due t o the complex n a t u r e of the t o p s u r f a c e boundary c o n d i t i o n , an a n a l y t i c a l s o l u t i o n t o the e q u a t i o n s i s i m p o s s i b l e . T h e r e f o r e a f i n i t e d i f f e r e n c e approach was used. The e q u a t i o n s g e n e r a t e d by the f i n i t e d i f f e r e n c e t e c h n i q u e a r e s o l v e d u s i n g the a l t e r n a t i n g d i r e c t i o n i m p l i c i t t e c h n i q u e s p l i t t i n g each time s t e p i n t o f o u r s m a l l e r time s t e p s t o reduce the e r r o r a s s o c i a t e d w i t h the n o n - l i n e a r n a t u r e of the r a d i a t i o n boundary c o n d i t i o n . 

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