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Mass transfer aspects of A.C. electroslag remelting Fraser, Michael E. 1974

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MASS TRANSFER ASPECTS OF A . C . ELECTROSLAG REMELTING by MICHAEL.E. FRASER B.Sc. McMaster U n i v e r s i t y , 1968 M.Sc. McMaster U n i v e r s i t y , 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of METALLURGY We accept th is thesis as conforming to the required standards THE UNIVERSITY OF BRITISH COLUMBIA January, 1974 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i lab le for reference and study. I further agree that permission for extensive copying of th i s thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i cat ion of this thesis for f inanc ia l gain shal l not be allowed without my written permission. The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Depa rtment i i ABSTRACT Previous attempts at a quantitative description of alloy losses i n e lec tros lag remelting (ESR) operations have not adequately re f lec ted the true nature of th i s process. In the current work a mass t ransfer model was developed which takes into account four react ion s i t e s and mass t ransfer of a l l species at each s i t e . An experimental program was subsequently devised to tes t extensively the provisions of the model. The overa l l system was s i m p l i f i e d by avoiding complex a l loys and multicomponent slags and also by excluding a i r . Care was taken to deal only with species whose thermochemistry was known in ,both slag and metal phases. The electrode material used was mild steel in which manganese was the o n l y o x i d i z a b l e species . Slags used throughout the melt program were CaV^ - 20% CaO, f o r which system the a c t i v i t y data for FeO, MnO were a v a i l a b l e . The melt program provided data on Mn losses i n a var ie ty of ESR operating conditions including steady and unsteady s ta te , l i v e and insulated mold conf igurat ions . The ef fects of melting i n a i r and of d i r e c t current operation were also invest igated . Several melts under-taken with Armco i ron were designed to permit d i r e c t c a l c u l a t i o n of c e r t a i n mass transfer c o e f f i c i e n t s . The mass transfer model was applied to each melt and the predict ions compared to the experimental data. There was general ly good agreement between the two, confirming that the manganese losses were contro l led e n t i r e l y by mass t rans fer e f f e c t s . Further study of the model i i i showed mass transfer of species in the slag phase to be the predominant rate c o n t r o l l i n g step. The r e l a t i v e roles of the various reaction s i tes in contr ibuting to the overa l l mass transfer were also elaborated by th is ana lys i s . In order to assess the f u l l potential of th is mass transfer approach, the model was extended to the manganese, sulphur reaction system and to l a rger , commercial-sized ESR furnaces. Projections of a l l o y losses based on th is scale-up study appeared to be consistent with the sparse information that i s current ly a v a i l a b l e . i v TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES i x LIST OF FIGURES x i ACKNOWLEDGEMENTS x iv Chapter 1. INTRODUCTION 1 1.1 Introduction 1 1.2 Nature of the Problem 1 1.3 Previous Work 4 1.3.1 Equi l ibr ium Models 4 1.3.2 Single Stage Reactor Models 10 1.3.3 Descriptive Mass Transfer Analysis 13 2. THE MASS TRANSFER MODEL 18 2.1 Introduction 18 2.2 Mathematical Formulation 20 2.3 Evaluation of Mass Transfer Mode 1 Parameters . . . 27 2.3.1 Bulk Concentrations 27 2.3.2 Molar Equi l ibr ium Constant 27 2.3.3 Reaction Sites Considered i n Modelling . . . 33 2.3.4 Mass Transfer at Electrode Tip/Slag Interface 34 V Chapter Page 2.3.5 Mass Transfer at Metal Pool-Slag Interface • 43 2.3.6 Mass Transfer at F a l l i n g Drop/Slag Interface 45 2.4 Area/Volume Ratio Estimates 54 2.4.1 Area/Volume Ratio of Drop 55 2.4.2 Area/Volume Ratio of Film . 56 2.4.3 Area/Volume Ratio of Pool 57 2.5 Numerical Solution of D i f f e r e n t i a l Equations for Mass Transfer 59 2.5.1 Estimation of Drop Fa l l Time 60 2.6 Summary . . . 62 3. EXPERIMENTAL PROGRAM 63 3.1 The U.B .C . E lectros lag Furnace 63 3.2 Slag Sampling Device 63 3.3 Materials 67 3.3.1 Electrode Material 67 3.3.2 Armco Iron Electrodes 68 3.3.3 Slag Materials 68 3.4 Atmosphere Control 70 3.5 Melting Conditions 70 3.6 Melt Records 71 3.7 Slag and Metal Analysis 71 3.8 Melt Program 73 vi Chapter Page 3.8.1 General Comments 73 3.8.2 Experimental Determination of Mass Transfer Coef f i c ients 73 3.8.3 Steady State 75 3.8.4 Unsteady State Recovery 79 3.8.5 Live Mold 79 3.8.6 Insulated Mold 83 3.8.7 Steady State Melts i n A i r and with DC Power 85 4. RESULTS . . 91 4 .1 . Analysis of Rate Contro l l ing Steps 91 4.2 Results of Experiments to Determine Pool Mass Transfer Coef f ic ients 95 4.2.1 Experimental Determination of Mass Transfer Coef f ic ients kM 2+, k F e2+ .95 4.2.2 Experimental Determination of the Mass Transfer Coef f i c ient k^2- 99 4.3 Boundary Conditions for Solution of Mass Transfer Model 104 4.4 Use of F e 2 + "Potent ia l s " 105 4.5 Choice of y M n 0 106 4.6 Modelling Results 108 4.6.1 Steady State Results 108 4.6.2 Unsteady State Results I l l 4 .6.3 Results of Live Mold Experiments 117 4.6.4 Insulated Mold Results 124 v i i Chapter Page 4.6.5 Results of Steady State Melting i n the Absence of Inert Gas Cover 128 4.6.6 Direct Current Results 131 5. DISCUSSION 5.1 Model Parameters 138 5.1.1 Mass Transfer Coef f ic ients 138 5.1.2 Molar Equi l ibr ium Constant 146 5.13 Melt Rate 149 5.14 Rate of Slag Loss 151 5.2 The Mass Transfer Model 152 5.2.1 Model Results 152 5.2.2 S o l i d i f i c a t i o n Effects . 156 5.3 Model Predict ions 157 5.3.1 Relat ive Contributions of Reaction Interfaces 158 5.3.2 Nature of Steady State Mass Transfer . . . . 159 5.3.3 Extension of Model to Combined Manganese and Sulphur Transfer 162 5.3.4 Scale-up Predict ions 171 5.3.5 Control of A l l o y Losses 180 5.4 Electrochemical Phenomena 182 5.5 Conclusions 183 5.6 Suggestions for Future Work 184 v i i i Page PRINCIPLE SYMBOLS 186 BIBLIOGRAPHY 189 APPENDICES 200 APPENDIX I - DETERMINATION OF AUXILARY PARAMETERS FOR MASS TRANSFER MODEL 200 A . I . I Calculat ion of Average Melt Rate 200 A . 1 . 2 Drop Weight Calculat ion 200 A . 1 . 3 Rate of Slag Loss 201 APPENDIX II - ANALYTICAL METHODS 204 A . I I . l Determination of Total Iron 204 A.11.2 Determination of Manganese 205 2+ A . I I . 3 Determination of Fe 206 APPENDIX III - DERIVATIONS OF ADDITIONAL MASS TRANSFER EXPRESSIONS 208 A . I I I . l Desulphurization of Fe-S A l l o y during ESR Using CaF2~20%CaO Slags 208 A . I I I . 2 Desulphurization of Resulphurized Steel i n CaF2-CaO Slags 21:3 2+ A . I I I . 3 Derivation of Expressions for (Mn ) . , ( F e 2 + ) i ] . . 220 APPENDIX IV - COMPUTER PROGRAM FOR SOLVING MASS TRANSFER MODEL 225 APPENDIX V - THERMOCHEMICAL AND HEAT TRANSFER CALCULATIONS . . 231 A . V . I Oxidation Potent ia ls of Mn, S i i n 1018 Steel Melting i n CaF2-20%Ca0 Slag 231 A.V.2 Calculat ion of Temperature of Electrode Film Using Derived Heat Transfer Co-e f f i c i e n t at S lag/Fi lm Interface 235 A . V . 3 Calculat ion of Temperature of Drop a f ter F a l l through Slag 236 ix LIST OF TABLES Table Page I . Summary of Theoretical Mass Transfer C o e f f i c i e n t s , Average Area/Volume Ratios and Approximate Residence Times for 76 mm (j) ESR Furnace 58 I I . (a) (b) Operating Parameters and Slag Analysis - Runs #60,61 ,62 76 I I I . (a ) Operating Parameters 77 (b) Slag Analysis - Run #27 . . . . . . . (c) Slag Analysis - Run #53 (d) Ingot Analysis - Run #53 78 IV.(a) Operating Parameters 80 (b) Slag Analyses - Runs #30,31 , (c) Slag Analysis - Run #34 81 (d) Slag Analysis - Run #37 (e) Slag Analysis - Run #42 82 (f) Ingot Analysis - Run #42 . . V.(a) Operating Parameters 84 (b) Slag Analysis - Run #47 . . . , (c) Slag Analysis - Run #59 VI. (a) Operating Parameters 86 (b) Slag Analysis - Run $40 (c) Ingot Analysis - Run #40 V I I . (a) Operating Parameters 87 (b) Slag Analysis - Run #46 (c) Slag Analysis - Run #58 VII I . (a ) (b) Operating Parameters and Slag Analysis - Run #50 . . 89 IX.(a) Operating Parameters 90 (b) Slag Analysis - Run #52 . . . . . . (c) Slag Analysis - Run #55 X. Evaluation of Terms i n Rate Contro l l ing Step Analysis 94 XI . Transport Control Data at Ingot Pool for Melts #60,61 96 X Table Page XII . Summary of Data for Experimental Evaluation of X I I I . Data for Experimental Evaluation of k<.2- 102 XIV. Heat Transfer Coef f ic ients i n Slag at S lag/Liquid Metal Boundaries 141 XV. Summary of Data for Modelling Combined Mn, S Losses . . . 168 XVI. Summary of Data Used i n Scale-Up Modelling of 510 mm cj) ESR Furnace 173 xi LIST OF FIGURES Figure Page 1. ESR Reaction Sites 3 2. Comparison of Theoretical Equi l ibr ium Model P r o f i l e s with Experimental Sulphur Concentration P r o f i l e s . 3. Use of F ic t ious "equi l ibr ium" Temperature 7 4. Comparison of Theoretical and Experimental A l l o y Losses from an A286 Steel 9 5. Comparison of Theoretical and Experimental Ti Losses using Single-Stage Reactor Model 12 6« [ S ] D , [ S ] T , [S ] p and [S]<- as a Function of Time . . . . 15 7. Schematic Representation of Fluxes 20 8. Schematic Concentration Gradients at Slag/Metal Phase Boundary 22 9. YpeQ vs N p e 0 for Constant Mole Ratio at 1450°C 32 10. Y M n Q vs wt % MnO i n Various CaF2-CaO Slags at 1500°C 32 11. Thickness of Liquid Metal Film on Conical Electrode Tip 36 12. Schematic Flow Pattern for Typical ESR Furnace 41 13. Surface Area and Amplitude Coef f i c ient of E l l i p s o i d 14. Gas Cap and Sampling Unit 65 15. Schematic Diagram Depicting ESR Configuration 66 2+ 16. Experimental Determination of Rate of Rise of Mn - Melts #60,61 98 2-17. Experimental Determination of Rate of Rise of S - MeTt #62 101 x i i Figure Page 18. Y M n Q Values F i t ted by Computer Program for CaF^ -20% CaO Slag 107 19. Comparison of Theoretical and Experimental Results - Steady State Melt #27 109 20. Comparison of Theoretical and Experimental Results for Steady State Melt #53 (a) Slag (b) Ingot 110 21. Comparison of Theoretical and Experimental Results for Unsteady State Melts #30,31 112 22. Comparison of Theoretical and Experimental Results f o r Unsteady State Melt #34 113 23. Comparison of Theoretical and Experimental Results for Unsteady State Melt #37 114 24. Comparison of Theoretical and Experimental Results for Unsteady State Melt #42 (a) Slag (b) Ingot 115 25. Comparison of Theoretical and Experiment Results for Live Mold Melt #47 118 26. AC Waveform - Mold Current - Live Mold Melt #47 . . . . 119 2+ 27. Rate of Rise of Fe When Melt ing Armco Iron - Live Mold Melt #59 122 28. AC Waveform - Mold Current - Live Mold-.Melt #59 . . . . 123 29. Comparison of Theoretical and Experimental Results for Insulated Mold Melt #40 (a) Slag (b) Ingot 125 30. Comparisons of Theoretical and Experimental Results for Fe^ , M n 2 + i n Insulated Mold (a) Melt #46 (b) Melt #58 127 31. Comparison of Theoretical and Experimental Results for Melt #50 ( i n A i r ) 129 32. Comparison of Theoretical and Experimental Results for DC-ve Melt #52 133 33. Comparison of Theoretical and Experimental Results for DC+ve Melt #55 134 x i i i Figure Page 34. P o l a r i z a t i o n Character is t ics of Iron i n CaF ?-CaO Melts (a) Anodic (b) Cathodic 136 2+ 35. Estimation of Mn Lost by Reduction at Electrode Tip - Melt #61 140 36. Effects of Varying Mass Transfer Coef f ic ients on Model Predict ions 143 37. Equi l ibr ium and Mass Transfer Model Comparison for Steady State Melt #27 154 38. Equi l ibr ium and Mass Transfer Model Comparison for Unsteady State Melt #42 . . . . . . . 154 39. Extrapolation of Theoretical Results for Melt #27 . . . . 161 40. Comparison of Theoretical and Experimental Results for Desulphurization i n ESR . 169 41. Effect of ESR Furnace Size on Desulphurization Character is t i cs 174 42. Pool Geometry as a Function of Ingot Height 177 43. Carbon D i s t r i b u t i o n Along Axis of an ESR Ingot of High C-Cr Steel 179 A . I . I Osci l lographic Current Trace 202 A.3.1 Schematic Representation of Fluxes i n Desulphurization. . 209 A . I I I . 2 Schematic Representation of Fluxes i n Manganese Desulphurization 215 A . I V . l Algorithm Outl ining Computer Program 226 A . V . I Free Energy - Temperature P lo t Showing Rotation of Mn/MnO Line 234 x i v ACKNOWLEDGEMENTS The author would l i k e to express h i s gratitude to Dr. A. M i t c h e l l f o r h i s help and guidance throughout the course of t h i s work. Much appreciation i s also given f o r the many h e l p f u l discussions with fellow graduate students and other f a c u l t y members. The author would also l i k e to thank Dr. J.K. Brimacombe for h i s h e l p f u l c r i t i c i s m of the d r a f t manuscript. The assistance of Mr. A. Thomas i n designing and construct-ing the experimental apparatus i s g r a t e f u l l y acknowledged. Special thanks are due also to Mr. R. Palylyk for drawing the f i g u r e s i n the t h e s i s . The author would l i k e to thank h i s wife, Deborah, f o r her constant encouragement over the duration of t h i s work. The f i n a n c i a l assistance of the American Iron and S t e e l I n s t i t u t e (Grant #190) and of the National Research Council are g r a t e f u l l y acknowledged. 1 CHAPTER 1 INTRODUCTION 1.1 Introduction The analysis of rate phenomena i n process metallurgy has received increasing at tent ion over the past decade. This in teres t has evolved c h i e f l y as a r e s u l t of the demands for increased automation to reduce costs and for increased process e f f i c i e n c y . In turn the necessity of developing the appropriate equipment and controls has placed more emphasis on a precise understanding of the deta i led transport processes occurring i n metal lurgical processes. The appl i ca t ion of mathematical modelling techniques i s now common i n ferrous and nonferrous operations a l i k e , from blast furnace to soaking p i t ; from roaster to r e f i n e r y . 1.2 Nature of the Problem The increasingly widespread use of the e lec t ros lag remelting process for the production of high q u a l i t y a l l o y s , espec ia l ly forging a l l o y s , has resulted i n numerous invest igat ions into the nature of th i s process. These studies may be c l a s s i f i e d as fo l lows : (1) the heat flow problem; (2) the chemical r e f i n i n g problem. To date the preponderance of work has been concerned with the mathematical and experimental analysis of heat flows for the purpose of c o n t r o l l i n g the s t ructural properties of ESR forging ingots.2.24,56,86 2 These research programs have played an important ro le i n the design and construction of larger and larger ESR f a c i l i t i e s (>100 tons) . However, i t has only recently been recognized that substantial degrees of r e f i n i n g involving essential a l l o y i n g elements may also occur. On comparing ESR to i t s major competitor, vacuum arc remelt-ing (VAR), an enhanced potential for chemical r e f i n i n g and composition control i n ESR becomes evident. In VAR, the environment, consist ing of water cooled copper mold and vacuum, i s iner t to the metal being remelted. Any r e f i n i n g e f fec t i s consequently l i m i t e d to pressure sens i t ive reactions and the f l o a t i n g out of i n c l u s i o n s . The s i t u a t i o n i n ESR i s very d i f f e r e n t . From the schematic diagram of a t y p i c a l ESR uni t (Figure 1) one observes a number of reaction s i t e s involv ing s o l i d / gas, s o l i d / l i q u i d , l i q u i d / l i q u i d and g a s / l i q u i d inter faces . This p r o b a b i l i t y of chemical in terac t ion suggests the d e s i r a b i l i t y of i n v e s t i -gation into the possible contr ibut ing fac tors . That spec ia l ty steel makers should be concerned with losses of highly ox idizable a l l o y i n g elements i s not s u r p r i s i n g . In some a l l o y s , where only small concentrations of essential a l l o y i n g elements are present, there may be a c r i t i c a l s e n s i t i v i t y to composition f l u c t u a t i o n s . For instance, a maraging steel containing 0.8% ti tanium i s reported to s u f f e r a d e c r e a s e i n y i e l d s t r e n g t h of 10^ p s i on l o s i n g 0.1% t i t a n i u m J Many examples of th i s type, involving losses of t i tanium, aluminium, 2 3 4 5 molybdenum, s i l i c o n , e t c . , have been reported i n the l i t e r a t u r e . ' ' ' Attempts have been made to correct or compensate for such 3 losses by say, using an i n e r t gas cover or descaling the electrode. Others have suggested the use of "balanced" slags with additions of the © ELECTRODE/ATMOSPHERE © SLAG/ATMOSPHERE ® ELECTRODE TIP/SLAG © DROPLET/SLAG © METAL POOL/SLAG © METAL POOL/INGOT Pig. 1—ESR reaction sites. 4 oxides of the react ive elements. In such a system TiO^ and A ^ O ^ would be added to slags used for remelting a l loys containing Ti and A l . However, there i s some doubt as to whether t h i s approach of s h i f t i n g the chemical equi l ibr ium i s r e a l l y e f f e c t i v e . The main cause of oxidat ion i s not the low a c t i v i t y of the oxidizable species i n the slag but rather the r e l a t i v e l y high oxidation potential of the s l a g . I t appears, from previous i n d u s t r i a l and experimental evidence, that losses of these important, but react ive a l l o y i n g elements cannot be completely prevented. As a r e s u l t , i f the production of commercially acceptable ingots i s to be ensured, ESR producers must be able to ant i c ipate these losses and compensate the electrode feed stock accord-i n g l y . At the present time, the correct ions are made on a s t r i c t l y empirical basis by observation of previous melts. This procedure i s necessary, due to the i m p r a c t i c a l i t y of sampling and analysis of the remelted metal during the process operation. Such a t r i a l and error approach becomes p r o h i b i t i v e l y expensive for commercial ESR operations producing ingots of high a l l o y steel i n the 60-150 ton range. I t i s imperative, therefore, that a l ternate methods of predic t ing a l l o y losses be developed. One approach, which has much potential i n th i s regard, i s the mathematical modelling of the composition change. Several attempts have been made to formulate such a model, but these have met with l i t t l e success. The reasons for th is lack of success and the p o s s i b i l i t i e s for improvement are discussed below. 5 1.3 Previous Work Many instances of composition changes i n ESR processed material have been reported in the l i t e r a t u r e . References (1) through (22) are representative of these. Attempts to explain the a l l o y element losses quant i ta t ive ly are found much less frequently , however. These studies may be c l a s s i f i e d as fo l lows : 1. Equi l ibr ium models. 2. Single stage reactor models. 3. Descript ive Mass Transfer ana lys i s . 1.3.1 Equi l ibr ium Models Hoyle 7 has treated the ESR process as a closed system fol lowing an equi l ibr ium re la t ionship between the slag and metal such that each element of molten metal was assumed to a t t a i n complete equi l ibr ium with the s l a g . With this approach, he predicted, for example, that the rate of sulphur exchange decreased progressively during the melt. This treatment led to a re la t ionship i n which the sulphur content of the metal depended exponential ly on time. These ca lculat ions were compared to experimental resul ts on steels containing sulphur. However, at the time th i s work was done (1962-1966) i n s u f f i c i e n t data were ava i lab le on thermodynamic a c t i v i t i e s l n (1.1) 6 i n CaF2-based slags to assess properly the value of K, the reaction equi l ibr ium constant. This f a c t , i n addi t ion to poor experimental p r e c i s i o n , prevented an exact comparison of the model and the experimental r e s u l t s . 17 28 29 Hawkins, Davis and co-workers ' ' presented much the same model. In th i s same p a p e r h o w e v e r , a f a i r l y comprehensive study of a c t i v i t i e s and phase e q u i l i b r i a in CaF2"based slags was given. These data allowed a more precise statement of the equi l ibr ium model but nevertheless agreement with published data was not good showing only that the model predicted the correct desulphurization trend (Figure 2). I t i s in teres t ing to note Hawkins' suggestion that k inet i c ef fects may be of some importance i n the ESR process. 22 Subsequently, Hawkins et al_. introduced the concept of a f i c t i t i o u s "equi l ibr ium" temperature. Calculat ion of th is value was accomplished v ia the re la t ionship AG = AG° + RTlnK (1.2) by subs t i tu t ing the appropriate a c t i v i t y coe f f i c i en ts and experimentally determined concentrations for K = " a p R 0 D U C T S . (1.3) REACTANTS Thus each reaction occurring in a system w i l l not necessari ly have the same "equi l ibr ium" temperature. E s s e n t i a l l y , then, th i s exercise was an attempt at curve f i t t i n g and, as shown i n Figure 3, there was some improvement i n the agreement between experimental and modelled p r o f i l e s o 0 z z I/) 1 1 1 1 1 1 1 0 -0 Ilnrtigl wt%S_ D _^ I -in dertrxxJe^x^ / ° / X / 0 _ _ / 5wt-%CaO _ / 1 i i i i i i l i i l i I ~l — Theoretical line -assuming a F e Q fixed by CO/CO2 gas mixture (CO/C02-99/1), -equilibrium at 1823°K Experimental ° results from Cooper18 Weight of metal -m= melted — "Initial wV%S - in electrode S * / X 0 £ o o 10wt-%CaO i i i i i i Initial weight of slag _ 0 0 ~ ^^20wt-%GaO~ 1 1 1 I I 0 8 m 12 0 8 m 12 Figure 2 Comparison of Theoretical Equil ibrium Model P r o f i l e s with Experimental Sulphur Concentration P r o f i l e s of Cooper 9 9 —7~ Calculated from(S) and (FeO) at 1786 °K 4 6 8 10 12 14 WEIGHT OF INGOT REMELTED.Kg 18 Figure 3 Use of f i c t i t i o u s "equi l ibr ium" Temperature i n Predict ing Sulphur Concentration P r o f i l e s 2 2 8 over the previous e f f o r t . , 21 Knights and Perkins have adopted e s s e n t i a l l y the same approach in t rea t ing losses of T i , A l , Si from an A286 iron base a l l o y melted through a CaF 2 - 10% CaO - 5% A l 2 0 3 - 5% T i 0 2 - l%Si0 2 s l a g . This attempt at modelling the ESR process as an "isothermal" system necessitated the assumption of unreasonably high temperatures (>1900°C) i n order to f i t the data. The agreement between model and data was reasonably good as i s shown i n Figure 4. One fundamental weakness of these two modelling attempts i s that mass transport was assumed to be so fas t as to not be rate l i m i t i n g . Another serious weakness shown c l e a r l y i n these resul ts i s the necessary assumption of a unique temperature throughout the react ive region of the ESR system. One of the most obvious features of ESR i s the progressive mel t ing , heating, coo l ing , and freezing cycle undergone by the metal phase. What i s not so obvious i s that the metal , once melted, does not heat up instantaneously to the maximum observed temperature. In th is process, heat t ransfer plays a very important ro le i n determining the heating and cooling cyc le . This has been the subject of many reports i n the l i t e r a t u r e 23 and has been well summarized by M i t c h e l l , Szekely and E l l i o t t . Of in teres t here i s t h e i r conclusion that the l i q u i d metal f i l m on the electrode t i p cannot superheat more than 20 - 30°C. This conclusion i s undoubtedly a r e s u l t of the thinness of the metal f i l m (-100 y) and the r e l a t i v e l y high melt ra te . Any large temperature differences between l i q u i d and s o l i d metal are accommodated instantaneously by a change in melt ra te . Such observations are commonplace in ESR operations where a change in the depth of electrode immersion immediately resul ts in a new S 2 O O O 2 1 9 0 0 - I 8 0 0 h ui u ce. ° = - 1 7 0 0 1 -£ 1 6 0 0 a. 2 1 5 0 0 ui »-L I S O T H E R M A L E S R A T I 6 0 0 ° C 2 0 Initial e l e c t r o d e level T I T A N I U M i-oh T h e full l ine assumes a variable tempera ture as ind icted above S I L I C O N e s sent ia l l y independent of temperature init ial e lectrode level 7 hj / I 9 0 0 ° C A N D 1 6 0 0 ' A L U M I N I U M in i t ia l e l e c t r o d e leve l ~\ 5 0 0 I O O O 1 5 0 0 W E I G H T O F E L E C T R O D E W E L T E D ( K G ) 2 0 0 0 Figure 4 Comparison of Theoretical and Experimental A l l o y Losses from an A286 S t e e l . ESR Operation Treated as an Isothermal System 2 1 10 melt ra te . On the other hand, heat transfer to the drop i s s u f f i c i e n t l y fas t that drops entering the metal pool are thermally equi l ibra ted with 24 the s l a g . Thus there can be a large temperature difference between l i q u i d metal f i l m and the drops. 1.3.2 Single Stage Reactor Models 5 1 A somewhat more r e a l i s t i c approach has been adopted by Etienne. ' In th i s case, while the model assumed the f a m i l i a r exponential r e l a t i o n s h i p , the condit ion that equi l ibr ium was not established between bulk slag and metal phases was incorporated. The mathematical formulation of th is 13 concept was f i r s t set down by stelnmetz and i s based on the exchange of material between a mobile and a s t a t i c phase. Electroslag furnaces were considered by Etienne, i n his adaptation of Steinmertz 1 model, to be c l a s s i f i e d with processes where the contact between phases i s " t r a n s i t -i o n a l " (slag s ta t ionary , metal f l o w i n g ) , without homogenization of the metal phase subsequent to chemical react ion . These concepts are embodied i n the re la t ionship (1.4) [X] [Xl — = 1 expC-CH (1.4) [X ] Q [X] Q where 5 = P S K L A / W m Y - W m / « s rxl i s the X stoichiometric equivalent of the oxidant concentration e i n i t i a l l y i n the slag 11 n MW W e = " -nr-rr (° x i d >o • m oxid Equation (1.4) described the oxidat ive loss of element X from the metal component under the fo l lowing condit ions : ( i ) the oxidized element was at a concentration i n the metal where d i f f u s i o n i n the metal phase did not control mass t rans fer ; ( i i ) the oxidant was present i n a l i m i t e d quantity i n the s l a g ; ( i i i ) the system was f u l l y described by the slag and metal components and interact ions with the atmosphere were excluded. The use of the lumped mass transfer c o e f f i c i e n t , k^, i n (1.4) means the ESR process has been treated as a s ingle stage reactor . In applying th i s model i t was found that the predictions agreed with the experimental concentration time data obtained on AC melts for what was e s s e n t i a l l y an i r r e v e r s i b l e react ion (large equi l ibr ium constant and low a c t i v i t y of oxidized species i n the slag).(Figure 5) However, no sa t i s fac tory agreement was achieved f o r the corresponding DC melts using ident i ca l mater ia l s . The reason for th is lack of agreement i n the l a t t e r cases apparently wasthat, in the DC mode, the reactions occurring at the electrode t i p / s l a g and ingot pool /s lag interfaces were governed by electrochemical phenomena a r i s i n g from concentration p o l a r i z a t i o n 25 e f fects at each reaction s i t e . The consequences of electrochemical control of these reactions was that d i f f e r e n t reactions occurred at the o-DATA THEORETICAL LINE 0 / / / AC-/ 4 6 Y 8 10 Figure 5 Comparison of Theoretical and Experimental Titanium Losses using Single-Stage Reactor Model1"! 13 e lec t ropos i t ive and electronegative poles (corrosion of iron at the e l e c t r o p o s i t i v e s i t e and deposition of Ca or Al at the electronegative 25 s i t e ) . The lumped mass transfer c o e f f i c i e n t approach taken by Etienne dealt with only one reac t ion , i . e . , the oxidat ive loss of an a l l o y species and thus could not be adapted to the DC case. These electrochemical ef fects were far less obvious i n the AC mode and there was no reason to assume that the same oxidat ion reaction did not occur at each reaction s i t e . The lumped mass transfer c o e f f i c i e n t could , therefore, be evaluated for AC melt ing. This model has provided evidence of mass transfer control i n AC ESR, but the real physical processes a f fec t ing mass transfer were not made apparent and factors contr ibut ing to scale-up problems were not readi ly i d e n t i f i e d . There i s some improvement i n the predict ions of the s ingle stage reactor model, but, nevertheless, a f i t t e d parameter, the lumped mass transfer c o e f f i c i e n t , i s s t i l l required. 1.3.3 Descriptive Mass Transfer Analysis I t has been shown that the previous attempts to model the chemical composition changes occurring i n ESR have met with l i m i t e d success. None of them has shed much l i g h t on the p a r t i c u l a r nature of mass t ransfer processes c o n t r o l l i n g these changes. Nevertheless, there have been a few descr ipt ive reports , some supported by data, of these mass t ransfer effects .9 ,10,14,16,18 m Q S t i m p 0 r t a n t of these i s the 14 work of Cooper et al_. who presented a c lear and precise analysis of the ESR reaction s i tes undergoing desulphurizat ion. Four reaction s i tes were i d e n t i f i e d q u a n t i t a t i v e l y : 14 ( i ) the electrode f i l m / s l a g in ter face ; ( i i ) the metal drop/slag in ter face ; ( i i i ) the slag/metal pool in te r face ; ( iv) the slag/atmosphere in ter face . The experimental resul ts of th is work (Figure 6) showed that the drop/slag sitewas of l i t t l e importance and that the greatest composition changes occurred at the f i l m / s l a g in ter face . Poss ib ly , the most in teres t -ing feature of t h e i r experimental resul ts however, was that the reaction occurring at the metal pool /s lag interface could be a reversion react ion. This observation has important consequences, espec ia l ly i n the area of i n c l u s i o n formation. From the resul ts of the analysis of the temperature 24 regimes of the metal phase, i t i s possible now to speculate on the reasons for th is behaviour. At the electrode f i l m , the temperature i s of the order of 150 - 200°C lower than at the metal pool . In general terms, since AG° = A + BT = -RTlnK we have InK = y - + B' (1.5) An increase in T w i l l cause a decrease i n K,(A,B<0 ; for desulph, K = 4.07) which s h i f t s the equi l ibr ium from r i g h t to l e f t . Therefore the thermo-dynamic d r i v i n g force i s reduced. T h i s , coupled with the fact that the reactant has been somewhat depleted at the f i l m , leads us to an t i c ipa te a reduced tendency to chemical change. In spi te of these r e s u l t s , there i s s t i l l no real information 18 to be learned on the mass transfer processes from th i s work. Crimes T i 1 1 1 1 1 r POOL SULPHUR INGOT SULPHUR 753x10"3+ 2 49x1Cr5t OO 9 0 " 8 0 -30 1 1 1 1 1 1 1 r DROP SULPHUR L S J D = M 6 x 1 0 ' 2 - 2 49x1(T5t - l r - i 1 1 r SLAG SULPHUR (s ) s rAeixicrVa-eexio^t J L 0 100 200 300 400 500 600 700 600 900 T I M E , sec Fig. 6—[S]n, [ S]i, [S]r, and (S)s as a function of time. CaF--20% CaO. 30 v, 2750 A. Melt rate 13.7 g/s. Slag cap 3784 g. Pool volumes 890 and 699 ml. 16 has used information obtained from aqueous mass transfer to l i q u i d metal drops to speculate on the mass transfer behaviour of drops i n ESR desulphurizat ion. His resul ts showed that a large degree of desulphur-i z a t i o n i n ESR could be a t t r ibuted to losses from the f a l l i n g droplet . This r e s u l t i s i n d i r e c t opposit ion to the experimental f indings of 14 Cooper. The reason for th is i s probably that Crimes has ignored the f i l m react ion s i t e which removes a large amount of S leaving l i t t l e ava i lab le during drop f a l l . He has, nonetheless, showed that the drop reaction i s subject to very high rates of mass t ransfer . The reasons why the drop s i t e i s not of p a r t i c u l a r importance i n ESR are connected with the short residence time of the drop in the slag and the low area/volume r a t i o . These ef fects w i l l be discussed i n de ta i l l a t e r . Probably the most l u c i d account of mass transfer ef fects in g e lec tros lag processing, has been given by Patchett and Mi lner . This work has covered a wide var ie ty of experimental conditions during ES welding and has provided some in teres t ing conclusions. In t h e i r i n i t i a l d i scuss ion , i t was noted that several d i f f e r e n t elements appear to approach e q u i l i b r i a at d i f f e r e n t temperatures, thus providing i n d i r e c t evidence that mass transport effects control the overa l l extent of chemical change. In t u r n , experimental evidence was presented to show that , while metal melted from the electrode stock appeared to approxi-mate equi l ibr ium behaviour with the bulk s l a g , metal melted from the sides the weld plate did not. This observation suggested that mass transfer to the droplets from the electrode was very rapid as one might expect -1 9 from the r e l a t i v e l y high slag v e l o c i t i e s (=100 cm sec ) and the intense c i r c u l a t i o n observed on the melting electrode t i p . The metal 17 pool , containing material melted from the weld p la te , had, on the other hand, a r e l a t i v e l y low reaction area/volume r a t i o and was much less turbulent , both these factors great ly a f fec t ing the overa l l mass transfer e f f i c i e n c y . I t i s th is evidence, then, which has prompted th is study of e lec t ros lag r e f i n i n g k i n e t i c s . The previous attempts to account for a l l o y element loss i n ESR have indicated that both thermodynamic and mass transfer effects are important. However, knowledge of the s p e c i f i c mass transfer processes governing the overa l l rate i s conspicuously l a c k i n g . The progressive manner i n which metal melts and i s subsequently refrozen and the d e f i n i t i v e react ion s i t e s would appear to make the ESR process an ideal one for modelling purposes. While these features have been exploi ted i n the heat flow a n a l y s i s , very l i t t l e has been done to develop any rigorous mathematical model dealing with k ine t i c aspects of composition c o n t r o l . The object ive of the present work i s therefore to out l ine the necessary procedure for developing such a model. The proposed quant i ta t ive mass t ransfer model w i l l be tested using experimental data and conclusions drawn as to the a p p l i c a b i l i t y of the model to commercial ESR processing. 18 CHAPTER 2 THE MASS TRANSFER MODEL 2.1 Introduction The preliminary requirement i n any descr ipt ion of mass transfer i s to i d e n t i f y the species involved. I t i s evident from some of the 16 21 previous work ' that systems of excessive complexity were chosen for 21 experimental purposes. For example, Knights and Perkins studied the oxidat ive losses from A286 a l l o y containing 2% T i , 0.23% Al and 0.7% Si melted through a CaF 2 - 10% CaO - 5% A 1 2 0 3 - 5% T i 0 2 - l%"S i0 2 s l a g . Part of the lack of agreement between t h e i r "equi l ibr ium" model and the experimental resul ts may c e r t a i n l y be ascribed to the complexity of th is system and the subsequent uncertainty of the appropriate a c t i v i t i e s . For th i s study, a much simpler system was chosen. Following a preliminary i n v e s t i g a t i o n , i t was found tha t , for the melting of AISI 1018 through CaF 2 - CaO s lags , manganese was the only element to be oxidized from the metal at the levels of FeO commonly found in ESR processing. I t was apparent that th i s was the sole reaction occurring and hence this 'was a sui tably simple system on which to test a- mass transfer model. In a d d i t i o n , the thermochemistry of the CaF 2 - CaO - MnO and CaF 2 - CaO -FeO systems were s u f f i c i e n t l y understood to provide reasonable estimates of the necessary a c t i v i t y c o e f f i c i e n t s . The reaction to be modelled then i s the oxidat ion of Mn in steel by FeO i n CaF 9 - CaO s lags . 19 I t i s generally agreed that CaF2~based fused sa l t s are com-7 17 27 30 p l e t e l y ion ic i n nature. ' ' ' Therefore, we can consider FeO and 2+ 2+ 2-MnO i n terms of the ion ic species involved, namely, Fe , Mn , 0 . Oxidation may then be treated as a simple exchange react ion : [Mn] + (Fe 2 + ) "* Fe(£) + (Mn 2 + ) (2.1) As pointed out in 1.3.1 the ESR process i s , of necessi ty , a heat exchanger. A question one might ask in consideration of the mass transport phenomena i s "To what extent does the heat transfer regime interact with the mass transfer processes?" I t has been shown, for example, that the temperatures of the phase boundaries are not a l l equal (1 .3 .1) . One must decide whether or not these temperatures remain r e l a t i v e l y unchanged throughout the ent i re course of ingot production. There i s no evidence, at present, to suggest that the phase boundary temperatures do change during a melt on the small 3" <j> furnace used here. ' The major e f fect of increasing ingot length is to increase 56 the depth of the l i q u i d metal pool , but th is i s a t t r ibuted to the displacement of the axia l heat sink (baseplate cooling) with respect to the ingot top and not to an increase i n slag temperature. We may, there-f o r e , consider the temperatures of these interfaces to be referred to a quasi-steady s ta te , i . e . , constant for our purposes. The above arguments h ighl ight an important aspect of mass t ransfer modelling in the ESR system, which i s the coupling of the heat and mass transfer processes. The coupled behaviour of these two phenomena w i l l be made more apparent as we proceed with the development of the mass transfer model in which i t w i l l be shown that heat transfer 20 analysis provides information which i s essential to a successful mass transfer model. 2.2 Mathematical Formulation An exchange reaction involves a sequence of steps including transport processes to and from the phase boundaries and chemical reaction 26 at the boundary. The s i tua t ion i s presented schematically i n Figure 7. METAL [Mn] UJ o & or UJ i -SLAG (Fe8*) (Mn2*) +ve direction (AC >O) Figure 7 Schematic Representation of Fluxes 21 I t i s assumed here that equi l ibr ium between the species [Mn], (Mn ) and (Fe ) exis ts at the phase boundary since the reaction of (6) and any adsorption or desorption steps are very fas t compared to the transport process i s considered as a possible rate c o n t r o l l i n g step. E s s e n t i a l l y , i t i s th is assumption that permits a descr ipt ion of the ent i re mass transfer process by using phenomenological f l u x equations. Since l i q u i d iron i s the metal solvent in the system, the transport of iron i n the metal can be discounted. The remaining three f l u x equations may be w r i t t e n , such that the concentration d r i v i n g forces are always greater than zero and transfer from metal to slag i s defined as p o s i t i v e . rate of transport to the in ter face . 31 In th i s a n a l y s i s , no s ingle For the transport of [Mn] to the slag/metal in ter face : n Mn = k M R ( [ M n ] b - [Mn].) (2.2) A for transport of (Mn )away from the in ter face : k M n 2 + « M n >1 " ( M n V (2.3) A and for transport of (Fe ) to the in ter face : k F e2+ ( ( F e ' T ) b - (Fe^) n . ) (2.4) A The symbols are described at the end of the thes i s . 22 In a d d i t i o n , we have the equi l ibr ium re la t ionship for fast i n t e r f a c i a l react ion: (Mn 2 + ) [ M n ] i ( F e 2 + ) i 1 (mol cm" 3 ) " 1 (2.5) I f we consider, now, that the process as a whole occurs at steady state then, n» = n M 2+ = - n r 2+ = n Mn Mn Fe (2.6) It i s apparent from a simple schematic representation of the o_i_ concentration p r o f i l e s of [Mn], (Mn ), (Fe ) , (Figure 8) that the i n t e r f a c i a l METAL [Mn] SLAG L U O cr L U (Mn8*), Figure 8 Schematic Concentration Gradients at Slag/Metal Phase Boundary 23 concentrations of these three species are unknown. They must there-fore be solved f o r , l i k e f i , using the four equations (2.2 - 2.5) i n four unknowns. Etienne 5 assumed that metal phase transport was not rate c o n t r o l l i n g i n evaluating his lumped mass transfer c o e f f i c i e n t but there i s no basis for th is assumption evident at th i s stage of the analysis pe and so we must attempt to solve for a l l of these i n t e r f a c i a l quant i t i es . Rearranging (2.5) and subst i tut ing into (2.4) we obta in : nF p2+ 2 + ( M n 2 + ) . F e " k F e 2 + [ (Fe 2 + ) 1 ] A r e u Ji[Mn]. n[Mn] ; mult ip ly ing by k F e 2 + n c|+n[Mn]. o. p+ F e jr-^- = 0 [ M n ] 1 ( F e 2 + ) b - ( M n 2 + ) i (2.7) A ^Fe' From (2.3): n M n 2 + 1 _ , M 2+. , M 2+ A k M n 2 + ( M n ^ ) . - ( M n t T ) b (2.8) Upon adding (2 .7) , (2.8) and using r e l a t i o n (2.6) W = ^ [Mn] , (Fe 2 + ) . - (MnZ +) A I k F e 2 + k M n 2 + j 24 Therefore: n fi[Mn].(Fe2+)b - ( M n 2 + ) b fi[Mn]. 1 k F e 2 + k M n 2 + Equating (2.2) and (2.9) W ^ b " W ^ i « [ M n ] i ( F e 2 + ) b - ( M n 2 + ) b n[Mn] i 1 k c 2+ kM 2+ Fe Mn | n[Mn]. 1 M u l t i p l y i n g by I + t k F e 2 + k M n 2 + % n C M n ] b [ M n ] i ^ [ M n ] 2 k M n [ M n ] b k M n [Mn] i k F e 2 + k F e 2 + k M n 2 + k M n 2 + = fi[Mn]i(Fe2+)b - ( M n 2 + ) b (2.9) Col lec t ing terms i n [Mn]^: " 0*0? • ' Ph M b " 7 T T K Fe* 1 KFe^ D KMrT + r% [ M n ] b + ( M " 2 + ) b Mn D D -k 2+ F i n a l l y , mul t ip ly ing by — we obtain: ^ kMn 2 k F e 2 + k F e 2 + 2+ L ^ i + { QFST + T<— ( F e ) b " Wn] 1 " K MrT KMn D ^ 2 + [Mn], ( M n 2 + ) b , k 2+ k ' fi KMrT KMn This i s a quadratic equation of the form a[Mn] 2 + b[Mn]. - c = 0 where a = l b = + - I § _ ( F e 2 + ) . - [Mn] fikMn2+ kMn 26 k c 2+ [Mn]. k M n 2 + (Mn 2 + ) . 'Mn (2.11c) Therefore, -b ± (b 2 + 4 c ) 1 / 2 [Mn], = — (2.11a) Equation (2.11a) i s a re la t ionship giving the value of [Mn]n-i n terms of measurable parameters: bulk concentrations, mass transfer c o e f f i c i e n t s and equi l ibr ium constants. Equation (2.11a) can be substituted into (2.2) to y i e l d (2.12) n M -b ± (b2 + 4 c ) 1 / 2 f - kMn { M b " " \ } ( 2' 1 2 ) The rate of change of [Mn], , etc . with time f i n a l l y can be obtained from a mole balance on the respective species as follows Rate in Rate out = Accumulation for [Mn] b : - n M n = Vmd[Mn]/dt for ( F e 2 + ) b - (-nF e2+) = V s d ( F e 2 + ) d / t (2.13) for ( M n 2 + ) b nMn2+ = V $ d ( M n 2 + ) / d t 27 The resu l t ing d i f f e r e n t i a l equation wri t ten for Mn i s : d[Mn] -A -b ± (b 2 + 4 c ) 1 / 2 = - k M n { [Mn] b - [ ] ) (2.14) d t vm m This expression is v a l i d for any l i q u i d metal/slag in ter face . 2+ 2+ The concentrations (Mn ) , and (Fe ) , may also be found by r e l a t i o n (2.15). d ( M n 2 + ) b = - d ( F e 2 + ) b = - d f M n ^ ^ (2.15) ^s 2.3 Evaluation of Mass Transfer Model Parameters 2.3.1 Bulk Concentrations Bulk concentrations are not, s t r i c t l y speaking, parameters of the model. Nevertheless, one must have some knowledge of these variables 2+ 2+ for the arguments to f o l l o w . Any values of [Mn], (Mn ), (Fe ) used below represent t y p i c a l levels found during the experimental melt program (Chapters 3, 4) . 2.3.2 Molar Equi l ibr ium Constant The molar equi l ibr ium constant, fi, for the reaction (2.1) has been defined i n terms of molar concentrations: 28 ( M n 2 + ) . 1 (2.5) [ M n ] i ( F e 2 + ) i fican be expressed as a function of the normal equi l ibr ium constant, K, for the reaction (2.1) where •^W.^FeO^. YMnO^XMnO^ "YMn^Mn^JFeO^FeO^ (2.16) By taking into account a c t i v i t y coe f f i c i en ts and factors to convert units of concentration, we have: for d i l u t e solutions of species i : wt % i M w . { w t J U } 100 x MW. A l s o , wt % i = [c . ] 1 (2.17) 1 p b u l k After subs t i tu t ion in (2.16) and cancel la t ion YMnO P F e ( M n 2 + ) i W F e O 5 5 ' 8 5 : M n ] i ( F e 2 + ) i (2.18) 29 Solving for the Q group (Equation 2 .5) : 5 5 ' 8 5 K V F e O (2.19) pFeYMnO 32 Values for K may be found from the l i t e r a t u r e . Bodsworth (p. 397) quotes a number of expressions for K i n the form, a.. ~ A • i _ M n 0 - B (2.20) a F e Q % Mn T 34 The most commonly used values for A, B are those of Chipman 33 et a l . , where K1 = 6440/T - 2.95 (2.21) 32 It i s assumed in 2.21 that Mn, in d i l u t e solutions of Fe, behaves i d e a l l y (y M n = 1 ) . From a comparison of (2.20) to (2.16) % Mn K = K' *Mn and from (2.17) %_Mn _ MWMn  XMn .5585 30 K = K' ^ ,5585 On subst i tut ion into (2.19) K' Y F e 0 ( 1 0 0 x MWMn) n = (2.22) pFe YMnO Q, i s now a function of the variables K 1 , YpeQ and Y ^ g - In assessing the value of K ' , one must specify the temperature of the interface involved. In th i s instance, we are concerned with three d i f f e r e n t reaction s i t e s . In Section 1.3 .1 , i t has been shown that the electrode f i l m does not superheat more than 20 - 30°C. The interface temperature was, therefore, taken to be 1525°C, fol lowing the arguments of Appendix V.2 . The temperature at the metal pool/s lag interface has been measured in the 35 U.B.C. ESR u n i t , under s i m i l a r experimental conditions and found to be =1675°C. Hence, t h i s value was used to ca lcula te K1 at th i s s i t e . This temperature was assumed constant over the whole in ter face , although 56 rad ia l var ia t ions of ±50°C may e x i s t . The f a l l i n g droplets were assumed to equi l ibra te quickly with the slag (cf . Appendix V.3) and an average drop/slag interface temperature i s used, equal to the metal pool/s lag temperature. Thermodynamic studies have been carr ied out on the systems CaF 2 - CaO - F e O 3 6 ' 2 7 and CaF 2 - CaO - MnO. 3 7 In both cases a strong pos i t ive deviation from i d e a l - s o l u t i o n laws have been found, with MnO 31 showing a larger deviat ion . With increasing lime content, th i s tendency to i m m i s c i b i l i t y i s found to decrease s u b s t a n t i a l l y , r e f l e c t i n g a greater a f f i n i t y of FeO and MnO for CaO than for CaF 2 . The measured a c t i v i t i e s also appear to be only s l i g h t l y affected by temperature, espec ia l ly at higher lime l e v e l s . The value of Yp e g used in th is work is taken from the work 27 of Hawkins and Davies. Although there i s some s l i g h t deviation from Henry's Law behaviour in CaF 2 - high lime melts apparent from Figure 9, the solutions of FeO in CaF 2 - 20% CaO can be considered to obey Henry's 36 Law. The value used f o r Yp e g i s , therefore, 3.0, midway between the 20 mole % and 30 mole % CaO l ines of Figure 9. A consideration of Figure 10, however, shows that th is behaviour may not, i n f a c t , be true for solutions of MnO i n CaF 2 - CaO. Although no data i s avai lable for 20% CaO in CaF,,, i t i s evident that Y ^ Q m a y not approach Henry's Law ( Y M n g = const = Y ^ Q ) e v e n i n d i l u t e s o l u t i o n . The values to be used w i l l be somewhat lower than those reported for 15% CaO. Because of th is uncertainty, values of Y ^ Q used in the model have been f i t t e d from experimental data. The values of Y m n used in the model must be based on the MnO 2+ i n t e r f a c i a l concentration, (Mn ). (Equation 2.16). In model runs on the computer, some arbitrary i n i t i a l value of Y ^ Q w a s read in with the data. The value of Q, was calculated by Equation 2.22 using the i n i t i a l value of Y ^ Q - A n addit ional ca l cu la t ion of Q, using Equation 2.5 was then performed using the expressions derived for the i n t e r f a c i a l quant i t i es , 2+ 2+ [Mn].j (Equation 2 . 1 l a ) , (Mn ) i and (Fe )• (Appendix I I I . 3 , Equations A3.36 and A3.38 respec t ive ly ) . A new value of Y M n g » Y M N 0 ' w a s obtained by 32 4-Oh Figure 9 y FeO v e r s u s NFeO f o r Constant Mole Ratio at 1450°C 2 7 0 OK) 0 20 030 0O5 0-15 0-25 NF. 0.fl730 40 1 • 5%CoO IO%CoO !5%CaO - i i i_ Figure 10 Y M n Q versus wt % MnO i n Various CaF2~CaO Slags at 1500°C. (Replotted from Ref.37) 0 I -2 -3 -4 5 -6 7 8 -9 IO wt.% MnO 33 d i v i d i n g the f i r s t value of tt ca l cu la ted , by the l a t t e r and mul t ip ly ing th i s quotient by the o r i g i n a l value of Y ^ Q J such that K'Y F e 0 (100 x MWMn) y , MnO (Mn2+7 X YMn0 v ; i [Mn] i ( F e 2 + ) . This procedure (subroutine OMT, Appendix IV) was carr ied out i t e r a t i v e l y u n t i l Y' M NQ d i f f e r e d from y M NQ by a prescribed tolerance ( ± . 1 ) . General ly, only one or two i tera t ions were needed for convergence af ter the f i r s t i t e r a t i v e sequence. In th is way, the data for Figure 18 (Chapter 4) was obtained. The values obtained for Y M n g a s a function wt % MnO appear to show a high degree of internal consistency ( i . e . a small degree of scatter over a large number of data points . They also are consistent with the expected values of Y M n g at CaO levels in excess of 15% CaO. 2.3.3 Reaction Sites Considered in Modelling The mass transfer Equation (2.14) has been derived for any l i q u i d metal/slag phase boundary. In order to evaluate the phenomeno-l o g i c a l mass transfer c o e f f i c i e n t s k ^ n , ^ 2 + , kp g2+,consideration must be made of the p a r t i c u l a r react ion interfaces which are operative in the ESR system used in the experimental program. The work of Cooper e_t a l _ . 1 4 has shown (1.3.3) that there are four major reaction s i t e s , namely: pFe YMnO 3 4 1 . electrode metal f i l m / s l a g interface 2 . drop/slag interface 3 . metal pool/s lag interface 4 . slag/atmosphere in ter face . However, throughout the experimental program (Chapter 3 ) an iner t gas atmosphere was used, and thus eliminated the necessity of considering ( 4 ) above, as contr ibut ing to the overa l l rate of mass t ransfer . I t i s necessary, now, to consider the nature of the mass transfer processes involved at each of the remaining s i t e s . This i s complicated by the dependence of mass transfer c o e f f i c i e n t s on flow conditions adjacent to the inter faces . 2 . 3 . 4 Mass Transfer at the Electrode Tip/Slag Interface The nature of flow on the electrode t i p has been considered by 5 Etienne. He has derived an expression for the thickness of the metal f i l m , 6, on conical electrode t i p s , where flow i s contro l led by grav i ta t iona l forces . The in terac t ion effects of the slag (momentum, surface tension) are not known.with s u f f i c i e n t precis ion to be expressed mathematically and thus are not included in the ana lys i s . We have, then 2 i r ( p m - P„)d x sin9cos9 * m s 2 2 x cos 0 , 1 / 3 ( 2 . 2 3 ) 35 where . 2 _ i W1 i s the volumetric melt rate cm sec" m 0 i s the cone angle x i s the length of cone edge i n contact with the s l a g . Film p r o f i l e s calculated from th i s equation have been plotted i n Figure 11. Typical constants used are: G = 5° - 50° y m = 0 .05 poise _3 P m = 7.2 gm cm P s = 2 . 6 3 8 gm c m - 3 R = 1.91 cm W1 = 0.36 cm 3 sec _ 1 m The f i l m thickness i s seen to vary from 50 - 200 y over most of the surface of the cone i f the cone angle i s approximately 45° (as i s usual ly observed on 1 1/2" (j> e lectrodes) . This magnitude of f i l m thickness 40 41 i s consistent with numerous observations. ' We may draw some conclusion as to the type of flow that might be expected with such f i l m thickness using the Reynolds Number for a 39 f a l l i n g f i l m on a v e r t i c a l wall and the physical constants above: 4 6 3 g Re = 2— (2.24) 3 v for 5 = 200 y Re = 205 S = 50 y Re = 3.2 This indicates the l i k e l i h o o d of r i p p l i n g on the electrode 39 f i l m during laminar f low. Since the nature of th i s r ippled surface i s Figure 11 Thickness of L iquid Metal Film on Conical Electrode T i p 5 37 not known, a smooth interface w i l l be assumed. In order to describe the mass transfer across th is interface we must adopt some model in order to evaluate the mass transfer c o e f f i c -ients.The two c l a s s i c a l models for the mechanism of mass transfer between 42 43-46 two phases are the f i l m theory and the penetration theory. The f i l m theory assumes that there i s a region in which steady state transfer i s c o n t r o l l i n g ; the pentetration theory assumes that the i n t e r -face i s continuously renewed by eddies of fresh material from the bulk l i q u i d and that unsteady state transfer into or from these eddies i s c o n t r o l l i n g i n th i s region. In the absence of any other resistances the f i l m theory predicts a f i r s t order dependence of the transfer rate on d i f f u s i v i t y while the penetration theory predicts a square root dependence. The average mass transfer c o e f f i c i e n t s for these models are; D -1 Film Theory: kp = - (cm sec ) (2.25) where L i s the thickness of the region i n which molecular transfer is c o n t r o l l i n g . Penetration Theory: k p = 2 ( D / n e ) 1 / 2 (cm s e c - 1 ) (2.26) where t i s the time that an eddy i s in contact with the inter face . Which of these models i s most applicable to describe mass transfer at the electrode t i p / s l a g interface can be established using 47 the c r i t e r i o n obtained by Toor and Marchello, i n which the penetration theory holds for short contact times and vice versa for the f i l m theory. 47 A short contact time i s said to ex i s t when the group t 07L 2 i s small In other words the penetration theory holds when t e « L 2 / D (2.27) and the f i l m theory holds when. t » L 2 / D (2.28) An expression for t on the electrode f i l m has been derived 5 by Etienne with the fo l lowing resu l t for the free f i l m flow discussed above. 2TT COS 0 i /"3 ^ c/o t = 3.35 ( -. ) d / 3 ( - ) l / 3 ( ) b / i (2.29) e 3 W gp s in 0 cos 0 m 3 For the typica l conditions described i n the evaluation of 6 , the f i l m thickness (Equation 2.23) we f i n d that t - .95 sees e Since the metal f i l m was assumed to be i n laminar f low, and i s very t h i n , the mass transfer boundary layer cannot be expected to be much less than the actual metal f i l m thickness. The d i f f u s i o n 39 . -4x48 49 2 -1 c o e f f i c i e n t of Mn in i ron has been estimated to be 1.(10 ) ' cm sec _p Taking L to be 2(10 ) cm, we have, from Equation 2.27 t e « 2 (10" 2 ) 2 / (10- 4 ) « 4 Since the estimated t „ for t h i s case was found to be .95 sec e the mass transfer c o e f f i c i e n t on the metal side of the electrode t i p / s l a g interface ( i . e . k M n ) can be ca l cu la ted , assuming the penetration theory to hold , by Equation 2.26. For a f i l m thickness of only 50 y , th is 2 inequal i ty does not hold since L /D i s equal to .25 sec. But, as can be seen from Figure 11, the proportion of electrode covered by th i s thickness of f i l m i s smal l . Therefore, kMn,FILM = 2 ( D A t e ) 1 / 2 =2(2(10-*)/7T x . 9 5 ) l / 2 kMn,FILM = °- 0 1 2 cm sec" 1 For the slag side of the in ter face , for which the coe f f i c ients kpg2+, kMn2+ must be evaluated, we must determine the nature of the flow i n the s l a g . I t has been observed that the slag v e l o c i t y varies -1 5 50 from 5 - 10 cm sec in ESR units ranging from 76 mm § to 300 mm <j>. The major d r i v i n g force for slag motion i s not thermal convection, but rather forced convection due to the c o n s t r i c t i o n of the current path as the current passes through the electrode t i p and then broadens out to 40 9 51 take the path of least res istance. I t was shown by Maecker that , where such a current path e x i s t s , there occurs a Lorentz - force induced, high v e l o c i t y motion away from the c o n s t r i c t i o n . The maximum ve loc i ty of the f l u i d i s expressed by 2 I 2 Vmax = < — ^ ) V 2 ( ^ s e c - 1 ) (2.30) p ^ R where I = absolute current R = radius of the electrode In the U.B.C. ESR u n i t , currents i n the order of 1000 amps for the 76 mm <j> furnace and a 38 mm cj) e lectrode: 2(100) 2 1/2 V - ( ) - 25 cm sec max K 0 ' 2.6^ x (2r The approximate pattern produced by th i s force i s shown schematically i n Figure 1 2 . ^ Although slag flow rates of th i s magnitude have not been observed in ESR, v e l o c i t i e s close to the maximum are seen i n ES welding. In ESR, the imposed c i r c u l a t i o n pattern of the slag would be 38 opposed by the thermal gradients found i n the system which p a r t i a l l y accounts for the lower observed flow rates . I t i s reasonable to use a value of 10 cm s e c - 1 for the mass transfer c a l c u l a t i o n s . The contact time, t g , of an element of slag with the metal f i l m w i l l be the time taken for that element to traverse the face of the electrode t i p . For this system that time i s about .2 sec. I t i s possible to estimate the mass transfer boundary layer 41 C O O L B O U N D A R Y L A Y E R H O T S L A G C O O L B O U N D A R Y L A Y E R M E T A L P O O L S O L I D I F I E D INGOT Figure 12 Schematic Flow Pattern for Typical ESR Furnace thickness , L , by considering the electrode t i p to be a f l a t plate . Assuming that flow is laminar and that d i f f u s i o n controls the rate of t ransport , the hydrodynamic boundary layer can be shown to have the 52 thickness A(x) 3.46 ( 4^- )1/2 (2.31) where x i s the distance from the leading edge of the f l a t plate and v^ i s the bulk v e l o c i t y . Since v = y_/p_ where u - 75 centipoise o 38 - 3 3 8 -1 and p s - 2.6 gm cm , v^ = 10 cm sec 42 12 x .75 1/2 , / 9 n A(x) = ( ) y}U = .59 x / d (cm) (2.32) 2.6 x 10 The re la t ionship between the hydrodynamic boundary layer thickness and the mass transfer boundary layer thickness for d i f f u s i o n 52 control led transport processes i s given by - = S c " 1 / 3 (2.33) A where L i s the mass transfer boundary layer thickness i n cm and Sc = = Schmidt # pD 2+ 2+ Since the d i f f u s i o n coe f f i c i en ts of Mn and Fe in CaF 2 - CaO -5 2 -1 are not known, a value of 5.(10 ) cm sec w i l l be used for both, th i s 34 being the value used for Dpg2+ i n basic open hearth s lags . The consequences of th i s are that kpg2+ w i l l be equal to kMn2+ . Evaluating the Schmidt Number and subst i tut ing (2.33) into (2.32) we obtain 3.4(10" 2) x 1 / 2 (cm) (2.34) If x i s approximately equal to 2, then the mass transfer boundary thickness, L = 4.8(10 ) cm. Recal l ing the i n e q u a l i t y , 2.27, t e « L 2 / D 43 we have t e « (4 .8 (10~ 2 ) ) 2 /5 (10 - 5 ) or t « 45.9 sec (2.35) e I t was shown above that t - .2 sec which s a t i s f i e s (2.35) and again the penetration theory may be used to evaluate the mass transfer c o e f f i c i e n t s , kMn 2 +> k F e 2 + kMn2+ = 2 ( D / T r t e ) 1 / 2 = 2(5(10 _ 5)/Tr x . 2 ) 1 / 2 k M n 2 + , F I L M = k F e 2 + , F I L M = • 0 1 8 c m s e c _ 1 2.3.5 Mass Transfer at the Metal Pool/Slag Interface The treatment of the mass transfer at th is reaction s i t e follows that of 2.3.4. Assuming again the slag c i r c u l a t i o n pattern of Figure 12, i t i s evident that flow may be considered to be a toroidal r o l l c e l l . The contact time of an element of slag with the metal pool i s , therefore, the time taken to traverse a distance equal to the radius , i . e . 37 mm. At a ve loc i ty of 10 cm s e c - 1 , we have: 3.7 I = = 3^7 s e c # e 10 Since the hydrodynamic regime at the slag/metal pool interface i s s i m i l a r to that of the slag/electrode t i p interface the mass transfer 44 f i l m thickness i s given by 2.34 Hence, L = 3.4(10" 2) x 1 / 2 I f x = 37 mm, L = 6 .6 (10 - 2 ) cm R e c a l l i n g , again 2.27 t e « L 2 / D or t « (6 .6(10" 2 ) 2 /5(10~ 5 ) « 87.1 sec Since t = .37 sec, the inequal i ty holds and we may once again use the penetration theory to ca lculate kM n2+, kpg2+ . D 5 (10~5) 1/2 k 2+ = 2 ( ) = 2 ( ) M N TT t e TT (.37) k Mn 2 + ,P00L = k F e 2 + , P 0 0 L = -013 cm sec" 1 The l i q u i d metal pool of the ESR unit has been shown to be 35 5 well s t i r r e d . Etienne has estimated that t for the metal side i s about .5 sees. Flow v e l o c i t i e s i n the l i q u i d metal pool are unknown but the condit ion of good mixing would suggest that the boundary layer i s s u f f i c i e n t l y th in to permit the use of the penetration theory for estimating k^ n . Thus, 45 D 1/2 kMn = 2 { - T )  11 le 2( 10" 4 1/2 TT X .5 kMn,POOL = - 0 1 6 c m s e c _ 1 2.3.6 Mass Transfer at the F a l l i n g Drop/Slag Interface There has been more speculation about the importance of droplet transfer related mass transfer than e i ther of the other two interfaces 10 14 18 20 53 54 » » » » » . I t i s generally agreed that the drop plays a very minor ro le i n the overal l mass transfer mechanism, but from a heat transfer 55 56 point of view, i t i s undoubtedly of some s i g n i f i c a n c e . ' There i s some need, therefore, for c r i t i c a l examination of transport phenomena occurring during drop f a l l . As a f i r s t approximation, the end effects of drop formation and subsequent coalescence w i l l be ignored. This i s not to say that these effects are not important but at present, there i s i n s u f f i c i e n t information about drop formation and coalescence i n ESR permit adequate a n a l y s i s . A number of comprehensive surveys of heat and mass transfer 57 58 59 60 to drops have appeared in the l i t e r a t u r e . ' ' ' There are a large number of corre lat ions that have been used i n a wide var ie ty of l i q u i d -l i q u i d systems involving drops. The major theoret ica l basis for any corre la t ion revolves around the hydrodynamic behaviour of the f a l l i n g drop 46 and of the surrounding f l u i d . For the dispersed phase (drop) mass transfer these corre lat ions may be subdivided into two categories ; those based on the assumption of a r i g i d sphere and those assuming internal c i r c u l a t i o n wi th in the f a l l i n g drop. The r i g i d drop model may be d i s -counted i n the ESR case since i t has been found to hold only i n the case 57 of very small drops. Also i t i s quite l i k e l y that drops f a l l i n g through the slag in the ESR process have a high degree of internal c i r c u l a t i o n 9 caused by electromagnetic s t i r r i n g . In a d d i t i o n , one might expect the droplets to o s c i l l a t e while f a l l i n g . When a drop i s forming and begins to neck i t i s subjected to a pinching e f f e c t 6 7 with the r e s u l t that , when the droplet detaches, i t i s s t i l l elongated. The forces of surface tension act ing on the drop w i l l attempt to return i t to i t s 67 spherical equi l ibr ium shape and hence set up an o s c i l l a t i o n . The magnitude of the o s c i l l a t i o n i s a function of drop s ize and interphase surface tension. There have been r e l a t i v e l y few accounts of dispersed drop phase mass transfer coe f f i c i en ts i n the meta l lurgica l 1iteratureJ8,62,68,74 For example, i n spi te of the abundance of work done on droplet/gas mass transfer by l e v i t a t i o n m e l t i n g 6 8 ' 6 9 ' 7 0 ' 7 1 and i n f r e e f a l l s i t u a t i o n s 7 5 most of the experiments were set up so that condit ion of gaseous phase mass t ransfer prevai led . More of in teres t for present purposes, however, i s the work concerning mass transport phenomena found in l i q u i d metal drop-fused s a l t systems. This technology has received some attent ion of la te due to i t s pot ntial in recovering spent fuel from fast breeder nuclear r e a c t o r s . 7 6 ' 7 7 The r e s u l t of th i s e f f o r t has led to some successful correlat ions for both internal (drop p h a s e ) 1 8 ' 6 2 ' 7 4 ' 7 7 and 4 7 external ( sa l t phase) mass t ransfer c o e f f i c i e n t s J ^ , 6 1 , 6 2 , 7 4 , 7 7 Considering again the internal or dispersed phase c o e f f i c i e n t s , 7 2 i t has been found that the c i r c u l a t i n g drop model of Handlos and Baron has s a t i s f i e d the upper l i m i t of internal mass transfer c o e f f i c i e n t i n i n cp JA each study. ' ' However, these experiments were a l l conducted with r e l a t i v e l y control led drop formation and release charac ter i s t i cs compared to the s i t u a t i o n preva i l ing i n the ESR system. Drops observed i n the above experiments rare ly o s c i l l a t e and the degree of internal turbulence i s not great. Crimes has, however, worked on systems i n which the drops were observed to o s c i l l a t e and i n th is case, a model proposed by Angelo 7 3 et a l_ . , incorporating aspects of drop o s c i l l a t i o n into the penetration theory, has proved to be s a t i s f a c t o r y . In view of the expected s i m i l a r i t y i n behaviour of droplets i n the e lec tros lag process to the conditions described by the o s c i l l a t i n g drop case, we s h a l l use th i s model here. I t i s suggested, therefore, that the value of kMn,rjR0P may 7 3 be predicted by the fol lowing r e l a t i o n s . 4Dco(l + e n ) 1 / 2 k Mn , D R 0 P = { — ] < 2 - 3 6 > where co i s the o s c i l l a t i o n frequency ( s e c - 1 ) and Eg i s an amplitude correct ion fac tor . Both oo and EQ are dependent on the amplitude of o s c i l l a t i o n and can be related to the maximum and minimum drop diameter 1 g compared to a sphere of equivalent volume. A l s o , crb 1 9 2 w • = ( -o ) ( 2 . 3 7 ) d 2 p m + 2 p s 48 where b = 1 -d - d . max min 2d (2.38) SPHERE and d = diameter i n cm. EQ = E + .375 £ (2.40) where Amax _ ^SPHERE "SPHERE A = area i n cm From Appendix I , i t i s seen that a t y p i c a l drop weight calculated from the melt rate i s about 2.5 gms. Taking a steel density -3 3 of 7.2 gm cm , the corresponding volume i s 0.35 cm and the drop diameter 8.8 mm. 18 Crimes has developed an empirical technique for determining d and d . . The r a t i o of these quanti t ies i s plotted versus b and £ max min 18 in Figure 13 where th i s r a t i o i s given empir ica l ly by max mi n = 2 ( d s p H E R E / . 2 9 4 ) 1 / 2 (2.41) If d SPHERE .88 d / d . = 2 H J X ) 1 7 2 = 3.46 max min ^.294' 50 From Figure 13, b - .18; e - .82 •'• e 0 = .82 + .375(.82) 2 = 1.07 I f the value of a i s set at 800 dyn cm and p g and p m have t h e i r usual values, then: 800 x .18 x 192 1/2 , \ _ / i n - I co = ( 5 ) = 40 sec .88^(3 x 7.2 + 2 x 2.6) This value of 40 sec-'-is very reasonable and agrees with previously 67 reported data on the o s c i l l a t i o n frequencies of iron drops. We are -4 2 -1 now able to evaluate, ; where D M n =10 cm sec . Mn 4(10~ 4) x 40 x (1 + 1.07) 1/2 kMn,DR0P = - 1 1 2 c m s e c _ 1 There i s , f o r t u i t o u s l y , not the same lack of information about the continuous phase or external mass transfer c o e f f i c i e n t . In almost every treatment of s a l t phase mass transfer res is tance , a penetration theory-based model has been used with s u c c e s s . 6 1 - 6 4 ' 7 6 ' 7 7 The major change i n the r e l a t i o n s h i p , 2.26, where. 1/2 kMn2+ = 2 x ^ " 1 / 2 (^-) (2.26) e 51 -1/2 has been to modify the const 2 x IT to account for the ef fect of the wake behind the f a l l i n g drop. In the wake region a lower rate of mass transfer w i l l be observed since the surface concentration i s not being renewed i n the same fashion as in the forward sec t ion . The -1/2 78 new c o e f f i c i e n t , replacing 2 x ir i s given as .69 and thus kMn2+ = .69 ( — ) 1 / 2 (2.42) t e The time, t , i s taken as the time for a drop to f a l l through a distance equal to i t s diameter. Equation 2.42 becomes uD , , , , k M n 2 + = ' 6 9 < i d ^ < 2- 4 3) where u i s the drop v e l o c i t y . C r i m e s 1 8 and Warren 7 4 have both found th is r e l a t i o n to f i t continuous phase mass transfer data. We must use some method to estimate the v e l o c i t y of the drop 79 in order to use Equation 2.43. Hu and Kintner have developed a generalized and widely u s e d 1 8 ' 7 4 ' 5 9 corre la t ion for the v e l o c i t y of f a l l i n g drops. The general i ty of th is corre la t ion stems from the fact that i t i s based exc lus ive ly on the physical properties of the system. Using dimensional analysis arguments, they were able to es tab l i sh that the essential dimensionless groups for the c o r r e l a t i o n were the Reynolds number, Re, the Weber number, We, and a physical properties group, P, (inverse Morton number) defined as 52 2 3 C (2.44) g(P D - P C ) y c 4 where the subscript c defines a continuous phase property and the subscr ip t , d , a dispersed phase property. I t was found that the p l o t t i n g of R e c X = .75+ (2.45) 4 d 2 g ( P . - p ) P 0 - 1 5 vs . Y = 9 £ (2.46) 3 a on logarithmic coordinates correlated the drop v e l o c i t y rate for a great quantity of data taken from a number of systems. From curve f i t t i n g , Hu and Kintner present the fo l lowing equations: 4 1 ?75 Y = — X 1 - * / 0 2 < Y < 70 3 (2.47) Y = 0.045 X 2 - 3 7 Y > 70 Evaluating P, using the appropriate data (2.34) (2 .6 ) 2 (800) 3 P = 4 980 x 4.6 x (.75r 2.43 (10 6) 53 From th i s c h a r a c t e r i s t i c value of P, we can calculate Y, from 2.46; i . e . Y 4 (.88) 2 980 (4.6) (2.43 ( 1 0 6 ) ) ' 1 5 3 x 800 52.79 From the f i r s t of the r e l a t i o n s , 2.47; Y = _ x1.275 3 i 07c 52«79 x 3 •*• x 1 ' ^ 0 = = 39.59 17.91 But from (2.45), X 75 + Re P -.15 17.16 = dup. 1 y c ( 2 . 4 3 ( 10 6 ) ) ' 1 5 u = 17.16 x .75 x ( 2 . 4 3 ( 10° ) ) ' 1 3 / (.88 x 2.6) = 51 cm sec This value of u i s t i e terminal v e l o c i t y of the drop. A drop f a l l i n g i n the ESR slag bath s t a r t s , obviously , from a rest pos i t ion accelerat ing u n t i l i t reaches i t s maximum speed. For the purposes of c a l c u l a t i n g the mass transfer c o e f f i c i e n t , i t w i l l be assumed the u = u 54 This v e l o c i t y w i l l r e s u l t i n an overestimate of kM n2+, but the consequences are small as w i l l be discussed l a t e r . If we include also the v e l o c i t y added to the drop by bulk motion of the slag (=10 cm s e c - 1 ) i n the d i r e c t i o n of f a l l , u becomes 61 cm s e c - 1 61(5 x 10" 5) 1/2 •'• k Mn 2 + ,DR0P ' k Fe 2 + ,DR0P = -023 cm sec" 1 Although we have used the terminal v e l o c i t y in t h i s c a l c u l a t i o n , the actual drop f a l l contact time was determined considering the accelera-t ion of the drop. (2 .5 .1) . We have now estimated a l l nine mass transfer c o e f f i c i e n t s needed for the model ca lculat ions and these are summarized in Table I . Before the model can be used, we must also estimate the area to volume ra t ios at the three mass transfer inter faces . 2.4 Area/Volume Ratio Estimates During the experimental program, as much care as possible was taken to run each melt under the same condit ions . As a consequence of t h i s , the area/volume r a t i o s of the f i l m , drop and pool remained approxi-mately constant, al lowing us to ca lcula te an average value for each s i t e . 55 2.4.1 Area/Volume Ratio of Drop 3 In 2 .3 .6 , the average drop volume was found to be .35 cm . I o To estimate the area of an o s c i l l a t i n g drop we must use the formula T T d 2 mi d . ln(e r t +{en - 1 ) 1 / 2 ) /\ = max + max min v 0 \ 0 ' ' ^ 48) ^ " 2 2(e2 - l ) 1 ' 2 In 2 .3 .6 , e was found to be 1.07 but we must ca lculate d . d . from I H C I A 111 I I I 2.38. Recal l ing 2.48 d - d . ^ _ max mm 2dSPHERE 2 d ( l - l ) = - d m a x + d m i n but from 2.41 max min 3.46 i f d = .88 and b = .18, then 1.76 x (-.82) •3.46 d . + d . min min min •1.44 •2.46 59 cm 2.04 cm 56 Now, from 2.48 >2 Tr(2.04r Tr(2.04)( .59)ln(1.07 + / ( 1 Q7)Z_i  ADR0P = ~ + 2 ^(1.07) 2 - 1 8.38 cm2 Area/volume r a t i o of drop = 8.38/.35 = 24.0 cm"1 2.4.2 Area/Volume Ratio of Film Since the electrode t i p can be considered to be a cone (2 .3 .4) , the area and volume ca lculat ions are greatly s i m p l i f i e d . I f the average f i l m thickness i s taken as lOOuandthe cone angle as 45° , the surface area for a 38 mm <j> electrode t i p i s given by T r r 2 T T (1 .924) 2 A F T 1 M = = = 16.45 (2.49) l " i L M cose .707 where r = R + r a d i a l component of f i l m = R + (.01/cos 45) = 1.924 The volume of a cone i s V C Q N E = ^ r 2 h (2.50) The f i l m volume i s taken to be the difference between a cone of 1.91 cm radius and that of 1.92 cm radius . Since 9 = 45° h = r . Therefore, 57 V F ILM = ^ (1 -924 3 - 1.91 3 ) .16 cm3 16.45 , the area/volume r a t i o of the f i l m = = 102.8 cm" .16 2.4.3 Area/Volume Ratio of Pool In the ingots produced during the melt program, the ingot pool was generally observed to be v i r t u a l l y c y l i n d r i c a l , i . e . with a f l a t top and bottom. The area/volume r a t i o i s given, then, as. A Trr 2 1 ( - ) = o = - (2.51) V POOL -rrr h h The approximate average depth of the pool was 15 mm and thus A -1 ( - ) = 1/1.5 = .67 cm 1 V POOL If the average slag sk in thickness i s equal to 1 mm. then, : the pool volume = TT x (3.81 - .1) x 1.5 = 65 cm3 The above information i s included i n the summary of Table I . TABLE 1 Summary of Theoretical Mass Transfer C o e f f i c i e n t s , Average Area/Volume Ratios and Approximate Residence Times for 7 6 mm <j> ESR Furnace ^ \ L o c a t i o n Film Reaction DROP Reaction Pool Reaction S i te S i te S i te Parameter^. Mn _-L (cm.sec. ) 0 . 0 1 2 0 . 1 1 2 0 . 0 1 6 (cm.sec. ) 0 . 0 1 8 0 . 0 2 3 0 . 0 1 3 F E -1 (cm.sec. ) 0 . 0 1 8 0 . 0 2 3 0 . 0 1 3 A/V • (cm - 1 ) 1 0 3 . 2 4 . . 6 7 Residence Time (Sec) .9.5 . 1 2 * 80 Residence time in pool - (rate of r i s e of ingot x pool depth) From Appendix I - rate of r i s e of ingot = w m / ^ • For a pool depth of 1.5 cm and a volumetric melt rate of 0.36 cm 3 sec" l V^ 0 0 1 = ( Q ; ( 6 3 . 8 1 ) 2 5 ) ~ 1 = 8 0 s e c 59 2.5 Numerical Solution of D i f f e r e n t i a l Equations for Mass Transfer The d i f f e r e n t i a l Equation 2.14, may now be evaluated subject to boundary conditions which must be determined experimentally. The equation, d[Mn] A b±(b 2 + 4 c ) 1 / 2 dt V k M n { [Mn], - [ ] } (2.14) i s applied to each reaction s i t e i n the fol lowing manner. Numerical so lut ion of th i s non l i n e a r , f i r s t - o r d e r d i f f e r e n t i a l equation was rd accomplished using a 3 order Runge-Kutta technique. The d e t a i l s of th is numerical method are found i n numerical analysis t ex t s , ( for 80 example, Lapidus, " D i g i t a l Computation for Chemical Engineers" ) and w i l l not be discussed here. The programming d e t a i l s are presented i n Appendix IV, and the general method is out l ined as fo l lows . Calculations were carr ied out simultaneously at the f i l m and pool s i tes for the period of drop formation, and then for the duration of the drop f a l l . The experimentally obtained frequency of drop production determined the in terva l of drop formation, th i s time being divided into 10 smaller increments for the purposes of accuracy. The step frequency for the Runge-Kutta routine was of the order of .1 sec for a drop frequency of 1 sec. This was adjusted for each run according to the measured frequency. The interva l s ize of .1 sec was found to be s u f f i c i e n t l y low to produce an accurate r e s u l t , without excessive expense. Since the drop f a l l time (see 2.5.1) i s only s l i g h t l y greater than .1 sec, the step s i z e was, of necessi ty , much shorter during t h i s per iod; - .01 sec. 60 In addi t ion to the numerical so lut ion of 2.14 at the 3 s i tes i t was necessary to keep s t r i c t account of the number of moles transferred to and from the various in ter faces . This was accomplished through the use of 2.15 at the end of each subinterva l : ingot concentrations was obtained. The resul ts of these modelled p r o f i l e s were compared to the appropriate experimental p r o f i l e s (Chapter 4) . 2.5.1 Estimation of Drop F a l l Time So f a r , we have not discussed how the drop f a l l time was 81 estimated. The procedure i s taken from Lamb's book on Hydrodynamics and i s based on a consideration of a spherical droplet f a l l i n g under the influence of gravi ty at steady s ta te , such that the grav i ta t iona l force i s exactly balanced by the drag resistance of the surrounding f l u i d . The so lut ion i s given as: A ( M O b = - A ( F e ^ ) b = -A [Mn] b (f) (2.15) Hence, a continuous composition vs time p r o f i l e of slag and S = a ln cosh ( ) (2.52) a U = U n tanh ( (2.53) a where S i s the distance f a l l e n during accelerat ion per iod, t , to reach terminal v e l o c i t y , U Q ; u i s the v e l o c i t y during accelerat ion and 61 2 d 2p . + p a = _ ( -A c ) ( 2 > 5 3 ) 3 UD p where i s the drag c o e f f i c i e n t such that 4 A p dg 3 p U 0 i f p = 2.6, pcj = 7.2, then a reduces to 2 p d + p c „ 2 2g A p U 2 ( 7 . 2 ) + 2.6 p U0 2 x 980 x 4.6 1.97 ( 1 0 " 3 ) U Q 2 The slag bath i n the 76 mm <j> ESR furnace i s approximately 4.5 cm deep. From 2.52, se t t ing UQ = 51 cm/sec (2.3.6) - 9 51 t 4.5 = 1.97 ( 1 0 ~ J ) ( 5 1 ) In cosh ( * 9 1 . 9 7 ( 1 0 " ^ ) ( 5 1 ) ^ t 4.5 cosh ( ~ ) = exp ( * o ) 1 . 9 7 ( 1 0 _ C ! ) ( 5 1 ) 1.97(10"J)(5ir .10 c o s h ' 1 {exp [8 .78(10 - 1 ) ] } 62 = .1 x 1.52 sec •*• Time to f a l l 4.5 cm during accelerat ion period = .15 sec. The v e l o c i t y at th i s time i s , from 2.53 U t u = U Q tanh ( -f- ) .15 = 51 tanh ( ) .1 46.3 s e c - 1 The drop would appear to be accelerat ing throughout i t s f a l l through the s l a g . The bulk slag motion would add another 10 cm/sec to the i n i t i a l v e l o c i t y and reduce the overa l l contact time by about 20%. Therefore the drop f a l l time i s of the order of .2 sec. This time has been estimated by others working i n E S R 1 8 ' 5 4 to be approximately 0.1 sec so th i s value is not unreasonable. 2.6 Summary A mathematical mass t ransfer model has been developed to describe the k inet i cs of manganese oxidation by FeO under contro l led conditions i n e lec tros lag remelt ing. The essential parameters have been i d e n t i f i e d and evaluated by various means, ranging from experimental observations to established theories for mass transfer c o e f f i c i e n t s . However, i n order to val idate the model, i t i s necessary to compare the resul ts to actual experimental condit ions . The experimental program designed for th is purpose i s developed i n Chapter 3 and the re-sul tant data compared to the mass transfer model in Chapter 4. 63 CHAPTER 3 EXPERIMENTAL PROGRAM 3.1 The U.B .C . Electroslag Furnace The experimental program designed to tes t the mass transfer model developed in Chapter 2 was carr ied out on the U.B.C e lectros lag u n i t . The design of th i s unit i s s p e c i f i c a l l y adapted to the require-ments of a range of research projects and has been described i n d e t a i l 5 by Etienne. The p a r t i c u l a r configurat ion used throughout th is work employed a short 3" <j> mold capable of making ingots 9" long over a period of approximately 1300 seconds. 3.2 Slag Sampling Device In order to fol low the progress of mass transfer as accurately as poss ib le , i t was necessary to take a number of slag samples during the course of a run. The majority of experiments were conducted under an iner t atmosphere and the more common techniques of slag sampling (such as dipping a copper rod into the slag bath) proved to be i m p r a c t i c a l . Investigations were then carr ied out on a novel method for c o l l e c t i n g slag samples with the fo l lowing r e s u l t . I t was discovered, i n a series of t r i a l s , that a quantity of s lag sui table for chemical analysis could be obtained by dipping a h e l i c a l c o i l of Mo wire into a CaF^-based fused s a l t . Samples weighing from 100 - 600 mg were co l lec ted i n 1" long c o i l s wound from .020" <j> 64 Mo wire on a .125" <j> rod using 22 turns to the inch . This technique was then incorporated into a device for use on the ESR furnace. The r e s u l t i n g piece of equipment i s shown i n Figure 14. Besides being used for sampling, t h i s unit also provided an integral part of the iner t gas c losure . B a s i c a l l y , i t consisted of a s ta in less steel tube 2 3/8" OD and 17" high. On top and bottom were flanges of s ta in less steel and c o l o r l i t h , respec t ive ly . 1" c o l o r l i t h was used to e l e c t r i c a l l y insulate the gas cap from the mold. In t h i s bottom f lange, 18 s ta in less steel guide tubes were press f i t into holes d r i l l e d as c lose ly as possible to the inner edge. Steel wires of - . 1 " <J), and s t i f fened by swaging, were inserted into these guide tubes and the Mo sampling c o i l s were then attached to the bottom ends. To ensure gas t ightness , small rubber stoppers with small holes in the centres were placed on top of the guide tubes. Since the gas cap as a whole became very hot during a melt , i t was necessary to place a few turns of 1/8" <J> copper tubing for water cooling purposes at the level of these stoppers. Once the Mo samplers had been attached to th i s u n i t , i t was placed on top of the mold, clamped into place and a 6" <j> neoprene rubber bellows clamped onto the top flange (Figure 15). The top of the bellows was subsequently secured to the water cooled electrode holder. Inert gas was flushed through the system, entering the gas i n l e t pipe i n the gas cap and leaving through an out le t at the top of the bellows. The gas cap sampling uni t was also f i t t e d with explosion windows covered with aluminium f o i l and s i l i c o n e rubber sealant . In-cluded also was a port which was sealed with a rubber stopper and Figure 14 Gas Cap and Sampling Unit Used i n Experimental Program 66 RUBBER INSULATION WIRE SAMPLER WATER COOLING SAMPLER GUIDE TUBES COLORLITH INSULATION COOLING WATER OUTLET MOLTEN METAL SOLIDIFIED INGOT COOLING WATER • INLET NEOPRENE BELLOWS CONSUMABLE ELECTRODE STAINLESS STEEL TUBE RUBBER STOPPER GAS SEAL ADDITION PORT COIL OF 0-020 Mo WIRE MOLTEN SLAG SOLIDIFIED SLAG SKIN WATER JACKET COPPER MOLD COPPER BASEPLATE Figure 15 Schematic Diagram Depicting ESR Configuration Designed for Slag Sampling under Inert Atmosphere 67 through which various materials could be dropped into the s l a g . In order to f a c i l i t a t e sample taking an AC voltmeter was hooked up so that when the Mo wire touched the surface of the s l a g , a voltage reading appeared on the instrument d i a l . The sampler was then pushed deeper into the slag and removed at once. To enable a precise timing of these samples, the voltage was recorded on a chart recorder with a chart speed of 1" per min and hence an accurate record of the time of each sample was obtained. Once a l l the samples had been obtained and the melt concluded, the uni t was removed and the Mo wires cut o f f . The slag contained i n each of these c o i l s was c o l l e c t e d , separately labe l led and stored for chemical ana lys i s . I t i s worth noting that although t h i s technique has been successful ly used to sample the l i q u i d metal i n the ingot p o o l , the f a i l u r e rate i s high. A l s o , simultaneous slag and metal sampling operations are not compatible unless very long c o i l s are used. 3.3 Materials 3.3.1 Electrode Material In mass transfer experiments i t i s desirable to control as many variables as poss ible . With th i s i n mind, the system that was chosen for th i s study was the oxidat ive loss of Mn from mild s t e e l , namely Cl018 i n 1 1/2" cj> bar form. The nominal analysis of 1018 i s : C Mn P S .15 - .2 .6 - .9 .04 .05 68 In order, however, to ensure that Mn was the only reacting species given the ant ic ipated slag FeO level of - .5%, the exact concentrations of manganese and s i l i c o n must be known. These varied from .61% to .79% for Mn and averaged,.1% for Si without much deviat ion i n the l a t t e r case. This r e l a t i v e l y low concentration of Si as well as the small value of y°. (where log y°. = 1.21 - 6100/T ( ° K ) ) 3 2 resul ts in an extremely low a c t i v i t y of Si i n the metal r e l a t i v e to Mn. Although the a c t i v i t y of S i 0 2 i n the slag was also very low and that of MnO 37 very high, one can show (Appendix V . l ) that even with the lowest observed leve ls of Mn, i t w i l l be the manganese that i s p r e f e r e n t i a l l y o x i d i z e d . This would indicate that 1018 i s a sui table steel for study of manganese mass t ransfer . 3.3.2 Armco Iron Electrodes 1 1/4" tj> Armco i ron was used as a r e l a t i v e l y pure i ron i n experiments to determine mass transfer coe f f i c i en ts at the molten s lag/ ingot pool in ter face . Of p a r t i c u l a r note are the low Mn and low S concentrations. The overal l assay i n the as-received condit ion was: C Mn P S Si 0 Fe .012 .017 .005 .025 trace .070 b a l . 3.3.3 Slag Materials The majority of melts were performed using CaF2-20% CaO s lags . This s lag was chosen for several important reasons. F i r s t l y , the CaF,,-20% CaO slags permitted stable and reasonably reproducible melting 69 conditions from run to run - - a necessary prerequis i te for adequate control over the experimental condi t ions . A l s o , these slags dissolve r e a d i l y i n hot HC1 a f te r sui tab le crushing, i n contrast to CaF^-based slags with an equivalent percentage of A l ^ O ^ . These l a t t e r slags require prefusing i n platinum crucibles with sodium borate and subsequent d i s s o l u -t i o n i n aqua-regia. This procedure i s very time consuming and extremely hard on the platinum c r u c i b l e s . The f i n a l , but most important reason concerns the appropriate thermochemical information. Of in teres t here are the ternary systems CaF 2 - CaO - F e O , 2 7 ' 3 0 ' 3 6 CaF 2 - CaO - M n O 3 0 ' 3 7 and CaF 2 - CaO - C a S . 1 7 ' 3 0 Fortunately, the thermochemistry of these systems has been investigated s u f f i c i e n t l y to be of use i n determining the necessary a c t i v i t y values required by the mass transfer model. Very l i t t l e work has been published on any other CaF 2-based systems. The calcium f l u o r i d e used i n these experiments was obtained from Eldorado Nuclear and i s a high pur i ty by-product of the production of zirconium. The majority impurity i s therefore zirconium as zirconium f l u o r i d e which i s quickly removed from the slag in the s tart -up operations i n the ESR u n i t . Very t r i v i a l amounts (up to 10 ppm) of Fe, Mn were found to be present i n th i s mater ia l . Recrys ta l l i zed calcium oxide of 99.5% pur i ty was used to make the balance of the s ta r t ing s l a g . This was supplied by Dynamit-Nobel, Germany. 70 3.4 Atmosphere Control As described i n 3.2, a system of neoprene rubber bellows and a gas cap provided the iner t gas enclosure. The i n e r t gas used i n these experiments was helium of normal commercial q u a l i t y . At a flow rate of 4 I m i n " 1 , the system could be flushed out in approximately 30 minutes. Checks were made on the out le t gas using a gas chromatograph and only traces of 0 2 were detected on the highest s e n s i t i v i t y a f ter one half hour. 3.5 Melting Conditions For the most par t , ingots produced i n the melt program were made under stable melting conditions which were approximately reproducible from run to run. Each melt was begun using the DC power supply with electrode negative and then switched to AC when stable melting conditions had been achieved. In several cases, ingots were made using a " l i v e " mold. This was done by connecting a separate lead d i r e c t l y to the bottom of the mold, thereby making i t a p a r a l l e l e l e c t r i c a l path to the normal ingot and base plate route. This configurat ion permitted measurement to be made of the proportion of current f lowing through the mold r e l a t i v e to that passing through the ingot . A few runs were made i n which the mold was completely insulated from e l e c t r i c a l f low. This was accomplished by painting the i n t e r i o r surface of the mold with 2 coats of boron n i t r i d e paint . The type of BN used was Type A, supplied by Carborundum. Since BN i s an excel lent thermal conductor (approximately equivalent to Fe) as well as an excel lent i n s u l a t o r , i t proved to be the ideal material for th i s a p p l i c a t i o n . 71 3.6 Melt Records During the melting of an ingot , detai led accounts were kept of a l l the important operating parameters. A Sargent Model SR M i l l i v o l t recorder was used to keep a continuous record of operating current . Once the electrode was melting in a stable fashion ( i . e . when the slag was completely molten), the fo l lowing data were recorded at su i tab le i n t e r v a l s : Volts - process voltage Amps = process current t = time i n seconds from some f ixed s tar t ing point P = t o t a l electrode travel i n mm from beginning of melt MS = speed of electrode travel drive motor AT = temperature difference in °C between i n l e t and out le t mold cooling water. In a d d i t i o n , the time and electrode pos i t ion were recorded whenever a slag sample was taken. Certain other data were co l lec ted a f te r the melt had been made and the ingot str ipped from the mold. These were slag cap height and weight and the average slag skin thickness. From these parameters, a l l of which are not useful for any given run, such information as average melt ra te , drop s ize and rate of s lag loss may be rout ine ly calculated (cf . Appendix I ) . 3.7 Slag and Metal Analysis Once the slag samples had been removed from the Mo sampler c o i l s , they were separately crushed to a f i n e powder with mortar and 72 pes t l e , weighed and dissolved i n hot 50% HC1 s o l u t i o n . These solutions were subsequently made up to a standard volume of 100 ml with d i s t i l l e d water. Al iquots of these solutions were then taken and analyzed for manganese and i ron according to the procedures out l ined i n Appendix I I . In cases where sulphur analyses of the slag samples were required, a small portion of the powdered slag was analyzed immediately a f te r crushing, using a semiautomatic "Leco" sulphur analyzer. Wet chemical analysis techniques were used throughout for s lag analysis in preference, say, to x-ray fluorescence or atomic absorption methods since i t was desired to measure small differences i n the low levels of Mn and Fe (occasionally as low as several parts per m i l l i o n in the sample s o l u t i o n ) . I t was f e l t that the techniques used, namely, the permanganate method for Mn and the orthophenanthroline method for Fe, were best suited for these analyses i n terms of accuracy, r e p r o d u c i b i l i t y and freedom from interference by other ions . The maximum r e l a t i v e uncertainty of i n d i v i d u a l analyses may be estimated as ± 5%. In many cases th is i s an overestimate of the e r r o r , but, due to the nonlinear nature of the Beckman "B" spectrophotometer absorbance sca le , the error associated with reading low absorbance levels i s assumed. Other sources of error are pr imar i ly due to inaccuracies i n measuring so lut ion quanti t ies during analysis but these tend to be systematic errors ( i . e . consis tent ly high or low) of small importance and are hence ignored. In order to draw error bars about the data points a value of 5% of .5 wt % or ± .025 wt % has been used. This w i l l roughly accommo-date the p r o b a b i l i t y of larger r e l a t i v e error at lower concentrations of the measured quant i t i e s . 73 A l l ingot metal analysis was done by spark spectrographs equipment (Baird-Atomic Spectromet Model HA3). In a d d i t i o n , only the ingot centre l ine was analyzed, usual ly at in terva ls of 1/4". This spacing was then converted to a time in terva l through knowledge of the average melt rate . Manganese and s i l i c o n leve ls could be measured to wi th in ± .01 wt % using th is equipment. I t should be noted that a l l ingot analysis was done as close to the centre as poss ible . 3.8 Melt Program 3.8.1 General Comments The major purpose of the experimental program was to provide deta i led information for the tes t ing of the mass transfer model. Part of the experimental data, such as melt ra tes , s lag cap data, e t c . , were used d i r e c t l y as input parameters for the computer program. Other input to the model involved establ ishing the i n i t i a l conditions with respect to i ron and manganese levels at some a r b i t r a r y time, t = 0. I t often happened that several samples were taken during a run before a p a r t i c u l a r set of experimental conditions were achieved. Since these samples represented no useful data and bore no relevance to the work at hand, they w i l l not be shown here. The melt program and resultant data are given below. 3.8.2 Experimental Determination of Mass Transfer Coef f ic ients In Chapter 2, a number of mass transfer coe f f i c i en ts were evaluated by large ly theoret ica l means. Due to the number of parameters 74 involved i n the mass transfer model of Chapter 2, i t i s necessary to v e r i f y , independently, as many of these values as poss ible . We have already used measured quanti t ies to ca lculate the various area/volume r a t i o s , but experimental assessment of the mass transfer coe f f i c i en ts would allow us to place more or less f a i t h i n the modelled estimates of Chapter 2. I n i t i a l l y , we can ignore the p o s s i b i l i t y of measuring mass transfer c o e f f i c i e n t s involving the drop reaction s i t e since i t i s v i r t u a l l y impossible to sample the drops or to observe t h e i r behaviour. 5 In a d d i t i o n , the modelling attempts of Etienne have resulted in a reasonable degree of confidence i n the f i l m mass transfer c o e f f i c i e n t s and so we are l e f t with the determination of the pool c o e f f i c i e n t s . There are several techniques possible here. I n i t i a l l y , the use of radioact ive tracers was considered. This would involve adding 55 radioact ive Mn (Mn ) to the metal pool and then analyzing the s lag samples on a s c i n t i l l a t i o n counter. There are a number of d i f f i c u l t i e s 55 with th is idea , not the least of which i s that Mn has a h a l f l i f e of only 2.57 hrs . In any case, what proved to be a most sa t i s fac tory and unambiguous technique was developed as an a l t e r n a t i v e . The basic idea of th is a l ternate method was to melt r e l a t i v e l y pure i ron (Armco iron) through the CaF 2 - CaO slag i n the ESR furnace and then to dope the metal pool with a cer ta in amount of manganese metal. The samples taken wi th in a short period of time a f te r the doping would contain MnO that came from the pool alone and hence might provide some information about mass transfer c o e f f i c i e n t s involv ing transport to and/or from the in ter face . In f a c t , the quantity of Mn (Granular +325 75 mesh 99.99+, supplied by Research Organic/Inorganic Chemical Corporation) added was s u f f i c i e n t to el iminate most resistance to metal phase transport control (see Chapter 4) and allow c a l c u l a t i o n to be made of ^ 2 + pgQ|_, k Fe 2 + ,P00l_-This experiment was repeated twice with 4 and 12 gms of Mn being dropped into the pool i n a small copper f o i l pouch (to prevent contamination of the s lag) . The f o i l pouch also contained a small amount of tungsten powder to mark the pool p r o f i l e i n the ingot . Samples were taken as quickly as possible a f ter the addi t ion of Mn. The data from these two experiments (60, 61 respect ively) are given i n Table I I . Also of in teres t was the determination of the sulphur mass transfer c o e f f i c i e n t at the metal pool/s lag in ter face . An ident i ca l experiment to those of 60, 61 was performed but the Mn i n the f o i l pouch was replaced by 4 gm of FeS (technical grade). The resul ts of the slag sample analyses from Run #62 are given i n Table I I . 3.8.3 Steady State Runs 27 and 53 were done i n order to es tab l i sh the extent to which chemical changes might occur during normal ESR processing of 1018 using a CaF 2 - 20% CaO s l a g . The system was allowed to es tabl i sh i t s steady state behaviours over as long a period as poss ib le . The operating parameters and the resul ts of s lag and ingot analysis are given i n Table I I I . Run # Average Volts Average Runs Amps AT(°C) Slag Skin Thickness (mm) Average Melt Final Slag Rate (g s e c - 1 ) Weight (g) 60 23.5 1200 16.5 .9 2.5 600 61 23.5 1250 17 .6 2.5 590 62 (Armco) 23.2 1250 16 .8 2.8 520 TABLE II-a OPERATING PARAMETERS Run # Time of Sampling (sec) Wt % Fe2+ Wt % Mn2H 60 304 .32 .08 (4 gm Mn added 333 .39 .12 to pool at 341 .35 .14 <\, 300 sec) 349 .34 .15 61 157 .52 .18 (12 gm Mn added 166 .53 .20 to pool at 174 .54 .21 150 sec) 182 .55 .23 , 193 .54 .24 Wt % S 62 120 - .08 (FES added 1*6 .40 .25 to -pool at 152 .43 .28 * 130 sec) 159 .45 .30 166 .46 .33 174 .48 .34 300 - .35 TABLE I l -b SLAG ANALYSES - RUNS #60, 61, 62 77 Run # Average Volts Average Rms Amps AT(°C) Slag Skin Thickness (mm) Average Melt Rate (gm sec" 1 ) Final Slag Weight (g) 27 (1018) 22.5 1175 15 - 2.6 -53 (1018) 23.2 1225 17.5 .8 2.7 540 TABLE III - a OPERATING PARAMETERS e T ] l i e ° f , Wt % F e 2 + Wt % M n 2 + Sampling (sec) Time of w t % F e 2+ w t % M n Sampling (sec) 191 .34 .28 304 .35 .33 388 .37 .32 432 .43 .31 497 .40 .32 568 .40 .35 723 .42 .34 798 .44 .36 881 .46 .39 943 .48 .39 1034 .51 .40 1096 .50 .43 165 214 291 400 468 559 648 706 774 844 904 980 1024 1083 1146 1230 1273 .34 .35 .35 .38 .41 .41 .41 .43 .43 .45 .46 .46 .48 .48 .48 .49 .50 .20 .26 .27 .26 .27 .30 .31 .29 .32 .31 .32 .33 .36 .33 .34 .38 .33 III - c SLAG ANALYSIS - RUN # 53 (1018 Electrode - .61% Mn) II I - b SLAG ANALYSIS - RUN # 27 (1018 Electrode - .75 % Mn) Time After Deoxidizing Wt % Mn (sec)' 118 .601 216 .592 334 .598 492 .572 531 .563 590 .570 755 .559 807 .569 . 866 .580 1004 .574 TABLE III - d INGOT ANALYSIS - RUN # 53 79 3.8.4 Unsteady State Recovery I t i s evident, from an examination of the slag and ingot concentration h is tor ies during a normal melt , that the composition changes are smal l . In order to t r y and produce larger changes in composition, i t was decided to deoxidize the slag at some point a f ter stable conditions had been achieved when melting in the AC mode. The r e s u l t of such an operation would be expected to produce a perturbation in s lag composition followed by a period of transient behaviour as the system returned to steady s ta te . The procedure for deoxidizing the slag was very simple. F i r s t , a small quantity of tungsten powder (3-5 g) was added to indicate the pos i t ion of the freezing interface at a precise time. This was immediately followed by the addi t ion of calcium metal shot (< 1/4" . A f t e r a short period of time sampling was started and continued at various time in terva ls u n t i l the conclusion of the melt. A number of melts (30, 31, 34, 37, 42) were carr ied out i n th i s fashion and the resul ts are presented i n Table IV. 3.8.5 Live Mold I t has previously been reported that the mold may form part 2 85 of the current path during AC electros lag remeltmg. ' This observa-t i o n i s probably the resul t of mold-ingot contact at regions where the slag skin i s imperfect, p a r t i c u l a r l y when cold slag s tar ts are used, as in small ESR u n i t s . However, i t i s v i r t u a l l y impossible to measure current flow through the mold during normal AC remelting due to the 80 Run # Average Volts Average Rms Amps AT(°C) Slag Skin Thickness (mm) Average Melt Rate (gm s e c - 1 Final Slag ) Weight (g) 30 (1018) 23 1100 14.5 2. 2.-6 480 31 (1018) 22.5 1200 14.5 2. 2.7 530 34 (1018) 23 1175 17. - 2.7 510 37 (1018) 22 1175 16.5 1. 2.5 470 42 (1018) 22.5 1350 19. 1.7 3.0 500 TABLE IV - a OPERATING PARAMETERS Run # Time of Sampling (sec) Wt % F e2 + Wt % M n 2 + 30 280 .11 .07 Deoxidized 320 .18 .08 at 250 sec 348 .28 .10 368 .36 .11 400 .40 .09 469 .37 .14 498 .37 .17 537 .41 .19 617 .39 .20 31 350 .11 .06 Deoxidized 375 .15 .08 at 320 sec 413 .24 .06 454 .37 .12 495 .42 .11 TABLE IV - b SLAG ANALYSIS - RUNS # 30, 31 (1018 Electrodes - .75% Mn) 1 ^ J ^ W c o r l W t % F e 2 + W t % M " 2 + Sampling (sec) 326 .24 .13 359 .44 .16 384 .63 .12 448 .68 .18 477 .62 .16 515 .53 .20 559 .55 .20 625 .47 .16 669 .43 .21 772 .37 .17 829 .35 .21 891 .34 .25 954 .35 .29 1025 .38 .27 1102 .38 .30 1201 .41 .31 1286 .43 .29 TABLE IV - c SLAG ANALYSIS RUN # 34 (1018 Electrode - .68% Mn; Deoxidized at 300 sec) 81 Time of w t % F 2+ w t % M n2+ Sampling (sec) 55 .17 .09 96 .29 .06 115 .32 .09 141 .38 .11 172 .48 .12 208 .57 .14 259 .73 .18 304 .92 .16 356 .87 .19 441 .78 .20 508 .75 .23 611 .69 .22 698 .60 .27 792 .60 .27 908 .47 .30 1003 .40 .32 1103 .38 .33 1244 .36 .29 TABLE IV - d SLAG ANALYSIS RUN # 37 (1018 Electrode - .79% Mn; Deoxidized at 0. sec) 82 Time of Sampling (sec) Wt % F e2 + Wt % M n 2 + 319 .18 0.04 405 .21 .05 455 .24 .07 522 • .31 .10 570 .33 .10 632 .41 .14 705 .52 .18 817 .42 .24 942 .39 .29 992 .35 .30 TABLE IV - e SLAG ANALYSIS RUN # 42 (1018 Electrode - .70% Mn) Time After Deoxidizing (sec) Wt % Mn 0 .669 62 .673 275 .626 330 .610 378 .594 446 .593 590 .574 673 .599 741 .595 TABLE IV - f INGOT ANALYSIS RUN # 42 83 small potential drop along the copper mold. Nevertheless, i t i s quite feas ib le to de l ibera te ly pass current through the mold as described i n 3.5. Once th is e l e c t r i c a l c i r c u i t has been set up, current may be measured in the usual fashion by means of a ca l ibrated shunt. In th i s way then, the passage of current through the mold during normal AC melting is simulated. In Run #47, the same procedure was used as described in 3 .8 .4 , the only change being the use of the l i v e mold c i r c u i t r y . The resul ts are tabulated i n Table V. In a d d i t i o n , the mold current and voltage were monitored on a Tektronic Type 564 storage osc i l loscope . In order to determine the consequences on slag composition of using l i v e mold on AC when no chemical reactions were involved, melt #59 was undertaken using Armco iron i n place of 1018. The data obtained are presented in Table V. In a d d i t i o n , several slag samples from the l i v e mold runs were 3+ examined for Fe . The a n a l y t i c a l procedure i s given in Appendix I I . 3+ E s s e n t i a l l y , the approach taken was to analyse for Fe and then to compare th i s analysis to the to ta l i ron obtained by the al ternate method. 3.8.6 Insulated Mold In conjunction with the tests on l i v e mold behaviour, the ef fects of completely insu la t ing the mold were also invest igated. The mold i n s u l a t i o n was accomplished by the boron n i t r i d e coating procedure out l ined i n 3.5. I t i s worth noting that th is BN paint , when properly 84 Run # Average Volts Average Rms Amps AT(°C) Slag Skin Thickness (mm) Average Melt Rate (g sec" 1 ) Final Slag Weight (g) 47 (1018) - 1500 21 1.2 2.6 510 59 (Armco) 23.5 1200 18 .9 2.7 530 TABLE V- - a OPERATING PARAMETERS Time of Sampling (sec) Wt % F e2 + Wt % M n 2 + 347 .14 .05 360 .13 .06 387 .18 .05 402 .20 .04 423 .22 .07 462 .28 .07 521 .33 .10 552 .35 .12 599 .43 .13 654 .49 .17 722 .54 .17 782 .55 .22 964 .47 .28 1006 .44 .31 1063 .40 .33 1138 .38 .38 Time of Sampling (sec) Wt % F e2 + 92 .06 323 .36 393 .45 457 .55 494 .58 572 .67 678 • 7 9 773 .93 851 1.01 919 1.09 TABLE V - c SLAG ANALYSIS - RUN # 59 TABLE V - b SLAG ANALYSIS - RUN # 47 (1018 Electrode - .70% Mn; deoxidized at 320 sec) 85 a p p l i e d , i s so durable that a wire brush must be used to remove i t on the completion of a run. Run #40 was carr ied out exactly as those i n section 3 .8 .4 ; i . e . with W powder markers and calcium d e o x i d i t i o n , followed by sampling. The r e s u l t s , as shown i n Table VI , are very d i f f e r e n t , however, from those recorded previously . As a d i r e c t consequence of the resu l t obtained i n Run #40, two addit ional experiments were done using the insulated mold. In Run #46, a quantity of barium f e r r i t e (Alpha Products, Beverly, Mass.) was added to the deoxidized slag and slag samples taken. Barium f e r r i t e was used i n place of FeO as i t appeared to dissolve i n the slag much more r e a d i l y . Levels of FeO > 2% in CaF^ - CaO slags r e s u l t i n two immiscible l i q u i d s and t h i s impairs the rate of d i s s o l u t i o n of FeO. A s i m i l a r procedure was followed i n Run #58, except this time with the replacement of barium f e r r i t e by MnO. This MnO was made from Mn02 by passing cracked ammonia over the Mn02 at approximately 600°C for = 24 hours. The resultant product was a c h a r a c t e r i s t i c l i g h t green colour and was further checked by taking an X-ray powder d i f f r a c t i o n 37 pattern. Although MnO does have the same tendency to i m m i s c i b i l i t y as FeO in CaF 2 - l ime s lags , i t nevertheless appeared to dissolve more r e a d i l y . The operating data and chemical analysis for these two experiments a r e found in Table V I I . 3.8.7 Steady State Melts i n A i r and With DC Power In order to test the mathematical mass transfer model f u r t h e r , three addit ional runs were carr ied out. 86 Run # Average Volts Average Rms Amps AT(°C) Slag Skin Thickness (mm) Average Melt Rate (g s e c - 1 ) Final Slaq Weight (g) 40 (1018) 22.5 1225 18 1. 3.0 570 TABLE VI - a OPERATING PARAMETERS Time of Sampling (sec ) Wt % F e2 + Wt % M n 2 + Time a f ter Deoxidizing (sec) Wt % Mn 252 .08 .03 93 .777 282 .08 .06 302 .773 353 .08 .06 418 .760 417 .09 .05 534 .764 514 .09 .04 650 .770 612 .09 .05 720 .759 710 .11 .08 859 .767 841 945 .12 .12 .08 .08 TABLE VI - c INGOT ANALYSIS 984 .12 .09 RUN # 40 TABLE VI - b SLAG ANALYSIS - RUN # 40 (1018 Electrode - .78%Mn; deoxidized at 200 sec) 87 Run # Average Volts Average Rms Amps AT(°C) Slag Skin Thickness (mm) Average Melt Rate (g s e c - 1 ) Final Slag Weight (g) 46 (1018) 23 1275 18 1.5 2.6 570 58 (1018) 23.4 1150 16.5 1. 2.6 600 TABLE VII - a OPERATING PARAMETERS V cJ^TJu^ Wt % F e 2 + Wt % M n 2 + Sampling (sec) 406 .52 .12 454 .47 .20 509 .37 .26 555 .34 .32 609 .30 .32 TABLE VII - b SLAG ANALYSIS RUN # 46 (1018 Electrode - .74% Mn; B a F e 1 2 0 1 9 added 350 sec) Time of Sampling (sec) Wt % F e2 + Wt % M n 2 + 328 .31 .65 395 .38 .55 419 .39 .55 468 .43 .53 529 .53 .47 TABLE VII - c SLAG ANALYSIS RUN # 58 (1018 Electrode - .68% Mn; MnO added at 280 sec) 88 Run #50 was e s s e n t i a l l y a duplicate of #27, 53 (3.8.3) but the i n e r t gas enclosure was removed, leaving the slag surface and electrode open to the atmosphere. The ant ic ipated resu l t would be a higher leve l of FeO i n the s l a g , c h i e f l y as a resu l t of electrode ox idat ion . The resul ts of the slag analysis and operating data are given in Table V I I I . A l l the above melts have been performed using AC power. One expects to f i n d a greatly a l tered pattern of behaviour i n DC operation where the major e f fect i s held to be electrochemical i n t e r -25 act ion between metal and slag and not the chemical phenomena we have considered to th i s point . One run was done for each of the DC power modes; Run #52 on electrode -ve and Run #55 on electrode +ve. Both were carr ied out under helium atmosphere using the CaF 2 - 20% CaO slag as i n a l l the previous cases. The data obtained are to be found in Table IX. 89 Run # Average Volts Average Rms Amps AT(°C) Slag Skin Thickness (mm) Average Melt Rate (g s e c - 1 ) Final Slag Weight (g) 50 (1018) 23.5 1200 17.5 1. 3.0 540 TABLE VIII - a OPERATING PARAMETERS Time of Sampling (sec) Wt % F e2 + Wt % M n 2 + 222 .22 .11 272 .35 .16 398 .41 .20 493 .42 .28 608 .50 .28 703 .59 .34 808 .69 .42 879 .77 .47 938 .79 .51 994 .83 .56 1041 .88 .60 TABLE VIII - b SLAG ANALYSIS - RUN # 50 (1018 Electrode - .75% Mn; AC - a i r atmosphere) 90 Run # Average Volts Average Rms Amps AT(°C) Slag Skin Thickness (mm) Average Melt Rate (g sec" 1 ) Final Slag Weight (g) 52 (1018) 20.5 1400 16 1. 2.1 532 55 (1018) 23.5 1250 16 .7 2.0 578 TABLE IX - a OPERATING PARAMETERS Time of Sampling (sec) Wt % F e2 + Wt % M n 2 + 740 .41 .36 800 .42 .37 900 .36 .36 1000 .39 .37 1100 .33 .31 1200 .32 .36 1300 .26 .30 1410 .25 .30 1500 .26 .31 1690 .27 .26 1795 .23 .24 1980 .17 .21 TABLE IX - b SLAG ANALYSIS RUN # 52 (1918 Electrode - .72% Mn; DC-ve) Time of Sampling (sec) Wt % F e2 + Wt % M n 2 + 186 .60 .19 224 .67 .21 263 .68 .20 334 .71 .23 417 .74 .23 502 .79 .25 577 .82 .26 621 .81 .31 653 .83 .29 725 .83 .31 815 .86 .32 903 .87 .31 973 .91 .34 TABLE IX - c SLAG ANALYSIS - RUN # 55 (1018 Electrode - .72% Mn; DC+ve) 91 CHAPTER 4 RESULTS 4.1 Analysis of Rate Contro l l ing Steps P r i o r to modelling the data obtained in the previous chapter, i t i s worthwhile to invest igate the p o s s i b i l i t y that one or more of 2+ 2+ the three transport steps ([Mn] in the metal phase, (Fe ) , (Mn ) i n the slag phase) may be rate determining. Should th is be the case, our equations would be greatly simplified. Consider the case where mass transfer i s accompanied by a reaction whose equi l ibr ium constant i s very large . K . !£_f j l + . ( 4 1 ) By inspect ion, i t would seem that e i ther or both of the terms a^, a^ 2+ w i l l tend to zero for a f i n i t e numerator i n 4 .1 . Thus, for f i n i t e values of a c t i v i t y c o e f f i c i e n t , the concentration of reactant in e i ther phase may go to zero. The question of which reactant concen-t r a t i o n tends, to zero i s dependent e n t i r e l y on the rate of supply to the reaction interface (assumed to be in equi l ibr ium and with instantaneous reaction r a t e ) . But, for the oxidation of manganese dissolved in the 2+ metal by Fe in the s l a g , the equi l ibr ium constant i s approximately equal to only 4. This value prevents us from making any assumptions about reactants tending to zero. In f a c t , what we must attempt to show 92 i s the reverse p o s s i b i l i t y ; i . e . , which, i f any, i n t e r f a c i a l reactant concentration does not deviate s i g n i f i c a n t l y from the bulk value. The procedure i s as f o l l o w s . Recal l ing Equation 2.11, we have -b ± ( b 2 + 4 c ) 1 / 2 [Mn], = (2.11) 1 i where k r 2+ kc 2+ 0 , b = + ( F e 2 + ) . - [Mn]. V kMn and [Mn], + ( M n 2 \ ]  a k M n 2 + kMn If we consider in b, that k F 2+ e i ther [Mn], » — (4.2) fikMn2+ or [Mn] b » kp 2+ 9 + ( F e 2 + ) , kMn (4.3) and in c , that [Mn] b » k M n 2 + ( M n 2 + ) b 'Mn (4.4) 93 Taking (4.3) and (4.4) to be true for the moment and solving 2.11 we have " k F e 2 + k F 2+ 2 k F 2+[Mn]. „ 2[Mn]. = + [Mn] .± (( )- 2 ^ + [Mn]£ fikMn2+ V fikMn2+ + 4 k F e 2 + ^ M n ] b } 1 / 2  Q k M n 2 + - k F 2+ k F 2+ 2 2k F 2+[Mn] 2[Mn], = + [Mn].± (( ) + ^ n k M n 2 + aW* flkMn2+ 2 1/2 + [Mn] b ) taking the pos i t ive root kcJ+ k P 2+ 2[Mn]. = - h e + [Mn]. + h e + [Mn]. fikMn2+ n k M n 2 + Thus, [Mn] i = [Mn] b (4.5) If the i n t e r f a c i a l concentration of the metal phase reactant i s approximately equal to the bulk value, t h i s implies that the rate of supply of [Mn] in the metal i s s u f f i c i e n t l y fast to prevent i n t e r f a c i a l deplet ion. We therefore conclude that , should the inequal i t i es (4 .2) , 94 (4 .3) , (4.4) be shown true for our system, then mass transfer w i l l be contro l led by slag phase transport . Table X shows the resul t s of evaluating these inequa l i t i e s at each of the reaction s i tes using typica l concentration l e v e l s , and mass transfer coe f f i c i en ts from Table I . TABLE X Evaluation of Terms i n Rate Contro l l ing Step A n a l y s i s . ( F e 2 + ) b = 2.3(10" 4) mole c m - 3 (.5 wt %) ( M n 2 + ) b = 1.9(10" 4) mole cm" 3 (.4 wt *) Q calculated from (2.22) ( Y M n r r 3 - 5 ) Reaction S i te [Mn] b © (.76 wt %) k F e 2 + ( F e 2 + ) h ( 2 > k F e 2 + © k M n 2 + ( M n 2 + ) b ® kMn kMn Electrode Film (fi=3371) 1(10" 3) 3.5(10" 4) 3.0(10" 4) 2.9(10" 4) DROP (fi=1785) 1(10~3) 4.3(10" 5 ) 5.6(10" 4) 3.6(10" 5) Ingot POOL (Q=l785 1(10" 3) 1.9(10"4) 5.6(10" 4) 1.5(10" 4) On comparing the tabulated values i n column Q) with those i n © , and that i n (D as required by the i n e q u a l i t i e s (4 .2-4 .4) , i t i s seen that the values in Q are higher than ei ther @ , or Q) and higher than (5). This i s a c lear indica t ion of the predominance of the slag transport processes in determining the overa l l rate of mass t rans fer , but also i t shows that we cannot ignore the contr ibut ion of 9 5 the metal phase resistance. We must, therefore, use the mass transfer d i f f e r e n t i a l Equation (2.12) as derived i n the modelling of the data of Chapter 3. 4.2 Results of Experiments to Determine Pool Mass Transfer Coef f ic ients 4.2.1 Experimental Determination of Mass Transfer Coeff ic ients k M n 2 + ' k F e 2 + In l i g h t of the above phase transport control a n a l y s i s , i t i s evident from examination of the resul ts of th is analysis at the ingot pool /s lag interface that by increasing the concentration of Mn i n the metal phase that one might e f f e c t i v e l y lower the metal phase mass t ransfer resistance to such an extent that i t could be ignored. Experiments 60, 61 were carr ied out with th is i n mind (3 .8 .2) . The e f f e c t i v e [Mn]^ leve ls were estimated at 0.8 wt % for Run #60 (ingot pool - 500 g) and 2.3 wt % for Run #61 (ingot pool 2520 gm). Consideration 2+ 2+ of these levels of [Mn] and the i n i t i a l levels of (Fe )^ and (Mn )^ of Table l i b for the respective runs, with respect to the i n e q u a l i t i e s 4 .2 , 4.4 gives the resul ts of Table XI (for the pool on ly , since Armco iron was used and no reaction i s assumed at the electrode t i p and at the drop s i t e s ) . 96 TABLE XI Transport Control Data at Ingot Pool for Melts # 60, 61 Run # [Mn] b 1 k F e 2 + ( F e 2 + ) b 2 k M n 2 + < M n 2 \ 3 kM Mn kMn 60 1.05(10" 3) 2.3(10" 4) 3.1(10" 5) 61 3.01(10" 3) 2.0(10" 4) 6.9(10" 5) I t i s apparent from the resul ts of Table XI that while the term 3 i s much smaller than 1 , the other i s not. Nevertheless i n our subsequent c a l c u l a t i o n we s h a l l assume [Mn]^ * [Mn] b (from the r e s u l t (4.5)) and examine the consequences of th i s assumption l a t e r . From r e l a t i o n 2.9 we have the equation nMn2+ n U t a ^ F e * ^ - ( M n 2 \ A (2.9) n[Mn]. 1 + k F e 2 + k M n 2 + We have assumed above that [Mn]. - [Mn] b and also ( in 2.3.4) that k p e2+ = kM n2+. Since d(Mn 2 ) n M 2+ = V c (2.13) M n s dt 97 we have d(Mn 2 + ) dt ^ n f r } { ^ n ] ( F e 2 + ) b (Mn 2 + ) b } (4.6(a)) From the data of Table l i b we obtain the plots given in Figure 16 2+ showing the rate of r i s e of (Mn 2 + ) in the slag ( i . e . ^5—h immediately a f ter manganese was added to the metal pool . The slopes were obtained by l i n e a r regression analysis of the ava i lab le points . The only remaining unknown, now, i n Equation 4.6(a) i s kM n2+. Rearranging 4.6(a) 2+ k M n 2 + = (d(Mn tT)/dt)Vs(n[Mn]b+l) A ( f i [ M n ] b ( F e 2 + ) b - ( M n 2 + ) b ) (4.6) 2+ The values of d(Mn )/dt determined from Figure 16 and the remaining parameters of (4.6) evaluated at t = 0 from data i n Table l i b and Table I are summarized i n Table XII below. We use the i n i t i a l time. TABLE XII Summary of Data for Experimental Evaluation of kM 2+ ( a l l concentrations i n moles cm" ) Parameter Run #60 Run #61 A / V s [Mn]b [ M n 2 + ) b ( ^ 2 + ) b d(Mn 2 + ) /d t 2500 .624 3.03 (10~3) 8.77 (10 - 5 ) 1.49 (10" 4) 8.06 (TO - 7 ) 2575 .624 1.05(10 - 3 ) 3.79(10- 5) 2 .04(10 - 4 ) 7.58(10" 7) 98 030i — i 1 1 r 025h r*(Mn8*)6l = l-7(IO"3)wt.%sec"' 11 I i i 1 0 10 20 30 4 0 50 TIME (SEC.) Figure 16 Experimental Determination of Rate of Rise of Mn2+ in Slag During Oxidation of Mn i n Ingot Pool Only (Melts #60,61) 99 t = 0, since th i s i s the only time that [Mn]^ i s known (assuming 35 56 complete mixing in the l i q u i d metal pool) ' without p r i o r knowledge of 1^2+. 9, i s evaluated at 1675°C, y ^ g being taken from Figure 20. The values of k^n2+ (and hence kpg2+) obtained were 0.01.1 cm sec" 1 for Run #60 and 0.009 cm sec" 1 for Run #61. The t h e o r e t i c a l l y modelled value of k^n2+(kpe2+) was 0.013 cm s e c - 1 . The agreement i s good in view of the assumptions made with respect to metal phase mass transfer res istance. Also some manganese i s undoubtedly being reduced at the electrode t i p and to a lesser extent at the f a l l i n g drops, thereby lowering the slag Mn l e v e l . Thus we might ant ic ipate obtaining a low value from the procedure used in t h i s instance. 4.2.2 Experimental Determination of the Mass Transfer Coef f i c ient k^2-Although we have not discussed the mass transfer model for desulphurization to date, a comprehensive mathematical treatment i s given i n Appendix I I I . The major purpose of the experiment to determine k<.2-(#62), was to demonstrate further that the mass transfer coe f f i c i en ts as calculated from the penetration theory etc . are r e a l i s t i c . By using the desulphurization react ion [S] + F e { £ ) - (Fe 2 + ) + (S 2~) (A3.3) as described in Appendix I I I , we have al tered several aspects of the mass transfer phenomena. F i r s t l y , the equi l ibr ium constant i s great ly reduced ( in f a c t , by almost 2 orders of magnitude (cf . Appendix I I I ) . Also the d i f f u s i o n c o e f f i c i e n t !of (S ) i n the slag i s somewhat higher 100 2+ 2+ than that used for (Mn ) , (Fe ) and hence the ant ic ipated mass transfer c o e f f i c i e n t w i l l be correspondingly higher. From Appendix I I I , we have the fo l lowing r e l a t i o n s h i p , s i m i l a r to that used in .the previous section (cf . Equation 4 .6) . k 2-(V / A ( d ( S 2 " ) / d t § — (A.3.12) ns[s]1 (s2-)b Is f d ( S 2 ' h _ L _ + ( F e 2 + ) A ' dt k F e2+ + ^ e j b 1.22 Exp(2330/T - 4.07) p $ where fic = — (A. 3.15) 'S YCaS YFeO P M It w i l l be assumed again that the e f fec t of adding a quantity of FeS to the metal pool was s u f f i c i e n t to reduce metal phase mass transfer res istance. Hence [S]^ becomes [S]^ i n (A.3 .12) . In Figure 17, the resul t s of Run #62 are plotted using the data from Table l i b . I t was apparent in th is run that a large quantity of FeS transferred ins tant ly to the slag without d i s so lv ing i n the metal pool . Taking a mass balance on the s lag we f i n d that since the i n i t i a l l eve l of s lag sulphur was approximately .08 wt % and f i n a l was .35 wt %, the change in sulphur level of .27 wt % times the f i n a l s lag volume of 520 gms gives us 1.40 gms of sulphur. A to ta l of 1.38 gms of S was added (4 gms FeS at 34.5 % S) and thus we have a reasonably sound overa l l mass balance. However, when the f i r s t sample was taken, 10 sees a f t e r introduct ion of the FeS, the s lag sulphur level had r i s e n to .25 wt %. This would appear to have been caused by incomplete so lut ion of the FeS i n the metal and thus we must ca lculate an ingot 0 4 0 i 0 3 5 0 3 0 LU O LU Q_ S2 LU 0 2 5 0 2 0 [ - T R A N S I E N T - * R U N 6 2 - I S * " ) n(S 2")62 = 3 3(,0~3)wt-o/<> sec" -1 0 1 5 0 1 0J x 2 0 4 0 6 0 8 0 100 T I M E ( S E C . ) 120 140 .2-160 Figure 17 Experimental Determination of Rate of Rise of S i n Slag a f ter FeS Addit ion to Ingot Pool 180 102 sulphur level at the time of the f i r s t sample. The i n i t i a l increase was from .08% to .25%, a to ta l of .17 wt % or in terms of the tota l change of .27 wt %, 68%. Thus only 32% remained i n the ingot pool . The pool weight was estimated to be - 500 gms, y i e l d i n g a S level of .32 x 1.38 or .088 wt % 500 The time of f i r s t sample marks the beginning of a short period of l inear 2- 2-rate of r i s e of (S ). The measured d(S )/dt from the graph of - f i - o _1 Figure 17 i s 4.23(10" ) mol cm" sec" . The values of the remaining parameters needed to estimate k<.2- are given i n Table XI I I . The 2+ 2-concentrations (Fe (S )^ were taken from Table l i b . The value of YpeQ was the usual one of 3.0 and that of YQ a $is estimated to 2 . 4 1 7 . A temperature of 1675°C is used for ca lcu la t ing ft<~. TABLE XIII Data for Experimental Evaluation of k^-2 Parameter Value Used V S/A 1.602 2.28(10" 3) d ( S 2 " ) / d t 4.23(10" 6) 1.86(10" 4)mol cm" 3 (s2")b 2.03(10" 4) " Mb 2.02(10" 4) " 103 The experimentally determined value of k<.2-is found to be 0.016 cm s e c - 1 . Using the penetration theory as applied in Chapter 2 90 2-and the d i f f u s i o n c o e f f i c i e n t for S ( in fused sa l t s ) where D $ 2- = 3.8(10~ 4) cm2 s e c - 1 we have 3.8(10" 4) 1/2 , k<.2- = 2 ( ) = .036 cm sec 5 TT x .37 The experimentally determined value i s about one hal f of the modelled value. There are several possible explanations for t h i s . One i s that the d i f f u s i o n c o e f f i c i e n t of S had been overestimated. Using -4 a value of D<.2- = 10 , k<-2- becomes .019, i . e . much nearer the value calculated from the data. Another ra t ional might be that our assumption of [S].j = [ S ] b i s hot t rue . I t i s obvious from the expression for k<-2-, that decreasing [S] i n the equation w i l l ra ise the value of k<,2-. However, i t i s u n l i k e l y that the i n t e r f a c i a l value of [S] i s much less than [S]^ because of the high d i f f u s i o n c o e f f i c i e n t of S i n the metal. These two experiments to calculate mass, transfer coefficients from experimental data show that our modelled values are indeed quite reasonable. Hence, having established some confidence in our modelled mass transfer c o e f f i c i e n t s , we proceed with, the modelling of the remainder of the experimental data for the runs described i n Chapter 3 using the values of mass transfer coe f f i c ients given i n Table I . 104 4.3 Boundary Conditions for Solution of Mass Transfer Model In e f fect we must solve the d i f f e r e n t i a l equation of mass transfer A - b ± ( b 2 + 4 c ) 1 / 2 d[Mn] = - T T - kM {[Mn] b - [- ] } (2.14) M 2 k F 2+ k F 2+ 9 where b = + (Fe* ). - [Mn]. Ok 2+ k Mn KMn c = ^ { J ™ b + [ M n 2 + ] b ]  fi k M n 2 + kMn for 3 concentrations v ia the steady state assumption d ( M n 2 + ) b = - d ( F e 2 + ) b = -d[Mn] f a ^ (2.15) In other words, the d i f f e r e n t i a l equation contains three var iable concen-2+ 2+ t rat ions (Fe ) b , (Mn ) b and [Mn] b > Each of these must, therefore, be spec i f ied in the i n i t i a l condit ion as ( F e 2 + ) , (Mn 2 + ) , [ M n ] b j Q at t = 0 Since we are applying (2.14) to three reaction s i tes and solving these simultaneously as described i n 2 .5 , i t i s necessary to add that [Mn] b Q must be spec i f ied i n the electrode t i p ( i . e . the bulk electrode value) and at the ingot pool s i t e (assumed equal to the bulk electrode value at t = 0) . I t i s further assumed that a l l three react ion s i tes see the 2+ 2+ same concentration of (Fe ) b and (Mn ) b at any time. The value [Mn] b Q 105 for the drop s i t e i s taken as the value of [Mn]^ i n the f i l m a f te r a short period corresponding to the formation of one drop. With the exception of the l a t t e r concentration, a l l other i n i t i a l values are determined from the experimental data for each indiv idua l melt . 4.4 Use of F e 2 + "Potent ia l s " Examination of the data given i n Tables II through IX showed 2+ that (Fe ) concentrations range from a low of .03 wt % to a high i n excess of 1 wt %. I t became evident that the rates of generation and 2+ loss of Fe were usual ly quant i ta t ive ly unpredictable. Without s p e c i f i c knowledge of these rate terms we cannot model the concentration changes 2+ for Fe . This lack of information was, of course, the ra t ionale behind the experiments using l i v e and insulated molds and the resul ts of these experiments w i l l be d u l y discussed. Nevertheless, i n s u f f i c i e n t 2+ information was obtained' on the Fe production phenomenon and we were forced to adopt a d i f f e r e n t approach. 2+ The concept of using an imposed Fe potential was selected. This involved the very simple operation of drawing a series of one or 2+ more s t r a i g h t l ines through the Fe data points . The slopes of these l ines were then incorporated into the computer program as production 2+ 2+ or loss terms for Fe . Hence the concentration of Fe , termed the 2+ "Fe p o t e n t i a l , " was known at any time and was used in the subsequent so lut ion of the d i f f e r e n t i a l equations. The model i s obviously l i m i t e d now to predict ing concentration changes of only manganese i n the ingot and s l a g . 106 The plots of the data and of the modelled s lag and ingot p r o f i l e s (where applicable) show generally three types of l i n e . The 2+ f i r s t of these i s an imposed Fe potential l i n e , fo l lowing the data 2+ obtained for Fe i n each experiment. Secondly, there i s a l i n e showing 2+ the model predicted (Mn ) composition and f i n a l l y , a series of 3 l ines g iv ing the contributions of the various react ion s i tes (electrode f i l m , f a l l i n g drop and ingot pool) towards the overa l l 2+ change i n s lag (Mn ) l e v e l . These l a t t e r l ines were normalized at zero time to zero percent contr ibut ion . 4.5 Choice of Y M n Q In Figure 10, the known data for y ^ g versus wt % MnO i n CaF 2 - CaO melts was presented. I t was observed (2.3.2) that Y M n g does not, i n a l l l i k e l i h o o d , obey Henry's Law even i n d i l u t e s o l u t i o n . Furthermore, the data given i n Figure 10 does not extend to the 20% CaO in CaF,, system and i t was suggested (2.3.2) that i n applying the mass transfer model that the approach we would adopt would be to f i t the model with s e l f consistent values of Y ^ g - I t was found that a l l of the modelling work could be successively correlated using a l i n e a r v a r i a t i o n of Y ^ Q as shown i n Figure 18, where the dotted l ines indicate the degree of scatter of values from run to run. The value 2+ of Yjv]ng for each level of (Mn )^ was calculated by the subroutine OMT (Appendix IV) and thus internal consistency was maintained. I t i s also worth noting that these " f i t t e d " values are i n l i n e with those one might ant ic ipate with the increased CaO content of the s l a g . 107 0 1 -2 -3 -4 - 5 - 6 -7 - 8 - 9 1 0 wt% MnO Figure 18 y ^ g Values F i t ted by Computer Program for CaF2-20% CaO Slag 108 4.6 Modelling Results A l l of the necessary parameters and boundary conditions for use i n the mathematical model have now been d e t a i l e d . I t was found that taking the pos i t ive square root i n the quadratic part of the d i f f e r e n t i a l Equation (2.14) provided meaningful r e s u l t s . No s i g n i f i -cance is attached to the negative root since i t leads to a negative value of concentration. The predicted composition changes are presented below i n graphical form and compared to the experimental data. 4.6.1 Steady State Results Figures 19 and 20a (Runs 27, 53, respect ively) exhib i t very 2+ nearly i d e n t i c a l patterns of behaviour. The (Fe ) levels r i s e from - .3 wt % to - .5 wt % during the course of each melt , the rate of r i s e 2+ of Fe being s l i g h t l y higher i n the l a t t e r . 2+ The predicted (Mn ) concentration change i n the slag i s i n good agreement with the experimental data for both of these runs. In Figure 20b, the predicted ingot p r o f i l e for [Mn] i n Run #53 i s compared with the ingot a n a l y s i s , again with sa t i s fac tory agreement. The overa l l a x i a l v a r i a t i o n i n manganese content i s 6.5% which, although i t may not be a serious loss for 1018, may represent a substantial loss for another a l l o y system. Of p a r t i c u l a r interest here are the r e l a t i v e contributions to mass transfer at each reaction s i t e . The f i l m s i t e and the drop 0 6 i 0 - 5 DATA P O I N T S A - (Fe 2 * ) IN S L A G O - ( M n 8 * ) IN S L A G t 0 4 h 0 lk I M P O S E D (Fe 2 *) P O T E N T I A L 2+, ( M n ' ) P R E D I C T E D F R O M M O D E L C O N T R I B U T I O N S F R O M E A C H R E A C T I O N S I T E 5 -S + 0-4 O JO > N m o o FILM l+0«3 55 c 1*0-2 $ m > m -* LLC POOL DROP 100 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 T I M E ( S E C . ) 8 0 0 9 0 0 1 0 0 0 1100 Figure 19 Comparison of Theoretical Mass Transfer Calculat ions to Experiment Results for Slag Composition vs Time P r o f i l e for Steady State Melt #27 1 2 0 0 o CD 110 x o w 02h 01 IMPOSED (Fe ) POTENTIAL (Mn2*) PREDICT. FROM MODEL CONTRIBUTIONS OF EACH REACTION SITE  R L J J _ POOL - U • 0-4 N m O o TO • 0*3 5 *oi > a X CO • 01 15 m "1 100 200 300 400 500 TIME (SEC.) 600 700 800 900 (a) X • 054h 0 52« • - [Mn] IN INGOT PREDICTED ; PROFILE FOR [Mn] _1_ _l_ -I—-J: J L 1000 100 200 300 400 500 TIME (SEC) 600 700 800 900 (b) Figure 20 Comparison of Theoretical Mass Transfer Calculat ions to Experimental Results for Melt #53 (a) Slag Composition vs . Time P r o f i l e (b) Ingot Composition vs. Time P r o f i l e I l l s i t e exhib i t very nearly i d e n t i c a l from run to run. The major difference between the runs i s the enhanced rate of Mn oxidation at the ingot pool/ 2+ 2+ slag interface in run #53, where the f i n a l level of (Mn ) and (Fe ) i n the slag are somewhat greater than in Run #27. I t would appear that the ingot pool reaction is more sens i t ive to changes in the oxidation potent ial of the slag than are the other two s i t e s . Should th is be the general case, one might expect a much more obvious trend to be observed i n the unsteady state condi t ion . Of note also i s the very minor ro le that i s played by droplets i n the overa l l contr ibut ion to mass transfer (- 1% i n #53; i . e . .005% i n to ta l change of .415%). This confirms the evidence of Cooper et a l , 1 4 which indicated that droplets were of l i t t l e s igni f i cance i n determining the overa l l rate of desulphurizat ion. 4 .6 .2 Unsteady State Results The resul ts of modelling the unsteady state melts #.'s 30,31,34, 37,42 are compared to the experimental data in Figures 21 through 24. Again the calculated composition changes are i n reasonable agreement with the slag analyses i n each case and also for ingot [Mn] p r o f i l e of Run #42 (Figure 24b). I t i s very apparent from these resul ts that , fol lowing the deoxidation of the s lag with calcium at some time <0 (see Table IV) , 2+ 2+ the system reacts quickly to produce (Fe ). The level of (Fe ) bui lds up very rap id ly and then decays i n some fashion which i s not p a r t i c u l a r l y consistent from melt to melt. I t i s cases such as these 2+ where the necessity of using the imposed (Fe ) potential becomes 112 A - ( F e 2 V R U N 3 0 A - ( F e 2 + ) - R U N 31 0 - ( M n 2 + ) - R U N 3 0 > D A T A P O I N T S > - ( M n 2 + ) - R U N 31 J - I M P O S E D F e 2 + P O T E N T I A L ( M n 2 + ) P R E D I C T E D F R O M M O D E L C O N T R I B U T I O N S F R O M E A C H R E A C T I O N S I T E 05 0 Figure 21 100 200 300 TIME (SEC.) 400 Comparison of Theoretical Mass Transfer Calculat ions to Slag Composition vs . Time P r o f i l e for Melts 30, 31 T DATA POINTS A - ( F e 2 + ) 0 - ( M n 2 + ) IMPOSED (Fe 2 +) POTENTIAL (Mn 2 + ) PREDICTED BY MODEL - - CONTRIBUTIONS OF EACH REACTION SITE FILM "POOL 0 . — =u —t- .-DROP 7 0 0 8 0 0 9 0 0 1 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 T I M E ( S E C ) Figure 22 Comparison of Theoretical Mass Transfer Calculat ions to Slag Composition vs Time P r o f i l e for Run #34 z o JO > IM •04 £ o o JO •0*3 QQ c •0 2 $ m > o X n , CO + 01 Zi m CO DATA POINTS IMPOSED (Fe2*) POTENTIAL (Mn2*) PREDICTED BY MODEL --CONTRIBUTIONS OF EACH REACTION SITE FILM POOL DROP 3 > •o* 5 n o 7 + 0 - 3 5 • 0 2 $ m > a co • 01 3 300 400 500 600 700 TIME (SEC.) 800 900 1000 1100 1200 (Mn*>. Figure 23 Comparison of Theoretical Mass Transfer Calculat ions to Slag Composition vs Time P r o f i l e for Melt #37 115 0.6 0.5 0.4 c Oi o \_ a> Q. 0.3 - C o< 0.2 4-A 2 + A Fe (experimental) 0 Mn 2 + 2 + Imposed Fe potential 2 + Mn calcd. from math, model Contributions of various reaction sites o? • — - ^ " - ' ^ - • " i Pool . . . Film — • -J • . D r ° P • 2 0 0 4 0 0 6 0 0 8 0 0 Time (sec) (a) 070r 0 68t • -[Mn] IN INGOT P R E D I C T E D INGOT PROFILE 100 200 300 (b) 400 TIME (SEC.) 500 600 700 800 Figure 24 Comparison of Theoretical Mass Transfer Calculat ions to Experimental Results for Melt #42 (a) Slag Composition vs . Time P r o f i l e (b) Ingot Composition vs. Time P r o f i l e 116 2+ clearer. The turnover i n slope of the (Fe ) level appears to be 2+ connected with an increase i n rate of change of (Mn ) which is what one would expect on a q u a l i t a t i v e basis . Also apparent i s that the 2+ i n i t i a l rate of generation of (Fe ) i s not constant, fur ther f r u s t r a t i n g any attempts at r a t i o n a l i z i n g th i s mode of behaviour. In one run, however, (# 37), the recorded current trace exhibited a large number of d i s c o n t i n u i t i e s soon a f ter deoxidiz ing . I t i s known that these i r r e g u l a r i t i e s are associated with arcing to the OC mold w a l l . An arc may be used to r e c t i f y an AC signal and t h i s leads us to the conclusion that a cer ta in amount of r e c t i f i c a t i o n of the AC current was probably occurring in the only part of th is run. The end r e s u l t of th is i s a net DC component of the to ta l current which could 2+ 25 very l i k e l y lead to production of Fe . In order to v e r i f y th i s observation the experiments on l i v e and insulated molds (3.85, 3.86) and those on DC power (3.87) were undertaken. The resul ts of these invest igat ions are given i n d e t a i l l a t e r i n the next sect ions. Recal l ing the observations made regarding the r e l a t i v e contributions of the three react ion s i t e s , i t was s tated, (a) that the droplets appeared to have l i t t l e e f fec t on the to ta l amount of mass t ransfer and (b) , that the rate of oxidat ion of Mn at the electrode f i l m / s l a g interface seemed to be constant, with the major response to 2+ the changing Fe potential a r i s i n g from the ingot pool react ion . The 2+ transient behaviour of the Fe potential i n the unsteady state melts reinforces th is view i n every respect. Not only does the contr ibut ion of the drops remain r e l a t i v e l y t r i v i a l and the f i l m rate remain nearly 117 constant and consistent with the steady state values, but also the 2+ rate of transfer at the ingot pool seems to r e f l e c t the Fe potential var ia t ions with a cer ta in time lag e f f e c t . This i s espec ia l ly notice-able i n the longer time in terva l experiments, #'s 34, 37 and 42, (Figures 22-24) and p a r t i c u l a r l y i n the ingot p r o f i l e of Run # 42 (Figure 24b). We also notice here, for the f i r s t t ime, the appearance of a reversion reaction (Figures 22 and 23) where rate of Mn oxidat ion at the ingot pool drops to zero and subsequently i s reversed to a reduction 2+ of (Mn ) back into the ingot. These observations have important ramif icat ions i n the understanding of inc lus ion formation in ESR ingots which w i l l be developed at a l a t e r stage. 2+ I t i s c l e a r , then, that the use of an imposed Fe potential 2+ has enabled us to overcome the problem of unknown rates of Fe production and consumption i n our e f f o r t s to model the composition changes r e l a t i n g to manganese t ransfer . The mass transfer model appears to predict these concentration changes in both slag and ingot with a good degree of accuracy and with a minimum of curve f i t t i n g . I t i s pert inent , therefore, that we continue with a detai led examination of the exper i -2+ ments done to invest igate the nature of the Fe production and subsequent experiments designed to further test the model. 4 .6.3 Results of the Live Mold Experiments 2+ The data of Run # 47 are compared to the predicted (Mn ) composition changes in Figure 25 with reasonable agreement between them. The most s t r i k i n g feature of th is p l o t , however, i s the s i m i l a r i t y with T 1 1 1 1 T DATA POINTS T I M E ( S E C . ) Figure 25 Comparison of Theoretical Mass Transfer Calculat ions to Slag Composition vs Time P r o f i l e for Melt #47 (Live Mold) 119 the unsteady state melts of the previous sec t ion . The former exper i -ments were performed with the normal mold configurat ion and the l a t t e r with l i v e mold. This suggests that in actual pract ice the normal mold configuration has, i n f a c t , behaved as a l i v e mold, i . e . , some current did pass through the mold. The question we must ask now 2+ i s "how does th is re la te to the problem of Fe generation?" During the course of Run #47 s the mold current was monitored on an osci l loscope as mentioned previously . Although i t was, unfor-tunately, impossible to assess the magnitude of th is amperage with any degree of accuracy, the amperage trace i t s e l f provides some interest ing q u a l i t a t i v e information. Figure 26 shows a t y p i c a l AC waveform of the mold current on the l i v e mold run. Figure 26 AC Waveform-Mold Current-Run #47 ( l i v e mold) (Scope set to t r igger on the +ve slope) 120 I t i s immediately obvious that d i f f e r e n t processes are occurring on each half cyc le . The precise nature of the processes i s not known (but possibly associated with conduction of ions in the frozen slag skin adjacent to the mold wal l ) and is unimportant as f a r as th is analysis i s concerned. However the ef fects of these processes are important. I t i s apparent that the areas A l , A2 under the curve of Figure 26 are not equal on each half cyc le . In f a c t , i t was found that , using a planimeter, the area Al under the electrode pos i t ive half cycle ( i . e . mold and ingot negative) i s subs tant ia l ly larger than the corresponding area, A2. Since these areas represent the to ta l number of coulombs passed through the mold wall per half cycle and, since they are not equal, there must be some r e c t i f i c a t i o n of the AC current due to passage of some port ion of th i s current through 2+ mold w a l l . The explanation as to how t h i s resul ts in Fe production i s as fo l lows . When the electrode was negative, the current through the mold wall was less than when the electrode was p o s i t i v e . However, the mold wall d i d not carry the t o t a l current but rather only some f r a c t i o n of the to ta l with the balance passing through the ingot . Since the t o t a l current passing through the ingot plus mold wall was held constant on each hal f c y c l e , i t follows that a greater proportion of current was passing through the ingot when the electrode was negative than when i t was p o s i t i v e , i . e . , the reverse of what we observed at the mold w a l l . When the electrode was negative and the 2+ ingot p o s i t i v e , there was an electrochemical generation of Fe at 25 the ingot pool surface. I f the same amount of current had passed 121 through the ingot pool on the other half c y c l e , one would have expected 2+ that the Fe generated i n the previous 1/120 second would be reduced. However, we have jus t shown that , due to current r e c t i f i c a t i o n at the 2+ mold w a l l , more current passed through the ingot on the Fe production cycle (electrode negative, ingot pos i t ive ) than vice versa. Hence 2+ there was a b u i l d - u p o f Fe at the ingot pool interface and subsequently i n the s l a g . I t i s in teres t ing to note that the electrode had the exact same current passing through i t per hal f cyc le and thus there was 2+ no net electrochemical generation of Fe at t h i s s i t e . In order to assess the ingot pool rate one must accurately determine the current through the mold w a l l . As stated previously , t h i s was not possible with s u f f i c i e n t r e l i a b i l i t y to warrant any c a l c u l a t i o n s . In a d d i t i o n , the anodic p o l a r i z a t i o n charac ter i s t i cs of 25 high lime -CaF2 based slags are not a v a i l a b l e , without which any 2+ Fe interface a c t i v i t y ca lculat ions are meaningless. 2+ The ef fect causing the rapid Fe build-up a f ter deoxidat ion, thus, would appear to be due to d i r e c t passage of current through the mold wall as i n l i v e mold operation and the subsequent r e c t i f i c a t i o n of th i s component of the AC current . In normal AC operat ion, the 2+ varying rates of Fe build-up are probably due to d i f f e r e n t proportions of current leakage through the mold from melt to melt . This leakage would be affected by such factors as slag skin thickness , s lag depth and depth of electrode immersion. The r e c t i f i c a t i o n analysis above i s consistent also with the conclusions made with respect to the arc induced r e c t i f i c a t i o n suspected in Run #37 (4 .6 .2) . A - (Fe ) IN S L A G L I N E O F S L O P E n F e2+ n F a 2 * = l-2(IO" 3)wt%sec"' 4 0 0 5 0 0 T I M E ( S E C . ) Figure 27 Rate of Rise of Fe Mold (Melt #59) 2+ in Slag when Melt ing Armco Iron With Live 123 In order to es tabl i sh the r e p r o d u c i b i l i t y of the r e c t i f i c a t i o n phenomenon, a melt was conducted with r e l a t i v e l y pure Armco iron and the l i v e mold conf igurat ion. The slag analyses showing the Fe b u i l d -up occurring during th i s melt (# 59) are presented in Figure 27. The AC waveform observed on the osc i l loscope trace of mold current i s shown i n Figure 28. Figure 28 AC Waveform of Mold Current in Armco Live Mold, Run # 59 This waveform, although of d i f f e r e n t character, shows a s i m i l a r im-balance between current passed on the opposing hal f cycles and 2+ supports our previous arguments as to the cause of Fe generation. 124 2+ Due to the lower a l l o y content of the Armco i r o n , however, the Fe continued to increase at a constant rate throughout the course of the run. As mentioned in Chapter 3, some samples taken i n run # 47 2+ were divided up and analyzed for Fe and compared to the values obtained for to ta l i ron on the same slag samples. There was no detect-able difference in the values found by each method and hence, i t i s 3+ l i k e l y that there was no Fe i n these slags under these operating condit ions . 4.6.4 Insulated Mold Results The procedure used i n performing the insulated mold experiment # 40 was i d e n t i c a l to the unsteady state t r i a l s with 1018 of the previous sect ions. Yet i t i s evident from Figure 29(a),(b) that the r e s u l t i s 2+ not equivalent. The Fe level rose only from .08% to 1.2% i n 730 sec-onds. This was a smaller rate of increase (by a factor of - 5) than i n even the steady state runs of Figures 19, 20. In l i g h t of the previous discussion concerning passage of current through the mold w a l l , i t i s evident that the boron n i t r i d e coating applied to the mold wall has indeed provided a good e l e c t r i c a l l y insu la t ing barr ier next to the slag s k i n . In th is case, current flow was r e s t r i c t e d to passing only through the electrode t i p and the ingot pool surface and the r e c t i f i c a t i o n ef fects observed i n the l i v e model case were e f f e c t i v e l y e l iminated. I t 2+ i s suggested that the s l i g h t r i s e i n Fe concentration may be associated + 0 2 ^  m > a 7L + 01 Zi m ( M n 2 \ z o > r N m o o o 5 CO c 400 TIME (SEC.) 500 600 (a) g 0-80f or UJ o- 0-781 2 0-76h U J 074« PREDICTED I INGOT PROFILE X X X 100 200 300 400 TIME (SEC.) 500 600 700 800 (b) Figure 29 Comparison of Theoretical Mass Transfer Calculat ions to Experimental Results for Insulated Mold - Melt #40 ^ (a) Slag Composition vs . Time P r o f i l e tn (b) Ingot Composition vs . Time P r o f i l e 126 with the difference i n current densi t ies occurring at the two e lec t ro -act ive poles of the system, the electrode having about 1/3 the area of the ingot. The predicted composition-time p r o f i l e s for slag and ingot also agree with the data in th i s case. However, th is instance i s the only one so far i n which we observe the contr ibut ion to mass transfer at f i l m s i t e to be subs tant ia l ly a l tered from the previous, very consistent behaviour. A l l the mathematical modelling so far has been done using the 2+ imposed Fe potential as indicated by the data for each indiv idual experiment. This device has enabled us to cope with the unknown features 2+ of the electrochemical production of Fe ions . In the case of the experiment jus t discussed, we have shown that i t i s possible to reduce 2+ the rate of Fe build-up subs tant ia l ly by e l e c t r i c a l l y insu la t ing the inner surface of the mold. Recognizing th i s feature of the insulated mold, experiments #46, and #58 were designed to test not only the model 2+ 2+ predict ions of [Mn] and (Mn ) changes but a lso those of (Fe ) i n the 2+ s l a g . This was done, as previously described (3 .8 .6) , by adding Fe 2+ as barium f e r r i t e and Mn as MnO i n Runs #'s 46 and 58 respect ive ly . These experiments were of short duration i n order to minimize any 2+ extraneous contributions of Fe other than that added or produced by chemical react ion . The predicted and experimental resul ts are i n reasonable agreement in both cases as shown i n Figure 30(a) , (b) . The resul ts are very nearly mirror images of one another, even insofar as the contributions of the reaction s i tes are concerned, 127 1 DATA POINTS A-(Fe 2 + ) •(Mn2+) PREDICTED COMPOSITIONS CHANGES 7 R u - S T 6 i C 0 N T R I B U T I 0 N 0 F E A C " REACTION SITE WITH (Mn2 ) — o - , . — INVERSE CONTRIBUTION OF EACH SITE WITH (Mn2*) J -,POOL / _ . FILM vL^--\-~— r-DR0P o 2 > i + 04 3 o o z -\ 1+0*3 65 c 1 + 0-2 % m > o X 1+0, g m X 0 100 200 T I M E (SEC.) ( a ) - R U N 4 6 100 200 300 T I M E (SEC. ) ( b ) - R U N 5 8 ( M n * \ Figure30 Comparisons of Mass Transfer Model Predictions of Fe 2 + , Mn 2 + vs Time with Experimental Composition vs. Time Profiles 128 thereby confirming the r e p r o d u c i b i l i t y of the mass transfer behaviour of the system. Once more i t i s observed that the ingot pool /s lag interface appeared to react more to the stimulus of changing reactant species in the slag than did the other s i t e s . I t appears, then, that the mass transfer model can accurately 2+ predict (Fe ) changes as well as manganese related composition var iat ions under control led experimental condit ions . These runs also demonstrate the precis ion of the mass balance maintained by the computer model which was not noticeable before. 4.6.5 Results of Steady State Melting i n the Absence of Inert Gas Cover The data and model calculated composition p r o f i l e s for Run # 50 are presented i n Figure 31. I t i s readi ly seen that there was 2+ a considerable change i n the behaviour of the Fe content of the slag i n comparison with the steady state runs of Figures 19, 20. Since the only dif ference between the former and present melts was the absence of the iner t gas cover in Run # 50 we can a t t r i b u t e th i s behaviour to the presence of a i r . The most obvious cause of th i s dif ference i s the oxidat ion of the electrode p r i o r to melt ing. 2+ The rate of Fe build-up was of the same order of magnitude as that caused by the apparent r e c t i f i c a t i o n effects discussed previously . This observation leads us to the conclusion that both rates , as observed by slag a n a l y s i s , were control led by mass transfer rates of 2+ 2+ Fe at respective phase boundaries where Fe was generated. Also 129 0 100 200 300 400 500 600 700 800 TIME (SEC.) Figure 31 Comparison of Theoretical Mass Transfer Calculat ions to Slag Composition vs . Time P r o f i l e for Melt #50, Performed Without Inert Gas Atmosphere \ 130 apparent from th is r e s u l t , i s that there i s no tendency here for the 2+ Fe build-up to diminish . On re-examining the unsteady-state runs 2+ (Figures 21 through 24), i t i s observed that , in every case, the Fe level at the end of the run had returned to the same level as had been found i n the steady state melts ; i . e . 0.3% - 0.5%. I t would appear, then, that i n the absence of a i r , the electrochemical production of 2+ Fe adjusted to some level which i s r e l a t i v e l y consistent . This may be re lated to the rate of oxidat ion of manganese as i t was noted e a r l i e r 2+ (4.6.2) that the turnover point i n the Fe rate appeared to be 2+ associated with the maximum rate of (Mn ) build-up i n the s l a g . In the presence of 02 from the a i r , th is apparently did not happen. Certa inly the addit ion of FeO from the electrode oxidation i s an important reason for th i s d i f ference , but .the further oxidation 2+ 3+ of Fe to Fe may have contributed to the continuation of i ron b u i l d -up i n the s l a g . In th i s p a r t i c u l a r instance, the possible existence 3+ 53 of Fe was not examined, but such a reaction is known to occur. In Targe commercial ESR units where the electrode f i l l r a t i o (diameter electrode/diameter mold) i s about .9 (compared to .5 here) FeO levels i n the slag are commonly 0.3 to 0.6 wt %. The area of s lag exposed to the atmosphere i s proportionately smaller i n these larger furnaces, 2+ 3+ suggesting that any contr ibut ion of the Fe to Fe reaction i s reduced and the system behaves more l i k e the iner t atmosphere melting s i t u a t i o n . The r e s u l t s , nevertheless, demonstrate the effectiveness of the i n e r t gas cover i n reducing overa l l a l l o y losses . In a d d i t i o n , 2+ the calculated (Mn ) changes predict the experimentally measured 131 changes quite reasonably. The predicted contr ibut ion of the three reaction s i t es are also what we might ant ic ipate from previous experience, but there does appear to be a small increase in the mass transfer at the f i l m and drop s i t e s . This was undoubtedly due to the 2+ increased d r i v i n g forces* at the higher Fe levels 4.6.6 Direct Current Results Composition changes i n d i r e c t current melting are generally considered to a d i rec t function of the electrochemical processes 11 25 53 91 occurring at the various e lectroact ive inter faces . ' ' ' The p r i n c i p a l anodic electrochemical react ion occurring in 25 DC ESR processing of an i r o n base a l l o y i s the corrosion of Fe (Equation 4.7) leading to Fe - F e 2 + + 2e (4.7) 2+ an Fe saturated layer on the anode surface at s u f f i c i e n t l y high current d e n s i t i e s . The corresponding cathodic reaction for CaF 2 - CaO slags i s the deposit ion of meta l l i c calcium by the faradaic react ion (4.8) . C a 2 + + 2e + Ca (4.8) The net r e s u l t of these two reactions i s the blocking of the e lec t ro -negative interface to loss of a l l o y elements by oxidat ion due to the zero a c t i v i t y of oxidant of th is s i t e . I t should, i n p r i n c i p l e be possible to explain the oxidat ive loss of a l l o y elements in terms of 132 only the e lec t ropos i t ive reaction s i t e . Runs #'s 52 and 53 were carr ied out in the DC electrode negative and DC electrode pos i t ive modes respect ive ly . The experimental data as given i n Table IX i s presented graphica l ly in Figures 32 and 33. The character of th is data i s very d i f f e r e n t from the behaviour we have observed during AC" melt ing. In the case of the electrode negative melt (Figure 32) the mass transfer model was applied considering only reactions occurring 2+ at the ingot pool /s lag interface and using the Fe potent ial as derived from the data as was done i n the majority of previous cases. The rest of the input parameters were consistent with the AC modell ing. The electrode pos i t ive run was modelled by reversing the act ive reaction s i t e from the pool to the electrode f i l m . In both cases the drop react ion was ignored as i t appeared to contribute a t r i v i a l amount i n a l l the foregoing modelling e f f o r t s . The resul ts i n each case are as good as were obtained i n the AC melts and consistent with the e a r l i e r discussion of the blocking of the electronegative reaction s i t e in DC melt ing. I t i s unfortunate that the anodic p o l a r i z a t i o n data for the anodic corrosion of iron in CaF^ - CaO slags has only been extended to 2 wt % CaO. Without th i s data we are unable to make any reasonably 2+ accurate assessment of the a c t i v i t y of Fe at the anodic in ter face . From the l i m i t e d data avai lab le (Figure 34) we can see that the concen-t r a t i o n overvoltages expected for 20% CaO slag would be s m a l l , even at r e l a t i v e l y high current d e n s i t i e s . For the current densi ty , I ' Q , on the electrode t i p of approximately 100 amp cm , the estimated over-0 - 5 h fe D A T A P O I N T S A - ( F e 2 * ) 0 - ( M n 2 * ) I M P O S E D (Fe 8 *) P O T E N T I A L (Mn 2 *) P R E D I C T E D A S S U M I N G C H E M I C A L R E A C T I O N AT INGOT P O O L O N L Y X 0 1 0 0 Figure 32 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 T I M E ( S E C . ) 8 0 0 9 0 0 1 0 0 0 1100 1 2 0 0 Comparison of Theoretical Mass Transfer Calculat ions to Slag Composition vs. Time P r o f i l e for DC-ve Melt #52 GO 0 0 134 0-9\ 0-8 07 g<>6f UJ CL 5 o s ! 0-4 03 02e£ / DATA POINTS A - (Fe2*) 0 - ( M n 2 * ) IMPOSED (Fe2*) POTENTIAL AT FILM (Mn2*) PREDICTED BY MODEL CONSIDERING ONLY ELECTRODE FILM CHEMICAL REACTIONS -\ (Mn2*) PREDICTED ASSUMING FeO SATURATION ON ELECTRODE TIP 100 200 300 400 500 T I M E ( S E C S . ) 600 700 800 Figure33 Comparison of Theoretical Mass Transfer Calculat ions to Slag Composition vs . Time P r o f i l e for DC+ve Melt #55 135 voltage would therefore be in the range of 10 to 20 mv. I f the knee 2+ m the curve for 0.5% CaO represents Fe saturation and hence unit . . 2+ a c t i v i t y , i t i s evident that the a c t i v i t y of Fe at the electrode t i p in DC + ve melting w i l l be very smal l . I t i s therefore suggested that 2+ the use of the bulk slag compositions of Fe to represent the oxidation potential for the mass transfer process i s probably reasonable. The e f fect of current density i s quite apparent on comparing the two r e s u l t s . The higher current density at the electrode t i p (by a factor of - 2) i s seen (from Figure 34) to cause a s l i g h t l y higher level of concentration p o l a r i z a t i o n at th i s interface for the anodic 25 react ion. Beynon has shown the same to be true for cathodic p o l a r i z -a t ion but also he has demonstrated that a higher l i m i t i n g current density i s required i n cathodic p o l a r i z a t i o n to produce saturat ion . The net 2+ resu l t of th i s on electrode pos i t ive was that more Fe was being produced at the electrode t i p than was being reduced by react ion with calcium metal produced at the ingot pool . Therefore we observe a net 2+ increase i n bulk Fe l e v e l . 2+ In the DC -ve run, we observe a lower level of Fe in the slag throughout the melt. A l s o , there i s tendency for th is level to 2+ decrease, although at a lower rate than the r i s e of Fe i n the reverse p o l a r i t y s i t u a t i o n . This points up both features of anodic and cathodic p o l a r i z a t i o n mentioned above. F i r s t l y , the current density at the ingot 2+ pool was lower leading to a reduced a c t i v i t y of Fe at the anodic surface. However, the higher current density at the electrode t i p , while increasing the extent of cathodic p o l a r i z a t i o n , did not have as 136 Figure 34 P o l a r i z a t i o n Character is t i cs of Iron in CaF2-CaO Melts.(25) (a) Anodic P o l a r i z a t i o n Curves, (b) Cathodic Po lar iza t ion Curves. Dotted l i n e -approximate l i n e for 20 % CaO. 137 pronounced an ef fect on the a c t i v i t y of Ca at the in ter face . Thus, the s i t u a t i o n was apparently not completely r e v e r s i b l e , providing some rat ionale for the differences i n behaviour between DC + ve and DC - ve melt ing. 2+ As a matter of i n t e r e s t , the e f fect of unit a c t i v i t y of Fe ( total saturation) on the electrode t i p on the oxidat ion of manganese was compared to the resul ts of the experiments i n which only local 2+ saturation (low a c t i v i t y ) of Fe was observed. The resu l t shown i n Figure 33 i s , as ant i c ipa ted , a greatly enhanced rate of manganese transfer from metal to s l a g . I t has been shown, then, that DC ESR processing does e f f e c t i v e l y block the electronegative interface from oxidat ion reactions and that losses of Mn from the metal may be s a t i s f a c t o r i l y explained by applying the mass transfer model at the e lec t ropos i t ive in ter face . 138 CHAPTER 5 DISCUSSION 5.1 Model Parameters 5.1.1 Mass Transfer Coef f ic ients . The accuracy of a mathematical mass transfer model depends c r i t -i c a l l y on the accuracy of the mass transfer coe f f i c i en ts used. In developing the mass transfer model for the ESR process we have t r i e d to use mass transfer c o e f f i c i e n t s which r e f l e c t , as c lose ly as poss ib le , the real physical processes a f fec t ing interphase transport . In p a r t i c u l a r , we have been concerned with the hydrodynamic regimes in the v i c i n i t y of each reaction s i t e , t h i s information being gathered from both theoret ical 5 considerations (e .g. flow on electrode t i p ) and d i r e c t observation CC (e .g. slag flow ). Consequently, we have been able to j u s t i f y use of the penetration theory o r , some modif icat ion of i t , in obtaining theoret-i c a l estimates of the mass transfer c o e f f i c i e n t s at each phase boundary. In a d d i t i o n , the mass transfer coe f f i c i en ts estimated at the ingot pool/s lag interface have been investigated experimentally (4 .2 .1) . The resul ts of th is work y ie lded lower values than given by the penetration theory. However, i t was f e l t that a major part of the 2+ discrepancy could be accounted for by reduction of Mn at the Armco iron electrode t i p . Calculat ion of th i s loss was accomplished by considering 2+ the electrode f i l m i n the model to be affected by both (Fe ) and 2+ (Mn ) potentials as given by Tables l i b for run # 61. The r e s u l t of 139 -8 - 3 -1 th i s c a l c u l a t i o n showed that approximately 6 x 10 mole cm sec were, l o s t to the f i l m (Figure 35). In comparison to the rate of r i s e of 2+ (Mn ) i n the slag for Run #60 (Figure 16), there was approximately 9% difference (— 6^ 1 0 8 ^ x 100). The experimentally calculated value 7.58(10 _ / ) _, was thereby corrected to 0.012 cm sec" which agrees c lose ly with the theoret ical value of 0.013 cm s e c - 1 . Another method of tes t ing the estimates of mass transfer coef f ic -ients i s to check the consistency of the heat transfer c o e f f i c i e n t , analogues. The fo l lowing r e l a t i o n (Reference 39, p. 671) i s given for the analogy between heat and mass transfer at low transport rates . h = kv P C n h r - ) 1 / 2 c a 1 c m " 2 s e c " l o C _ 1 ( 5 . 1 ) where h i s a heat transfer c o e f f i c i e n t , Cp i s heat capaci ty , a i s thermal d i f f u s i v i t y and p i s the density of the bulk phase. This expression i s held to be true in th i s case since contact times are short (<.5 sees), thereby reducing the ef fects of reactant depletion during unsteady state d i f f u s i o n . I f C p 3 8 - .2 cal g m " l o C _ 1 ; a h - .02 cm2 s e c " 1 ; D = 5(10" 5 )cm 2 sec _ 1 -3 -2 -1 -1 - I 3 8 PSLAG = 2.6 gm cm ; K = 10 cal cm °K sec then h = 10 k . The values of h<, and kMn2+ for the slag phase are given in Table XIV. 140 0-6 0 4 U J o ul0-3 Q_ 0 1 DATA POINTS A-(Fe2*) 0-(Mn24) IMPOSED POTENTIALS (Fe2*) (Mn2*) (Mn2*) LOST TO FILM 4£15-L 4 4(IG-5)wt.%sec-1 a t 10 20 30 TIME (SECS) Figure 35 Estimation of (Mn 2 + ) Lost by Reduction at Electrode Tip during Melt #61 141 TABLE XIV Heat Transfer Coeff ic ients i n Slag at S lag/Liquid Metal Boundaries (as derived from corresponding mass transfer coe f f i c i en ts ) Phase Boundary k M n 2 + hS electrode t i p / s l a g .018 .18 drop/slag .023 .23 ingot pool /s lag .013 .13 The value of h<- at the electrode t i p of .18 cal cm" sec" sec 1 °K _ 1 i s somewhat greater than the value of .07 cal cm" 2 s e c ~ 1 o K _ 1 24 assumed by Mendrykowski et a]_. i n t h e i r work on electrode temperature gradients i n ESR. Nevertheless, the ca lcu la t ion of superheating of the electrode f i l m (Appendix V.2) shows that very reasonable values were obtained by using the derived value of h s from Table XIV. In a d d i t i o n , the value of h $ for the pool /s lag interface (;13) agrees -2 1 -1 3F. with the value of .1 cal cnf sec" °K obtained by Jackson i n his study of thermal i n s t a b i l i t i e s i n ESR. In Appendix V.3 a ca l cu la t ion i s presented showing the temperatures of the f a l l i n g drop as i t passes through s l a g . The resul ts obtained indicate that each drop w i l l indeed reach the bulk temperature of the slag during drop f a l l . This had been assumed to happen by other workers but has not previously 142 been shown to be true . The heat transfer analogy has, therefore, provided another check on the v a l i d i t y of the theoret ica l mass transfer c o e f f i c i e n t s , with the added benefit of v e r i f y i n g some of the previous assumptions 23 56 current ly used in heat transfer modell ing. ' To further increase our confidence i n the values of these mass transfer coe f f i c i en ts by experimentation would be extremely d i f f i c u l t since th is would involve examination i n greater de ta i l of the f l u i d flow patterns. In view of the degree of confidence that we have developed to date i n the modelling of the mass transfer c o e f f i c i e n t s , i t i s doubtful whether such an invest igat ion would be worthwhile. I t i s in teres t ing to examine the effects of changing the mass transfer coe f f i c ients by substantial amounts on the predicted concentration changes. In 4 . 1 , an analysis of potential rate con-t r o l l i n g steps showed conclusively that transfer of species i n the slag phase had the predominant e f f ec t . I f , i n f a c t , metal phase transport of Mn i s unimportant, a large increase i n the mass transfer c o e f f i c i e n t should not a f fec t the predicted r e s u l t s . This i s exactly the case and i s demonstrated i n Figure 36 ( l ines B) where we have increased the mass transfer c o e f f i c i e n t of k^ n i n the metal at each 2+ s i t e by 100%. The to ta l change i n the predicted (Mn ) i s i n s i g n i f i c a n t for th i s case. However, on r a i s i n g the values of kM n2+ by 35%, we 2+ - • observe quite a substantial increase in (Mn ) as expected. While th i s change amounts to about 10% i t nevertheless passes through a 2+ large percentage of the error bars on the data points for (Mn ) . 0 6 SEE FIG 24(a) DESCRIPTIONS OF POINTS, LINES A-REGULAR MASS TRANSFER COEF T B-k.. INCREASED 1 0 0 % Mn C - k l j 2 + , k n 2 + INCREASED 3 5 % Mn re Z o > Figure 36 300 400 TIME (SEC) Effects of Varying Mass Transfer Coef f i c ients in Mass Transfer Model on Calculated Slag Composition vs Time P r o f i l e s of Run #42 (Mn*% OJ 144 I t i s important to note that the t o t a l change of 10% i s less than the change i n the mass transfer c o e f f i c i e n t s . The theoret ical mass transfer model i s seen, therefore, to be c r i t i c a l l y dependent on the values of the mass transfer c o e f f i c i e n t s . This observation then j u s t i f i e s the p r i o r preoccupation of es tabl i shing as much confidence as possible i n the modelled values of the mass transfer c o e f f i c i e n t s . As indicated i n 2 .3 .6 , i t i s l i k e l y that over-estimates have been used for drop related mass transfer c o e f f i c i e n t s . In spi te of these overestimates, the contr ibut ion of the drop to a l l o y loss remains very small (< 5%). I t i s probable therefore that a r e l a t i v e l y small uncertainty would be introduced by ignoring droplet mass transfer altogether. I t i s obvious from a consideration of the penetration model that the major sources of uncertainty i n estimating kMn2+ and kpg2+ are the d i f f u s i o n coe f f i c i en ts and the contact time. The slag flow 5 9 65 5 50 -1 pattern ' ' and v e l o c i t y ' of 7-10 cm sec have been f a i r l y well established i n ESR furnaces up to 20" <\>. The value of t , the contact time, i s therefore probably accurate to wi th in ± 10%. However, the 2+ 2+ d i f f u s i o n coe f f i c i en ts of Mn and Fe i n the CaF£ - CaO slag system -5 2 - 1 are v i r t u a l l y unknown and the value of 5(10 ) cm sec used is very approximate and the uncertainty conservatively estimated at ± 50%. F o r t u i t o u s l y , k i s a funct ion of t g and D raised to the - ^- and ^-92 powers respect ively. The percentage error i n k i s then St ( ^ x l 0 0 = 1 ( ^ X 100) +1 ( ^ x 100) = \ (50 + 10) = 30% 145 However, i t was shown e a r l i e r i n th i s section that th i s level of uncertainty appears to f a l l wi th in the error bounds of the experimental data points . Thus ± 30% i s an acceptable l i m i t of error i n k since a r e l a t i v e l y large uncertainty in k i s re f lec ted as a smaller e f fec t 2+ in the (Mn ) p r o f i l e s . There i s a p o s s i b i l i t y that the mass transfer coe f f i c i en ts may be influenced by spontaneous i n t e r f a c i a l motion generated by local changes i n i n t e r f a c i a l surface tension. I t i s well known that the surface ac t ive so lutes , oxygen and sulphur, cause dramatic reductions i n interphase surface tension even at r e l a t i v e l y low levels 93 and, on a loca l sca le , large surface v e l o c i t i e s may be generated. Although th i s e f fec t has not been d i r e c t l y observed i n slag/metal systems, the ESR system i s a potent ial candidate for such occurrences due to the low v i s c o s i t y of both slag and metal. Also the surface tension of pure iron i s known to be in excess of 1800 dym cm"1 dropping to leve ls of = 800 dyn cm"1 at several hundred ppm oxygen l e v e l . The i n t e r f a c i a l tension of i ron in a t y p i c a l ESR system has been estimated - l 1 8 at 800 dyn cm , the low value due probably to the high oxygen ion concentration in the s l a g . This would tend to reduce the potential ef fects of mass transfer involv ing oxygen but not necessari ly where desulphurization i s occurr ing. No means have yet been devised to i s o l a t e such an ef fect and therefore we can only ant ic ipate that any increase i n surface v e l o c i t y due to i n t e r f a c i a l turbulence would r e s u l t i n higher levels of mass t ransfer . 146 5.1.2 Molar Equil ibrium Constant In 2.3.2 i t was shown that fi, the molar equi l ibr ium constant was a function of (a) K, the equi l ibr ium constant for the exchange react ion (2 .1) ; (b) Yp e g and (c) Y M n Q- The level of uncertainty i n K can be wholly a t t r ibuted to errors i n estimating the temperature of react ion since the expression for K has achieved wide spread 34 acceptance. In Appendix V.2 and V . 3 , we have demonstrated that the temperatures of 1525°C and 1675°C chosen for the f i l m interface and the pool interface respect ively are very reasonable. The error bound i s estimated at ± 25°C. The r e l a t i v e errors i n Yp e g and Y ^ Q are d i f f i c u l t to assess. The value of Y p e 0 n a s been evaluated in the CaF,, - 20% CaO - FeO system ' from a good deal of experimental data and the value is reasonably well establ ished. The estimated values of Y^nO' however, were chosen as that set of values which best f i t a l l of the experimental runs of t h i s work (Figure 18) and although these are consistent with the published values for the system CaF 2 - CaO - MnO at lower levels of CaO (2.3.2) no cer ta in error bound can properly be associated with them. Also we have no idea of the in terac t ion effects of FeO and MnO in CaF 2 - CaO based slags but since they do appear to behave in a s i m i l a r 30 fashion (large pos i t ive deviations decreasing with CaO content ) one might ant ic ipate a cooperative e f f ec t . K YFeO Since fi a YMnO we can estimate the r e l a t i v e contributions of these three factors the error bound of fi. Taking logarithms, we have An fi = &n K + £n y. FeO - An Y, MnO An K = 4' - B' (see Equation 2.21) where A' = 2.303 x A B = 2.303 x B Therefore In fi = T - B' + An Y FeO - An Yi MnO By i m p l i c i t d i f f e r e n t i a t i o n we obta in , using the f i n i t e di f ference notation for d i f f e r e n t i a l s Sfi A' X T . 6 y FeO 6 yMnO _ = 6 T + 1 YFeO YMnO To estimate the upper error bound, we must use pos i t ive quant i t i e s . Since A = 6440 and i f T = 1525, ST = 25 then. contribute ± 9% to the to ta l uncertainty in fi. I t i s l i k e l y that the uncertainty in Y F E 0 A N D YMnQ i s a t l e a s t t n i s l a r 9 e a n d n e n c e we would ant ic ipate fi to be accurate to no less than ± - 25%. .09 M u l t i p l y i n g by 100, the error i n temperature i s seen to 148 However, we assume that our f i t t e d values for Y^nQ are reasonably v a l i d as they s a t i s f y a large number of experimental resul ts and are s e l f consistent . On r a i s i n g y ^ g by 25% i n the input data for modelling Run #42, a p r o f i l e nearly i d e n t i c a l to that of l i n e C in Figure 36 was obtained. As th is r e s u l t was again wi th in the experimental error l i m i t s , we can say that even th i s large uncertainty i n fi w i l l not change the in terpre ta t ion of the mass transfer modelling r e s u l t s . Recal l ing from Chapter I the previous attempts at modelling the ESR system from an equi l ibr ium standpoint, i t i s easier now to understand the d i f f i c u l t i e s encountered i n such an approach. For example, without the complete thermochemical information concerning the a c t i v i t i e s of a l l species in slag and metal one i s led to make a number of assumptions. This necessitates use of a somewhat a r b i t r a r y equi l ibr ium constant which can lead to puzzl ing conclusions. The best example of th i s i s shown by the use of d i f f e r e n t "equi l ibr ium temperatures" for 22 separate reactions occurring i n the same system. This problem i s not so great i n the mass transfer model since we define the interface and hence equi l ibr ium temperatures i n advance, thereby leaving only the a c t i v i t i e s to be sorted out. The greatest aid to resolving th i s problem i s to use as simple a system as possible to reduce the number of species involved. Of course, i n th is instance the prime object ive was to es tabl i sh that a l l o y losses i n general are mass transfer con-t r o l l e d . I t i s very evident from the error analysis of the equi l ibr ium constant, fi, that any large discrepancy i n a c t i v i t y coe f f i c i en ts w i l l 149 have a substantial e f fect on the predicted a l l o y losses . This s i t u a t i o n i s compounded by the addit ion of more reacting species. Nevertheless, i t i s f e l t that the mass transfer model can be used to resolve this problem partially, through i t e r a t i v e c a l c u l a t i o n of the equi l ibr ium constant as shown by the subroutine OMT (Appendix IV) . 5.1.3 Melt Rate Var ia t ion of the melt rate in the e lectros lag process i s known to produce a number of e f fects in the system as a whole. These effects are a l l re lated to the increase i n temperature as melt rate i n c r e a s e s . 3 8 ' ^ 1 The most pronounced dif ference at high melt rates i s an increase in the depth of the l i q u i d metal pool . The related mass transfer ef fects w i l l include an increased residence time of an element of metal i n the molten pool thereby increasing the closeness of approach to s lag/ metal equi l ibr ium. In a d d i t i o n , the temperature gradients in the pool are a l tered so that the metal flow may be d i f f e r e n t . Generally a higher temperature gradient resul ts i n high flow rates , but since the gradients are related to pool geometry, i t i s d i f f i c u l t to assess the exact e f fect of increasing the pool depth. In any event, we have shown metal phase transfer to be unimportant and therefore the major r e s u l t of high melt rate for th is part of the system w i l l be the increased residence time in the pool . The higher slag temperatures, which are a necessary prerequis i te of high melting rates , w i l l cause some change in the interface temperatures and consequently i n the equi l ibr ium constants. I t i s 150 antic ipated that the major e f fec t would occur at the ingot pool /s lag boundary since an increase i n melt rate would probably not ra ise the electrode f i l m temperature appreciably. Temperatures of the order of 1800°C are observed at high melt rates thus reducing the equi l ibr ium constant at the ingot pool surface s u b s t a n t i a l l y . This decrease would enhance the l i k e l i h o o d of reversion reactions and might, i n f a c t , reduce the overa l l a l l o y losses . Since higher melt rates are accomplished by passing more current through the system, i t i s probable that the loca l slag v e l o c i t y would be increased. This increased v e l o c i t y would reduce the time, t g , used i n the penetration theory model for mass transfer coe f f i c i en ts and resu l t i n an enhanced mass transfer rate of the slag phase species. Again, th i s e f fec t tends to bring a closer approach to equi l ibr ium for the ingot pool react ions . The ef fect of high melt rates on the electrode t i p i s a c t u a l l y the converse of the case for the pool . The f i l m thickness w i l l tend 5 to decrease thereby increasing the area to volume r a t i o . Also an element of molten metal w i l l spend less time on the f i l m . These effects appear to oppose one another and thus i t i s suggested that the major e f fect w i l l be due to increased bulk slag flow as above. Other minor changes such as the thinning of the slag skin w i l l occur at high melt rates as w e l l , but these a f fec t mass transfer i n a negl ib le fashion. In the r e l a t i v e l y small ESR furnace used throughout th is work, large var iat ions in melt rate are not observed (and, i n f a c t , were avoided - ± 10% cf. Tables 11-1X). However, on large commercial units var ia t ions of up to 50% are readi ly achieved and, indeed, are necessary for diverse operations such as hot topping and s t a r t i n g . The potential effects of melt rate on mass transfer are , therefore, not apparent i n th i s modelling work but should be readi ly observed on, say, a 20" $ furnace. 5.1.4 Rate of Slag Loss During the course of an ESR melt , slag i s continuously being l o s t to due formation of a s o l i d crust or s lag skin on the mold w a l l . In the small furnace used here i t was not necessary to replace th is s lag as stable melting conditions could be maintained throughout. I t was necessary, therefore, to include th i s loss (which amounts to up to 30%) i n the computer program (Appendix IV) . Some v a r i a t i o n in slag skin thickness i s observed from melt to melt and even over the course of a s ingle melt . To obtain the amount of slag at the s tar t of each experiment i t was necessary to back-calculate from the f i n a l s lag volume to the appropriate i n i t i a l time. This approach must be taken since a disproportionate and var iable amount of s lag i s l o s t during s t a r t i n g and switch over to AC from DC. The ef fect of slag loss may be observed i n the steady state runs #'s 53 and 40. The ingot concentration p r o f i l e s (Figure 20b and Figure 29b respect ively) show a non l inear decrease over the course of the run. However, the corresponding slag p r o f i l e s (Figure 20a, 2+ 29a) show a l inear rate of r i s e of Mn . This i s probably due to the diminishing volume of the slag compensating for the gradual 152 l e v e l l i n g o f f of mass transfer rate as a constant concentration of Mn i n the ingot i s approached. These experiments are rather short and an 2+ upward trend i n the Mn of the slag would be observed at longer times (cf . 5 .2 .1) . In commercial p r a c t i c e , fresh slag i s added at the same rate slag sk in i s formed. Therefore, i n t reat ing th i s s i t u a t i o n one would have to al low for slag freezing out with the current bulk levels of 2+ 2+ Mn , e t c . , but entering with no Mn , etc . 5.2 The1Mass Transfer Model 5.2.1 Model Results As was mentioned i n the Introduct ion, the ESR system i s i d e a l l y suited to mass transfer analysis pr imar i ly due to the progressive, stage-by-stage fashion in which melting and freezing occur. I t was also convenient that a number of important parameters such as electrode f i l m thickness , ingot pool volumes and slag flow were known or could be calculated with reasonable c e r t a i n t y . A l l of these factors have con-t r ibuted to the success of the proposed model i n predict ing the observed composition changes of Mn in both slag and ingot . (Figures 19-33) 2+ The use of the imposed Fe potent ial has allowed us to proceed i n the 2+ majority of instances where the sources and/or sinks of Fe were not accessible to accurate (or even inaccurate) measurement. I t has been stated repeatedly that the agreement between the predicted and experimental composition changes was good. However, no reference has been made of how one might expect an e q u i l i b r i u m -153 type model (such as discussed i n 1.3.1) to behave. Figures 37, 38 provide such a comparison. In Figure 37, the equi l ibr ium model predict ion for the 2+ (Mn ) i n the slag during normal or steady operation i s shown along with mass transfer p r e d i c t i o n . The mass transfer predic t ion was previously shown to agree well with the data (Figure 19) and thus i t i s evident that the equi l ibr ium predicted l i n e does not. Since the ingot pool i s the l a s t contact metal has with the s l a g , the temperature of the ingot pool /s lag interface was used in c a l c u l a t i o n of the equi l ibr ium tempera-ture i n th is model. Use of th is temperature, however, resul ts i n somewhat high pred ic t ion . By subs tant ia l ly r a i s i n g t h i s temperature i t would be possible to make the equi l ibr ium f i t the data, but we have previously shown (Appendix V . 2 , V.3) that the temperature of 1675°C i s e n t i r e l y reasonable. This problem, then, i s the root of the dilemma faced with the equi l ibr ium model and the cause of the f i c t i t i o u s and 21 22 sometimes anomalous "equi l ibr ium" temperatures. ' Although the steady state equi l ibr ium model may be adjusted to coincide with the data, i t i s doubtful that any amount of temperature adjustment would permit the unsteady state equi l ibr ium modelling attempt i n Figure 38 to achieve sa t i s fac tory agreement with the data. The shape of the curve i s not at a l l s i m i l a r to that of the mass transfer predic t ion and changing the equi l ibr ium temperature could not a l t e r the basic shape. Therefore, i t i s c lear that these unsteady state experiments are a much more r e a l i s t i c test of the mass transfer model than the steady state tests where the changes predicted by e i ther k ine t i c 0 - 6 154 2 0 0 4 0 0 6 0 0 8 0 0 T I M E (SEC) 1 0 0 0 1200 Figure 37 Comparison of Equil ibrium and Mass Transfer Model Calculat ions for Steady State Melt #27 200 300 400 TIME (SEC.) 500 600 700 Figure 38 Comparison of Equil ibrium and Mass Transfer Model Calculat ions for Unsteady State Melt #42 t 155 or equi l ibr ium model are b a s i c a l l y l i n e a r . (The transient in Figure 37 i s due to the s t a r t i n g point being d i f f e r e n t from the "equi l ibr ium" composition). The a b i l i t y of the mass transfer model to predict successful ly 2+ the (Mn ) leve ls i n the slag during DC processing (4 .6 .6 , Figures 32, 33) i s another example of the super ior i ty of the mass transfer model in handling the wide range of operating conditions that can occur in ESR 25 processing. These model resul ts also corroborate the resul ts of Beynon who proposed the concept of blocking of the electronegative interface to a l l o y losses i n DC ESR. He was able to i n f e r such behaviour from gross composition differences between ingot and electrode, but lacked any k i n e t i c information on a l l o y losses . The mass transfer model over-comes th i s d i f f i c u l t y . There can be no doubt that manganese losses i n the system studied are mass transfer c o n t r o l l e d . A l l the evidence points to a considerable departure from equi l ibr ium type behaviour, as was suspected at the outset . The equi l ibr ium constant of the manganese-iron exchange react ion i s quite low and one might well inquire of the s i t u a t i o n i n v o l v -i n g , say, aluminum oxidation where the equi l ibr ium constant i s of the order of 1 0 1 2 . 2+ In t h i s case, one or other of the reacting species (Fe )^ or [Al].j would tend to zero. Since we have shown that s lag phase transport 2+ i s predominant, i t would be reasonable to assume that (Fe ). would 2+ tend to zero and the dr iv ing force for mass transfer of (Fe ) would be increased. The overa l l ra te , of course, would be strongly affected by 156 the a c t i v i t y of alumina in the s l a g . In spi te of the increased d r i v i n g force , i t i s l i k e l y that mass transfer control would s t i l l l i m i t the 98 approach to equi l ibr ium. In f a c t , Boucher has shown that substantial losses of Al do occur even i n high A i 2 o 3 s lags . The true nature of the steady state melting conditions w i l l be discussed s h o r t l y , but b a s i c a l l y the reason equi l ibr ium cannot be achieved i s because of the temperature differences at the two major reaction interfaces ( f i l m , pool ) . In 21 a d d i t i o n , i t appears from the work of Knights and Perkins that i t was necessary to adopt isothermal temperatures of the order of 1900°C to explain losses of more react ive a l l o y elements. This i s c lear evidence of mass transfer control as i l l u s t r a t e d e a r l i e r i n th i s sec t ion . 5.2.2 S o l i d i f i c a t i o n Effects We have t o t a l l y excluded any effects of s o l i d i f i c a t i o n phenomena from our model. The only e f fec t of s igni f i cance to mass trans-fer i s , of course, macrosegregation which i s a function of s o l i d i f i c a t i o n ra te , s t i r r i n g i n the l i q u i d metal and the solute under considerat ion. In f a c t , the parameter in which we are interested i s the d i s t r i b u t i o n c o e f f i c i e n t kg ( ra t io of concentrations of solute i n s o l i d and l i q u i d at equil ibrium) or more properly , the e f fec t ive d i s t r i b u t i o n c o e f f i c i e n t , k g , where k e = CS 1 C 0 being the concentration of s o l i d at any time formed from a l i q u i d of average composition, CQ . k g i s a function of the rate of advancement of the s o l i d i f i c a t i o n f r o n t , the extent of f l u i d flow and value of k n 157 95 for each p a r t i c u l a r solute . Chalmers goes through a treatment of these factors on the value of k g , but his arguments are based on the assumption of a planar in ter face . The ESR s o l i d i f i c a t i o n front i s d e f i n i t e l y dendr i t i c and thus, we cannot use his arguments. For the s o l i d i f i c a t i o n of ESR ingots , 14 35 56 i t has been shown ' ' that k g i s apparently independent of the solute for the purposes of predict ing macro segregation, for solutes with kg values ranging from .08 for sulphur, to .7 for t i n . The value 56 of k g was invar iab ly found to be i n the range of .8 . I t i s considered that k e l eve ls of th i s order r e s u l t in very low degrees of macro-segregation i n ESR and, thus, our assumptions in regards to ignoring such effects remain v a l i d . 5.3 Model Predict ions We have so far only al luded to the prospects of improving our knowledge of the real nature of the ESR process through use of the predic t ive aspects of the mass transfer model. I t was pointed out i n the Introduction and i n the discussion of previous modelling e f for ts that one of the chief drawbacks of the equi l ibr ium and s ingle stage reactor models was t h e i r f a i l u r e to reveal any new information about the process. Since th i s a b i l i t y i s the essence of successful mathematical modell ing, i t also i s one of the best c r i t e r i a for evaluating the worth of the modelling exercise . 158 5.3.1 Relat ive Contributions of Reaction Interfaces Accompanying each- model c a l c u l a t i o n in Chapter 4 (Figures 19-31) the contr ibut ion of each reaction s i t e to the overa l l mass transfer e f fec t was p l o t t e d . Several important trends were observed from the addi t ional information predicted by the model. F i r s t l y , there was a cons is tent ly low and almost negligible contribution^ i%) of the drops to the to ta l a l l o y loss and t h i s , i n spi te of our apparent overestimation of the mass transfer coe f f i c i en ts of slag and metal species (2 .3 .6) . Previous ly , droplets had been the subject of some controversy; on the one hand, the to ta l amount of a l l o y loss being a t t r ibuted to mass transfer from the drop and on the other , only a small f r a c t i o n of the to ta l being considered due to droplet related l o s s e s . T h e l a t t e r view i s in accordance with current f indings and i s generally accepted although one might express some doubt in the arguments and techniques previously used to a r r i v e at t h i s , c o n c l u s i o n . The reason the drop contributes so l i t t l e i s , of course, the extremely short residence time (^ .1 sec) i n the s lag . (Table 1) Droplets are therefore never l i k e l y to be of much s igni f i cance from the mass transfer point of view, even when mul t ip le droplets are f a l l i n g from large electrodes. From the heat transfer standpoint, however, we now know that droplets leaving the electrode t i p with l i t t l e superheat do heat up almost to the bulk slag temperature. (Appendix I I I ) . This con-t r i b u t i o n to the sensible heat of the molten ingot pool is of great s igni f i cance i n a f fec t ing the geometry of the s o l i d i f i c a t i o n front 56 and hence the overa l l ingot q u a l i t y . 159 The second important observation made with respect to the contributions of the ind iv idua l reaction s i tes was the way in which the 2+ electrode f i l m / s l a g and ingot pool/s lag s i tes reacted to changes i n Fe in the s l a g . From run to run the electrode f i l m mass transfer rate appeared to be very nearly constant. Only when there was a great excess 2+ of Fe , as i n the melt in a i r (Figure 31), did th is rate change s i g n i f i c a n t l y . Conversely, i t was the ingot pool s i t e which reacted to 2+ the stimulus of the Fe var ia t ions and hence the s i t e which determined the i n d i v i d u a l character of each run. The reason for th is behaviour i s s i m i l a r to that given above for the negl ib le droplet e f f e c t s . The residence time of an element of metal on the electrode t i p was estimated to be i n the order of 1 sec compared to that i n the pool of about 80 sec (Table I ) . However, the short residence time on the electrode t i p i s p a r t i a l l y o f f se t by the extremely high area/volume r a t i o of the l i q u i d metal (- 100 cm - 1 ) and therefore the overa l l a l l o y element loss i s s i g n i f i c a n t . The long residence time of a metal element i n the pool , on the other hand i s tempered somewhat by the much lower area/volume r a t i o (- .6 cm - 1 ) i n t h i s instance. The primary e f f e c t , nonetheless, s t i l l would seem to the residence time at each s i t e since th i s ul t imately controls the extent of approach to slag/metal equi l ibr ium. 5.3.2 Nature of Steady State Mass Transfer Probably the most consis tent ly misinterpreted phenomenon related to slag/metal reactions i n ESR processing i s the steady state operation in which there would appear to be no slag/metal reaction occurr ing. 160 Figure 39 i s the resu l t of an extrapolation of the model for 2+ 2+ Run #27 (Figure 19) at constant Fe . I t i s seen that the Mn level in the slag approaches a constant value a f ter some 1200 addit ional sees. Although th i s constant level has been interpreted previously ' as equi l ibr ium behaviour, i t i s obvious from the net r e l a t i v e c o n t r i -butions of the f i l m and the pool s i tes that chemical react ion and mass transfer are occurring at each s i t e . Note again that Mn oxidation at the f i l m i s occurring at the same rate as observed in pr ior runs. The more in teres t ing observation, however, i s that the rate of Mn reduction at the higher temperature pool s i t e due to the reversion reaction i s very nearly ident i ca l to the oxidat ion rate . Thus, the net rate of Mn loss i s v i r t u a l l y zero. I t i s also of in teres t to r e c a l l Figure 37, in which an equi l ibr ium model f o r Run #27 was compared to the mass transfer r e s u l t s . 2+ 2+ The constant level of Mn predicted for constant Fe of .5 wt % i n Figure 39 i s .05% higher than the equi l ibr ium model c a l c u l a t i o n . This observation confirms the conclusion that steady state i s not an 2+ equi l ibr ium s i t u a t i o n , the higher level of Mn in the steady state being due to the balance of mass transfer ef fects at the d i f f e r e n t temperatures of the f i l m and pool react ion in ter faces . Reversion reactions may, i n f a c t , produce undesirable effects on the ingot q u a l i t y . In Figure A5.1 (Appendix V . l ) , i t is c lear that , should there be any s i l i c a i n the s l a g ; the S i / S i ^ l i n e would rotate ant ic lockwise . In a d d i t i o n , one might ant ic ipate reduction of s i l i c a in the slag by Mn i n the metal. This e f fect would increase with temperature as the Mn/MnO and S i / S i 0 9 l ines diverge r with increasing Figure 39 Extrapolat ion of Theoretical Mass Transfer Predictions f o r Steady State Melt #27 to 2400 Seconds at Constant F e 2 + 162 temperature. As a r e s u l t of the reversion reaction putting manganese back into the ingot pool , i t i s seen that s i l i c o n may also revert to the metal in the pool by reduction with manganese. Since s i l i c o n i s an inc lus ion forming element and since inclusions are known to prec ip i ta te 94 101 on freezing ' of the metal , i t would be possible to introduce inclusions into the ESR ingot . These inclusions may possibly be of a type not found i n the electrode. i t i s known that the inc lus ion content of ESR ingots i s affected 96 by the slag materials used. In th i s example we have necessari ly dealt with only a simple case of the possible reactions that might occur. In the multicomponent slags used i n commercial ESR (e .g . CaF 2 - A 1 2 0 3 - CaO - MgO - S i 0 2 - T i 0 2 ) , the 1ikelihood of such reactions leading to inc lus ion formation i s probably increased. Inclusions present i n the electrode material may thereby be replaced i n the ingot by inclusions bearing elements previously present only in the s l a g . 5.3.3 Extension of the Model to Combined Manganese and Sulphur Transfer To date, we have shown that manganese losses are control led by mass t ransfer effects and that the model can provide addit ional useful information about the k inet i cs of the ESR process. This section w i l l deal with the extension of the model to combined manganese and sulphur t rans fer . This system has a much lower equi l ibr ium constant (Appendix A.111.2) . and has the addit ional benefit again of known thermochemical data. We have previously evaluated k^2- (4 .2 .2) , the mass transfer c o e f f i c i e n t of (S ) i n the slag at the pool s i t e by 163 both theoret ica l and experimental methods with sa t i s fac tory r e s u l t s . Other values for the mass transfer c o e f f i c i e n t s may be determined as in Chapter 2. 14 Cooper et al_. have published a very complete set of data ( including a l l the parameters necessary for our mass transfer model) for the melting of AISI 1.117 through a CaF 2 - 20% CaO s l a g . This data i s presented in Table XV. In addi t ion to the operating data, complete slag and ingot analyses were given, along with experimentally obtained assessments of the r e l a t i v e rates of desulphurization at the ingot pool , electrode t i p and drop s i t e s . I t w i l l therefore be a worthwhile exercise to apply the mass transfer model to th is system and compare the theoret ica l predict ions to Cooper's data. The formulation of the mathematical mass t ransfer re lat ionships for the desulphurization reaction Mn + S + (S 2~) + (Mn 2 + ) i s given i n Appendix A . I I I . 2 . Although Cooper et a l _ . 1 4 have only con-97 sidered the behaviour of S i n t h e i r paper, Olsen has shown that manganese and sulphur losses are related and therefore desulphurization should be considered as a cooperative phenomenon. Also in Appendix A . I I I . 2 , 2- 2+ i t has been shown that the desulphurization constant of (S )(Mn ) i s 2- 2+ 2+ greater than that for the couple (S )(Fe ) and thus the Fe level does not enter into th is desulphurization react ion . Cooper et al_. used an ESR furnace twice the diameter of the machine considered i n a l l the previous c a l c u l a t i o n s . Therefore, in 164 addit ion to the change i n react ion species we must introduce some sca le -up c r i t e r i a . Several of the model parameters may be evaluated d i r e c t l y . We have no reason to assume any d i f f e r e n t temperatures of the reaction interfaces and thus we shal l use 1525°C and 1675°C for the f i l m and pool respect ively as before. The electrode diameter of 3" i s s t i l l small enough to form only s ingle drops. I t i s well known that the drop 2 38 s ize does not vary appreciably with electrode diameter in ESR ' and the increase i n melt rate observed i s due e n t i r e l y to the increase i n the number of drops formed per uni t time. Therefore we can consider that the drop f a l l time (for s i m i l a r height of s lag bath) and area/volume r a t i o remain the same as before. The area/volume r a t i o at the pool i s , d i f f e r e n t , however, but may be e a s i l y calculated from Cooper's data on pool volumes. Calculat ion of the f i l m volume and areas as in 2.3.4 and 2.4 shows that the area/volume r a t i o remains approximately constant as the increase i n volume due to a th icker f i l m compensates for the increase i n area. This e f fect i s a c t u a l l y maintained to reasonably large electrode sizes as long as the electrode exhibi ts a conical t i p . The rate of s lag loss i s e a s i l y calculated from Appendix 1.3 since the slag skin thickness remains constant (- 1mm) with mold diameter. The number of drops/sec i s found as before by d i v i d i n g the melt rate by average drop weight. This information is summarized i n Table XV. The remainder of the information needed now i s the mass transfer c o e f f i c i e n t of each species at the three react ion in ter faces . A t o t a l of twelve parameters are required. The information needed for 165 the evaluation of the mass transfer c o e f f i c i e n t s i s the d i f f u s i o n co-e f f i c i e n t s of sulphur, manganese in both metal and slag phases and the contact times for the penetration theory model. We have already -4 estimated the manganese d i f f u s i o n c o e f f i c i e n t s (D M n p g * 10 , DMn2+ S L A G = 5(10" 5)) and the d i f f u s i o n c o e f f i c i e n t s of S 2 " (D 2- = 10" 4 ) , -4 2 -1 The value of D<- p g given by Darken and Gurry i s 1.2(10 )cm sec . We are l e f t , therefore, to estimate the values of t . Consider f i r s t the droplet mass transfer coe f f i c i en ts for S, S (kfvin* k M n 2 + a r e 9 i v e n i n Table I for the drop and remain the same). Using Equation 2.36 for k<. and 1.2(10~ 4) for D<- F g , we have 4 x 1.2(10" 4) x 40 x (1+1.07) 1/2 kS,DR0P = ( ) = ' 1 1 3 S i m i l a r l y using 2.43 for k<,2-61(10" 4) 1/2 k-2- = .69 ( ) = .032 5 TT .88 The rest of the mass t ransfer coe f f i c i en ts are estimated from the penetration theory as before where D X 1/2 k x = 2 ( — ) The problem, as mentioned above, i s the evaluation of t . I t i s assumed that the slag v e l o c i t y remains the same at 10 cm s e c - 1 since th is value has appeared i n the l i t e r a t u r e for furnaces up to at least 12" 50 in diameter. This resu l t s i m p l i f i e s the problem considerably and we 166 can, i n f a c t , derive a general formula for as a function of diameter of electrode or mold diameter, depending on the choice of reaction s i t e . Since t i s the time taken for an element of slag to pass over the metal surface and i f R i s the radius of electrode or mold and u i s the slag ve loc i ty R t . e Since u i s constant Subst i tut ing into the expression for k^, we have D x 1/2 R, V 2 or ( k x ) 2 = ( k x ) 1 (5.2) Size ef fects may be corre la ted , then, by the square root of the r a t i o of the r a d i i of the electrode and mold. Thus, for the manganese transfer 1/2 c o e f f i c i e n t s , we simply mul t ip ly by (.5) ' or .71 since the electrode 14 and mold are exactly twice as big in Cooper's equipment. I t i s found, however, on using the expression for t (2.29), that the calculated value of .95 sec i s ident i ca l to the case of the 1 1/2" electrode. This 167 i s due to the higher melt rate in the larger system. p j ^ i s the same, therefore, as before. The sulphur mass transfer coe f f i c i en ts can be obtained d i r e c t l y from the penetration theory using the appropriate values for t consistent with the reaction interface under considerat ion. A l l of the above information i s summarized in Table XV. The l a s t b i t of information needed i s the a c t i v i t y c o e f f i c i e n t 2- 17 of S in CaF2"20%CaO. From the data of Hawkins et a l_ . , the value of YCaS o f 2 , 4 i s o b t a i n e d f o r t n e l e v e l o f s 2 ~ o f ~ - 5 % i n C a F 2 " 2 0 % CaO s l a g , y ^ g values were found as previously described. The computer ca lculat ions were carr ied out using a program s i m i l a r to that given i n Appendix IV and the resul ts were compared to the experimental data of Cooper e j t a J L 1 4 for sulphur loss (There is no data for Mn) (Figure 40). I t i s evident that there i s very reasonable agreement here between the experimental and calculated sulphur levels i n both ingot and s lag . The predicted resul ts for the ingot do, however, appear to be s l i g h t l y high. The reason for th i s i s probably due to the fact that the experiments were carr ied out open to the a i r and a cer ta in amount of sulphur was l o s t to the a i r . The model does not take th i s into account. The predict ions are , nevertheless, consisted with melting under iner t atmosphere as one would ant ic ipate a lesser degree of desulphurization than the system is open to a i r . 14 The main purpose of the experimental work, as pointed out i n the in t roduct ion , was to invest igate the nature of the reaction s i tes i n ESR. Consequently, rates of sulphur loss at the electrode f i l m and at the ingot pool surface were measured experimentally using TABLE XV Summary of Data for Modelling Combined Mn, S Losses Parameter Film Pool Drop kMn.(cm sec" 1 ) .012 .011 .112 k M n 2 + .013 .009 .023 .020 .014 .113 .012 .013 .032 V(cm3) .53 800 .35 A/V(cm - 1 ) 100 •19 24 Volume of Slag (cm3) -Rate of loss 1456 - , -.2(crri sec ) melt Rate (gm sec~l) 13.7 Slag Type CaF2-20%Ca0 Electrode cb (inches) 3 Electrode Analysis 1.34%Mn .154%S Mold <b (inches) 6 1 M CO UJ O or LU OL t-X e> LU 005 y 0 04| g 0 031-0 POOL A -INGOT POL SULPHUR O - SLAG SULPHR MASS TRANSFER MODEL PREDITION EQUIL. MODEL PREDICTION CONTRIBUTIONS OF EACH SITE 100 2 0 0 3 0 0 400 TIME (SEC.) 500 Figure 40 Comparison of Theoretical Mass Transfer Model Calculat ions to Ingot and Slag Sulphur vs . Time P r o f i l e s Obtained by Cooper et a l . U 4 ) t o 170 a technique involving catching the f a l l i n g drops. The rate of desulphurization found by Cooper ejt al_. at the -2 -1 electrode t i p was 1.8 (10 ) gms sec . From the slope of the l i n e for the f i l m contr ibut ion to overa l l mass t rans fer , we f i n d that the mass -2 -1 transfer model predicts a rate of 1.9 (10 ) gms sec . There i s very good agreement between the two values. However, the rate of loss of the pool s i t e , as calculated from the experimental evidence i s -1.1 (10 ) gm sec 1 whereas the mass transfer model predicts a reversion reaction with -3 -1 a rate of 2.1 (10 ) gm sec . This difference, by a factor of two may be ascribed to the loss of sulphur to the atmosphere i n the exper i -ments. The assessment of reaction with a i r was considered by Cooper to be extremely inaccurate and thus we cannot make any quant i tat ive statements regarding the differences i n the reversion rates . In both cases the contr ibut ion of the drops to the overa l l e f fect was found to be t r i v i a l . The resul ts of th i s modelling exercise and the apparent success of the model i n interpret ing the nature of the mass transfer phenomena at the various reaction s i tes i s very encouraging. The impact of these resul ts i s even greater when one considers that we have changed not only the chemical reaction involved, but also the s ize of the furnace. Since resul ts for desulphurization of a s i m i l a r steel are ava i lab le for a very 99 small ESR u n i t , namely 16 mm <b, i t i s extremely in teres t ing to extend our model t o , say, a 20" cb unit (510 mm) and then compare the extent of desulphurization over th i s wide range of ESR processing equipment. 171 5.3.4 Scale-Up Predict ions The choice of a 510 mm cb ESR unit was based on the populari ty of th is s ize of commercial uni t and therefore the a v a i l a b i l i t y of data such as melt rates , pool volumes, etc . The fo l lowing procedures were used to estimate some of unknown physical parameters. The f i l l r a t i o of larger ESR furnaces i s of the order of .9 . The electrode diameter was therefore taken to be 460 mm. This s ize of electrode i s observed to melt by forming a l i q u i d metal f i l m approaching 1000 microns t h i c k . Although i t retains the conical shape, the ef fect i s less pronounced and metal i s observed to leave the electrode by formation of droplets at a number of points . Calculat ion of the area/ volume r a t i o gave a value s i m i l a r to that used in the previous examples. Examination of an electrode t i p of th i s s ize shows that the drop s i t e density i s about one for every 45 - 50 sq. cm of electrode surface area. For the electrode under consideration (Area ~ 1660 cm ), there would then be about 35 drop s i t e s . Since the drops are known to be approx-2 30 imately the same s ize regardless of electrode t i p dimensions, ' we would expect about 50 drops/sec for a melt rate of 150 gm/sec (1200 l b s / h r ) . The residence time of any element of metal on the electrode t i p was then estimated by d i v i d i n g the number of drop s i tes into the number of drop per sec. This procecure resulted i n a residence time of about 1.4 sec. op The ingot pool was assumed to be hemispherical and the 3 resultant volume of molten metal was approximately 35,400 cm . The area to volume r a t i o was therefore .06 c m - 1 . The quantity of slag present was calculated by assuming the typ ica l slag bath depth of 5 172 3 inches and the r e s u l t was 25,950 cm . A l s o , since slag skin thickness is r e l a t i v e l y constant for a p a r t i c u l a r slag regardless of furnace 3 -1 s i z e , we calculated a rate of slag loss of about .7 cm sec . The mass transfer c o e f f i c i e n t s in the slag phase were found with the a id of r e l a t i o n (5 .2) , where ( k x ) 2 = ( k x ) 1 x ( R 1 / R 2 ) 1 / 2 For R.j = 6 , R 2 = 20, the conversion factor for ^ 2 + , k<-2- at the ingot pool /s lag boundary was .54 and for R^  = 3 and R 2 = 18 at the electrode t i p , the slag t ransfer c o e f f i c i e n t s conversion factor was .41. The drop mass transfer c o e f f i c i e n t s remained the same as before and remaining metal phase c o e f f i c i e n t s were calculated from the local residence times. We assumed here that the flow regimes i n the slag and metal d id not change subs tant ia l ly with the increase i n s i z e . A l l of the above data necessary for the scale-up modelling i s summarized in Table XVI. The resul ts of the mass transfer model ca lculat ions are pre-sented i n Figure 41, being plotted as ingot sulphur by electrode sulphur level vs time. There i s a very d e f i n i t e trend from the 16 mm to the 510 mm diameter furnace and th i s trend i s an enhancement of desulphur-i z a t i o n with increasing s i z e . This r e s u l t cannot be a t t r ibuted to increased mass transfer rates since a comparison of Tables I and XVI shows that the mass transfer c o e f f i c i e n t s are generally lower for the larger u n i t s . The decrease i n area/volume r a t i o at the ingot pool /s lag boundary would also appear to .contradict the observed trend. However, TABLE XVI Summary of Data Used i n Scale-up Modelling of 510 mm <b ESR Furnace Parameter Film Pool Drop kMn , (cm sec" ) .005 .006 .112 k M n 2 + .010 .007. .023 k s .008 .008 .113 k s 2 - .010 .010 .032 V (cm3) 19.1 35400 .35 A/V (cm-l) 100 .06 24 Volume of Slag -Rate of loss 25950 cm3 - .7 cm3 sec" ' melt rate (gm s e c _ l ) 150 Slag Type CaF2-20%CaO Electrode diameter (mm) 460 Electrode Analysis 1.34%Mn .154%S Mold diameter (mm) 510 T .99 0 4 • -DATA"* 'FOR 16mm 0 E.S.R. MOLD O -DATA 1 4 FOR 150mm.0 E.S.R. MOLD ASSUMED LINE FOR I6mmj2r MODEL PREDICTION FROM FIG. 4 0 MODEL PREDICTION FOR 510 mm. E.S.R r l6mm.jeT MOLD 03 o Too. S 3 0-2 o 150mm.0 MOLD O i k 0 o o o o *-5l0mmjer MOLD 0 Figure 41 100 200 300 400 500 600 TIME (SEC) Effect of ESR Furnace Size on Desulphurization Character is t i cs of the ESR Process -F5. 175 on r e c a l l i n g the reaction occurring at th i s p a r t i c u l a r s i t e , we f i n d that i t i s most l i k e l y to be a reversion react ion (5 .3 .3) . Therefore, the e f fec t of decreasing the area/volume r a t i o i n the mass transfer equation i s to l i m i t the ef fect of the reversion reac t ion . I t was also pointed out e a r l i e r i n th i s section that the residence time of an element of metal on the electrode t i p was about one and hal f times greater i n the largest uni t compared to the 150 mm (6") u n i t . This residence time led to an increase i n the amount of desulphurizat ion occurring at th i s stage. Thus, t h i s argument and the one preceding at least par t ly explain the observed r e s u l t s . The model appears to be adaptable to use i n scale up predic-t i o n s . The desulphurization levels of 81% (.134 - .025%) for the 16 mm <J> mold and 89% (.154 - .034%) for the 150 mm 4> mold are consistent with unpublished work at U.B.C. showing that 84% (.256 - .040%) i s obtained i n a 76 mm furnace. However, a l l of the above work was done with very high lime slags which have an unusually high capacity for sulphur. Nevertheless, removal e f f i c i e n c i e s of 40 - 70% are reported in remelting steels of low i n i t i a l sulphur levels (.007% - .037% respect ively) i n large ESR furnaces (1480 x 640 mm = 40 tons) using complex,multicomponent s l a g s . ^ This level of desulphurization observed i n large units tends to confirm our model ca lculat ions for the 510 mm mold. An addit ional aspect of e lectros lag remelting i n reasonably large units (>150 mm <f> x 1 metre high) i s the change i n pool volume during the course of a melt. A t y p i c a l example of th is i s shown in 176 Figure 42. The increase i n pool volume as the ingot grows i s caused by the greater separation of the heat source (slag bath) and the predominant heat sink which i s the base p la te . The mold w a l l s , however, exert a stronger e f fec t as the ingot grows thereby increasing the rad ia l heat flow component. This pool volume growth i n large ingot manufacture might be expected to exert some influence on the mass transfer charac ter i s t i c s of the ingot pool /s lag interface since the residence time of an element of metal in the pool i s d i r e c t l y proportional to the pool volume and inversely proportional to the melt rate (at steady s ta te ) . General ly , however, commercial ESR units are operated on the basis of constant melt ra te . Therefore, the residence time w i l l progressively increase with ingot height, thereby tending to enhance a l l o y losses (assuming no reversion reac t ion) . On the other hand, the area/volume r a t i o w i l l be decreased at larger pool volumes o f f s e t t i n g somewhat the increase i n residence t ime. Thus, at constant mel t ,rate, one might anticipate l i t t l e or no e f fect of the increasing pool volume on the pattern of a l l o y losses during ingot production. That th i s i s so i s demonstrated i n Figure 43 i n which i s observed the f a m i l i a r t ransient culminating i n steady state behaviour for the ax ia l d i s t r i b u t i o n of carbon i n a 14 ton ingot . Although the f i n a l , steady state level of carbon in th i s ingot was influenced by the effects of increasing pool volume, the overa l l behaviour i s consistent with our p r i o r observations i n other systems. 177 DIAMETER Figure 42 Pool Geometry as a Function of Ingot Height in a Large ESR Furnace (102) 178 I t i s in teres t ing to speculate b r i e f l y on some other aspects of large ingot manufacture. The melting program generally followed i n very large ESR operations (>100 tons) i s to melt i n i t i a l l y at an abnormally high rate to b u i l d up a deep l i q u i d metal pool . Since such ingots are often not much t a l l e r than twice the mold diameter, th i s l i q u i d metal pool may comprise 50% or more of the ingot up to the halfway point . High melt rates as mentioned previously (5.1.3) may r e s u l t i n enhanced slag mass transfer c o e f f i c i e n t s due t o , say, an increase i n the slag v e l o c i t y . I t i s l i k e l y , however, that the increased temperature at the ingot pool /s lag i n t e r f a c e , r e s u l t i n g from the higher bulk s lag temperatures necessary to sustain high melt ra tes , would have the major e f fec t by increasing the l i k e l i h o o d of reversion react ions. This , i n t u r n , w i l l r e s u l t i n more reduction of potential inc lus ion forming elements from the slag into the ingot pool , th i s e f fec t again being assisted by the higher temperatures (5 .3 .2) . Lower a l l o y losses may be achieved but at the expense of an increase in inc lus ion content. The process of completing the upper portion of the ingot may be considered simply as a hot topping operation wherein the power is gradually reduced, cooling the system slowly and maintaining good ingot s t ructure . At these lower melt ra tes , however, one can ant ic ipate higher a l l o y losses , but lower inc lus ion levels by using the converse of the above arguments. Some evidence of the v a l i d i t y of th i s speculation i s 104 to be found i n the l i t e r a t u r e . Myzetsky et al_. show that sulphur 179 Figure 43 Carbon D i s t r i b u t i o n along the Axis of an ESR Ingot of High Carbon Chromium S t e e l . 0.89% - Mean C Content of Forged Electrode. 180 losses to the slag i n a 40 ton ESR unit are decreased at high melt rates in accordance with our predict ions with respect to higher temperatures and enhanced slag mass transfer c o e f f i c i e n t s . In a d d i t i o n , they have shown that the rate of alumina loss from the slag i s s u b s t a n t i a l l y decreased at lower melting rates . The above discussion deals with only a few examples of the usefulness i n applying the present mass transfer model. The fact that we can see the r e l a t i o n between model parameters and known ESR phenomena shows very d i s t i n c t l y the benefits of mass transfer a n a l y s i s . Important also has been the constant l i n k with the heat transfer ana lys i s . For example our analysis of the mass transfer phenomena would have been much more d i f f i c u l t and, l i k e l y impossible, without s p e c i f i c knowledge of interface temperatures and pool volumes. 5.3.5 Control of A l l o y Losses I t i s pertinent at th is time, then, to attempt to es tabl i sh some guidelines for control of a l l o y losses i n ESR processing. A number of p r a c t i c a l suggestions have already been made and a good many of these are in day to day use as mentioned in 1.2. These ideas include such practises as (a) melting under argon instead of a i r , (b) use of balanced slags and (c) surface preparation of the electrode material p r i o r to use. Melting under argon, however, would appear to decrease s l i g h t l y the e f f i c i e n c y of the desulphurization reaction but th i s problem might be overcome by increased lime content of the s l a g . 181 Corrective additions of material into the slag to ra ise the a c t i v i t y and suppress the reaction could . indeed backf i re , as we have shown, by r e s u l t i n g i n a increased tendency for reduction of inc lus ion forming elements into the ingot pool . A few d i f f e r e n t ideas a r i s i n g out of th i s work are given below. F i r s t l y , we know that slag composition c lose ly r e f l e c t s the extent of the a l l o y losses . Sampling the slag i s a much simpler operation than sampling the metal. Therefore, provided some rapid analysis technique were avai lab le i t should be possible to monitor the a l l o y losses as they are occurring and perhaps make correct ive addi t ions . In any event, once one run was made a correct ion could be made to the electrode composition to produce an ingot of required s p e c i f i c a t i o n s . This method would appear to be preferable to correct ing the s l a g . We have shown (4.1) that mass transfer i s control led by the slag transport steps. Since these mass transfer coe f f i c i en ts are functions of the slag v e l o c i t y , i t seems reasonable that any measures that we take to reduce the flow rate of s lag would be benef ic ia l from a mass transfer point of view. The most prac t i ca l method of doing th is would appear to be the use of as large an electrode as poss ible . This would have the benef ic ia l e f fect of reducing the Lorentz-force induced v e l o c i t y since th i s v e l o c i t y i s inversely proportional to the electrode radius . The use of external controls on the slag v e l o c i t y has not yet been shown to produce a benef ic ia l e f f e c t . A large electrode/mold r a t i o w i l l have other secondary benefits as w e l l . These effects include removing a large amount of s l a g / a i r contact surface and improving the 56 geometry of the ingot pool . 182 F i n a l l y , i t appears from our resul ts and from the published data that smaller furnaces seem to produce smaller changes in a l l o y content from electrode to ingot . However, accompanying th i s trend may be the increased number of inclusions in the f i n a l ingot due to reduction react ions . In large furnaces, therefore, i t i s suggested again that electrodes be made up with higher than s p e c i f i c a t i o n levels of essential a l l o y elements to avoid substandard ingot q u a l i t y . Since i t seems that large furnaces are less prone to producing high leve ls of inclusions due to the cooler slag bath temperatures, there i s a f i n i t e probab i l i ty of producing superior ingots in large ESR i n s t a l l a t i o n s . A l s o , i f low sulphur levels are important, the use of large units would be h e l p f u l . 5.4 Electrochemical Phenomena We have establ ished, i n our discussions i n Chapter 4, that r e c t i f i c a t i o n of the AC waveform by current leakage through the mold wall 2+ i s responsible for the production of Fe i n the s l a g . Unfortunately, the actual measurement of the rate of th i s electrochemical reaction could not be assessed due to d i f f i c u l t i e s i n measuring the necessary currents . Nevertheless, the existence of the phenomenon would appear not to be confined to our small furnace for there are numerous •> r e p o r t s 2 ' 5 3 ' 9 6 ' 1 0 3 ' 1 0 4 on levels of FeO i n ESR slags s i m i l a r to those observed i n th is work (.1 - 1%). We have, however, found that insu la t ing the mold very e f f i c i e n t l y reduces t h i s e f fec t and thus we w i l l l i m i t ourselves to suggesting a method for preventing current leakage i n the mold. I t i s impractical 1 8 3 and, undoubtedly, very expensive to use boron n i t r i d e paint i n commercial prac t i ce . Conveniently, however, a great number of ESR furnaces use a withdrawal mold, much l i k e a continuous casting operation. In such a conf igurat ion , i t may be possible to inser t some type of insula t ing gasket in the mold at some point below the slag surface and above the ingot metal pool . Since the slag is not subject to much v e r t i c a l t r a n s l a t i o n , th i s device could e f f e c t i v e l y prevent current leakage. At present we know nothing of the nature of th i s leakage and th i s would appear to be a pro f i tab le area of future research because of i t s impor-tant consequences. 5.5 Conclusions The oxidat ive loss of manganese has been successful ly i n t e r -preted by the proposed mass transfer model i n a wide var ie ty of ESR condit ions . This success may be a t t r ibuted to r e l a t i v e s i m p l i c i t y of the system chosen for th is study, the a v a i l a b i l i t y of the appropriate thermochemical information and to the well defined mass transport pro-cesses a f f e c t i n g the l o s s o f a l l o y elements. The model has been used to provide s p e c i f i c , quant i tat ive information on the nature of the mass transfer phenomena occurring in ESR processing. Previous assumptions of isothermal behaviour have been disproved while those concerning the n e g l i g i b l e contr ibut ion of the f a l l i n g droplets to a l l o y losses have been confirmed. In a d d i t i o n , much new information has been rea l ized as a consequence of the model predic t ions . 184 We have ascertained that mass transfer i s predominantly contro l led by slag phase transport and that the a l l o y losses at the ingot pool/molten slag interface determine the ultimate extent of composition change. It has been possible to i d e n t i f y the c r i t i c a l parameters, such as slag flow ve loc i ty and ingot pool volume, which exert major influences on the overa l l mass transfer behaviour of the system. In t u r n , these parameters have been used as the basis for predict ion of a l l o y losses in scaled-up ESR furnaces. Such ca lculat ions have shown that the a l l o y losses observed on small scale ESR are not, i n f a c t , representative of that which i s to be expected i n the l a r g e r , commercial operations. Certain information r e l a t i n g to the nature of inc lusions found i n ESR ingots can also be derived from the model predic t ions , which indicate the close connection between slag composition and inc lus ion composition. An addit ional major source of oxidant i n AC electros lag processing has been i d e n t i f i e d . The electrochemical generation of ferrous ions has been found to be due to r e c t i f i c a t i o n of the AC waveform as a r e s u l t of current leakage through the mold. A procedure to eliminate th is harmful reaction in commercial pract ice was o u t l i n e d . F i n a l l y , i t was concluded that , with proper recognit ion and control of the factors a f fec t ing composition change, i t should be possible to manufacture repro-ducibly good ingots in larger commercial e lectros lag furnaces. 5.6 Suggestions for Future Work Possible avenues of future e f f o r t to extend th is work may be divided approximately into three categories. I n i t i a l l y , as a pre-r e q u i s i t e of extension of the model to the study of losses of the more 185 react ive elements ( S i , C r , T i , A 1 ) , one must es tabl i sh a complete body of thermochemical information r e l a t i n g to the a c t i v i t i e s of these components i n the various CaF^^based s lags . The need for th i s data has been shown to be c r i t i c a l for successful implementation of the mass transfer model. I t would undoubtedly be benef ic ia l to continue with tests on the small laboratory ESR unit to evaluate mass transfer behaviour of these more react ive elements. In p a r t i c u l a r , i t i s suggested that the apparent close re la t ionship between inclusions and slag composition be studied extensively from a mass transfer point of view. The electrochemical aspects of AC ESR may also best be i n v e s t i -gated further on the laboratory u n i t . Such a program should include a invest igat ion into the nature of e l e c t r i c a l conduction through the s lag skin and accurate measurement of the mold current component. F i n a l l y , there appears to be an increasing necessity for experimentation on the larger commerical ESR furnaces. Although such invest igat ions would undoubtedly be c o s t l y , there i s great lack of knowledge of the s p e c i f i c deta i led behaviour of these u n i t s . The parameters about which information should be obtained have been outl ined i n the t ex t . In a d d i t i o n , there are great potential benefits to be r e a l i z e d i n the heat transfer modelling of the large furnaces and such information has been shown to be indispensible not only to the successful scale-up of the heat transfer behaviour but also to the scale-up of the mass transfer model. 186 PRINCIPAL SYMBOLS _ o [A] concentration of A i n metal phase mole cm [A]g i n i t i a l concentration of A in metal _3 phase mole cm (A) concentration of A in slag phase mole cm ( A ) ^ , ^ ] ^ bulk concentrations of A in s l a g , _3 metal phases respect ively mole cm ( A ) . j , [ A ] . i n t e r f a c i a l concentrations of A i n _3 s l a g , metal phases respect ively mole cm ? A. area of reaction interface j cm 2 A m a x maximum area of o s c i l l a t i n g drop cm a.j a c t i v i t y of species i c heat capacity cal g m ~ l o K - 1 d drop diameter cm d ,d . maximum, minimum diameters of o s c i l l a t i n g UlaX Fill M drop cm 2 -1 D.j d i f f u s i v i t y of species i cm sec AG free energy (AG° = A + BT) cal mol" 1 _2 g accelerat ion due to gravi ty cm sec -2 -1 -1 h heat transfer c o e f f i c i e n t cal cm sec °C I absolute current ab amperes K equi l ibr ium constant k.j mass transfer c o e f f i c i e n t of species i cm s e c - 1 k^ lumped mass transfer c o e f f i c i e n t cm s e c - 1 L mass transfer boundary layer thickness cm MW. molecular weight mass f l u x mole cm~3sec~ 2 3 p g a Physical Properties p g( P -P )y 4 G r o u p dimensionless q heat f l u x cal s e c - 1 Q latent heat cal gm 1 R radius cm Re Reynolds number Du p/u dimensionless Sc Schmidt number y/pD dimensionless T temperature °C, °K t time sec contact time sec u v e l o c i t y cm s e c - 1 V J volume of phase j cm3 V weight of metal gm weight of slag gm melt rate gm s e c - 1 WM volumetric melt rate 3 -1 cm sec We 2 Weber number ^ - ^ — -a. dimensionless mole f r a c t i o n 188 Greek Symbols -2 -1 thermal d i f f u s i v i t y cal cm sec y.j a c t i v i t y c o e f f i c i e n t of species i 6 electrode t i p f i l m thickness cm A(x) hydrodynamic boundary layer thickness cm e, eg amplitude correct ion factor for o s c i l l a t i n g drop dimensionless 6 cone angle of electrode t i p degrees K thermal conduct ivi ty cal cm _ 1 sec~ 1 o K y m , y s v i s c o s i t y of metal , slag respect ively poise 2 -1 v y/p kinematic v i s c o s i t y cm sec E, dimensionless mass transfer function P s k c A M _3 P m , p g density of metal , slag respect ively gm cm o surface tension dyn cm - 1 a) o s c i l l a t i o n frequency sec 1 _3 9 molar equi l ibr ium constant mol cm 189 BIBLIOGRAPHY 1. 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Metals (1972), TOO, 97. 92. BAIRD, D.C. "Experimentation." P r e n t i c e - H a l l , 1962. 93. BRIMACOMBE, J . K . " I n t e r f a c i a l Turbulence in Liquid Metal Systems" i n "Richardson Conference on "The Physical Chemistry of Process Meta l lurgy. " Proc. to be published. 94. MITCHELL, A. "Oxide Inclusion Behaviour during Consumable Electrode Remelting" to be published. 95. CHALMERS, B. " P r i n c i p l e s of S o l i d i f i c a t i o n , " Wiley, 1964. 96. MITCHELL, A. and BELL, M. Can. Met. Quart. (1972), 1J_, 2, 363. 97. OLSEN, V. i n "Physico-Chemical Aspects of Process Metallurgy" ed. A .M. Samarin, Nanka, Moscow, 1973, pp. 135-141. 98. BOUCHER, A. "Appl ica t ion of ESR." i n " B u l l e t i n de Centre des Etudes de Meta l lurgie" (1972), ]_2, 229-263. 199 99. COOPER, C.K. and KAY, D.A.R. JISI (1970), 208, 9, 856. 100. ELTXOV, K.S . et al_. " P e c u l i a r i t i e s of Large Plate Ingot . . . ESR" i n "Conf. on Special Electrometal lurgy," ed. B .E . Paton, Kiev, 1972, V I , 23-32. 101. VOLKOV, S .E. "On Mechanism of Removal of Inclusions . . . " I b i d . , 12-32. 102. PATON, B .E . et al_. " Invest igat ion of Temp. F ie lds of Large ESR Ingots . . . " I b i d . , 144-157. 103. VASILJEV, Ya. M. et a l_ . , I b i d . , 3-12. 104. MYZETSKY, V . L . et a]_. "Qual i ty of 22K ESR Steel',' I b i d . , 119-126. 105. FORTUNE, W.B. and MELLON, M.G. Ind. Eng. Chem. Anal . (1938), 10, 60. 106. CHARLOT, G. "Nouveau Tracte de Chimie Analyt ique , " Masson et. a l_ . , P a r i s , 1961, p. 786. 107. DEVUYST, E. Pr ivate Communication. 200 APPENDIX I DETERMINATION OF AUXILARY PARAMETERS FOR MASS TRANSFER MODEL A . I . I Calculat ion of Average Melt Rate From the motor speeds given in Tables 11-VII, we obtain an electrode travel rate from the appropriate c a l i b r a t i o n curve. For example, taking the average motor speed from Run #27, Table I I , the -2 -1 corresponding electrode travel rate 3.1 (10 ) cm sec . The volumetric melt rate for the 3.81 cm <f> electrode commonly used here i s then T T d 2 TT(3.81) 2 ? x ETR = x 3.1(10"*) (A.1.1) W^ = 0.36 cm3 s e c - 1 M u l t i p l y i n g by p p g , we obtain the melt rate i n mass sec -1 or = 2.6 gm sec" A . I . 2 Drop Weight Calculat ion The regular ly occurring amperage peaks found on the recorded amperage trace have been shown to correspond to the detachment of 35 drops from the electrode t i p . I t was a r e l a t i v e l y simple matter to 201 count these peaks for a given number of seconds to obtain the number of drops per sec. (Figure A . I . 1 ) D i v i d i n g the melt ra te , W^, by th i s quantity y i e l d s the average drop weight. Experience has shown that th i s 5 weight averages about 2.5 gm for lime slags i n melts on AC power. The melt rate i s , then, a d i r e c t ind ica t ion of the number of drops/unit time passing through the s l a g . On a large ESR u n i t , where mult iple drop s i tes are found, the average drop s ize may be calculated from the estimated number of s i tes and the average melt rate although the frequency of drop formation i s unknown. A . I . 3 Rate of Slag Loss The amount of s lag l o s t to the slag skin was important in determining overa l l mass balances. Although the slag skin was often not uniform i n thickness over the length or circumference of an ingot , we have quoted an average of these values i n the Tables 11-VII. Since the slag bath necessari ly rose at the same rate as the ingot grew, the slag skin frozen was proportional to the melt rate . Volume of slag sk in of height h = ^ ( R ^ o l d - R i n g 0 t ) n (A.1.2) Rate of r i s e of the ingot , RI, may be estimated by RI = Volumetric Melt Rate/Cross Sectional Area of Ingot 7 2 -1 = Wl, I Trr cm sec igure A l . l Osci l lographic Current Trace Obtained While Melting I V • C1018 in CaF2-20% CaO using 3" <b Mold. Chart Speed 5" min" 1 Subst i tut ing RI for h i n A . I . 2 and since the rate of ingot r i s e i s equivalent to the rate of r i s e of slag s k i n , we have the rate 1 of slag l o s s , VL, where = / p F g (melt rate (gm sec )/density = V L = A r n o l d " -"ingot* x R I x ^slag (R 2 - R 2) x MR ~9 p s l a g R I PFe RM 2 Rate of slag loss = .36 MR { ( ^ ) - 1} ( A . l R I For a slag sk in thickness of 1 mm and melt rate of 2.6 gm sec Rate of slag loss = .36 x 2.6 x ( ( y y j - ) 2 - 1) .05 gm s e c - 1 . •1 204 APPENDIX II ANALYTICAL METHODS A . I I . l Determination of Total Iron Iron i n the dissolved slag sample was estimated by a c o l o u r i -metric method using the orange-red complex of orthophenanthroline. Ferr ic i ron has no e f fec t on th is react ion and hydroxylamine hydrochloride was added to ensure that a l l the i ron in the solut ion was reduced to the ferrous s tate . Buffering of the so lut ion at pH 4.5 was obtained with a sodium acetate-acetic : acid buffer s o l u t i o n . A composite reagent was prepared by mixing the fo l lowing so lu t ions . Buffer So lut ion : 816 gm anhydrous sodium acetate dissolved i n - 1 12 1. of water with heat. Add 810 ml g l a c i a l acet ic a c i d . Cool and make up to 3 1. with water. 3+ 2+ Reducing So lut ion : (Fe -> Fe ) 10 gm hydroxylamine hydro-chlor ide dissolved i n 1 1. of water. Colour Producing So lut ion : , 1 1/2 gm orthophenanthroline dissolve i n - 200 ml water with enough HC1 to jus t d i s s o l v e . Made up to 1 1. with water. The analysis was carr ied out using 5 ml a l iquots of the slag sample solut ion with 90 ml of the reagent so lut ion made up to 100 mis with water. The opt ica l density of each so lut ion was measured on a 205 Beckman Model B spectrophotometer, using l i g h t of wavelength 510 my. The concentration of i ron was read d i r e c t l y from a c a l i b r a t i o n curve prepared by using standard iron so lut ions . Fluorine i s known to in ter fere i n the orthophenanthroline method but only when the amount of f l u o r i n e i s 50 times greater than the ferrous i r o n . Although th i s s i t u a t i o n may ex is t i n the s o l i d slag sample, by the time the sample i s d i sso lved , a large amount of f l u o r i n e has been driven of f as HF. This i s evidenced from the etching of the 2+ glass during even a s ingle ana lys i s . Also Fe standards run with comparable levels of CaF 2 to the slag samples showed no ef fect on the accuracy. Therefore we have assumed no interference from F i n the analysis of i r o n . A . I I . 2 Determination of Manganese M n 2 + was oxidized to MnO^ by potassium p e r i o d a t e . 1 0 6 Phosphoric ac id addi t ion prevented p r e c i p i t a t i o n of Mn02 or manganese periodate and decolourized the f e r r i c so lut ion while s t a b i l i z i n g MnO^. It was also necessary i n th i s case to heat the 10 ml a l iquot of the slag sample so lut ion with a small amount (- 5-10 mis) of n i t r i c ac id to ensure complete oxidat ion of ferrous i r o n . A composite reagent was prepared by mixing 100 mis of concen-trated sulphuric a c i d , 100 mis of concentrated phosphoric acid and 800 mis of 5% potassium periodate s o l u t i o n . The 10 ml a l iquot of sample so lut ion was evaporated with HNOo to - 5 ml . To th i s 90 mis of reagent so lut ion were added and heated to 90°C for f i v e minutes to ox id ize Mn to MnO^. Af ter complete ox idat ion , the cooled solut ion was made up to 100 ml with water. The concentration of manganese was determined from the spectrophotometer absorbance for l i g h t of 524 my and a c a l i b r a t i o n curve obtained from pre-pared manganese standard so lu t ions . Some ions do in ter fere in th i s a n a l y s i s , espec ia l ly Mo(VI), V(V), Ti ( IV) and Cr(VI) . Since the slag samples were taken with Mo wires , there was a possible source of Mo interference. The fo l lowing simple test was performed to show that no s i g n i f i c a n t amount of Mo dissolved i n the slag sample. Two slag samplers of Mo wire were weighed and one dipped in a CaF^ - CaO - FeO s l a g . Both were then placed in hot and 50% HC1 so lut ion u n t i l the slag had d isso lved . Upon reweighing the Mo wires , i t was found that the weight loss of each was ident i ca l at .1 mg. The to ta l weight loss would thus appear to be due to the d i s s o l u t i o n of Mo in the acid so lut ion and not i n the s l a g . Mo i n t e r -ference was, then, not a factor i n Mn a n a l y s i s . A . I I . 3 Determination of F e 2 + . A portion of the slag sample was dissolved i n 20% F^SO^ under Helium to protect any ferrous from o x i d a t i o n . The dissolved sample was then t i t r a t e d with .004299 N C e C S O ^ with a known amount of orthophenan-t h r o l i n e ferrous sulphate complex. The end point was a very pale green co lour , the o r i g i n a l so lut ion being l i g h t amber. The concentration of 2+ Fe was calculated from the reac t ion . 2 0 7 r~4+ . c 2+ r 3+ . c 3+ Ce + Fe -»• Ce + Fe 4+ and the number of equivalents of Ce used i n the t i t r a t i o n . A 2+ correct ion was then made for the Fe i n the ferrous sulphate complex. 208 APPENDIX III DERIVATIONS OF ADDITIONAL MASS TRANSFER EXPRESSIONS A.111.1 Desulphurization of Fe -S A l l o y During ESR Using CaFp - CaO  Slags Desulphurization is undoubtedly the most benef ic ia l chemical react ion occurring i n ESR processing of many commercially ref ined a l loys and, as such, i t has been a common area of interest amongst previous 2 14 15 17 ?R ?Q workers i n the f i e l d ' ' ' ' < ^ ° " L : \ The exchange reaction i s normally wr i t ten as (0 2~) + S -+ (S 2~) + 0 (A.3.1) In order to include the e f fect of FeO on th i s reac t ion , we combine the reaction Fe + 0 + (FeO) (A.3.2) 2+ 2-where (FeO) may be wri t ten as (Fe ) + (0 ) for a completely ionic , 30 s l a g . The r e l a t i o n that w i l l be used i n the formulation of the mass transfer model for desulphurizat ion i s S + Fe(£) -> (S 2~) + ( F l + ) (A.3.3) obtained by combining (A.3.1) and (A.3.2) 209 M E T A L [ S ] LU O LU S L A G (Fe2*) ( S 2 R ) POSITIVE FLUX (AOO) Figure A.3.1 Schematic Representation of Mass Fluxes i n Desulphurization Reaction 5 + F e ( & ) + {Fq2+) + ( s2- } The procedure followed i s i d e n t i c a l to that of Chapter 2 for the mass transfer processes occurring during the oxidat ion of Mn by FeO, using the assumptions of steady state and instantaneous i n t e r f a c i a l reac t ion ; i . e . i n t e r f a c i a l equi l ibr ium. The f l u x equations are nS X = k S { L s ] b " L^V (A.3.4) 210 ^ - = k s 2 - { ( S 2 " ) i - (S 2 ' ) b > (A.3.5) ^ = k p e2+ { ( F e 2 + ) i - (Fe 2 + ) b > (A.3.6) at steady s ta te , we have, by conservation of mass. n $ = n $ 2- = n F g2+ (A.3.7) The existence of instantaneous i n t e r f a c i a l react ion normally assumed i n pyrometallurgical mass transfer studies does not always 89 apply to sulphur transfer s i t u a t i o n s . With increasing s i l i c a content of a s l a g , the overa l l t ransfer rate appears to become chemically 89 c o n t r o l l e d . Although Ward was unable to pinpoint an exact mechanism for sulphur transfer i n general from his experimental evidence, he d id show that slags with CaF 2 additions exhibited no traces of chemical control behaviour with respect to sulphur t ransfer . Since the slags dealt with here are 80% CaF^, the o r i g i n a l assumption of fas t i n t e r f a c i a l reactions leading to i n t e r f a c i a l equi l ibr ium i s s t i l l v a l i d . Hence, ( F e 2 + ) . ( S 2 - ) . QC = • L = molar e q u i l . Const. (A.3 .8 . ) S [S] . This series of equations can now be used to develop an expression for k<-2- in terms of quant i t ies that have been measured i n Run #62. Therefore, taking (A.3.5) and subst i tut ing (A.3.8) for ( S 2 " ) i we obtain 211 n s 2 " k s 2 - { ¥S ] i ( F e 2 + ) . " ^ b } (A.3.9) but from (A.3 .6 ) , we have ( F e2 + ) = ! ! f £ _ i _ + ( Fe 2 + ) ( F e J i A k F e 2 + ( ) b (A.3.10) From the steady state assumption, we know that n F e2+ = n $ 2 - = V s d ( S 2 " ) / d t (A.3.11) Subst i tut ing (A.3.11) and (A.3.10) into (A.3.9) and solving for k $ 2 - , we obtain — d ( S 2 " ) / d t A k 2 - = (A.3.12) n s [ s ] i _ ( s 2 - ) b VS d(S 2 " ) 1 + f | r 2-K A dt k F e2+ [ r e >b This i s the r e l a t i o n used i n IV.2.2 for ca l cu la t ion of k<»2-. The expression for the evaluation of fi<- i s derived as f o l l o w s . The equi l ibr ium constant for reaction (A.3.3) i s ( a r,c ) ( a F p n ) K F ^ = ^ ^ — (A.3.13) h e : > [Wt%S] 212 where lx\ K F e S = ^ ° - - 4 . 0 7 3 2 (A.3.14) for the react ion wri t ten with S i n the 1 wt% standard state but aCaS " YCaS XCaS (wt %'S2') Y C a S ( for CaF2-20%CaO Slag) * s 2 - (7O8 + llW 9 7 100 x MW-2-since wt % S + (S ) x — p Slag (S 2 ") x 100  aCaS YCaS (1.38) P s l a g S i m i l a r l y a F e Q = Y p e 0 j F e j _ j 2 + N x 100 p Slag wt % S = [S] x 1 0 0 x 3 2 pmetal Subst i tut ing these expressions into (A.3.13) and solving for ft<- ( A . 3 . 8 ) , we get ( F e 2 + ) i ( S 2 " ) i K F e $ x 64 x (1 .38) 2 x p $ n s " = ^ 1 1 0 0 YCaS YFe0 P M 213 Replacing K by the appropriate form of A.3 .14 , 1.22 exp(2330/T - 4.07) p-n s = - • (A.3.15) YCaS YFeO P M A.111.2 Desulphurization of Resulphurized Steel i n CaFp-CaO Slags ResuTphurized steels are generally known for t h e i r free machin-ing q u a l i t i e s . The cause of th i s machinabi l i ty i s the presence of a r e l a t i v e l y large number of MnS inclusions due to t y p i c a l sulphur levels of 0.1 - 0.25 wt % and manganese levels of 0.8 - 1.4%. At steelmaking temperatures, MnS dissolves in the molten i ron and we must consider the 32 species Mn and S in our ana lys i s . I t i s readi ly shown that the equi l ibr ium constant, K, for the reaction Mn + S Z (Mn 2 + ) + (S 2~) (A.3.16) i s given by KMnS = e x p t 1 ^ 5 . _ 10.86 } (A.3.17) for the react ion wri t ten i n terms of 1 wt % standard state for [S] and [Mn] i n the metal. Also KM Q = ( a C a S ) ( a M n 0 ) (A.3.18) m b [% S] [% Mn] 214 Using the factors given in A . I I I . l for conversion of Kpe<, to the molar form, fy., (A.3.18) becomes i / _ YCaS YMnO , 7.2 ,2 ( S 2 ' ) ( M n 2 + ) f A o l q x TlnS 32 x 54.9 1 2.6 x 1.381 ; [SJLMnJ ^ M . J . i y ; ( S 2 " ) , ( M n 2 + ) . i f QM = 1- (A.3.20) M [ S ] 1 [Mn]. = molar equil const for Reaction (A.3.16) then solving A.3.19 for 9^ and subst i tut ing (A.3.17) 438 x exp(17160/T - 10.86) fiMnS = v Y ( A ' 3 - 2 1 ) 1 ^ YCaS YMnO In order to demonstrate that th i s reaction of manganese and sulphur i s r e a l l y the react ion we must consider we perform the fo l lowing operations. Combining the expressions (A.3.20) and (A.3.21) we can solve for a 2- 2+ desulphurization constant (S )(Mn ) with the r e s u l t o o. [S][Mn] 438 exp(l7160 /T-l0.86) (S 2 " ) (Mn 2 + ) = YCaS YMnO S i m i l a r l y for (A.3.8) and (A.3 .15) , we obtain 215 Taking the fol lowing values of the oarameters [S] = .13%; [Mn] = 1.3%; Y C A S = 3; Y F E 0 = 3.0; Y M n 0 = 7 we f i n d that for 1525°C and 1675°C, the desulphurization constant (A.3.22) has- values of 2.78(10" ) and 1.33(10" ) respect ively compared with 7.43(10" 7) and 6.72(10" 7) respect ively for (A.3.23) . Therefore we are , indeed, looking at the Mn_ + S desulphurization reaction for a typ ica l resulphurized steel melted through a CaF,, - 20%CaO s lag . The development of the mathematical model i s done exactly as i n the two previous cases, but with the addit ional complication of an extra f l u x term. M E T A L LMn] o [S ] S L A G (Mn 2 * ) (s 2 - ) P O S I T I V E F L U X ( A O O ) Figure A.3 .2 Schematic Representation of Mass Fluxes in Manganese Desulphurization Reaction (A.3.16) 216 The phenomenological f l u x equations are X = k S { [ S ] b " [ S ] i } (A.3.24) nMn ^ = k M n { [Mn] b - [Mn]. } (A.3.25) k s 2 - { (S 2 -) n . - ( S 2 _ ) b } (A.3.26) n 2"^~ = k M n 2 + { ( M n 2 + ) i - ( M n 2 + ) b } (A.3.27) at steady s ta te , we have: n S = nMn = n M n 2 + = n S 2 " = n ( A ' 3 ' 2 8 ) by conservation of mass, we also have •Vm d[S] = -Vmd[Mn] = V s d ( S 2 " ) = V s d ( F e 2 + ) (A.3.28(a)) A l s o , we have the condit ion of i n t e r f a c i a l equi l ibr ium ( M n 2 + ) . ( S 2 - ) . ^ M = ! 1 (A.3.20) M [Mn]i [S ] i 217 El iminat ing the i n t e r f a c i a l concentration [Mn]^ from equation (A.3.25) ^ - k M n { [Mn], - [Mn] 1 } ( M n 2 + ) . ( S 2 " ) , Rearranging th i s expression n M n n „ [ S ] 1 _ [ M n ] b ^ [ S ] i , A k M n ( M n 2 + ) . ( M n 2 + ) i ( S t _ ) . (A.3.29) From A.3 .24 , we have T o r " <s2">i " <s2"'b Upon adding t h i s r e l a t i o n to A.3.29 and solving for ^ n ^ M [Mn] b [S ] i - ( S 2 " ) b ( M n 2 + ) i Also A (Mn 2 + ) 1 ^ [ S ] . —k ?- + ~k KSC KMn n n S i= x ~ ks { [ s ] b " [ S V f r o m A - 3 - 2 4 Equating the above expressions for j^ -k s [ S ] b ( M n 2 + ) i k sQM [S] 2 ^ [ S ^ S ] . ^ k S 2 " kMn kMn k s [ S ] , ( M n ^ ) . 2 _ 2 + k 2 - 1 • ^ [ M n] bES] r (S 2 ) b ( M n 2 + ) i Col lec t ing terms and mul t ip ly ing by k ^ / k ^ [ S ] 2 + [S]1 ( ^ [Mn] b + ^ ( M n 2 + } i - [S], } ( M n 2 + ) i k (S2-). [ S I KS KS Equation A.3.30 s t i l l contains two i n t e r f a c i a l concentrations and thus one of them must be el iminated. At steady state n M n 2 + n S Therefore kMf)2+ [ ( M n 2 + ) . - ( M n 2 + ) b ] = k $ ( [ S ] b - [S].) Solving for (Mn ). 219 ( M n 2 + ) . = JXj- ( [ S I - [S].) + (Mn 2 + ) . (A.3.31) On subst i tut ing A.3.31 into A.3.30 and s i m p l i f y i n g by c o l l e c t i n g terms i n [S] . and d i v i d i n g by kMn k c we obtain a quadratic i £ "in o j Vs2' k M n 2 + equation i n [S] . of the form a [ S ] 2 + B [S ] i + Y = 0 where a = 1 3 = W - k M n 2 + ( W M n V k S ^ kS < f i M k S 2 " k Mn 2 + " W Y = k M n [ k s [ S ] b + k M n 2 + ( M n 2 + ) b ] [ k s [ S ] b + k s 2 - ( S 2 - ) b ] k s ( f i M k $ 2-k M n 2+ - k M n k $ ) Subsequently B ± ( 3 2 - 4 Y ) 1 / 2 [S] . = (A.3.32) i o Subst i tut ing (A.3.32) back into (A.3.24) and employing (A.3.28(a)) we obtain the d i f f e r e n t i a l equation for mass transfer of sulphur from metal to slag when reacting with Mn i n the metal 220 d(S 2 ") n - B ± ( 3 2 - 4 y ) 1 / 2 = — k.,2- { [S]. - [ ] } (A.3.33) dt V $ * D 2 This r e l a t i o n i s used i n Chapter 5 i n the modelling of s lag and ingot sulphur p r o f i l e s for the AC ESR melting of C1117 i n CaF 2 - 20%Ca0 s lags . 2+ 2+ A.111.3 Derivation of Expressions for (Mn ) . , (Fe ) . This sect ion i s a continuation of Section 11.2 and is concerned with the formulation of expressions for the i n t e r f a c i a l concentrations of 2+ 2+ the slag species (Fe ) , (Mn ) . The method i s e n t i r e l y analogous to that used i n I I .2 and begins with the f l u x equations (2 .2) - (2 .4) , the i n t e r f a c i a l equi l ibr ium of (2.5) and the steady state assumption (2.6) . Combining (2.4) with (2.5) and e l iminat ing the i n t e r f a c i a l 2+ concentration (Fe ) . we obtain n F 2+ [Mn]. ( M n 2 + ) . " - if" ( -2T- ) = M i 2 + - 1 k 2+(Fe' +) 1 9(Fed+) From (2 .2) , we have ^ (i^-) = [Mn] b - [Mn] Mn Upon adding and invoking the steady state assumption ( M n 2 + ) . [Mn]. 1 A [Mn]. -1 1 + 1 k F e 2+ ( F e 2 + ) b kMn Sett ing (2.2) = (2.3) and solving for [Mn]. 221 b ^ ( F e 2 + ) h n _ — (A.3.34) k 2+ [Mn]. = [Mn.]. - - j ^ — { ( M n 2 + ) . - (Mn 2 + ) . } (A.3.35) Mn 1 D Set t ing , now, (A.3.34) = (2.3) and subst i tut ing (A.3.35) 2+ for [Mn]^, we obtain the fol lowing expression i n (Mn ). (Mn 2 + ) 2 + M b - 7 7 7 W - = { k M n 2 + ^ M n ) i " ( M n V } x ft(Fe ) b 1 . [ M n ] b k M n 2 + 2-K # l i _ 2 + , kMn k F e 2 + ( F e 2 + ) b k F e 2 + k M n ( F e 2 + ) b 1 b This expression i s r e a d i l y reduced to the quadratic of (A.3.36) , M n 2 V (M2+, f k F e 2 + kMn , ^ M n ] b kMn + ^ ^ + 2 ( , n 2 \ > M M n 2 ^ (continued overleaf) 222 , kMn W ^ y ^ ' b , ' "A k F e 2 + ' F e 2 + » b < k M n 2 + ' 2 k M n 2 + + k M n ^ b < " " 2 \ . „ ( f l . 3 . 3 6 )  k M n 2 + I t i s eas i ly shown that (A.3.36) i s the correct expression for (Mn ). under the conditions where Q i s large and [Mn] b i s smal l . (A.3.36) becomes ( M n 2 + ) 2 - ( M n 2 + ) . ( 2 ( M n 2 + ) b + > • ^X ( H n 2 + ) b k F e 2 . ( F e 2 + ) b  k M n 2 + I t i s readi ly shown that under the conditions stated (large Q and [Mn] b « ( M n 2 + ) b ) that (Mn 2 +).j = ( M n 2 + ) b . The expression 2+ for (Fe ) i s obtained l i k e w i s e . Froma combination of (2.2) and (2.4) we f i n d that 2+ n M n ^(Fe )• o+ ? + -f- ( - r ) = n (Fe 2 + ) . [Mn] - (Mn 2 + ) Mn from (2.3) 223 A kMrT 1 b Upon adding these two expressions J, _ ^ ( F e 2 + ) i [ M n ] b - ( M n 2 \ x ~ oz (A.3.37) ^ ( F e 2 + ) . 1 + —:— kMn k M n 2 + Sett ing (A.3.37) equal to (2.4) we obtain n ( F e 2 + ) 1 fi(Fe2+) [Mn] - (Mn 2 + ) = k 2 + ( ( F e 2 + ) b - ( F e 2 + ) . ) ( - v -• + r—*z) i o D 1 kMn KMn* This equation reduces to the quadratic (A.3.38) ( F e 2 + ) 2 + (Fe 2 + ) { + . ( F e 2 + ) } kMn + _ n 1 k„_2+ k r 2+ * ~ u "Mn" "Fe' Fe 2+ 2+ i t i s e a s i l y shown that (Fe ) . = (Fe ) b and thus that (A.3.38) i s correct . 224 Since the value of a l l the i n t e r f a c i a l concentrations may be found independently, we may use these as a check on the value of fi that i s used i n each step of the computer computation. This i s the purpose of the subroutine OMT i n Appendix IV. By using th i s rout ine , we are able to es tabl i sh the internal consistency of the f i t t e d values of Y M n f l vs % MnO which are given i n Figure 18. APPENDIX IV COMPUTER PROGRAM FOR SOLVING MASS TRANSFER MODEL Figure A . I V . l Algorithm Outl ining Computer Program on Following Pages C C O M B I N E D MASS T R A N S F E R ? 9 7 D I M E N S I O N S ( 1 0 0 0 ) , W T ( 1 0 0 0 ) , W F ( 1 0 0 0 ) , W D ( 1 0 0 0 ) , W P ( 1 0 0 0 ) , W F E ( 1 0 0 0 ) c c l COMMON O H E G F . O H E G P , A V F , A V D , A V P COMMON K F B N , K D M N , K P M N , K F F E , K D P B , K P F E , K F M I , K D M I , K P M I R E A L K F M N , K D M N , K P M N , K F F E , K D F E , K P F E , K F M I , K D H I , K P M I , I T E R E X T E R N A L S Q R T , F , D , P C O M M O N / D E B U G / F L A G L O G I C A L F L A G C C E C ( A ) I S S T A T E M E N T F U N C T I O N T H A T C A L C U L A T E S E Q U I L I B R I U M C O N S T A N T A S F ( T E M P ) C E C ( A ) = E X P ( 2 . 3 0 3 * ( 6 4 4 0 . / A - 2 . 9 5 ) ) F L A G = . T R U E . C A L L P L O T S C C T F = T E M P OF F I L M C T P = T E M P OF POOL C A V F , A V D , A V P A R E A R E A / V O L U M E R A T I O S OF F I L M , D R O P , P O O L C RUN I S RUN NUM B E H / / I T E R I S # OF I T E R A T I O N S C K F M N , K F F E , K F M I A R E MASS T R A N S F E R C O E F F . OF MN ( I R O N ) , F E ( S L A G ) , M N ( S L A G ) AT F I L M C S I M I L A R L Y KDMN E T C ARE DROP MT C O E F F . / K P B N ARE POOL MT C O E F F . C CM I S I N I T I A L C O N C . OF M E T A L (WTS) I N E L E C T R O D E C V F , V D , V P A R E V O L U M E S OF F I L M , D R O P , P O O L C H I S S T E P S I Z E FOR I N T E G R A T I O N AT F I L M , P O O L I N T E R F A C E S / H D FOB DBOP F A L L T I M E C C S I N I T I A L C O N C . MN I N S L A G (WT«)/CFE I N I T I A L C O N C . F E I N S L A G (WT*) C VS I S V O L U M E OF S L A G / V L I S R A T E OF C H A N G E OF V S / Y P = I N I T . C O N C . MN I N POOL C X M F , X M D , X B P - C H A N G E I N S L A G COMP DUE TO S I T E F , D , P IN C U R R E N T I T E R C T O L L I M I T OF A C C OF GAMMA MNO C X T F , X T D , X T P - C U M U L A T I V E CHANGE I N S L A G COMP CUE TO S I T E F , D , P C READ ( 7 , 1 0 1 ) T F , T P , A V F , A V D , A V P , G A M F , G A M P , G M X F , G M X P , R U N , I T E R READ ( 7 , 2 0 0 ) K F M N , K F F E , K F M I , K D M N , K D F E , K D M I , K P M N , K P F E R E A D ( 7 , 2 0 0 ) K P M I , C M , V F , V D , V P , H , C S , C F E , H D , V L , V S , Y P , G F E O C A L L A X I S ( 0 . , 0 . , ' T I M E I N S E C O N D S ' , - 1 5 , 1 4 . , 0 . , 0 . , 1 0 0 . ) C A L L A X I S ( 0 . , 0 . , * WEIGHT P E R C E N T * , 1 4 , 1 0 . , 9 0 . , 0 . , . 1) C A L L S Y M B O L ( 1 . , 9 . 6 , . 2 8 , • T H E O K E T I C A L MODEL « , 0 . , 1 7 ) C A L L NUMBER ( 5 . 5 , 9 . 6 , . 2 8 , R U N , 0 . , 0 ) 1 Y L = C M * 1 . 3 1 E - 3 X K = C S * 4 . 7 4 E - 4 XMI=XM X M F = 0 . XMD=0 . X M P = 0 . X T F = 0 . X T D = 0 . X T P = 0 . F E = C F E * 4 . 6 5 E - 4 F F = F E T I B E = 0 . MM=20 1 I = I T E K / (H»10.) S (1) =.T IM E WT (1) =XM WF ( 1 ) = X M F W D ( 1 ) = X B D W P ( 1 ) = X M P WFE (1) = F E * 2 . 15E»4 L=2 M=10 C GF = C U R R E N T V A L U E OF GAMMA 0 F I L M C GP = C U R R E N T V A L U E O F GAMMA 0 POOL C U S E S A M E GAMMA 0 FOR DROP AS FOR POOL Y P = Y P * 1 . 3 1 E - 3 C T = T F + 2 7 3 . C A L L O M T ( K F M N , K F F E , G A M F , Y L , X M . F E . T , T O L ) GF=GAMF O H E G F = 7 6 4 . 7 7 * G F E O * E C ( T ) / G F T = T P * 2 7 3 . C A L L O M T ( K P M N , K P F F . , G A M P , Y P , X P I , F E , T , T O L ) GP=G A MP O M E G P = 7 6 4 . 7 7 * G F E O * E C ( T ) / G P WHITE ( 6 , 4 0 ) - H ( ! N , O M F G F , T F , G F , O N E G P , T P , G P , G B X F , G M X P , G F E O WRITE ( 6 , 3 0 0 ) KFMN , K F F E , KFM I , A V F , K D M N , K C F E , K D M I , A V D , K P M N , K P F E , K P f l I 1 , A V P WRITE ( 6 , 3 3 3 ) C B , Y P . C S , V F , V D , V P , V S , V L , H , H C WHITE ( 6 , 3 ) WRITE ( 6 , 2 ) X B , T I M E , X M F , X M D , X B P , Y F , Y C , Y P C B E G I N I T E R A T I O N S C DO 1 1 = 1 , 1 1 T = T F + 2 7 3 . L C G GF=GA K F C A L L O M T ( K F M N , K F F E , G A M F , Y L , X M , F E , T , T O L ) ONEGF = 7 6 4 . 7 7 * U F E O * R C ( T ) / G F T = T P * 2 7 3 . GP=G A f! P C A L L OMT ( K P M N , K P F E , G A M P , Y P , X M , F E , T , T O L ) O K E G P = 7 6 4 . 7 7»GFEO*EC ( T ) / G P Y F = Y L C C F U N C T I O N S F , P , D , C O N T A I N D I F F E H E N T I A L E Q U A T I O N S TO BE I N T E G R A T E D BY S C L V M N C S O L V K N OSES T H I R D ORDER RUNGE KUTTA C C DROP F O R M A T I O N DO 10 J = 1 , 1 0 C A L L R K 3 ( F , H , Y F , X M , F F . , X N F , V F , X T F , V S , T 1 H E ) F E = F F C A L L RK3 ( P , t l , Y P , X M , F E , X M P , V P , X T P , V S , T I H E ) C ADD DROP TO POOL I F ( J . E Q . 2 . A N D . I . N E . 1 ) Y P = K P * ( Y C - Y P ) * V C / V P V S = V S - V L * H 10 F F = F E C DROP F A L L YD = YF DO 1 1 J = 1 , 1 0 11 C A L L R K 3 ( D , H D , Y D , XP! , F E , XMD, V C , X T D , V S , T I M E ) F E = F F TIME=TIME»10.*H I F ( I . G T . 1 . A N D . I . L T . I I ) GO TO 6 WRITE ( 6 , 2 ) X P 1 , T I M E , X M F , X M D , X M P , Y F , Y C , Y P GO TO 12 6 I F ( I . L T . 11) GO TO 1 WHITE ( 6 , 2 ) X f l , T l H 5 , X t t F , X M D , i M P , Y F , Y D , Y P K=M + P!K 12 S ( L ) - T I M E / 1 0 0 . WT (L ) =Xt1*2 . 1 1 E * 4 KF ( L ) = X f F * 2 . 11E + 4 . WD ( L ) = X T D * 2 . 1 1 E * 4 WP ( L ) = X T P * 2 . 1 1E + 4 WFE (L ) = F E * 2 . 1 5 E * 4 L=L+ 1 1 C O N T I N U E L = L - 1 LM=L C A L L L I N E ( S , W T , L , 1 ) C A L L L I N E ( S , W F E , L , 1 ) C A L L L I N E ( S , W F , L , 1 ) C A L L L I N E ( S , W D , L , 1 ) C A L L L I N E ( S , W P , L , 1) C A L L S Y M B O L (S (LM) , W F E ( L B ) , . 1 4 , • I R O N • , 0 . , 4 ) C A L L S Y M B O L ( S ( L M ) , W T ( L M ) , . 1 4 , 'MN T O T A L * , 0 . , b ) C A L L S Y M B O L (S (LM) ,WP (LM) , . 1 4 , • P O O L • , 0 . , 4 ) C A L L S Y M B O L (S ( L M ) , W F (LM) , . 1 4 , ' F I L M ' , 0 . , 4 ) C A L L S Y M B O L (S ( L M ) , W D ( L H ) , . 1 4 , ' D R O P ' , 0 . , 4 ) C A L L P L O T N D 2 FORMAT ( 1 X , 8 ( 1 P E 1 2 . 5 , 3 X ) / ) 3 FORMAT ( 1 X , ' ( M N 2 + ) S L A G ' , 6 X , ' T 1 M E • , 2 7 X , ' S L A G • , 4 1 X , • M E T A L ' / 2 2 X , 1 2 ( 1 1 X , ' F I L M ' , 1 1 X , ' D R O P * , 1 1 X , ' P O O L ' ) / ) 4 0 FORMAT (1 H1 , 1 X , ' M O D E L # * , F 5 . 0 / / 1 0 X , ' M O L A R E Q U I L C O N S T A N T ( F I L M ) =• 1 . F 5 . 0 , 2 X , ' G A M M A UNO ( ' , F 5 . 0 , • ) = ' , F 6 . 2 / 1 0 X , • M O L A R E Q U I L C O N S 2 T A N T ( P O O L , D R O P , ) = ' , F 5 . 0 , 2 X , 'GAMMA MNO ( • , F 5 . 0 , • ) = ' , F 6 . 2 / / , | 3 1 X , ' G A M M A - - F F N L = ' , F 5 . 1 , / 1 X , , • G A M M A - - P F N L = ' , F 5 . 1 / / , 1 X , ' G A M M A 4 F E O = ' , F 5 . 1 / / ) ' j 101 FORMAT ( 2 F 5 . 0 , 5 F 1 0 . 3 . 4 F 5 . 0 ) I 2 0 0 FORMAT ( 8 F 1 0 . 6 ) 3 0 0 FORMAT ( 1 X , ' F I L M T R A N S F E R CATA'/1X,«KMN = ' , F 1 0 . 6 , ' ; K F E 2 + = ' , F 1 0 . 6 1 , ' ; KMN2 + = ' , F 1 0 . 6 , ' ; A R E A / V O L U M E R A T I O = ' , F 1 0 . 3 / / 2 I X , ' D R O P T R A N S F E R D A T A ' / I X , ' K M N = « , F 1 0 . 6 , ' ; K F E 2 + = ' , F 1 0 . 6 3 , ' ; KMN2 + = ' , F 1 0 . 6 , ' ; A R E A / V O L U M E R A T I O = ' , F 1 0 . 3 / / 4 1X,»POOL T R A N S F E R D A T A ' / I X , 'KMN = ' , F 1 0 . 6 , ' ; K F E 2 + = « , F 1 0 . 6 5 , ' ; KMN2+ = ' , F 1 0 . 6 , ' ; A R E A / V O L U M E R A T I O = ' , F 1 0 . 3 ) 3 3 3 FORMAT ( / / / / 1 X , ' I N I T I A L C O N D I T I O N S ' / / / 1 X , ' W T % MN I N E L E C T R O D E = • , 1 F 1 0 . 6 / 1 X , ' W T % fit) I N POOL = « ,F10. 6 / 1 X , • W T * MN IN S L A G = ' , F 1 0 . 6 / / / 2 / 1 X , ' F I L M VOLUME = • , F 1 0 . 6 / 1 X , ' D R O P VOLUME = ' , F 1 0 . 6 / 1 X , ' P O O L VOLUME 3 = ' , F 1 0 . 6 / / / / I X , ' I N I T I A L S L A G VOLUME = • , F 1 0 . 6 / 1 X , • R A T E OF S L A G LO 4 S S = ' , F 1 0 . 6 / / / 1 X , ' S T E P S I Z E ( S E C S ) FOR P O O L , F I L M I N T E G R A T I O N S = ' , F 1 5 0 . 6 / 1 X , ' S T E P S I Z E ( S E C S ) FOR DROP I N T E G R A T I O N = ' , F 1 0 . 6 / 1 H 1 ) S T O P END _ 229 SUBROUTINE OMT ( X M , X S , G , C M , C M S , C F , T , T C L ) ROOT (A , B) = ( -B + SQRT ( B * B -U.* A) ) / 2 . EC (X)=EXP ( 2 . 3 0 3 * (6U4 0 . / X - 2 . 9 5 ) ) T=T*273. CM=CM*1.31E-3 CMS=CMS*U.7UE-U GG=G C F = C F * U . 6 5 E - 4 G=GG EQ=2753.*EC (T)/G BF=XM/ (EQ*XS)•XM*CMS/XS-CF AF=- (XM/EQ)* (CF/XS+CMS/XS) F I=ROOT(AF,BF) BM=1./EQ+XS+CF/XM-CM A M - - (XS/EQ) * (CM/XS+CMS/XB) XMI=ROOT(AM,BM) B K S - - (XM/EQ+CH*XB/XS+CF+2.*CMS) AMS=CMS*CMS+XN*CM*CF/XS+CMS*CF+XM*CM*CBS/XS XMSI=ROOT (AMS,BMS) GN=XMSI/ (XMI*FI) GG= (EQ/GN)*G IF (ABS (GG-G) .GT .TOL) GO TO 10 RETURN END SUBROUTINE RK3 ( A U X , H , Y , X M,F E , X M M , V , X T M , V S , T I M E ) XK1 = AUX (Y ,XM,FE ) XK2=AUX(Y+H*XK1/2. ,XM+H/2. ,FE) XK3 = AUX (Y + 2 .*H*XK2-H*XK1,XM+H,FE) YN = Y + (H/6.) * (XK1+4.*XK2*XK3) XMM = (Y-YN)*V/VS XT M= XT M +X MM XM=XM+XMM Y = YN RETURN END 230 FUNCTION F ( Y , X M , F E ) REAL K F M N , K D M N , K P M N , K F F E , K D F E , K P F E , K F M I , K D M I , K P M I COMMON O M E G F , O M E G P , A V F , A V D , A V P COMMON K F M N , K D M N , K P M N , K F F E , K D F E , K P F E , K F M I , K D M I , K P M I A = K F F E / (KFMI*OMEGF) E=KFFE/KFMN B=A+E*FE-Y C=A*Y+E*XM/OMEGF G=-B*SQRT (B*B + «».*C) F = - A V F * K F M N * (Y-G/2.) RETURN END FUNCTION D ( Y , XM , FE) REAL K F M N , K D M N , K P M N , K F F E , K D F E , K P F E , K F M I , K D M I , K P M I COMMON O M E G F , O M E G P , A V F , A V D , A V P COMMON K F M N , K D M N , K P M N , K F F E , K D F E , K P F E , K F M I , K D M I , K P M I A=KDFE/(KDMI+OMEGP) E= KDFE/KDMN B = A * E * F E - Y C=A*Y*E*XM/OMEGP G=-B + SQRT ( B * B * I * . * C ) D=-AVD*KDMN* (Y-G/2.) RETURN END FUNCTION P ( Y , X M , F E ) REAL K F M N , K D M N , K P M N , K F F E , K D F E , K P F E , K F M I , K D M I , K P M I COMMON O M E G F , O M E G P , A V F , A V D , A V P COMMON K F M N , K D M N , K P M N , K F F E , K D F E , K P F E , K F M I , K D M I , K P M I A=KPFE/ (KPMI*OMEGP) E= KPFE/KPMN B = A * E * F E - Y C=A*Y*E*XM/OMEGP G = - B + S Q R T ( B * B *4 . *C) P = - A V P * K P M N * ( Y - G /2 . ) RETURN END 231 APPENDIX V THERMOCHEMICAL AND HEAT TRANSFER CALCULATIONS A . V . I Oxidation Potentials of Mn, Si in 1018 Steel Melting in CaFp-20% CaO Slag N - ? log h = log F* + E e^ (wt%y) + log (wt%x)^ A X -j A where h i s the Henrian a c t i v i t y of species x i n so lut ion of i ron with X other s o l u t e s , ( y ) . The major solutes in 1018 are C, S, Si at 0.15, 0.105, 0.04 wt % respect ively i n addi t ion to manganese at - .57% (the 32 lowest observed value) . From Tables i n Bodsworth eMn - °' eSn - - 0 ' 0 0 9 ' eMn = - ° ' 0 4 5 ' eMn = 0 e^] = 0.32; e^n = 0; ej?. = 0.20; e^. = 0.057 Therefore, log h M n = (0 x .57) -( .009 x .15) - .045 x .04 + 0 x .04 + log = -.0014 - 0.0018 - .2441 hMn = - 5 7 also log h s i = .105 x .32 +.15 x 2 + .04 x .057 + log .04 h C l . = an t i log (T.0871) = .122 232 The Raoultian a c t i v i t y , a , i s given by X a = h y° .5585/MWY X X X A Y^- = ant i log (1.21 - 6100/T) at 1525°CY5 i = 6.56 (TO - 3 ) at 1675°CY5i = 1-2 (10 - 2 ) a S i 1525 = , 1 2 2 x 6 - 5 6 ( 1 0 " 3 ) x -5585/28.09 = 1.59(10" 5) a S i 1675 = J 2 2 x 1 - 2 ( 1 0 " 2 ) x -5585/28.09 = 2.9(10~ 5) The standard free energies for the reactions A, B are given by: (A) Si(A) + 0 2 ( ) 1 ( S i 0 2 ) £ AF° = -223, 800 + 46.08 T 8 2 cal mole" 1 (B) 2Mn + 0 2 ( ) 1 2 ( M n ° ) s A F ° = " 1 9 5 > 1 0 0 + 5 1 - 3 3 j 8 2 c a l mole" 1 a S i 0 ? AF°. = RT in P n - RT in ( -) bi u 2 a s i AF° = RT in P 0 - RT £n (^P_) " n u 2 nMn aMn0 = 5 ( 1 0 ) " typica l value from experimental resul ts (Chapter 4 and (37)) a S i 0 = 5 ( 1 0 " 4 ) " m " i n i n i u m expected value i n CaF2-CaO s l a g 8 3 a t 1550°C(1823°K). 233 Solving for RT Jin P n for Mn/MnO, we have u 2 RT In P n = -195100 + 51.557 + 2 RT Zn (5^°7 ^ Solving for P n at 1525°C and 1675°C. u 2 ( P02 )Mn/MnO,1525°C = 2 - 7 0 ° ~ 1 7 ) a t m ( P02 )Mn/MnO,1675°C = 1 ' 8 ( 1 0 " 1 5 ) a t m for S i / S i 0 2 , we have RT Zn P n = -223800 + 46.08 T + RT Jin ( - ^ l^ l l l - ) U 2 a S i J Therefore, ( P 0 2 } S i / S i 0 2 , 1 5 2 5 ° C = 2 - 3 ( 1 0 " 1 6 ) a t m ^ P 0 2 ^S i /S i0 2 , 1675°C = 1 .6 (10 - 1 4 ) atm. The a c t i v i t y effects are c l e a r l y shown i n Figure A . V . I . The S i / S i 0 2 l i n e does not move due to low a c t i v i t i e s of Si and SiO,,. However, the low level and thus, low a c t i v i t y , of MnO in the slag produces a clockwise ro ta t ion of the Mn/MnO l i n e . This shows that at both 1525°C and 1675°C that Mn w i l l ox idize p r e f e r e n t i a l l y to S i . Figiirei/Af. V.1'- ;„* Free Energy - Temperature P l o t Showing Rotation of Mn/MnO Line r-o 235 A.V.2. Calculat ion of Temperature of Electrode Film Using Derived Heat Transfer Coef f i c ient at S lag/Fi lm Interface At steady s ta te , a simple heat balance at the electrode t i p i s 24 made up of three parts . Rate of heat i n at slag/metal boundary = Rate of heat loss by conduction up the electrode + Rate of heat loss by melt ing. The net heat flow into the electrode t i p , q , i s given by A F h s (T S -T M ) = Q x WM + A x s k ( g ) z = Q (A.V.2.1) or S^c-} = A F hS ( W " Q x WM " A Y , k ( £ ) 7 = f  "XS * v dz'z=0 z = 0 = electrode t i p melting surface h $ = .18 cal c m - 2 s e c " 1 o K - 1 (V- l -1) T $ = s lag temp = 1675°C TM = melting point = 1520°C Q = 65.65 cal gnf = 2.63 gm sec -1 k = 0.074 cal cm"1 s e c - 1 °K _ 1 A p = 16.2 A x s = T r ( 1 . 9 i r ( g ) z = Q = 260°C cm"1 24 temp grad at electrode t i p in electrode at steady state q = 0, and solving for \ 236 L x W. M XS T M = T S " dzyz=0 (A.V.2.2) Subst i tut ing i n the above values. 1675 65.65 x 2.6 TT(1 .91 ) 2 X .074 x 260 .18 x 16.2 " .18 x 16.2 = 1540°C We have assumed only a small degree of superheating of the electrode f i l m i n our analysis and the value obtained using the derived heat transfer c o e f f i c i e n t from V . l . l seems to confirm t h i s . I t i s possible that T c . Siag may be lower that 1675°C since the slag in contact with the electrode t i p i s coming from the surface of the slag bath (see Figure 12). Thus the value of 1525°C used for the metal f i l m temp in c a l c u l a t i n g the equi l ibr ium constant i n the mass transfer model i s e n t i r e l y reasonable. This conclusion has also been shown to be v a l i d for the unsteady state case where metal i s flowing on the electrode t i p and the electrode and ingot 38 pool surface are moving. A . V . 3 Calculat ion of Temperature of Drop a f te r F a l l through Slag Heat input to drop i s given by 237 H x w A h s W (T S -T M ) dT, (A.V.3.1) A W 8.4 cm 2.5 gm (II .4) (Appendix 1.6) .2 cal gm"1 °K _ 1 ,23 T c = 1675°C, T M (Table XIV) = 1520°C Rearranging (A.V.3.1) and integrat ing we have r M dT M T„l A h $ W dt T C -T M 2 A h c U l n ( J _ J L ) = s _ t W Cp AhQ W or TM2 = T s - (T S-TM2) exp (- t) (A.V.3.2) If the drop f a l l time i s .12 sec (Table 1) then TM2 = 1675 - (155) exp ( . 8 . 4 x .23 x 2.5 x .12 } = 1675 - (155 x .055) = 1666°C Therefore, the drop i s very nearly thermally equi l ibrated with the slag by the time i t reaches the ingot pool surface. 

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