NO9605289 I BIRGER ANDRESEN NEI-NO--671 PROCESS MODEL FOR CARBOTHERMIC PRODUCTION OF SILICON METAL

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DOKTORINGENI0RAVHANDLING 1995:84 METALLURGISKINSTITUTT NTH TRONDHEIM UNIVERSITETET I TRONDHEIM NORGES TEKNISKE H0GSKOLE Mi-rapport 1995:34

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Process model for carbothermic production of silicon metal

Dr. ing. thesis 1

Birger Andresen Division of Metallurgy The Norwegian Institute of Technology University of Trondheim N-7034 Trondheim, Norway

September 12,1995

1This thesis is submitted in partial fulfillment of the requirements for the degree Doktor Ingeni0r.

NTH-Tiykk 1995 r DISCLAIMER

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Preface

The carbothermic silicon metal process is rather complex and is carried out in submerged arc furnaces at temperatures up to more than 2000°C. The high temperatures involved and the chemically aggressive conditions in the interior of the furnace make it extremely difficult to carry out measurements and to monitor the situation in this crucial part of the process. The metallurgy of the process is therefore only partly known and the conditions inside the furnace are mostly estimated from secondary variables or from educated guessing. Typical silicon recoveries for large scale industrial furnaces are around 85%. This means that there is room for considerable improvements in furnace operation and control. Further knowledge about the metallurgy of the process and the involved physical phenomena are important for improving furnace control. The Elkem company has developed several models for the silicon metal process. The stoichiometric model (section 2.1) and the unidimensional dynamical model focusing on the furnace shaft (section 2.2) have both proven useful. No advanced model has so far been developed for the high-temperature part of the system. In 1989 the Norwegian ferroalloy industry formed a research association (The Norwegian Ferroalloy Producers Research Association) to sponsor research in various fields of com­ mon interest for the members and to educate highly qualified research personnel for their industry. The members decided to sponsor research on the high-temperature part of the silicon metal process as one of their objectives, and this research was to be carried out within the framework of a Dr. Ing. study at The Norwegian Institute of Technology, Trondheim, Norway. The work was started in the autumn of 1989 with the primary goals to gain additional knowledge about the high-temperature part of the carbothermic silicon metal process and to develop a prototype dynamical model for this part of the process. The work with the model turned out to be much more complicated than first anticipated. The presented model should be seen in light of the long-term goal to develop a model suitable for educating furnace operators and for conducting research. In its present state it illustrates important effects in the high-temperature part of the process, but it suffers from obvious weaknesses that must be removed before the long-term goal is met. The model will hopefully be thoroughly evaluated by research personnel and experts from the industry in the near future so that the necessary improvements can be made. The initial plans for carrying out experiments to verify the model were soon put aside as it became apparent that proper experiments are difficult to design. Verification by IV PREFACE experiments would have be easier for a model covering the entire process. It was at early stages of the project expressed a wish from those representing the industry that the model should be designed such that it could easily be merged with the unidimen ­ sional Elkem model. The two models are, however, so basically different in their structure that such coupling seems difficult. In fact, any thought of merging the two models was in reality abandoned when it was decided to apply the computational fluid dynamics code FLUENT as a basis for the present model. Also, both models are in themselves so com­ plicated that it is probably an unproductive sidetrack to extend any of them significantly. It seems more productive to improve each of the models and thus have" one model for the high-temperature zone and another one for the furnace shaft. The author sees no fundamental problem in extending the present model to cover also the chemical reactions in the upper parts of the furnace. Preparations for such extentions have to some extent been made in the present model, but major changes are even so needed to implement the chemical reactions taking place in these areas. It is, however, unlikely that such extentions will be carried out soon. This means that the author sees no bright prospects for a complete and detailed dynamical model for the overall carbothermic silicon metal process in the near future. The model may prove useful as a basis for mathematical models for other metallurgical processes.I

I would like to express my thanks to present and former colleagues at The Norwegian Insti ­ tute of Technology and SINTEF Materials Technology, Departement of Process Metallurgy and Electrochemistry for kind assistance during the work with this thesis and for creating a nice atmosphere at work. The financial and administrative support from The Norwegian Research Counsil, The Norwegian Ferroalloy Producers Research Association and SINTEF is gratefully acknowledged. Above all I am in debt to Professor Johan Kr. Tuset whom through his guidance and support has brought me a long step forward as a researcher. Dr. Stein Tore Johansen and Dr. Trend Bergstrgm has been indispensable for my understanding of the many well hidden secrets of the FLUENT code. Their valuable guidance while designing and implementing new code and while writing this thesis is also very much appreciated. I also thank Professor Thorvald Abel Engh, Professor Jon Arne Bakken, Mrs. Hilde Lgken Larsen, Mr. Arne E. Amtsberg and representatives from the ferrosilicon industry for giving valuable suggestions and for fruitful discussions during the work with this thesis. Mr. Amtsberg implemented the original code for the electric arc. I express my deep gratitude to my cohabitant, Hilde S0derholm, for continuous encourage ­ ment, patience and understanding. PREFACE v

Parts of this thesis was published in the proceedings from the 7’th International Ferroalloy Congress (INFACON 7), 1995 :

Andresen, B. and J. Kr. Tuset, Dynamical Model for the High-temperature Part of the Carbothermic Silicon Metal Process., INFACON 7, Trondheim, June 11- 14 1995. pp. 535-544.

Trondheim, September 1995

/B>izve^n Clv\c2heaavv Birger Andresen VI Abstract

An advanced dynamical two-dimensional cylinder symmetric model for the high-temper- ature part of the carbothermic sihcon metal process has been developed and encoded in a computer program. The situation close to that which is believed to exist around one of three electrodes in full-scale industrial furnaces is modelled. This area comprises a gas filled cavity surrounding the lower tip of the electrode, the metal pool underneath and the lower parts of the materials above. The most important phenomena included are :

• heterogeneous chemical reactions taking place in the high-temperature zone (above 1860°C). • evaporation and condensation of silicon. • transport of materials by dripping. • turbulent or laminar fluid flow. • DC electric arcs. • heat transport by convection, conduction and radiation.

The most important simplifications are :

• assuming that all carbon from the raw materials are converted to SiC in the upper parts of the furnace before entering the high-temperature zone and that no free carbon is present in the high-temperature zone. • assuming that the metal pool straight below the electrode consists of pure sihcon and that the metal pool is a rigid body rather than a fluid under ­ going continuous movement.

• neglecting the direct reaction between SiC and SiC>2- • assuming that condensate formed by cooling of Si'O (g)-rich gas in the low-temperature zone enters the high-temperature zone as pure silica. • assuming that the charge is non-permeable. • assuming inert furnace walls and inert electrode. • neglecting electric currents and any effects thereof outside the electric arc.

The results from the calculations, such as production rates, gas- and temperature distri­ butions, furnace- and particle geometries, fluid flow fields etc, are presented graphically in an informative way. ABSTRACT vu

The model is in its present state a prototype. The most important conclusions from the simulations are :

• Reasonable average production rates are calculated for the high-tempera- ture zone. The changes in the average net silicon production are quali­ tatively as expected for a transient period starting from a relatively cold furnace. The calculated average silicon production is closest to the ex ­ pected values for higher power inputs. • The calculated chemical reaction rates and the heat fluxes to particles are sensitive to local changes in the flow field when the geometry of the crater cavity changes. This gives irregular dripping and it sets up fluctuations in the instantaneous production rates around the average value. The average silicon production rate changes gradually despite the fluctuations in the instantaneous production rates. Local variations in the flow field seem not to be essential for the overall production rates. • The crater cavity expands qualitatively as expected. Higher temperatures are calculated for SiC than for Si02, and higher temperatures than rea­ sonable are calculated in the gas and in the central parts of the metal pool. • The heat fluxes to the central parts of the metal pool are overestimated, and those to the rest of the system are underestimated accordingly. Strong evaporation is calculated from the pure silicon melt underneath the elec­ trode. This evaporation and the following condensation is essential for the heat transport from the central parts of the metal pool to the rest of the system. Most of the SiO(g) is produced by silicon condensing on SiC>2-particles. The evaporation would be less dominating and the model become considerably more realistic if SiC and Si02 were present under ­ neath the electrode. • The DC electric arc generates a strong recirculation zone in the crater cavity. The flow is outwards along the metal pool, upwards along the outer edge of the crater cavity and inwards along the materials in the furnace shaft and underneath the electrode. The calculated flow field is not realistic for an AC electric arc moving around in the crater cavity. Calculating a realistic flow field for such situations is considered almost impossible. • Strongly overestimated condensation is calculated in areas with low flow velocities. This gives too high SiO(g)-concentrations close to SiOa-parti- cles in such areas. • A huge number of iterations is necessary to calculate an accurate flow field in cases with low flow velocities in the outer parts of the crater cavity. viii ABSTRACT

Further simulations are required to determine the sensitivity of the model to changes in the model parameters and model geometry. The model may prove useful for educational purposes and research when its major weaknesses are removed. The model is, however, unlikely to be suited for online automatic process control in many years. It is recommended first to evaluate the present model thoroughly together with experts from the industry. The main goal should be to evaluate the potential of the model, to identify its major weaknesses and to estimate the work required to remove these weaknesses. The present work indicates that the following changes will improve the model considerably :

1. Include transport of reacting materials to the area underneath the elec­ trode. 2. Improve the model for condensation of Si(g). 3. Change the algorithm for distribution of radiation from the electric arc.

It is recommended to address these items before considering the other problem areas which are identified and discussed in this thesis. The calculations are extremely time-consuming even on powerful computers. The overall complexity of the process is formidable, and vital physical data are still unavailable. It is not recommened to extend the model to include the chemical reactions in the furnace shaft. Contents s.

Preface...... iii Abstract ...... vi Nomenclature ...... xiii Abbreviations ...... xxiii

1 Introduction. 1 1.1 Background and main goals...... 1 1.2 Outline of the thesis ...... 2 1.3 An overview of the carbothermic silicon metal process ...... 3

2 Existing simulation programs. 11 2.1' The stoichiometric Elkem model...... 12 2.2 The dynamical unidimensional Elkemmodel ...... 15

3 Description and discussion of the model. 19 3.1 General description and main assumptions ...... 19 3.2 Governing equations ...... 22 3.2.1 The conservation equations ...... 23 3.2.2 Turbulence modelling ...... 25 3.2.3 Heat transfer in solid materials...... 27 3.2.4 Energy transfer at gas/liquid- and gas/solid-interfaces ...... 28 3.2.5 Numerical solution of the governing equations ...... 29 3.3 Chemical compounds ...... 32 3.3.1 The furnace gas...... 32 3.3.2 Carbon ...... 33 3.3.3 Silicon carbide ...... 35 3.3.4 The condensate ...... 35 3.3.5 Silica...... 38 3.3.6 Silicon...... 39 3.3.7 Material composition in the furnace shaft ...... 39 3.4 Chemical reactions and reaction rates...... 41 3.4.1 Transport of gas through boundary layers...... 42 3.4.2 The reaction SiC>2 + C = SiO + CO...... 46 3.4.3 The reaction 2SiC>2 + SiC = 3SiO + CO...... 48

IX X CONTENTS

3.4.4 The reaction Si02 + Si = 2SiO...... 51 3.4.4.1 Reaction with liquid silicon...... 51 5.4.4.2 Reaction with silicon condensing from the gas...... 55 3.4.5 The reaction SiO + SiC = 2Si + CO...... 59 3.4.6 The reaction SiO + 2C = SiC + CO...... 64 3.5 Phase transformations and transformation rates...... 66 3.5.1 Melting and solidification of Si02...... 67 3.5.2 Melting and solidification of Si...... 67 3.5.3 Evaporation and condensation of Si...... 68 3.6 The furnace walls...... 77 3.7 The electrode...... 77 3.8 The electric arc...... 77 3.8.1 The Prescribed Current Distribution model...... 80 3.8.2 Source terms in the in the energy equation for the gas...... 84 3.8.3 Plasma properties ...... 84 3.8.4 Radiative heat transfer to solids and liquids...... 86 3.8.4.1 The view factors...... 87 3.5.4.2 Numerical evaluation of the integrals for the view factors. 90 5.8.4.3 Obstacles blocking the radiation ...... 91 3.8.5 Heat exchange in the cathode and anode regions...... 92 3.9 The furnace shaft including the crater wall...... 94 3.9.1 Electric currents ...... 95 3.9.2 Particle representation ...... 95 3.9.3 The specific surface area for particles ...... 97 3.9.4 Dripping ...... 98 3.9.5 Chemical conversion...... 103 3.9.6 Updating the particle geometry...... 104 3.10 The metal pool ...... 113 3.10.1 The basic structure of the metal pool ...... 113 3.10.2 Electric currents ...... 114 3.10.3 Particle representation ...... 114 3.10.4 The specific surface area for particles ...... 114 3.10.5 Chemical conversion...... 115 3.10.6 Updating the materials in the metal pool ...... 115 3.11 The crater cavity...... 123 3.12 Heat transfer caused by evaporation and condensation ...... 126 3.13 Energy and mass exchange for chemical reactions ...... 127 3.13.1 Reactions at gas/SiC-interfaces ...... 127 3.13.2 Reactions at Si(l)/Si02^-interfaces ...... 128 3.13.3 Reactions at gas/S'i02-interfaces ...... 130

4 The simulated furnace and default parameters. 133 4.1 The furnace and its representation in the model...... 134

Va 'Vv -T" X-- i CONTENTS xi

4.2 The detailed model geometry...... 137 4.3 The computational grid...... 139 4.4 Convergence criteria and speed of convergence ...... 142 4.5 Default input parameters and data...... 144

5 Results and discussion. 149 5.1 Simulation number 1...... 149 5.1.1 Input parameters and data...... 149 5.1.2 Power level and radiation ...... 150 5.1.3 Silicon production ...... 151 5.1.3.1 Expected production rates...... 151 5.1.3.2 Calculated average production rates...... 153 5.1.3.3 Calculated instantaneous production rates...... 154 5.1.4 The flow field...... 159 5.1.5 The geometry of the particles ...... '...... 160 5.1.6 The temperature distribution ...... 166 5.1.7 Chemical production/consumption and the gas composition...... 170 5.2 Simulation number 2...... 176 5.2.1 Input parameters and data...... 176 5.2.2 Power level, radiation and arc temperatures ...... 176 5.2.3 Silicon production ...... 179 5.2.4 The flow field...... 181 5.2.5 The geometry of the particles ...... 181 5.2.6 The temperature distribution ...... 185 5.2.7 The gas composition ...... 186 5.3 Simulation number 3...... 187 5.3.1 Input parameters and data...... 187 5.3.2 Power level, radiation and arc temperature ...... 187 5.3.3 Silicon production ...... 188 5.3.4 The flow field...... 189 5.3.5 The geometry of the particles ...... 191 5.3.6 The temperature distribution ...... 192 5.3.7 The gas composition ...... 193 5.4 Simulation number 4...... 194 5.4.1 Input parameters and data...... 195 5.4.2 Power level and radiation ...... 195 5.4.3 Silicon production ...... 195 5.4.4 The flow field...... 198 5.4.5 The gas temperature in areas with low flow velocities ...... 200 5.4.6 The gas composition ...... 200 5.5 Summary of the main results...... 201

6 Final discussion and conclusion. 203 xii CONTENTS

6.1 General discussion of the model...... 203 6.1.1 Effects related to the electric arc...... 203 6.1.2 Effects related to chemical reactions ...... 207 6.1.3 Effects related to the physical structure of the model...... 210 6.1.4 Miscellaneous...... 211 6.2 Conclusions ...... 212 6.3 Recommendations ...... 213 Bibliography ...... 215 Appendix ...... 220 I. Evaluating the inner intergrals for the view factors...... 220 II. Thermochemical data, transport properties, volumetric radiation densities. 223 XUl Nomenclature

The following notation is used in the list of symbols :

cell Computational cell (control volume) in the FLUENT code (see Figure 3.4, page 30). chemical compound Si, Si02, SiC, SiO, CO etc. drip Dripping of materials from the crater wall to the metal pool. operator The simulation operator (executing the computer program). particle Condensed materials as physically represented in the model. pool Metal pool (see Figure 1.2, page 5). wall Crater wall (see Figure 1.2, page 5) or gas/solid- and gas/liquid interfaces in general.

General notation : v : v-vector. x : Value of variable ’x ’ in bulk gas. x : Time averaged value of variable ’x ’. V : Nabla operator. 6 : Element in set.

Latin letters : a Molar flow rate in the stoichiometric Elkem model [mole/s]. a.i Chemical activity of compound i (dimensionless). A Surface area [m2]. Ai Surface area of particle i [to 2]. Aeff,j Effective surface area of particle j [to 2]. Acell Surface area of cell [m2]. Acell,pool Gas/cell surface area for cells in the metal pool [to 2]. Acell,wall The part of the gas/cell surface area where condensation of Si(g) can take place for cells in the crater wall [to 2]. Ai,rad The part of the surface area of cell i which is exposed to radiation [to 2]. The part of the surface area of particle i where chemical XIV NOMENCLATURE

reactions can take place [m2]. Atotjtool Sum over all Aceii,p 00i [m2]. Atot^wall Sum over all ACeii,viaii [m2]. A(z) Maximal current density in the electric arc at distance z from the cathode spot [A/m2]. b Molar flow rate in the stoichiometric Elkem model [mole/s]. B Magnetic field [T]. Br Magnetic field in the r-direction [T]. Bz Magnetic field in the z-direction [T], Be Magnetic field in the 6-direction [T]. c Molar flow rate in the stoichiometric Elkem model [mole/s]. C Expansion coefficient for the electric arc (dimensionless). C Mass concentration [kg/m3]. C+ Dimensionless mass concentration in turbulent boundary layer. Q Constant in the k-e turbulence model [1.44]. c2 Constant in the k-e turbulence model [1.92]. Cp Specific heat capacity [J/kgK\. Cp Constant in the k-e turbulence model [0.09]. cT Shear concentration in turbulent boundary layer [kg/m3]. D Mass diffusion coefficient [m2/s]. dA Surface area [m2]. dr Differential radial distance [m]. dT Melting interval [K], dV Volume [m3]. dz Axial displacement [m]. dz Height of a particle or part thereof [m]. dzn _j Height of particle j at time step n [m]. dzsiC,dripj Height of material which is removed from SiC-particle j due to dripping [m]. dzsi02,drip,j Height of material which is removed from SiCVparticle j due to dripping [to ]. dZtransition Particle height at which a given particle changes its geometry due to changes in its mass [to ]. du Differential angular distance [rad\. eo Electron charge [1.602 x 10~19 C]. Ei Relative error (dimensionless). F View factor (dimensionless). fa Fraction of condensing silicon that must react chemically on an NOMENCLATURE xv

SiOa-surface to give neutral heat flux to the surface from the combined condensation and reaction (dimensionless). fb,j Operator defined correction factor for blocked/inactivated surface area for chemical compound j (dimensionless). /«gas Operator defined parameter determining the minimum fraction of the total evaporated mass and energy that can be distributed to the gas (dimensionless). Fi View factor to surface i (dimensionless). Fj,i View factor from cell j to cell i (dimensionless). F(Ni) Calculated view factor (dimensionless). jpool Operator defined parameter determining the maximum fraction of the total evaporated mass and energy that can be distributed to the metal pool (dimensionless). fr Integrand for evaluation of angle factor (dimensionless). Fr Radial Lorentz force in momentum equation [kg/m2s2\. Fr View factor to a vertical cylinder (dimensionless). fsj Operator defined correction factor for specific surface area for chemical compound j (dimensionless). fviail Operator defined parameter determining the maximum fraction of the total evaporated mass and energy that can be distributed to the furnace wall (dimensionless). fz Integrand for evaluation of angle factor (dimensionless). Fz Axial Lorentz force in momentum equation [&y/m2s2]. Fz View factor to a horizontal circular surface (dimensionless). Fe Azimuthal Lorentz force in momentum equation [kg/m2s2]. g{y+,arg) Function determining the shape of the concentration profile (arg = Sc) or temperature profile (arg = Pr) in turbulent boundary layer (dimensionless). h Specific enthalpy [J/kg\. hi Specific enthalpy for compound i [J/kg]. I Electric current [A]. i Integer counter- or index variable (dimensionless). i Segment number in the unidimensional Elkem model (dimensionless). j Current density [A/m2]. j Integer counter- or index variable (dimensionless). Ji Molar flux of chemical compound i to a surface [mole/m 2s]. j Mass flux to a surface [kg/m?s]. Jr Radial current density [A/m2]. jz Axial current density [A/m2]. XVI NOMENCLATURE

jo Azimuthal current density [A/m2]. k Kinetic energy of turbulence [m2/s2]. k Thermal conductivity \W/mK\. h Backward reaction rate constant [mole/m 2s]. kB Bolzmann ’s constant [1.381 x 10-23 J/K]. ks Forward reaction rate constant [mole/m 2s]. kg Mass transfer coefficient for gas through a boundary layer [mole/m 2s]. Ki Equilibrium constant for chemical reaction i (dimensionless). ki Thermal conductivity for material i \W/mK]. Kinetic energy of turbulence evaluated at the node position closest to the surface (wall) [m2/s2]. Thermal conductivity at the wall surface [W/mK], Lc Characteristic length [m]. m Mass [kg]. M Total number of view factors (dimensionless). m Mass flux [kg/m2s]. m Volumetric mass production rate [kg/m3s]. M Mass production rate [kg/s]. ^■cellygas Condensation rate for cells in the gas [kg/s]. Mcell,pool Condensation rate for cells in the metal pool [kg/s]. MCelltwall Condensation rate for cells in the crater wall [kg/s]. 'WlchemJ Mass consumed from particle j [kg]. WldripJ Mass dripping down from particle j [kg]. Tftdripj,! Mass dripping down from cell i of particle j [kg]. Mi Molar mass of compound i [kg/mole or g/mole ]. 77li Mass flux for compound i [kg/m2s]. 77li Volumetric mass production rate for compound i [fcg/m3s]. Mass of particle j at time step n [kg]. Tft>SiC,chem,tot Total mass consumed from SiC-particle j by chemical reactions in a time step [kg]. mSiCtdrip ttot Total mass dripping down from SiC-particle j in a time step [kg]. TH'Si02,chem,tct Total mass consumed from SiCVparticle j by chemical reactions in a time step [kg]. TH'Si02,drip,tot Total mass dripping down from Si02-particle j in a time step [kg]. Mtot Total evaporation rate [kg/s]. n Time step number (dimensionless). n Integer (dimensionless). NOMENCLATURE xvn

n Unit normal vector [m]. W'drip Operator defined model parameter used in the dripping model for Si02 (dimensionless). Ni Integer (dimensionless). P Pressure [Pa]. P* Pressure in the SIMPLE algorithm [Pa]. P° Pressure at 1 atm [1.013 x 10s Pa].

Pi Partial pressure of gas species i (dimensionless). Pi,vi Partial pressure of gas species i at gas/solid- or gas/liquid interfaces (walls) (dimensionless). Pr Prandtl number (Cp fi/k) (dimensionless). Prt Turbulent Prandtl number (dimensionless). Ptot Relative total pressure (dimensionless). Q Heat flux [VP/m2]. q Power density [VP/m3]. Q Power [VP]. Qall Combined heat flux from condensation and chemical reaction when all condensed Si(g) reacts at the surface [VP/m2].

Qanode Heat flux from the electric arc to the anode [VP/m2]. Qavail Heat flux available for chemical reactions [W/m2]. Qcell,gas Power (from condensation) to cells in the gas [W], Qcell,pool Power (from condensation) to cell in the metal pool [W]. Qcell,wall Power (from condensation) to cell in the crater wall [VP].

Qcond Heat flux caused by condensation of Si(g) [VP/m2]. Qconv Heat flux caused by convection [VP/m2]. Qelec Heat flux caused by electron transfer from the electric arc to the anode [VP/m2]. Qelecyarc Power density due to electron drift inside the electric arc [VP/m3]. Qevap Heat flux caused by evaporation of Si(g) [VP/m2]. Qgas Heat flux caused by heat transfer from gas to solids or liquids [VP/m2]. Qheat Part of the heat flux to a surface from condensation of Si(g) used for heating the surface [VP/m2].

Qi Heat flux to a surface by conduction through material i [VP/m2]. Qi Total radiation power to cell i [VP]. Qj,i Radiation power from cell j to cell i [VP].

Qohm,arc Power density by ohmic heating inside the electric arc [VP/m3]. Qrad Radiation flux to a surface [VP/m2]. XVU1 NOMENCLATURE

Qrad,arc Power density by radiation loss inside the electric arc [W/m3]. Qradj Radiation power from cell j [W].

Qreac Part of heat flux to a surface from condensation of Si(g) consumed in chemical reactions [W/m2]. Qtot Total enthalpy of evaporation [W]. Q tot,gas Total enthalpy of condensation to the gas [W]. r Radial coordinate [m]. r Reactivity number in the stoichiometric Elkem model (dimensionless). r Radius [m]. R Radius [m]. R Length of R [mj. R The universal gas constant [8.31434 J/K mole]. R Vector from radiation source to surface receiving radiation [m]. R#i Chemical reaction rate for reaction number # to the left as written [mole/m2s, or mole/m 3s]. R# r Chemical reaction rate for reaction number # to the right as written [mole/m?s, or mole/m 3s]. R&r,langmnir Limiting rate of evaporation for Si(g) (Langmuir) [mole/m 2s]. Rsr,xfr Mass transfer controlled evaporation rate for Si(g) [mole/m2s]. TO Radial posistion of radiation source [m]. Rc Radius of the cathode spot of the electric arc [m]. Rc Chemical reaction rate in the unidimensional Elkem model [mole/m 3s]. Re Reynolds number (= LcVcp/fi) (dimensionless). n Radial position [m]. Ri Chemical reaction rate for reaction i [mole/m2s, or mole/m 3s]. TmasSjnorm Mass ratio of SiC to S1O2 in the furnace wall (dimensionless). Tnode. Radial node position for cell [m]. Trad Effective radiation radius of the electric arc [m], B(z) Radius of the electric arc at a distance z from the cathode [m]. s Thermodynamical/kinetic parameter in the stoichiometric Elkem model (dimensionless). S The set of all cells contributing to dripping for a specified particle (dimensionless). Sc Schmidt number (y/D) (dimensionless). Set Turbulent Schmidt number (dimensionless). Sh Source term for production of enthalpy (in the energy equation for fluids) [IV], Sh,.trans Source term for production of enthalpy caused by heat transfer NOMENCLATURE xix

over a gas/liquid- or gas/solid-interface [W]. Sk Source term for production of kinetic energy of turbulence [kg/ms3]. ST Volumetric radiation density [W/m3]. St Source term for production of enthalpy (in the energy equation for solid materials) [W). Sc Source term for production of turbulent energy dissipation [kg/ms4]. s$ Source term for production of scalar quantity $ (depends on 0) Su,$ Constant part of source term for production of scalar quantity 5 (depends on 0) Sp,$ Linear part of source term for production of scalar quantity 0 (depends on 0) t Time variable [s]. T Temperature [K\. Tl,e Operator defined model parameter used for determining the fraction of condensing silicon that also reacts chemically [K\. T2,e Operator defined model parameter used for determining the fraction of condensing silicon that also reacts chemically [K\. Tz, a Operator defined model parameter used for determining the fraction of condensing silicon that also reacts chemically [K\. Tl,i Operator defined model parameter used for determining the actual mass- and energy transfer to a surface due to condensation of Si(g) \K\. T2li Operator defined model parameter used for determining the actual mass- and energy transfer to a surface due to condensation of Si(g) \K\. Ti Temperature of chemical compound i, cell i, or at position i [K\. Tlangmuir Parameter used for determining the evaporation rate of silicon [K\. Tlim Operator defined model parameter used for determining the fraction of condensing silicon that also reacts chemically [.K]. Tm Melting temperature [K]. tn Value of the time variable at step number n [s]. Tnode Temperature of cell (node) [K\. TTeac Reaction temperature [K]. Tsi,r Temperature of the silicon reservoir [K\. Ts Operator defined model parameter used in the dripping model [K]. Tsifioii Boiling point of silicon [3504.616RT]. TSurf Surface temperature [K]. U Molar flow rate in the stoichiometric Elkem model [mole/s]. U Velocity [m/s]. Ui Velocity in the i’th coordinate direction [m/s]. UT Friction (shear) velocity [m/s]. XX NOMENCLATURE

y : Molar flow rate in the stoichiometric Elkem model [mole/s]. V : Volume [m3]. Vanode ■ Anode fall voltage [V]. Vc : Characteristic volume [m3]. Vceu : Volume of cell [m3]. Vgas : Total gas volume [m3]. Vj : Volume of cell j [m3]. x : Molar flow rate in the stoichiometric Elkem model [mplefs]. x : Silicon recovery (dimensionless). Xi : Coordinate in the i-direction [m]. y : Molar flow rate in the stoichiometric Elkem model [mole/s\. y : Distance from a gas/solid- or gas/liquid interface [m]. y+ : Dimensionless distance from liquid or solid surfaces to. yn : Distance from liquid or solid surfaces to the node position of the gas cell closest to the surface [m]. z : Molar flow rate in the stoichiometric Elkem model \mole/s\. z : Axial coordinate [m].

zq : Axial posistion of radiation source [m]. zi : Axial position [m]. Znode : Axial node position for cell [m].

Greek letters : a : Stoichiometric coefficient (dimensionless). a : Radiative absorptivity (dimensionless). ai : Distribution factor for enthalpy of condensation (dimensionless). ai : Scaling factor (dimensionless). <*£,r : Operator defined radial Lorentz force correction factor (dimensionless). &L,z : Operator defined axial Lorentz force correction factor (dimensionless). at : Turbulence indicator (1 = turbulent case, 0 = laminar case) (dimensionless). OtT : Function used in the dripping model for Si02 (dimensionless). P : Parameter used for determining the evaporation rate of silicon (dimensionless). Pi : Number of cells that receive radiation from cell j (dimensionless). Pt : Parameter used in the dripping model for SiOz (dimensionless). r : Diffusivity [kgjms]. Te// : Effective thermal diffusivity [kg/ms]. 7 (T) : Function used for determining the condensation rate of Si(g) (dimensionless). : Diffusivity of # [kg/ms]. NOMENCLATURE xxi

Sij : Kronecker delta (equals 1 if i = j, 0 otherwise) (dimensionless). A Hi : Enthalpy of reaction for reaction i [J/mole or J/kg]. AHr : Enthalpy of reaction [J/mole or J/kg]. Atfrip : Operator defined model parameter used in the dripping model for Si02 [s]. ATdrip : Operator defined model parameter used in the dripping model for Si02 [K]. ATrg : Operator defined model parameter used for determining the evaporation rate of silicon [K]. e : Dissipation rate of turbulent energy [m2/s3]. 9 : Angle [rod]. 6 : Azimuthal coordinate [rod], fi : Molecular viscosity [kg/ms]. p 0 : Magnetic permeability of vacuum [4tt x 10~7 H/m]. Hejf : Effective viscosity [kg/ms]. Ht : Turbulent viscosity [kg/ms], v : Kinematic viscosity (= fi/p) [m2/s]. vt : Turbulent kinematic viscosity [m2/s]. u? : Dimensionless turbulent kinematic viscosity. p : Mass density [kg/m?]. Pi : Mass density for particle i [kg/m3]. p nj : Mass density of particle j at time step n [kg/m3], a : Electrical conductivity [S/m]. <7C : Constant in the k-e turbulence model [1.30]. <7k : Constant in the k-e turbulence model [1.00].

tw : Wall shear stress [N/m2]. # : Passive scalar quantity (depends on <$>). (/an ode '■ Work function of the anode material [K]. ui : Angle [rod]. cjo : Azimuthal posistion of radiation source [rod].

Superscripts :

x : Fluctuating part of variable ’x ’. xxn NOMENCLATURE

Subscripts : b : Bulk. chern :: Chemical reaction. cell : : Computational cell in the FLUENT program. cond : Condensation. drip : : Dripping from the crater wall to the metal pool. eq :: At equilibrium. gas :: In/of the gas. new :: At the next (end of the current) time step. node : : At the node position of a cell. old : : At previous (old) time step. pooZ :: Metal pool. r : : In the r-direction. roc? : : Radiation. reac : : Chemical reaction. surf : : Surface of a solid or liquid materials. fot :: Total. w :: At gas/solid- or gas/liquid interface (wall). wall :: Crater wall. z :: In the z-direction. e :: In the ^-direction. XX111 Abbreviations AC Alternating Current. CFD Computational Fluid Dynamics (page 22). DC Direct Current. LTE Local Thermodynamic Equilibrium (page 80). NTH Norges Tekniske H0gskole (Norwegian Institute of Technology). PCD Prescribed Current Distribution model (page 80). QUICK Quadratic Upstream Interpolation for Convective Kinematics (page 30). SIMPLE Semi-Implicit Method for Pressure Linked Equations (page 30). SINTEF Stiftelsen for Industrie!! og Teknisk Forskning ved Norges Tekniske H0gskole (The Foundation for Scientific and Industrial Research at the Norwegian Institute of Technology). XXIV Chapter 1

Introduction.

1.1 Background and main goals.

The basic motivation for designing and using simulation programs is usually either to try to extract information that is impossible to obtain by other means or to get such infor ­ mation in a less expensive way than other methods allow. Comparing the actual power consumption of the carbothermic silicon metal process as it is carried out today with the theoretical power demand indicates that there is room for significant improvements. The complexity of the process and the limited knowledge about the metallurgy and important data indicate that simulation programs describing the main physical and metallurgical as­ pects of the process may be useful when searching for these improvements. Major problems involved in monitoring important parts of the process point in the same direction. There are several reasons why it is difficult to monitor this process which is carried out in large submerged arc furnaces. One reason is the practical difficulties involved in mounting suitable sensors and devices for continuous measurements deep inside the furnace. Such devices are in most cases exposed to chemically aggressive surroundings. Their lifetimes are consequently short and the devices are too expensive to be of practical use. This means that essential parts of the process are inaccessible to direct measurement. Important parame ­ ters that could be used to characterise the state of the process are therefore unavailable. Examples are local temperatures, compositions and the geometry of the interior of the fur­ nace. Instead, these must be estimated from measurements of other observable variables or from educated guessing. Large time delays caused by slow transport of materials in the furnace and sometimes by slow responses in measured variables to other changes add to the difficulties involved in monitoring the state of the process. The aspects mentioned above accentuate the need for simulation programs which may help the metallurgist in chosing the correct strategy for furnace operation. Such simulation

1 2 CHAPTER 1. INTRODUCTION.

programs are quite different from the computer control systems widely applied in the ferrosilicon and silicon metal industry at present. These control systems do not model the actual physical or chemical behaviour of the process, but instead help the operator to monitor the process and equipment by collecting and processing historical data and then presenting them in a quick and suitable way. They also regulate some input variables automatically to keep the furnace state close to that which is believed to be the most favourable. Such systems have been applied for several decades and have paid off mainly by making it easier to keep the furnace operation steady and running. It has become possible to control larger furnaces due to these control systems. Even with the improvements achieved in recent decades, there is still room for significant improvements. Simulation programs modelling some of the most prominent physical and chemical phenomena of the process may contribute to this even though such programs are not likely to be applied directly in furnace control in the near future. Their main contribution will instead probably be to gain additional knowledge about the process and to provide a tool for checking how the process, or parts of it, is likely to respond to certain changes in the control parameters. Better furnace operation may be obtained from this. The model presented in this thesis is such a simulation program. As mentioned in the preface, two explicit goals were formulated for the present work :

1. To gain additional knowledge about the high-temperature part of the car- bothermic silicon metal process. 2. To develop a prototype model for this part of the process.

The long-term goal is to develop a model suitable for educating furnace operators and for conducting research.

1.2 Outline of the thesis.

The high overall complexity of the carbothermic silicon metal process, the motivation for designing simulation programs for it and the goals for the work has already been addressed in chapter 1. An overview of the process is given in the rest of this chapter. Relevant existing simulation models are described in chapter 2. Chapter 3 contains the detailed description of the model. The process itself is described to the extent needed for discussing the various assumptions and simplifications. A general description including a summary of the main assumptions is offered in section 3.1. Readers interested in the detailed algorithms and discussion of the assumptions and simplifications will find these in the later sections in this chapter. The various chemical compounds, chemical reactions and phase transformations involved are described before giving a de­ tailed description of each separate part of the model. 1.3. AN OVERVIEW OF THE CARBOTHERMIC SILICON METAL PROCESS. 3

The model is applied to simulate the conditions around one of the electrodes in a 21MW industrial furnace. These conditions and the model geometry are described in chap ­ ter 4. The computational grid, convergence criteria and model parameters are also pre ­ sented and discussed in the same chapter. Chapter 5 contains the results from the calculations. The results for each test case are discussed separately and the results for different simulations are compared. The main results are summarised in section 5.5. A final discussion of the model is given and the main conclusions are drawn in chap ­ ter 6. Suggestions for further improvements and extentions are given. Some calculations plus thermochemical data, transport properties and radiation properties for the gas are presented in the appendix. The computer code is not listed because this would cover several hundred pages.

1.3 An overview of the carbothermic silicon metal process.

Schei [1] and Schei and Larsen [2] give excellent and thorough descriptions of the metallurgy and other important aspects of the carbothermic ferrosilicon and silicon metal processes. These descriptions form the metallurgical basis for the simulation model presented in this thesis. An overview of the silicon metal process is given below. The relevant details are described together with the simulation model in chapter 3. Metallurgical grade silicon is produced by carbothermic reduction of silica (SiOz) in sub ­ merged arc furnaces charged with a raw material mixture of lumpy quarts or quartzite and carbon materials (coke, coal, charcoal and woodchips). Industrial furnaces are operated on AC-current 1 supplied through three carbon electrodes positioned at the comers of an equilateral triangle as illustrated in Figure 1.1. The furnace body has the shape of a shallow cylindrical crucible with an internal diameter of 5-8 meters and a depth of 2-3 meters. The power and current levels are 10-25MW and 50-90kA, respectively. With an energy consumption of ll-13MWh per tonne silicon metal produced this corresponds to production rates in the range 1-2 tonnes per hour. This metal is drained and tapped continuously or at regular intervals through holes in the sidewall lining at the bottom of the furnace crucible (not shown in Figure 1.1). The furnace crucible is under operation filled up to its rim with a more and less permeable

1 Although AC-current is exclusively applied at present, the consept of using DC-current and single electrode furnaces also for silicon metal production has been tried both on a pilot plant and a semi­ industrial scale. Reported results are encouraging for small scale tests (Dosaj, May and Arvidsson [3] and Ksinsik [4]), but appear to be less so for large scale tests [3]. 4 CHAPTER 1. INTRODUCTION.

=1

u u

Viewed from the side Viewed from above

Figure 1.1: Electrode positions, carbothermic production of silicon metal. bed of charged materials. The main reason for using woodchips in the charge is to improve the permeability of this bed. Fresh raw materials are distributed on top of it. These materials enter directly into the furnace through a closed system of feeding tubes from silos located above the furnace hood or are charged batchwise by use of especially designed vehicles. In the ideal case one would expect these materials to descend evenly in counter current with CO(g) formed in the hot bottom zone where metal is produced as predicted from the ideal gross reaction of the process : Si02{s) + 2C(s) = Si(l) + 2CO(g) (1.1) The chemistry of the process is, however, more complex than equation 1.1 expresses. One complication is that the compound SiC(s) appears as a stable phase instead of free carbon at temperatures above 1514°C2. The next is that a gaseous suboxide of silicon (SiO(g)) forms and starts to appear in significant concentrations beside CO (g) in the same temper ­ ature range. The reaction pressure of SiO is in fact quite high at the high temperatures required for liquid silicon to appear as a stable phase at an operating pressure of 1 atm3. Thus, the CO-gas leaving the high-temperature zone where metal forms is carrying a high load of SiO(g). A fairly large fraction of this SiO(g) reacts with preheated and accessible carbon under formation of solid SiC and CO-gas on its way up through the bed of charged materials. There is, however, always a surplus of SiO that condenses in the colder parts of the bed. Some SiO(g) even escapes with the CO(g) leaving at the top of the bed as expressed in the following and more realistic gross reaction for the process : SiOzW + (1 + z)C(s) = rg*(Z) 4- (1 + a)CO(g) + (1 - z)S:C(g) (1.2) where x denotes the silicon recovery (typically 0.85-0.90).

2The triple point for coexistence of SiC2, C and SiC at 1 bar calculated by Halvorsen [5] from the most recent data from JANAF Termochemical Tables [6]. 3The maximum value of 0.67 is reached at the triple point for coexistence of Si02, SiC and Si (1811°C [5]). 1.3. AN OVERVIEW OF THE CARBOTHERMIC SILICON METAL PROCESS. 5

The condensation reactions are highly exothermic and the quantity of SiO{g) that can be recovered as a condensate in the charge is in reality limited by the heat capacity of the charge and possible contributions from heat consuming reactions 4 that may take place at temperatures lower than required for condensation of SiO(g)5. The condensate is found to consist of an intimate mixture of glassy silica, tiny droplets of silicon and occasionally also some SiC- and C-particles. This is in itself a sticky material that tends to glue the solid constituents of the charge together and thus prevents them from descending by gravity. As a consequence of this and the way the supplied electric energy is dissipated in the furnace, a gas filled cavity forms around the lower end of each electrode. This cavity is also denoted the crater. Direct observations on bench scale furnaces in operation as well as excavations on industrial furnaces after shuting them down have confirmed that the situation around one of the three AC-electrodes may be pictured as shown in Figure 1.2.

Raw materials

Fairly inactive charge

Chemically \ Fairly inactive Crater wall Electrode active charge charge

Gasfilledt cavity \ J

Metal pool / —-Electric arc Material dripping down from the Silicon metal with SiC and Silica crater wall Furnace lining

Figure 1.2: The situation around one out of the three electrodes ([1]).

There are reasons to believe that these cavities are not physically connected, but separated by walls of relatively inactive charge. The cavities are formed because the materials in the bed due to bridging and sticky condensate are descending at a slower rate than actually

4Examples of heat consuming reactions are evaporation of humidity in the feed, gasification of volatiles in coals and woodchips and direct SiC formation caused by a reaction between intimately mixed and agglomerated fines of silica and carbon materials. 5Less SiO(g) needs to be recovered when smelting feixosilicon since the iron in the melt lowers the silicon activity. Metal production in the high-temperature zone then takes place at lower concentrations of SiO in the gas and less SiO{g) enters the furnace shaft. 6 CHAPTER 1. INTRODUCTION. consumed in the areas facing the upper parts of the cavities. These areas are in this thesis referred to as the crater wall. Temperatures of about 2000°C have been recorded on the inner crater wall during bench scale operations [1, ch. 5, p. 24]. The volume of the bed located straight above each cavity is referred to as the furnace shaft and is believed to confine the most active part of the charge. Excavations have shown that the SiOz entering with the raw materials mixture with a typical sizing of 5-20 cm is reaching down to the crater wall practically unchanged. Carbon materials added as coke or char with a typical sizing of l-5cm are also retaining their original shape, but convert gradually to SiC as they approach the crater wall. If coals are applied, they are subject to rapid coking when entering the furnace and thereby change their shape, but will thereafter behave as the other carbon materials. Varying amounts of condensate are found in-between the other materials in the shaft. This material has been deposited from the gas phase as this has been cooled from above 1800°C down to 1600°C. The majority is formed in the warmest and lowest parts of the shaft. It appears to become increasingly glassy as it approaches the crater wall where it consists of nearly pure silica. The silicon originally formed in the condensate as tiny droplets has therefore to a large extent left the condensate and dripped down to the area underneath the crater cavity. Direct observations on a 50 kW furnace have shown that fairly large amounts of viscous silica clusters with SiC'-particles attached are also dripping from the crater wall. Thus, molten silica as well as solid SiC are found together with liquid silicon in the area referred to as the metal pool in this thesis. There are local variations in the relative amounts of the various constituents in the shaft in all directions. Such variations affect the permeability of the charge and thereby the gas distribution through the shaft. This enhances inhomogenity and gas channels may develop. The furnace is said to be blowing if one of these channels extends all the way up to the top of the charge releasing SiO-rich crater gas directly into the off-gas system. The risk for this to happen increases as the crater cavity expands in the vertical direction and the thickness of the materials in the shaft decreases. Significant losses of SiO{g) may be the consequence if this development is allowed to continue. To prevent this from happening the crater walls and the bridging materials surrounding it are broken down by use of mechanical stokers at regular intervals. Fresh raw materials are then immediately added on top of the old materials. This gives the process a pronounced semi-periodic behaviour with the interval between two successive stokings defining the length of the cycle. Local stoking at irregular intervals may also be necessary to break the blow-pipes that occasionally forms at earlier stages of the cycle. The stokers are manipulated pneumatically or by use of the same types of vehicles as applied for charging. In the latter case, it is a necessity that the furnace top is accessible. For that reason silicon metal furnaces are open or semi-open. This means that off-gases are exposed to air and burn when escaping through the top layer of the charge. Some of the charged carbon may also be ignited due to air entrainments in this area. It is believed that as much as 80-90% of the electric energy input to the furnace is dissipated 1.3. AN OVERVIEW OF THE CARBOTHERMIC SILICON METAL PROCESS. 7 in the transferred AC-arcs extending from the lower edges of the electrodes to the metal pools underneath or to the lower parts of the crater walls. Apart from a small fraction that enters the charge directly due to stray currents in-between the electrodes, the rest ends up as ohmic heating of the electrodes and the metal pool. The majority of the energy input is thus dissipated in the crater area where it supports the strongly endothermic reactions associated with the formation of SiO(g) and the direct metal production. As much as 93% of the energy required for the reduction of Si02 to Si(l) is in fact used in the first reduction step to SiO(g). Loosing SiO(g) from the process is thus almost as bad as loosing silicon metal. It affects the silicon yield as expressed in equation 1.2, but more seriously, it increases the specific energy consumption and thereby decreases the productivity of furnaces operated on fixed power ratings. Furnace control basically aims at minimising the long-term losses of SiO(g) through the off-gas and at preventing significant amounts of SiC from building up over time in the high-temperature zone. Too much carbon in the raw materials or unfavourable process operations may cause SiC to build up in the metal pool. Too little carbon in the raw materials may cause the SiO content in the crater gas to increase beyond the levels that can be recovered in the shaft. The carbon coverage is therefore important in furnace control. The process responds quite slowly to changes in the raw materials since it takes hours before the materials enter the crater wall and the metal pool. Also, the amount of SiC in the metal pool changes over days, weeks and even months rather than hours. Such changes are not easy to detect at an early stage. Other important parameters in furnace control are the power level, the electrode positions, the stoking and the distribution of fresh raw materials. The raw materials applied in silicon metal production are quite pure. Non-volatile com­ ponents involving other elements than Si, C and O (Al2Os, CaO, Fe20%, Ti02) can be disregarded when discussing the chemistry of the process. For the three-component system in question Kolbeinsen [7] has listed as many as 12 re­ actions that may be of importance when discussing the reaction kinetics of the process. According to Schei [1] and others, however, only five of these are actually needed to describe the thermodynamics of the system when assuming that SiO and CO are the only gas com­ ponents that matters under the prevailing conditions. These reactions are all monovariant and are the following :

Si02 + C = SiO + CO (1.3) 2Si02 + SiC = 3 SiO + CO (1.4) Si02 + Si = 2 SiO (1.5) SiO + SiC = 2 Si + CO (1.6) SiO + 2 C = SiC + CO (1.7) 8 CHAPTER 1. INTRODUCTION.

The following phase transformations are also important :

Si02(s) = Si02{l) (1.8) Si(s) = Sill) (1.9) Si(l) = Si(g) (1.10)

The equilibrium pressure of the phase combinations defined by equations 1.3-1.7 are shown in Figure 1.3. According to the thermodynamics of the system reaction 1.3 is the preferred

co 10

1400 1500 1600 1700 1800 1900 2000 2100 Temperature (°C)

Figure 1.3: Equilibrium pressure above the phase combinations C—SiC, Si02—C, Si02— SiC, Si02 — Si and SiC — C at total pressure 1 bar (Halvorsen, Schei and Downing [8]). one when a mixture of Si02 and C is heated in an environment of CO(g) of 1 atm pressure. A low, but noticable SiO(g) pressure builds up and reaches at 1514°C a level where reaction 1.7 may proceed and SiC is formed. Equations 1.3 and 1.7 add up to the equation

Si02 + ZC = SiC + 2CO (1.11) which may be considered a gross reaction of certain importance if agglomerates of in ­ timately mixed Si02 and C are used as a part of the feed. Even then the reaction is expected to proceed over a series of elementary reactions where the gas phase plays an 1.3. AN OVERVIEW OF THE CARBOTHERMIC SILICON METAL PROCESS. 9

active role. This is due to the formation of SiC-layers on the reacting surfaces and lack of solid state diffusion. This reaction is of no importance when lumpy raw materials are used. The stable combination of condensed phases above 1514°C is Si02 and SiC when Si02 is present in surplus of what is required from equation 1.11. These react according to equation 1.4. The equilibrium pressure of SiO(g) for this phase combination increases strongly with increasing temperature. It reaches a level where liquid silicon may form as expressed by equation 1.6 at 1811°C which is the triple-point for coexistence of Si02, SiC and Si. This is the lowest temperature possible for direct formation of silicon at Psio + Pco = Ptot = 1. The SzO-pressure required for the silicon to form according to equation 1.6 decreases with increasing temperature as shown by the equilibrium curve in Figure 1.4.

Ptnt = Pi

Temperature, °C

Figure 1.4: The equilibrium between SiO and CO for reaction 1.6 giving Si ([5]).

Direct metal formation therefore requires that the mole fraction of SiO in the gas phase for example must exceed 0.67 at 1811°C and 0.48 at 2000°C. In practice, the SiO-pressure needs to be somewhat higher than given by this curve in order to speed up the reaction rate. The SiO(g) needed for silicon to be produced in this high-temperature zone is mainly formed by the reaction between liquid Si02 and Si from the metal pool or from the gas according to reaction 1.5, but possibly also by a direct reaction between Si02 and SiC as expressed by equation 1.4. The reaction pressure of these two reactions reach 1 atm at 1859 and 1813°C, respectively. Their reaction rates are believed to increase rapidly above these temperatures. The SiO-producing reactions (equations 1.4 and 1.5) are both strongly endothermic reactions whereas the metal forming reaction 1.6 is less so. The warm SiO-rich gas leaving the high-temperature zone has three options when entering the low-temperature zone. The SiO(g) can : 10 CHAPTER 1. INTRODUCTION.

1. react and form SiC according to equation 1.7 if free carbon is accessible. This is the most favourable reaction from a thermo dynamic point of view.

2. react with CO(g) and form an intimate mixture of S1O2 and SiC according to re­ action 1.4 proceeding from right to left. This is the second most favourable reaction from a thermodynamic point of view.

3. condense to an intimate mixture of SiO% and Si according to reaction 1.5 proceeding from right to left. This is the least favourable reaction from a thermo dynamic point of view.

The first option (reaction 1.7) leads to the formation of a carbide layer on the C-particles and the reaction rate slows down. It is known from experiments and observations that some unreacted carbon may reach the metal producing zone. This is believed to be unfavourable (page 33), and lumpy carbon materials of relative small sizes and with high reactivity with respect to SiO(g) are preferred in the raw materials. The second option appears to be of little importance in practice and is for kinetic reasons excluded as a possibility if proceeding from right to left as written in equation 1.4. Direct condensation of SiO(g) as postulated in the third option is however more likely to occur. This agrees with observations showing that the condensate mainly forms by the reversal of reaction 1.5 as the temperature drops from above 1800 to about 1600°(7. Most of the silicon produced by condensation is believed to separate out from the rest of the condensate upon heating. It then flows rapidly down to the metal pool. It is unknown to what extent this silicon undergoes chemical reactions while flowing down to the metal pool. It is reasonable to believe that the temperatures are so low and the residence times in the lower part of the shaft so short that most of it enters the metal pool unreacted. In brief, the silicon metal process may be described as a two stage process comprising :

1. A high-temperature zone surrounding a crater where the majority of the supplied electric energy is dissipated and effectively consumed by SiO (^-producing reactions and direct metal formation. 2. A low-temperature zone above the crater where the SiO{g) in the hot gases leaving the high-temperature zone is recovered by reacting with accessible free carbon or by condensing.

These zones are reflected in the stochiometric model developed by the Elkem company (section 2.1). The knowledge of the reaction kinetics and the thermodynamics of the reactions involved at these elevated temperatures is limited. The same is true also for important data on properties affecting the exchange of mass and heat. Chapter 2

Existing simulation programs.

Few computer programs exist for simulation of the silicon metal process. The most system­ atic and complete analysis of the carbothermic silicon and ferrosilicon processes have been carried ut by the Elkem companies in Norway and USA together with SINTEF Materials Technology and NTH in Norway. This joint effort has resulted in several non-dynamical models and also in the only successful dynamical simulation model for the silicon metal process known to the author; the dynamical unidimensional Elkem model. Key scientists during the development of these models have been Anders Schei, James H. Downing and Svenn A. Halvorsen (Elkem), Leiv Kolbeinsen (SINTEF) and Ketil Motzfeldt (NTH). The stoichiometric model arose from a proposal by K. Motzfeldt [9] in 1961. The model was further developed by the Elkem company and the implementation of it was completed in 1991. The work with the unidimensional model started in 1984. The model was formulated rather rigorously as a set of partial and ordinary differential equations. Serious numerical difficulties were encountered when starting to test a general preliminary version of the model [10]. Meanwhile, another model was developed and programmed at Elkem’s R&D Center in USA. This model was based on a previous study by Downing and Leavitt [11]. It had many simplifications, but produced some results consistent with furnace operation. The present version of the unidimensional Elkem model is a result of merging features of the preliminary model with the one developed in USA. It was completed in 1991/92. The most important Elkem models; the non-dynamical stoichiometric model and the dy­ namical unidimensional model are described in the following. The Elkem model developed in USA is not described further since no reference to it exists in the literature. Also, its basic features are included in the unidimensional model. A forth model for the carbothermic silicon metal process was developed by Johansson and Eriksson in the early eighties. This model is based on equilibrium calculations utilising the SOLGASMIX free energy minimiser. The silicon process is divided into a number of

11 12 CHAPTER 2. EXISTING SIMULATION PROGRAMS.

segments in the vertical direction, and SOLGASMIX computations are carried out in each segment. Reaction kinetics are included utilising socalled "bypass factors ” (which allow some of the reactant gas to bypass segments) [12], [13] and [14]. The SOLGASMIX approach implicitly assumes that the reactions will run straight towards equilibrium. For the (ferro)siiicon process this does not seem to be the case. When fast and slow reactions proceed simultaneously, the system will try to get closer to the equilibrium, but it does not necessarily move in the direction of the equilibrium. Furthermore, the (ferro)silicon process seems to operate far from equilibrium. Thus, the kinetic description in the SOLGASMIX approach does not seem to be relevant for the carbothermic silicon metal process. This model is therefore not described further.

2.1 The stoichiometric Elkem model.

Important references for the stoichiometric Elkem model are Schei and Larsen [2], Schei and Halvorsen [15] and Halvorsen [5]. The following outline of the model is mainly based on [5], whereas [15] is the more detailed of the listed references. The average stationary state of the silicon metal process is simulated. The following fundamental elements are considered/included :

• Thermodynamics • Kinetics • Material balance • Heat (enthalpy) balance

Basically, the furnace is split in two spatially separated zones, a lower zone (the furnace hearth) with high temperatures (2200-2300K) and an upper zone with comparatively low temperature (2000K). Condensation reactions and conversion of carbon to SiC take place in the upper zone while the direct metal production takes place in the lower zone. The temperature of each zone is assumed to be uniform and known. This original model has been expanded by splitting the upper zone into two separate chambers to allow for inhomogenities in the incoming charge and the distribution of outgoing gas. Both the incoming charge and outgoing gas from the lower zone can be distributed as desired in the two chambers. Gas may also be removed from the lower zone without interacting with either chamber, and materials may be added directly into the lower zone. A simplified picture of the model is shown in Figure 2.1 and the expanded model is shown in Figure 2.2.

The minimum content of SiO in the gas required for metal production in the lower zone is defined by the equilibrium of equation 1.6 (SiO(g) + SiC(s) = Si(l) + CO{g)) which is shown in Figure 1.4 on page 9. The restriction on the SiO content in the gas is in the 2.1. THE STOICHIOMETRIC ELKEM MODEL. 13

Si02 C

Low temperature zone SiO CO 2 C + SiO = SiC + CO 2 SiO = SiQz+ CO Si02 C SiC Si A A III! 1 1 V V 1 V SiO CO —__ a SiO; + b SiC + cC =xSi + y SiO + z CO + u SiC ; vX SiC . SiC : Si High temperature zone (hearth)

Figure 2.1: Simplified picture of the stoichiometric Elkem model ([5]).

Quarts Carbon materials Silicon carbide Agglomerates Straight to SiOj C SiC SiOz + v C inner zone

Chamber A Chamber B Carbon : 2 C + SiO = SiC + CO Carbon : 2 C + SiO = SiC + CO Agglom: Si02+ 3 C = SiC + 2 CO Agglom: Si02+ 3 C = SiC + 2 CO Condensation : 2 SiO = Si02+ CO Condensation : 2 SiO = Si02+ CO

SiOj C SiC Si SiOz C SiC Si

Reaction in the inner zone : aSiOz + bSiC + cC =xSi + ySiO + zCO + uSiC

Figure 2.2: The expanded stoichiometric Elkem model with reactions and material flows ([5])- 14 CHAPTER 2. EXISTING SIMULATION PROGRAMS.

model related to a thennodynamical/kinetic parameter s such that m.=—P*o ..->8 (2.i) Plot PSiO + Pco The parameter s is assumed to be constant for a given furnace and a given mode of operation, and is chosen by the simulation operator. Let the flows [moles fs\ of SiOi, SiC and C to the lower zone be denoted o, b and c, respectively. The overall reaction in the hearth zone can then be described by :

a SiC>2 + b SiC + cC = x Si + y SiO + z CO + u SiC (2.2) u represents the amount of SiC that is deposited in the lower zone. The material balances for Si, C and O provide three independent equations for the four unknown coefficients x, y, z and u. The fourth equation is deduced by observing that the ratio between the gas pressures must be equal to the ratio between the gas flows :

PSiO 55 (2.3) PSiO + Pco The equations are solved, requiring non-negative values for the unknown coefficients in 2.2. Two situations may occur :

• a true stable solution with no SiC build-up (u = 0) and where the SiO-pressure generally is higher than the limit given by equation 2.3. • a solution where SiC is deposited in the hearth (u > 0) and where the SiO-pressure is given by the equality sign in equation 2.3.

The enthalpy balance is used to compute the energy requirement when the hearth temper ­ ature is known and the s-parameter is fixed. A reactivity number r specifies the fraction of carbon that reacts in the upper zone ac­ cording to reaction 1.7 (2C(s) + SiO(g) = SiC(s) + CO(g)) provided a sufficient amount of SiO{g) is available. The main condensation reaction is assumed to be the strongly exothermic reaction 2SiO -4 Si+Si02 (equation 1.5 proceeding backwards). The reaction takes place in a temperature range around 2000K (determining the reaction temperature of the upper zone). The raw materials are in the model first heated to 2000K, and then react at this tempera ­ ture before the products are transferred to the hearth zone. The enthalpy balance for the upper zone determines the amount of SiO that condenses. The overall conversion for the entire furnace is found by an iterative procedure; first solving the equations for the upper zone, assuming certain (guessed) SiO- and CO-flows from the 2.2. THE DYNAMICAL UNIDIMENSIONAL ELKEM MODEL. 15

hearth. The solution for the upper zone provides the material flow needed for solving the equations for the hearth zone, which in turn provides corrected gas flows as input to new calculations for the upper zone. Such iteration cycles are repeated until converging within a sufficient accuracy. The model is implemented in Excel. Most of the calculations have been entered in a normal spreadsheet. The iterations, input checks, dialogues, and some special evaluations have been programmed as Excel macros. The stoichiometric model has been applied for educating process metallurgists and furnace operators in the Elkem company. It has proven useful for transferring ideas and information between process metallurgists and researchers/theoretical experts. It has, although being a stationary model, been useful also for some dynamic discussions. A situation with SiC gradually building up in the furnace hearth, leading to decreased active hearth volume, may for example be simulated by increasing the s-parameter. The resulting gradual change in the process conditions can be visualised. The model can also be applied to describe production of iron-silicon alloys containing from about 60 to 100% Si (since the silicon chemistry is dominating under these conditions).

2.2 The dynamical unidimensional Elkem model.

The dynamical unidimensional model, developed and used by Elkem a/s (Halvorsen, Down ­ ing and Schei [8], [16] and Halvorsen [10]), is the only successfully implemented dynamical simulation program for the silicon metal process known to the author. The following de­ scription is based on [8] which is the most important reference. Most formulations are taken directly from this reference. The calculations are carried out for a vertical shaft extending from a coarsly described high-temperature zone (the furnace hearth) up to the top of the furnace where fresh raw materials are charged. The model describes the height variations in the active part of the furnace for a lm2 cross section area. The shaft is partitioned in the vertical direction as shown in Figure 2.3. In the order of 10 segments are normally used. No gradients exist in other coordinate directions. Focus is on the metallurgical aspects of the upper part of the furnace (the furnace shaft). Interaction between electrical conduction and the metallurgical process are neglected. The electrode is represented by a constant heat source. Only the pure components C, Si02, SiC, Si, SiO(g) and CO(g) are considered. The condensate (denoted Sz'O-condensate or ”SiO”) is assumed to be \Si + \SiO2 and is treated as if it were a separate species. For each segment the model keeps track of the enthalpy and the local amounts of solids (C, Si02, SiC and ”SiO”), while the amounts of C, SiC and molten ”SiO” are chosen as state variables for the hearth. It is assumed that a certain amount of silicon is always 16 CHAPTER 2. EXISTING SIMULATION PROGRAMS.

Furnace shaft

Power ;\ Furnace heart

Figure 2.3: Furnace hearth and ’’shaft ” partitioned into segments in the dynamical uni ­ dimensional Elkem model ([8]). present in the hearth. Silicon produced is continuously removed (tapped) at the hearth temperature. In addition to computing the time evolution of the state variables, some additional variables are integrated. These are the heat and material balance variables which keep track of all material and energy that enters or leave the furnace : amount of C and Si02 charged, net production of silicon, SiO(g) and CO(g), heat content in tapped Si, etc. Remaining variables like gas pressures, gas fluxes, reaction rates etc. can be computed when the state variables are known. The chemical reactions 1.3-1.7 and 1.11 have been considered together with melting. Reac­ tions 1.5-1.7 proceeding as written are included in the furnace hearth, while reactions 1.5 (proceeding only from right to left), 1.7 (proceeding both ways) and 1.6 (proceeding both ways) are included in the shaft. Reactions 1.11 and 1.4 together with an "agglomer­ ate solid” are included when charging agglomerates consisting of intimately mixed C and Si02 in the charge (Halvorsen [17]). Reactions 1.3 is excluded both in the hearth and in the shaft. Kinetic rate laws are based on deviation from equilibrium and the gross rates are propor ­ tional to this and the concentration of the reacting material. The reaction rate [moles/m 3 s] for reaction 1.7 in the shaft is listed as a typical example :

Rc — a cc Ap e ElRr (2.4)

where 2.2. THE DYNAMICAL UNIDIMENSIONAL ELKEM MODEL. 17

a = Proportionality factor. cc = Local concentration of carbon [moles/mzs\. Ap = Deviation from equilibrium pressure. E = Activation energy in Arrhenius factor [J/mole] R = Molar gas constant [J/K mole] T = Local temperature \K\.

For the hearth reactions concentrations are replaced by the amount of material per m2 cross section area (as the height of the furnace hearth is not defined in the model). Partial differential equations are formulated for vertical mass- and energy transport. Mass conservation is formulated separately for each liquid, solid and gas component. Energy conservation adds one single partial differential equation since all materials are assumed to have the same temperature. The equations for the material- and heat balances in the shaft constitute a coupled set of non-linear partial differential equations, while the hearth is described by a set of non-linear ordinary differential equations. The two sets of equations are linked through material- and heat exchange (radiation and convection) across the shaft-hearth interface. The time derivatives in the gas flow equations and in the equations for the liquid species are neglected and the gas pressure is assumed to be 1 atm throughout the furnace. The concentrations of the solid components and the enthalpy in the shaft together with the amounts of solids, ”SiO” and enthalpy in the hearth constitutes the state variables in the model. These are time integrated and the remaining quantities (reaction rates, gas fluxes and liquid fluxes etc.) are computed from them. The model has two operational modes; stoking cycles mode and continuous feed mode. In the stoking cycles mode the top level of the shaft is kept at a constant position. The shaft shrinks when voids are created due to reaction 1.6 and/or melting and the shaft-heart interface moves upwards. Stoking is performed when certain criteria are met. At stoking, the materials in the shaft are pushed down. A specified fraction of them enters the hearth. Cold, fresh charge mix is added at the top or is more or less mixed with the old warm materials. Cold materials can also be stoked directly into the hearth. Then a new cycle is started. The model parameters can here be regarded as true physical parameters. The stoking cycles mode is the only approach that can resolve variations within the stoking cycle. In the continuous feed mode the height of the furnace shaft is kept constant. The voids created in the shaft due to chemical conversion and melting are immediately filled from above. Fresh charge mix is added to maintain the top level. This mode is suited for simulating long term dynamics .(more than one day) and for finding stationary states. The model parameters are here average values over the stoking cycle. 18 CHAPTER 2. EXISTING SIMULATION PROGRAMS.

The model has been used by the Elkem company and has proven useful as a basis for educational material. Its predictions should be interpreted qualitatively and not quantita ­ tively. The model is consistent with the stoichiometric Elkem model. Both models predict a steadily increasing silicon recovery when increasing the C/Si02-ratio for undercoked charge mixes. Above a certain (optimal) ratio depending on the height of the charge, the availabil ­ ity/reactivity of SiC in the hearth etc., the silicon recovery decreases when increasing the carbon content in the charge. SiC builds up in the hearth when applying high C/SiOr- ratios, leading to reduced active hearth volume. The silicon recovery is predicted to increase with the charge height as expected. However, difficulties of mechanical and electrical nature occur when increasing the depth of the charge. For instance, larger depth

increases reactance. increases the possibility of conduction between the electrodes, increases the probability of electrode failure, increases the difficulty of stoking, increases the difficulty of furnace rotation.

An optimum charge height seems to exist in practice, but cannot be predicted by the model. The unidimensional model has been applied to predict the long-term dynamical responses to overcoking and undercoking. Examples with a stable period followed by one week with high carbon content in the charge and then one week with low carbon content are presented in [8]. The silicon recovery drops slightly at the start of the overcoked period before assymptotically approching a somewhat higher value than in the inital (stable) period (86% compared to the initial value of 85%). The recovery increases rapidly to about 90% within a day after entering the undercoked period. It then starts to decrease at an increasing rate to 87% within two days of undercoking before settling at 80% recovery within the next 12 hours. The results are discussed in [8]. Short term dynamics are predicted by the stoking cycles mode. Calculations showing typical cyclic variations of :

• the temperature distribution • the amount of SiO{g) recovered/captured in each segment • the amount of SiO(g) leaving at the furnace top are presented and discussed in [8] for two different stoking intervals. Chapter 3

Description and discussion of the model.

It is obvious that not all details of a complicated process like the carbothermic silicon metal process can be included in a simulation model. A number of simplifications are necessary to reduce the overall complexity. The background for these simplifications and their expected consequences are presented and discussed in this chapter. Detailed descriptions of the process are given to the extent needed for these discussions.

3.1 General description and main assumptions.

The dynamical unidimensional Elkem model (page 15) focuses on the area from the crater wall up to the top of the furnace where fresh raw materials are added. The high-tempera ­ ture zone underneath the crater wall is coarsly described. The dynamical model presented in this thesis focuses on this high-temperature area including the gas filled cavity, the metal pool and the lower parts of the furnace shaft. The most important chemical reactions taking place in these areas at temperatures higher than approximately 1860°C are modelled. The colder parts of the process are also modelled, but chemical reactions that are assumed to be important only below 1860°C' are not included. The conversion of C to SiC and the strongly exothermic condensation of SiO-rich gas in the upper parts of the furnace are the most important chemical reactions that are excluded. The temperature limit of 1860°C is not arbitrarily chosen. The SiO-pressure for reac­ tion 1.5 (Si02(l) + Si(l) = SiO(g)) becomes 1 atm at 1859°C. This means that the structure of the condensate, at least in theory, becomes conveniently simple above this temperature (page 36) and the model simplifies considerably by treating the condensate as pure Si02 with exactly the same properties as the Si02 from the raw materials.

19 20 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

The situation around one of the three electrodes is modelled as a two-dimensional cylinder symmetric system. Figure 3.1 shows this situation to the left and the same situation as represented in the simulation model to the right.

Axis of symmetry Raw materials (Si02 & C) j Si02& SiC

Cylindrical Inactive particles charge

Inert furnace wall Furnace Model

Figure 3.1: The furnace and the model seen from the side.

In the model all the carbon from the raw materials is assumed to have reacted with SiO(g) to SiC and CO(g) before entering the high-temperature area. Any condensate formed in the upper and colder parts of the furnace is assumed to be present as pure SiOz in the high- temperature area. No silicon is present in the furnace shaft in the model, and any silicon produced there by the chemical reactions included in the model is moved instantaneously down to the metal pool. The materials in the furnace shaft are represented in the two-dimensional model by in ­ dividual cylindrical particles (concentric rings) appearing as vertical rods in the figure. These particles are not permeable, and they may or may not be separated by vertical gas channels. Each particle in the furnace shaft has a particle of the same type straight be ­ low it in the metal pool. Materials drip vertically down from the shaft to the metal pool upon melting as explained in section 3.9.4. No materials are transported sideways in the model so materials from the furnace shaft are never transported to the area underneath the electrode. The materials in the shaft are assumed to be stationary except for the dripping. Energy is mainl y supplied to the system by the AC electric arc burning in the gas-filled crater cavity that evolves around the lower tip of the electrode. In the model the AC electric arc is replaced by a DC arc and the electrode serves as the cathode. The details concerning the electric arc are explained in section 3.8. The metal pool is believed to consist of a porous bed of silicon carbide at the bottom. Silicon metal may cover all or parts of this bed, and quartzite may either rest on the 3.1. GENERAL DESCRIPTION AND MAIN ASSUMPTIONS. 21 carbide bed or float in the liquid silicon. Liquid SiOz and liquid silicon are in practice immiscible liquids. In the model both the quartzite and the silicon carbide floats with their upper surfaces exactly level with the metal pool surface. This means that they can react both with the gas and with the silicon metal in the metal pool. The silicon metal and the other materials in the metal pool are all represented by solid materials. No convection takes place in these materials in the model. The electrode and the furnace walls are assumed to be inert. The model combines submodels for the electric arc, fluid flow, energy transport and het ­ erogeneous chemical reactions. Such a model has not previously been developed for the silicon metal process. Energy transfer by radiation from the electric arc to the metal pool and to the crater walls is included (section 3.8.4), whereas radiation between surfaces of different temperatures is neglected. Electric currents through the solid materials and possible effects thereof (in ­ cluding ohmic heating) are excluded both in the furnace shaft and in the metal pool. Heat is exchanged between the gas, solids and liquids as explained in sections 3.2.3 and 3.2.4. The basic features included in the model are shown in Figure 3.2.

Chemical reactions, Heat transfer (convection, evaporation and conduction and diffusion) condensation Fluid flow

Ohmic heat generation

Radiation

Heat conduction Dripping from wall Inert furnace wall

Figure 3.2: Basic features included in the model.

The chemical reactions occur at gas/liquid- or gas/solid-interfaces, but also between silicon metal and the other materials in the metal pool as explained in section 3.4. Evaporation of silicon takes place from the upper surface of the metal pool. The vapour may condense at all horizontal surfaces and in the gas phase as explained in section 3.5.3. 22 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

The chemical reactions and evaporation/condensation are associated with mass- and energy- transfer as described in sections 3.12 and 3.13. The various assumptions and simplifications are discussed later in this chapter. The period between two successive stokings (page 6) is simulated. The crater wall is expected to move upwards during the simulation period as a result of dripping and chemical reactions taking place at the crater wall or just above it. The simulation period is initiated by defining the particle sizes, their positions and initial values for all essential variables (such as the temperature fields, the gas composition and the flow field etc.). The state of the furnace at any given time is then calculated under the assumptions and simplifications applied in the model. Some of the most interesting outputs are the temperature distribution, gas composition, flow field, local chemical conversion, overall silicon production and geometric changes of the crater wall and the metal pool. These results are presented for the simulation operator by means of illustrative colour graphic plots. Such results are presented in chapter 5, but then only as black and white plots. Colour plots could have been used, but those would not be suited for reproduction in black and white. The model is implemented in FORTRAN-77 code applying the commercial 3D computa ­ tional fluid dynamics (CFD) code FLUENT [18], version 2.97 as a basis. The simulations are run on a HP-Apollo 9000/755 computer.

3.2 Governing equations.

The governing equations are conservation of mass, momentum, energy and turbulent quan ­ tities. These equations are presented in this section after being rearranged into the following general form : (3.1) where $ is a scalar quantity, p is the density, a, and «; are the coordinate and the velocity component in the i’th direction. The terms are from left to right : accumulation with time, convection, diffusion and production (sources). F$ represents the diffusivity of . The equations are solved numerically by a modified version of the commercial CFD code FLUENT which is described briefly in section 3.2.5. 3.2. GOVERNING EQUATIONS. 23

3.2.1 The conservation equations.

The conservation equations applied in the present work (cylindrical symmetry) are listed in the following. uz and ur denote the velocities in the z- and r-direction, respectively.

Conservation of mass. Mass is conserved according to the equation of continuity ($ = 1) :

+ (3.2) dt az r or where to is the volumetric production rate [kg/m3s] which represents the mass produced by chemical reactions.

Conservation of momentum. Momentum transfer is according to the Navier-Stokes equation :

r-direction ($ = ur) :

m (PUr) + Tz {PU* Ur) + r Jr (rpUrUr) =

- 3F 4- & [war (3-3)

+ [2rM^ “ zrPeffiy • %)] - OLt §+ Fr

z-direction (# = uz) :

S (PUZ) + Wz (Pu* uz) + (rPuruz) =

- W+ (# 4" M (3 4)

+ A [2m1t - (v • u)] - at l^(pk) + Fz

P is the pressure and yn e# is the effective viscosity (equation 3.14). FT and F, are the Lorentz forces exerted on the gas by the electric arc (page 83). at = 1 for turbulent flows and zero for laminar flows, k is the kinetic energy of turbulence (equation 3.10). V • u is given by : 24 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

L0ken Larsen [19] found recently that one term in each of the above equations was not implemented in the code despite that the documentation of the FLUENT code indicated otherwise. These terms were : d ( duz in equation 3.3.

and 1 d ( dllr in equation 3.4.

The terms were implemented and their significance for situations similar to those in the cavities of submerged arc furnaces producing silicon metal were examined [19]. It was concluded that these terms were not significant in such situations. They are not included in the simulations presented in this thesis.

Conservation of energy. The energy equation is given by (# = h) :

+ Sh (3.5) where h is the specific enthalpy of the fluid and Teg its effective thermal diffusivity :

(3.6) k is the thermal conductivity of the fluid, Cp is its specific heat capacity and /m is the turbulent viscosity (equation 3.12). The turbulent Prandtl number, Prt, is assumed to be a constant, and the default value of 0.7 is used in the present work. The simulation operator may change this value at any time. The source term [W] includes the ohmic heating, the electron drift, the radiation losses (page 84) and the enthalpy content of the gas removed from or added to the gas phase due to chemical reactions :

jz and jr are the axial and radial current densities [A/m2], a is the thermal conductivity of the gas [S/m]. is Bolzmann ’s constant and eo is the electronic charge. ST(T,rTad) is the volumetric radiation density for the gas mixture [W/m3] as described on page 85. m. and h{ are the rate of production of gas species i [kg/m/s] and its specific enthalpy [J/kg], respectively, m, is calculated from the reaction rates for each reaction. V is the volume of the considered computational cell (Figure 3.4, page 30). 3.2. GOVERNING EQUATIONS. 25

3.2.2 Turbulence modelling.

The conservation equations for turbulent flow are obtained from those for laminar flow by using a time averaging procedure known as Reynolds averaging. Scalar quantities are here assumed to be a sum of a time averaged mean value and a randomly fluctuating turbulent part : $ = + <&' (3.8) The time averaged value of the fluctuating part ($') is equal to zero. The turbulent versions of the conservation equations are obtained by substituting this equation into the conservation equations for the laminar case and integrating over a suf­ ficiently large time interval. The resulting equations are of the same form as those for laminar flows, but with each quantity represented by its time averaged value and a new term (u-i1') that multiplied by the density represents the transport of $ due to turbulent fluctuations. In the momentum equations this correlation becomes uj-u'-. This corresponds to the Reynolds stresses when multiplied by the density. These are obtained from the k-e model which is an eddy-viscosity model based on the Boussinesq hypothesis; that is: the Reynolds stresses are assumed to be proportional to the mean velocity gradients and the constant of proportionality is the turbulent eddy viscosity, p t. This results in the following expression for the Reynolds stresses (which are analogous to the shear stresses in the laminar case) :

(3.9) where k is the kinetic energy of turbulence [m2/s2] :

(3.10) and the Kronecker delta is defined by : 1 when i = j (3.11) 0 otherwise

The turbulent viscosity is given by : _ &2 At = PUp — (3.12) e is the dissipation rate for turbulent energy [m2/s3] as defined by equation 3.13. is a constant (Table 3.1, page 27). ______du'i du'i (3.13) dxj dxj 26 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL. v is the kinematic viscosity [m2/s] of the fluid. The turbulent momentum equations remain formally identical to their laminar counterparts except that the molecular viscosity is replaced by an effective viscosity given by the sum of a molecular and a turbulent part :

Peff = fi + Mt (3.14)

The extra terms 7 {pu'iu'j) dx. are embedded in the turbulent part of the effective viscosity and in the terms multiplied by at in equations 3.3 and 3.4. The velocities denote the time averaged velocities. The k-e model requires that the two transport equations for k and e are solved : Turbulent kinetic energy ($ = k) :

§i (Pk) + Tz (rpurk) = (3.15) - & + r3F

Turbulent energy dissipation (4? = e) : §i (P£) + Tz (P^£) + rT (7’PUre) = (3.16) - £(S£)+&(••££)+* where Sk = Vt ■(£) +#) +eMiMr) (3.17)

C, ,2 St = C1ef-C2p e- (3.18)

The constants Cu C2, ok and

The k-e model may give inaccurate results when applied to complex flows where buoyancy, swirl, strong streamline curvature or density gradients are important phenomena (Holt [20, page 201]). 3.2. GOVERNING EQUATIONS. 27

Table 3.1: Parameter values used in the k-e model [18].

C, Ci c2 <7fc 0.09 1.44 1.92 1.00 1.30

Based on estimations of the Reynolds number, the simulation operator decides if the lam­ inar or turbulent versions of the equations shall be applied for each simulation run. The Reynolds number is given by : LcVcP Re = where Lc is the characteristic length scale of the system and Vc its characteristic velocity. Turbulence modelling applies to most cases since the implemented model for the DC electric arc (section 3.8.1) sets up high flow velocities in the electric arc and along the anodic metal pool. The algebraic stress turbulence model is available [18], but is not applied in the present work.

3.2.3 Heat transfer in solid materials.

The equation for energy transfer in solid materials is similar to the energy equation for fluids (equation 3.5), but it is solved with respect to the temperature and not the enthalpy. The equation for the solids is derived using the relation dh/dT = Cp and canceling the terms containing the flow velocities since there is no convective mass transport in the solids :

(3.19) kw is the thermal conductivity for the solid (wall). Its value at the interface between successive computational cells is estimated by a harmonic mean value : k\k 2 kw = 2 (3.20) ki + &2 where ki and k2 are the thermal conductivities for the materials at the node points of the cells as shown in Figure 3.3.

The source term, St [W], represents the energy added by chemical reactions as described in section 3.13. 28 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

Solid materials (wall cells) possibly of different types

Figure 3.3: Definition of variables used when estimating the thermal conductivity at the interfaces of solid materials.

3.2.4 Energy transfer at gas/liquid- and gas/solid-interfaces.

The heat flux across the gas/liquid- and gas/solid-interfaces are included as a source term comprised of three terms :

S>h,trans — (jZrad "b Qanode “b Qgas'} Acen (3.21) where

• Aceii is the surface area where the heat transfer takes place. • Qrad is the radiation flux from the electric arc as given by equation 3.123 on page 86. • Qanode is the heat flux from the electric arc to the anodic metal pool underneath the electric arc as given by equation 3.141 on page 92. The heat transfer between the electrode (the cathode) and the electric arc is not included (section 3.8.5). •

• qgas is the heat flux from the gas to the liquid/solid, and is calculated applying the wall functions for transport of energy through a turbulent boundary layer presented by Johansen [21] : (3'22)

where p is the density of the gas, cj> is the heat capacity of the gas, uT is the friction (shear) velocity as given by equation 3.33 on page 44. y+ is a dimensionless distance from the interface as defined by equation 3.34 on page 44 and g(y+, Pr) is a function describing the temperature profile in the turbulent boundary layer close to the inter ­ face (equation 3.40, page 45). The second argument to this function is the Prandtl

j 3.2. GOVERNING EQUATIONS. 29

number (Cpn/k ) and not the Schmidt number which is applied for mass transfer. The turbulent Schmidt numbers should likewise be substituted by the turbulent Prandtl number in the appropriate equations. T(yn) is the temperature at the node position of the near wall node (yn) and Tw is the temperature at the interface (the surface of the wall).

3.2.5 Numerical solution of the governing equations.

The numerical solution of the governing equations are based on version 2.97 of the 3D computational fluid dynamics code FLUENT [18]. FLUENT was primarily developed for fluid flow simulations at low and moderate temper ­ atures, and the current version is implemented in FORTRAN-77 code. The original code has been changed in several ways by research personell at SINTEF/NTH over the last decade to make it applicable to high-temperature metallurgical processes, but also to im­ prove several of the models implemented in the original code. Further modifications were necessary while developing the present model. The major contributions from the present work are the introduction of

• Heterogeneous chemical reactions. • Evaporation and condensation of silicon. • Changes of particle geometry due to chemical reactions, evaporation and condensation. • Dripping of materials from one particle to another due to melting. • Modifications of models for the electric arc and radiation from it.

Version 2.97 of the FLUENT code is not particulary modular and it is rather poorly documented. It is therefore difficult to understand the interactions between the different parts of the code or the significance of the numerous variables defined in the program. This makes it difficult to implement new models or to improve already existing models. The more recent versions seem to be considerably better in this respect, and they have much more powerful and quicker numerical routines for solving the resulting system of linear equations. Some recent versions also support body-fitted coordinates and unstructured grids. Transferring models or changes from older versions to more recent versions is cumbersome since the basic structure of the versions are quite different. This is why version 2.97 is applied even though new versions have been released during the present work. The partial differential equations for conservation of mass, momentum, energy and turbu ­ lent quantities are solved numerically by applying a control volume based finite difference method on a staggered grid following the ideas presented by Patankar [22]. The equations 30 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL. are reduced to their finite difference analogues by integrating over the control volumes of the computational grid. First order approximations are used for differentials, and a fully implicit formulation is applied. The source terms are linearised :

Si> — SUA, — Sp t$$

The details are described in [18]. As previously mentioned, 2D cylindrical symmetry is assumed in the present work. The control volume (the computational cell) is in this case as illustrated in Figure 3.4. The grid is generally non-uniform, and the cell faces are located midway between successive node points.

A z Upper cell face Outer cell face

Inner Node position

Lower cell face

Figure 3.4: Computational cell (control volume) in 2D cylindrical symmetry.

All dependent variables except the velocity components are stored at the node position. The velocity components are evaluated at the cell boundaries using a staggered grid. The radial velocity is evaluated at the inner cell face (r = rm) and the velocity in the z-direction is evaluated at the lower cell face (z = zm). A first-order Power Law scheme or the QUICK (Quadratic Upstream Interpolation for Con ­ vective Kinematics) scheme is used for interpolation [18]. The system of linear equations is solved by applying an iterative line-by-line matrix solver. The SIMPLE algorithm (Semi-Implicit Method for Pressure-Linked Equations) is applied for calculating of the flow field (Patankar [22, p. 126]). The important operations, in order of their execution, are : 3.2. GOVERNING EQUATIONS. 31

1. Guess the initial pressure P*. 2. Solve the momentum equation to obtain the velocity componentes. 3. Solve the pressure correction equation derived from the continuity equa­ tion to obtain an improved estimate for the pressure. 4. Apply the velocity-correction formulae for obtaining improved estimates for the velocity components. 5. Solve the conservation equations for other quantities (such as concentra ­ tions, energy/temperature and turbulence quantities). 6. Treat the corrected pressure as a new guessed pressure P*, return to step 2 and repeat the whole procedure until a converged solution is obtained.

The pressure correction equation and the velocity-correction formulae are presented in Patankar [22, pp. 123-126].

Progressing from one time step to the next. Including production and consumption of solids and liquids has resulted in one important change (step 3 in the algorithm presented below) when progressing from one time step to the next. Assume that the estimated situation at a certain point in time (time step number n) is known. The following sequential steps are then carried out to estimate the situation at the next time step :

1. Start with the solution for the previous time step as the initial guess for the new time step. 2. Improve this solution iteratively until convergence including all effects except mass transfer in solids and liquids1. 3. Account for changes in the mass of solids and liquids due to chemical reactions, evaporation, condensation and dripping as described in sec­ tions 3.9.6 and 3.10.6. 4. Accept this solution as the final solution for this time step and proceed to the next by returning to step 1 above with this solution as the initial guess.

Applying this method, assures that the geometry of the system is completely unchanged during the iteration process on each time step (step 2 above).

1The effects of mass- and heat transfer with the gas phase and the heat transfer to/from the solids and liquids caused by chemical reactions are, however, accounted for in this iteration process. 32 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

Comment to the enthalpy/temperature relation. The enthalpy/temperature relation is in the standard FLUENT code defined by :

h = CpT

This is not useful when the heat capacity varies strongly with the temperature. The en ­ thalpy/temperature relation is instead defined by tabulated pairs of enthalpy and tempera ­ ture values. Linear interpolation is applied to find the values at intermediate temperatures. The corresponding specific heat capacity is evaluated by : dh Cp = dT

This modification was first implemented by Holt [20].

3.3 Chemical compounds.

Si02(s, l), C(s), SiC(s), Si(s,l,g), CO(g), SiO(g) and a condensate comprised of Si02, Si and possibly some SiC and C are the chemical compounds believed to be important for the metallurgy of the carbothermic silicon metal process. These materials and their significance for the process and the model are discussed in the following

3.3.1 The furnace gas.

SiO(g), CO(g) and Si(g) are the gas species considered in this thesis. Some additional neutral compounds and atoms plus several ionic compounds are important for the be ­ haviour of the electric arc. They are, however, believed to be of minor importance for the metallurgy of the process and are neglected in the model. SiO(g) and CO(g) are produced or consumed in heterogeneous chemical reactions. Si(g) enters the gas phase by evaporation of liquid silicon, chemical reactions or dissosiation at high temperatures. Production of Si(g) by chemical reactions and dissosiation are excluded. Si(g) is present in significant amounts above approximately 2800K only [23]. It either reacts with other materials at elevated temperatures or condenses to Si(l) when arriving at colder locations. Chemical reactions involving silicon vapour may be important in the crater, and especially so close to the surface of the metal pool as discussed by Muller, Olsen and Tuset [24]. Evaporation and condensation of silicon are included in the model, but the Si(g) concen­ tration is not calculated (section 3.5.3). Si(g) is therefore not included in the gas phase. 3.3. CHEMICAL COMPOUNDS. 33

Homogeneous reactions in the gas phase are in general neglected and the gas species are assumed to have identical temperatures at the same location.

3.3.2 Carbon

Carbon is present as lumpy and highly irregular porous carbon particles from the raw materials, compact carbon materials in the electrode and possibly as small particles in the condensate. Carbon in the condensate is described in section 3.3.4. The temperatures are high in the area where the electric arc attaches to the electrode. Significant electrode errosion by evaporation is observed in this area. Evaporation of carbon from the electrode keeps the temperatures of the electrode surface down and leads to more carbon entering the crater. Also, the electrode surface may be subject to chemical reactions. Assuming that the electrode is inert is therefore dubious, but is done to simplify the model. Equation 1.7 (SiO(g) + 2C(s) = SiC(s) + CO(g)) is as explained in section 1.3 the most important reaction for consuming carbon in the raw materials. Carbon may in principle also react according to equation 1.3 (SiC>2(l) + C(s) = SiO(g) + CO(g)), but this reaction is believed to be slow since the carbon is covered by an SiC-layer from reaction 1.7 at T > 1787K. Reaction 1.3 is for this reason excluded in the model (section 3.4.2). All the carbon in the raw materials is consequently assumed to react according to equation 1.7. Reaction 1.7 is topochemical, meaning that the reacting carbon particle preserves its orig­ inal shape and size during the reaction. The particle is gradually converted to SiC from its outer edge. New SiO(g) is transported to the reacting C/SiC-interface by diffusion through the porous SiC product layer. The reaction front moves inwards with time, grad­ ually slowing down the reaction rate. A typical carbon particle may after a certain time be as indicated in Figure 3.5. The reaction rate depends on the gas diffusion through the SiC-layer. The size of the particle, its porosity as well as the shape of the pores are important parameters in this respect. It is favourable for the process if as much carbon as possible reacts to SiC by reaction 1.7 high up in the furnace. This implies that :

* more SiO(g) is consumed in the low-temperature zone, thus reducing the amount of SiO(g) that must condense there. • less carbon enters the high-temperature zone because some of it is con ­ verted to CO(g) by reaction 1.7 in the low-temperature zone and leaves with the off-gas.

The latter also reduces the amount of SiO(g) that must condense in the low-temperature 34 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

Gas, somewhat depleted in SiO

Unreacted carbon

SiO(g) +2C(s) SiC(s) + CO(g) A

SiO-rich gas

Figure 3.5: Topochemical conversion of porous carbon particles to SiC. The reaction front moves inwards while the outer surface of the SzC’-particle remains un ­ changed. zone. The reason is that free carbon entering the metal producing zone must necessarily leave again as CO-gas (or be deposited as SiC). The gas leaving this zone has a high content of SiO(g) as explained in section 1.3. A decreasing amount of carbon entering this zone therefore leads to less CO(g) and thus also less SiO(g) leaving it. Less SiO(g) must consequently be recovered in the upper parts of the furnace. Fairly small carbon particles with high reactivity with respect to SiO(g) are therefore normally applied to achieve large conversion to SiC in the upper parts of the furnace 2. Observations indicate that some unreacted carbon despite this enters the metal producing zone, but it is assumed in the model that all of the carbon in the raw materials is indeed converted to SiC before entering the high-temperature zone. This means that carbon is not present as a chemically active agent, and the model simplifies considerably. About 50% of the original amount of carbon in the raw materials is substituted by SiC in the model. This value is deduced directly from the stoichiometry of reaction 1.7 which shows that only 50% of the carbon ends up as SiC. The remaining 50% leaves the furnace as CO(g) with the rest of the off-gases. A somewhat smaller fraction is converted to SiC if some carbon is also consumed by reaction 1.3. Carbon entering the high-temperature zone by evaporation from the electrode or from other sources has the same effect as described above for unconverted charged carbon.

2There is a lower practical limit, to the size of the carbon materials since too small particles are found to have an unfavourable influence on the permeability of the charge. 3.3. CHEMICAL COMPOUNDS. 35

3.3.3 Silicon carbide.

Solid SiC forms from the carbon in the raw materials as explained in section 3.3.2. Some SiC may also be generated by the reactions 3SiO(g) + CO{g) -> 25z02(Z) + SiC(s) and 2Si(l) + CO(g) -¥ SiO(g) + SiC(s) (reactions 1.4 and 1.6 both proceeding from right to left). SiC from both these reactions joins the condensate as described in section 3.3.4. In the high-temperature zone, SiC reacts with SiO(g) according to reaction 1.6 as de­ scribed in section 3.4.5. SiC may also react with Si02 according to reaction 1.4. This reaction is excluded in the model as explained in section 3.4.3. SiC remains solid until it decomposes to Si(g) and C(s) at approximately 2800K. This reaction is neglected since it is not likely that the SiC reaches 2800K anywhere in the process under normal process operation. The SiC is assumed to be of the P modification in all parts of the furnace.

3.3.4 The condensate.

The condensate mainly consists of solid or liquid silicon and Si02 (Schei and Sandberg [25]). Additional small amounts of solid carbon and SiC may be present. The condensate is formed when gases from the metal producing zone are cooled in the upper parts of the furnace. They react to form solid or liquid products according to equations 1.3,1.4 and 1.5, all proceeding from right to left. Reaction 1.5 (2SiO(g) ~ SW2(Z) + Si(l)) is for kinetic reasons by far the most important condensation reaction and is believed to take place in a temperature interval from 1800-2000K. The condensation is believed to take place at the surface of solid or liquid materials. The silica in the condensate is amorphous and it appears as a glassy substance even at temperatures below the melting point of cristobalite. All parts of the condensate is solid below the melting point of silicon (1685K). However, such solid condensate is almost non ­ existent during normal process operation since the equilibrium pressure of SiO for the reaction 2SiO —> Si02 + Si is less than 10-2 atm at the melting point of silicon. Most of the condensate is thus generated at temperatures above the melting point of silicon, possibly by rapid cooling to an amorphous metastable condensate which decomposes into Si(l) and glassy silica when heated. It may also disproportionate directly into these prod ­ ucts. Independent of the way it forms, the metal appears as small silicon droplets in a silica matrix together with small particles of solid carbon and silicon carbide as indicated in Figure 3.6 and described by Schei and Sandberg [25]. The C- and SiC-particles are extremely small since they are formed in small amounts together with large amounts of Si02 and are thus immediately trapped in the silica matrix. 36 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

Si-droplets Tiny SiC or C particles

Figure 3.6: Condensate structure.

The reaction rates for the various condensation reactions change differently with the tem­ perature. This implies that the Si/Si02-ratio, the SiC/Si02-ratio and the C/SWg-ratio in the newly formed condensate vary with the temperature at which the condensation takes place. This enhances concentration gradients inside the condensate. Liquid silicon has low viscosity, and the tiny silicon droplets may coalesce and form bigger droplets. Some of the silicon may eventually separate out from the condensate and flow rapidly down towards the metal pool as proposed by Schei [1, ch. 4, p. 40]. The carbon, silicon carbide and the remaining silicon droplets in the condensate may undergo chemical reactions when heated. They are in intimate contact with Si02, and the reactions are probably fast when the total reaction pressure for the given reaction exceeds the ambient pressure of approximately 1 atm. The product gases are under such conditions easily purged out of the condensate and into the bulk gas provided that the condensate is not too compact in its structure. The pressure of SiO for the reaction between silicon and silica reaches 1 atm at 2132K, whereas the SiC gives a reaction pressure of 1 atm at 2086K when reacting with silica. Carbon, on the other hand, reacts with Si02 to SiC and CO of 1 atm at 1787K. The details are given in the description of the chemical reactions in section 3.4. The consequence is that the condensate is gradually depleted with respect to silicon, carbon and silicon carbide as the temperature increases. From a thermodynamical point of view, only pure silica remains above 2132K (= 1859°C). The condensate may, however, for kinetic reasons contain other compounds than Si02 also above this temperature. The condensate forms on the surface of silicon carbide and silica particles (and on other condensates). It is highly viscous and probably covers parts of the solid particle surfaces with a condensate film as indicated in Figure 3.7. The thickness of the condensate layer varies on different particle types and at different par ­ ticle locations due to temperature differences, competing chemical reactions and different surface conditions. 3.3. CHEMICAL COMPOUNDS. 37

Figure 3.7: Particle partly covered by condensate. The condensate may slow down chem ­ ical reaction on the blocked surface.

The condensate film may be important for the kinetics since gas penetration through the condensate film is probably slow. It may thus inactivate some of the surface area that would otherwise participate in chemical reactions. In the model this is compensated for by reducing the surface area that participate in chemical reactions accordingly. Correction factors for such blockages are defined explicitly by the simulation operator as explained in section 3.9.3. Estimating the fraction of the surface that is inactivated for a given particle is difficult. The condensate gradually becomes less viscous as the temperature increases. This implies that the physical shape of the condensate at high temperatures becomes more dependent on the wetting conditions of the condensate towards silicon carbide and silica particles. The condensate may remain as a film even at moderate viscosities at good wetting conditions, but probably forms relatively large lumps between the particles at poor wetting conditions. The wetting conditions for SiC and Si02 at conditions similar to those prevailing in the carbothermic silicon metal process are not known. In the model the chemical reactions producing condensate are all excluded since they take place below 1860°C only. Also, the condensate is excluded as a separate substance in the model. It is instead present as pure Si02 with the same geometry and the same physical and chemical properties as Si02 from the raw materials. This seems reasonable since the condensate becomes pure Si02 from a thermodynamical point of view above 1859°C as already explained. The Si02 produced by the condensation reactions is in practice accounted for in the model by adjusting the SiC/Si02-r&tio of the materials in the furnace shaft before starting a simulation as explained in section 3.3.7. Si produced by condensation is not accounted for in the model. This has no serious chem­ ical consequences provided that the silicon that precipitates out from the condensate does 38 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL. not react significantly with Si02 in the lower parts of the furnace shaft. Otherwise, addi­ tional SiO(g) is produced and the reacting surfaces are cooled by this strongly endothermic reaction. The residence times for this silicon in areas where reaction 1.5 can proceed are probably so low that excluding it from the model is a reasonable simplification. Excluding it implies that silicon produced in the low-temperature zone is not included in the total production rates calculated by the model. In the model any effects of possible SiC and C inside the condensate are excluded.

3.3.5 Silica.

The Si02 in the raw materials is compact, irregular lumps with low reactivity with respect to the surrounding materials and gases at low and moderate temperatures. It reaches down to the crater wall practically unchanged. Molten silica is highly viscous and is treated as solid particles in the furnace shaft. Ex ­ cavations performed after shuting down furnaces indicate that this may be a reasonable assumption. Treating silica in the metal pool as solid particles is more dubious. However, molten silica and silicon mix poorly, and the silica floats in a possible pure silicon metal pool. Moreover, it is the chemical conversion of silica in the metal pool that is important. The contact area between the silica and silicon is the essential factor in this respect, not whether the silica is liquid or solid. Treating the silica in the metal pool formally as a liquid thus probably adds little to the model, and it is for convenience treated as solid particles just like the silica elsewhere. The shape, size and location of the particles may, however, be important for the chemical conversion. Si02 mainly reacts with Si in the high-temperature zone and produces SiO(g) according to equation 1.5. This reaction is included at Si/S'fOa-interfaces in the metal pool and at gas/5i02-iiiterfaces both in the metal pool and in the crater wall. The reaction at gas/SiC>2-interfaces takes place with silicon condensing from the gas. Any reaction between Si02 and Si flowing down through the furnace shaft after precipitating out from the condensate is excluded. Reaction 1.4 is excluded both in the furnace shaft and in the metal pool. Si02 is assumed to be quartz below 1079K and high cristobalite above this temperature until melting at 1996K 3. Silica produced by condensation is described in section 3.3.4.

3Ttidymite is the thermodynamically stable modification in the temperature range from 1140K to 1743K [1, ch. 2, p. 6]. However, according to the same reference, the transformation to tridymite is normally slow. Which modifications that are present at low and moderate temperatures is believed to be of minor importance since it is assumed to be chemically inactive below the melting point. 3.3. CHEMICAL COMPOUNDS. 39

3.3.6 Silicon.

Silicon melts at 1685K. Liquid silicon is produced by condensation of SiO-rich gas in the low-temperature zone and by reaction 1.6 (SiO(g) + SiC(s) -> 2Si(l) + CO(g)) in the high-temperature zone. Liquid silicon has low viscosity, and any silicon in the furnace shaft probably flows rapidly down to the metal pool unless it is trapped in the SiOs-matrix (page 36). This silicon may react with silica from the raw materials or in the condensate on its way down to the metal pool. It also exchanges energy with the other materials in the shaft. Silicon from the furnace shaft enters the metal pool either by flowing along the sides of the crater wall or it falls as droplets across the crater cavity. Some of it may gather in small £z(Z)-pools resting on top of SzC^-particles and react there. In the model silicon produced by condensation of SzO-rich gas is excluded as mentioned in section 3.3.4. Silicon is therefore only produced by reaction 1.6 in the furnace shaft. This silicon is assumed to move instantaneously down to the metal pool without interacting with the other materials in the shaft in any way. It is added directly into the rest of the silicon in the metal as described in section 3.10.6. Thus, no silicon gathers on top of SiC^-particles in the metal pool and reacts there. Instead, it reacts at Si/Si02-interfaces down in the metal pool, evaporates, remains in the pool or is transferred to the tapping ladle. Some evaporated silicon reacts on gas/SiC>2-interfaces in the shaft or in the metal pool after condensing there. Leaving out the reaction with silicon gathering on top of Sz02-particles in the metal pool may be dubious. The heat fluxes to these gas/Sz02-interfaces are large compared to the heat fluxes at Si/Sz02-interfaces down in the metal pool. The reaction at gas/Si02- interfaces may therefore be important. On the other hand, most of the heat that is not consumed by reaction 1.5 at gas/5zC>2-interfaces as it should be is transported by conduc ­ tion to the Si/SiOa-hiterfaces where the reaction is included. The net chemical conversion may possibly be quite similar in the two situations, but the temperature profiles are dif­ ferent.

3.3.7 Material composition in the furnace shaft.

The simulation operator defines the geometry of the furnace shaft before starting a simula­ tion. Only SiC and SiC2 are present except for inert materials. There are no restrictions to the composition, but it is natural to choose compositions that are close to those found in the lower parts of the furnace shaft during normal operation. The average composition of the materials entering the high-temperature zone varies with the silicon recovery and with the fraction of indirect silicon production by condensation (2SiO(g) —> Si(l) + Si02(l)) in the furnace shaft as shown in Figure 3.8. It is assumed 40 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

g 0.90 85% recovery

h 0.80 100% recovery 3 0.70

S 0.60 75% recovery

r7 0.50

fraction of Si produced by condensation (2Si0(g) = Si(I) + Si02(I))

Figure 3.8: The composition of the materials entering the high-temperature zone of a fur­ nace at various silicon recoveries and fractions of indirect silicon production by condensation (2SiO(g) = Si(l)+Si02(l)) in the furnace shaft when all car­ bon in the raw materials reacts according to SiO(g)+C(s) = SiC(s)+CO(g) before entering the high-temperature zone. 3.4. CHEMICAL REACTIONS AND REACTION RATES. 41 in these calculations that all carbon in the raw materials reacts according to equation 1.7 (SiO(g) + C(s) = SiC(s) + CO{g)) before entering the high-temperature zone. 80-90% silicon recovery and about 30-35% of the silicon being produced by the given condensation reaction are typical values for normal furnace operation. This corresponds to mole ratios of SiC to Si02 in the range 0.70-0.75 according to Figure 3.8. Silicon carbide is expected to accumulate in the metal pool at larger SiC-values. The total mass ratio for the materials in the furnace shaft is different from the mass ratio in the crater wall since the particle heights differ (Figure 3.1, page 20). Calculating the mass ratio after normalising all particles to a common height gives a better estimate for the composition of the materials entering the furnace shaft. This ratio is given by :

where pi is the density of particle number i and A; is the cross section of its lower end. In the model the various materials are represented by a number of separate pure particles as shown in Figure 3.1 on page 20. This means that the composition of the materials actually entering the high-temperature zone depends on how the geometry of the crater wall changes with time. Enhanced removal of materials from areas with much SiC results in more SiC entering the high-temperature area than given by rmasS]norm and so on. Despite this, it seems reasonable to use values for Tmas^nm-m in the range 0.70-0.75 when defining the initial state of the system.

3.4 Chemical reactions and reaction rates.

The chemical reactions believed to be important at the conditions prevailing inside the furnace were as mentioned in section 1.3 : Ri Si02 + C = SiO + CO (1.3)

r2 2Si02 + SiC = 3 SiO + CO (1.4) Rz Si02 + Si = 2 SiO (1.5) Ri SiO + SiC = 2Si + CO (1.6) Rz SiO + 2C = SiC + CO (1.7) 42 CHAPTER 3. DESCRIPTION AND DISCUSSIONOF THE MODEL.

These reactions are in principle reversible and are characterised by rate constants for the reactions in both directions. The forward rate (kf) is to the right and the backward rate (kb) is to the left. Reactions to the right are defined to be positive. In the model reactions involving gases are assumed to take place only after absorption on solid or liquid surfaces. Thus, the reactions are all heterogeneous and are in general controlled by mass transport to and from the reacting interface or by the intrinsic rate of the reaction. The gas transport to/from this interface then takes place through laminar or turbulent boundary layers, possibly combined with diffusion through porous product layers. The intrinsic reaction rate depends on the temperature, and so does the mass transfer due to the influence of temperature on the partial pressures at the reacting surfaces. The heat flux to the surface is therefore important, and is in practice rate controlling for boiling and for reactions such as equation 1.5 above a certain temperature as explained in section 3.4.4. The general form of chemical reaction rates under different kinetic conditions are described by Kolbeinsen [7], Halvorsen [26] has used this to set up chemical reaction rates for some of the listed reactions. Reactions 1.3-1.7 are analysed in the following and reaction rates are derived for those included in the model. First, however, general equations for transport of gases through laminar and turbulent boundary layers as applied in the model are presented.

3.4.1 Transport of gas through boundary layers.

A common approach is to assume that transport of a gaseous component i to a surface (a wall) is governed by the following simplified expression :

ji = kg(pi-pi tW) (3.24)

where Ji = Mass flux [moles/m 2s] of species i to the surface. kg = Mass transfer coefficient for the gas through the boundary layer surrounding the particle [moles/m 2s]. Pi,w = partial pressure of component i at the surface. Pi = partial pressure of component i in the bulk gas. It is difficult to determine the parameters needed for applying this expression. Johansen [21] has developed expressions for heat and mass transfer through turbulent boundary layers. These expressions are applied in the model and are presented in the following together with an expression for the laminar case. 3.4. CHEMICAL REACTIONS AND REACTION RATES. 43

Transport in turbulent boundary layers involves complicated transport processes where pressure effects, turbulent dispersion and molecular transport interact. Several methods are available, but many of them (for instance as described by Nagano and Hishida [27] and Yakhot, Orzag and Yakhot [28]) require that the turbulent boundary layer is resolved by the computational grid and thus demand large computer resources. Methods that do not require such fine grid have thus been developed. The standard technique is to relate the values computed for the first computational cell outside the reacting surface (the near wall node) to the values at the reacting surface (the wall) by socalled wall functions. Johansen [21] points out that good wall functions are difficult to derive and that they are often very complicated, especially for heat- and mass transport. The work of Loughlin, Abul-Hamayel and Thomas [29] is listed as an example. Johansen further addresses the need for simple wall functions suitable for numerical applications, and offers such functions where the pressure gradient effects are neglected. The expressions for heat transfer are presented explicitly. The expressions for mass transfer are obtained from these by replacing the Prandtl number with the Schmidt number in these expressions. The wall functions thus obtained are applied in the present model. The equations for mass transport are listed below. The mass flux [kg/m 2s] to the reacting surface is given by :

j = D (3.25) where D is the diffusion coefficient for the transported species [m2/s], C(y) is its mass concentration [kg/m 3 ] at a distance y [m] from the wall and the subscript w indicates that the value is to be evaluated at the wall. The reaction rate at the surface [kg/s] is calculated by multiplying with the total area of the reacting surface :

R = jA (3.26)

Dimensionless quantities for distance y+, mass concentration C+ and turbulent kinematic viscosity v? are defined as follows :

y+ = ; wT = \Jrw/p (3.27)

c%+) = ^ ; cW/tfr (3.28) II

£ (3.29) uT is the friction (shear) velocity [m/s], rw is the wall shear stress [JV/m2], p is the den ­ sity of the gas and v and vt its kinematic viscosity and its turbulent counterpart [m2/s], respectively. 44 CHAPTER 3. DESCRIPTION AND DISCUSSIONOF THE MODEL.

The equation for diffusive transport of gas through the boundary layer

(3.30) is then transformed to dimensionless form and solved to arrive at a relation between the concentration at the wall and at a distance y+ away from the wall ([21, equation 20] after correcting a printing error) :

a(y+,Sc) = c+(y+)-c: = £ (3'31)

C+ = C+(0.0) is the concentration at the wall, Sc and Set are the laminar and turbulent Schmidt numbers. F contains both the molecular diffusivity and the turbulent transport of gas. The flux to the surface and thus the reaction rate can now be calculated if the concen­ tration at a given distance y is known. Substituting C+(y+) = uTC(y)/j (equation 3.28) into equation 3.31 and solving for the mass flux j gives : •. _ ur(C(y) — Cw) 3 ~ 9{y+, Sc)

The evaluation of g(y+, Sc) is not straight forward, and only the final result is reproduced here. The following expressions are used in the model [21] :

v~r = yfo.3k w (3.33) kw is the kinetic energy of turbulence [m2/s2] evaluated from the k — e model for the node position of the near wall node. . Ur P^t y = vyn=TVn (3.34) yn is here the distance from the wall to the node position of the near wall node. v(= y/p) is also evaluated at yn . y is the molecular viscosity for the gas [kg/ms].

(y+/11.15)3 when y+ <3.0 (y+/11.4)2 - 0.049774 when 3.0 < y+ < 52.108 (3.35) 0.4y+ when 52.108 < y+

Set = 0.7 + (Sc • Sc=5 (3.36) The turbulent Schmidt number is evaluated at the node position of the near wall node (v? — v?(yn)) and is assumed to prevail throughout the entire boundary layer while integrating equation 3.31. Sc is evaluated at the wall (y = 0.0). An analytic expression for the concentration profile is then obtained : 3.4. CHEMICAL REACTIONS AND REACTION RATES. 45

V+ < 3.0 :

x,+i A 9i{y+, Sc) = C+(y+) - C+ 11.15{S

where

11.15

3.0 < y+ < 52.108 :

g2(y+, Sc) = C+(y+) - C+(3.0) = 11.4 (~^j f(y+) (3.38)

where P 1.0 - 0.049774Sc/Sc t

tan -1(a:2(y+)) — tan^(%(3.0)) ; /? > 0.01

11.425Cf (| - ^r) f(y+) \P\ < 0.01

i i ( (%(;/+)—1)(%(3.0)+1) \ (%(y+)+i)(%(3-o)-i) ) 18 < -0.01

(_Sc_) 0-5 JlL x2 (y+) ysctpj 11.4

52.108 < y+ :

g3(y+,Sc) = C+{y+) - C+(52.108) = ^ln( ) (3‘39 )

The function g(y+, Sc) thus becomes :

gi(y+,Sc) when y+ < 3.0 g(y +, Sc) g1(3.0,Sc) + g2{y+,Sc) when 3.0 < y+ < 52.108 (3.40) 52(52.108, Sc)+ 53(y+, Sc) when 52.108 < y+ 46 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

The mass flux to the wall (the reacting surface) is calculated from equation 3.32 and assumptions about the concentration at the surface Cw. In the model Cw is assumed to be equal to the equilibrium concentration for the reaction. This corresponds to a situation where the intrinsic rate of the reaction is so large compared to the gas transport to/from the surface that equilibrium prevails at the surface. Mass fractions are used in the FLUENT code instead of mass concentrations :

C(y) = p{y)X{y) (3.41) where X is the mass fraction of the given species. Equation 3.32 then transforms to :

(3.42) where it is assumed that the density of the gas is constant from the wall to the node position (y = yn). Xw = Xeq is the equilibrium mass fraction at the wall.

In the laminar case the following expression replaces equation 3.42 :

(3.43)

3.4.2 The reaction 310% + C = SiO + CO.

Ri Si02{s,l) + C(s) = SiO(g) + CO(g) (1.3)

The equilibrium constant for the reaction is given by :

PSiO,eqPCO,eq asio 2ac Assuming unit activity for C and Si02 gives :

Hi = PSiO,eqPCO,eq (3.44)

Reaction to the right. Wiik [30] examined the reaction kinetics for the reaction between solid Si02 and C in the absence of SiC. His work indicated, although not conclusive, that the reaction took place with C02(g) as an important intermediate compound through the following mechanism : Si02(s) + CO(g) —> SiO(g) + C02(g) (3.45) C(s)+ (%%(<;) 2CO(g) 3.4. CHEMICAL REACTIONS AND REACTION RATES. 47

These two reactions add up to the considered reaction. The equilibrium pressure of CO2 is very low in the process. This means that the reaction mechanism proposed by Wiik gives slow conversion unless the two reactants (C and Si02) have large contact areas as in intimately mixed briquettes or pellets. The reaction between highly viscous liquid Si02 and solid carbon is also slow unless the specific contact area between the reactants are large. The irregular and lumpy particles in the raw materials do not meet with these requirements, and the reactants are further separated as a result of the SiC(s)-layer formed around the carbon particles by equation 1.7 (Figure 3.5, page 33). The reaction between carbon in the raw materials and Si02 is thus expected to be slow, and it is neglected in the model. Similar arguments are used for disregarding the same reaction in the dynamical unidimensional model developed by the Elkem company (section 2.2). The tiny carbon particles inside the condensate are, on the other hand, in intimate physical contact with the Si02 in the condensate. This means that the conditions are favourable for the reaction to proceed to the right inside the condensate. The reaction can proceed here at a significant rate provided that the total reaction pressure of the product gases exceeds the external pressure of approximately 1 atm. The product gas is in this case effectively purged out of the condensate provided that the condensate is not too compact in its structure. Transport of product gases through the condensate to the ambient gas is believed to be slow when the equilbrium pressure for the reaction is below the ambient pressure. It seems reasonable to assume that the gas from the surroundings interferes little inside the condensate when the reaction pressure exceeds the ambient pressure. This implies that the gas composition close to the reacting surface is given by the stoichiometry of the reaction (psio = Pco = 0.5ptot)- Ptot is the local total relative reaction pressure. Solving equation 3.44 for the total relative reaction pressure at equilibrium gives :

Thus, the total reaction pressure exceeds 1 atm if and only if K\ > 1/4 (that is : at temperatures above 1680°C). Any carbon in the condensate therefore reacts to gas at temperatures well below those considered in the model (1860°C and above). The reaction to the right is consequently neglected in the model.

Reaction to the left. This reaction is believed to proceed when SiO(g) and CO(g) is absorbed at solid or liquid surfaces. The absorbed gases may then react chemically at these surfaces. The reaction may proceed to the left if and only if psioPco > Ki- Assuming that SiO and 48 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

CO are the only gases present {jpsio +Pco = Ptot) and eliminating pco from the inequality gives : PSiO ~ PtotPSiO + Hi < 0 (3.46) The left hand side of this inequality has an absolute minimum for psio = 0.5pM - This minimum value is given by :

Inequality 3.46 cannot be satisfied if Ki > 0.25p%, t, and the reaction cannot proceed to the left at any gas composition above 1680°C and a total pressure of 1 atm. This is consistent with reaction chamber experiments carried out by Poch and Dietzel [31] which showed that the reaction takes place only at temperatures well below those considered in this thesis (1860°C). The reaction is therefore excluded in the model.

3.4.3 The reaction 2SiC>2 + SiC = 3SiO + CO.

R2 2 Si02(s,l)+SiC{s) = 3SiO{g) + CO(g) (1.4)

Si02 is molten at temperatures above 1996K. The interesting reaction is therefore :

Ra 2SiOz(l) + SiC(s) = 3 SiO(g) + CO(g)

Reaction to the right.

i?2r 2SiOz(l) + SiC(s) -> 3SiO{g) + CO(g)

According to Tuset [32], this gross reaction is slow below the melting point of Si02, but a significant increase in the reaction rate is observed above this temperature. The reaction probably takes place to a certain extent as written for temperatures considered in the model if there is physical contact between the reactants. The relative reaction pressure for this reaction is given in Figure 3.9. The figure shows that the reaction reaches a total reaction pressure of 1 atm at 2085.5K when unit activities are assumed for the reactants. From a thermodynamic point of view, it proceeds to the right above this temperature under the given assumptions and an ambient 3.4. CHEMICAL REACTIONS AND REACTION RATES. 49

2SiO20)j+ SiC(s) = 3SiO(g)> CO(g)

2050K 2060K 2070K 2080K 2090K 2100K Temperature

Figure 3.9: The relative reaction pressure for 2Si02{l) + SiC(s) = 3SiO(g) + CO(g) as­ suming stoichiometric gas composition. Data from JANAF Thermochemical Tables [6]. pressure of 1 atm. However, the direct reaction is unlikely to proceed at a high rate with highly viscous silica due to limited physical contact between the reactants. The following reaction mechanism has been proposed by Tuset [32] :

SiC(s) + SiO(g) -5- 2Si(g) + CO(g) 2 ( Si(g) + Si02(l) -+ 2SiO(g) )

The concentration of Si(g) is believed to be low at temperatures prevailing in the furnace shaft and the reaction is probably slow according to this reaction mechanism even if there is intimate contact between the reactants. An alternative mechanism involves Si(l) of low viscosity :

ac(,)+aok) -» 2a(z)+cob) 2 ( Si(l) + SiC2(l) -> 2SiO(g) ) { )

These reactions are described in section 3.4.5 and section 3.4.4, respectively. They are both included in the model in the furnace shaft; the former with the produced Si(l) being instantaneously transported to the metal pool and the latter with Si(l) from condensation of Si(g) on the crater wall. 50 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

The direct reaction between SiC and SiO% is, however, not included in the furnace shaft. It is not known to what extent this reaction actually takes place in this area, but it may well be important. Its rate depends on the temperatures, the gas composition and the physical contact between SiOz and SiC. It should be considered to include this reaction in future versions of the model. The direct reaction between SiC and SiOz is also excluded in the metal pool. This is dubious since the temperatures are so high in this area that the reaction surely takes place at a considerable rate if the reactants are in physical contact. Such contact is likely even though a physical separation of the two is possible because SiC sinks "and SiOz floats in a pure silicon melt. It is however unlikely that all SiC is covered by silicon metal because this inactivates the SiC from any reactions with SiOz and the gas. The only way to consume SiC is in this case by dissolution in the silicon metal followed by transport of dissolved carbon to the surface of the metal pool or to SiCVparticles where it can react. This process is likely to be much slower than the supply of new SiC from the crater wall. SiC consequently builds up in the metal pool and it eventually reaches the surface where it starts reacting with the SiO(g) or with SiOz dripping down from the crater wall. The reaction between SiC and SiOz according to equation 1.4 is therefore likely to take place in areas where SiOz rests on top of the SiC-bed in the metal pool. Dissolution of SiC in the silicon metal and reactions with dissolved carbon in it is excluded in the model even though this may account for at least some of the SiC-consumption in this area. The consequences of excluding the direct reaction at SiC/SzC^-interfaces in the metal pool are probably not serious. The gross reaction can instead proceed through the two reactions listed in 3.47. The energy that should have been consumed at SiC/SiOa-interfaces is trans ­ ported downwards through the materials by conduction, and is instead used for producing SiO(g ) at Si/Si02-interfaces by equation 1.5. The SiO(g) content of the gas increases and results in higher conversion according to equation 1.6 at the gas/SiC-interface. The net result may possibly be similar to the situation where reaction 1.4 proceed as written. The temperature profiles in the reacting materials will be somewhat different though, and that may be important. It should therefore be considered to include reaction 1.4 in future versions of the model. Reaction 1.4 is in principle similar to that between Si and SiOz at the Si(l)/SiOz{l)- interface which is assumed to be controlled by the heat flux to the reacting interface when the reaction pressure exceeds the ambient pressure of 1 atm (section 3.4.4). It seems reasonable to apply a similar rate controlling mechanism also for reaction 1.4. Including this reaction requires no major changes to the code since all basic mechanisms for calculating the reaction rates and handling the associated mass- and heat transfer are already included for the other reactions.

'z.'srv-r^- V-r 3.4. CHEMICAL REACTIONS AND REACTION RATES. 51

Reaction to the left.

i?2Z 2Si02{l) + SiC(s) <- 3SiO(g) + CO(g)

From thermodynamical considerations this is the most preferable condensation reaction. However, the probability for simultaneous collision between 4 gas molecules is small, and the reaction cannot proceed as written. It must instead take place via some other complex reaction mechanism comprising reaction 1.5 proceeding from right to left as one of the intermediate steps. Also, experiments done by Poch and Dietzel [31] (reaction chamber) show that this reaction only takes place at temperatures below 2073K, which is well below the temperatures considered in the present model. The reaction is therefore excluded.

3.4.4 The reaction SiOz + Si = 2SiO.

Si and Si02 are not present as solids above 1860°C and the interesting reaction is :

r3 Si02(l) + Si(l,g) = 2SiO{g) AHz (1.5) where AH3 is the enthalpy of reaction. The reaction involving liquid silicon and the reaction involving silicon vapour are described separately below.

3.4.4.1 Reaction with liquid silicon.

The reaction and the equilibrium constant are given by :

Rs Si02{l) + Si{l) = 2 SiO(g)

k3 = p2si °'eq 0-SiO2aSi

Unit activity for the condensed materials gives :

(3 43)

Assuming uni t activity for silicon in the metal pool above 1800°C is dubious as the amount of dissolved carbon in the melt becomes significant above this temperature. The activity of 52 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL. silicon decreases accordingly, but osi = 1.0 is despite this assumed in the model. Lowering the activity of silicon implies that the equilibrium is shifted to the left so that the pressure of SiO(g) at equilibrium is lowered at otherwise identical conditions. The relative equilibrium pressure of SiO(g) for asio 2 = 10 in the two cases os,- = 1.0 and as, = 0.85 is shown in Figure 3.10.

4.0

cin mL. — 'jitniTrS

1600K 1800K 2000K 2200K 2400K Temperature

Figure 3.10: The relative equilibrium pressure of SiO(g) for the reaction Si02(l)+Si(l) = 2SiO(g) (1.5). psio = 10 at T « 2132K when as, = 1.0. Data from JANAF Thermochemical Tables [6].

The reaction can proceed to the right when

(3.49) and to the left otherwise. The reaction is highly endothermic when proceeding to the right.

Reaction to the right.

Rsr Si02{l) + Si{l) -s- 2SiO(g)

Ryabchikov, Khrushcev and Shchedrovitskii [33] observed the weight losses from Si02, Si and Si02+Si heated in inert gases at 1 atm. They reported no production of SiO{g) below 3.4. CHEMICAL REACTIONS AND REACTION RATES. 53

1850°C, but a significant reaction was observed above this temperature. They proposed a reaction mechanism involving 0(g) :

Si(l) + 0(g) -»• SiO(g)

The results of Ryabchikov et. al. are to be expected since the total pressure for the reaction exceeds 1 atm at 1859°<7 according to JANAF data [6]. Above this temperature the produced gas is purged away from the reaction surface and into the bulk gas which has a somewhat lower pressure. This transport becomes easier at higher temperatures since the equilibrium pressure then increases. The reaction is not included in the model at temperatures below 1859°C even though a certain conversion may take place at lower temperatures. It is, however, included at Si/SiOa-interfaces in the metal pool above this temperature. It is excluded in the furnace shaft since liquid silicon is not present in this area in the model. It is assumed in the model that there are no kinetic hindrances for the reaction when the re­ action pressure is above the ambient pressure of 1 atm. The reaction rate [moles of Si/m2s] is then controlled by the net energy supply to the reacting Si/Si02-interface : {qtot/AH3 when qtot > 0 ; Psio,eq>Ptct (3-51)

0 otherwise where qtot is the net heat flux \W/m2} to the reacting surface and A i?3 is the enthalpy of reaction [J/mole Si].

The heat flux to the Si/SiOa-interface consists of a convective and a conductive term. In the model the convective term is zero since the silicon and silica are formally treated as solid materials. The lack of convective energy transport can, however, be partly compensated for by increasing the thermal conductivities of the materials. These values can be changed by the simulation operator at any time. Moreover, liquid silica is glassy and fairly transparent. Most of the radiation from the electric arc is therefore likely to pass through the silica and should consequently be credited to the reacting surface. In the model the radiation is instead credited to the gas/SiOz- interfaces. The conductive energy transport to the reacting surface, qtot, is calculated after assuming that its temperature is always 1859°C as long as the reaction proceeds. It may be argued that a slightly higher temperature should be used since the gas transport away from the surface is fast only if the SiO-pressure is significantly larger than the ambient pressure. This model parameter can easily be changed if wanted. 54 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

The thermal conductivity for silicon metal is much higher than for silica. It is therefore likely that most of the energy is transported to the reacting surface through the silicon metal. The conductive heat fluxes \W/m2\ are in cylindrical coordinates given by :

(3.52)

where hi is the effective thermal conductivity \W/mK\ of the material (Si or Si02).

Figure 3.11 illustrates the situation around a silica particle reacting with silicon metal in the metal pool.

Furnace gas

Silicon

T = 2132K Sifl) + Si02(l) = 2SiO(g); AH3

Figure 3.11: Si(l) reacting with Si02 in the metal pool. The reacting surface is assumed to be at a constant temperature of 2132K (corresponding to a reaction pressure of 1 atm) as long as the reaction proceeds, qua is the total heat flux to the reacting surface from the surroundings under this assumption and represents the energy that is available for production of SiO(g) at the given surface area. AHs is the enthalpy of reaction.

Reaction to the left.

Rzi Si02(l) + Si(l) 4- 2SiO(g) For kinetic reasons this reaction is the dominating condensation reaction taking place in the furnace shaft. At a total pressure of 1 atm the reaction can only proceed to the left below 3.4. CHEMICAL REACTIONS AND REACTION RATES. 55

2132K since its reaction pressure is above 1 atm at higher temperatures. The reaction is consequently neglected in the model.

S.4.4.2 Reaction with silicon condensing from the gas.

The reaction r3 Si02(l) + Si{g) = 2SiO(g) may proceed to the right in the model both in the metal pool and in the furnace shaft when Si(g) condenses on the surface of SiCVparticles. The reaction cannot proceed to the left producing Si(g), and this case is thus not discussed further. Silicon may evaporate from the metal pool as described in section 3.5.3. A fraction of this silicon is transported to the surfaces of SiC^-particles where some or all of it may react. This is assumed to take place by the following two successive steps in the model :

Si(g) -> Si(l) (3.53) aSi02(l) + aSi(l) —> 2aSiO(g) (3.54)

Adding these gives the following gross reaction :

Si(g) + ocSi02(l) ->(1- a)Si(l) + 2aSiO(g) (3.55) where a = [0.0, 1.0] represents the fraction of condensed Si that reacts chemically, a = 0.0 implies that no condensing silicon reacts while a = 1.0 implies that all reacts. The actual condensation (equation 3.53) provides a considerable energy flux (qcond) to the Si02-surface. Part of this flux (qheat) is used to increase the temperature of the Si02 while the rest (qreac) is consumed in the endothermic chemical reaction between Si02 and condensed Si : Qcond = Qreac ~b Qheat 41 (3.56)

Qreac — Qcond Qheat qcond is calculated from equation 3.98 on page 75 and qheat as described soon. The reaction rate [moles/m 2s] caused by the condensation is then uniquely determined from Qreac and AH3 in the same way as for the reaction at Si(l)/Si02(^-interfaces in the metal pool (equation 3.51, page 53) :

R3= Qreac/AH3 ; qreac > 0.0 (3.57)

The conversion never exceeds the amount of silicon that condenses. 56 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

n Net energy flux to the surface from condensation and reaction.

No Si reacts

Fraction of the condensed All Si reacts silicon that reacts at the surface (%n) r

Figure 3.12: The net heat flux to the SzC^-surface from the combined condensation of Si and associated chemical reaction as a function of the fraction (a) of the condensed Si that reacts chemically.

Qheat changes with a as shown in Figure 3.12. Pure condensation (a = 0.0) gives a significant net heating of the surface (qheat > 0.0). Qheat decreases when a increases since the reaction is endothermic, and it usually becomes negative (net cooling) at sufficiently high a values4. There is a net heating for a < fa and a net cooling for a > fa in the figure. Qaii = Qheat(& = 1.0) is the net heat flux to the surface from the combined condensation and chemical reaction when all condensing Si reacts :

Qall = Qheat(a = 1.0) = Qcond Qrcac(& ~ 1.0) (3.58) = Qcond (f^cond ^H%) JMsi where ihamd is calculated from equation 3.100 on page 75, A Hz is the enthalpy of reaction and Msi is the molar mass of silicon. The value of Qheat to be used in equation 3.56 is determined as shown and described in Figure 3.13. Tiim, Ti>e, and TziC are model parameters defined explicitely by the simulation operator. The consequences of different choices for these parameters can easily be checked out. The algorithm implies that

• No condensing silicon reacts chemically below a certain temperature (Ti,e or T3]e depending on the sign of Qau). All the energy of condensation is then used for heating the Si02- *

* qheat stays positive for all values of a if and only if the temperature of the condensing Si(g) is extremely high. This hardly happens in practice because the reaction with SiO^ is strongly endothermic. Even so, this case is included in the description. 3.4. CHEMICAL REACTIONS AND REACTION RATES. 57

No Si reacts Some Si reacts All Si reacts

Net heating (always)

No Si reacts Some Si reacts All Si reacts

Net heating Net cooling

Figure 3.13: Determining qheat from the temperature of the computational cell where the condensation takes place (Tsurf). The lower graph is applied if qau < 0.0 (the normal case) and the upper graph is applied otherwise. The value of qheat is determined directly from the piecewise linear graphs. 58 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

• The fraction of condensing Si that reacts chemically then increases linearly as the temperature increases to Tum. In the normal case (q^i < 0.0) the net heat flux to the surface caused by the combined condensation and reaction (qheat) is zero at This implies that all the energy of condensation is consumed in the reaction and the temperature of the surface is not affected. For qau > 0.0 all condensing silicon reacts at 2}im and above. The surface is then heated by the combined condensation and reaction (qheat > 0.0) even when all Si reacts. • In the normal case (q^i < 0.0) the fraction of condensing Si that reacts chemically increases further beyond Tum, but generally at a different linear rate. It reaches the maximum value (where all condensing silicon reacts) at 22,ej and stays at this level above the same temperature, qheat becomes increasingly negative and the surface is cooled stronger as the temperature approaches 2a,e from below. The cooling is kept at its maximum level above T2,e.

2132K is a logical value for Tum. This implies that the reaction rate adjusts to a level giving no net heat flux from the combined condensation and reaction (qheat = 0.0) when the equilibrium pressure of SiO(g) for step 2 (equation 3.54) is 1 atm. This expresses that there are no kinetic hindrances for the reaction above this temperature, and is consistent with how the same reaction is treated at Si(l)/Si02-interfaces in the metal pool (see page 53). As mentioned for the reaction at Si(l)/Si02-interfaces, it could be argued that a value slightly higher than 2132K should be used. At increasing deviation from 2}im the heating- (below Tum) or cooling effect (above Tj,-m) becomes stronger. The algorithm therefore tends to regulate the local Si02 temperature towards Ttim by adusting the fraction of the condensing silicon that reacts. This seems reasonable. Qcond represents the heat flux to the surface due to condensation (equation 3.53). It could be argued that the heat flux from radiation (qrad), the heat flux from the gas as such (qgas) and the heat flux from conduction through the reacting Si02 (qsio 2) should also be included in the algorithm, qcond would then be replaced by qavau in the arguments above, where qavau is given by :

Qavail — Qcond "b Qrad ~b Qgas "b QSiOr (3.59) All quantities are here positive if they add energy to the SiOa-surface. The heat flux through the Si02 is likely to give a negative contribution because the surface temperature is probably higher than the temperature inside the Si02. Energy can, however, in principle be transported to the surface from below (for example through warm S'iC'-particles next to the Si02). 3.4. CHEMICAL REACTIONS AND REACTION RATES. 59

Applying equation 3.59 means that the total heat flux to the surface is used for determinin g the value of qheat instead of the heat flux from the condensation only. The algorithm is otherwise unchanged. The conversion should still not exceed the total amount of Si that condenses. Equation 3.59 is not applied in the present version of the model.

3.4.5 The reaction SiO + SiC = 2Si + CO.

R* SiO{g) + SiC(s) = 2 Si(s,l,g) + CO{g) (1.6)

The reaction to Si(g) is neglected in the model since Si(g) is only present in significant amounts above approximately 2800K [23]. Si(s) does not exist above 1685K, and the interesting reaction is thus :

R* SiO(g) + SiC{s) = 2Si(l) + CO(g)

The equilibrium constant is given by :

Ki = (3.60) PSiO.eqO.SiC

Assuming unit activity for the condensed materials leads to equation 3.61.

Ki = (3.61) PSiO.eq

Assuming unit activity for silicon in the metal pool above approximately 1800 °C is dubious as the amount of dissolved carbon in the melt becomes significant above this temperature. The activity of silicon decreases accordingly, but os; = 1.0 is nevertheless assumed in the model. Lowering the activity of silicon implies that the equilibrium is shifted to the right so that the partial pressure of SiO(g) at equilibrium is lowered at otherwise identical conditions. The relative equilibrium pressure of SiO(g) for as,c = 1.0 in the two cases asi = 1.0 and asi = 0.85 at ptot = PsiO +pco = 1 is shown in Figure 3.14. The reaction can proceed to the right when

<=> PsiO — ~^r- > 0 (3.62) PSiO and to the left otherwise. 60 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

0.0 ------'------1------'------'------'------1------'------1500K 2000K 2500K 3000K 3500K Temperature

Figure 3.14: The relative equilibrium pressure of SiO(g) for the reaction SiC(s) + SiO(g) = 2Si(l) + CO(g) at p Sio + Pco = 1- Data from JANAF Ther ­ mochemical Tables [6].

Reaction to the right.

i?4r SiO(g) + SiC(s) -»• 2Si(l) + CO(g)

This reaction can proceed on the surface of SiC-particles both in the metal pool and the furnace shaft in the model when inequality 3.62 is satisfied. The reaction rate [moles of SiO consumed/m 2s] is equal to the net molar flux of SiO(g) to the reaction surface : Rir = Jsio (3.63)

Applying equation 3.24 (page 42) gives the following expressions for the molar fluxes [moles/m2s] of SiO(g) and CO(g) to the gas/SiC-interface :

JsiO = kg&sio - PsiO,w) (3.64) Jco = kg(Pco - Pco.w) (3.65) Pco,w and psio,v> are the partial pressures on the surface and kg is the mass transfer coefficient for SiO(g) and CO(g) in the binary gas mixture (CO(g) + SiO(g)). jsw > 0 and Jco < 0 when the reaction proceeds to the right. 3.4. CHEMICAL REACTIONS AND REACTION RATES. 61

The stoichiometry of the reaction implies that :

Jsio = -Jco (3.66)

Solving equation 3.64 with respect to psio.w gives :

PsiO,w = Psio ~ (3-67) Kg

Solving equation 3.65 with respect to pco and using equation 3.66 to eliminate Jco gives :

Pco,w = Pco + (3-68) Kg

The reversible chemical reaction is determined by a forward rate constant (&/) and a backward rate constant (fa). The flux of SiO(g) to the reacting surface (Js,o) is then given by : Jsio — kfPsio.w — fapco.w (3.69)

Solving equation 3.69 for fa at equilibrium (jSio = 0, pco,w = Pco,eq and psio,w = PsiO.eq) gives : 0 = kfPsiO,eq faPCO,eq

JJ- (equation 3.61)

fa — /Hq

Substituting this fa-value into equation 3.69 and solving for Jsio, using equations 3.67 and 3.68 :

jsiO = kfPsiO.w — JqPCO,w

1) (equation 3.68 and equation 3.67) (3.70) Jsio = kf(j5Si0 - ^) - j£(Pco + 4^) Jj- Jsio = (Psio~*iu)/(itj + k;(1 + lu)) Equation 3.70 substituted into 3.63 then gives the following final expression for the reaction rate : (PSiO ~ )/% + + ifc)) when PSiO ~ > 0

Rir = ' (3.71) 0 otherwise 62 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

The reaction rate on the surface of SzC-particles is here expressed by the equilibrium constant, the partial pressures in the bulk gas, the forward reaction rate constant and the mass transfer coefficient for transport of gas through the boundary layer around the particle. The parameters kf and kg and the thickness of the boundary layer must be determined from experiments or estimated, but are difficult to obtain. Assuming infini te intrinsic rate (kf —> oo) gives : (PsiO - )/(x7 + ^(1 + wj) = kgiPsiO - JC^)/(1 + j^)

= + +

= kg ( pSi0 — (Psio + Pco)/(1 + ^4) )

= kg( psio - ( (pSi0 + Pco)/(PSiO,eq + PCO,eg) ) PSiO,eq ) If only SiO(g) and CO(g) are present, then p Sio + Pco = PsiO,eg + Pco,eq = Ptot and the the equation reduces to :

(PsiO - )/+ + ^4)) = ka(PsiO - Psio,eq) (3.72)

Equation 3.71 then simplifies to

(kg(PSiO - PSiO,eq) when p SiQ - PsiO,eq > 0 (3.73)

0 otherwise Equation 3.71 is implemented in the model, but it has not been applied in the simulations presented in the thesis due to the difficulties in determining reasonable parameter values and in evaluating the bulk pressure. Instead, the equations for mass transport through a turbulent boundary layer as described in section 3.4.1 are used. According to these, the flux of SiO(g) to the surface [kg/m2s] becomes (equation 3.42, page 46) :

py-AXsioiVn) ~ XsiQ,w) 3siO = jsiO > 0 (3.74) g(y+,Sc)

XsioiVn) is the mass fraction of SiO at the node position of the near wall node (y„) and Xsio,w is the mass fraction of SiO(g) on the reacting surface (the wall). The expressions for uT and g(y+, Sc) are given in section 3.4.1. In the model it is assumed that the intrinsic rate is large compared to the transport of gas to and from the reacting surface (corresponding to kf » kg in equation 3.71). This 3.4. CHEMICAL REACTIONS AND REACTION RATES. 63

means that equilibrium is in practice established at the reacting surface and that Xsio,w is taken to be the equilibrium mass fraction oLSiO(g) at the prevailing surface temperature. The reaction rate is then fully controlled by the mass transport to the reacting surface :

1 pur (XsiO (Vn) -^SiO.eq) when XsioiVn ) > Xsio,eq MsiO 9(y+,Sc) (3.75) 0 otherwise

where Msio is the molar mass of SiO [kg SiO/mole SiO\. Comparing equations 3.73 and 3.75 shows that the two approaches are identical when the intrinsic rate is assumed to be infinite and the gas is comprised of SiO{g) and CO(g) only. Modification of equation 3.75 is necessary if these assumptions are not reasonable, or else equation 3.71 must be used.

Reaction to the left. Ra SiO(g) + SiC(s) <- 2Si(l) + CO{g)

This reaction can in principle proceed on the surface of liquid Si anywhere in the furnace. Liquid silicon that can react according to this reaction is in the model only found in the metal pool. The reaction can only proceed to the left if the partial pressure of SiO(g) is below the equilibrium values shown in Figure 3.14 on page 60. This relative equilibrium pressure of SiO(g) increases with decreasing temperature and reaches psio ~ 0.63 at T = 2132K. SiO(g) is produced in large amounts by reaction 1.5 above this temperature (psio = 1 at T = 2132K). It is difficult to imagine how the gas composition can reach a sufficiently high content of CO to make the reaction proceed to the left above 2132K as long as SiO2 and Si are present in the same area to produce lots of SiO(g). The reaction to the left is therefore excluded in the model. R41 = 0 (3.76)

It would, however, be possible for the reaction to proceed to the left in some parts of the metal pool if only SiC (and no Si02) were present together with Si. The reaction may then proceed to the right at SiC-surfaces in the central and warmest parts of the metal pool where the equilibrium pressure of SiO is very low. The gas may become strongly depleted in SiO, and the fraction of SiO(g) that is obtained in this way may well be lower than the equilibrium composition as defined by the much lower temperatures in the outer parts of the metal pool. The reaction then proceeds to the left when this gas arrives at these colder Si-surfaces unless other reactions add more SiO(g) in the meantime. The net result is reaction to the right in the central parts of the furnace and to the left further out. 64 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

This is associated with a net transport of SiC from the arc zone to the outer region of the metal pool. The presence of SiO%, however, adds sufficient amounts of SiO(g) to keep the gas composition above the curve in Figure 3.14. Thus, the driving force for the reaction is most probably always to the right.

3.4.6 The reaction SiO + 2C = SiC + CO.

i?5 SiO(g) + 2C(s) = SiC(s) + CO{g) (1.7)

Reaction to the right.

means that the reaction does not proceed to the right in the model and that the reaction rate is zero : (3.77)

This reaction must however be implemented if the model is extended to include also the low-temperature zone and/or free carbon in the high-temperature zone. The reaction is therefore described briefly below. The reaction rate [moles of C consumed/m 3s] applied in the dynamical unidimensional Elkem model [8] is given by equation 2.4 on page 16. This expression is somewhat different from the one proposed earlier by Halvorsen [26] who assumed spherical carbon particles and an idealised shrinking core model for the conversion of C to SiC as shown in Figure 3.15.

Here the reaction rate [moles of C consumed/m 3s] is given by : Rs = NcAar%J5 = %,4%Tg(^^ —Psio)/Ao (3.78)

where Nc = Number of C/SiC-particles per m3 [1/m3]. rc = Radius of unconverted carbon particles, [m], J5 = Flux density of SiO through the surface of C/SiC-particles [mole/m2s]. K5 = Equilibrium constant for the reaction. As = Conversion ’’resistance ” [m2s/mole]. 3.4. CHEMICAL REACTIONS AND REACTION RATES. 65

SIC

Figure 3.15: Spherical C partly converted to SiC [26].

The conversion ’’resistance ” [m?s/mole] is given by [26] : 1 _r* 2 (1 - r*)r* As = (1 + + rSic (3.79) rxs KgS D,'pS ]+h

where

r* = Tc/rsic kg5 = Mass transfer coefficient for the gas through the gas film surrounding the particle [mole/m 2s]. Dp 5 = Diffusion coefficient for the gas through the product layer \mole/ms\. k's5 = Intrinsic reaction rate at the interface [mole/m2s].

This reaction model assumes that the reaction takes place in an infinitely thin reaction zone at the interface between fully converted SiC and pure unconverted C given by the radius rc in Figure 3.15. According to this, the reaction takes place at an area given by the surface of a sphere of radius rc- A corresponding reaction rate constant referring to this calculated surface area must be estimated in the model. This model is relevant for a sufficiently thin reaction zone. However, the carbon material is highly porous (Raanes [34]), and the reaction consequently takes place in a reaction zone of a given volume rather than on a compact spherical interface. Using the reaction rate constant as defined above thus requires that the total surface area of the reacting volume is used instead of the area of the compact sphere of radius rc- Another way to solve this problem is to stick to the compact sphere model and make the necessary adjustments by increasing the chemical rate constant to take care of the increased effective reaction area. The overall chemical reaction rate can be measured experimentally for different carbon materials as described by Raanes and Tuset [35] or by Videm [36]. Equations 2.4 and 3.78 are similar in their form except for the Arrhenius factor in equa­ tion 2.4.

$• 66 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

Reaction to the left.

R51 SiO(g) + 2C{s) <- SiC(s) + CO(g)

The reaction does not proceed as written with the gas compositions existing in the high- temperature zone, and it is therefore neglected in the model (R5i = 0).

3.5 Phase transformations and transformation rates.

As mentioned in section 1.3, the following three phase transformations are included in the model : Re Si02(s) = Si02[l) (1.8) Rr Si(s) = Si(l) (1.9) Rs II % (1.10) £

The reaction rates for the melting reactions (equations 1.8 and 1.9) are for two reasons not important in the model. Firstly, both Si and Sz'02 melt far below 2133K. They are thus both molten in the temperature range covered by the model. Secondly, the energy required for melting is automatically taken care of by the enthalpy/temperature relations used for the various materials. The enthalpy of melting and other possible phase transformations of solids or liquids are included in these data. The data found in the literature for Si and Si02 states that the melting takes place at a distinct temperature. The enthalpy/temperature curve is discontinuous at this melting temperature, and the enthalpy of the substance is not defined in a unique way from its temperature at the melting point. This causes practical problems during calculations. There are different ways of handling this problem and the one used in the model is to introduce a small melting interval ([Tm — dT, Tm+dT}) around the actual melting point as shown in Figure 3.16. dT is a small value. The substance starts melting at Tm — dT and it is fully molten at Tm+dT. The enthalpy is assumed to be a linear function of temperature in this interval. The enthalpy is then defined in a unique way from the temperature also during melting. Introducing this small melting interval causes no problems since the actual melting temperature is not an important parameter for the chemical reactions or for other phenomena that are simulated. Evaporation of Si(l) is on the other hand important since it is associated with significant heat and mass transfer from one location to another. 3.5. PHASE TRANSFORMATIONS AND TRANSFORMATION RATES. 67

Data from the literature Data used in the simulation model „

Figure 3.16: The enthalpy as a function of temperature around the melting point (Tm). Data from the literature to the left (assuming melting at a distinct temper ­ ature, Tm), and data used in the model to the right (introducing a small melting interval [Tm—dT, Tm+dT] in which the enthalpy function is linear).

3.5.1 Melting and solidification of 310%. \ Re Si02(s) = Si02{l) (1.8)

The Si02 is assumed to be high cristobalite (page 38) prior to melting. High cristobalite melts at 1996K. The reaction rate is therefore zero both to the right and to the left at all interesting locations in the model : #6 = 0

3.5.2 Melting and solidification of Si.

Rr Si(s) = Si(l) (1.9)

Silicon melts at 1685K. The reaction rate is therefore zero both to the right and to the left at all interesting locations in the model :

#7 = 0

\ 68 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

3.5.3 Evaporation and condensation of Si.

Rs Si(l) = Si(g) (1.10)

The equilibrium constant is given by :

K& =

(3.80)

As described previously, assuming unit activity for silicon in the metal pool above 1800°C is dubious as the amount of dissolved carbon in the melt becomes significant above this temperature. The activity of silicon decreases accordingly, but a# = 1.0 is despite this assumed in the model. Lowering the activity of silicon implies that the equilibrium is shifted to the left so that the partial pressure of Si(g) at equilibrium is lowered at otherwise identical conditions. The relative equilibrium pressure of Si(g) for the two cases a# = 1.0 and asi = 0.85 is shown in Figure 3.17. Silicon boils at 3504.616K and 1 atm (JANAF Thermochemical Tables [6]). The reaction can proceed to the right when

PSi < Ks (3.81) and to the left otherwise. It was at first planned to exclude the evaporation of silicon in the model since observations reviewed by Schei indicate that there are only moderate amounts of Si(g) in the crater gas; ’’The gas in the crater cavity is only briefly examined by experiments, but it is verified that it is mostly comprised of CO and SiO, and possibly compounds with Si/O-ratios greater than 1” ([1, ch. 4, p. 40] translated into English). This may be correct if SiOi is present in all parts of the metal pool so that the strongly endothermic reaction between Si and SiO2 is able to consume sufficient amounts of energy in the central parts of the furnace to keep the temperatures below those giving significant evaporation (Figure 3.17). Muller, Olsen and Tuset [24] proposed that the evaporation of silicon and the transport of silicon vapour to the outer parts of the metal pool and especially to the crater wall is indeed important for the ferrosilicon process. The idea is that the heat generation in the warmest parts of the metal pool is so strong that lots of silicon must necessarily evaporate. Nucleation problems may cause the actual silicon vapour pressure to be considerably higher than the equilibrium value. This vapour brings large amounts of energy with it, and it may undergo chemical reactions if it condenses on the SiCVrich condensate or 5z02-particles. 3.5. PHASE TRANSFORMATIONS AND TRANSFORMATION RATES. 69

2400K 2800K 3200K 3600K Temperature

Figure 3.17: The relative equilibrium pressure of Si(g) for the evaporation of silicon. p Si = 1.0 at T = 3504.616K when asi — 1.0. Data from JANAF Thermo ­ chemical Tables [6],

This may explain observations indicating that the Si02 in the crater wall is attacked by an aggressive agent from the gas phase. This aggressive agent was proposed to be silicon vapour by Muller et. al.; ’’Until further proof can be offered it is the authors 1 contention that the process is largely sustained by a stream of vapour from the arc zone with a content of Si, C and Fe that varies with process conditions, and with a temperature in the range of 2500 — 2800°C7, to conform with the known vapour pressure of the species. ” The theory of a high content of Si(g) in the crater cavity is supported by several observations described in [24]. This analysis was carried out for the ferrosilicon process, but it is probably relevant for the silicon process as well. No sideways movement of Si02 or SiC is included in the model, and the central parts of the metal pool thus consist of pure silicon. The temperatures in these areas therefore become very high if evaporation of silicon is not included. The spatially fixed DC electric arc applied in the model adds to this problem because it results in strongly overestimated heat fluxes to the central parts of the metal pool as discussed in section 3.8. Thus, the evaporation of silicon becomes even more important for keeping the temperatures down in the central parts of the metal pool. The consequences of this problem were realised too late to implement advanced and physi ­ cally well-founded models for evaporation and condensation of silicon. The selected method 70 CHAPTER 3. DESCRIPTION AND DISCUSSIONOF THE MODEL. provides rough estimates of the heat- and mass transfer caused by evaporation and con ­ densation, and it should definitely be improved in future versions of the model. The evaporation is modelled in a reasonable way, but the transport of the Si(g) and its condensation is modelled by an ad hoc method based on the simple strategy that

• All evaporated silicon in the end condenses somewhere in the system before pene ­ trating into the materials in the furnace shaft. • The amount that condenses at a given surface depends only on the temperature of this surface and some distribution parameters defined explicitly by the simulation operator.

No kinetic arguments are included in the implemented algorithm for condensation of Si(g). The following description of the algorithm for evaporation and subsequent condensation may be useful :

• Silicon evaporates from the metal pool and the enthalpy of evaporation is removed from the surface. The surface is consequently cooled. The amount of silicon that evaporates is calculated from reaction kinetics. • The evaporated silicon is transported instantaneously, and without changing its tem­ perature, to the position where it condenses and gives off energy. How much silicon that condenses on a given surface and the associated energy fiux are estimated by the algorithm described on page 74-77. This represents a rough estimate of the mass- and energy fluxes caused by condensation of silicon vapour. • At Si02-surfaces, some or all of the condensing silicon may react chemically as given by equation 3.56 on page 55. The silicon that reacts is taken from the surroundings of the reacting surface as described in section 3.13.3 rather than from the condensing silicon. At other surfaces and in the gas, the condensation only delivers heat. • All of the condensed silicon is then instantaneously returned to its original position (from where it evaporated), still without changing its temperature.

Notice that no mass transfer involving Si(g) to/from the gas phase takes place as a result of the implemented method. Neglecting the physical presence of silicon vapour in the gas reduces the complexity of the algorithm considerably and is the motivation for applying this simple approach. It does, however, create the practical problem to decide from where the reacting silicon is taken (referred to as "the surroundings of the reacting surface” in the description above). The solutions to this problem and the associated heat transport are described in section 3.13.3. 3.5. PHASE TRANSFORMATIONS AND TRANSFORMATION RATES. 71

Reaction to the right (Evaporation).

R&r Si(l) -s- Si®

The reaction rate [moles/m?s] is equal to the net molar flux of Si(g) away from the reacting surface : Rgr = —Jsi (3.82) The negative sign is because fluxes to a surface are defined as positive.- Equation 3.24 gives the following expression for the molar flux [moles/m 2s] of Si(g) at the surface : —Jsi = kg(psi,w — Psi) (3.83) p Sl> and p Si are the partial pressures at the surface and in the bulk gas, respectively. kg is the mass transfer coefficient for Si(g). Solving equation 3.83 with respect to ps.> gives :

Psi,w = Psi ~ ~jr (3.84)

The reversible chemical reaction is determined by a forward rate constant (kf) and a backward rate constant (fa). The net production of Si(g) at the surface (which must be equal to the flux of Si(g) away from the reacting surface) is then given by : —Jsi — kj — kb psi,w (3.85)

Solving equation 3.85 for fa at equilibrium (j$i = 0 and psi, w = Psi,eq) gives :

0 = kf-fapsi,eq

D- (equation 3.80)

fa = kf/Ks

Substituting this fa-value into equation 3.85 and solving for JSi, using equation 3.84 :

~~Jsi = kf — (kf/ K%)psi,w

(equation 3.84) (3.86) —Jsi = kf ~ (kf/K8 )(p Si - 4*0 i). —Jsi = (K8 -p Si)/(jf + ft;) 72 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

Equation 3.86 substituted into 3.82 then gives the following final expression for the evap ­ oration rate : f {Hz — Psi)/(ff + jr) when K& — p Si > 0 R* r = \ (3.87) [ 0 otherwise

The evaporation rate is here expressed by the equilibrium constant, the partial pressure of Si(g) in the bulk gas, the forward reaction rate constant and the mass transfer coefficient for transport of Si(g) through the boundary layer at the reacting surface. The parameters kf and kg and the thickness of the boundary layer must be determined from experiments or estimated, but are difficult to obtain. Assuming infinite intrinsic rate (kf -5- oo) and applying equation 3.80 for the equilibrium constant gives :

kg (psi.eq - Psi) when psi, eg -p Si>0 Rsr = (3.88) 0 otherwise

Equation 3.87 is implemented in the model, but it has not been applied in the final simula­ tions due to the difficulties in determining reasonable parameter values and in evaluating the bulk pressure. Instead, the equations for mass transport through a turbulent boundary layer as described in section 3.4.1 are applied. According to these, the flux of Si(g) away from the surface [kg/m2s] becomes (equation 3.42, page 46) :

p UT (Xsityri) XSi,w) ~3Si = Xsi(yn) < XSi,i (3.89) g{y+,Sc)

XsiiVn) is the mass fraction of Si at the node position of the near wall node (yn), and Xsi,w is the mass fraction of Si(g) at the metal pool surface (the wall). See page 43-46 for further details. In the model it is assumed that the intrinsic rate is large compared to the transport of gas away from the surface (corresponding to kf » kg in equation 3.87). This means that equilibrium is in practice established at the surface, and that Xsi,w is taken to be the equilibrium mass fraction of Si(g) at the prevailing surface temperature. The evaporation rate [moles/m2s] is then fully controlled by the mass transport away from the metal pool surface : 1 PUr{Xsi,eci-Xsi{yn )) Msi g(y+,Sc) when XSi(yn ) < XSi,eq R%r,xfr — ' (3.90) 0 otherwise

Msi is the molar mass of Si [kg Si/mole Si]. 3.5. PHASE TRANSFORMATIONS AND TRANSFORMATION RATES. 73

The mass fraction of Si(g) at yn (Xsi(yn )) is unknown since the model does not calculate it. Its value (Xs,) is instead defined explicitly by the simulation operator from educated guessing. Comparing equations 3.88 and 3.90 shows that the two approaches are identical when the intrinsic rate is assumed to be infini te. Modification of equation 3.90 is necessary if this assumption is not reasonable, or else equation 3.87 must be used. The equilibrium pressure of Si(g) exceeds the ambient pressure of 1 atm at T = 3504.616.K" when as, = 1.0. It is not reasonable to assume that the evaporation rate is controlled by equation 3.90 well above this temperature. The equation developed from kinetic gas theory by Langmuir for the limiting (maximum) rate of evaporation (Dushman [37]) is instead applied above a certain temperature :

MSi R&r,langmuir — Jtfg. (PSifiq -Psi)P° (3.91) 2-kRT P° = 1.013 x 105 pascal. The partial pressure of Si in the bulk gas (p Si) is unknown since the model does not calculate it. Its value (pSi ) is instead defined explicitly by the simulation operator from educated guessing. A gradual change in rate controlling mechanism from mass transfer control by equation 3.90 to the limiting rate of evaporation by equation 3.91 is implemented in a temperature interval from the boiling point of silicon (Tsi,boii ) and above. The minimum of the two values from equations 3.90 and 3.91 are used below TSi,boU ■ The final rate expression becomes :

{min \Rsr,xfri R&r}lang-muir\ when Tstirf < Tsifioil (3.92)

(1.0 S') Rsr,zfr @Rsr,langmuir when Tsurf ^ Tsi,boil where /? is given by : 0.0 when Tsurf < Tsifioil

Tsurf “Tsi.boit when Tsurf Tiangmuir (3.93) p=< Tlangmuir ~~Tsifboil Tsifioil ^ — i O —l when Tsurf Tiangmuir

The model parameter T;a„smut> is defined by the simulation operator and can be changed at any time. It is defined implicitly by the value ATr& = Tian gmmr — The heat flux caused by the evaporation, qevap [W/m2] (negative since heat is removed from the surface) is given by :

Qevap — RsrMsihsiiTsvrf ) (3.94 ) 74 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL. where hsi(T £tirf) is the specific enthalpy of the evaporating silicon [J/kg]. RSr is given by equation 3.92 5. j6 is underrelaxed to avoid oscillations in the evaporation rate around Tsi,boii when solving the equations by an iterative numerical method. The value of the underrelaxation factor is defined by the simulation operator and can be changed at any time.

Reaction to the left (Condensation).

R&i Si{l) <- Si(g)

Condensation of Si(g) may in principle take place at all surfaces exposed to furnace gas when equation 3.95 is satisfied. Psi > K& (3.95)

The condensation rate is determined from the total evaporation rate (Mtot [kg/s]), the asso­ ciated power (Qtot [W]) and the local temperature of the surface at which the condensation takes place. Three model parameters define the fractions of the total evaporated mass and enthalpy of evaporation that is distributed to the various parts of the system :

fpooi = fraction to the metal pool. fWati = fraction to the crater wall. fgas = fraction to the gas phase.

They are defined by the simulation operator and can be changed at any time. They normally sum up to 1.0 even though smaller sums are also accepted. A sum less than 1.0 means that some of the energy applied for producing silicon vapour is not added back when the same silicon condenses. The motivation for allowing energy to disappear like this is explained on page 77. Sums larger than 1.0 are not accepted. /p 00i and fwan define upper limits to the amount of mass and energy that can be distributed to the liquid and solid parts of the furnace by condensation. The amounts that are actually transferred to them may be less, and the balance is then distributed to the gas phase thus adding more energy to the gas than defined by fgas . The algorithm for calculating the amount of energy and mass that condenses is in principle identical for Si-, SiC- and Si02-surfaces both in the metal pool and in the crater wall. Thus, the algorithm is explained for the metal pool only. Different parameter values are defined for the different materials, but that is not important for the description as such. The algorithm for the gas phase is slightly different and is described separately.

sJ2gr from equation 3.87 replaces Rsr,xfr in equation 3.92 if equation 3.90 in not applied. 3.5. PHASE TRANSFORMATIONS AND TRANSFORMATION RATES. 75

The power [W] that is distributed to a given computational cell in the metal pool is given by: - x Qcett,pool = \Qtotfpool/-Atot,poolJ (1-0 T(-^stxr/)) -^cell/pool (3.96)

Atot,pooi is the total surface area of the gas/metal pool interface. AceuiP00i is the same quantity for the computational cell and 7 is a function of the surface temperature (TSWf) :

0.0 when Tsurf < T1;i

7 {Tsurf) — " when ,i< Tsurf

1.0 when Tsurf > T2)i

Tu and T2j are model parameters (temperatures) defined by the simulation operator. Different pairs of parameters are defined for the different materials (Si,Si02,SiC). The expression inside the left parenthesis in equation 3.96 is equal to the average heat flux to the metal pool, and defines the maximum heat flux that any cell can receive from the condensation. The cell receives this maximum flux (maximum condensation rate) when the temperature is below Ti,;. The heat flux (and condensation rate) then drops linearly to zero as the temperature increases from T\, to T2«. No condensation takes place above

The average heat flux [W/m2] to the surface of the cell becomes :

Qcond — Q cell,pod/A CCU:P001 (3.98)

The mass transfer to the computational cell from the condensation is in the model related to the heat transfer by equation 3.99.

Mcell,pool Qcell,pool Mtot&ool Q tot,pool

Q tot,poo l — Qtotfpool a. Mtot,pOol — Mtotfpool

Qcell,pool ■ Mcell,pool — -Mtott (3.99) Qtot

Mu,t is the total amount of silicon [kg/s] that evaporates from the entire metal pool. The mass flux [kg/m2s] to the surface of the cell becomes :

Iftcond ~ Mcell,pool/Acell,pool (3.100)

This mass flux is used when qaii is calculated (equation 3.58, page 56). 76 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

The corresponding equations for the crater wall are obtained by substituting the subscript pool with wall in the above equations. Only the horizontal surfaces facing the metal pool are considered during these calculations. This means that the vertical sides of non-gaseous structures above the metal pool (the particles, the electrode and the furnace walls) do not receive any heat or react as a result of condensation. These surfaces are consequently excluded when calculating Atot,waii- The condensation rate [moles/m 2s] for materials in the metal pool and the crater wall then becomes : Rm = mcond /Mi ■ i e {Si, Si02, SiC} (3.101)

The total power to the furnace gas caused by condensation is given by :

Q tot,gas = Qtotfgas ”b AQpool ”b AQuiall (3.102)

AQpooi is here the difference between Qtot,Pooi and the amount of heat that is actually distributed to the metal pool (Z) Qceii$ooi)- AQpoo i > 0.0 if 7 > 0.0 for one or more cells and zero otherwise. AQwaa for the crater wall is calculated similarily. This energy is distributed evenly (same power density [W/m3]) to all parts of the gas regardless of the local gas temperature, the position and the flow field. Each cell thus receives an amount of energy per unit time given by :

Qcell,gas — 0tot,gas y (3.103) gas Vceu and Vgas are the volumes of the computational cell and the total volume of all gas in the system, respectively.

Qceii,gas should have been a function both of the local temperature, the flow field and the -^(^-concentration. This may be implemented in future versions of the model. As it is, large condensation rates can be calculated in areas with low flow velocities even though little condensation actually takes place there. In the model no mass is transferred to the gas phase as a result of the evaporation and condensation : Mcell,gas = 0.0 (3.104) The consequence of transferring heat to the gas without also adding the corresponding mass is that the temperature of the gas increases. This is the opposite of what really happens when the gas is warmer than the surface where the evaporation takes place. The evaporation then adds colder gas (Tsurj) to the gas phase and its temperature should consequently decrease locally. Condensation of the Si(g) later adds back considerable amounts of enthalpy to the gas, and the temperature of the gas may decrease slower than it does with no Si(g) present. The net result depends on where and when the condensation takes place, and is difficult to estimate without knowing pst and performing the appropriate 3.6. THE FURNACE WALLS. 77

calculations. Such calculations are not carried out in the model, and the only way to partly compensate for this is to adjust the sum of fpoo i, fwau and fga5 to a suitable value. This is in fact the motivation for accepting their sum to be lower than 1.0 as mentioned on page 74. The difficult task of finding suitable values for these parameters is left to the simulation operator.

3.6 The furnace walls.

The furnace walls are assumed to be inert in the model. They can either be at a specified constant temperature or be subject to conductive heating. The outer edge of a furnace wall must either be at a specified constant temperature or it may be subject to a specified heat flux to or from the surroundings.

3.7 The electrode.

The electrode is assumed to be inert in the model. Any part of the electrode can either be at a specified constant temperature or be subject to conductive heating. The upper edge of the electrode must either be at a specified constant temperature or it may be subject to a specified heat flux to or from the surroundings.

3.8 The electric arc.

Fauchais, Pfender and Boulos [38] give a thorough discussion of the properties of electric arcs in general, including the cathode and anode regions. A plasma is an ionised gas comprised of molecules, atoms, ions, electrons and photons. In contrast to an ordinary gas, a plasma contains free electric charges which are usually produced by the gas itself by a variety of ionisation processes. The composition of the plasma and its properties vary with temperature. Overall, a plasma is electrically neutral. The electrical conductivity of a plasma increases rapidly with temperature. The ionised plasma is normally referred to as the electric arc, and it is visible from the radiation it emits at visible wavelengths. AC electric arcs are usually applied in the carbothermic silicon metal process. It is a formidable task to simulate the highly complicated behaviour of such arcs. Models for spatially fixed AC electric arcs operating at conditions similar to those prevailing in the silicon metal process are despite this being developed at present by L0ken Larsen [39]. These models are in their present form too time-consuming for practical applications in the metallurgical model described in this thesis. The main problem is that their time scales 78 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

(~ 0.01ms) are extremely short compared to the time scales of the metallurgical model. Instead, they may give useful information about quantities averaged over one AC-period and the generated flow field. These can be used for improving the metallurgical model. Research personnel and Dr. Ing. students at SINTEF/NTH in Norway have over the last decade developed and implemented both fairly simple and highly advanced models for DC electric arcs. Holt [20] has implemented a three dimensional model for a reactor consisting of three plasma torches, and Gu [40] has developed an advanced model for the interactions between an electric arc, a gas filled cavity and a silicon metal pool. These models are implemented with the fluid flow simulation program FLUENT [18] as the basis. The models are time consuming, and they must generally be run on fast computers. In the model a vertical DC electric arc with its center at the axis of symmetry and with the electrode serving as the cathode is applied (section 3.8.1). This is the same model as applied to a more complex geometry by Holt [20]. The electric arc influences the process not only by generating energy, but also by setting up a flow field in the crater. The flow field induced inside and in the near vincinity of a DC electric arc like that applied in the model is expected to be roughly as shown in Figure 3.18.

Electrode (cathode)

Gas flow

Metal pool (anode)

Figure 3.18: The approximate flow field (stream lines) inside a spatially fixed DC electric arc and in its near vincinity.

For the simplified situation with a spatially fixed AC arc, the arc tries to establish a situation similar to that in the DC case during each of the half periods. This means that the flow field reverses 100 times per second for 50Hz arcs. The resulting flow field is complex and basically different from that for the DC case. The overall effect is not known in detail, but the flow field inside the arc probably responds quickly since the Lorentz forces are strong and act directly on the gas in these areas. The impact on the flow field outside the electric arc is less obvious. Gas is sucked inwards along the metal pool surface close to the electric arc when the metal pool serves as cathode and is purged outwards when 3.8. THE ELECTRIC ARC. 79

it serves as anode. The resulting dynamical flow in the area underneath the electrode, but outside the electric arc is difficult to predict without doing the actual calculations. It seems, however, likely that the high-velocity gas at the anode end of the arc is purged away along the surface in an effective way, but that the gas sucked in a half period later is taken from the near vincinity of the electric arc. The situation for a spatially fixed AC arc is according to this qualitatively as shown in Figure 3.19.

| - w Electrode

f Flow (+) ^

Net backflow A'" L.... FIow +__ _ ——— Net outflow

r

Figure 3.19: Tentative fluid flow close to an AC electric arc.

A net outwards flow seems likely along the electrode- and metal pool surfaces and a net backflow (inflow) is then needed in-between to supply new gas to the electric arc. The flow may be fairly chaotic in this area. The flow field induced by an AC electric arc is even more difficult to predict further away from it. The effects of the arc are here very indirect, and especially weak are the effects from the cathode end (inflow end). It is difficult to imagine how these weak, indirect and rapidly changing effects can set up a highly structured and stable flow field underneath the furnace shaft. Instead, a chaotic flow field is expected here. Another aspect, that is even more important for the overall gas flow, is that the electric arc is most certainly not spatially fixed. Instead, it moves irregularity around in the electrode/metal pool area due to instabilities. Jensen [41] has documented such behaviour by recording the situation around the electrode of a 50kW furnace producing ferro-silicon on video tape. The electric arc is seen to move quickly around in the crater zone and along the surface of the electrode in an apparently random way. The length of the arc changes considerably during these movements. Furthermore, an electric arc that stays close to the axis of symmetry all the time enhances electrode corrosion in this area. Observations on industrial furnaces do not show very enhanced corrosion at the center of the electrode and thus indicate that the electric arc also for industrial furnaces is drifting with its ends along the lower surface of the electrode and the metal pool. Significant and sudden movements of the electrodes for industrial 80 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL. furnaces when regulated on constant electrical resistance indicate that the arc suddenly changes significantly. Such movements and changes in the arc geometry creates a chaotic flow field in the crater cavity. A spatially fixed electric arc is likely to overestimate the heat flux to the central parts of the metal pool considerably compared to a drifting arc of the same power level. The heat flux to the other parts of the metal pool is underestimated accordingly. Also, a DC arc is likely to overestimate the heat flux to the metal pool compared to an AC arc since the heat flux is larger at the anode end where gas heated in the electric arc is constantly purged down towards the metal pool. An AC arc purges the warm gas onto the electrode and the metal pool alternately, and the net heat flux to the metal pool is lower than for a DC arc of the same power level. The net heat flux to the innermost parts of the electrode tip (the cathode) is also overes ­ timated in the model compared to a situation with a drifting cathode spot.

3.8.1 The Prescribed Current Distribution model.

As mentioned previously, calculation models for AC electric arcs are too complicated for ap ­ plication in the model, and it is impossible to calculate a realistic flow field for AC arcs from models based on DC arcs. It is therefore pointless to use advanced and time-consuming DC models for the electric arc. Instead, the simple formulation by Ramakrishnan, Stokes and Lowke [42] is applied as a first approximation (see also Backer and Szekely [43]). This model is referred to as the ’’Prescribed Current Distribution ” model (the PCD model). The equations were implemented in the FLUENT code by Arne Endre Amtsberg, Division of Metallurgy, NTH, Trondheim, Norway. Some modifications have been carried out at later stages. Local thermodynamic equilibrium (LTE) is assumed when the energy equation is formu­ lated. LTE essentially requires that the temperature for the electrons is not significantly higher than those for the heavier particles in the plasma. The current distribution in the entire arc column is prescribed and the temperature and ve­ locity fields are calculated from the self-induced magnetic field and governing conservation equations. High velocities are calculated, and turbulent flow is assumed. The azimuthal ve­ locity component, vg, is zero due to symmetry and the absence of tangential gas injections or gas exhausts. The geometry of the electric arc and the current distribution over the cross section of the arc is assumed to be known apriori. The radius of the arc is given by equation 3.105 :

R{z) = Rc 1.0 + C. (3.105) 3.8. THE ELECTRIC ARC. 81 where Rc is the radius of the cathode spot, C is a non-dimensional expansion coefficient and z is the distance from the cathode spot as shown in Figure 3.20. Values in the range of 0.8 to 1.2 are normally used for the expansion coefficient. Large values are usually used for high currents and low flow rates (Backer and Szekely [43]). The outline of the electric arc and important quantities described in the following are shown in Figure 3.20.

R=R(0) Electrode (cathode) - j— ----- 1

Heat generation' and transport .

Metal pool (anode) +

Figure 3.20: The electric arc and important quantities. F is the Lorentz force and j is the current density. Their radial and vertical components are also shown.

The radius of the cathode spot, Rc, can be estimated from assumptions for the average current density inside it. Jordan, Bowman and Wakelam [44] photographed electric arcs from graphite electrodes at current levels up to 10 kA. They found current densities of jc = 4kA cm-2 in the central core of the plasma jet at the graphite surface. This corresponds to a cathode spot radius of Rc = sjl/{nj c) % 2.5 cm for a 77kA electric arc. Using this value for Rc in the Prescribed Current Distribution model resulted in unreasonably high temperatures in the electric arc (well above 35000K at maximum). Instead, the cathode spot was determined empirically to give reasonable temperatures (maximum temperatures in the order of 25000K) and reasonable power levels.' The actual values for the cathode spot used in different simulations are presented in chapter 5. The Prescribed Current Distribution model was developed for high-current free-burning arcs which are characterised by strong self-magnetic driven plasma flows. Cylindrical symmetry is assumed. The model is expected to be suitable for arcs with cathode jets strong enough to dominate the anode region of the arc. The arcs of the silicon metal process are high-current arcs (more than 50kA) of relatively short lengths (normally assumed to be about 10-20cm). 82 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

The axial current density is assumed to be :

jz(r, z) = A(z) fl.O - (^y)2j ; r < R{z) where A(z) is determined by integrating jz(r, z) over the entire cross section of the arc (which must be equal to the negative of the total current (/) with the definitions of the coordinate directions used in Figure 3.20) : rR{z) I = -27r J jz{r,z)rdr

1)

, 21 A(z) “ -lm7)

Substituting this value for A(z) into the equation for j2(r, z) gives the following equation for the axial current density :

<3-106)

The equation of continuity for the current is :

£-GUr,2)) + ^Wr(r,z)) = 0

Substituting equation 3.106 into the equation of continuity gives :

d , .f „ -2IC / 2r3 sF(rJ-(r’z» = ^)VT(r-»5)

Integrating from r=0 to r on both sides gives the following equation for the radial current density :

-It 3r(r, z) = lo -w r

The current density in the azimuthal direction is neglected :

je(r,z) = 0 (3.108) 3.8. THE ELECTRIC ARC. 83

This assumption is reasonable as there are no electric fields present in the cylinder sym­ metric system that can set up any current in the azimuthal direction. The azimuthal magnetic field, Bg, is evaluated by integrating the axial component of Ampere ’s law : (V x B(r, z))z = Hojz(r, z) 4 \^irBs{r, z)) = W:(r, z)

u

The magnetic fields in the axial and radial directions become zero from similar calculations :

Bz(r,z) = 0 (3.110)

Br(r,z) = 0 (3.111)

The Lorentz forces, F = j x B, give source terms in the axial and radial momentum equations :

Fz(r, z) = jr(r,z)Bg(r,z) (3.112) Fr{r, z) = —jz(r, z)Bg(r, z) (3.113) Fg(r,z) = 0 (3.114)

These forces (Figure 3.20, page 81) cause gas to be drawn in towards the axis of the arc and away from the cathode to set up a flow field roughly like that shown in Figure 3.18 on page 78. Slightly modified versions of the equations for the Lorentz forces are applied in the model :

Fz{r,z) = ah , jr(r,z)Be(r, z) (3.115) Fr(r,z) = -aLiT jz(r,z)Be(r,z) (3.116) Fg(r,z) = 0 (3.117) aLr are model parameters (the Lorentz force correction factors) determined by the simulation operator. These parameters are introduced because the gas velocities resulting from the DC arc are obviously far too large compared to the AC situation. Parameter values below 1.0 reduce the flow velocities. 84 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

3.8.2 Source terms in the in the energy equation for the gas.

Three terms [W/m3] specific to the electric arc are included in the energy equation for the gas (equation 3.5, page 24) :

• _ 3r , 3z . if Ohmic heating qokm,arc-- + - + - (3.118) 5kB (. dT . dT . dT\ Electron drift qelec'arc== 2^[3rdir+3,,d^+3e’*89) (3J19)

Volumetric radiation heat loss ([rad,arc = Sr (T, Tradj (3.120)

where

3ri jzi JO Current densities, jg = 0.0 (equation 3.108) a The electrical conductivity. kB Bolzmann ’s constant. e0 The electronic charge. ('rad The effective radiation radius of the arc. Sr(T,rrad) The volumetric radiation density.

The electrical conductivity, <7, and the volumetric radiation density, Sr(T,rrad), are both strongly temperature dependent, and are described in section 3.8.3.

3.8.3 Plasma properties.

The electrical conductivity is calculated together with other transport properties and the- modynamic properties for the Ar-Si-O-C system in the temperature range T=[1000K- 30000K] and at a pressure of 1 atm by a method presented and implemented by Gu [45]. This involves calculations of the plasma composition where 31 molecular, atomic, ionic species and electrons are taken into account. Two of these species are neglected since they are believed not to be important. Values for pure Argon and gas mixtures of 60 mole% SiO and 40 mole% CO are available in the present version of the model. The simulation operator selects the data to be used in the different simulations. The available data is presented in Appendix II. It is in principle easy to add data for any gas mixture in the (Ar-)Si-O-C system based on the method of Gu. The simulation operator could then select the desired gas mixture, and run simulations to examine how this influences the system. It would also be possible, however considerably more difficult, to make the model automatically interpolate between data for several selected gas compositions to estimate the transport properties etc. in each computational cell individually. Such extentions may be carried out in future versions of the model. 3.8. THE ELECTRIC ARC. 85

The volumetric radiation density is also calculated from a method presented and imple ­ mented by Gu [40]. The optically thin approximation is used when calculating radiative energy transport. Absorption is roughly accounted for by introducing an effective radia­ tion radius, rTad, i.e. gTad,are = Sr(T,rrad) (Gu [40, page 58]). Radiation data for Si-Ar plasma containing 0, 25, 50 and 100 mole% silicon and with rTad = 1cm, 5cm and 10cm are available in the present version of the model. The simulation operator selects the data used in the different simulations. The model does not automatically choose radiation data corresponding to the actual radius of the arc. This may however be implemented in future versions of the model. The available data is presented in Appendix II. . The radiation properties of any gas mixture can in principle be calculated by the Integral Method of Partial Characteristics developed by professor Viktor G. Sevastyanenko at the Belorussian Polytechnic Academy, Minsk, Belarus (Soloukhin [46]). With this method, reabsorption of radiation is automatically accounted for. Estimating an effective radiation radius is thus avoided. All necessary data for the Si-O-C system was recently calculated by Sevastyanenko and applied to arc modelling by Sevastyanenko, Gu and Bakken [47]. The data was not available in time to be considered used in the present work. Automatic interpolation procedures similar to those described above for the thermodynam ­ ical properties and the transport coefficients may be implemented also for the volumetric radiation density in future versions of the model. Better approximations for an arbitrary and dynamic local gas mixture and local arc radius may then be obtained automatically from the data for a number of selected gas mixtures and absorption radii.

Comment to a. Holt [20] remarks that the Prescribed Current Distribution model generally gives unrea ­ sonably high ohmic heating rates in the outer parts of the arc. The explanation is believed to be that the current density is prescribed (page 82) and thus not linked to the electrical conductivity through the temperature field as it should be. Nothing therefore prevents high current densities in areas with low electric conductivities caused by moderate or low temperatures. Overestimating the current density in these areas (the outer parts of the arc) results in too high ohmic heat generation as found by Holt. This is partly compensated for when evaluating q0hm,arc (equation 3.118, page 84) by using the electrical conductivity at T=5000K also for locations with temperatures below 5000K. 86 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

3.8.4 Radiative heat transfer to solids and liquids.

Gas volumes at high temperatures radiate energy to their surroundings. It is assumed that each such volume is sufficiently small to be considered a point source with a unif orm temperature. The total amount of energy radiated from cell j per unit time [W] is given by: QradJ — Sr(Tj,rr

Qj,i — &Fj'iQrad,j = &FjtiSr(Tj, Traj) Vj (3.122) where a is the absorptivity of the material receiving the radiation and Fjt, is the view factor from the point source to the cell. No radiation is absorbed by the gas outside the arc, but reabsorption inside the electric arc is to some extent accounted for by the effective radiation radius (rrad) as mentioned on page 85. The total amount of radiation energy received by cell i equals the sum of the radiation received from each point source j : 01—yi Qj,x j

This corresponds to the following average heat flux \W/m2] at the surface of the cell: Qrad — Qi/AitTa(l (3.123) where 4,->rad is the total area receiving radiation. A given surface receives radiation if and only if no non-transparent materials obstruct the radiation. Such obstacles are described in section S.8.4.3. In the model the absorptivity is set to 1.0 for all materials even though some radiation is actually reflected. Gu [40] uses a = 0.28 for the silicon metal pool. This is the value for liquid steel from Guthrie [48]. Using a value lower than 1.0 requires that the reflected radiation must be distributed to the other parts of the system to prevent that energy is lost. Such calculations are not included in the model, and neither is wall to wall radiation as described by Holt [20]6. Thus, assuming 100% absorption at all surfaces is probably a

6Average values for a moderate number of large surfaces are used in [20]. Thus, the algorithms are rough estimators for the wall to wall radiation. 3.8. THE ELECTRIC ARC. 87

better approximation than using data from the literature and then neglecting the reflected radiation. The distribution of the radiation will be somewhat in error, though. In fact, accurate estimation of the heat transfer by radiation for absorptivities less than 1.0 is difficult. The calculations are in practice both complicated and time consuming since the radiation emitted and reflected from one surface is only partly absorbed when arriving at the next. The calculations must therefore be repeated again and again for every reflection until most of the radiation is absorbed to get an accurate estimate for the overall radiative heat transfer from the electric arc. a = 0.28 implies that (1.0 — 0.28)" x 100% of the original radiation is still not absorbed after hitting n surfaces. Thus, about 10% of the original radiation is still not absorbed after hitting solid or liquid materials 7 times. Also, the radiation is scattered by the generally irregular surfaces. The reflected radiation is thus difficult to predict. This indicates some of the complexity involved in such calculations. Assuming a = 1.0 provides a rough first approximation. Most of the radiation is distributed to the central parts of the metal pool and the electrode when assuming a = 1.0 for a spatially fixed arc like that applied in the model. As mentioned previously, the electric arc certainly moves around from one location to another in an irregular way. The radiation is in that case distributed much more evenly. Also, scattering and reflections of the radiation as described above implies more even distribution of the radiation. Also surfaces in the shadow from the electrode or other obstacles would receive a considerable amount of radiation if reflection and scattering are accounted for. Thus, the energy flux caused by radiation to the central parts of the metal pool and the electrode is definitely overestimated in the model. A fairly simple way of adjusting for this is to distribute a predefined fraction of the total ra­ diation from the arc as at present. The rest of the radiation could be distributed uniformly to all surfaces in the system. This probably gives a more realistic overall distribution of the radiation than at present. One important consequence would be that more energy is delivered directly in the areas where SiC and SiOz are located in the model. Less energy would be delivered to the central parts of the furnace where only Si(l) is present in the model.

3.8.4.1 The view factors.

The view factor from a point source to an arbitrary area is defined by :

where n is the outwards unit vector perpendicular to the infinitesimal element at the surface receiving radiation and R is the vector from the point source to the same infinitesimal element as shown in Figure 3.21. cos(9)dA represents the projection of dA onto the plane located at the intersection between n and R and which is perpendicular R. 88 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

dA

Radiation source

Figure 3.21: Calculation of view factors from a point source to a surface dA.

For cylindrical symmetry (coordinates r, w and z) and a point source located at (r0, cv0, z0), the situation is as shown in Figure 3.22.

Cylinder receiving Radiation source radiation \ /■>!

Axis of synunetry

Figure 3.22: Cylinder receiving radiation from a source with rQ

Then, equation 3.125 is derived for a horizontal circular ring at z\ with inner radius and outer radius r2 and equation 3.126 for a vertical cylinder of radius n extending from z\ to *2 : 1 /-2ir rr2 nz(r,w,z1)-R(r,u,z1) F,(r,*,*) = -1 (3.125) R=(r,w,zi)

1 /-2jr rz2 nr(ruuj,z) - R(n,w,z) ^n.u>,z) = -l jn ridzdcu (3.126) R?{ruw,z) where „ , v f k when z0> z n,(r,»,z) = whenZo

7v(r,

R(r, w, z) = (t-cos (w) — r0)?+ rsin(w)j + (z — z0)fc (3.129) R(r,w,z) = | jR(r, w,z) | = y/r2 — 2rr0cos(w) + rg + (z — z0)2 (3.130) i, j and k are the unit basis vectors in the cartesian coordinate system. cv0 is not important as the integration is carried out around the full circle (2tt). Thus, the result is the same for all cvq . Also notice that the restriction r0 < r in equation 3.128 is in practice always satisfied since the electrode has larger radius than the electric arc. It is therefore not necessary to consider the much more complicated case rQ> r where radiation to some parts of the cylinder is obstructed by the cylinder itself. The case r0 > r must be considered for surfaces perpendicular to the z-axis though, but this causes no similar problems. Equations 3.127, 3.128 and 3.129 give : nz(r,w,Zi) ■ R(r,w,zi) = -|zi -z0| (3.131) i%r(ri, w, z) • R(ri, tv, z) = r0cos(uj) — r* (3.132)

Substituting these expressions into equations 3.125 and 3.126 gives : [zi - zp| f/•z-2~ fmr2 ______rdr______Fz(r, cv,zi) = rdw (3.133) JO Jr i 4tt (r2 — 2ttqcos{w) + t"o + (zi — z0)2)2 1 9 f*2Z dz Fr(ru tv, z) = — Jo {r\ - nr 0cos(w)) J rdu) {r\ - 2rir0cos(uj) + rg + (z - z0)2)5 (3.134) The inner integrals of these equations are integrated analytically by using relations from Gradshteyn and Ryzhik [49]. The calculations are carried out in appendix I, and the results are,7 . Fz{r,uj,z j) = AM&v (3.135) 1 Fr(ruaj,z) = —Jo /r(cv)dtv (3.136) where \*i ~ zq\ rl + (zi - z0)2 - ri7~0 cos (tv) /zM = rgsm2(tv) + (zi - z0)2 V R(n, w, zi) r\ + (zi - zp) 2 - r2rocos(u ) (3.137) R(r2,v,zi) ff,A - ri ~ nr 0cos{uj) ( z2-z0 Zl-z0 \ ^ r?-2ri7'0cos(tv)+rg \J?(ri,cv,z2) R(ri,tv,zi)y

7Both integrands are symmetric about the line from tv = 0 to tv = n, so that the integrals are evaluated from 0 to 7T rather than from 0 to 2ir and then multiplied by 2. 90 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

It is sufficient to use only one of the two integrals 3.135 and 3.136. The view factor to the inner surface of a vertical cylinder z = [z%, z%] and r = r2 can for example be evaluated as the difference between the view factors to the two full circles of radius r = r2 at z = z\ and z = z2 using equation 3.135 with r\ = 0 :

Fr(r2,u,z) = Fz(r,u,Zi)\ri=0-Fz(r,u;,Z2)\ri=o ; zQ

A

z. 2

zo ;...... Radiation source

Figure 3.23: The connection between the view factors to surface A, B and C.

In terms of the surfaces defined in Figure 3.23, this corresponds to :

Fc — Fb — Fa where Fa, Fb and Fc are the view factors to surfaces A, B and C, respectively. All view fac­ tors can thus be calculated using equation 3.135 several times and then adding/subtracting view factors to carefully chosen full circles. This approach has been used by Holt [20, page 212-217]8 , but not in the present thesis.

3.S.4.2 Numerical evaluation of the integrals for the view factors.

The integrals for the view factors (equations 3.135 and 3.136) are both evaluated by numer ­ ical integration using Simpsons rule [50]. The integrand is evaluated at (M +1) different values of w. Ni is determined as follows :

The initial value is found by :

(3.139)

8 There are some printing errors on the referenced pages. Equation 5-38 in [20] is, however, identical to equation 3.136 with ri = 0 and with zq replaced by |zi — zo| in [20]. 3.8. THE ELECTRIC ARC. 91

where the function int(#) returns the integer closest to the argument x. Thus, max R and min R are the maximum and minimum distances from the radiation source to the surface receiving radiation. The integral is then evaluated for jV0 and M = 2N0, and the relative error is estimated by : F(Nj) - F(Nj-x) Ei = (3.140) F(Ni) where i = 1 and F(N{) is the view factor calculated when N = IV;. New evaluations are performed with higher values of Ni (Ni = 2Ari_1 , 3M-i or 4Ni-i depending on the value of E{) until the relative error is below a given tolerance limit (0.001) or a maximum number of attempts (10) has been made. There is also an upper limit (10000) to the value M- The simulation operator is informed if the final view factor does not satisfy the accuracy check.

A total of M = Ej Pj view factors are calculated (where 0j is the number of computational cells that receive radiation from point source number j). The number M may in some cases be quite large, and evaluating the M view factors may thus take considerable time9 . This is not critical since the calculations are performed only once for every time step. The view factors occupy considerable memory for computational domains involving many radiation sources and many cells receiving radiation.

3.S.4.3 Obstacles blocking the radiation.

As mentioned previously, the radiation reaches a given location (sink) if and only if other non-transparent structures do not obstruct the line of sight between the source and sink. Calculating such obstructions is rather complicated for radiation sources that are not po ­ sitioned on the z-axis (r0 ^ 0). The area that is obstructed then varies with the value of w. Minimum obstruction is found for w = tv0, and maximum for w = w0 + tt as shown for a simple situation with only two cylindrical particles in Figure 3.24. In the model the situation is simplified by assuming that the radiation sources are located at the axis of symmetry when calculating the areas obstructed by other structures. The obstructed area is then the same for all values of w, corresponding to the area at z> zs in Figure 3.24. The view factors to cells below z = zs are then calculated from equations 3.135 and 3.13610. The shadow typically starts inside a computational cell rather than at the interface between two such cells. The practical calculations are performed by superposition of the individual shadows from all structures that are defined to be non-transparent. 5z02- and SzC-particles are non-

9 In the order of 10 seconds for the simulations presented in this thesis. 10The radiation source is located at r = r

Axis of symmetry

Point source for radiation

Figure 3.24: Particle obstructing radiation from an off-axis radiation source (r0,cv0,zo). The shaded areas represent the radiation that is not obstructed by the innermost particle. The upper edge of the shaded area varies between the extreme values Zmin for w = wa + -it to Zmax for u> = w0. zs is the lower edge of the shadow when the radiation source is moved onto the axis of symmetry (r = 0). transparent. All other solid- and liquid materials are transparent unless the simulation operator explicitly defines them to be non-transparent. Non-transparent materials receive radiation while the radiation passes straight through transparent materials.

3.8.5 Heat exchange in the cathode and anode regions.

The anode region. The anode region is in the model identical to the metal pool. The total heat flux \W/m2} to the anode region is calculated by :

Qanode — Qgas “b Qrad "b Qelec “b Qcond "b Qevap Qreac (3.141) qgas is the heat transfer from the gas to the metal pool surface as given by equation 3.22 on page 28. qTad is given by equation 3.123 on page 86. The condensation (qcond), evap­ oration (qevap) and reaction terms (qTeac) are given by equations 3.98 on page 75, 3.94 on page 73 and 3.56 on page 55, respectively. qeiec represents the energy transfer caused by the flow of electrons from the arc to the anode. It consists of the electron thermal energy (Thomson effect), the electron potensial energy (work function) and the energy acquired 3.8. THE ELECTRIC ARC. 93

by the electrons across the anode fall region :

?elec = jz ^ gg (-^6 Twj + & an ode ~i~ Vanodc (3.142)

where jz = The current density at the anode surface. Ub = Bolzmann ’s constant, eo = The electronic charge. Tb = The temperature in the bulk gas above the anode. i Tw = The surface temperature of the anode. f anode = The work function of the anode material. Vanode = The anode fall voltage.

The anode fall data from the literature are inconsistent. Slightly negative anode fall volt ­ ages have been reported from experiments (Sanders and Pfender [51]). Other data refer­ enced by Sanders and Pfender [51] show anode falls in the range 3.2-9.0V. The work of Choi and Gauvin [52] indicates that the anode fall becomes less negative and eventually positive at increasing currents and arc lengths. Negative anode falls seem to be associated with diffuse arc attachment at the anode, which is not the case for the arcs of the carbo- thermic silicon metal process. It is therefore reasonable to assume a positive anode fall in the model. It is also unknown if or how the anode fall varies with arc length and current. The data found in the literature is normally for arcs of less than lkA rather than for arcs of several tens of kA as applied in the carbothermic silicon metal process. In absence of more accurate data, the value is set to the same value as used by Gu [40]. This value, 3.7V, is based on the studies of Szekely, McKelliget and Choudhary [53], Ushio, Szekely and Chang [54] and Hsu, Etemadi and Pfender [55]. The value of the work function, which is a characteristic property of the anode material, is set to 4.83V in accordance with the data for silicon given by Riviere [56]. The heat transfer is calculated for every computational cell in the anode region. jz is calculated from the model for the electric arc (equation 3.106) and is averaged over the closest computational cell above the anode surface cell. The temperature of the same cell is used as an estimate for Tb, and the temperature of the anode surface cell is used as an estimate for Tw.

The cathode region. As mentioned previously, the chemical reactions and evaporation of carbon from the elec­ trode surface are not included in the model. This implies that the cathode region can be modelled in a rough way with minimal consequences for the performance of the model. The surface of the electrode where the electric arc is attached is kept at a constant prede ­ fined temperature corresponding to the sublimation temperature for carbon. JANAF [6]

T 94 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL. indicates values in the range 3895K-4070K, and the value 4000K is used. This value can be changed by the simulation operator. The heat flux to the part of the cathode surface where the temperature is prescribed is irrelevant and is therefore not calculated. The gas temperature is, however, influenced by the temperature of the cathode surface. Holt [20] and Gu [40] discuss the cathode region in greater detail. See also Fauchais, Pfender and Boulos [38] for a comprehensive discussion of these regions.

3.9 The furnace shaft including the crater wall.

The situation in the furnace shaft may be as indicated at the top of Figure 3.25.

| =Si (T*) = Condensate

= SiC O=Si02 Fumaceigas < = C partly converted to SiC

Crater wall Axis of symmetry Furnace wall

Figure 3.25: The furnace shaft viewed from the side. The furnace is on the left and the model on the right. A closeup of the materials in the shaft is included. 3.9. THE FURNACE SHAFT INCLUDING THE CRATER WALL. 95

This situation is simplified considerably in the model. The background for these simplifi ­ cations is presented in sections 3.3, 3.4 and 3.5. Pure SiOa, pure SiC, inert materials and gas are the only materials left.

3.9.1 Electric currents.

An unknown, but moderate fraction of the total electric current is believed to pass through the materials in the furnace shaft. The current transfer mechanisms in the charge of submerged-arc furnaces are according to Urquhart, Jochens and Howat [57] and Channon, Urquhart and Howat [58] :

• Ohmic conduction at low voltages. • Arcing between the particles at high voltages.

They also found that there is a critical voltage at which the ohmic conduction starts to break down. The main part of the currents in the charge is probably present in the lower parts of the shaft where the amount of SiC is largest and the temperatures are highest. The electrode-to-electrode currents in submerged-arc ferrosilicon furnaces amounts to about 1% of the total electrode currents according to Schafer [59]. Further details are presented by Valderhaug [60, pages 106-113]. All currents in the furnace shaft and their effects are neglected in the model. The most noticeable consequence of this simplification is that heating of the materials in the shaft caused by such currents are lost. This is not critical if the results of Schafer are correct and also apply to carbothermic production of silicon metal.

3.9.2 Particle representation.

Condensed materials are represented by individual concentric rings of compact solid ma­ terials appearing as vertical rods in Figure 3.25. Each particle consists of a number of computational cells in the vertical and radial directions. The cell sizes change with z and r as illustrated in Figure 3.26. Modelling the raw materials as a large number of individual particles rather than as a comparably small number of long cylinders increases the overall complexity significantly for two main reasons. First, handling materials dripping down from the crater wall to the metal pool becomes much more complicated than the simple method described in section 3.9.4. Secondly, the calculations become extremely time-consuming since the num ­ ber of computational cells grows excessively large. This is because each individual particle must consist of a minimum of computational cells to resolve temperature gradients that are 96 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

r Particle in the V:L; ^ furnace shaft •T" , ; : ♦ t k c Computational cell ■? V Cell interfaces (, ML It ■' ? % j 1

T

Figure 3.26: Particle geometry and representation in the model. essential for the chemical reactions. Also, estimating the flow field around and in-between numerous non-permeable and randomly distributed particles requires a large number of computational cells. Thus, a fairly low number of individual particles is a necessity for keeping the overall computational time at an acceptable level. It is also doubtful if there is much to gain from introducing lots of small particles. Applying permeable/porous ma­ terials is probably better 11. This would allow gas to penetrate into all parts of the shaft, and it would in many respects be a good approximation even though the SiOa-particles are compact. This will be considered in future versions of the model. In fact, porous cells have been used in the model, but not for representing Si02 or SiC. Instead, they represented inert materials in the upper parts of the gas channels between SiOz and SiC. The motivation was to adjust the resistance to the gas flow in these channels to stop the unreasonable inflow of gas that was calculated for some of the channels. The DC arc causes high gas velocities along the crater wall. The crater gas either enters the channel or it continues along the crater wall as indicated in Figure 3.27. The viscous forces from the crater gas passing by a gas channel may result in gas flowing into the crater cavity through these channels. Such inflow of gas is unlikely for industrial furnaces where the velocities along the crater wall are probably moderate or low, and where the gas gradually penetrates into the permeable crater wall. It was hoped that introducing porous cells with carefully adjusted gas flow resistances in the gas channels should stop the inflow. The idea was to increase the flow resistance in channels with outflow. The pressure would increase and some of the gas would leave the system through channels with inflow in the first place. All such attempts failed, and it was decided to avoid the problem by filling up all gas *

nThe FLUENT code has built-in mechanisms for modelling permeable/porous materials. 3.9. THE FURNACE SHAFT INCLUDING THE CRATER WALL. 97

Gas channels between particles

Furnace gas

Figure 3.27: Gas flow in the channels of the furnace shaft. Gas is dragged into the crater cavity from channel C by viscous forces induced by the crater gas passing by- channels except one with compact inert materials. That the attempts all failed, does not necessarily mean that the desired effect can not be achieved by applying a suitable combination of resistances to the various gas channels 12. It seems reasonable that such a combination does indeed exist, but it may be difficult to find.

3.9.3 The specific surface area for particles.

A large compact body has a much smaller specific surface area than the same body cut up into several smaller bodies. Also, irregular bodies generally have larger specific surface areas than smoother bodies, and porous materials (like SiC) have larger specific surfaces tha n compact materials. Representing the charged materials by smooth and large cylinders consequently leads to underestimation of the specific surface areas. The specific surface area is important for the chemical conversion of those materials where the reaction rate is proportional to the surface area. Correction factors are introduced to compensate for this. One such factor is defined for each type of material in the system. Another set of correction factors is introduced to compensate for the fraction of the total surface that for reasons like those explained in section 3.3.4 is physically blocked from undergoing chemical reactions.

"Notice that all such attempts were made before the Lorentz correction factors (page 83) were introduced to reduce the unrealistically high velocities created in the crater cavity by the DC arc. The problem is less prominent and thus easier to solve when the flow velocities along the crater wall are lower. It was not possible to pursue this point within the scope of the present work. 98 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

It is likely that both the correction factors for specific surface area and for blocked surface change significantly with the position of a specific particle. Even so, identical values are used for all particles of the same type regardless of their positions. The effective surface area undergoing chemical reactions is therefore calculated by :

Aefftj = /5J-(1.0 - fbij )Areac ; j€{Si,SiC,Si02} (3.143)

where fsj = Correction factor for specific surface of particle type j. fb j = Correction factor for blocked surface of patricle type j. Areac = Reacting surface area calculated from the model geometry.

The model parameters (fsj (> 1.0) and fb j e [0.0,1.0]) are defined by the simulation operator, and they can be changed at any time. Default values of fsj = 1.0 and fb ,j = 0.0 are used unless the simulation operator explicitly changes the values. Determining these parameters is difficult.

3.9.4 Dripping.

SiOz drips down from the crater wall to the metal pool upon melting. The Si02 is believed to form viscous structures that hang down from the crater wall for considerable time before gravity finally tears them loose as shown in Figure 3.28. Dripping is therefore irregular.

Furnace shaft Furnace shaft

Figure 3.28: Dripping of materials from the crater wall. The situation is for a small part of the crater wall at time tn and a bit later at time tn+\ just before the droplet to the right drips down.

SiC and other materials present in the crater wall drip down together with Si02. 3.9. THE FURNACE SHAFT INCLUDING THE CRATER WALL. 99

In the model dripping is governed by the melting of Si02. The algorithms used for the different types of materials are described in the following. Materials that drip down are added on top of particles of the same type positioned straight below in the metal pool as described in section 3.9.6.

Dripping of Si02 The mass that drips down from an arbitrary computational cell i of an 5i02-particle j is calculated by : mdriPj,i = arPtPjdVi (3.144) where

or = -hp —*- : ar e [0-0,1.0] A-Ldrip ft = ; ft e [o.o, l.o] Atdrip PjdVi = the total mass of cell i

Ti is the temperature of the computational cell, Ts is the temperature at which Si02 starts to drip down from the cell. tn +1 and tn denote the time at time step n +1 and n, respectively. The model parameters Ts , ATdrip and AtdTip are defined by the simulation operator, and can be changed at any time. No mass drips down from the cell at temperatures below Ts. The fraction that drips down increases linearly in the temperature range Ts to Ts + ATdrip . All the mass in the cell drips down over a time period of Atdr{p when 2) > Ts + ATdriP. The total mass that drips down from a given particle, j, is equal to the sum of the mass that drips down from each individual cell:

Tfldrip,j — 'y ] Tfldrip,i :j (3.145) i€S where S represents all computational cells of particle j that contribute to the dripping. The set S is defined by a model parameter, n drip , which determines how many rows of computational cells counted from the bottom of the particle that contribute to the dripping. n driP = 3 means that the three lowermost rows of cells contribute, whereas all cells above this do not (Figure 3.29). This means that cells both in the crater wall and high up in the furnace shaft in principle can contribute to the overall dripping from the particle. The simulation operator can change the value of n drip at any time. It would be natural to select a low value as it is difficult to imagine how Si02 high up in the furnace can contribute directly to the dripping even if it for some reason is warmer than Ts. 100 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

0 = Cells not contributing to the dripping ■ = Cells contributing to the dripping dz Si02,drip,j ' | = Mass that actually drips down r

Figure 3.29: Cells contributing to the dripping from Sz"02-particles when nd riP = 3.

The total mass that drips down from a given particle is removed as an equally thick slice from the lower edge of the particle as shown in Figure 3.29. The thickness of this slice (dzSio 2,dripJ ) is given by : where pj is the density of the given material and dAj is the cross section of the lower end of the particle. dzsio 2,

Dripping of SiC SiC that drips down is also removed as an equally thick slice of materials from the lower end of the particle. The thickness of this slice is calculated from the amount of 5z02 that drips down, from the chemical reaction and from some additional assumptions to be explained soon. The simulation operator selects between two different strategies. One strategy is to demand that equally thick slices of SiC drip down from all SiC-particles. The other is to preserve the general geometrical shape of the crater wall according to certain rules. The former strategy was the one that was implemented first. It is very simple, but lacks the favourable property that much SiC also drips down in areas where much Sz02 drips down and less where little Si02 drips down. The overall result is that the geometry of the crater wall may become rather irregular after a time as shown to the left in Figure 3.30. 3.9. THE FURNACE SHAFT INCLUDING THE CRATER WALL. 101

□ =Si02 ■ =SiC = Removed mass

Equal dripping height Preserving the for all SiC-particles geometry of the wall

Figure 3.30: Available strategies for dripping of SiC. The original crater wall is given by the dotted lines and the situation after a while by the solid lines. The amount of Si02 that drips down is identical in the two situations.

The two algorithms are as follows :

1. Equal dripping height for all SiC-particles.

• Assume that mSic,oid and mSio 2,oid are the total mass of SiC and Si02 in the furnace shaft at the end of the previous time step. • The total mass of Si02 in the furnace shaft at the end of the current time step is then given by :

msi02,new = ™'Si02,old ~ mSi02,drip,tot ~ msi02,chan,tot

where mSio 2,drip,tot and mSio 2,chan,tot are the total amount of Si02 in the fur­ nace shaft that drips down and reacts chemically during the current time step, respectively. • The total mass of SiC in the furnace shaft at the end of the current time step is calculated by : msic,oid TftSiCtiiew — Tfl'Si02inew ™Si02iold This aims at preserving the SiC/SiC2-ratio in the furnace shaft. • The total mass of SiC that drips down from the furnace shaft is calculated by :

W-SiC,drip,tot = TTlSiC.old ~ mSiC,new — t^SiC,chan,tot j ™-SiC,drip,tot > 0.0 where mSic,chan,tot is the total amount of SiC in the furnace shaft that reacts chemically during the current time step. 102 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

Notice that the SiC/SiOz-ratio of the furnace shaft is not preserved if the chemical consumption of SiC is so large that the expression for msic,drip,tot becomes negative (no dripping of SiC).

• Finally, mstc,drip,tot is removed from the lower ends of the SYC-particIes as equally thick slices (dzsic,dHP,j) from each particle. 2. Preserving the geometry of the crater wall. • Assume that dzsio 2j is the change in the height of Sz02-particle number j due to dripping and chemical conversion, and that these value's are plotted as a function of the center position of the particle in the r-direction as shown at the top in Figure 3.31. • A piecewise linear curve is drawn between these values, and the dz-values for the SiC-particles are simply taken to be the values for this curve at the center positions (in the r-direction) for the SiC-particles. Constant dz-values are used inside and outside the innermost and outermost SiC2-particles as shown by the solid line in the figure. An alternative would be to extrapolate from the two nearest SiC2-values as indicated by the dashed lines in Figure 3.31. • dzSic,dripj is calculated by :

dZsiC,dripJ = dz(r) dzsic,chemj i dz§iC,drip,j ^ 0.0

where dzsic,chom,j is the change in height caused by chemical conversion. Sit­ uations may occur where dzSic,chemj > dz(r). More mass is then removed by chemical reactions than corresponding to dz(r). Thus, the geometry of the crater wall is not preserved as intended in these cases.

The mass that drips down from each SiC-particles is given by :

TTldripj ~ Pj dzsiC,dripj dAj (3.148)

’’Dripping ” of inert materials. Inert materials are also removed from the crater wall as a result of removing other materials. These materials are not transferred to the metal pool. Thus, mass conservation for inert materials is not satisfied. This is not believed to be essential since the inert materials in the furnace shaft are not important for the process as such. They are merely introduced in order to fill up the gas channels and by this prevent inflow of gas to the crater cavity as described on page 96. Ordinary materials (SiC2 or SiC) may be used for filling up the gas channels instead of inert materials. The difference is basically that no free silicon is then found between the 3.9. THE FURNACE SHAFT INCLUDING THE CRATER WALL. 103

dz(r)

Figure 3.31: Determining how much mass that is removed (dripping plus chemical reac­ tions) from the SzC-particles by algorithm number 2. A slice of thickness dz(r) is removed. The resulting crater wall profile is shown at the bottom. other particles in the metal pool since non-inert materials drip vertically down and block such surfaces. All inert materials not having other solid materials on both sides are removed. This implies that inert materials adjust their lower surfaces to the uppermost SiC- or SzC^-particle to their left and right as shown in Figure 3.32. However, an algorithm similar to that used for the SzC-particles would probably have been a better choice, giving a somewhat smoother crater wall. The geometry of inert particles are adjusted after the other particles in the furnace shaft.

3.9.5 Chemical conversion.

Particles in the furnace shaft undergo chemical reactions as described in section 3.4. The chemical conversion for each cell i of a particle j is calculated separately and all contribu ­ tions are added to get the overall conversion for the particle :

Wlckem,} = ^ ^chem,i,j (3.149) rrichemj is positive when materials are consumed from the particle. No S1O2 or SiC is produced by the included chemical reactions. Liquid Si is produced, but this does not add 104 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

Before adjusting inert particles After adjusting inert particles

□ = other material

Figure 3.32: Adjusting the lower ends of inert particles in the furnace shaft. All inert materials are removed that do not have other solid materials on both sides.

any silicon in the shaft since it is immediately distributed to the metal pool as described in section 3.10.6. Thus, solids and liquids are never added to the furnaces shaft. The total mass consumed in chemical reactions for a given particle is removed as an equally thick slice from its lower end regardless of where the material did actually react. Notice the similarity with the algorithm for removing mass that drips down (section 3.9.4). The details concerning the energy and mass exchange associated with chemical reactions are presented in section 3.13. Possible future extention of the model to also cover the chemical reactions in the low- temperature zone implies that solid and liquid materials are produced in the furnace shaft. The main challenge will be to distribute the produced condensate among the already existing materials and to update the particle geometry accordingly.

3.9.6 Updating the particle geometry.

As described in section 3.2.5, changes in mass of solids and liquids caused by chemical reactions are accounted for when progressing from one time step to the next. The detailed algorithms for the furnace shaft are described in the following. The mass for a given particle j changes due to dripping and chemical reactions. The sum of these is removed as an equally thick slice from the lower edge of the particle. This sum generally differs from the mass contained in the lowermost row of computational cells for the particle. A typical situation is shown in Figure 3.33 where 75% of the mass in the lowermost row of cells is removed. The upper edge of the particle is not changed. Each computational cell represents either solid material or gas, but not both at the same 3.9. THE FURNACE SHAFT INCLUDING THE CRATER WALL. 105

Particle in the a z furnace shaft

Computational cell

Cell interfaces

Dripping down

Chemical consumption

r ■55-

Figure 3.33: Material to be removed from a particle in the furnace shaft due to dripping and chemical reactions before progressing to the next time step. time. The density of the particle is therefore adjusted so that its new lower edge fits exactly with the boundary between two computational cells13. Occasionally, the height of the particle changes. The applied strategy is to adjust its height so that its density is as close as possible to its true density. The details are described soon. The situation where more mass is removed than what is present in the lowermost row of cells is in principle identical to the one shown in Figure 3.33. The only difference is that mass is removed from more than one row of cells. In fact, only a moderate fraction of the lowermost cells should be removed in one single time step 14. No automatic time step control is at present implemented to assure this. However, such automatic control is easy to implement once the exact criteria for adjusting the time step length are determined. These criteria are not necessarily easy to derive. Mass consumed by chemical reactions is physically positioned below the dripping mass as shown in Figure 3.33. There is no specific reason for choosing this arrangement rather than the opposite. Both probably do equally well, but the enthalpy of the materials that drip down is somewhat different in the two cases. The difference increases for large temperature gradients. Complicated algorithms are needed to handle this accurately. It is probably little

13 An alternative strategy is to define a parameter representing the fraction of a given cell that is actually filled with solid or liquid materials. This strategy is not used because large heat fluxes to almost empty cells would lead to rapidly increasing temperatures. 14No warnings are issued to inform the simulation operator if this is not the case. 106 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL. to gain from such algorithms since the mass is removed as an equally thick slice from the lower end of the particle rather than from the exact location where the chemical reactions take place or mass drips.

In the model the density of particle j at time step n (pnj) differs from the true density of the material (pj). The total mass of particle j at time step n + 1 after adjusting for dripping and chemical reactions is calculated by :

TT^n+lj — 77^-n j TTtchemj ?7Zdripj " Pn,jdznjdAj fflchemj ffidripj (3.150) where dznj is the height of particle j at time step n and dAj is the cross section of its lower edge. dAj is constant since materials are removed only from the lower edge of the particle and not from its sides. mchem,j is given by equation 3.149 on page 103 and mdrip

Alternative I Alternative n

be removed;

Mi: iCeii row izI : i i i i I Before adjusting Adjusting according to the Adjusting to lower Adjusting to upper J s true density of the material interface interface

Figure 3.34: Adjusting the lower surface for a particle in the furnace shaft. The final adjustment is in this case according to alternative II. Cells that change from representing the particle to representing gas are marked by ’X’.

The algorithm consists of the following sequential steps :

1. Calculate the new mass of the particle (m„+li3) by equation 3.150. 2. Calculate the height of the particle (dzj) using its true density (pj) :

dzj mn+l,j (3.151) Pj dAj 3.9. THE FURNACE SHAFT INCLUDING THE CRATER WALL. 107

dzj is inside cell row number Iz in the vertical coordinate direction as shown in Figure 3.34.

3. Calculate dz^ronstfion by .

<2dzioWerdzupper

dzi■transition — (3.152) dZlower "h dzuipper where d2Wr is the height of the particle if its lower surface is at the lower edge of cell row number Iz, and dzupper its height if its lower edge is at the upper edge of cell row number Iz. Equation 3.152 is derived later. 4. Calculate the new height of the particle by : f dzioujer when dzj ^ d^transition Or dz-upp 0.0 dZn+ij = (3.153) dzupper otherwise

5. Scale the particle in the vertical direction as described later and adjust its density to : mn+l,j (3.154) dA, 6. Update the variables in the FLUENT code to account for the changed situation as described later.

The mass that drips down is added to particles in the metal pool as described in sec­ tion 3.10.

Deriving the equation for dztransitim • The basic strategy when determini ng the geometry of the particle is that its density shall be as close as possible to its true density. Equation 3.152 is consistent with this strategy as explained in the following.

Assume that piower and p uppe r are the densities of the particle corresponding to heights dziower and dzupper in Figure 3.34, respectively. The relative errors in the density for the two cases are then given by :

Piower Pj Eiower Pi (3.155)

Pupper Pj E,'upper Pi 108 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL. dapper is used if Eupper < Elower. Otherwise, dziower is used. This inequality can be reformulated as follows :

Eupper ^ Elower

E:upper < E?1lower

(.Pupper Pj) < (Plower Pj)

Pj n (Popper *b Plower)

Substituting

mn+l,j Pj dzj dAj mn+l,j Plower dziower dAj t^n+l ,j Pupper dZuppcr dAj into this inequality gives :

1 1- + - 1 > 2 dzj dz,upper dzio

^ 2dZiowerdZwpper

dziower b dZy*-• upper This reduces to equation 3.152 when dzj is replaced by dztransition at the transition height (equality instead or inequality). Eupper and Eiower are small unless the particle consists of only a few cells in the z-direction (about 0.05 at maximum for a particle consisting of 10 equally high cells). The particles in the furnace shaft normally consist of far more than 10 cells in the vertical direction, and the lowermosts cells are generally smaller than those at the top. Thus, the difference between the true density of the material and the density used in the model is well below 5% in most cases.

Scaling particles. The practical work involved in scaling a particle consists of adjusting its temperature profile, updating its density and changing the cell types and data for those computational cells that change from representing solid materials to representing gas and vice versa. Four 3.9. THE FURNACE SHAFT INCLUDING THE CRATER WALL. 109 cells undergo this change in the example in Figure 3.34 where alternative II is assumed to be the final situation. These cells are marked ’X’. No cells change types in alternative I. The geometry of the particle then remains unchanged, but its density decreases slightly and the scaling of the remaining material is carried out as usual. The scaling is performed in one single operation from dz to dzn+i,j. A particle consists of several vertical columns of computational cells. The temperature profile for one such column is shown in Figure 3.35. Linear interpolation is assumed between the node points.

Node points

Particle

Figure 3.35: The temperature profile through a vertical column (shaded area) of compu ­ tational cells in a particle. Linear interpolation between the node points is assumed.

The temperature profile for a particle is updated by straight forward scaling of the tem­ perature profile for each of its vertical columns. The scaling factor, o%, is identical for all columns : a. = (3.156) Pn+lJ aj > 1.0 corresponds to expansion and otj < 1.0 corresponds to contraction. This very simple way of updating the temperature profile has the disadvantage that it does not fully preserve the enthalpy. The problem arises partly because the temperature profile is known only at a fixed number of points (the node points), and partly because the value at a node point represents the temperature corresponding to the average enthalpy of the cell rather than the temperature itself. 110 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

The temperature profile for the scaled particle is determined as illustrated graphically in Figure 3.36.

Old node temperatures, — but scaled to new z-values

Final T(z)-

Final node temperatures, interpolated from the scaled old profile.

B b

'Node points

Figure 3.36: Determining the temperature profile in the range z € [c, o] when scaling a contracting particle (a,- < 1.0). Up is to the left. Linear interpolation is used between T(A) and T(B) to find the new value T(a). T(A) is the temperature at z = a before scaling.

The positions of the node points do not change during the scaling, but the temperature profile is stretched or compressed by the factor a, so that the original node temperatures no longer fit with the node points. First, the node temperatures for the unsealed particle are plotted at their new z-positions (A, B and C in Figure 3.36)15. Then, the new node temperatures (T(a), T(b) and T(c)) are calculated by linear interpolation between the values on each side of the node point (T(a) from T(A) and T(B) and so on). The final temperature profile is represented by the solid line through T(a), T(b) and T(c). As mentioned previously, the enthalpy is not fully conserved when applying this algo­ rithm. It would not be conserved even with higher order interpolation methods since the temperature represents the average enthalpy of the cell rather than the temperature at the node position. It is difficult to design an algorithm that both conserves the enthalpy and preserves the temperature profile accurately. The applied method is believed to be a reasonable trade-off between accuracy and simplicity.

lsThe offset (|A-a|, |B-b| and |C-c|) increases with the distance from the fixed upper edge of the particle. 3.9. THE FURNACE SHAFT INCLUDING THE CRATER WALL. Ill

Alternative algorithm for particle scaling. A more advanced algorithm was first designed for updating the particle geometry. This algorithm added up the contributions from dripping and chemical reactions for each vertical column and then removed this mass from the lower edge of each column separately. The result before scaling was a particle with a smooth (unchanged) upper edge and an irregular lower edge as shown in Figure 3.37 for a particle consisting of four columns.

Moving mass Scaling to sideways grid interface

Removed mass Before updating Final result

Figure 3.37: Updating the particle geometry after dripping and chemical reactions ac­ cording to the old and rejected algorithm. The arrows in the figure on the left indicate mass being moved from long columns to short ones to smooth the lower edge (the dashed line).

The lower surface was then smoothed by moving mass sideways from the lower edges of long columns to the lower edges of short columns applying a complex algorithm. Finally, the particle was scaled using the same algorithm as described previously. The algorithm was implemented and tested, but the problems concerning conservation of enthalpy were more serious than for the current algorithm. Also, moving materials sideways influences the temperature profile near the reacting surface. Such changes are unfavourable since it is believed to be important to preserve the temperature profile in areas where chemical reactions proceed. It is therefore doubtful if this algorithm is better than the much simpler algorithm described earlier. The complex algorithm was eventually rejected and replaced by the simpler and less time consuming algorithm.

Adjusting variables when the particle geometry changes. The applied algorithm typically causes the geometry for a particle to stay unchanged for many time steps while its density slowly decreases as mass is removed. Thus, dzj decreases slowly. The lowermost row of cells is removed as described earlier when dzj eventually becomes less than dztran siti

B = Remaining cells □ = Cells changing to gass cells □ = Original gas cells

Figure 3.38: Initiating new gas cells after removing two rows of computational cells from a solid particle. Data from existing gas cells straight below the changed cells is used as initial values for the new gas cells as indicated by the arrows. to the surface of the particle. However, it violates the conservation of mass and energy and so on since gas is added in the new gas cells without simultaneously being removed from nearby locations. The amount of mass added in this way is, however, insignificant compared to the overall production of gas. Algorithms based on distribution of some of the mass from the surrounding gas to the new ’LIVE’ cells before progressing to the next time step would be an alternative to the one applied. As mentioned on page 105, the time step length should be reduced significantly if more than one row of cells is removed during a time step. The situation in Figure 3.38 is therefore not typical, but it illustrates what happens both in the normal case and if more than one row of cells is removed. 3.10. THE METAL POOL. 113 3.10 The metal pool.

The metal pool mainly consists of molten silicon, Si02 and SiC. Some unreacted carbon may also be present. Its detailed structure is, however, not known. Solid and liquid mate­ rials are transported to the metal pool by dripping from the furnace shaft or is produced by chemical reactions. The SiC is believed to form a porous bed at the bottom of the furnace. The liquid silicon metal flows easily down into this bed and covers it partly or completely depending on the amount of silicon metal in it. Si02 floats in the silicon metal or rests on the SiC bed. The situation in the metal pool may be as illustrated in Figure 3.39.

Axis of symmetry Inactive charge Iff % Metal pool

X/WUIIJO Silicon metal Furnace wall

0 = SiOz ® =SiC D =Sip2 D = SiC Furnace Model

Figure 3.39: The metal pool and the lower parts of the shaft viewed from the side. The furnace is on the left and the model on the right. The particles, appearing as vertical bars in the simulation model, are concentric rings with their centers at the axis of symmetry. The particles in the metal pool are positioned straight below those in the furnace shaft.

In the model the situation is strongly simplified. The SiC and Si02-particles are assumed to float with their upper surfaces alligned with the top of the silicon metal as shown in Figure 3.39. These are the only materials present in the metal pool.

3.10.1 The basic structure of the metal pool.

Material flow is set up in a liquid metal pool mainly by the temperature gradients and the self-induced magnetic fields from currents passing through it. Despite this, the metal pool is modelled as a rigid body. 114 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

Gu [40] has calculated flow fields for silicon melts in a small reactor for argon arcs of lengths 130-233mm and currents in the range 500-10Q0A. In the laminar case, these gave maxi ­ mum flow velocities in the metal pool of 0.15-0.18 m/s for metal pools with radius 14cm and depths varying between 8.5cm and 18.5cm. In the turbulent case, the corresponding maximum velocities were 0.21-0.29 m/s. The fluid flow in the metal pool gives a more uniform temperature distribution than cal­ culated for a rigid body with the same heat conductivity. In the model this is partly compensated for by increasing the heat conductivity for liquid silicon compared to values listed in literature. The simulation operator defines this value. It is not known to what extent the metal pool is actually a freely flowing melt. In a furnace being continuously tapped, the SiC bed may very well occupy most of the metal pool. In this case, treating it as a rigid body is probably more realistic than modelling it as a freely moving silicon melt. The surface of a melt is deformed by the pressure forces exerted by the high-velocity gases purged downwards by the electric arc. Such deformations are not included in the model.

3.10.2 Electric currents.

The electric current passing through the electric arc continues down through the metal pool. In the model all currents in the metal pool and any effects thereof are excluded. The most noticeable consequence is that heating caused by such currents are excluded.

3.10.3 Particle representation.

SiC- and SzC^-particIes in the metal pool are represented as solid, compact materials in exactly the same way as particles in the furnace shaft (section 3.9.2). Silicon metal fills up the rest of the metal pool and is also represented as a solid, compact material as mentioned in section 3.10.1.

3.10.4 The specific surface area for particles.

The same correction factors for specific surface area and blocked surface as applied for particles in the furnace shaft and described in section 3.9.3 also apply for particles in the metal pool. 3.10. THE METAL POOL. 115

3.10.5 Chemical conversion.

Particles in the metal pool undergo chemical reactions as described in section 3.4. Con­ sumption of SiC- and SiOa-particIes is calculated as for particles in the furnace shaft. Thus, equation 3.149 on page 103 applies also in the metal pool. Silicon is produced and consumed as described in section 3.4. The total consumption at the surface of each SiC^-particle is calculated by adding the contributions from each computational cell as shown in Figure 3.40. This silicon is removed as an equally thick slice

Figure 3.40: Chemical conversion at the interface between Si02 and Si (left) and the corresponding mass actually removed (right). of materials from the upper edge of the silicon underneath the SiC>2-particle as explained in detail in section 3.10.6. The total production of silicon is calculated by adding the contributions from all compu ­ tational cells in the system. This silicon is distributed to the metal pool as described in section 3.10.6.

3.10.6 Updating the materials in the metal pool.

In the model both the upper and lower surfaces of the metal pool are fixed. This reduces the movements of the materials in the metal pool to a minimum. The motivation is as for the furnace shaft (page 109); that any movement of materials in practice introduces small errors in the temperature distribution and in the enthalpy of the involved materials. 116 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

The upper surface of the metal pool is kept at a constant level by exchanging mass with a socalled "silicon reservoir ”. This reservoir can be regarded as a tapping ladle having the additional property that silicon can flow not only into it from the metal pool, but also back into the furnace if the metal pool surface for some reason would otherwise sink. The details are described on page 121. The upper surface of the metal pool is smooth since all particles are forced to float with their upper surfaces exactly level with the top of the metal pool. The motivation is to have simple flow conditions just above the metal pool. As explained in section 3.5.3, the silicon that evaporates from the metal pool is immediately returned to its original position after transporting energy to other parts of the system. It is thus not necessary to update the silicon mass because of evaporation and condensation as such. The geometry of the particles in the metal pool are updated applying the same basic algo­ rithms as described in section 3.9.6 for the furnace shaft. The situation in the metal pool is, however, slightly different from that in the furnace shaft. Some additional algorithms and adjustments were therefore needed to handle special situations in this area. Just as for the furnace shaft, the geometry of the metal pool is updated when progressing from one time step to the next. The Si02-particles and the S'iC-particles are updated before the silicon.

Updating the Si02- and SW-particles. Si02- and SiC-particIes are updated by an algorithm involving the following sequential steps :

1. Remove materials consumed in chemical reactions. An equally thick slice is removed from the lower edge of any SiOa-particle and from the upper edge of any SiC-particle. Si02 and SiC are treated differently because Si02 reacts with silicon in the metal pool, whereas SiC reacts with SiO(g) from the crater gas at the upper surface of the particle. Some Si02 also reacts at the upper surface of Sz02-particles due to conden ­ sation of Si(g) as explained in section 3.5.3. This Si02 should according to the same logic be removed from the upper edge of the 5z'02-particle, and not from its lower edge. In the model, however, this reaction takes place at the lower edge of the Sz02-particle as described in section 3.13.3. Remov­ ing the consumed Si02 from the lower edge of the particle is consistent with this. 3.10. THE METAL POOL. 117

2. Add materials dripping down from the crater wall.

This material is added as an equally thick slice on top of what is left of the particle after the chemical reactions. Each particle in the furnace shaft has a ’’mirror” particle of the same type and the same cross section straight below it in the metal pool. The dripping is therefore a straight forward vertical transport of materials as shown in Figure 3.41.

3. Move the particle in the vertical direction to allign its upper surface with the top of the metal pool. 4. Scale the particle in the vertical direction by adjusting its density so that its lower edge fits exactly with the interface between two computational cells.

The two last steps are in practice performed in one operation.

Furnace shaft

Dripping material Mass transfer Metal pool surface

Remaining mass after chemical reactions

Metal pool

Figure 3.41: Mass transfer from the furnace shaft to the metal pool. The mass is added on top of what is left of the particle in the metal pool after the mass consumed in chemical reactions is removed. Mass moves vertically, not sideways.

The algorithm applied for scaling particles in the metal pool is very similar to that applied in the furnace shaft. However, some adjustments are required since the dripping material is merged with mass already present in the metal pool. In the model the dripping material generally has a different density than that in the metal pool. Also, the temperature profiles of the two parts are independent. A special algorithm is therefore necessary to merge the materials together. A general situation after step 2 and before step 3 in the algorithm just presented is as shown on the left in Figure 3.42. 118 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

Before scaling and Final geometry vertical translation

Figure 3.42: Updating the particle geometry in the metal pool.

The dripping material and the old material are scaled individually to the same density by applying exactly the same algorithm as used for scaling particles in the furnace shaft (section 3.9.6). The scaling factors are different for the two parts, and are given by equa­ tion 3.156 on page 109 (<*i = pi/ps and a2 = Pi/ps)- The situation for the cell in which the dripping material meets with the rest of the particle is shown in Figure 3.43. There is a discontinuity in the temperature profile for the position where the two parts meet. The scaled temperature profiles in Figure 3.43 are both calculated as explained in Figure 3.36 on page 110. The new node temperature (Tno de) for the cell containing the discontinuity is calculated from the following equations : Tpdzp + Tmdzm (3.157) dzp + dzm Tm 0.5(Tdm + Tminus) TP 0.5(T

The variables are as shown in Figure 3.43. Linear interpolation and extrapolation is used to find the temperatures at the contact point ^discontinuity) when approached from below (Tdm) and from above (Tdp ). A better method that may be implemented in future versions of the model is to integrate the enthalpy content in each linear segment of the scaled temperature profiles (A-B, B- C, D-E and E-F in the figure) and then derive Tnode from the average enthalpy of these materials. 3.10. THE METAL POOL. 119

Cell boundaries

Metal pool surface

'discontinuity

After updating node points

Scaled temperature profile for dripping material \

; kJUUGU on** G profile for remaining ■l material in the ;i metal pool

minus 'discontinuity

Figure 3.43: Variables applied when calculating the temperature at the node point for the cell in which the dripping material merge with the remaining material in the metal pool. Tnoiz is the new node temperature. 120 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

Updating the silicon in the metal pool. Silicon metal consumed in the metal pool is removed from underneath the Si02-particle it reacts with as shown on the right in Figure 3.40 on page 115. The materials in the metal pool are updated in a three step procedure. The SiOi- and SiC-particles are updated first as just explained. The new geometries for these particles and the predefined upper and lower edges of the metal pool then define the volume which is left for the silicon. The silicon left in the metal pool after chemical conversion is then moved vertically upwards or downwards to regain physical contact with the newly updated particles floating in the pool. Only silicon underneath the particles needs to be moved since the silicon between the particles is not influenced by the chemical reactions, dripping or evaporation of silicon. Finally, silicon produced by chemical reactions are added to the metal pool. Some of this silicon may be transferred to the tapping ladle. Unlike the other materials, the density of the silicon is not modified. Depending on how the particles change their geometries and how much silicon that is consumed in chemical reactions, two different situations can occur at the bottom of the metal pool after moving the Si(l) into contact with the particle :

• empty ’’holes ” may be created if there is too little silicon left underneath the particle. • silicon may extend below the lower edge of the metal pool.

Both these cases are shown in Figure 3.44.

Particles (SIC or SiOj) Metal pool

Si production (crater \X Metal pool top wall or metal pool) / <

Metal pool bottom Refilling "holes' Outflow Inflow -l .Si

Silicon reservoir Silicon redistibution pool (tapping ladle)

Figure 3.44: The silicon flow between different parts of the furnace in connection with updating the geometry of the metal pool. 3.10. THE METAL POOL. 121

The silicon extending below the bottom of the metal pool is redistributed to the areas with empty ’’holes ” or to the silicon reservoir (the tapping ladle). This is done by introducing an imaginary ’’ladle” (the ’’silicon redistribution pool ”) where all the silicon extending below the metal pool is gathered before being redistributed. All silicon produced by chemical reactions in the current time step is also added to the this pool. Its total mass is mixed to a uniform temperature determined by the total enthalpy of the added silicon. The empty ’’holes ” in the metal pool are then filled with mass from the silicon redistribution pool. The surplus is transferred to the silicon reservoir (denoted ’’outflow ” in Figure 3.44). The total mass in the silicon redistribution pool may occasionally be less than required to fill up the ’’holes ”. Additional silicon is then added to it from the silicon reservoir before the "holes ” are filled (denoted ’’inflow ” in the figure). This inflow from the tapping ladle to the system does not necessarily mean that there has been a net consumption of silicon in the furnace during the time step. It may also be caused by contracting Si02- or SiC-particles (page 122). A detailed picture of the interaction between a particle in the metal pool, the silicon underneath it and the silicon redistribution pool is shown in Figure 3.45.

Metal pool top Metal pool top

Scaling Consumed in chemical reactions

SiO,/Si interface -- -- Si02/Si interface

Consumed in chemical reactions "Hole" filled by Si from /the redistribution pool. No scaling Metal pool bottom ------Metal pool bottom Before updating After updating

Silicon Redistribution Pool

Figure 3.45: Updating the geometry of a particle in the metal pool and the silicon below it. The empty ’’hole ” underneath the particle and remaining silicon after updating is filled by silicon from the silicon redistribution pool.

It may seem more logical to leave the remaining silicon underneath the reacting particle 122 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL. completely unchanged at the bottom of the pool and instead remove or refill silicon close to the SiOz/Si-interface. This implies that the silicon already present in the metal pool does not move at all (which is favourable as explained earlier). It is, however, considered more important to preserve the temperature gradients close to the reacting surface as good as possible because the temperatures here are important for the chemical reaction rates. The temperature of the silicon from the redistribution pool may be quite different from that of the SiOz/Si-interface. The best way to minimise changes in the temperature at this interface is therefore to move the remaining silicon upwards and then fill the "hole ” at the bottom with materials from the redistribution pool. The temperatures along the sides of the particles are not influenced much by updating the geometry since the mass is only removed from the upper or lower edges of the particles and from the silicon underneath 5202-particles 16. This is consistent with the way the chemical reactions are handled in the furnace shaft.

Comments to the silicon redistribution pool and the silicon reservoir. Both the silicon redistribution pool and the silicon reservoir are assumed to be perfectly mixed melts of pure silicon. This implies that there are no temperature gradients in either of them. The enthalpy of the materials added to them is conserved. The silicon redistribution pool is always empty after the metal pool is updated and the calculations for a new time step starts17. The temperature of this pool is thus fully determined by the materials added to it during the actual updating on the given time step. A simulation is started with a predefined amount of silicon of a predefined temperature in the silicon reservoir (normally close to the assumed tapping temperature of the silicon). The temperature then changes as new material is added to it. The silicon reservoir may in principle become empty since mass can flow out of it and back into the metal pool under special conditions described previously. New mass is in this case added to the reservoir at its latest recorded temperature, as if there exists an additional reservoir with infinite mass and a temperature always identical to the continuously changing temperature of the silicon reservior. The initial mass in the silicon reservoir is fairly small (10kg) so that the temperature in it after a short while is grossly determined by the materials added to it from the furnace itself rather than its initial temperature. This provides better conservation of energy for situations when silicon occasionally flows into the furnace. Silicon occasionally flows into the furnace from the reservoir. This is a natural consequence of the algorithm applied for updating the geometry of particles. The mass of a particle decreases when the chemical consumption from it is larger than the mass that drips down

16Some changes are due to the scaling procedure itself as explained in section 3.9.6. 17Any mass left after filling the "holes” is transferred to the silicon reservoir (the tapping ladle) as described on page 121. 3.11. THE CRATER CAVITY. 123 to it from the furnace shaft. Its geometry stays unchanged until its mass is so low that a full row of cells is removed from its lower edge. The ’’hole ” that is created from this may be greater than the net production of silicon in the furnace during the given time step. Net inflow from the silicon reservoir is then needed to compensate for the increased Si-volume. Such inflow of silicon from the reservoir should, however, take place only in short periods. The amount of silicon in the reservoir should generally increase over longer periods for a furnace that actually produces silicon. The amount of silicon in the redistribution pool may also increase even in periods with net consumption of silicon. This can for example occur due to enhanced dripping of materials from the crater wall. The particles in the metal pool may then grow and silicon underneath them are consequently ’’pushed ” out of the furnace since the total volume of the metal pool is unchanged by assumption. The total mass in the reservoir therefore gives an inaccurate estimate for the actual silicon production. Corrections are made to obtain the true production rate. Particles in the metal pool may in principle grow all the way down to the lower edge of the metal pool. This situation is automatically taken care of by the algorithm for adjusting the density of the material. It may, however, in extreme cases result in densities that are far higher than the true density of the material. The density may also become far less than the true density of the material when particles are almost fully consumed (extremely small). In this case, they occupy only a small fraction of the uppermost row of cells in the particle. No warnings are issued to inform the simulation operator about such situations.

3.11 The crater cavity.

The crater cavity is filled with gas, and its state is mainly calculated from the equations implemented in the standard FLUENT code (section 3.2.5, page 29) and from the equations for the electric arc (section 3.8, page 77). Included are also source terms in the mass- and enthalpy conservation equations to account for gas produced/consumed by chemical reactions as described in section 3.13. Adding these source terms in the enthalpy equation is straight forward. They simply represent the amount of energy [W] added to the given computational cell. The source terms for the mass balances are, however, not so easy to handle since the equations for conservation of mass are solved for mass fractions. Thus, the reaction rates [kg/s] must be linked to these so that the total mass fraction remains 1.0 also after the gas is removed or added to the computational cell. This is achieved by first introducing source terms (kg/s) in the conservation equations for each mass fraction, and then removing the same amount from the resulting gas mix. Con­ sumption of mass is handled in the same formal way, but an equivalent amount of the final 124 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

composition is then added to the cell to compensate for the removed mass. The principles are illustrated in Figure 3.46. Xsio , XCo and XineHs are the mass fractions of SiO(g),

a kg SiO added, b kg CO removed, (a-b) kg of the final gas mix ^ inerts,0 removed inerts,1 mkg mkg

Figure 3.46: Updating the mass fractions when SiO(g) is added and CO(g) is removed from a computational cell.

CO(g) and other gas species (inerts 18 ) present in the computational cell, respectively. The composition of the final gas mixture is unknown until the iteration process has converged, but an estimate (Xsio , Xco and Xinerts) is available as the iterative process progresses. The general form of the source term for the mass fraction is :

Sj = SUj-SPjXj ; j G {SiO, CO, inerts} (3.158)

Sj is positive when mass is removed from the cell. SPj must always be negative to as­ sure that the iterative procedure for solving the equations is numerically stable and thus converges. SUj can take any value (positive, negative or zero). Assume that :

mi = The amount [fcg/s] of SiO(g) produced in the cell by the reaction between Si and Sz02. —m2 = The amount [kg/s] of SiO(g) consumed in the cell by the reaction between SiO(g) and SiC. m3 = The amount [kg/s] of CO(g) produced in the cell by the reaction between SiO(g) and SiC.

These terms are all positive or zero.

18 In the model no inert gas is present (Xin„ts = 0). Inert gas is even so included to be consistent with the FLUENT description. 3.11. THE CRATER CAVITY. 125

The expressions for the source terms in the equation for conservation of mass fractions then become :

SUsiO — ~m 1 + 7772 XsiO SPsio = —mi + (7B2/Xsio) — m3

SUco = m2 Xco — m3 SPco = -mi — m3

SUinerts — ni2-^Qnerts

SFinerts = ~mi ~ 7773 where Xj is the best current estimate for the final mass fraction of gas species j in the cell. Applying equation 3.158 then gives the following overall source terms :

—mi — 7712 + (77I1 + 777-2 + mJjXsiO

—7773 + (?77i + 7772 *b 7773)

(777i + 7772 + m^Xinert

The source term for inerts is not zero even though no inerts are added or removed from the cell. This may seem strange at first sight, but it merely expresses that the mass fraction of the inerts changes when other gases are added or removed. For example, adding CO(g) to the cell causes lower concentrations of the other gases, and the mass fractions of these decrease accordingly. All source terms have a similar term, (mi + 7772 + mz)Xj, for the same reason. The total mass fraction does not change :

Stot — HsiO "b HCO "b Sinerts

— —(ml + 7772 + 7773)(1.0 — XsiO ~ XcO ~ Ajnert) •V- (1.0 — XsiO — Xco — Xinert = 0.0)

Stot = 0.0

In terms of added and removed mass, the overall result is that :

• 777i kg of SiO(g) and m3 kg of CO(g) is added to the cell, and the same amount (7771 + m3 kg) of its estimated final gas mixture is removed from it.

• tt72 kg of SiO(g) is removed from the cell, and the same amount (7772 kg) of its estimated final gas mixture is added to the cell. 126 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL. 3.12 Heat transfer caused by evaporation and con­ densation.

The rate of evaporation (equation 3.92, page 73) is calculated for every computational cell representing silicon metal at the upper surface of the metal pool. The temperature of the cell is used when evaluating the equilibrium pressure of Si(g) at the metal pool surface. The energy transport caused by the evaporation from a computational cell and the following condensation is shown in Figure 3.47.

Crater wall

Inert (S' furnace wall

Energy flux determined by local temperature and distribution key given by the operator

Remove enthalpy of evaporation from here

Figure 3.47: Energy transport in the system caused by evaporation and condensation of silicon from the metal pool.

The details concerning the evaporation is illustrated for a silicon cell next to a SiC-particle in Figure 3.48. on is here the fraction of the total enthalpy of evaporation distributed to the given cell. Its value depends on the local temperature of the cell and the distribution of the enthalpy to the different parts of the furnace as defined by the operator (page 74).

As described on page 70, no Si(g) is in the model transferred to the gas phase. The mass balance for the silicon metal in the gas phase or the metal pool is thus not influenced by evaporation. The enthalpy of reaction is, however, removed from the silicon cell, and it is redistributed to the other parts of the system as shown in Figure 3.47 and described in detail on page 74-77. The practical calculations are done by adding source terms (aiAHT) in the equations for energy conservation. 3.13. ENERGY AND MASS EXCHANGE FOR CHEMICAL REACTIONS. 127

FLUENT cell boundaries

Figure 3.48: Evaporation of silicon from one computational cell at the metal pool surface.

3.13 Energy and mass exchange for chemical reac­ tions.

The energy and mass exchange for chemical reactions are described in the following. The produced gases are distributed in different ways to the gas phase due to the different nature of the reactions. The enthalpy of reaction is also supplied in different ways.

3.13.1 Reactions at gas/SiC-interfaces.

The reaction at the interface between a computational SiC-ce11 in the metal pool and a gas cell is shown in Figure 3.49. The following description applies also to the furnace shaft where the gas cell is positioned below the SiC-cell or at one of its vertical surfaces. Tgas and Tsic are the temperatures of the gas cell and SiC-cell, respectively. AHr = msio&Hi is the total enthalpy that is consumed in the reaction in the time step (msio is the amount of SiO(g) consumed). The reaction is assumed to take place with all reactants and products having the same temperature as the SiC-cell except for the SiO(g) which enters with the same temperature as the rest of the gas. The energy needed to support the reaction is taken from the SiC-cell exclusively, and no energy is therefore removed from the gas cell to support the reaction. However, the reacting SiO(g) and its corresponding enthalpy are removed from the gas cell, and the mass and 128 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

SiO(g) + SiC(s) = Si® + CO(g)

Metal pool top

Reacting surface

Silicon redistribution pool SiC cell :

Figure 3.49: Mass and heat transfer, gas distribution and temperatures for the reactants and products from the reaction between SiO(g) and SiC(s). The enthalpy of reaction is taken from the SiC-cell. enthalpy of the produced CO{g) are added to the same cell. The temperature for the gas cell is in practice higher than that for the SiC-cell. The produced CO(g) is consequently colder than the consumed SiO(g) and the reaction results in a net cooling of the gas above the SzC'-particle. The practical calculations are carried out by adding source terms in the equations for mass- and heat conservation. The surrounding volumes (computational cells) are not influenced directly by the chemical reaction, but they are influenced indirectly by the changes in the mass- and heat fluxes over the boundaries of the cells. The reaction is assumed to proceed only to the right (producing CO(g)).

3.13.2 Reactions at Si(l)/SzC>2(Z)-interfaces.

The reaction between SiO2(0 and Si(l) at Si(l)/Si02(Z)-ihterfaces in the metal pool is different from the one between SiO(g) and SiC(s) because it takes place at the interface between two immiscible liquids rather than at the interface between a gas and a condensed material. There is no consumption of gas in this case, only production of SiO(g) as shown in Figure 3.50. Distribution of the product gas is straight forward for the reaction at the gas/SiC-interface where the logical choice is to add all the produced gas to the gas cell at the interface. The situation is different for a reaction that takes place between liquids down in the metal pool. The product gas must then move upwards along the surface of the particle before it enters 3.13. ENERGY AND MASS EXCHANGE FOR CHEMICAL REACTIONS. 129

Sid) + Si02(l) = 2SiO(g) ■/Distribution of / reaction gas

Gas phase Metal pool Reacting surface

Pe:Slatm

FLUENT cell boundaries Removed qSi Removed qg.^

Figure 3.50: Mass and heat transfer, gas distribution and temperatures for the reactants and products from the reaction between Si02{l) and Si(l) in the metal pool. the gas phase at the metal pool surface. Any energy transfer between the liquids and the gas associated with this gas movement is excluded in the model. It is not obvious how to distribute the produced gas above the metal pool. The applied strategy is to distribute it with an equal density [kg/m3] to all computational cells in the first row of gas cells above the particle as illustrated in Figure 3.50. This mass and its associated enthalpy is added to these cells as source terms in the equations for mass- and energy conservation. The produced gas enters the gas phase with a temperature identical to that of the interface where the reaction takes place (Treac in Figure 3.50). This temperature is assumed to be equal to the temperature which gives a reaction pressure of 1 atm for SiO(g) (page 53), and it is normally much lower than the temperature of the gas above the Sz'C>2-particIe. The furnace gas is consequently cooled. The heat flux [W/m2] to the reacting surface for the situation in Figure 3.50 is calculated by :

Tsio 2 — Tn TSj — Trt <7tot = qsi02 + QSi = ksio 2 • + k.■sr (3.159) ^reac TSi T'tc 130 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

where the thermal conductivities are calculated by :

n ksi(Treac)ksi(Tsi) ksi (3.160) ksi(TTeac) +

2 ksio2{Treac)ksio2{Tsio2) ksi02 (3.161) ksi02 (T-reac) + ksi02 {Tsi02)

The heat flux from the SiC^-side of the reacting surface (qsio 2) is removed from the Si02- celi and that from the Si-side (

3.13.3 Reactions at gas/SiCVinterfaces.

Silicon condensing from the gas phase after first evaporating from the warmer parts of the metal pool can react chemically at gas/Si02-interfaces in the metal pool and in the furnace shaft as described in section 3.5.3. In the model the evaporation and condensation of silicon are handled without including physical transport of silicon (page 70). The amount of silicon that condenses at a given gas/Sz"02-interface in the metal pool is estimated (equation 3.100, page 75), and the amount that actually reacts is calculated (equation 3.57, page 55). Similar expressions are used for the furnace shaft. The heat transfer associated with this reaction is handled in the same way in the metal pool and in the furnace shaft. The energy of condensation is in both cases added to the SiOa-cell where the condensation takes place (at the gas/5z02-interface) and the energy required to support the chemical reaction is removed from the same cell. A net cooling of this Si02-cell takes place if the amount that reacts consumes more energy than what the condensation provides. This occurs when the temperature of the SiC>2-cell is above a certain limit (page 53). A net heating of the same cell takes place if the cell is below the same temperature limit. The mass transfer is handled somewhat differently in the metal pool and in the furnace shaft. The situation around an Si02-particle in the crater wall is shown in Figure 3.51. In the model no silicon is physically present in the furnace shaft, and the reacting silicon is instead taken from the silicon reservoir (the ’’tapping ladle”). This silicon has a tem­ perature equal to that in the silicon reservoir. Si02 is taken from the Si02-cell where the condensation takes place and it reacts with a temperature equal to the temperature of this cell (Tsio 2 in Figure 3.51). The produced SiO(g) and its enthalpy are added to the gas cell at the gas/Si02-interface where the reaction takes place. The temperature of the 3.13. ENERGY AND MASS EXCHANGE FOR CHEMICAL REACTIONS. 131

The enthalpy of condensation is added to this cell Condensation and The enthalpy of reaction y reaction take place is removed from this cell f -. at these surfaces only SiOgis removed from this cell

IT X

n i lSi, reservoir . The produced SiO(g) at T = T^io / : : - is added to this cell

Si from the silicon reservoir

Figure 3.51: The reaction between condensing silicon and Si02 in the furnace shaft. produced gas is equal to the temperature of the Sz02-cell where the gas is produced (Tsio 2 in Figure 3.51). In the metal pool silicon is available underneath and possibly along the sides of the Si02-particIe where the condensation takes place. The reactants are here taken from the Si(l)/Si02 (Z)-interface straight below the place where the condensation actually takes place as illustrated in Figure 3.52. The SiO(g) produced at the Si(Z)/Si02(Z)-interface is distributed with an equal density [kg/m3] to the gas cells immediately above the Si02- particle in exactly the same way as for other SiO(g) produced at the same location (Fig­ ure 3.50, page 129). The SiO(g) is assumed to be produced with the same temperature as the Si(1)/Si02 (Z)-interface and it is added to the furnace gas with this temperature. Again, this is exactly as for other SiO(g) produced at the same location (section 3.13.2).

The energy consumed in the reaction is taken from the Si02 cell at the gas/Si02-interface while the reactants are taken from the Si(Z)/Si02(Z)-interface straight below it in the metal pool. 132 CHAPTER 3. DESCRIPTION AND DISCUSSION OF THE MODEL.

"Condensing" Si(g) and The reaction should enthalpy of condensation take place here y Furnace gas

X, FLUENT cell LlJ.lt boundaries The reaction takes place The enthalpy of reaction is removed from this cell

ijcly4; : fX^TMreactmgmaterialsare Sl® ■ = : > reinoyiedJrom these cells ....

Figure 3.52: The mass- and heat transfer caused by chemical reactions involving con­ densing silicon vapour at gas/SiOa-surfaces in the metal pool. Chapter 4

The simulated furnace and default parameters.

The model is in its present state a prototype. It has been applied to simulate a system which is close to the situation believed to exist around one of the electrodes in a full- scale submerged arc furnace producing silicon metal (section 4.1). The purpose of these simulations has been to prove that the model is working and to demonstrate its capabilities and weaknesses. It is not considered useful to seek an "optimal" combination for the model parameters at the present stage. This is a difficult and time consuming task that is better addressed after the model has been in operation for considerable time and the needed improvements has been implemented to it. Improving submodels and optimising model parameters re­ quire considerable effort and should involve both process metallurgists and scientists with detailed knowledge about the process and about numerical methods and modelling. Com­ paring simulations with observations, experiments and experience will be an important, but difficult part of this. As explained on page 139, the calculations are extremely time-consuming even when using fast computers. This has been a limiting factor in the present work. Lengthy simulations have been run and the code has been changed when undesired effects or errors have been discovered. New undesired effects have then been discovered and further modifications have been implemented before running the simulations yet again. The model still suffers from simplified submodels leading to unreasonable or non-physical effects. These are pointed out as the results are presented and discussed in chapter 5.

133 134 CHAPTER 4. THE SIMULATED FURNACE AND DEFAULT PARAMETERS. 4.1 The furnace and its representation in the model.

The simulated system is close to the situation which is believed to prevail around one of the electrodes in a furnace producing silicon metal at the Elkem Fiskaa plant in Kristiansand, Norway. The applied dimensions are as shown in Figure 4.1. The total radius (1.2

Furnace wall/ Electrode inactive charge

0.63m Active charge

1.1345m Crater cavity

Metal pool

0.295m

Figure 4.1: Geometrical dimensions of the simulated system (right half only). meters) is close to the estimated inner radius of the crater cavity around each electrode. The radius of the electrode is 0.63 meters and the total height of the furnace is 3.0 meters. The electric arc bums over a distance of 10cm. The typical load is about 7MW per phase and the current is about 77kA. About 90% of the energy (6.3MW) is assumed to be generated in the electric arc. This situation is represented in the model as shown in Figure 4.2. Closeups of two selected areas in the outer part of the furnace show the detailed structure of the furnace walls and their surroundings. One obvious difference from Figure 4.1 is the inert materials found above the charge in the shaft. These materials occupy a volume that used to serve as a manifold for gas from several vertical gas channels through the charge as shown in Figure 3.25 on page 94. This gas left the system through a common narrow exit at the top of the manifold. For reasons explained in section 3.9.2, all these gas channels except the one next to the electrode were later filled with inert materials (WE-cells, see Table 4.1). The manifold was simultaneously 4.1. THE FURNACE AND ITS REPRESENTATION IN THE MODEL. 135

Furnace wall ^ert materials

Electrode ^ZNt 0.67m X e :d 0.63m

\ 3.0m 0.1m Charge ■ = Wall type B (WB) „ , . . □ = Wall type C(WQ 1.1345m Metal pool Q = Wall type 3 (W3)

10.4m 0.295m z at 1.2m

Figure 4.2: Geometrical dimensions of the simulated furnace (right half only). Closeups of selected areas on the right. filled with inert materials1 (WC-cells). Gas leaves the high-temperature zone through the remaining channel. The data for different types of cells applied in the furnace walls, in the electrode and as inert materials in the furnace shaft are listed in Table 4.1. Wall cell types WB, WH and W3 all have predefined constant temperatures, whereas wall cell types WC and WE are conducting materials for which the temperature is calculated.

Table 4.1: Wall cell data.

Cell type Heat conduction T [K] k [1W/mK\ Cp [J/kgK] P [kg/m6] WB No 2100 - - - WC Yes - 2500.00 2083.00 2300.00 WE Yes - 1.60 749.38 2300.00 WH No 4000 - - - W3 No 2000 - - -

1It might just as well have been filled with raw materials (charge) instead of inert materials. 136 CHAPTER 4. THE SIMULATED FURNACE AND DEFAULT PARAMETERS.

Inert materials of type WC are applied next to the outer edge of the crater cavity. The thermal conductivity for this material is high. Energy is therefore transported effectively away from the cavity/wall-interface and into the rest of the wall which is at 2100K (type WB). This prevents extremely high temperatures in this area which is heated by the gas and by radiation without being cooled by chemical reactions. A closeup of the area around the electric arc is shown in Figure 4.3.

Furnace wall

Electrode

= Wall type H(WH) = Wall type E (WE) 0.63m

1.1345m

a 0.295m 0.4m Metal pool

Figure 4.3: Geometrical dimensions of the simulated furnace (right half only). Closeup of the area around the electric arc on the left.

The electrode consists of WE-cells except for a single layer of WH-cells along the innermost parts of the electrode tip as shown in Figure 4.3. This layer prevents extremly high temperatures from developing where the electric arc attaches to the electrode. The heat flux from the gas in the arc is large in this area, and high temperatures are calculated if the WH-cells are replaced by conducting materials. The temperature of the WH-cells is predefined to 4000K in the simulations presented in this work. This is close to the sublimation temperature for graphite as mentioned on page 94. 4.2. THE DETAILED MODEL GEOMETRY. 137 4.2 The detailed model geometry.

The same basic geometry (electrode size and position, furnace wall sizes and positions, metal pool surface etc) is applied for all simulations. The geometrical outline is shown in Figure 4.4. This figure should be compared with Figure 4.2. The vertical line at the

J31

IJJJUJUi 3 FLUENT, FLUENT, 3EEZ t • ua&a

Figure 4.4: Geometrical outline of the simulated system and the electric arc. Left : The I complete system. Right : The outline of the arc and the areas around the crater cavity.

center of the left figure is the axis of symmetry and has no physical significance. The electrode is the large structure in the central and upper part of the system. The SiC- and SiCVparticles are easily identified as vertical rods outside the electrode, but also straight below at the top of the metal pool. The metal pool extends all the way down to the lowermost horizontal line in the figure on the left. Inert materials are found below the metal pool and outside the particles in the furnace shaft. These are not visible in the figures. Inert materials are also found between and above the SiC- and Si02-particles in the furnace shaft. The inert wall at the outer edge of the crater cavity is rounded off to give more reasonable geometry in this area than indicated in Figure 4.2. The single vertical outlet gas channel is found next to the eletrode. 138 CHAPTER 4. THE SIMULATED FURNACE AND DEFAULT PARAMETERS.

Figure 4.5 shows where the different types of particles are located in the lower parts of the furnace shaft and the metal pool. Notice that the inert particles present in the shaft leave room for Si(l) in-between the particles in the metal pool.

Figure 4.5: The location of different types of materials in the simulation model. SiC = black, SiOz = hatched and inerts = dotted. The metal pool is at the lower edge, the inert furnace wall at the right edge and the electrode at the upper left corner.

The same initial positions and geometry for the particles are used in all simulations unless otherwise stated. The geometry of the particles generally changes with time due to chemical reactions and dripping. The total mass ratio of SiC to SiO^ in the furnace shaft is 0.466 for the situation shown in Figure 4.4 (and 4.5). This corresponds to a mole ratio of 0.698. The normalised mass ratio in the crater wall (w,^, page 41) is 0.482, corresponding to a mole ratio of 0.723. The composition of the materials entering the high-temperature part is therefore within the estimated limits for normal process operation (0.70-0.75) derived in section 3.3.7. 4.3. THE COMPUTATIONAL GRID. 139 4.3 The computational grid.

Selecting a numerical grid for the simulations is not straight forward. There are two basic motivations for using a large number of computational cells :

1. The various geometrical structures are better resolved. This is particulary important in areas where the gradients are large or important phenomena like chemical reactions take place. 2. The accuracy of the calculated results depends on the grid. For a first order discretisation technique like that used in the applied version of the FLUENT code, a fairly fine grid is required to get high accuracy. It is common practice to run a second set of simulations with the double number of computational cells in each coordinate direction to check if the original grid is sufficiently fine. The original grid is usually assumed to be OK if the two runs give almost similar results. Grid refinement is otherwise probably required.

The required computer resources (storage and computational time) increase dramatically with the number of computational cells. The required time to run simulations and the involved economic cost often determine the upper practical limit for the grid size. Another important aspect, that is sometimes not given enough weight when selecting the grid size, is the accuracy of the physical description of the modelled system. It is usually a waste of time and money to use an extremely fine grid for a system where important process details are either excluded or modelled in a rough way. The validity of the results is then limited by the physical description rather than the numerical solution and the results do not improve beyond a certain optimal grid size. It is also important to consider carefully what degree of accuracy that is actually needed for the results. Extreme accuracy is required for some types of problems while rough approximations are sufficient for others. This is obvious, but experience shows that extreme measures are sometimes taken to achieve marginal improvements to results that are already well within the needed accuracy. A trade-off between different arguments like those listed above is needed to select a useful grid size for a given problem. The simulations are time-consuming for the system at hand. The grid has for this reason been limited to 202 computational cells in the z-direction and 102 cells in the radial direc­ tion. This gives an effective number of 200 and 100 cells in the two directions since one cell at each edge is artificial and is consequently outside the physical domain. The simulations then typically progress 250 time steps of 2 seconds each in 4-6 hours for the simulations presented in this thesis when the simulation program is allowed to utilise all resources on 140 CHAPTER 4. THE SIMULATED FURNACE AND DEFAULT PARAMETERS. the HP-Apollo 9000/755 computer and the simulation is started from a well-converged ini ­ tial state. This corresponds to a situation where a minimum of 10 iterations are required per time step. A considerably finer grid is impractical for the simulation runs presented here. A 200 by 100 effective grid is expected to give sufficient numerical accuracy and resolution compared to the inaccuracies introduced by the matematical description of the various submodels and the prototype nature of the model. Refining the grid at the expense of much longer simulation times probably adds little to the validity of the conclusions drawn at the present stage. Refining the grid should be considered when the model is sufficiently improved in the future. The numerical grid used for all simulations is shown in Figures 4.6 and 4.7.

FINITE OlFFEigCE CgiO FINUE OIFFESENCE CtlD

Figure 4.6: Numerical grid in the furnace shaft. Left : The lower part of the charge. Right : The upper part of the furnace. The same scale is used for both figures, but the scale is different from the one used in Figure 4.7.

The grid is non-uniform and has smallest cells inside and close to the area where the electric arc is located. This is because the gradients are largest in this area, especially for the temperature and velocities. A fine grid is needed here to resolve the gradients. The grid should be fairly fine also at the gas/particle- and gas/metal pool interfaces where the chemical reactions take place. A much coarser grid can be applied in areas where the gradients are smaller and in less important areas. The lower part of the metal pool is one such area which is not particulary important to the overall behaviour of the system. The uppermost parts of the furnace is even less important since this area is far away from the high-temperature zone and because important phenomena like condensation of SzO-rich 4.3. THE COMPUTATIONAL GRID. 141

Figure 4.7: Numerical grid in the lower part of the furnace. Top left: around the electric arc. Top right : middle part of the gas/metal pool interface. Bottom left : Outer part of the gas/metal pool interface. Bottow right : Crater wall close to the electrode. All figures are in the same scale, but the scale is different from the one used in Figure 4.6. 142 CHAPTER 4. THE SIMULATED FURNACE AND DEFAULT PARAMETERS. gas are not modelled.

4.4 Convergence criteria and speed of convergence.

The accuracy of the solution is measured by the socalled residual for each variable. A small value for the residual usually indicates an accurate solution. The applied convergence criteria are listed in Table 4.2. The simulation operator can change these values at any time.

Table 4.2: Convergence criteria (residual limits) for different variables.

velocities turbulent quantities mass fractions other variables uz ur e k SiO(g) co(g) enthalpy 7.5e-4 7.5e-4 7.5e-4 7.5e-4 2.5e-4 2.5e-4 1.8e-5

The pressure residual is not listed because it is a relative quantity and is thus not very informative. The reference value for this residual is calculated from the variables at the start of a given simulation or at a selected time2. The pressure field is usually poor at the start of a simulation, and the value for the pressure residual therefore drops at a considerable rate. Its value then jumps back to almost unity if a new reference value is evaluated. Moreover, it stays close to unity for a long time if the solution was quite accurate when the latest reference value was evaluated. The pressure residual then gives the false impression of a poor (non-converged) solution, and it is practically useless as an indicator for the accuracy of the pressure field. The solution for a time step is considered converged when the reported value for each individual residual is smaller than its value listed in Table 4.2. All residuals must satisfy their individual convergence criteria simultaneously. In the standard FLUENT code the sum of all residuals is instead required to be below a certain limit. Implementing the new strategy was necessary because the pressure residual usually decreased slowly since the simulations were started from a fairly accurate solution derived from a common initial simulation run. The enthalpy residual should normally be in the order of a magnitude or two lower than the other residuals. This is due to the way the residuals are evaluated in the FLUENT code. The applied values are considered to be sufficiently strict for the level of accuracy needed in the present work.

2New reference values are for example evaluated automatically whenever entering the ’SETUP-1’ menu of FLUENT (when changing the geometry of the system, the boundary conditions or various other impor­ tant data). 4.4. CONVERGENCE CRITERIA AND SPEED OF CONVERGENCE. 143

The motivation for starting the simulations from a common and fairly accurate solution is that generating a well-converged solution from an arbitrary situation is extremely time- consuming. The flow field and other fields adjust slowly with the numerical method im­ plemented in the applied version of FLUENT. Information from one cell is only passed on to its closest neighbour in each coordinate direction for every time the equations are solved. The neighbours to these cells then ’’notice ” the effect the next time the equations are solved and so on. Many evaluations are therefore needed before the effect of a change in one computational cell causes a change in computational cells far away from it. Many more evaluations are required before an accurate solution is obtained. . The propagation of information through the computational domain is much quicker with more modem numerical techniques than those implemented in version 2.97 of FLUENT. Multigrid methods ([61]), where the basic idea is to solve the equations alternately on grids of different coarseness, is one class of such powerful methods. The information prop ­ agates quickly through the computational domain when coarse grids are applied, and the approximate solution from this grid is extrapolated to a finer grid and is then refined. This solution can in turn be transformed to even finer grids or be solved on coarser grids again if desired. The iterations on coarse grids are quick since few computational nodes are involved. The basic challenges are to develop effective algorithms for transforming solutions from one grid to another and to select the correct grids and number of iterations on each grid before moving on to the next. The strategies for solving these challenges are not obvious, and a strategy that works well on one specific problem may prove inefficient on others. Multigrid methods are implemented in some recent versions of FLUENT. For the simulations presented in this thesis, the first few time steps require a large number of iterations 3 to reach the residual limits listed in Table 4.2 and are extremely slow unless a good initial solution is available. Each iteration may involve several successive evaluations of the system of linear equations for each variable (h, uz, %. etc). It was impractical to start the simulations presented in this thesis from scratch with the large number of computational cells applied. Also, the objective for the simulations is achived equally well by starting from an already partly converged solution. A considerable amount of computer time is saved by this. Different simulations are usually started from the common initial situation after changing the geometry of the particles, changing some of the model parameters or after altering some of the fields by patching in new values for the temperature, gas composition and so on. The slow convergence when starting from poor initial solutions is also the main reason for using identical grid and furnace dimensions for the simulations. The simulations must be started from scratch if changing any of these. Setting up the geometry and constructing a suitable non-uniform grid for a given geometry is also an annoyingly time-consuming task.

3Several thousand iterations are common for the first time step. 144 CHAPTER 4. THE SIMULATED FURNACE AND DEFAULT PARAMETERS.'

Awkward grid generation is in fact a problem with most CFD codes and is usually avoided if there are no substantial arguments for doing them. Once a well-converged solution is achieved, only a moderate number of iterations (5-15) is usually required for each new time step to satisfy the convergence criteria in Table 4.2. This is because the situation in the furnace changes slowly with time. The old solution therefore provides an excellent initial guess for the next time step. One problem occurs when the geometry for at least one of the SiC- or SiOa-particles changes. Such changes are accompanied by changes both in the temperature of the particle and in the flow field close to it. These are important for the chemical reaction rates, and significant changes are found in such cases (section 5.1.3.3). However, even strong deviations from convergence in limited areas give only small contributions to the overall residuals. Thus, the overall solution may look accurate even if there are significant local deviations from convergence. Two parameters have for this reason been implemented to control the minimum number of iterations performed on each time step. One of these parameters defines the minimum number of iterations when no particles were updated since the previous time step. The other defines the same number when at least one particle changed geometry since the previous time step. These parameters take the values 10 and 20 iterations, respectively unless otherwise stated. Larger values should be considered if more accurate results are wanted. However, increasing these limits results in much slower simulations since the given convergence criteria are mostly met before the minimum number of iterations is performed.

4.5 Default input parameters and data.

The default input parameters for the simulations are given in Table 4.3-4.11. These are used for all simulations unless otherwise stated.

Comments to the condensation control parameters.

The condensation control parameters for Si02 (Table 4.6) are extremely large (Tltsio 2 = 4000.0K and T2,sio 2 = 5000.0K). No SzOa-surface ever reaches 4000K. Thus, the total amount of Si(g) available at a given SiC^-surface do actually condense as described in sec­ tion 3.5.3. Above T2fi = 2232K (Table 4.5), all of the condensing silicon reacts chemically with Si02 as described in section 3.4.4. This is similar to a situation where all available Si(g) reacts directly on the SzOa-surface without first condensing. There seems to be no reasons why the Si(g) shall not react directly with the SiC2 under such conditions. Condensation on other surfaces (SiC and Si) does on the other hand always give a positive heat flux to the surfaces since the condensed silicon can not react chemically on these 4.5. DEFAULT INPUT PARAMETERS AND DATA. 145

Table 4.3: Surface area correction factors (page 98).

Effective surface area Blocked surface area fs.Si fs.SiC fs,SiOz fb,Si fb,SiC fb,SiC>2 1.0 15.0 1.0 0.0 0.0 0.0

Table 4.4: Dripping control and data (page 99 and 101).

Model type Parameters Ts[K\ 'H'drzp ATdrip [K1 Afrfrip [s] Preserve crater wall shape 2 2140.0 200.0 50.0

Table 4.5: Distribution of enthalpy of condensation (page 56 and 74).

Control parameters Distribution factors Turn [K] Tl.e [K] T2,e IK] Ts, [K\ fpoal fwall fgas 2132.0 2032.0 2232.0 2032.0 0.65 0.25 0.10

Table 4.6: Condensation control parameters (page 75).

Material type Si SiC Si02 Ti,Si \K] T2.si [K] Ti.sic [K] T2,sic [K] Ti'Sio? [K\ T2,sio 2 [K\ 2500.0 2850.0 2600.0 3200.0 4000.0 5000.0

Table 4.7: Evaporation control parameters (page 73).

K Si Psi ATt8 [K] 0.0 0.0 150.0 146 CHAPTER 4. THE SIMULATED FURNACE AND DEFAULT PARAMETERS.

Table 4.8: Thermal conductivity.

Material type Si [W/mK] SiC [W/mK\ Si02 \W/mK\ 300.0 5.0 1.3

Table 4.9: Electric arc data (page 81 and 83).

Electric arc data Lorentz force corrections I \kA] Rc [cm] C 70.0 6.0 1.0 0.025 0.025

Table 4.10: Volumetric radiation density (page 85).

Gas composition [mole% Ar : mole% Si] Effective radiation radius [cm] 50.0 : 50.0 5.0

Table 4.11: Transport properties (page 84).

Gas composition 60 mole% SiO 40 mole% CO 4.5. DEFAULT INPUT PARAMETERS AND DATA. 147 surfaces4. The condensation should therefore decrease above a certain temperature.

Comments to the large correction factor for ffiC-surfaces. The correction factor for the effective surface area of SiC-particles is large (fs,sic — 15.0). This means that the effective surface area for the reaction between SiO(g) and SiC at gas/SiC-interfaces is 15 times the area calculated from the model geometry. The produc ­ tion rates decrease with smaller correction factors. Small reaction rates when applying small and moderate correction factors for the SiC- particles indicate that the mass transfer of SiO(g) to the reacting surface or the size of the SiC-surface itself are strongly underestimated. In the model the gas/SiC-interface in the metal pool is small. As mentioned in section 3.10, there may be a bed of irregular and porous StC-particles extending above the other ma­ terials in this area. The gas/SiC-interface is then strongly underestimated in the model and large surface correction factors for SiC-surfaces seem reasonable. However, compensating for a large porous SiC-bed by multiplying the surface area of the particles by large values for fStsic when calculating the reaction rates is a far from perfect method. The main reason is a less effective transport of reactant gas (SiO(g)) and energy to the Sz'C-surfaces than for a large SiC-bed :

• Increasing the surface area leads to larger reaction rate under otherwise identical conditions. Equilibrium is almost established for sufficiently large correction factors. However, transport of reactant gas (SiO(g)) to the surface is important for the reaction rate. In the model this gas is supplied from small areas close to the reacting SiC-particIes. The overall supply of SiO(g) to the particles is likely to be smaller than to a large SiC-bed. The conversion from a large SiC-bed is then larger than calculated by the model no matter how large correction factors are used for the SiC-surfaces. • The effect of increasing the correction factor is also reduced because the temperature of the reacting surface decreases5. The driving force for the reaction then decreases because the equilibrium pressure of SiO(g) for the reaction increases (Figure 1.4, page 9).

4On SiC-surfaces, this heat flux may increase the reaction rate for the endothermic reaction between SiO(g) and SiC(s) (reaction 1.6, page 41). 5The reaction rate increases with the correction factor. Thus, more energy is consumed from the surface by the endothermic reaction. The heat fluxes from the surroundings are, however, not compensated for the increase in effective surface area. The temperature consequently decreases since more energy is removed from a surface with constant influx of energy.

t- 148 CHAPTER 4. THE SIMULATED FURNACE AND DEFAULT PARAMETERS.

Thus, large values for fs,sic compensate only partly for a possible large SiC-bed in the metal pool since no similar corrections are made for the heat- and gas fluxes to the particles. It is not obvious how to include such corrections. As mentioned previously, SiC is in the furnace shaft present as quite small and highly irregular particles. Furthermore, in the model gas is not allowed to penetrate into the furnace shaft. The effective surface area for SiC in the furnace shaft is therefore strongly underestimated and large correction factors seem reasonable. Again, large values for fs,sic compensate only partly for the underestimated effective sur­ face area. The reasons are the same as just explained for the metal pool.

Comments to the thermal conductivities. The thermal conductivities for materials at temperatures relevant in the high-temperature part of the silicon metal process are difficult to measure or estimate. Data found in literature are thus highly uncertain. Gu [40, p. 98] used kst = ASWjmK. A larger value seems reasonable to compensate for the lack of convective movement in the metal pool, and k$i = 300W/mK is used in this thesis (Table 4.8). Kern, Hamill, Deem and Sheets [62] indicate values around 12W/mK for fi-SiC in the temperature range 1400 — 2000°C. Compared to compact SiC, a somewhat smaller value is reasonable for porous SiC-particles in the permeable charge or porous SiC in the metal pool. kSiC = SW/mK is used in this thesis (Table 4.8). The value given by Perry and Green [63, p. 3-263] (ksio 2 = l-'SW/mK) is used for silica in this thesis (Table 4.8). Chapter 5

Results and discussion.

The results are mostly presented by graphical plots generated from the FLUENT code. The black and white plots presented here are considerably more difficult to extract information from than the original colour plots. Some results are therefore presented in several black and white plots to illustrate the same information easily provided by a single colour plot. Colour plots could have been used, but those would not be suited for reproduction in black and white. It is often practical to present several related plots on the same page to clearly illustrate effects by comparing them. Some of these plots are therefore slightly smaller than they would otherwise have been.

5.1 Simulation number 1.

The most important responses are demonstrated from the results of this simulation which is run for 1800 time steps of 2 seconds each (an hour). The situation at selected intermediate times is described by graphical plots to show how the system changes with time and to visualise some prominent features of the process. This simulation is characterised by strongly reduced Lorentz forces in the electric arc.

5.1.1 Input parameters and data.

The default parameter values and data from Table 4.3-4.11 are applied. The initial temperatures in the furnace shaft and the metal pool are patched to 2000K and 2115K, respectively. The temperature in the metal pool is slightly below the temperature at which SiO? starts reacting with Si (2132K). The initial temperature distribution in

149 150 CHAPTER 5. RESULTS AND DISCUSSION. the electrode corresponds to the one calculated while determining the initial flow field and gas temperatures. The initial gas composition is patched to a mass fraction equal to 0.7 for SiO(g) and 0.3 for CO(g) in all gas-filled parts of the system. This corresponds to Psiof (psio +PCO) ~ 0.60. Figure 5.1 shows the initial temperature distribution for the most interesting parts of the furnace.

1.ZEH4 ------_____ 1.1KH4 1.06EH4 5. WH3 (.MEH3 //III \ ——'—. T.ece*c 6. WW3 'll (

4 S1H*1 / INITIAL STATE sm-i INIIIW. STATE FLUENT^ can cubs cf rerezATUF imviN i MAX. ■ 2.532716*04 I MIN. » 1.974025+03 MAX. ■ 2.S37716+04 1 HIM. • 1.974026403

Figure 5.1: Initial temperature distribution. Left : The crater cavity and nearby areas, temperatures below 5000K. Right : Close-up of the area around the electric arc, temperatures above 5000K. Highest temperatures are for contours closest to the electric arc.

5.1.2 Power level and radiation.

The ohmic production of heat inside the electric arc decreases from 5.71MW at the start of the simulation to 5.68MW at the end. In the same period, the radiative energy loss from it increases from 3.95 to 3.98MW. The changes are largest in the start. The calculations indicate that the computed solution for the electric arc has not fully reached the steady state condition, but the deviations from it are small and decreasing. A large fraction of the energy generated by ohmic heating in the electric arc is transported away from it by radiation. This is because the Lorentz-forces are reduced by a factor of 20 (o !l,z = 0.05 and aL,r = 0.05) compared to what the Prescribed Current Distribution model for the electric arc normally gives. The flow velocities in the arc are therefore 5.1. SIMULATION NUMBER 1. 151 strongly reduced. Thus, the convective heat transfer is reduced and radiation becomes more important. The reduced convective heat transfer in the electric arc gives much lower temperatures close to the axis of symmetry than for the case a^jZ = Qx, r = 1.0. Further comments are given in section 5.2.2. As just mentioned, the radiation changes little with time. Figure 5.2 shows its distribution to different locations after 30 minutes. Most of the radiation is received by the parts of the metal pool closest to the electric arc. Most of the crater wall receives no radiation since it is in the shadow from the electrode. The radiation to the electrode is disregarded. It might be interesting to redistribute this energy to the other parts of the system, but this is not included. The radiation energy fluxes to the innermost parts of the metal pool are definitely over ­ estimated and the fluxes to the other parts of the system are underestimated accordingly. The reasons are that

• the electric arc is modelled as a spatially fixed DC electric arc with the electrode serving as the cathode. The effects of this on the radiative energy transfer are discussed on page 80. • all radiation to liquid and solid surfaces is assumed to be absorbed by the surfaces (no radiation is reflected back from the surfaces). The effects of this on the radiative energy transfer are discussed on page 87. • wall to wall radiation is excluded (page 86). Wall to wall radiation implies a net radiation from the warmest parts underneath the electrode to the colder parts further out.

5.1.3 Silicon production.

5.1.3.1 Expected production rates.

Typical power consumption for silicon metal production is 12MWh/tonne produced silicon. This value corresponds to a silicon recovery of about 85% and a thermal efficiency of about 0.80. This value leads to an expected silicon production of 475 kg/hour (7.9 kg/minute) for a total average energy input of 5.7MW. Assuming that 65% of this silicon is produced directly from the reaction SiO{g) + SiC{s) = 2Si(l) 4- CO(g) in the metal pool and the crater wall gives an expected silicon production of 309 kg/hour (5.1 kg/minute) in the high-temperature zone. The remaining 166kg (35%) is then produced by condensation of SiO(g) in the low-temperature zone of the furnace shaft and is not accounted for in the model. 152 CHAPTER 5. RESULTS AND DISCUSSION.

TLFIjriPTqj J.

SIIM / 30 WHITES SW-1 / 30 MINUTES COfTOBS CF BOUT. SU TB86 IKAnS/X.SO. ) MAX. » O.OOOOOS+OO I MIN. » -5.31093E+Q7 MAX. ' O.OOOOCF4QO 1 MIN. ■ -5.3*095+07 *** wy)

SIH*I / 30 Ml Hires FLUENT, MAX. » O.OOOOOE+OO I MIN. ■ "5.3*095+07 MAX. ■ 0.000006*00 I MIN. « «S.3*093B07

Figure 5.2: Distribution of radiation from the electric arc to other parts of the system after 30 minutes. Negative values indicate radiation to the surface. Top left : Highest values. Top right : High to medium values. Bottom left : Medium to low values. Bottom right : Lowest values.

I 5.1. SIMULATION NUMBER 1. 153

As mentioned previously, the radiation received by the electrode is disregarded in this simulation. The temperatures in the electric arc and thus also the radiation from it is by far largest just below the electrode surface (Figure 5.1, page 150). This means that almost half of the radiation is received by the electrode. A small fraction of the radiation is also received by the vertical inert furnace wall at the outer edge of the crater cavity. This radiation is mostly transferred out of the system by conduction. Thus, about 50% of the total radiation does not contribute to the chemical production in the model. Recalculating the production rate assuming a total average useful energy input of 3.7MW (assuming that 2MW of the radiation is not accounted for) gives an expected silicon pro ­ duction of 200 kg/hour (3.34 kg/minute) in the high-temperature zone. This is a rough estimate for the expected production rate. It applies to a situation when the furnace has reached its steady state production level and not to the transient period starting from a fairly cold furnace as is the case for this particular simulation.

5.1.3.2 Calculated average production rates.

The calculated net production of silicon is 112.6 kg, corresponding to an average of 1.88 kg/minute. Figure 5.3 illustrates the accumulated net silicon production.

200.0 Theoretical production

150.0

100.0

Calculated production

Elapsed simulation time (min.)

Figure 5.3: Accumulated net silicon production for simulation 1.

There is a short period of 22 seconds with net consumption of silicon in the start of the simulation. This is because the initial temperatures of the S'iC-particles are too low to 154 CHAPTER 5. RESULTS AND DISCUSSION. give a sufficiently high rate for the silicon producing reaction to compensate for the silicon vapour reacting on gas/Si02-surfaces. The average net production rate increases gradually to about 1.8 kg/min after about 10 minutes and stays almost constant for the next 20 minutes. Evaporation of silicon from the metal pool followed by condensation at gas/SiCVinterfaces is believed to be important in these early stages of the simulation. This is simply because the condensing Si(g) brings large amounts of energy with it to support the production of SiO(g) by reaction 1.5 and to heat gas/SiC-interfaces where the SiO(g) later reacts by reaction 1.6 to produce Si(l). In the start of the simulation, the evaporation increases rapidly since the heat flux to the inner parts of the metal pool is large. It continues to increase towards a constant steady state value. The calculations indicate that the evaporation is close to this value after about 10 minutes when the production rate stabilises. After about 30 minutes, the average net production starts to increase faster again. It reaches 2.4-2.5 kg/minute towards the end of the simulation. The main reason for this increase is believed to be gradually increasing temperatures in the parts of the metal pool where the reacting materials are located and in the furnace shaft. The production of SiO(g) then increases at the Si(l)/SiOi(Z)-interfaces in the metal pool. The metal production at the gas/SYC'-interfaces increases accordingly as explained previously. Also, higher temperatures for the gas/SiC-interfaces shift the equilibrium for the metal produc ­ ing reaction towards higher silicon production (Figure 1.4, page 9). The changes in the particle geometry may also influence the production rates. Whether the production rate approaches the theoretical value of 3.34 kg/min in the long run is at present unknown. It is, however, unlikely that it will do so since this is a prototype model and no attempts have been made to find an optimal combination for the model parameters to give realistic production rates.

5.1.3.3 Calculated instantaneous production rates.

Figure 5.4 shows the net production rate for each time step. Significant fluctuations are found around the average production rate. These fluctuations are linked to changes in the geometry of the particles. Two important effects are identified : 1

1. Changes in the surface temperatures for the reacting particles caused by the algorithm applied for updating the particle geometry. 2. Changes in the flow field close to the reacting surface. This influences the transport of reactant gas to the surface and is essential for reaction 1.6 (SiO(g) + SiC(s) —> 2Si{l) + CO(g)). 5.1. SIMULATION NUMBER 1. 155 Time (min)

^Theoretical steady state production rate

Calculated production rate

600 1200 Iteration number

Figure 5.4: Instantaneous silicon production rates.

Changes in the surface temperatures for the reacting particles. Changing the geometry for a particle as described in sections 3.9.6 and 3.10.6 alters the temperatures of its computational cells. This is illustrated in Figure 5.5 for a test particle consisting of three rows of cells in the vertical direction and one single column in the radial direction. All three computational cells are assumed to have equal sizes. For simplicity, the temper ­ ature profile is assumed to be linear before updating the geometry. The particle is then compressed from three to two cells in the vertical direction. The new uppermost cell consists of all mass above the midpoint, zm. The new temperature for this cell becomes 2125K, and that of the new lowermost cell becomes 2275K. As implemented in the model, the temperature of the computational cell is applied when calculating the equilibrium for the reaction and the heat flux to the reacting surface. In the test case, the temperature for the uppermost cell changes from 2100K to 2125K due to the updating. The chemical reaction rate for possible reactions involving this cell consequently changes. The changes in the temperatures are largest for particles consisting of only a few rows of cells in the vertical direction. Applying the temperature at the reacting surface rather than the cell temperature when calculating the reaction rates would in most cases reduce the effect of the updating. In 156 CHAPTER 5. RESULTS AND DISCUSSION.

Before updating After updating

■<— 2050K <------2100K - - < 2125K < 2150K 2200K ---3»- • •••■ 2200K- Zm • 2275K -C-- 2250K 2350K 2300K- — -- 2350K

Figure 5.5: Changes in temperature caused by updating a single-column particle from three to two cells in the vertical direction. The node temperatures change while the surface temperatures remain unchanged. the test case, linear extrapolation gives 2050K for the upper surface both before and after updating. The reaction rates are therefore identical if this temperature is applied. For non-linear temperature profiles, the updating changes also the surface temperatures.

Changes in the flow field close to the reacting surface. Transport of SiO(g) to the SiC surface is important for the reaction rate of reaction 1.6 (SiO(g) + SiC(s) —> 2Si(l) + CO(g)). The flow conditions close to the reacting surface are therefore essential for the conversion. In the metal pool all particles float with their upper surfaces exactly level with the fixed upper edge of the metal pool. Thus, the flow conditions are not influenced by changes in the particle geometry here. The situation is quite different in the crater wall where the geometry is irregular. Fig­ ure 5.6 shows typical situations. The particles are here arranged in the same way as in the simulation. Only the innermost particles are shown, but no SiC is found further out (to the right). The transport of SiO(g) to the SiC-particles is poor in situation 2 because the SiC- particles are shielded by other materials in the upstream direction. Transport of produced CO(g) away from the surface is for the same reason slow and the reaction rate is smaller than for situation 1. Thus, removing materials from the lower edges of particles 1, 7 or 11 is likely to reduce the production of silicon metal for these particles instantaneously. This implies larger concentrations of SiO in the gas and the production rate may increase again after a while because of this. Removing cells from the lower edges for other particles is likely to increase the transport of SiO(g) to the downstream SiC-particles, thus increasing 5.1. SIMULATION NUMBER 1. 157

□ =SiC 0 =Si02 ■ = Inert material

Figure 5.6: Situations with poor and good gas flux to SiC-particles in the crater wall. the metal production. This is consistent with the results from the calculations. Figure 5.7 shows which particles are updated just before major changes are observed in the instantaneous production rates in the period 5-25 minutes. As expected, changing the SiC-particles in the furnace shaft decreases the production rate, while changing other particles close to the SiC-particles increases the production rate (especially particles 3, 9 and 13). Other particles than those marked in Figure 5.7 also change their geometry during the given period. In most cases, this gives insignificant changes in the net silicon production. One exception is one case where particle number 10 in the metal pool is compressed. In this case, Figure 5.7 shows that the silicon production first increases and then soon decreases again to a level slightly below that prior to the change. No other particles change geometry until particle number 13 changes as shown in the figure. This indicates that the changes in the silicon production is induced by the changes in particle number 10. These changes must be caused by changes in temperature due to the updating as explained previously or by reduction in the Si/SzOa-interface when the particle reduced its size. Smaller Si/Sf(^-interface leads to an instantaneous decrease in the SzO-production under otherwise identical conditions. Less Si is therefore consumed just after the updating and a larger net silicon production is calculated. Later, the decreased SiO-production leads to less production of silicon for the downstream SiC-particles as explained previously. Thus, the net silicon production rate decreases again after a while. This may be the reason for the calculated changes. However, particle number 10 also changes geometry after 312 iterations. This time it gives only minor changes in the pro- 158 CHAPTER 5. RESULTS AND DISCUSSION.

Time (min) 5 10 15 20 25 1,7,11: SiC-particles in the crater wall 3,9,13: S102-particles in the crater wall 10: Si02-particle in the metal pool (straight below particle 9)

Iteration number

Figure 5.7: Instantaneous production rates from 5 to 25 minutes for simulation 1. The numbers show which particle is updated just prior to major changes in the production rate. The particle positions in the crater wall are as defined in Figure 5.6. 5.1. SIMULATION NUMBER 1. 159 duction rates (Figure 5.7). The detailed calculations must be examined carefully to reveal the exact reasons for these changes and why the responses are different in the two cases. The instantaneous production rate obviously changes significantly due to changes in the geometry of the SiC-particles in the crater wall or other particles close to them. However, the average silicon production rate increases smoothly as seen from Figure 5.3 on page 153. This indicates that the geometry of the individual particles are not essential for the long ­ term production rate calculated by the model. The fluctuations in the production rates are likely to be smaller if finer grids are applied or if each particle is split in several particles in the radial direction. This leads to a smoother crater wall, and the transport of gas and energy to/from the reacting surfaces becomes less sensitive to updating the geometry. Also, the changes in the particle size would be less for each updating if a finer grid is applied in the vertical direction. This would also reduce the fluctuations.

5.1.4 The flow field.

Figure 5.8 shows the flow field in the crater cavity after 10 minutes.

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Figure 5.8: The flow field in the crater cavity after 10 minutes. Left : Stream functions. The flow is counter clockwise in the cavity and upwards through the gas exit channel. Right : Velocity magnitudes 1-39 m/s. The highest velocities are found close to the electric arc.

There is a pronounced recirculation zone in the cavity. This is set up by the Lorentz forces, which in the case of a DC electric arc suck in gas at the electrode end (the cathode) and 160 CHAPTER 5. RESULTS AND DISCUSSION.

purges it downwards onto the metal pool (the anode). The gas molecules are strongly accelerated by the Lorentz forces on their way through the electric arc. The maximum velocity inside the arc is about 520 m/s. The velocities just above the metal pool decreases from about 230 m/s in the innermost parts to about 17 m/s above the innermost particle and to 1-2 m/s at the outer edge of the crater cavity. Some of the gas reenters the electric arc and the rest leaves the system through the gas exit channel next to the electrode. New gas is produced by chemical reactions along the surfaces of the metal pool and the crater wall. The overall flow field changes little with time while adjusting to the gradually changing geometry of the crater wall. This is because it is dominated by the Lorentz forces for the stable DC electric arc. The local flow close to particles in the crater wall can change significantly when these are updated.

5.1.5 The geometry of-the particles.

Figure 5.9 illustrates the long-term changes in the particle geometry. The crater cavity is seen to expand upwards with time as expected. This is partly caused by chemical reactions proceeding in the furnace shaft, and partly by dripping of materials from the furnace shaft to the metal pool upon melting. In the start the dripping is largest from the outermost particles in the furnace shaft. The heat fluxes to these particles are large in this period because the gas flow is mainly directed upwards in the outer zones of the crater cavity1. This gives more effective heat transfer than the almost tangential flow along the particles further in. Also, radiation from the electric arc heats these particles as shown in Figure 5.2 on page 152. The large dripping and the moderate chemical conversion in this period implies that the outermost particles in the metal pool increase significantly in the initial stages of the simulation as seen in Figure 5.9. Later, both the radiation to these particles and the heat flux to them from the gas decreases. The radiation decreases because the crater wall moves upwards. More of the radiation to these particles is therefore blocked by the electrode. The heat flux from the gas decreases mainly because the permanent outer furnace wall next to the particles deflects the gas inwards and away from the outermost particles in the crater wall12. This implies less dripping from these particles. Simultaneously, the chemical conversion increases for the particles underneath them in the metal pool. Consequently, the outermost particles in the metal pool stop growing or even start shrinking towards the end of the simulation period as seen in Figure 5.9.

1This is a result of the applied DC model for the electric arc, which as pointed out in section 3.8, gives a fundamentally different flow field than an AC arc. The crater wall develops differently in the AC case. 2A straight, vertical outer crater wall would not deflect the gas flow inwards like this. 5.1. SIMULATION NUMBER 1. 161

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Figure 5.9: Geometrical changes. Top left : Initial state. Top right : after 20 minutes. Bottom left : after 40 minutes. Bottom right : after 60 minutes. 162 CHAPTER 5. RESULTS AND DISCUSSION.

The innermost particle in the shaft (SiC) receives considerable amounts of energy from the warm gas flowing through the exit gas channel. This energy is transported through the SiC-particle and into the SiOz-particle next to it as shown in Figure 5.10. The SiOz- particle responds by dripping down (melting) at an increasing rate. Electrode Gas outlet Inert material

Figure 5.10: The innermost particles in the furnace shaft. Considerable amounts of en ­ ergy is supplied to particle number 1 from the warm gas in the outlet gas channel. Some of this energy is transported by conduction into particle number 3 as indicated by the arrows. The SiCVparticle is heated, espe ­ cially in area A.

Considerable amounts of mass drip down from the innermost SiC-particle together with the SiOz since the dripping mode preserving the general shape of the crater wall (page 102) is selected. The other particles are heated less effectively, and they drip down more slowly 3. The innermost particles in the metal pool (where most of the SiO(g) is produced) decrease in size all the time (Figure 5.9). This decrease is largest early in the simulation when the materials in the crater wall are comparatively cold and little mass drips down to the metal pool. The dripping from each particle varies with time. This is illustrated for the three innermost 5'i02-particles in the furnace shaft in Figure 5.114. The particle numbers in Figure 5.11 correspond to those in Figure 5.10. The dripping changes significantly when the geometry of the particle changes (when the

3 The situation with one single gas outlet channel is quite similar to a situation when the furnace is allowed to blow (page 6) undisturbed for a long time. In a way this simulation therefore predicts how a blowing furnace develops. 4The reason for selecting the SiOa-partides is that the dripping for all materials is calculated from the data for the SiOa-particIes as explained in section 3.9.4. 5.1. SIMULATION NUMBER 1. 163

— Innermost Si02-partide (No. 3) — Second innermost Si02-particIe (No. 5) — Third innermost Si02-particle (No. 9)

20.0 41 Time (minutes)

Figure 5.11: Dripping from the three innermost Si02-particles. The particle numbers and relative particle positions are as defined in Figure 5.10. lowermost row of cells is removed). Only the two lowermost rows of cells contribute to the dripping (n drip = 2).

Explaining the calculated dripping. The temperatures in the cells contributing to dripping change when removing a row of cells. Figure 5.5 on page 156 shows a situation when the temperature is highest at the lower edge. The temperature in the two lowermost cells are lower after the updating because colder materials enter these cells when the particle is compressed in the vertical direction. Less materials then drip down until the temperature increases to its former values again. This should give a sharp decrease in the dripping just after the geometry of the particle is updated. It should be followed by a gradual increase in the dripping towards a constant level. New changes in the particle geometry may occur before this constant level is reached. A significant decrease in the dripping is found for particles 5 and 9 after updating their geometry. However, this decrease takes place over a considerable time. Changes in tem­ perature caused by the updating itself are therefore only part of the explanation. The subsequent gradual decrease after the updating is believed to be caused mainly by changes in the flow field around the particle. 164 CHAPTER 5. RESULTS AND DISCUSSION.

As explained in section 5.1.3, the flow field close to a particle in the crater wall changes considerably when the particle changes geometry. The consequences for the mass flux and the reaction rates were discussed in the same section. The heat transfer from the gas to the surface is equally sensitive to changes in the flow field as is the mass transfer. In fact, the mass- and heat fluxes are calculated from analogue equations. This implies that the heat flux to the surface decreases whenever the mass flux to it decreases. Applying the same arguments as for the mass flux (section 5.1.3) leads to the conclusion that the heat flux from the gas to a given particle in the crater wall decreases considerably when its lower row of cells is removed. If so, the temperatures in the cells contributing to dripping may continue to decrease for a while after the particle geometry is changed. As for the mass flux, the heat flux increases again if nearby particles in the upstream direction are updated. The heat flux and the dripping then increases again. This is believed to explain the main trends in the calculated dripping for particle 5 and 9 just after updating their geometry (Figure 5.11) :

• First, the dripping decreases due to the updating itself. • It continues to decrease for a while because the new flow field reduces the heat flux to the particle. • Finally, the dripping increases again when particles in the upstream direc­ tion are updated and change the flow field to increase the heat flux.

Other effects complicate the situation. The most important of these are believed to be :

1. Heat is consumed by chemical reactions from the surface of reacting par ­ ticles. The reaction rate depends on the flow field around the particle and its temperature. Large chemical conversion implies that a smaller fraction of the total en­ ergy flux to the surface is available for heating the materials. Thus, drip ­ ping decreases. 2. The vertical part of the gas/particle-interface changes when the particle itself or its closest neighbours change geometry. The heat flux to the particle from the gas changes accordingly. 3. Heat is also transferred by heat conduction between the particles in the crater wall. This is important if the temperatures in the cells contributing to dripping are influenced significantly by this heat transfer. 5.1. SIMULATION NUMBER 1. 165

Item 2 from the list is believed to explain the long-term increase in the average dripping for particle 5. The particle inside it (particle 3) is removed at a higher rate, and the vertical gas/particle-interface therefore increases with time as shown in Figure 5.9. This allows the gas to heat a larger part of the particle. The temperature consequently increases above the two rows of cells contributing to dripping. Thus, dripping increases when these preheated materials enter the lower parts of the particle. Similar long-term increase in dripping is not found for particle 9. This indicates an almost constant average heat flux to this particle. This is reasonable since the crater wall changes little with time in this area except that it moves slowly upwards. Almost the same surface area is therefore exposed to the crater gas at all times and the average flow field around it changes little. Thus, the heat flux and dripping become fairly constant. Item 3 from the list is believed to explain the calculated dripping for particle number 3. For this particle, the dripping increases rather than decreases after removing the lowermost row of cells. The details are as follows :

As explained on page 162, a significant part of the energy supplied to this Si02-particle is transported through the SzC-particIe inside it rather than by direct heat exchange with the crater gas at its lower end. The Sz02-particle is therefore warmest next to the lower end of the SzC-particle (area A in Fig­ ure 5.10, page 162). Some of this warm SiO% enters the two lowermost cells when a row of cells is removed from it. Thus, the dripping from the particle increases rather than decreases when its geometry is updated 5. The time-averaged dripping from particle 3 clearly increases with time. This is mainly because the particle is gradually heated by conduction through the SzC-particle. Its interface towards the gas changes little with time.

As explained in section 3.9.4, the dripping is expected to be highly irregular. Lumps of material are expected to drip down at irregular intervals while little drips down in-between (Figure 3.28, page 98). Thus, the calculated variations in the dripping are both explainable and acceptable.

Comments to the distribution of dripping materials. The materials drip vertically down from a particle in the furnace shaft to a particle of the same type underneath it in the metal pool (section 3.9.4, page 98). This method was implemented to have a simple algorithm for transporting materials from the shaft to the metal pool. One side-effect is that there is often a mismatch between the amount of material that drips down to a given particle and the chemical conversion from it. Distributing the

5It is verified that the temperature is indeed higher just above the two cells contributing to the dripping for particle number 3. 166 CHAPTER 5. RESULTS AND DISCUSSION. dripping materials more in accordance with the local chemical conversion in the metal pool may be better. This requires additional computer code to handle sideways movement of dripping mass. Such algorithms could also be applied to move reacting materials to the area underneath the electrode.

5.1.6 The temperature distribution.

The temperatures in and around the crater cavity after 10 and 60 minutes are shown in Figure 5.12 and Figure 5.13, respectively.

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Figure 5.12: The temperature distribution after 10 minutes. Highest temperatures clos­ est to the electric arc. Left : Temperatures below 2500K. Right : Tem­ peratures in the range 2500K-6000K.

Gas temperatures. The temperatures inside the electric arc are slightly lower than for the initial state with the maximum temperature decreasing with about 400K to 24935K. This is because slightly different model parameters were applied when generating the initial state. The maximum temperature changes very little from 10 minutes to 60 minutes. The gas temperatures in the crater cavity increase slightly except for the areas where gas is produced by chemical reactions. The produced gas is colder than the crater gas and therefore cools the crater gas locally. This can be seen especially above the particles in the 5.1. SIMULATION NUMBER 1. 167

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Figure 5.13: The temperature distribution after 60 minutes. Highest temperatures clos­ est to the electric arc. Left : Temperatures below 2500K. Right : Tem­ peratures in the range 2500K-6000K. metal pool in the plot on the right in Figure 5.13, but is more easily observed in Figure 5.39 on page 200. The temperatures both in the central parts of the metal pool and in the crater cavity are higher than expected. This is probably related to the lack of reacting materials underneath the electrode, the lack of physical transport of silicon vapour to the gas and the spatially fixed DC electric arc :

The lack of reacting materials underneath the electrode.

No materials from the furnace shaft are ever transported to the area underneath the electrode. No energy-consuming chemical reactions involving these mate­ rials therefore proceed in this area. As explained in section 3.4.4, reaction 1.5 (Si(l) + Si02(l) —F 2SiO(g)) proceeds more or less without kinetic hindrances at temperatures above 2132K. It is therefore reasonable to assume that the tem­ perature of the silicon in the metal pool does not elevate significantly above this temperature in areas where Si(l) and Si02(l) are both present. For a pure silicon melt underneath the electrode, as assumed in the present model, the temperature of the metal pool surface is limited by the evaporation of silicon. The boiling point of silicon is 3504.616K, and the metal pool surface adjusts to a temperature close to this if the heat flux to the area is sufficiently high. 168 CHAPTER 5. RESULTS AND DISCUSSION.

This implies that the temperatures in the metal pool are far higher in the model than expected for a situation where SiOi is present in the central parts of the furnace. It is not known to what extent reactants are actually present in these areas during normal furnace operations, but it reasonable to believe that SiO^ indeed floats into these areas. One consequence of too high temperatures in the central parts of the metal pool is that the gas exchanges heat with a too warm surface. The gas temperatures are therefore likely to be too high as well.

The lack of physical transport of silicon vapour to the gas.

The lack of physical transport of silicon vapour to the gas has no direct conse ­ quences for the temperatures in the metal pool. It is, however, important for the temperature distribution in the gas. The evaporation is expected to cool the warmer gas above the metal pool. Instead, the applied model for evaporation gives increasing gas temperatures because some of the enthalpy of condensation is distributed to the gas without simultaneously adding the corresponding mass (page 70). One obvious remedy is to disregard also the energy transfer to the gas phase, but this implies that some of the enthalpy of condensation is lost from the system. As it is, the gas temperatures increase and some of the associated energy is later transferred back to the metal pool, walls and particles in the downstream direction. The rest leaves with the off-gas.

The spatially fixed DC electric arc.

The applied spatially fixed DC electric arc overestimates the heat flux to the innermost parts of the metal pool and underestimates the heat flux to the outermost parts where the reacting materials are located (page 80). In the abscense of reacting materials in the central parts of the metal pool, the temperatures in these areas are determined by the evaporation of silicon. Thus, increasing the heat flux mostly results in higher evaporation rates and not so much in higher temperatures in the metal pool. However, increased evaporation implies further overestimation of the gas tem­ peratures in the model as just explained. The high flow velocities induced by the DC electric arc make the heat transfer from the gas to solids and liquids less efficient. This probably prevents the 5.1. SIMULATION NUMBER 1. 169

gas from giving off much of its energy before leaving the crater cavity. It may partly explain the high gas temperatures calculated in this area.

Temperatures in solids and liquids. The temperatures in the metal pool increase gradually with time as energy is transported into the initially rather cold metal pool. This is easily seen by comparing the 2500 Kelvin isoterms of the leftmost plots of Figures 5.12 and 5.13 (the innermost isoterms in the metal pool). The temperatures for the particles in the furnace shaft also increase with time from their initial low values, especially for the innermost particle (SiC) which is effectively heated by the warm gas in the outlet gas channel. The cooling of the Si(l) / SiOzil)-interfaces caused by the reaction between the two ma­ terials is apparent in Figure 5.14. The temperature of the interface stays slightly above

Figure 5.14: Temperatures for the Si(l)/Si02(Z)-interfaces after 30 minutes. Left : All particles in the metal pool, T = 2120K-2155K. Right : The innermost particles in the metal pool, T = 2125K-2140K. Highest temperature for the leftmost isoterm in the metal pool.

2132K which gives psio = 1 for reaction 1.5 (Si(l) + Si02(l) = 2SiO(g)). The cooling effect is best seen underneath particle number two and three from the left (Si02-particles). The innermost particle is SiC. The temperatures for the SiOa-particles are generally lower than those for SiC-particles. The former are effectively cooled at all its surfaces by the strongly endothermic reaction 1.5. 170 CHAPTER 5. RESULTS AND DISCUSSION.

The latter are only cooled at surfaces facing the gas, and then by a less endothermic reaction. The temperature difference is clearly visible in Figure 5.15.

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Figure 5.15: Temperatures for SiC-particles (number 1, 4 and 6 from the left) and Si02- particles in the metal pool after 30 minutes. T = 2150K-2600K. The low­ ermost isoterm represents the lowest temperature (2150K).

This effect with warmer SiC-particles may well be real, but is probably less pronounced than calculated in the model. If so, it eases the process since psio,eq for the metal producing reaction 1.6 (SiO(g) + SiC(s) = 2Si(l) + CO(g)) decreases with increasing temperature (Figure 1.4, page 9). The reaction shifts to the Si-side and less SiO(g) needs to be recovered in the upper parts of the furnace when the SiC-surfaces are warm.

5.1.7 Chemical production/consumption and the gas composi ­ tion.

Figure 5.16 shows the consumption of silicon in the metal pool and SiC both in the metal pool and in the furnace shaft. Figure 5.17 shows the consumption of Si02.

The reaction between Si02 and condensed silicon takes place with Si02 supplied from the lower end of the SiCVparticle rather than at the gas/SiOa-interface (section 3.13.3). This is why there is no consumption of Si02 at the upper surface of the SiOa-particles in the metal pool. It also explains the large consumption from the lower edges of the Si02-particles. Figure 5.18 shows the corresponding production of gas (SiO(g) and CO{g)). The con­ sumption of SiO(g) is proportional to the production of CO(g). 5.1. SIMULATION NUMBER 1. 171

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Figure 5.17: Consumption of Si02 after 30 minutes. Values in the range 0.35-6.65 kg/m?s (maximum « 22.6kg/m3s). The lowest value (0.35) corresponds to the outermost contour in each area where consumption takes place. 172 CHAPTER 5. RESULTS AND DISCUSSION.

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Figure 5.18: Production of gas after 30 minutes. Top left : SiO(g), 8.0 kg/m3s and below. Top right : SiO(g), 8.0 — 14.0 kg/m3s. Bottom left : CO(g), 2.0 kg/m3s and below. Bottom right : CO(g), 5.0—15.0 kg/m3s. The lowest values correspond to the outermost contour in each area where production takes place. 5.1. SIMULATION NUMBER 1. 173

The total production of SiO(g) is a result of the reaction with silicon in the metal pool ■ at the Si(l)/Si02(Z)-interface and the reaction with silicon condensing at the gas/SiC2- interface. The general trend is that the production of SiO(g), and thus also the consump ­ tion of SiC2, decreases outwards along the metal pool surface. This is to be expected since the production increases with the temperature, and the temperature generally decreases considerably with the distance from the center of the furnace. The production of CO(g), and thus also the consumption of SiO(g), decreases outwards in the metal pool. The difference in the production rates for the two innermost SiC-particles (number 1 and 4 from the left) is small. Figure 5.19 shows the mass fraction of SiO(g) above the innermost particles in the metal pool after 30 minutes.

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Figure 5.19: Mass fraction of SiO(g) above the innermost particles in the metal pool after 30 minutes. Particles 1, 4 and 6 from the left are SiC. The others are SiC2. In-between and underneath is Si(l). Left : 0.605-0.686. The lowest value (0.605) corresponds to the contours closest to the SiC-particles. The highest value (0.686) corresponds to the uppermost contours. Right : 0.696-0.794. The lowest value (0.696) corresponds to the uppermost contour. The highest value (0.794) corresponds to the contour closest to the SiOa-particles.

SiO(g) is supplied to the innermost SiC-particle by gas arriving from the inner parts of the metal pool. The SiO(g) concentration decreases considerably above the SiC-particle due to reaction 1.6 as shown on the left in Figure 5.19. Large amounts of SiO(g) is produced by the two Si02-particles to the right of this SiC- 174 CHAPTER 5. RESULTS AND DISCUSSION. particle (number 2 and 3 from the left). The SiO(^-concentration is therefore high also above the downstream SzC-particle (number 4 from the left) and considerable conversion takes place at its upper surface. Figure 5.20 shows the gas composition for the entire crater cavity 30 minutes after starting the simulation. The high mass fractions of SiO(g) just above the outermost particles in the metal pool are caused by low flow velocities in this area in combination with the oversimplified treatment of condensation of Si(g) (pages 74-77). The basic problem is that the amount of silicon that ’’condenses ” is independent of the flow of Si(g) to the surface. The flux of Si(g) is instead calculated from a predefined distribution to the different par ­ ticles and from the local temperature where it condenses (page 74). Considerable amounts of silicon may therefore condense even when the gas velocities close to the surface are almost zero. For SiC^-particles, some of this silicon reacts and produces SiO(g). This SiO(g) is added to the gas phase next to the SiOa-surface and is transported away mainly by diffusion since the gas is almost stagnant. The calculated SiO{g) concentrations are therefore unreasonably high in these areas. The algorithm for condensation should be changed in possible future versions of the model. The obvious solution is to include physical transfer of silicon vapour to the gas phase and to calculate condensation rates from kinetic arguments just as for the other rate expressions. The condensation rates then depend on the flow field, the mass fraction of Si(g) close to the surface and its temperature as it should. It was decided not to address this problem during the present work. No major fundamental changes are necessary to implement this since the code is already designed for handling a third gas species in addition to the two that are already included (SiO(g) and CO(g)). 5.1. SIMULATION NUMBER 1.

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Figure 5.20: Mass fractions of SiO(g) after 30 minutes. Particles 1, 4 and 6 from the left in the metal pool are SiC. The rest are Si02- Same particle types above them in the shaft (inerts in-between). Top left : 0.450-0.700. Highest value (0.700) for the contour extending far into the cavity. Decreasing values towards the SiC-surfaces. Top right : 0.700-0.750. Lowest values (see top left plot). Highest values (see bottom left plot). Bottom left : 0.750- 0.950. Lowest value (0.750) closest to the center of the cavity. Increasing values towards SiOa-surfaces. Bottom right : 0.710-0.715. Lowest and highest values are found by comparing with top right plot. 176 CHAPTER 5. RESULTS AND DISCUSSION. 5.2 Simulation number 2.

Simulation number 1 was carried out with strongly reduced Lorentz forces which gave low flow velocities compared to what the Prescribed Current Distribution model usually gives. The main purpose of simulation number 2 is to show the responses to increasing Lorentz forces. Also, the parameters for the dripping model are changed during the simulation to increase the dripping towards the end. The simulation runs for 1800 time steps of 2 seconds each (an hour). The results are compared with those from simulation number 1. Important changes are discussed.

5.2.1 Input parameters and data.

The default parameter values and data from Table 4.3-4.11 (page 145-146) are applied with the following changes :

• Lorentz force correction factors : CiL,z and Qfz,,r both change from 0.05 to 0.20 for the entire simulation. The Lorentz forces are now 20% of what the Prescribed Current Distri­ bution model normally gives. The velociti es inside the electric arc are expected to increase by a factor of about ^0.20/0.05 = 2.0 compared to simulation number 1. This is a rough estimate since the conditions inside the electric arc are complicated. Pressure effects and changes in the density of the gas also affect the flow field. • Dripping model parameters (Table 4.4, page 145) : Ts is changed from 2140K to 2130K after 40 minutes of elapsed simulation time. AtdriP is changed from 50 to 30s after 40 minutes of elapsed simulation time. These changes give considerably larger dripping in the last 20 minutes.

The initial geometry, temperatures, flow field and gas composition are identical to those for simulation number 1 (section 5.1.1).

5.2.2 Power level, radiation and arc temperatures.

The ohmic production of heat inside the electric arc decreases from 6.61MW at the start of the simulation to 5.53MW at the end. In the same period, the radiative energy loss 5.2. SIMULATION NUMBER 2. 177 from it increases from 3.41 to 3.46MW. The reason for these changes are the same as for simulation 1 (section 5.1.2). The electric arc has responded to the larger Lorentz forces by increasing the average ohmic heating from about 5.7MW to about 6.6MW and by decreasing the radiation loss from the arc from about 4.0MW to 3.4MW. This indicates a lower average temperature level in the electric arc since the geometry of the arc is unchanged (prescribed). This seems reasonable since larger velocities inside the electric arc lead to larger convective heat transfer away from it. The average temperature of the arc therefore decreases under otherwise identical conditions. The situation is, however, more complicated than this as discussed in the following. Decreasing temperature in a given gas volume gives lower radiation losses and larger ohmic heating. The former is obvious since the volumetric radiation density decreases with de­ creasing temperatures. The increase in the ohmic heat production is explained from the equation for the ohmic heating (equation 3.118, page 84) :

qohm,arc = £ + - + - i j? S 0 a a a The current densities (jr, jz and jg) are unchanged since they are independent of the conditions inside the arc in the Prescribed Current Distribution model. The value for the electrical conductivity (cr), on the other hand, decreases with decreasing temperature (Appendix II, page 226). Lower temperature therefore leads to higher ohmic heating. In fact, the major weakness of the Prescribed Current Distribution model is that the current densities are independent of the temperature field. Instead, lower temperature in one area should result in lower current density because the electrical conductivity decreases. More of the current would then pass through areas with higher temperatures. The arc would respond by contracting, and by this establish higher temperatures near the center of the arc. No such automatic regulation of the arc radius is included in the Prescribed Current Distribution model. Figure 5.21 shows the calculated temperatures inside the electric arc after 60 minutes 6 for simulations number 1 and number 2. The temperatures in the outer parts of the arc are lower for simulation 2 except for the area close to the anodic metal pool. The central parts of the arc are warmer for simulation 2. The maximum temperature has increased from 24935K for simulation 1 to 25506K for simulation 2. Higher flow velocities lead to more gas being dragged into the electric arc at the cathode end. This gas is much colder than the gas inside the arc, and it seems reasonable that the temperatures in the outer parts of the arc decreases as a result. This effect is to some

6The temperature distribution inside the electric arc changes little with time. The situations after 60 minutes therefore represent the entire period. 178 CHAPTER 5. RESULTS AND DISCUSSION.

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Figure 5.21: The temperature distribution in the electric arc after 60 minutes of elapsed simulation time. The outermost isoterm corresponds to 7000K and the innermost to 22000K. Left : Simulation number 1. Right : Simulation number 2. extent reduced by lower radiation losses and by larger ohmic heat generation due to the decreasing local temperature as just explained. The net result is lower arc temperatures, larger ohmi c heat generation and lower radiation losses in the outer and upper parts of the electric arc. The ohmic heat generation is strongest close to the electrode since lots of cold gas (giving low u-values) is here entering an area with high current densities. Increased flow velocities increase the convective heat transfer to the downstream areas, and the temperatures close to the center of the arc increase as a result7. This effect is to some extent reduced by larger heat losses from the arc by radiation and by lower ohmic heat generation due to the increasing local temperature. The net effects are :

» Higher temperatures in the central parts of the arc and close to the anode. • Lower temperatures in the upper and outer parts of the arc. • Larger overall ohmic heating in the arc.

7The temperatures dose to the metal pool are dominated by the convection of warm gas from the central parts of the electric arc. The convection is stronger in simulation number 2, and higher temperatures are thus expected dose to the anode even in the outer parts of the arc. 5.2. SIMULATION NUMBER 2. 179

• Lower overall radiation losses from the arc.

It is not obvious that the overall result is larger ohmic heating and lower radiation losses since the opposit effect is found in the central parts of the arc where the temperatures increase. This part do, however, constitute only a small fraction of the total arc volume, and the effect of cooling in the outer parts is stronger. Still higher temperatures are expected in the central parts of the arc if the Lorentz forces are increased further. The temperatures in the outer parts of the arc are likely to decrease further. The ohmic heat generation is likely to increase and the radiation would probably decrease. The temperature distribution, the ohmic heat generation and the radiation losses from the arc also depend strongly on the radius of the electrode spot and the properties of the gas (electrical conductivity, volumetric radiation density etc).

5.2.3 Silicon production.

The same method as applied for simulation number 1 is applied for estimating the average useful energy input from the electric arc and the expected steady state silicon production in the high-temperature zone. The applied input data are :

• Average ohmic heat generation : 6.6MW. • Average radiation loss from the arc: 3.4MW (of which 50% is estimated to be added back to the system as useful energy as described in section 5.1.3). • Power demand : 12MWh/tonne produced silicon (corresponding to a sili­ con recovery of about 85% and a thermal efficiency of about 0.80). • Amount of silicon produced in the high-temperature zone : 65% (the rest being produced by condensation of SiO(g)-rich gas in the upper parts of the furnace shaft according to the reaction 2SiO(g) —> Si(l) 4- SiOz(l)).

The average useful energy input for simulation number 2 then becomes 6.6MW - (0.5 x 3.4MW) = 4.9MW, and the expected silicon production becomes (4.9MW / 0.012MWh/kg) x 0.65 = 265 kg/hour (= 4.42 kg/minute). This rough estimate for the expected production rate applies to a situation where the furnace has reached its steady state production level and not to the transient period starting from a fairly cold furnace as in this particular simulation. The calculated net production of silicon is 231.7 kg, corresponding to an average of 3.86 kg/minute. Figure 5.22 illustrates the accumulated net silicon production. The average net production rate develops qualitatively in almost the same way as for simulation number 1. It is negative at the start of the simulation, but increases rapidly to 180 CHAPTER 5. RESULTS AND DISCUSSION.

250.0 Theoretical production l(4.42kg/min) I

------1—yt ——

I Calculated production

Elapsed simulation time (min.)

Figure 5.22: Calculated and theoretical accumulated net silicon production for simula­ tion number 2.

3.3 kg/min after about 6 minutes as the evaporation of silicon adjusts to the heat supply in the central parts of the metal pool. It increases slowly in the next 15 minutes, and then starts to increase faster as the temperatures in the outer parts of the metal pool and the crater wall increases. The average net production rate is about 4.8 kg/min at the end of the simulation. This is 8.6% above the estimated theoretical steady state production. The calculated production is closer to the theoretical production level for this simulation than for simulation number 1. This is probably because the temperatures in the outer parts of the metal pool and in the crater wall adjust faster to the steady state since the overall energy input to the system is larger. The changes in the parameters for the dripping model after 40 minutes of elapsed simulation time do not seem to have any pronounced effect on the average net production rate. It does, however, increase the frequency of the fluctuations in the production rate around the average level as shown in Figure 5.23. The explanations for these fluctuations are the same as for simulation number 1 (page 154); changes in the local reaction rates induced by changes in the geometry of the particles. The increased frequency of the fluctuations is reasonable since the increased dripping implies more frequent changes in the particle geometry. 5.2. SIMULATION NUMBER 2. 181 Time (min)

Theoretical steady state production rate

Calculated production rate

00 1200 Iteration number

Figure 5.23: Calculated and theoretical instantaneous silicon production rates for simu­ lations number 2.

5.2.4 The flow field.

Figure 5.24 shows the flow fields in the crater cavity for simulation number 1 and simulation number 2 after 30 minutes of elapsed simulation time. The velocities are considerably higher for simulation number 2 since the Lorentz forces are 4 times higher than for simulation number 1. The maximum velocity has increased by a factor of 2.09 (from 520 m/s to 1085 m/s). This is slightly larger than the estimated value of \/4 = 2.0 (page 176). For simulation number 2, the velocities just above the metal pool decreases from about 470 m/s in the innermost parts to about 45 m/s above the innermost particle and to about 10 m/s above the outermost particle. The corresponding values for simulation number 1 were : 230 m/s, 17 m/s and 1-2 m/s. The increased flow velocities result in larger heat- and mass transfer to the reacting surfaces, and thus also to larger chemical conversion.

5.2.5 The geometry of the particles.

Figure 5.25 illustrates the calculated long-term changes in the geometry of the metal pool and the furnace shaft. The crater cavity is seen to expand upwards with time as expected. The dripping is larger 182 CHAPTER 5. RESULTS AND DISCUSSION.

3SIM-?, 30 MINUTES (EIFW Z fluent ^ CCKIMS CF STOW RJCUCN tKlLOOWre/SEC 1 ru#€» i MAX. ■ ?.RSS4?F-0? 1 HlN. ■ -t.S90«3H)l

Figure 5.24: The flow field in the crater cavity after 30 minutes. Top : Simulation number 1. Bottom : Simulation number 2. Left : Stream functions. The flow is counter clockwise in the cavity and upwards through the gas exit channel. Right : Iso-velocity curves in the range 2-38 m/s. The highest velocities are closest to the electric arc. 5.2. SIMULATION NUMBER 2. 183

SIM-2 / JMITIAL STATE SIH-2 / 20 HIHJTES

-wr

'-L

U..

T.

r J.

SIH-2 / 40 WHITES «I0 1 • Z rr 1 fMT - MINUTES • i 1 LULU 1^ CEMI6ICAI OUTLIlt PUH WO ***c Wj)

Figure 5.25: Geometrical changes throughout the simulation. Top left : Initial state. Top right : after 20 minutes. Bottom left : after 40 minutes. Bottom right : after 60 minutes. 184 CHAPTER 5. RESULTS AND DISCUSSION.

than for simulation number 1, and it is again largest close to the outlet gas channel next to the electrode. Large dripping is calculated in the start for the outermost particles in the furnace shaft. The reason is the same as explained previously for simulation number 1. The innermost particles in the metal pool (where most of the chemical conversion takes place) decrease in size for the first part of the simulation. They decrease slower as higher temperatures in the crater wall lead to larger dripping. Only small changes are found for any particle in the metal pool from 20 minutes to 40 minutes. The consumption is almost equal to the dripping in this period. The dripping increases considerably as expected after 40 minutes when changing the parameters of the dripping model. The dripping then becomes larger than the consumption for all particles in the metal pool, and they all expand. Figure 5.26 compares the geometry of the system after 40 minutes for simulations number 1 and 2. The parameters for the dripping model are identical for the two cases until this

X

4J-JLJJLU L[[ Ol SIM-2 / 40 MINUTES 021 ENT • 2 fluent ^ GEOereiCM. 0UIL1IC EVfE • 1 *<** 'Ey) 1 T.4.7SEH3

Figure 5.26: The geometry of the system after 40 minutes. Left : Simulation number 1. Right : Simulation number 2. Parameters are identical except for the Lorentz correction factors. point in time. Any difference between the two cases are therefore caused by changes in the chemical conversion and the temperature levels in the furnace shaft. The crater has opened up more for simulation number 2. This is reasonable since the useful power input is larger in this case. The chemical conversion and dripping is likely to increase due to this. 5.2. SIMULATION NUMBER 2. 185

5.2.6 The temperature distribution.

Figure 5.27 shows the temperatures in and around the crater cavity after 60 minutes of elapsed simulation time for simulations number 1 and number 2.

S1M-I / 60 MINUTES conoes op iBfgmus tcaviw 2.*9itg*0* I HIM. 2.SS0S2£t0i 1 MIN.

Figure 5.27: Isoterms in the range 2500-6000K after 60 minutes. Highest temperatures are closest to the electric arc. Left : Simulation number 1. Right : Simulation number 2.

The temperature just above the innermost particle in the metal pool is about 4400K for simulation number 1 and 4700K for simulation number 2 after 60 minutes of elapsed simulation time. At the same time, the temperature is about 2500K for simulation number 1 and 3700K for simulation number 2 just above the outermost particles in the metal pool. The cooling effect from the gas production in the metal pool is less pronounced for simu­ lation number 2 despite that the gas production is in fact larger. The reason is that the convective gas transport along the metal pool has increased more than the gas production at the interface. The produced gas therefore contributes less to the total mass in these areas for simulation 2. The relative cooling effect is consequently weaker. The isoterms in Figure 5.27 show that the temperatures in the metal pool are generally higher for simulation number 2. This is reasonable since there is a larger overall heat generation in simulation number 2. The temperature of the gas entering the outlet gas channel is higher for simulation number 2. The values are 4100-4200K for simulation number 1 and 4400-4500K for simulation number 2. The reason is believed to be that the increased heat generation leads to higher enthalpy content in the gas and that higher flow velocities make it more difficult for the 186 CHAPTER 5. RESULTS AND DISCUSSION. gas to transfer its energy to the surroundings before leaving the crater cavity. The temperature of the gas and in the metal pool are unreasonably high as discussed in section 5.1.6.

5.2.7 The gas composition.

The chemical conversion follows the same general pattern as discussed for simulation num ­ ber 1 in section 5.1.7. The higher flow velocities in the outermost parts of the metal pool do, however, give a more realistic gas composition in this area than for simulation number 1. This is clearly seen by comparing the results after 40 minutes for both simulations (Figure 5.28).

/ 40 HlHJTES" SIM-1 / 40 WHITES CCNTOUS OP S10 CAS (01»BS10H£SS f CmTflfiS CF SIO CAS tDlieSlCKFSS > MAX. ' 9,807516-01 I HIM, ■ 4,161606-01 .531906-01 1 MIN. * 4.913286-01

Figure 5.28: Mass fractions of SiO(g) after 40 minutes. 0.700-0.850. Particle types are as defined in Figure 4.5 on page 138. Lowest mass fraction (0.700) is represented by the contours nearest to the center of the cavity. The SiO- content increases towards SiOa-particles. Left : Simulation number 1. Eight : Simulation number 2.

The strongly overestimated production of SiO(g) in areas with low flow velocities caused by the oversimplified model for the condensation still gives high SiO((^-concentrations in the outermost parts of the metal pool. However, they are not as extreme as for simulation number 1 (page 174). The maximum mass fraction of SiO(g) is 0.98 for simulation number 1 and 0.85 for simulation number 2. The minimum mass fraction of SiO(g) has increased from 0.42 for simulation 1 to 0.49 for 5.3. SIMULATION NUMBER 3. 187 simulation 2. It is lower in the main parts of the crater cavity for simulation number 2 (0.68 compared to 0.70 for simulation number 1).

5.3 Simulation number 3.

Simulation number 3 demonstrates the effects of decreasing the effective surface area for SiC-particles. The simulation runs for 1800 time steps of 2 seconds each (an hour). The results are compared with those from simulation number 2 and important changes are discussed.

5.3.1 Input parameters and data.

The parameter values and data from simulation number 2 (section 5.2.1) are applied with the following changes :

• Correction factor for SiC-surfaces (Table 4.3, page 145) :

The correction factor for effective surface area of SiC (fs,sic) is changed from 15.0 to 10.0. This leads to lower rates for reaction 1.6 (SiC(s) + SiO(g) -* 2Si(l) + CO(g)), and thus to lower silicon production rates.

The initial geometry, temperatures, flow field and gas composition are identical to those for simulation number 1 and 2 (section 5.1.1).

5.3.2 Power level, radiation and arc temperature.

The conditions inside the electric arc are for all practical purposes identical to those for simulation number 2 (section 5.2.2). This is reasonable since the plasma properties are assumed to be independent of the gas composition. Changes in the reaction rates therefore influence the electric arc only by their effect on the temperature of the gas entering the arc. These changes are, however, so small that no measurable differences in the conditions inside the arc (the flow velocities, temperatures, heat generation and radiation losses) are detected when comparing simulations number 2 and 3. 188 CHAPTER 5. RESULTS AND DISCUSSION.

5.3.3 Silicon production.

As explained in section 5.3.2, the total ohmic heat generation and the radiation are for all practical purposes indentical for simulations number 2 and 3. Thus, the average useful energy input and the expected steady state silicon production for the high-temperature zone are also identical. Their values are 4.9MW and 265 kg/hour (=4.42 kg/minute), respectively. The calculated net production of silicon is 171.5 kg, corresponding to an average of 2.86 kg/minute. The average production rate was 3.86 kg/minute for simulation number 2. The calculated accumulated net silicon production for simulations number 2 and 3 are shown together with the theoretical value in Figure 5.29.

Theoretical production | (4.42 kg/min)

S 200.0 Calculated production Simulation number 2 S, 150.0 Simulationnuinber 3

S 50.0

Elapsed simulation time (min.)

Figure 5.29: Calculated and theoretical accumulated net silicon production for simula­ tions number 2 and 3.

Figure 5.30 shows the instantaneous silicon production rates for simulations number 2 and 3 together with the theoretical steady state production rate. The silicon production develops in the same qualitative way in both cases, but the produc ­ tion is lower for simulation number 3 as expected. 5.3. SIMULATION NUMBER 3. 189 Time (min)

Theoretical steady state production rate

Calculated production rate ' Simulation number 2 ' Simulation number 3

00 1200 Iteration number

Figure 5.30: Calculated and theoretical instantaneous silicon production rates for simu­ lations number 2 and 3.

The silicon production decreases less than the effective surface area for SiC&. The reason is that the driving force for the metal producing reaction (the deviation from equilibrium for reaction 1.6) decreases as the reaction rate increases because the gas transport to and from the surface is limited. The effects of this are discussed in section 4.5.

5.3.4 The flow field.

Figure 5.31 shows the flow fields in the crater cavity for simulations number 2 and 3 after 30 minutes. No significant differences in the flow field are found for the two cases. 8

8 The ratio of the average production rates (2.86/3.86 = 0.74) is higher than the ratio of the correction factors (10.0/15.0 = 0.67). 190 CHAPTER 5. RESULTS AND DISCUSSION.

SIM*? / 30 HIHJTF5 SlM-2 / 30 MINUTES FLUENT, COCOES 0= ST5EAM fUCTICH tKILCCSAfS/SEC ) MAX. ■ 2.6SS12F-0? 1 MIN. • -I.5S043E-01 1.085016+03 I MIN.

30 WHITES FLUENT, ■FLUENT, MX. » 1.bg006+03 t HIM. « 3.18228^08

Figure 5.31: The flow field in the crater cavity after 30 minutes. Top : Simulation number 2. Bottom : Simulation number 3. Left : Stream functions. The flow is counter clockwise in the cavity and upwards through the gas exit channel. Right : Iso-velocity curves in the range 2-38 m/s. The highest velocities are closest to the electric arc. 5.3. SIMULATION NUMBER 3. 191

5.3.5 The geometry of the particles.

Figure 5.32 illustrates the long-term changes in the geometry of the metal pool and the furnace shaft for simulation number 3.

JkL.

L‘-_ 411 J L LU-ILMLM L[JJ

SUfS Z 20 MIMES ctoereiw. ojnofc i 1 LULNIrY^ eni **ec

Figure 5.32: Geometrical changes throughout simulation number 3. Top left : Initial state. Top right : after 20 minutes. Bottom left : after 40 minutes. Bottom right : after 60 minutes. 192 CHAPTER 5. RESULTS AND DISCUSSION.

The shape of the crater wall develops basically as for simulation number 2 (Figure 5.25, page 183). However, the lower chemical reaction leads to slower expansion of the crater cavity as seen from comparing the situations after 40 minutes in Figure 5.33. KEY LL

—L y T

LH U-i L| 4 S1H-3 / 40 N1NJTES tRIBO • Z ri i PLU€ • i 1 LI CE06IBICAL OUTUie T • 4.7SEHJ **2"#

Figure 5.33: The geometry of the system after 40 minutes. Left : Simulation number 2. Right : Simulation number 3.

The SiC-particles in the metal pool (number 1, 4 and 6 from the left in Figure 5.33) are larger (particles number 1 and 4) or equal (particle number 6) for simulation number 3. The chemical conversion for particles 1 and 4 has therefore decreased more than the dripping from the crater wall. The Sz'Oa-particles are, on the other hand, smaller or of equal size for simulation number 3 except for the innermost Si02-particle (number 2 from the left). The chemical conversion is almost identical for these particles in the two cases. The changes are therefore mainly caused by differences in dripping. The chemical conversion is, however, lower for the innermost SzOa-particle in the metal pool for simulation number 3. This is mainly because the SzC'-particIe inside it is larger. The Si(l)/SiOa-interface is therefore smaller for simulation number 3, and the reaction at the inner (left) vertical surface of the SzOg-particle decreases. The reaction is here significant since the temperatures increase towards the center of the metal pool.

5.3.6 The temperature distribution.

Figure 5.34 shows the temperature distributions in and around the crater cavity after 60 minutes for simulations number 2 and 3. No significant differences in the temperatures are found for the two cases. 5.3. SIMULATION NUMBER 3. 193

FLUENT, COncUS CP TBfgATlg KELVIN KAX. ■ "2.'Sa62E+W I MIN. . 1.9337*6+0?

Figure 5.34: Isoterms in the range 2500-6000K after 60 minutes. Highest temperatures are closest to the electric arc. Left : Simulation number 2. Right : Simulation number 3.

5.3.7 The gas composition.

Figure 5.35 shows the mass fractions of SiO(g) after 40 minutes for simulations number 2 and 3. The maximum mass fraction of SiO(g) is 0.85 for simulation number 2 and 0.88 for simu­ lation number 3. The corresponding minimum values are 0.49 for simulation number 2 and 0.55 for simulation number 3. It is higher in the main parts of the crater cavity for simu­ lation number 3 (0.70 compared to 0.68 for simulation number 2). The calculated changes are reasonable since the chemical conversion at gas/SiC-interfaces is lower for simulation number 3. 194 CHAPTER 5. RESULTS AND DISCUSSION.

^LPTjJLU Lf SIM-2 / 40 MiWJTES SIM-3 / <0 MINUTES FLUENT, CCNIDUgS CP S10 CAS loitesitKess )" CENTCURS CP S[Q CAS IOI»6SKH£SS ) I.S3190P-01 I HIM. ■ <.913226-01 MAX. ■ 6.7^5-01 I MIN. « S.4ffing5T

Figure 5.35: Mass fractions of SiO(g) after 40 minutes. 0.700-0.850. Particle types are as defined in Figure 4.5 on page 138. Lowest mass fraction (0.700) is represented by the contours nearest to the center of the cavity. The SiO- content increases towards SiC^-particles. Left : Simulation number 2. Bight : Simulation number 3.

5.4 Simulation number 4.

Simulation number 4 demonstrates the effects of extremely low Lorentz forces. The calculations for this simulation illustrate that :

• the temperature of the crater gas decreases in areas where colder gas are added as a result of chemical reactions. • the flow field is difficult to estimate in areas with low flow velocities. This influences the production rates significantly because the applied expres ­ sions for the reaction rates are sensitive to the flow field. • the algorithm for condensation of Si{g) gives unreasonably large mass fractions for SiO(g) in areas with low flow velocities.

The simulation runs for 1800 time steps of 2 seconds each (an hour). 5.4. SIMULATION NUMBER 4. 195

5.4.1 Input parameters and data.

The parameter values and data from simulation number 1 (section 5.1.1) are applied with the following changes :

• Lorentz force correction factors (Table 4.9, page 146) :

q>l,z and ax, r both change from 0.05 to 0.025. The Lorentz forces are now only 2.5% of what the Prescribed Current Distribution model normally gives. The velocities insid e the electric arc are expected to decrease by a factor of about ^0.05/0.025 « 1.41 compared to simulation number 1. This is a rough estimate since the conditions inside the electric arc are complicated. Pressure effects and changes in the density of the gas also affect the flow field.

The initial geometry, temperatures, flow field and gas composition are identical to those for simulation number 1 (section 5.1.1).

5.4.2 Power level and radiation.

The ohmic production of heat inside the electric arc decreases from 5.47MW at the start of the simulation to 5.43MW at the end. In the same period, the radiative energy loss from it increases from 4.10 to 4.16MW. The reason for these changes are the same as for simulation 1 (section 5.1.2). The electric arc has responded to the lower Lorentz forces by decreasing the average ohmic heating from about 5.7MW to about 5.5MW and by increasing the radiation loss from the arc from about 4.0MW to 4.1MW. These changes are consistent with the discussion of the interaction between the Lorentz forces and the electric arc in section 5.2.2.

5.4.3 Silicon production.

The same method as applied for simulation number 1 is applied for estimating the average useful energy input from the electric arc and the expected steady state silicon production in the high-temperature zone. The applied input data are :

• Average ohmic heat generation : 5.5MW. • Average radiation loss from the arc: 4.1MW (of which 50% is estimated to be added back to the system as useful energy as described in section 5.1.3). • Power demand : 12MWh/tonne produced silicon (corresponding to a sili­ con recovery of about 85% and a thermal efficiency of about 0.80). 196 CHAPTER 5. RESULTS AND DISCUSSION.

• Amount of silicon produced in the high-temperature zone : 65% (the rest being produced by condensation of SiO(g)-rich gas in the upper parts of the furnace shaft according to the reaction 2SiO(g) —> Si(l) + JSriO2(0)-

The average useful energy input for simulation number 4 becomes 5.5MW - (0.5 x 4.1MW) = 3.45MW, and the expected silicon production becomes (3.45MW / 0.012MWh/kg) x 0.65 = 187 kg/hour (= 3.12 kg/minute). This rough estimate for the expected production rate applies to a situation where the furnace has reached its steady state production level and not to the transient period starting from a fairly cold furnace as in this particular simulation. The calculated net production of silicon is 74.5 kg, corresponding to an average of 1.24 kg/minute. This is only 40% of the theoretical steady state production rate. The produc ­ tion rates are about 1.9 kg/minute towards the end (61% of the theoretical value). Figure 5.36 illustrates the accumulated net silicon production.

Theoretical production !

100.0

Calculated production

20.0 40.0 Elapsed simulation time (min.)

Figure 5.36: Calculated and theoretical accumulated net silicon production for simula­ tion number 4.

The production rates are expected to adjust slower to the steady state production rate in this case than for the other simulations because the net useful energy input to the system is lower. It therefore takes longer time to reach the steady state temperature levels. This is only part of the explanation for the calculated low production rates. The low flow velocities are also believed to be part of the explanation. These lead to low mass transfer 5.4. SIMULATION NUMBER 4. 197 to the reacting surfaces. The reaction rate for the metal producing reaction is calculated from equations for mass transfer through a turbulent boundary layer. This is probably a poor method with the low gas velocities induced by the electric arc in this simulation (section 5.4.4). The average production rate increases considerably after a short period with negative production at the start. This is consistent with the results from the other simulations. The average production starts decreasing after about 15 minutes. It continues to decrease for about 15 minutes before increasing again towards the end. This is also seen from the instantaneous production rates shown in Figure 5.37. Time (min)

Theoretical steady state production rate

Calculated production rate

600 1200 Iteration number

Figure 5.37: Calculated and theoretical instantaneous silicon production rates for simu­ lation number 4.

The fluctuations in the instantaneous production rates around the average value are ex ­ plained in section 5.1.3.3. Decreasing production rates from about 15 minutes to about 30 minutes are illogical be ­ cause the flow velocities should change little (since the electric arc is stable) and the tem­ peratures should increase during this period. The geometry of the particles also changes little. An increasing average production rate is therefore expected. The explanation for the decreasing production rates is that the calculations of the flow field are not sufficiently accurate for this simulation (section 5.4.4), and that the chemi­ cal reaction rates are so sensitive to these changes that the chemical conversion changes significantly. 198 CHAPTER 5. RESULTS AND DISCUSSION.

As discussed in section 5.4.4, this problem occurs only for low flow velocities. It can be solved either by applying a huge number of iterations for the first time steps 9 , or by calculating the chemical reaction rates from expressions that are less sensitive to the flow field. The former solution is probably not practical since it gives extremely slow simulations.

5.4.4 The flow field.

Figure 5.38 shows the flow velocities in the crater cavity for simulation number 4 after 20, 30, 40 and 60 minutes. The maximum velocity has decreased by a factor of 1.46 (from 520 m/s to 356 m/s) compared to simulation number 1. This is slightly larger than the estimated value of 1.41 (page 195). More important, the velocities just above the particles in the metal pool decrease consid­ erably throughout the simulation period despite stable conditions in the electric arc. The reason is believed to be :

The initial flow field was calculated with stronger Lorentz forces than applied in simulation number 4. This implies higher initial gas velocities than those corresponding to the new Lorentz correction factors. The velocities should decrease instantaneously when changing the correction factor. However, weak forces act on the gas above the particles in the metal pool in this particular case. A huge number of iterations are therefore required to calculate an accurate flow field in this area10.

No similar problems occur for the other simulations presented in this thesis. For simulation number 1, the reason is that the initial data was calculated with the same Lorentz correction factors as applied for the simulation itself. The flow field was therefore accurate from the start. For the other simulations, the increased Lorentz forces are strong enough to dominate the flow field also in the crater cavity. The flow field therefore adjusts quickly to the changed situation. 9 In the order of 10 000 iterations seems to be necessary to obtain a good estimate for the steady state flow field for simulation number 4 (see section 5.4.4). 1011325 iterations were performed during the first 30 minutes of elapsed simulation time for simulation number 4, and the changes in the flow field were still significant for the average production rates. The rate of convergence for the applied numerical method is discussed in section 4.4 5.4. SIMULATION NUMBER 4. 199

SIM-4 / 30 MINLfTFS coaoesi* wmn woaiuE oenE&sc i fluent ^ MAX. • 3.563535+02 1 MIN. ■ 3.162285-08

ST

V n

.4 SIM-4 / 60 MIHJIES FLUENT^ FLUENT, caucus cf laocm wocinte oeno/sc t K.&uocf+o) 1 Mix.3.is?zse-oa MX. ■ 3.Ss5l3M2 1 MIN. . 3.I6228E-08

Figure 5.38: The flow velocities in the crater cavity (1-9 m/s). The highest velocity (9 m/s) is closest to the electric arc. Top left : after 20 minutes. Top right : after 30 minutes. Bottom left : after 40 minutes. Bottom right : after 60 minutes. 200 CHAPTER 5. RESULTS AND DISCUSSION.

5.4.5 The gas temperature in areas with low flow velocities.

Figure 5.39 shows the gas temperatures above the outermost particles in the metal pool after 50 min utes for simulation number 4.

SIH-* / 50 MINUTES FLUENT, CWT0U3 CF TBfgATUE II3-VIH hax - . z.msxioi I mu. * 1.99973E+03

Figure 5.39: Gas temperatures (2200-3200K) above the outermost particles in the metal pool after 50 minutes for simulation 4. The highest temperature (3200K) corresponds to the contour closest to the center of the crater cavity.

In this area where the gas is almost stagnant the gas temperatures are lower than elsewhere in the crater cavity. Little gas is transported to this area by convection. Considerable amounts of gas is, however, produced by chemical reactions at the gas/FiC^-interfaces as explained in section 5.1.7. The temperature of this gas therefore determines the gas temperatures. This explains the low temperatures in this area.

5.4.6 The gas composition.

Figure 5.40 shows the mass fraction of SiO(g) after 50 minutes for simulation number 4. Unrealistically high mass fractions of SiO(g) are calculated above the outermost particles in the metal pool. As discussed in section 5.1.7, this is caused by low flow velocities in this area (section 5.4.4) in combination with overestimated SiO(g) production caused by the oversimplified algorithm for condensation of Si(g). 5.5. SUMMARY OF THE MAIN RESULTS. 201

SIH-A / SO HIMJIES conggs cpsio cas (0i>e6i(w^ss > 1.000006*00 I MIN. « 3.7MS3E»01

Figure 5.40: Mass fractions of SiO(g) after 50 minutes. The lowest values (0.790 and 0.990, respectively) are represented by the contours nearest to the center of the cavity. The SiO-content increases towards Si02-particles. Left : Values from 0.790 to 0.990. Right : Values from 0.990 to 1.000.

5.5 Summary of the main results.

Results presented earlier in this chapter show that reasonable average production rates are calculated for the high-temperature zone for situations similar to those around one of the electrodes of a full-scale industrial furnace. The changes in the average net silicon production are qualitatively as expected for a transient period starting from a relatively cold furnace. Large correction factors for the surface area of SiC are required to get realistic production rates. This indicates that the total gas/SiC-interface is underestimated in the model or alternatively that the calculated reaction rates at gas/SfC-interfaces are underestimated. The general trend is the same for all simulations; First, there is a short period with net silicon consumption. The net silicon production then increases rapidly before it becomes almost constant for a while. It increases significantly again towards the end. The calculated chemical reaction rates and the heat fluxes to particles are sensitive to local changes in the flow field (see section 5.1.3). This gives irregular dripping (Figure 5.11, page 163) and it sets up fluctuations in the instantaneous production rates around the average value (Figures 5.4, 5.30 and 5.37). The average silicon production rate changes gradually despite the fluctuations in the instantaneous production rates. Simulation number 4 indicates that the calculated silicon production is considerably lower 202 CHAPTER 5. RESULTS AND DISCUSSION. than the theoretical value for particulary low flow velocities close to reacting surfaces. The crater cavity expands upwards with largest expansion in the area close to the outlet gas channel next to the electrode. The expansion is also large in the outermost parts of the furnace shaft in the early stages of the simulations 11. The heat transfer is large in these areas and both dripping and chemical reaction rates become large. Higher temperatures than reasonable are calculated in the gas and in the central parts of the metal pool. The heat fluxes to the central parts of the metal pool are overestimated, and those to the rest of the system are underestimated accordingly. Gas/Si02-interfaces are effectively cooled by the endothermic reaction with condensing silicon. Si(l)/SiOa-interfaces stay close to 2132K due to the same reaction. Gas/SiC- interfaces are cooled only by the less endothermic reaction with SiO(g), and thus become warmer in the model. The DC electric arc generates a strong recirculation zone in the crater cavity. The flow is outwards along the metal pool, upwards along the outer edge of the crater cavity and inwards along the materials in the furnace shaft and underneath the electrode. Some gas also leaves through the outlet gas channel. Strong evaporation is calculated from the pure silicon melt underneath the electrode. This evaporation and the following condensation is essential for the heat transport from the central parts of the metal pool to the rest of the system. Most of the SiO(g) is produced by silicon condensing on SWz-particles. Strongly overestimated condensation is calculated in areas with low flow velocities. This gives too high SiO (^-concentrations close to SiOi-particles in such areas. Simulation number 4 shows that a huge number of iterations is necessary to calculate an accurate flow field in cases with low flow velocities in the outer parts of the crater cavity. Similar problems are not found for the other simulations. 11

11The heat fluxes to the outer areas decrease with time because the gas flow is deflected more towards other areas when the crater cavity expands in these particular simulations (page 160). Chapter 6

Final discussion and conclusion.

6.1 General discussion of the model.

A general discussion of the model is presented in this section. The attention is focused on the most important simplifications and their consequences for the performance of the model.

6.1.1 Effects related to the electric arc.

The electric arc is essential for generating energy in the crater zone and for transporting energy and mass in the system. The heat transfer is especially important because it determines where the endothermic chemical reactions take place and where the dripping is largest. It thus also determines the geometry of the crater cavity. The carbothermic silicon metal process is usually carried out applying AC electric arcs1, while a DC electric arc is assumed in the model.

Heat- and mass transfer. The flow field in the crater cavity changes rapidly both because it reverses frequently in the AC case and because the electric arc continuously drifts with its ends along the electrode and the metal pool/crater wall (section 3.8). This complex flow field is not reproduced accurately by the applied DC arc model. Instead, the DC electric arc generates a strong and well structured recirculation zone in the crater cavity (Figure 5.8, page 159). The flow is outwards along the metal pool, upwards along the outer edge of the crater cavity and inwards along the materials in the furnace shaft and underneath the electrode. Some gas 1DG has been tried as reported by Dosaj, May and Arvidson [3] and Ksinsik [4].

203 204 CHAPTER 6. FINAL DISCUSSION AND CONCLUSION. leaves through the outlet gas channel. The rest circulates in the cavity or is dragged in towards the electric arc. More realistic models, both for moving DC arcs and for spatially fixed or moving AC arcs are too complex for practical applications in the model. Instead, improvements may result from ongoing research as mentioned on page 77 and from calculating more accurate data for transport coefficients and plasma properties as explained in section 3.8.3. It is however not obvious that such extentions improve the model. In fact, the flow field may not be essential for the overall production rates even though the.local rates for the various chemical reactions depend both on the mass transfer and on the heat transfer. The reason is that energy not leaving the high-temperature zone by conduction or with the gas is eventually consumed in chemical reactions there 2. The calculated net production rate is therefore mainly determined by the overall heat supply to the high-temperature zone, and not so much by the detailed flow field. The distribution of energy do, however, influence the temperature distribution in the high-temperature zone, and thus determines where the different reactions take place. The flow field is important for the convective part of this energy transport and also for reaction kinetics. Changes in the flow field may therefore both change the composition of the gas leaving the high-temperature zone and the overall heat transfer to the high-temperature zone. The results presented in chapter 5 indicate that the detailed flow field may not be essential for the overall production rates. Significant fluctuations are calculated in the instantaneous production rates around the average production (Figures 5.4, 5.30 and 5.37)3. However, the average silicon production seems to be influenced little by these fluctuations (Fig­ ures 5.3, 5.29 and 5.36). The reason is believed to be that instantaneous reduction in the conversion leads to reduced heat consumption from the surface. Thus, more energy is later available here or at other surfaces4, and the overall conversion soon increases again. In the model the flow velocities can be reduced by changing the Lorentz forces exerted on the gas inside the electric arc. This gives flow fields more similar to those expected for a moving DC-arc and even also for the AC case. The silicon production seems to adjust faster to the expected rates for simulations with large flow velocities (small reduction in the Lorentz forces). The reason may be that strong flow fields are favourable to the kinetics and/or the heat distribution in the model. The reason may also be that the transient period becomes shorter since the net useful energy input increases for stronger flow fields. Further simulations are required to find this out. As discussed on page 80, the spatially fixed DC electric arc overestimates the heat flux to

2Some of this energy, however, heats the materials during the initial transient period. 3These fluctuations are, as discussed in section 5.1.3.3, mainly caused by changes in the flow field close to SiC-particles in the crater wall. The changes in the flow field are due to changes in the geometry for the particle itself or other nearby particles as described in the same section. 4 The energy transfer from the gas to a given surface also changes with the flow field as explained on page 164. Thus, the energy flux to the nearby locations chances. 6.1. GENERAL DISCUSSION OF THE MODEL. 205 the central parts of the furnace. The applied method for distributing the radiation from the electric arc also overestimates the heat flux to these areas (section 3.8.4). Figure 5.2 on page 152 verifies this. The heat transfer to the other parts of the system is underestimated accordingly. A proposal for how the radiation can be distributed in a more realistic way is described on page 87.

The geometry of the crater cavity. The flow field induced by the applied DC electric arc is quite different from that in the AC case. The convective mass- and heat transfer are likely to be larger in the outer parts of the cavity in the DC case. Thus, the dynamic changes in the crater geometry are probably different. A broader cavity, probably with larger chemical conversion in the outermost, upper parts is expected for the DC case. The calculations presented in chapter 5 seem to be consistent with this. As mentioned in section 5.1.5, strong expansion of the crater cavity closest to the outer furnace wall is calculated for the early stages of the simulations. The outermost particles drip down faster than those in the middle of the shaft. This indicates large heat fluxes to these areas. It is caused mainly by the strong upwards flow of gas in this area. An AC arc would probably not give enhanced dripping here. This makes it dubious to apply crater dimmensions estimated for an AC furnace for simulations with DC electric arcs as in the present work. Later, the situation changes because the applied geometry of the fixed outer inert furnace wall deflects the flow inwards, and thus away from the outermost particles as the cavity expands. The heat flux decreases to the outermost areas of the furnace shaft, and the dripping from these areas decreases as pointed out in section 5.1.5. This decrease would probably not have occured if the outer inert furnace wall had been straight instead of ’’hollow ” as shown in Figure 5.9, page 161. It is dubious to assume an inert outer furnace wall both in the AC case, and especially in the DC case when the convective mass- and heat transfer are considerable to this area5. The work required to include chemical reactions here depends mainly on how the mass of the reacting charge is updated. The following algorithm serves as a first approximation, and requires little extra code since all complicated algorithms are already implemented :

• Represent the vertical part of the crater wall by a particle identical to those representing the materials in the furnace shaft. Its lower edge is level with the surface of the metal pool, and inert materials, SiOz{l) or Si(l) are underneath it.

5Remember that this furnace wall represents the charge between the electrodes or close to the furnace lining as shown in Figure 1.2 on page 5, and not the furnace lining itself. 206 CHAPTER 6. FINAL DISCUSSION AND CONCLUSION.

• Calculate the chemical conversion from the particle applying the same basic algorithms as already applied for the other particles. • Remove the consumed materials from the lower edge of the particle just as for the other particles in the shaft. • Move the particle downwards to allign its lower edge with the metal pool surface. At the same time scale it applying the same algorithms as already applied for the other particles.

Evaporation from the metal pool. In the model the metal pool is pure silicon underneath the electrode. Evaporation of silicon is therefore the only option for consuming the large amounts of energy transferred to this area. This implies that evaporation of silicon is essential both for heat transfer and for production of SiO(g) in the model. It is essential for the heat transfer because the evaporation consumes energy from the metal pool and delivers the same amount of energy elsewhere when it condenses. It is essential for the production of SiO(g) because a considerable fraction of the condensing Si(g) reacts with SiOz- Evaporation becomes even more important since the heat flux from the electric arc to the central parts of the metal pool is overestimated in the model as explained previously. The calculated temperatures for the metal pool surface closest to the electric arc soon stabilises close to the boiling point for silicon as shown in Figure 5.12 on page 166. The local evaporation rate is calculated from the local surface temperature. The total evaporation rate consequently adjusts to an almost constant value as the temperatures in the metal pool gradually adjust to the heat flux from above and the heat conduction in the metal. The rapidly increasing evaporation is believed to account for the sharp increase in silicon production at the start of the simulations. The initial sharp increase is followed by a considerably slower increase as seen from Figures 5.4 (page 155) and 5.30 (page 189) 6. It is belived that the evaporation of silicon has almost reached its steady state level when the sharp increase in the production rate stops. At this point most of the Si(l) is probably formed from SiO(g) produced from condensing Si(g). The increase in silicon production later on is belived to be caused mainly by larger SiO{g) production at Si(l)/SiC^-interfaces down in the metal pool. This increase is driven by the general increase in the temperatures in and close to the particles in the metal pool. The calculations indicate, however, that the evaporation dominates the SiO(g) production throughout the entire simulation period. Evaporation is therefore essential both for the heat transport and the chemical conversion in the model. 6Figure 5.37 on page 197 shows a decrease in the production rates after the initial sharp increase for simulation number 4. The rates then increase again. This is related to inaccuracies in the calculations of the flow field as explained on page 197. 6.1. GENERAL DISCUSSION OF THE MODEL. 207

A metal pool with considerable amounts of Sz02 and SiC (and possibly some C) under ­ neath the electrode is more realistic than a pure silicon melt in the same area. Strongly endothermic reactions between silicon and these materials then consume most of the sup ­ plied energy. The temperatures in the area decrease considerably, and evaporation becomes less important. The overall conditions in the metal pool and the crater wall would also change considerably. New algorithms are required to include transport of materials to the area underneath the electrode. This can be achieved either by moving materials sideways in the metal pool or directly by dripping from the crater wall. Such algorithms should definitely be implemented in future versions of the model. The work required to implement them depends on the complexity in the distribution of materials and on the accuracy demanded in the enthalpy conservation.

6.1.2 Effects related to chemical reactions.

The chemical reactions were to some extent discussed in section 6.1.1. Further effects are discussed in this section.

The silicon producing reaction. The silicon producing reaction is equation 1.6 (SiO(g) + SiC(s) —¥ 2Si(g) + CO(g)). SiO{g) is assumed to be transported to the SiC-surface through a turbulent boundary layer. This transport controls the reaction rate which becomes sensitive to the flow field close to the gas/SiC-interfaces. Low reaction rates are found for low flow velocities (sim­ ulation number 4). The flow field changes significantly when the particles in the crater wall change geometry as illustrated in Figure 5.7, page 158. The method for updating the geometry also leads to changes in the temperature of the particles. The reaction rate changes accordingly. These effects are discussed in section 5.I.3.3. In the simulations presented in chapter 5, each particle contains several computational cells in the radial direction. Splitting each particle in single column particles is likely to give a smoother crater wall. Each column would then be updated separately, and the consumption would be largest for columns exposed to the largest heat- and mass fluxes. Also, the changes for each updating would be less since a single computational cell is removed each time instead of a number of cells. There is no specific reason for avoiding such splitting. The simulations become marginally slower since a larger number of particles are updated. The crater wall would be even smoother if a finer computational grid is applied. Also, the 208 CHAPTER 6. FINAL DISCUSSION AND CONCLUSION. changes in the flow field are smaller when removing small computational cells. Finer grids, however, dramatically increase the time-consumption on each time step. This defines an upper practical limit to the grid size as discussed in section 4.3. Reaction 1.6 is sensitive to the correction factor for SiC’-surfaces. Low production rates are calculated unless large correction factors are applied. This indicates that the surface area for SiC is strongly underestimated in the model. This is pointed out and discussed thoroughly in section 4.5. In the crater wall, the main reason is that the charge consists of numerous individual small and irregular particles. These have considerably larger specific surfaces than the large particles representing them in the model. In the metal pool, the reason is different. A large SiC-bed exposed to the gas probably exists here. This bed has much larger interface to the gas than the particles representing SiC in the model, and the gas/SiC-int erface is consequently underestimated. Increasing the correction factor for SiC-surfaces partly compensates for this. However, limitations in mass- and energy transport prevents this correction from being fully efficient as explained in section 4.5. Densities corresponding to compact SiC and SiOz are applied in the simulations presented in chapter 5. The bulk density for both materials are lower than these values. The deviation is especially large for the relatively small and porous FiC'-particles in the furnace shaft. Reducing the applied densities therefore seems reasonable, and the density for SiC should be reduced comparatively more than for SiOz- This increases the volume fraction of SiC relative to SiOz both in the furnace shaft and in the crater wall. It gives more realistic ratios between the volumes for these materials and between their specific surface areas. The mass- and energy transfer also become more realistic. Changing the densities requires no additional code and should be made to some extent. The correction factor for SiC is still, however, important to compensate for the highly irregular and porous SiC-particles in the shaft and for the possible SiC-bed in the metal pool.

The temperature for reacting materials. The temperatures of reacting materials are determined by the heat flux to their surfaces (including the contribution from condensing Si(g)), the conductive heat transport through them and the consumption of energy from them due to endothermic chemical reactions. As mentioned previously, condensation of silicon is essential for the heat transport in the model and contributes with a large fraction of the heat flux to the reacting materials. SiOa-particles are effectively cooled by the endothermic reaction with silicon both at gas/SiOz-hiterfaces and at Si(l)/SiOz(l)-iateTiaces, while SiC-particles are found to be warmer (Figure 5.15, page 170). The temperature difference between SiC- and SiOz- particles is likely to decrease if the evaporation of Si(g) decreases. The difference also 6.1. GENERAL DISCUSSION OF THE MODEL. 209 changes if the condensation control parameters (Table 4.6, page 145) or the condensation distribution parameters (Table 4.5, page 145) change. The effective heat conductivities for the materials also affect the temperature distribution in the particles. Further simulations are necessary to determine how sensitive the calculations are to changes in these model parameters.

Condensation of Si(g). The amount of Si(g) that condenses on various gas/solid- and gas/liquid interfaces is calculated without taking into account the flow field or the concentration of Si(g) close to the interface (section 3.5.3). In fact, the concentration of Si(g) is not calculated in the model. The applied algorithm overestimates condensation in areas with low flow velocities. Some condensing silicon reacts with Si02, and the produced SiO{g) is transported slowly away since the gas is almost stagnant. This implies too high SiO(g) concentrations if Si02 is present in such areas (Figure 5.40, page 201). New algorithms should be implemented for condensation in future versions. These should include calculation of silicon vapour in the gas. Expressions analogue to those applied for reaction 1.6 can then be implemented for condensation. Such algorithms require moderate extentions/changes to the existing computer code. Evaporation (and thus also condensation) is believed to be strongly overestimated in the model since the metal pool is pure silicon underneath the electrode (page 206). This implies that considerable amounts of silicon react at gas/Si02-interfaces in the metal pool. Large chemical conversion at these interfaces are reasonable because considerable amounts of Si(l) drip down from the crater wall after being produced by condensation of SiO(g) in the upper parts of the furnace shaft. Some of this may gather on top of Si02 in the metal pool and react there. This transport of Si(l) to the upper surface of Sz"02-particles in the metal pool is excluded in the model. However, the condensing silicon has the same basic effect at gas/Si02-interfaces in the metal pool even though the mass- and heat transfer are different. Thus, overestimating condensation of Si(g) may partly compensate for the missing Si(l) transport from the crater wall to these interfaces.

Reactions between SiC and Si02.

Reaction 1.4 (2Si02(l)+SiC(s) -> 3SiO{g)+CO{g)) is excluded in the model. This reac­ tion may be important, especially in the metal pool (section 3.4.3). Instead, the included reactions 1.5 and 1.6 add up to reaction 1.4. However, this is not equivalent to including reaction 1.4 because no reaction takes place at SiC/Si02-interfaces in the model. These interfaces are therefore not cooled by any chemical consumption, and their temperatures 210 CHAPTER 6. FINAL DISCUSSION AND CONCLUSION.

become unreasonably high. The energy is transported elsewhere in the system where it may be consumed in other chemical reactions as discussed on page 50.

6.1.3 Effects related to the physical structure of the model.

Porous materials in the crater wall. Representing the permeable charge in the furnace shaft by compact materials is rather dubious. This prevents gas from penetrating into the crater wall. The heat exchange between the gas and the crater wall becomes inefficient. The flow field is also different and the effective surface areas for reacting materials are underestimated. The latter is partly compensated for by the correction factors as discussed previously. The heat transfer and the flow field are not corrected for. As mentioned in section 3.9.2, porous cells are available in the FLUENT code. Applying these instead of compact materials probably gives a more realistic situation in the furnace shaft. Also, there will be no need for a separate outlet gas channel. Applying porous cells requires changes in the algorithms for chemical reactions and in other algorithms developed especially for the carbothermic silicon metal process. The overall complexity of these algorithms are not known.

Composite materials. Reacting materials are represented as pure SiC, SiO% or Si in the model. The particles are relatively large. Allowing these particles to contain Si02, SiC and Si (and even other materials) at the same time gives a more realistic charge. It is then necessary to keep track of the composition of each computational cell. Chemical reaction rates for each particle can be calculated from the relative volume fractions for each chemical compound. Including such composite materials in the model probably involves considerable changes to the present code. It should, however, be considered since it gives a more realistic structure of the charge. It also allows the various chemical reactions to proceed simultaneously at the surface of all particles. This may be important to the kinetics since SiO(g) produced by reaction 1.5 is consumed by reaction 1.6 in the same small volume. Thus, the conversion becomes less sensitive to the flow field. It also gives a more homogeneous gas composition compared to the present situation where SiO(g) is added to the gas in some locations, and is consumed in others. Furthermore, it would probably reduce the possible consequences of excluding the direct reaction between SiC and SiO^ (reaction 1.4). Applying composite materials also gives smaller differences in temperature for different particles because the materials are more homogeneous. It is possible to calculate different 6.1. GENERAL DISCUSSION OF THE MODEL. 211

temperatures for SiC and Si02 in the same computational cell. The motivation would be to allow the SiC to be somewhat warmer than the Si02, as may well be the case since these materials are present in the charge as individual lumps. This would, however, require complicated algorithms. Preferably, the introduction of composite materials should be combined with the introduc ­ tion of porous materials.

The structure of the metal pool. Representing the metal pool as a liquid instead of a rigid body is probably a complicated change. The motivation would mainly be to include convective heat transfer, and by this achieve more realistic heat transport. At present, this is compensated for by increasing the conductivity of the materials in the metal pool. It is, however, likely that there exist a large S'iC'-bed in the metal pool. Representing the metal pool as a rigid body is then reasonable. An SiC-bed can be added in the metal pool. A solution involving composite materials as just described is probably preferable. Reaction 1.4 is probably important in a large SiC-bed, and this reaction should definitely be implemented if an SiC-bed is added.

Electric currents in the crater wall and in the metal pool. Heat generation is the most important consequence of electric currents in the crater wall and in the metal pool. These effects can easily be accounted for by introducing source terms in the energy equation for solid materials. Only minor changes to the code are required to implement this provided that the current distribution and electrical conductivity are known. These are, however, difficult to calculate. Rough corrections can easily be included, but excluding these effects probably means little for the overall performance of the model.

6.1.4 Miscellaneous.

Dripping.

The dripping control parameters (Table 4.4, page 145) determine how much Si02 and SiC that drip down from the furnace shaft at various temperatures. These parameters are basically applied for adjusting the dripping to the chemical conversion in the metal pool so that the amount of mass in the metal pool stays reasonable. The dripping model is simple, and materials drip vertically down. This implies that ma­ terials may build up in some areas of the metal pool while the consumption is larger than the dripping in other areas as described for simulation number 1 in section 5.1.5. 212 CHAPTER 6. FINAL DISCUSSION AND CONCLUSION.

Further simulations are required to determine if the overall results are sensitive to dripping. Moderate changes in dripping probably give minor changes in the production rates as long as materials are present in the metal pool. The overall silicon production' then basically depends on the heat fluxes to the high-temperature zone.

6.2 Conclusions .

The main goals for the work were to gain additional knowledge about the high-temperature part of the carbothermic silicon metal process and to develop a prototype dynamical model for this part of the process. These goals have been achieved. A comprehensive matematical/numerical model has been developed for the high-tempera ­ ture zone around one of the electrodes of the full-scale carbothermic silicon metal process. The model is dynamical, and cylindrical symmetry around the vertical axis of the elec­ trode is assumed. The model includes heterogeneous chemical reactions, evaporation and condensation, mass- and heat transfer, turbulent fluid flow, heat generation in DC electric arcs and radiation. The conservation equations for mass, momentum, energy and turbulent quantities are solved numerically by a modified version of the computational fluid dynamics code FLUENT. Materials drip down from the furnace shaft to the metal pool. The most important conclusions are :

1. The model gives reasonable overall silicon production rates when applied to situations similar to those found around the electrodes of a 21MW industrial furnace. 2. Even though the model gives reasonable overall production rates, calcu­ lations have shown that some submodels are too simple to give realistic results. 3. Calculations indicate that the detailed flow field is not essential to the overall production rates. Instead, these are mainly determined by the total heat flux to the high-temperature zone. 4. The local reaction rates are sensitive to the local flow field and the local heat fluxes as demonstrated by their responses to changes in the geom­ etry of the crater wall. This is due to the reaction kinetics applied for reaction 1.6. 5. The spatially fixed DC-electric arc gives unrealistic flow fields and overes ­ timation of the heat flux to the central parts of the metal pool. 6. The distribution of radiation from the electric arc overestimates the heat flux to the central parts of the metal pool. 7. The absence of other materials than Si(l) underneath the electrode gives too high temperatures in this area and too strong evaporation. 6.3. RECOMMENDATIONS. 213

8. Evaporation and condensation dominate both the heat transfer and the production of SiO(g). This is unrealistic, and evaporation/condensation would be less important if SiC and S1O2 were present underneath the electrode. 9. Too high gas temperatures are calculated. The main reasons are believed to be the models for evaporation/condensation, the unrealistic flow field and the absence of SiC and SiOz underneath the electrode. 10. Significantly higher temperatures are calculated for SiC-particles com­ pared to Si02-particles. 11. The applied method for condensation gives unrealistic condensation rates especially in areas with particulary low flow velocities. The reason is that the condensation rate is not linked to the presence of Si(g) in the gas phase or the flow field.

Except for the unrealistic flow field, the listed undesired effects can all be reduced con­ siderably by moderate changes in the computer code. Calculating a realistic flow field is, however, difficult even with advanced models. Further simulations are required to determine the sensitivity of the model to changes in different model parameters. The results show that the model has a great potential for illustrating the calculated sit­ uation. Calculated temperature distributions, flow fields, gas compositions, geometrical changes and production rates are among the data that can be visualised in an informa ­ tive way applying colour graphics. The model may therefore prove useful for educational purposes and research when its major weaknesses are removed. The model is, however, unlikely to be suited for online automatic process control in many years. The calculations are extremely time-consuming even on powerful computers. The overall complexity of the process is formidable, and vital physical data are still unavailable.

6.3 Recommendations.

The long-term goal is to develop a model suitable for educating furnace operators and for conducting research. A number of improvements may be required to reach this goal. It is recommended first to evaluate the present model thoroughly together with experts from the industry. The main goal should be to evaluate the potential of the model, to identify its major weaknesses and to estimate the work required to remove these weaknesses. This work has been started within the scope of this thesis, and the following changes are regarded as the most appropriate : 214 CHAPTER 6. FINAL DISCUSSION AND CONCLUSION.

1. Change the algorithm for dripping so that materials from the crater wall are also moved to the area straight underneath the electrode. 2. Calculate the concentration of silicon vapour in the gas, and calculate the condensation rates from this and the flow field by equations analogue to those applied for reaction 1.6. 3. Change the algorithm for distribution of radiation from the electric arc as proposed on page 87.

It is recommended also to consider the following additional changes :

1. To include the direct reaction between SiC and SiO% (reaction 1.4). 2. To allow some carbon to enter the high-temperature zone unreacted. 3. To allow evaporation of carbon from the electrode and chemical reactions involving the electrode. 4. To change the reaction kinetics. 5. To introduce porous materials in the furnace shaft. 6. To introduce composite materials both in the furnace shaft and in the metal pool. 7. To introduce a large SzC-bed in the metal pool. This can be achieved by a combination of distributing dripping materials underneath the electrode and introducing composite materials.

The model should be reevaluated after implementing some/all of these and other changes which are considered necessary. When doing so, it is recommended to focus attention on the sensitivity of the model to various submodels, model parameters, model geometry and computational grids. New changes and reevaluations should be made to the extent considered necessary. Finally, optimal combinations of the model parameters should be sought, and calculations with finer grids and higher accuracy should be considered. The overall complexity of the model is already formidable. It is recommended to apply simple algorithms until proven inefficient or unsatisfactory, and not to extend the model to include the chemical reactions in the furnace shaft. Animation may be suited for visualising time dependent effects in the calculated results. Successful animations based on results from the FLUENT program have been carried out by L0ken Larsen [64] for simulations of AC electric arcs. Bibliography

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[2] Schei A. and K. Larsen, A Stoichiometric Model of the Ferrosilicon Process. 39th Electric Furnace Conf., Houston, USA, 1981 (Publ. by The Iron and Steel Soc. of AIME). [3] Dosaj V. D., J. B. May and A. N. Arvidson, Direct Current, Closed Furnace Silicon Technology. From "Silicon for the chemical industry II, Loen - Norway, 8-10 June 1994 ”. Eds : H. A. 0ye, H.M. Rong, L. Nygaard, G. Schussler, J. Kr. Tuset., Tapir Forlag, Trondheim, Norway, 1994. pp. 9-24. [4] Ksinsik D., Silicon Metal Production in an Open DC Arc Furnace - is this the way towards cheaper silicon metal production costs ? From ’’Silicon for the Chemical In ­ dustry II, Loen - Norway, 8-10 June 1994 ”. Eds : H. A. 0ye, H.M. Rong, L. Nygaard, G. Schussler, J. Kr. Tuset., Tapir Forlag, Trondheim, Norway, 1994. p. 25. [5] Halvorsen S. A., Mathematical Modelling - An Integrated Part of Elkem’s Contin ­ ued Education for Process Metallurgists. Proceedings from INFACON 7, Trondheim, Norway, 1995, pp. 729-738. [6] JANAF Thermochemical Tables, 3rd. ed. by M. W. Chase et. al., J. Phys. Chem. Ref. Data, Vol. 14, 1985. [7] Kolbeinsen L., Model of the Si-process, Mass Transfer and Chemical Reactions. SIN- TEF report STF34 F86036, Trondheim, 1986. [8] Halvorsen S. A., A. Schei and J. H. Downing, A Unidimensional Dynamic Model for the (Ferro)Silicon Process. Electric Furnace Conference Proceedings, ISS/AIME, Vol. 50, Atlanta, USA, 1992. pp. 45-59. [9] Motzfeldt K., En Vurdering av Ferrosilisiumprosesens Kjemi, med spesielt sikte pa mu- lighetene for a 0ke Silisiumuthyttet. (in Norwegian) Inst. Silikatforskning/ Inst. Uorg. Kjemi, NTH, 1961. pp. 4-5, 16.

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Evaluating the inner intergrals for the view factors. Equation 3.130 on page 89 :

J?(r, tv, z) = yjr2 — 2 rr0cos(u>) + r% + (z — z0)2

The inner integral of Fz. The integral to be evaluated is (equation 3.133 on page 89) :

rdr Iz = (6.1) (r2 — 2rr0cos(tv) +rg + (zz — zq )2)5

Integrals of this form are tabulated by Gradshteyn and Ryzhik [49, page 83, eqn 2.264-6] :

f rdr _ 2(2a + br ) J UHr)~~A-y/u(r) with

U(r) = cr2 + 6r-fa A = 4ac — 62

This is the exact form of equation 6.1 with

R(r,u,zi) = \/U(r)

a = r\ + (zi - z0)2 b = —2r0eos(cv) c = 1 which gives :

2(2a + b - r) 2(2a + 6 • 7"i) 2(2a + 6 • r%) Iz = A.yiTOx) " A-iJufc) 2(2 a + 6 * T"i) 2(2a + 6 * A * jR(ris w, zi) A • i?(r2?a;, z%) APPENDIX I 221

Notice that A = 4ac — 62 = 4(ro + (zi ~ *o) 2) - (—2r0cos(w)) 2 = 4?o(l - cos2(cv)) + 4(zi - z0)2) = Arlsin 2{w)) + 4(z; - z0)2)

2(2o + 6 • r) = 4o + 26 • r = 4(rg + (zi — z0)2) — 4r0rcos(o;) which gives the following final expression for the inner integral of equation 3.133 : 7~o + (zi - z0)2 - nrpcosfa) Iz = rlsin 2{ w) + (za — z0)2 R{ri,u,zi) ro + (zi ~ zo)2 ~ r2r0cos(u>) (6.2) R{r2,u),zx)

The inner integral of Fr. The integral to be evaluated is (equation 3.134 on page 89) : dz

(r2 — 2r1r0cos{w) + r% + (z — z0)2)2

4)

dv 7, - / (6.3) "(2i) (r2 — 2rir0cos(u) + r% + u2)2 where the variable substitution u(z) = z — zo has been used. Integrals of this form are tabulated by Gradshteyn and Ryzhik [49, page 83, eqn 2.264-5] : r dv _ 2(2 co + 6) J Ui{v) A • \fU(v) with U{v) = cv2 + 6v + a A = 4ac — 62 This is the exact form of equation 6.3 with R(ri,w,z) = \JU{v{z)) a = r2 — 2rir0cos(tv) + r%

6 = 0 c — 1 222 APPENDIX I which gives (since A = 4ac — 62 = 4a) :

i «(%) 4 v _ _4 / v(z2) \ It A \y/U(v{z2))

1 f V(z2) v(zi) \ a \B(r%,w,%) R(ruu,zi)J

The following final expression for the inner integral of equation 3.134 is obtained from this :

1 Z2 ZQ zi-zp \ It = (6.4) r\ — 2r1r0cos(tu) + rg R(r1,u,z2) R{ri,w,zi)J APPENDIX II 223 APPENDIX II.

Thermochemical data, transport properties, volumetric radiation densities. All the data presented in appendix II are calculated by the method developed and imple ­ mented by Gu [40] and Gu [45]. These are the data that are applied in the simulations in the present work. The simulation operator can select among the given data for the volumetric radiation densities. The specific heat capacity of the gas is calculated from the relation :

dh Cp=dT

60 mo!e% SiO, 40 moIe% CO

10000K 20000K 30000K Temperature 224 APPENDIX II (W/mK)

conductivity

Thermal

Temperature

0.015 /s) (m coefficient

Diffusion

Temperature APPENDIX II 225

60 mole% SiO, 40 mole% CO

10000K 20000K 30000K Temperature

1.4e-04 60 mole% SiO, 40 mole% CO

1.0e-04

Jd 8.0e-05

6.0e-05

£ 4.0e-05

2.0e-05

0.0e+00 10000K 20000K 30000K Temperature 226

Radiation density (W/m ) Electrical conductivity (S/m) le+04 le+05 le+07 le+08 le+09 le+06 le+10 le+11 1.0e+01 1.0e+03 1.0e+04 1.0e-03 1.0e-02 1.0e-01 OK

•

10000K 10000K 60 mole% Temperature Temperature

Pure

SiO, 20000K 20000K

Argon

40

mole% r„,

=

10cm

CO 30000K 30000K APPENDIX

II APPENDIX

II Radiation density (W/m3) Radiation density (W/m3) le+10 le+11 le+06 le+07 le+08 le+09 le+04 le+05 le+04 le+05 le+06 le+09 le+07 le+08 le+10 le+11 OK

-

10000K 10000K 50 Temperature Temperature

25 mo!e%

mole% 00K 30000K 20000K

Si,

20000K Si,

50

75

mole r„„ r„„ r„H

moiej%

= = = % iOcm

1cm 10cm

Ar 30000K 227 228

Radiation density (W/m ) Radiation density (W/m) le+04 le+05 le+06 le+07 le+09 le+10 le+11 le+04 le+05 le+06 le+07 le+08 le+09 le+10 le+11 OK OK

10000K 10000K Temperature Temperature

00K 30000K 20000K At-Si jure 20000K

gas Silicon

100 mixtures 50

mole% mdle% =

10cm 30000K

Si Si

APPENDIX

II