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Kinetics of lead concentrate oxidation in a stagnant gas reactor Salomon de Friedberg, Adam Maciej 1987

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K I N E T I C S O F L E A D C O N C E N T R A T E O X I D A T I O N I N A S T A G N A N T G A S R E A C T O R by ADAM MACIEJ SALOMON DE FRIEDBERG B.A.Sc, QUEENS UNIVERSITY, 1984 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Metals and Materials Engineering) We accept this thes"is~as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA ^November, 1987 ® ADAM MACIEJ SALOMON DE FRIEDBERG, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia 1956 Main Mall Vancouver, Canada Department V6T 1Y3 Date ? W > /ft? DE-6G/81) ABSTRACT The behaviour of lead concentrate particles oxidizing in a stagnant gas reactor has been examined and a mathematical model which predicts the kinetics of galena particles developed. The effects of oxygen concentration, particle size, furnace temperature and concentrate composition were studied. The results showed that the concentrates all exhibited sharply defined ignition points. The ignition points were found to be strongly dependent on oxygen concentration. Reductions in ignition temperature of up to 100 K were observed when the concentrates were reacted in pure oxygen rather than in air. Iron composition was found to have a similar effect on ignition temperature. The modelling results predict short reaction times for ignited particles (less than 100 milliseconds). Particles which ignited in air attained predicted temperatures in excess of 2600 K. In oxygen, particle temperatures were calculated to be greater than 2800 K. Good agreement was found between experimental results and model predictions. ii TABLE OF CONTENTS ABSTRACT ii LIST OF TABLES vi LIST OF FIGURES vii TABLE OF NOMENCLATURE xi ACKNOWLEDGEMENTS xiv Chapter 1. INTRODUCTION 1 1.1. Smelting of Lead Sulphide: History 1 1.2. Flash Smelting 2 Chapter 2. LITERATURE REVIEW 5 2.1. Oxidation Studies of Sulphide Particles 5 2.2. Oxidation Studies of Lead Sulphide 11 2.3. Summary 11 Chapter 3. OBJECTIVES AND SCOPE 14 Chapter 4. EXPERIMENTAL 16 4.1. Experimental Apparatus: The Stagnant Gas Reactor 16 4.1.1. Furnace and Temperature Control 16 4.1.2. Reaction Tube 19 4.1.3. Feed Apparatus 20 4.1.4. Collector and Gas Supply 20 4.1.5. Observation Slit 21 4.2. Procedure 21 4.2.1. Feed Preparation 23 4.2.2. Streak Photography Tests 23 4.2.3. Mass Loss Experiments 25 4.2.4. Range of Variables Studied 26 4.2.5. Checks Conducted on Operating Conditions 26 4.2.6. Analysis of Feed and Products 29 Chapter 5. EXPERIMENTAL RESULTS 30 5.1. Streak Photographs of Reacting Particles 30 5.2. Mass Loss Tests 36 5.3. Powder Diffraction Patterns 41 5.4. Particle Size Distribution 41 5.5. S.E.M. Photography 46 5.5.1. S.E.M. Photography of Sectioned Particles 46 5.5.2. S.E.M. Photography of Whole Particles 53 Chapter 6. DISCUSSION OF EXPERIMENTAL RESULTS 63 6.1. Ignition Temperature 63 6.2. Effect of Bulk Oxygen Concentration 64 iii 6.3. Effect of Fe + Zn Sulphide 66 6.4. Rate Limiting Processes 74 6.5. Overall Reaction for Galena 75 6.6. Summary of the Reaction Sequence 81 Chapter 7. M A T H E M A T I C A L M O D E L A N D PREDICTIONS 87 7.1. System Under Consideration 87 7.2. Model Formulation: S T A G E I 92 7.2.1. Assumptions 92 7.2.2. Heat Balance 96 7.2.3. Rate of Reaction 98 7.3. Model Formulation: S T A G E II 101 7.3.1. Assumptions 101 7.3.2. The Heat Balance 102 7.3.3. Rate of Reaction 103 7.4. Model Formulation: S T A G E III 106 7.4.1. Assumptions 106 7.4.2. Heat Balance 107 7.4.3. The Mass Balance 107 7.5. Model Formulation: S T A G E IV 108 7.5.1. Assumptions 109 7.5.2. The Heat Balance 109 7.5.3. The Mass Balance 109 7.6. Calculations of Gas and Particle Properties 110 7.6.1. Diffusion Coefficients 110 7.6.2. Viscosities I l l 7.6.3. Thermal Conductivity I l l 7.6.4. Boundary Layer Thickness I l l 7.6.5. PbS Thermal Conductivity and Emissivity 112 7.6.6. Thermodynamic Data 113 7.7. Numerical Solution of the Model 113 7.8. Model Fitting 113 7.9. Typical Model Predictions 114 7.10. The Sensitivity of the Model to the Heat-Transfer Coefficients 120 7.11. Comparison of Model Predictions and Experimental Results ... 123 7.11.1. Model Predictions vs. Streak Photographs 123 7.11.2. Model Predictions vs. Mass Loss Curves 127 7.12. Model Predictions 129 7.12.1. The Effect of Particle Size 129 7.12.2. The Effect of Oxygen Concentration 134 7.12.3. The Effect of Furnace Temperature 134 Chapter 8. Conclusions 137 8.1. Further Work 138 References 139 i v APPENDIX I: Calculation of Particle Velocities 141 APPENDIX II: Biot Number Calculation 145 APPENDIX III: Mathematical Model - Computer Program 147 v LIST OF TABLES Table 4.1: Assays of Brunswick, Sullivan and Galena Concentrates 22 Table 5.1: Ignition Temperatures Observed in Streak Photographs 35 Table 5.2: Transition Temperatures Obtained From Mass Loss 39 Curves Table 5.3: X-Ray Crystallographic Analyes of Feeds and Products 42 Table 6.1: Comparison of Streak Photograph Ignition Temperatures 65 And Mass Loss Curve Transition Temperatures Table 6.2: Enthalpies of Oxidation for ZnS, FeS, FeS 2 and PbS 67 Table 6.3: Comparison of Theoretical and Experimental Final Masses 68 Table 7.1: Galena Feed Size Distribution Mass Fraction 126 vi LIST OF FIGURES Figure 2.1: Shrinking Core Model 6 Figure 4.1: Photograph of Experimental Apparatus 17 Figure 4.2: Experimental Apparatus 18 Figure 4.3: Particle Size Distribution of Feeds 24 Figure 4.4: Temperature Profile of Reactor With Hot Zone at 700 °C 28 Figure 5.1: Streak Photographs of Brunswick Particles Reacting in Air 31 Figure 5.2: Streak Photographs of Sullivan Particles Reacting in Air 32 Figure 5.3: Streak Photographs of Galena Particles Reacting in Air 33 Figure 5.4: Streak Photographs of Galena Particles Reacting in Oxygen 34 Figure 5.5: Loss of Mass From Lead Concentrate Particles Reacting in Air as a Function of Furnace Temperature 37 Figure 5.6: Loss of Mass From Lead Concentrate Particles Reacting in Oxygen as a Function of Furnace Temperature 38 Figure 5.7: Size Comparison Between Brunswick Feed and Particles Reacted in Oxygen 43 Figure 5.8: Size Comparison Between Sullivan Feed and Particles Reacted in Oxygen 44 Figure 5.9: Size Comparison Between Galena Feed and Particles Reacted in Oxygen 45 Figure 5.10: Brunswick Concentrate Reaction Sequence in Air Particle Size: 53-75 Micron, Mag: X400 47 Figure 5.11: Sullivan Concentrate Reaction Sequence in Air Particle Size: 53-75 Micron, Mag: X400 48 V l l Figure 5.12: Galena Concentrate Reaction Sequence in Air; Particle Size: 75-150 Micron, Mag: X200 Figure 5.13: Brunswick Reacted Particles, Concentrate Size: 53-75 Micron, Furnace Atmosphere: Air, Furnace Temperature: 800 °C, Mag: X400 Figure 5.14: Sullivan Reacted Particles, Concentrate Size: 53-75 Micron, Furnace Atmosphere: Air, Furnace Temperature: 800 °C, Mag: X400 Figure 5.15: Galena Reacted Particles, Concentrate Size: 75-150 Micron, Furnace Atmosphere: Air, Furnace Temperature: 800 °C, Mag: X80 Figure 5.16: Brunswick Reacted Particles: X-Ray Maps, Concentrate Size: 53-75 Micron, Furnace Atmosphere: Air, Furnace Temperature: 900 °C, Mag: X400 Figure 5.17: Sullivan Reacted Particles: X-Ray Maps, Concentrate Size: 53-75 Micron, Furnace Atmosphere: Air, Furnace Temperature: 900 °C, Mag: X400 Figure 5.18: Galena Reacted Particles: X-Ray Maps, Concentrate Size: 75-150 Micron, Furnace Atmosphere: Air, Furnace Temperature: 900 °C, Mag: X200 Figure 5.19: Comparison of Whole Particles of Brunswick Feed and Reacted Product, Mag: X400 Figure 5.20: Comparison of Whole Particles of Sullivan Feed and Reacted Product, Mag: X400 Figure 5.21: Comparison of Whole Particles of Galena Feed and Reacted Product, Mag: X200 Figure 5.22: Fume Sample From Mixed Concentrate, Furnace Atmosphere: Air, Furnace Temperature: 800 °C Figure 5.23: Fume Covered Reacted Particle. Furnace Atmosphere: Oxygen, Furnace Temperature: 800 °C Figure 6.1: Equilibrium Phase Diagram of the Fe-S-0 System at 1100 Degrees Kelvin Figure 6.2: Equilibrium Phase Diagram of the Zn-S-0 System at 1100 Degrees Kelvin viii Figure 6.3: Equilibrium Vapour Pressures of Pb, PbS and PbO as 73 Functions of Temperature Figure 6.4: Equilibrium Phase Diagram of the Pb-S-0 System at 76 900 Degrees Kelvin Figure 6.5: Equilibrium Phase Diagram of the Pb-S-0 System at 77 1100 Degrees Kelvin Figure 6.6: Equilibrium Phase Diagram of the Pb-S-0 System at 78 1300 Degrees Kelvin Figure 6.7: Equilibrium Phase Diagram of the Pb-S-0 System at 79 1500 Degrees Kelvin Figure 6.8: Equilibrium Phase Diagram of the Pb-S-0 System at 80 1700 Degrees Kelvin Figure 6.9: Galena Particle Reaction Sequence 82 Figure 6.10: Lead Concentrate (Fe and Zn Bearing) Reaction 83 Sequence Figure 7.1: Stage I - Chemically Controlled Surface Reaction 89 Figure 7.2: Stage II -- Mass-Transfer Controlled Reaction Taking 91 Place at a Well Defined Interface Figure 7.3: Stage III -- Boiling, Expanding and Reacting Particle 93 Figure 7.4: Stage IV - Shrinking, Reacting PbS Gas Cloud 94 Figure 7.5: Fitting of Model Predictions to Experimental Results 115 Figure 7.6: Effect of the Activation Energy on The Mass Loss 116 Curve of a 100 Micron Particle Reacting in Air, Furnace Temperature = 978 K Figure 7.7: Effect of the Pre-Exponential Constant on The Mass 117 Loss Curve of a 100 Micron Particle Reacting in Air, Furnace Temperature = 978 K Figure 7.8: Temperature vs. Time Curve for a 100 Micron 118 Particle Reacting in Air, Furnace Temperature = 978 K Figure 7.9: Mass Loss vs. Time Curve for a 100 Micron Particle 119 Reacting in Air, Furnace Temperature = 978 K ix Figure 7.10: Comparison of Radiative and Convective Heat Transfer 121 Coefficients for a 100 Micron Particle Reacting in Air Figure 7.11: Effect of the Convective Heat Transfer Coefficient on 122 the Mass Loss Curve of a 100 Micron Particle Reacting in Air, Furnace Temperature = 978 K Figure 7.12: Particle Temperature vs. Distance Fallen Through the 124 Reactor for a 100 Micron Particle Reacted in Oxygen Figure 7.13: Particle Temperature vs. Distance Fallen Through the 125 Reactor, Furnace Atmosphere = Oxygen, Furnace Temperature = 978 K Figure 7.14: Comparison of the Mass Loss Curves (Predicted and 128 Experimental) for Particles Reacted in Air and Oxygen Figure 7.15: Effect of Particle Size on Mass Loss Curves, Reaction 130 in Air, Furnace Temperature = 978 K Figure 7.16: Effect of Particle Size on Mass Loss Curves, Reaction 131 in Air, Furnace Temperature = 978 K Figure 7.17: Effect of Particle Size on Particle Temperature, 132 Reaction in Air, Furnace Temperature = 978 K Figure 7.18: Effect of Oxygen Concentration on the Temperature of 135 a 100 Micron Particle, Furnace Temperature = 978 K Figure 7.19: Effect of Oxygen Concentration on the Temperature of 136 a 100 Micron Particle, Furnace Temperature = 978 K x T A B L E O F N O M E N C L A T U R E 2 A = particle surface area (m ) a = exponential constant (EQN(7.9)) Bi = Biot number Cp = gas heat capacity (J/kgK) (EQN(7.5)) Cp(T) = heat capacity of the particle as a function of temperature (J/kgK) C p = total molar concentration at the mean film temperature (moles/m )^ ^ T l = tota* m o * a r concentration at the particle temperature (moles/m^ ) D = particle diameter (m) D» = diffusion coefficient of the gas at mean film temperature T f (m2/s) Ea = activation energy (J/mole) EQN(7.9) « E = evaporation rate of PbS (moles/s) 2 g = gravitational acceleration (m/s ) H = particle enthalpy (J/kg) H Q = particle enthalpy at reference temperature (J/kg) H(T) = particle enthalpy with temperature (H - H 0 ) (J/kg) A H r e a c t = enthalpy of the oxidation reaction (J/mole) A H J u y = enthalpy of sublimation at T (J/kg) h = heat-transfer coefficient for natural convection around a sphere (Watts/m2 K) k = pre-exponential constant (moles/m s) EQN(7.9) kj. = heat conductivity of the furnace gas at the mean film temperature (T + T)/2 = mass-transfer coefficient (m/s) x i kp = PbS thermal conductivity (W/mK) M. = molecular mass of species i (kg/mole) m = particle mass (kg) m = mass diffusion rate (kg/s) mQ V = the overall rate of PbS transport (kg/s) Nu = Nusselt Number « n = molar diffusion rate of species A (moles/s) A n Q V = overall molar rate of PbS transport (moles/s) PbS n . = number of moles of species i P = vapour pressure of PbS at the particle surface (Pa) b P— = bulk pressure of O- (Pa) «2 2 P. = interfacial pressure of species i (Pa) s P n = pressure of 0« at particle surface (Pa) °2 2 P,j, = system pressure in (Pa) P* = equilibrium vapour pressure of PbS (Pa) R = gas constant (J/moleK) Re = Reynolds Number • R = reaction rate of PbS (moles/s) • Rp^g = overall reaction rate for PbS (moles/s) r = radius (m) rg = boundary layer radius (m) r = radius of reaction interface within the particle boundary layer (m) L i Tp = the particle radius (m) Sc = Schmidt Number xii Sh = Sherwood Number T = particle temperature (K) Tp = furnace wall temperature (K) T = mean film temperature = (T g + T)/2 (K) T = furnace gas temperature (K) t = time (seconds) Vj = diffusion volume of species i (m^ ) Vp = particle volume (m^ ) = particle terminal velocity (m/s) X. = mole fraction of species i in the gas phase OC = a correction factor for the fraction of molecules which are able to escape from the particle surface. Under ideal conditions (i.e. a clean surface and a good vacuum, CC = D £ = particle emissivity = kinematic viscosity of the gas (m 2 /s) = gas viscosity (kg/ms) p = PbS particle density (kg/m^ ) Pf 2 = gas density (kg/m 2 4 O* = Stefan Boltzmann constant (W/m K ) xiii ACKNOWLEDGEMENTS I would like to take this opportunity to thank Drs. Keith Brimacombe and Greg Richards for their invaluable guidance and support throughout this project. The efforts of Cominco Laboratories in their assaying of several samples and the cooperation of Brunswick Mining and Smelting were greatly appreciated. The financial assistance of NSERC is gratefully acknowledged. I would also like to thank all the faculty, staff and students of the Department of Metals and Materials Engineering who were kind enough to help me in this endeavour. x i v CHAPTER 1. INTRODUCTION 1.1. SMELTING OF LEAD SULPHIDE: HISTORY Up until the end of the nineteenth century, lead sulphide smelting was conducted primarily in two types of furnaces, the reverberatory and the ore-hearth. In the reverberatory furnace, high grade galena ores were smelted using a two stage 1-3 process which was repeated several times for each charge. In the first stage, crushed ore (<4 or 5 mesh) was spread in a layer three to four inches deep over the hearth of the furnace and heated to between 500 and 600 °C. A roasting step was then carried out which oxidized some of the galena to PbO and PbSO^ . The second stage involved further heating of the charge to 800 °C where unreacted PbS combined with the oxidized products to yield liquid lead and sulphur dioxide. These two stages were then repeated until all of the PbS was used up. This process had several disadvantages. It was slow, difficult to control, and required large quantities of high grade fuel. However, lead losses were relatively low and precious metal (notably silver) recovery was good. In the ore-hearth, the charge was smelted on a bath of liquid lead. The reactions were similar to those in the reverberatory furnace. However, a portion of the PbO was directly reduced by carbon from the fuel. The ore-hearth had a smaller capacity than the reverberatory furnace and poorer metal recoveries but it was easier to run, was more energy efficient and used lower grade fuels. Therefore, in the smelting of ores which did not contain silver, the ore-hearth managed to hold its own. However, both the reverberatory and ore-hearth 1 INTRODUCTION / 2 processes had important weaknesses. Only high grade ores (>58% Pb for the reverberatory and > 65% Pb for the ore-hearth) could be smelted in these furnaces. In addition, impurities such as antimony, arsenic and zinc could not be separated from the lead using these processes. Another smelting technique was required as soon as high purity ores became scarce. And so, near the turn of the century, the lead blast furnace became the mainstay of the industry. The smelting of lead in the blast furnace requires that the galena first be 1-5 roasted to PbO and PbSO^ on a sintering strand. The sinter is then crushed, sized and charged at the top of the furnace along with coke and flux similar to the iron blast furnace. Reduction of most of the metallic oxides is carried out by the carbon and CO. Operating conditions are maintained such that ZnO is not reduced but reports to the slag and is often recovered by zinc slag fuming. The lead produced is known as base bullion and must be further refined as it typically contains small amounts of antimony, arsenic, bismuth, copper, gold, silver and zinc. To this day, lead is primarily smelted using the blast furnace. 1.2. FLASH SMELTING As has been the case in other sectors of the non-ferrous industry, recent economic trends have forced the lead industry to look at new technology. The price of base metals in the last decade have fallen to the point where lead smelting using the blast furnace has become financially unattractive. A cheaper, more efficient and cleaner technique for smelting lead is. needed. One of the technologies being seriously considered as an alternative is flash smelting. Since INTRODUCTION / 3 the mid 1960's, flash smelting has been the technology of choice in the copper 7 industry. The nickel industry also makes extensive use (the INCO process) of g this type of smelting. In this process, fine, dry sulphide concentrates are mixed together with preheated air, oxygen enriched air, or oxygen and blown into a hot furnace. These concentrates then react rapidly with the oxidizing gas resulting in partial or complete oxidation of the sulphides as well as the liberation of large quantities of heat. Flash smelting has several advantages over reverberatory furnaces and blast furnaces. Firstly, the process makes use of the energy which is available from oxidizing sulphides thereby reducing fuel costs. In some cases, the process is autogeneous. Secondly, the off gas produced is rich in SO2, the leading villain in acid rain, which can be easily and efficiently removed as liquid SO 2 or sulphuric acid. Thirdly, flash smelting presents fewer problems with respect to industrial hygiene since harmful fumes are contained in a single, concentrated gas stream. Fourthly, throughputs are high due to the rapid reaction rates. This allows for smaller furnaces and overall space savings. In short, the economic and environmental reasons for choosing flash smelting are compelling. However, even though flash smelting is an established and successful method of producing copper and nickel, lead presents some special problems which require examination. Perhaps the most serious of these problems is the volatility of lead species at typical flash smelting temperatures (flame temperatures are near 1400 °C). Because of this, it is expected that large quantities of lead bearing fume would be produced. This requires extensive recycle facilities. INTRODUCTION / 4 On a fundamental level, very little is known about the behaviour of galena particles under flash smelting conditions. Only a limited amount of work has been done on the oxidation of sulphides in the shaft of a flash furnace. Virtually no work has been done on lead sulphide. In examining any metallurgical process, it is useful to attempt to characterize the process on the basis of fundamental concepts, specifically heat and mass-transfer, chemical kinetics, and thermodynamics. It is for these reasons that this thesis, a preliminary step in the mathematical modelling of the oxidation kinetics of galena at high temperatures, exists. It is hoped that this work will aid in the understanding, successful modelling and implementation of lead flash smelting. C H A P T E R 2. L I T E R A T U R E R E V I E W 2.1. O X I D A T I O N S T U D I E S O F S U L P H I D E P A R T I C L E S Although the body of literature relating to the modelling of the oxidation kinetics of lead sulphide is very small, similar studies have been conducted on other sulphide minerals, specifically sphalerite, pyrite, pyrrhotite and chalcopyrite. 9 In 1975, Fukunaka et al. studied the oxidation kinetics of sphalerite in a fluidized bed reactor. The authors reacted small quantities of 60-100 mesh sphalerite particles at temperatures from 800 to 910 °C. A large fraction of fused alumina particles were used to keep the lead temperature constant. The reaction time of the particles did not exceed 140 seconds. A mathematical model, based on a shrinking core model (see Fig. 2.1), was developed to investigate the rate controlling mechanisms. Chemical kinetics, pore diffusion and gas film mass transfer were examined as the possible rate limiting processes. The model predictions were fitted to the experimental results by adjusting the reaction rate constant. From their calculations, Fukunaka et al. concluded the following: 1. At lower temperatures, the oxidation rate was controlled by both chemical reaction kinetics and gas film mass transfer. 2. At higher temperatures, gas film mass transfer was rate controlling. 3. At no time did diffusion through the product layer present a significant resistance to oxidation. Extensive experimental work on the combustion of pyrite under simulated flash smelting conditions was presented in 1980 and 1981 by Jorgensen.^ ^ The 5 LITERATURE REVIEW / 6 F i g u r e 2 . 1 : S h r i n k i n g Core Model LITERATURE REVIEW / 7 material studied was Mount Morgan Pyrite concentrate (>99% FeS2) r a n&* nS m size from 37 to 105 microns. The concentrate was screened into narrow size fractions and reacted in a laminar-flow furnace. Temperature measurements of the reacting particle streams were made using a two-colour pyrometer. From these experiments, Jorgensen was able to report several important observations concerning ignition temperatures, reaction times, maximum particle temperatures and the characteristics of reacted particles. The author found that the oxidation of pyrite occurred in three distinct stages: 1. Heating to ignition of the particles (600 °C) 2. Decomposition of the remaining pyrite to pyrrhotite (700 °C) 3. Formation of a Fe-S-0 melt together with a rapid rise in particle temperature. Reaction rates were found to be extremely fast. Particles 37-53 microns in size oxidized completely in air in 70 milliseconds. The maximum particle temperatures recorded varied depending on furnace temperature and oxygen pressure but values as high as 2300 °C were observed. Microscopic examination of fully reacted particles revealed the presence of hollow cenospheres of magnetite and hematite. These cenospheres usually contained some particulate debris. 11 12 Jorgensen also produced two mathematical models ' based on heat balances on a pyrite particle and assuming either SO or SOj as the gaseous products of oxidation. The rate limiting step in the models was assumed to be diffusion of oxygen to the particle surface. Model predictions of maximum particle temperatures were generated and compared to the experimental results. The SO. model was found to predict temperatures 500 to 600 °C higher than those LITERATURE REVIEW / 8 observed in the experiments. The SO model predictions gave much better agreement (higher by roughly 100 °C than the experimental temperatures). For this reason, Jorgensen concluded that the primary gaseous product was SO rather than SOj- However, thermodynamics predict that at 2200 °C, the oxygen partial pressure throughout the boundary layer would have to be less than 4 x 10(-5) atm. in order to favour SO formation. In 1984, an attempt at modelling the oxidation of pyrrhotite particles in a 13 laminar-flow furnace was made by. Asaki et al. The oxidation of 51 and 88 micron particles was studied over a wide range of furnace temperatures (424 to 742 °C). The authors developed a fairly sophisticated model, based on the shrinking core concept, which took into account the melting of the unreacted sulphide core inside the oxide shell. As with sphalerite,^  chemical reaction kinetics, mass transport through the oxidized product and boundary layer mass-transfer were included as potential rate determining steps. The pre-exponential rate constant was adjusted to fit the model results to the experimental data. The experiments revealed that pyrrhotite particles posessed a definite ignition temperature (680 °C) above which the temperature rose sharply. Below ignition, the 51 micron particles were seen to oxidize more rapidly than the 88 micron particles. Above the ignition temperature, no difference was found. Examination of reacted particles confirmed that, after ignition, the particles reached temperatures high enough for melting to occur. The model results showed reasonable agreement with experimental observations and also revealed that the ignition temperature LITERATURE REVIEW / 9 was determined by the heat balance. Once the reaction liberated more heat than was lost to the furnace, the particle temperature began to rise sharply resulting in a decrease in the chemical resistance to oxidation (which was reported as being rate controlling up to >—> 1080 °C) and a faster reaction rate. 14 12 15 Jorgensen and Sengit, and Jorgensen, ' respectively, presented studies on chalcopyrite oxidation very similar to the previously mentioned investigation of pyrite.** The authors reported an ignition temperature for chalcopyrite of 480 °C and maximum reaction temperatures ranging from 1930 °C in air to 2300 °C in oxygen. Photomicrographs of sectioned, ignited particles confirmed that melting had taken place. Also, the presence of hollow spheres was noted. 15 The heat balance model developed for pyrite was also applied to chalcopyrite. Jorgensen concluded from his modelling work that the oxidation reaction was heat-transfer controlled. 16 Chaubal and Sohn conducted a study to determine the intrinsic kinetics of chalcopyrite oxidation in the absence of heat and mass transfer effects. The authors concluded from their study of the literature that an "ignition" temperature is not an intrinsic property of the concentrate but rather is subject to the combined influence of heat and mass-transfer in addition to intrinsic reaction kinetics. In order to eliminate heat and mass-transfer effects, the particle temperature had to be controlled and oxygen had to be supplied to the particle surface at a rate greater than that provided in a laminar-flow furnace. In order to achieve these objectives, a stationary-bed reactor was constructed which held LITERATURE REVIEW / 10 the particles suspended in mineral wool, while the oxidizing gas was blown at high velocities through the bed. The study was carried out at temperatures up to 877 °C using Transvaal concentrate which was analysed to be pure chalcopyrite. Arrhenius plots were generated from the experimental results and activation energies calculated. The authors developed a pore-blocking model which considered the slowing of the reaction by the formation of product larger in molar volume than the reactant. By fitting this model to the experimental data, Chaubal and Sohn were able to identify three temperature ranges of interest. Below 600 °C, the pore-blocking model was found to be applicable. Activation energies of 215 kJ/mol and 71 kJ/mol were calculated for temperatures below 481 °C and between 481 °C and 600 °C respectively. Above 600 °C, the reaction was controlled by sulphur vaporization which obeyed power law kinetics and had an activation energy of 208 kJ/mol. The most recent and detailed model of chalcopyrite oxidation under flash smelting 17 conditions is the work of Kim and Themilis. In their model, the effects of a volatile species (in this case labile sulphur), phase transformations, and particle fragmentation are included. It is assumed that the high temperature reaction of liquid sulphide particles is controlled by oxygen transport to the molten sulphide core of the particle through the boundary layer and the oxidized shell. The authors report several important conclusions. Firstly, the calculated maximum particle temperatures exceed 1850 °C. This result agrees reasonably well with 15 the findings of Jorgensen (1930 °C in air). Secondly, the reaction proceeds LITERATURE REVIEW / 11 very slowly at low temperatures where it is totally controlled by chemical kinetics. Thirdly, the model predicts an ignition temperature for chalcopyrite of 380 °C for the 25 micron particles (roughly 100 degrees lower than that 15 reported by Jorgensen ). Fourthly, once the particles have ignited, the temperature soon rises to the point where chalcopyrite begins to decompose and oxidation of labile sulphur becomes the predominant reaction. Finally, the authors attach particular significance to the melting of the sulphide core (assumed to occur at 1230 °C) which changes the reaction mechanism from gas-solid to gas-liquid. Further evolution of sulphur then causes porosity, formation of cenospheres and fragmentation in the case of larger (> 50 um) particles. 2.2. OXIDATION STUDIES OF LEAD SULPHIDE The work done to date regarding the oxidation kinetics of galena particles under flash smelting conditions is sparse. Besides experimental work done by 12 Jorgensen and flash smelting pilot plant triads conducted in the mid-1960's by 18 19 12 Outokumpu Oy, ' very little work has been done on this subject. Jorgensen reported ignition temperatures for galena in air of 700 °C. He also observed the presence of large quantities of fume (primarily PbSO^ ) produced above the ignition temperature during the galena experiments. 2.3. SUMMARY It is obvious that a great deal of work remains to be done before our understanding of the oxidation kinetics of pure metal sulphide, let alone complex L I T E R A T U R E REVIEW / 12 metal sulphide concentrate is complete The problem is complicated by phase transformations, changing reaction control mechanisms, and the presence of volatile species. A research methodology which adequately addresses all of these concerns appears to be mandator} .^ Finally, it is useful to review the basic features of particulate sulphide oxidation briefly, as they are currently understood: 1. At temperatures below the ignition point, the oxidation rates are slow and appear to be controlled by chemical kinetics. In this regime, both shrinking-core and pore-blocking models can be fitted well to the experimental results. 2. The phenomenon of ignition appears to be common to all sulphide concentrates. The ignition temperature varies depending on several factors but can be defined as the temperature at which the heat balance becomes unstable and the heat liberated by the reaction overwhelms heat loss from the particle by convection and radiation. An extremely rapid rise in particle temperature results. 3. Above the ignition temperature, oxidation of sulphide particles becomes exceedingly complex. Changes in rate controlling mechanisms, phase changes, evolution of sulphur (or other volatilte species), formation of cenospheres and fragmentation can all occur and the order of the events is not known. Eventually, mass-transport becomes rate controlling and particle temperatures attain values well above the melting point. The reaction rate during this stage is very rapid. 4. The entire process described above is usually completed in a fraction of a L I T E R A T U R E REVIEW / 13 second. 5. The reaction products are usually small spheres or cenospheres consisting of metal oxide solutions and small quantities of sulphur. The gaseous product is either SO or SO . In some cases, temperatures of reacting particles are high enough to volatize metal compounds and produce fume. CHAPTER 3. OBJECTIVES AND SCOPE From a review of the literature, it is apparent that little is known about the oxidation kinetics of lead concentrates. Since the application of flash smelting technology to lead processing is a relatively recent development, a fundamental understanding of the kinetics involved is essential if this process is to be fully exploited and optimized. Therefore, . it is the objective of this project to experimentally examine the oxidation kinetics of lead concentrates at elevated temperatures and to mathematically model the reaction kinetics of galena particles in air and oxygen. In order to achieve this objective, an approach was chosen involving experiments in a stagnant gas reactor and the development of a mathematical model based on transport phenomena and experimental results. The experimental work carried out entailed the dropping of lead concentrate particles of varying composition through a stagnant, oxidizing atmosphere at different furnace temperatures. The effects of concentrate composition, furnace temperature, and oxygen concentration of the reactor gas were examined. Several studies were performed on the concentrates and reacted products. Particle size analyses and scanning electron micrography were conducted to characterize changes in the shape and sizes of particles undergoing reaction. X-ray powder diffraction patterns were made to identify phase's present before and after oxidation. Photographs of oxidizing particle streams (streak photographs) were 14 O B J E C T I V E S A N D SCOPE / 15 taken to estimate ignition temperatures. Finally, particle mass loss was measured as a function of reactor temperature. The experimental results obtained from these tests were combined to provide a framework for the mathematical model. The modelling study was directed specifically at the oxidation kinetics of pure galena. Although some work has been done on the oxidation kinetics of other metallic sulphides such as chalcopyrite, the high vapour pressures of lead species at process temperatures complicates the modelling of galena. Modelling of the oxidation kinetics of concentrates consisting of more than one sulphide, specifically galena mixed with pyrite . and sphalerite, is far more complex and beyond the scope of this project. Therefore, the mathematical model developed is intended to be a first step towards the modelling of lead concentrate flash smelting in general. Because so little is known about this process, the integration of. the numerous experimental techniques described and mathematical modelling provides information which is both new and fundamental to the understanding of the kinetics of lead concentrate oxidation. CHAPTER 4. EXPERIMENTAL 4.1. EXPERIMENTAL APPARATUS: THE STAGNANT GAS REACTOR The stagnant gas reactor employed in the oxidation experiments is shown in Figs. 4.1 and 4.2. This particular reactor was chosen over a laminar flow or a turbulent flow reactor for two main reasons. Firstly, the stagnant gas conditions eliminated the need for extensive preheating of a gas stream which would be introduced along with the concentrate. This greatly simplified the temperature control and design of the reactor. Secondly, Stokes Law calculations (see Appendix I) revealed that the particles quickly achieved a reproducible terminal _ velocity owing to the small size of the concentrates. This allowed simple calculation of particle residence times. A description of the reactor is given below. 4.1.1. Furnace and Temperature Control The reactor was heated by a clamshell electric furnace with a cylindrical hot zone 100 mm in diameter and 450 mm in length as shown in Fig. 4.2. Heating elements were arranged in a cylindrical pattern with their longitudinal axes running parallel to that of the furnace hot zone. The heating coils were insulated by 100 mm of chrome-magnesite brick. The furnace was split axially into two halves which were hinged in order to 16 EXPERIMENTAL / 17 - Furnace C o n t r o l l e r Reactor Chart Recorder F i g u r e 4.1: Photograph of Experimental Apparatus EXPERIMENTAL / 18 Primary fttd tub* Plug Stcondary fttd tubt Colltctor support Plug Htating tltmtnts Outsidt T / c Sample collector 0 2 supply F i g u r e 4.2: E x p e r i m e n t a l A p p a r a t u s E X P E R I M E N T A L / 19 facilitate opening of the furnace. Both windings of the furnace had a resistance of 15 ohms. The furnace half sections were electrically connected in parallel to a 220V line source which provided a power output in excess of 6 KW. A Wheelco furnace controller, seen in Fig. 4.1, coupled to an unshielded chromel-alumel thermocouple, located between the wall of the reaction tube (detailed in Section 4.1.2) and the heating elements, provided temperature control. The thermocouple was placed in the axial center of the hot zone of the furnace. A second unshielded chromel-alumel thermocouple inserted inside the quartz tube was used to monitor the reactor temperature. This thermocouple was connected to a potentiometer/chart recorder. An unshielded thermocouple was used inside the reaction tube for two reasons. Firstly, it was felt that an unshielded thermocouple would provide a more accurate measurement of the temperatures to which the particles were exposed in the reactor. Secondly, a short response time was desired. The temperatures recorded inside the reactor were consistently 5 to 10 °C lower than those recorded by the control thermocouple. 4.1.2. Reaction Tube The vessel in which the oxidation reactions were effected was a fused quartz tube 71 mm in diameter (LD.) and 850 mm long. As shown in Fig. 4.2, the tube was placed in the furnace so that the top 450 mm was heated by the furnace elements and the remainder extended below the furnace. E X P E R I M E N T A L / 20 4.1.3. Feed Apparatus The feed apparatus, shown in Fig. 4.2, consisted of three parts. The primary-feed tube, located at the top of the reactor, was a 150 mm long pyrex tube which tapered from a 5 mm L D . to a 1 mm L D . The reason for the taper was twofold. Firstly, the narrow opening at the top of the reactor prevented a significant loss of the hot gas thus conserving stagnant gas conditions inside the reactor. Secondly, the concentrate particles were guided into a narrow, confined stream which was easily isolated from the walls of the quartz tube. The second part of the feed assembly was a rubber stopper which mated the two feed tubes together. The final part of the feed assembly was a 200 mm long mullite tube with a 14 mm L D . This tube extended through the insulation at the top of the furnace just into the top and center of the quartz tube. The purpose of the mullite tube was to prevent the relatively low melting pyrex tube from being exposed to the hot zone of the furnace. 4.1.4. Collector and Gas Supply Located at the bottom of the reactor, a 670 mm diameter watch glass mounted on a small, steel three-legged support served as the sample collector. The size of the watch glass was chosen in order to cover most of the quartz tube cross-sectional area while allowing the passage of gas around it. E X P E R I M E N T A L / 21 A large rubber plug sealed the lower half of the reactor. A 5 mm L D . pyrex tube, inserted through the plug, was connected via flexible P V C tubing to an oxygen cylinder to flush the reactor between runs with fresh oxygen. In the case of the experiments carried out in air, the plugs at the top and bottom of the reactor could be removed to allow natural convection to flush the reactor. 4.1.5. Observation Slit The clamshell furnace was left slightly ajar in order to expose a 30 mm wide observation slit over the length of the fused quartz tube to allow the direct observation of reacting particles. The gaps created between the furnace and the -top and bottom of the reaction tube by leaving the furnace ajar in this manner, were filled with FIBERFRAX blanket insulation. A removable wedge of this insulation was also used to close off the observation slit during the heat up of the reactor and during experiments not requiring direct observation. This arrangement can be seen in Fig. 4.1. 4.2. PROCEDURE Experiments were conducted on three feed materials, whose assays are given in Table 4.1, at furnace temperatures ranging between 400 and 900 °C in both air and oxygen atmospheres. The procedures applied in preparing the feed and conducting the experimental work are described below. EXPERIMENTAL / 22 Table 4.1: Assays of Brunswick, S u l l i v a n and Galena Concentrates Concentrate %S %Pb %Fe %Zn %S as SQf. Brunswick (53-75um) 31 .9 27.1 19.5 13.4 1.3 S u l l i v a n (53-75um) 21.5 52.0 13.8 9.2 0.1 Galena (75-l50um) 14.1 84.3 0.6 1.2 <0.1 E X P E R I M E N T A L / 23 4.2.1. Feed Preparation The feed materials used in the experiments were lead concentrates obtained from Brunswick Mining and Smelting and Cominco. These concentrates were all prepared in the following manner: 1. The concentrates were first wet screened (using a number 325 M E S H or 43 micron sieve) in order to remove fines. 2. The coarser fractions were then dried for 4 hours at 150 °C. The drying time and temperature were selected to avoid oxidation of the concentrate. 3. Finally, the concentrates were dry screened in a R O - T A P for 20 minutes in order to obtain the appropriate size fraction. Fig. 4.3 shows the size distributions of the three concentrates. 4.2.2. Streak Photography Tests The first set of experiments carried out on the concentrates involved the photographing of oxidizing streams of lead concentrate particles. Small quantities of concentrate (approximately 0.25g) were hand-fed into the primary feed tube and allowed to fall, under gravity, through the preheated reactor. The furnace temperature was taken to be that recorded by the thermocouple inside the reaction tube. As the concentrate was dropped, the blanket insulation covering the observation slit was removed and a picture taken of the falling, reacting particles. A 35 mm S.L.R. Y A S H I C A FX-3 camera mounted securely on a stand 1.2 m EXPERIMENTAL / 24 HKWSWtCX P U D f GALENA r U O Figure 4.3: P a r t i c l e Size D i s t r i b u t i o n s of Feeds E X P E R I M E N T A L / 25 away from the hot quartz tube was used to take the photographs. The laboratory was darkened to reduce background. light which obscured the image of the igniting particles. Colour slides and film (400 ASA) were taken at an f-s'top of 4 and the exposure time was approximately three seconds. Between test runs, the reactor was flushed with either pressurized oxygen or via natural convection for the experiments performed in air as described earlier. Approximately 10 minutes was then allowed for the gas to reach the desired temperature. 4.2.3. Mass Loss Experiments The second set of experiments was concerned with measuring the mass loss of oxidizing particles. These tests also included the collection of reacted products for further analysis. In these experiments, pre-weighed 0.25g samples of concentrate were hand-fed into the reactor. The oxidized product, which deposited on the watch glass at the bottom of the furnace, was collected and immediately weighed. These particles were then stored in glass vials for later analysis. The same flushing procedures and temperatures as used in the streak photography tests were applied. However, during these experiments, the observation port was kept sealed. During those tests which were conducted at sufficiently high temperatures to cause significant particle reaction, large quantities of white fume were generated. Samples of this fume were collected after the tests during flushing of the reactor E X P E R I M E N T A L / 26 by placing a glass slide in the off-gas at the top of the reactor. The fume, which filled the reaction tube at the end of a test, then slowly deposited on the cold glass. 4.2.4. Range of Variables Studied The oxidation experiments described in this chapter were carried out on all three concentrates described in Section 4.2.1. In each case, runs were conducted over a range of reactor temperatures between 400 and 900 °C. All of the tests were repeated for both air and oxygen atmospheres. Typical samples of feed and products were collected from these experiments and later analysed as described in Section 4.2.6. 4.2.5. Checks Conducted on Operating Conditions Three important assumptions were made concerning the operating conditions under which the test work was performed. Firstly, it was assumed that the temperature measurements taken during the experiments were accurate and consistent. Secondly, the temperature in the heated section of the reaction tube was assumed to be uniform. Thirdly, it was assumed that the atmosphere in the oxygen tests was pure oxygen. In order to check these assumptions, a series of tests was carried out under various furnace operating conditions. In order to check the first assumption, temperature readings were taken, with the observation slit closed, from the outside and inside thermocouples. The outside E X P E R I M E N T A L / 27 thermocouple consistently registered temperatures 5 to 10 degrees higher than those given by the inside thermocouple. The higher reading given outside the quartz tube and nearer the heating elements was expected. The fact that the two separate temperature readings were so close and that the outside thermocouple always registered slightly higher temperatures indicates the temperatures recorded were accurate to within approximately 10 °C. During experiments with an open view slit, the inside temperature did not change by more than 5 °C. In cases where the oxidation reactions were rapid, the thermocouple registered a slight increase in temperature of up to 3 °C. This confirms that the reactor temperature did not change significantly during the tests. In order to check the second assumption, a temperature profile, shown in Fig. 4.4, was taken from top to bottom in the reactor with the reactor hot spot at 700 °C. As shown in Fig. 4.4, for the first 350 mm of the reactor, the temperature varied only 10 °C. Below this point, the temperature dropped off sharply and end effects became pronounced. At the level of the sample collector, the temperature never exceeded 100 °C , even with the reactor at 900 °C. To test the third assumption, a check was made on the leakage of air into the reactor during the pure oxygen tests. Three gas samples were syringed from the top of the reactor and analysed with a gas chromatograph. In all three cases, the nitrogen levels detected were less than 0.5%. EXPERIMENTAL / 28 3QVci9llN33 S33cJ03Q :3dniVa3dW31 EXPERIMENTAL / 29 4.2.6. Analysis of Feed and Products The feed materials and reaction products were analysed as follows: 1. Particle size distributions were determined with a Coulter Counter. 2. X-ray powder patterns were taken on a Phillips X-ray crystallography machine in order to determine the phases present in the particles. 3. S.E.M. photomicrographs were taken, on an HITACHI model S570 S.E.M., of whole particles and particle sections. For the former, particles were glued to glass slides and gold coated. For the latter, the particles were mounted in a liquid quick-set resin. The samples were then polished to a one micron finish and carbon coated. 4. Assays of the feeds and products were performed at Cominco Laboratories in Trail, B.C. Pb, Zn and Fe were measured using X-ray emission spectroscopy. Sulphur was measured using a Leco Sulphur Analyser and SO. was measured by gravimetric wet analysis. CHAPTER 5. EXPERIMENTAL RESULTS 5.1. STREAK PHOTOGRAPHS OF REACTING PARTICLES The photographs in Figs. 5.1 to 5.4 show the appearance of reacting particle streams in air for Brunswick and Sullivan concentrates (Figs. 5.1 and 5.2 respectively) and both air and oxygen for Galena (Figs. 5.3 and 5.4). In each of the figures, the photographs of the reacting streams progress from lower reactor temperatures (a), where there is little or no visible evidence of reaction, to higher temperatures (b), (c) and (d) where bright streaks signify ignition of the concentrates and rapid reaction. The actual temperatures reached by the particles during reaction have been estimated by the mathematical model described later in the thesis. The phenomenon of ignition can be seen in all four sets of photographs. The concentrates appear to reach critical temperatures at which they ignite and begin to react rapidly. A possible mechanism for the ignition is discussed in Chapter 6. Furthermore, the transition from slow reacting to ignited particles occurs over a narrow temperature range, for example, roughly 30 °C in the case of Brunswick concentrate in air. A summary of the ignition temperatures obtained for all three feeds in both air and oxygen is presented in Table 5.1. The three feeds all begin to react at temperatures roughly 100 °C lower in oxygen than in air. The narrow range over which the ignition points of the concentrates appear remains sharply defined. Increased oxygen activity therefore lowers the ignition temperatures of the concentrates. The effects of composition are illustrated by comparing the ignition 30 EXPERIMENTAL RESULTS / 31 F i g u r e 5.1: Streak Photographs of Brunswick P a r t i c l e s R e a c t i n g i n A i r EXPERIMENTAL RESULTS / 32 F i g u r e 5.2: Streak Photographs of S u l l i v a n P a r t i c l e s R e a c t i n g i n A i r EXPERIMENTAL RESULTS / 33 F i g u r e 5.3: Streak Photographs of Galena P a r t i c l e s R e acting i n A i r EXPERIMENTAL RESULTS / 34 F i g u r e 5 . 4 : S t r e a k P h o t o g r a p h s o f G a l e n a P a r t i c l e s R e a c t i n g i n Oxygen EXPERIMENTAL RESULTS / 35 T a b l e 5.1 I g n i t i o n Temperatures Observed i n S t r e a k Photographs C o n c e n t r a t e %Fe I g n i t i o n Temperature ( d e g r e e s C) In A i r In O t B r u n s w i c k 19.5 550 390 S u l l i v a n 13.8 590 520 G a l e n a 0.6 700 610 E X P E R I M E N T A L R E S U L T S / 36 temperatures of Brunswick and Sullivan concentrate and galena with their respective assays in Table 4.1. The Brunswick concentrate ignites at the lowest temperature and the galena at the highest. The major impurities in the Brunswick and Sullivan concentrates are iron and zinc sulphide. The ignition temperatures observed decreased with increasing iron and zinc concentrations. 5.2. MASS LOSS TESTS Figs. 5.5 and 5.6 show the results of the mass loss tests for all three concentrates in air and oxygen respectively. It was found that approximately 2.5% of the mass was lost even at room temperature (collection inefficiency). Therefore, the curves in Figs. 5.5 and 5.6 have been adjusted to account for this loss. These tests were repeated in order to estimate the accuracy of the results. The values obtained were consistent within a range of 4%. There are several features of these curves which should be noted. Firstly, the basic shapes of all the curves are similar and have four distinct parts. In the initial stages of reaction, at low furnace temperatures, the mass lost by the particles is small and follows a roughly linear relation with temperature. The second stage in these curves is a narrow transition zone where the mass loss curves acquire steeper slopes. The transition temperatures determined from the mass loss curves are given in Table 5.2. A comparison of this table and Table 5.1 shows that the transition temperatures coincide quite closely with the ignition temperatures seen in the first set of experiments. The third part of these curves is again linear but of a far steeper slope than that in the first EXPERIMENTAL RESULTS / 37 < OS o z o u u £ (A 5 **» or < Z H -ui z z u ° 5 a o z ° 3 < < O • i o O O o < 4 o o < o < O o o o or o U J o L J or o o o 00 I o SO 1^  o o CN O O c • m 4J U (Q « « 3 in «j « o a — 6 •u tl u H (Q a « u « a •« e U 3 * J h. e (I <M o o e o e o o <0 U <u c J 3 e o (0 u b. in « in in u X < «-> C O — 01 in O J in u 3 CP SSVW Q33J IVNIOIdO JO % 7 T S n A Q • o A A A • o Legend • A . • A BRUNSWICK CONCENTRATE O SULLIVAN CONCENTRATE • GALENA CONCENTRATE o o o o A • A d A , 1 1 1 1 1 — ° — 300 400 500 600 700 800 900 FURNACE TEMPERATURE: DEGREES C F i g u r e 5.6: L o s s o f M a s s P r o m Lead Concentrate P a r t i c l e s R e a c t i n g i n O x y g e n a s a P u n c t i o n of F u r n a c e T e m p e r a t u r e EXPERIMENTAL RESULTS / 39 T a b l e 5.2 T r a n s i t i o n Temperature O b t a i n e d From Mass Loss Curves C o n c e n t r a t e %Fe T r a n s i t i o n Temperature (degrees C) In A i r In 0 2 B r u n s w i c k 19.5 560 400 S u l l i v a n 13.8 600 530 G a l e n a 0.6 710 610 E X P E R I M E N T A L RESULTS / 40 part. This shows evidence of more rapid and complete reaction. The final part of these curves is typified by a levelling off of the mass loss. In the case of higher reactor temperatures, the final mass loss is slightly larger. The very large mass loss represented by parts three and four of these curves cannot be accounted for by loss of sulphur alone. Furthermore, during these experiments, it was noted that large volumes of white fume were generated when the particles lost significant mass. Some of the fume produced reported to the final product. Unfortunately, it was not feasible to separate the reacted particles from this fume. Therefore, the mass loss curves report values which represent minimum losses. The presence of this fume suggests that lead species are being volatilized from the particles. This is consistent with the known high vapour pressure of lead compounds at temperatures in excess of 1300 °C. Analysis of this fume is given later. The mass loss curves also followed the trends observed in the streak photographs. The curves for Brunswick and Sullivan concentrates have transition zones at lower furnace temperatures than those of galena. The higher concentrations of iron and zinc sulfides in these feeds coincides with lower transition temperatures. Higher oxygen partial pressure lowered the transition temperature for all three feeds as seen in Fig. 5.6. The transition zones observed in these figures occurred at furnace temperatures close to the ignition temperatures listed in Table 5.1. EXPERIMENTAL RESULTS / 41 5.3. POWDER DD7FRACTION PATTERNS Table 5.3 summarizes the results of the X-ray powder patterns obtained from the feed concentrates and products. Of interest are the large quantities of lead, iron and zinc oxides in the reacted Brunswick and Sullivan concentrate particles. The presence of these oxides verifies that extensive oxidation has taken place. However, the analysis also revealed the presence of unreacted sulphides in these particles indicating that the reactions did not go to completion. In the case of galena, oxides were not seen in the product although lead sulphate was. The fume collected was found to be pure PbSO^ . Thus, it may be inferred that the particles lost only lead and sulphur during reaction and not iron or zinc. The presence of lead in the fume suggests that the particles attained temperatures high enough to evaporate lead species after they ignited. 5.4. PARTICLE SIZE DISTRIBUTION Figs. 5.7, 5.8 and 5.9 compare the size distributions between feed and reacted particles for Brunswick and Sullivan concentrates and galena respectively. It can be seen that concentrates bearing iron and zinc experienced overall increases in size despite the mass losses sustained (Figs. 5.7 and 5.8). Only the galena particles shrank with increasing mass loss, Fig. 5.9. A possible explanation for this phenomenon is presented later in the thesis. EXPERIMENTAL RESULTS / 42 T a b l e 5.3 X-Ray C r y s t a l l o g r a p h i c A n a l y s e s of Feeds and P r o d u c t s M a t e r i a l P r i n c i p a l Phases P r e s e n t B r u n s w i c k C o n c e n t r a t e PbS, ZnS, FeS, F e S 2 , PbSO^, F e 2 0 3 S u l l i v a n C o n c e n t r a t e PbS, ZnS, FeS, F e S a , PbSO^, F e a 0 3 G a l e n a C o n c e n t r a t e PbS, ZnS Br u n s w i c k R e a c t e d P a r t i c l e s : 900*C PbSO^, PbO, FejjO,, F e j O * , ZnO PbS, ZnS, FeS, F e S z S u l l i v a n R e a c t e d P a r t i c l e s : 900*C PbSO^,, PbO, Fe xOj., F e s 0 4 , Z n 0 PbS, ZnS, FeS, F e S 2 G a l e n a R e a c t e d P a r t i c l e s : 900*C PbS, P b S 0 4 Fume PbS, PbSOj EXPERIMENTAL RESULTS / 43 IT -Lillll! FEED HA •i i J0 7» fr:::::::::::::::-4 • " ' o JO 7. Furnace Temperature » 900 C Figure 5.7: s i z e Comparison Between Brunswick Feed and P a r t i c l e s Reacted i n Oxygen EXPERIMENTAL RESULTS / 44 , I •" • l l J l i i x J i i E j I l i I l l l i ^ l J l l X ^ Furnace Temperature • 900°C F i g u r e 5.8: S i z e Comparison Between S u l l i v a n Peed and P a r t i c l e s Reacted i n Oxygen EXPERIMENTAL RESULTS / 45 5 r " i i > < » F u r n a c e Temperature » 800 C F i g u r e 5 . 9 : S i z e Compar i son Between G a l e n a Feed and P a r t i c l e s R e a c t e d i n Oxygen E X P E R I M E N T A L R E S U L T S / 46 5.5. S.E.M. PHOTOGRAPHY 5.5.1. S.E.M. Photography of Sectioned Particles As previously noted, feed examples as well as samples of reacted products were collected throughout the experiments. These particles were mounted, polished and examined on an HITACHI So70 S.E.M. Typical photomicrographs of the sectioned particles are presented in Figs. 5.10 to 5.18. In Figs. 5.10 to 5.12, reacted particle sections corresponding to the streak photographs shown in Figs. 5.1 to 5.3 are seen. As the reactor temperature increases (from a) to (d) in Figs. 5.10 to 5.12), evidence of high temperature (due to melting) is seen through the . progressive rounding of the particles. The temperatures at which the particles show rounding of edges coincides well (to within 20 °C) with the ignition temperatures and transition temperatures listed in Tables 5.1 and 5.2. The highest temperature photographs show porous particles of reacted Brunswick and Sullivan concentrates. The photomicrographs of galena exhibit no evidence of this. These photomicrographs offer some explanation of the discrepancies observed in the size distributions of the concentrates before and after reaction. As the Brunswick and Sullivan concentrate particles react, they lose mass but maintain their size as they become porous and apparently inflate. Figs. 5.13 to 5.15 show sections of Brunswick and Sullivan concentrate and galena particles, respectively, which have been reacted at furnace temperatures well above the ignition temperatures. The porous appearance of the Brunswick EXPERIMENTAL RESULTS / 47 F i g u r e 5.10 Brunswick Concentrate Reaction Sequence i n A i r P a r t i c l e S i z e : 5 3 - 7 5 micron Mag: X400 EXPERIMENTAL RESULTS / 48 - N A (c) Furnace Temp 610°C (d) Furnace Temp » 620°C Mfl& x ZoO F i g u r e 5.11 S u l l i v a n Concentrate Reaction Sequence i n A i r P a r t i c l e S i z e : 53-75 micron Mag: X400 EXPERIMENTAL RESULTS / 49 F i g u r e 5.12 Galena Concentrate Reaction Sequence i n A i r P a r t i c l e S i z e : 75-150 micron Mag: X200 EXPERIMENTAL RESULTS / 50 F i g u r e 5.13 Brunswick Reacted P a r t i c l e s Concentrate S i z e : 53-75 micron Furnace Atmosphere: A i r Furnace Temperature: 800°C Mag: X400 EXPERIMENTAL RESULTS / 51 S u l l i v a n Reacted P a r t i c l e s Concentrate S i z e : 53-75 micron Furnace Atmosphere: A i r Furnace Temperature: 800°C Mag: X400 EXPERIMENTAL RESULTS / 52 F i g u r e 5.15 Galena Reacted P a r t i c l e s Concentrate S i z e : 75-150 micron Furnace Atmosphere: A i r Furnace Temperature: 800°C Mag: X80 E X P E R I M E N T A L RESULTS / 53 and Sullivan products is even more pronounced in these photomicrographs. Evidence of reaction and melting is clear since the particles have taken oh spherical shapes. In addition, the reacted Bruncwick and Sullivan particles are obviously hollow. The creation of these hollow spheres, or cenospheres, appears to be unique to the iron and zinc bearing particles which have achieved ignition. Figs. 5.16 to 5.18 show element traces of Pb-S, Fe and Zn, obtained from X-ray back scattered patterns using the ED A X Image Analyser on the S .E.M. , in concentrate particles reacted at high furnace temperatures. Lead and sulphur are reported as a single element trace in these figures because their scattering peaks coincide with one another making accurate separation of the two difficult. In every case, particles high in Fe, Zn and Pb + S formed cenospheres or porous products whereas those containing only one of the three did not. Thus, the formation of cenospheres appears to be linked to particles of mixed sulphide composition. 5.5.2. S.E.M. Photography of Whole Particles Whole particles of feed and product were also mounted and examined under the HITACHI S570 S . E . M . Figs. 5.19 to 5.21 show the difference in size, shape and-surface characteristics between feed and reacted particles of Brunswick, Sullivan and galena respectively. In the case of Brunswick and Sullivan particles, Figs. 5.19 to 5.20, the feed particles are angular, approximately 50 to 75 microns in size, and have smooth surfaces. The reacted particles appear as spheres exhibiting a rougher surface. Those spheres are about the same size as the feed EXPERIMENTAL RESULTS / 54 Reacted P a r t i c l e s P b , S Fe Zn F i g u r e 5.16 Brunswick Reacted P a r t i c l e s : X-Ray Maps Concentrate S i z e : 53-75 micron Furnace Atmosphere: A i r Furnace Temperature: 900 6C Mag: X400 EXPERIMENTAL RESULTS / 55 F e Zn F i g u r e 5 . 1 7 S u l l i v a n R e a c t e d P a r t i c l e s : X - R a y M a p s C o n c e n t r a t e S i z e : 5 3 - 7 5 m i c r o n F u r n a c e A t m o s p h e r e : A i r F u r n a c e T e m p e r a t u r e : 9 0 0 ° C M a g : X 4 0 0 EXPERIMENTAL RESULTS / 56 Fe Zn F i g u r e 5.18 Galena Reacted P a r t i c l e s : X-Ray Maps Concentrate S i z e : 75-150 micron Furnace Atmosphere: A i r Furnace Temperature: 800°C Mag: X200 EXPERIMENTAL RESULTS / 57 F i g u r e 5.19 Comparison of Whole P a r t i c l e s of Brunswick Feed and Reacted Product Mag: X400 EXPERIMENTAL RESULTS / 58 (b) Furnace Atmosphere: A i r Furnace Temperature: 800°C F i g u r e 5.20 Comparison of Whole P a r t i c l e s of S u l l i v a n Feed and Reacted P r o d u c t Mag: Z400 EXPERIMENTAL RESULTS / 59 particles. In the case of galena, Fig. 5.21, the feed particles are cubic, range in size between 75 and 150 microns on a side, and have very smooth faces. The reacted particles are again spheres with rough surfaces but they are noticably smaller than the feed particles. Fig. 5.22 shows particles of fume which are small (less than 2 microns in diameter), roughly spherical and, as previously mentioned, pure PbSO^ . A closer examination of Figs. 5.19 to 5.21 reveals that the deposits roughening the surfaces of the reacted particles appear to be small spheres which look very much like the fume. This is shown more clearly in Fig. 5.23 which is a higher magnification photomicrograph of a reacted particle with a surface heavily covered by these deposits (a). In (b), a closeup of these deposits clearly reveals their similarity to the fume particles in Fig. 5.22. EXPERIMENTAL RESULTS / 60 (b) Furnace Atmosphere: A i r Furnace Temperature: 800°C F i g u r e 5.21 Comparison of Whole P a r t i c l e s of Galena Feed and Reacted Product Mag: Z200 EXPERIMENTAL RESULTS / 61 F i g u r e 5.22 Fume Sample From Mixed Concentrate Furnace Atmosphere: A i r Furnace Temperature: 800°C EXPERIMENTAL RESULTS / 62 (b) Closeup of P a r t i c l e Surface i n (a) Mag: X10000 F i g u r e 5.23 Fume Covered Reacted P a r t i c l e Furnace Atmosphere: Oxygen Furnace Temperature: 800°C CHAPTER 6. DISCUSSION OF EXPERIMENTAL RESULTS The experimental results presented in Chapter 5 reveal five important characteristics of the oxidation kinetics of lead concentrates. These characteristics are: 1. The ignition temperature. 2. The effect of bulk oxygen concentration. 3. The effect of Fe + Zn sulphides. 4. The rate limiting processes. 5. The overall reaction for PbS. 6.1. IGNITION TEMPERATURE The presence of a sharp transition from slowly reacting to rapidly reacting particles is clearly shown in Figs. 5.1 to 5.4, 5.10 to 5.12 and the mass loss curves in Figs. 5.5 to 5.6. The streak photographs in Figs. 5.1 to 5.4 span a range of only 30 °C over which the particle streams change from little reaction to bright flares which illuminate the entire reaction tube. A similar transition is observed in Figs. 5.10 to 5.12 from the progressive rounding (evidence of melting) of the particles over the same temperature range. The mass loss curves (Figs. 5.5 and 5.6) show transition points which coincide with the observed ignition temperatures. Below the ignition point, mass loss progresses very slowly with reactor temperature. Above that point, there is a dramatic increase in the slopes of the curves and very rapid mass loss. The fact that the amount of mass loss is greater than can be accounted for by sulphur oxidation alone 63 DISCUSSION OF E X P E R I M E N T A L RESULTS / 64 coupled with the generation of the lead sulphate fume suggests that, over a very narrow range in furnace temperature, a very large change in particle temperature takes place. Once ignited, particles attain temperatures high enough to volatilize lead species. Table 6.1 lists the ignition temperatures discussed above. As is shown in this table, the results all agree within approximately 10 °C and are therefore inside the confidence level for temperature given in Section 4.2. Thus, it may be concluded that the concentrate particles have flashpoints or ignition temperatures which are sharply defined. 6.2. EFFECT OF BULK OXYGEN CONCENTRATION The second characteristic of the oxidation kinetics involves the effects of bulk oxygen concentration on the reaction. In every case, increased oxygen partial pressure resulted in a drop in the observed ignition point. A comparison of the air and oxygen experiments clearly demonstrates that the kinetics were accelerated at lower temperatures when pure oxygen was used. This would be expected if either oxygen mass-transfer or chemical kinetics were rate controlling (provided the reaction was not zero order with respect to oxygen). A higher oxygen concentration would increase the driving force for the reaction and for oxygen transport to the particle and an increase in reaction rate would result. DISCUSSION OF EXPERIMENTAL RESULTS / 65 Table 6.1 Comparison of Streak Photograph I g n i t i o n Temperatures and Mass Loss Curve T r a n s i t i o n Temperatures Concentrate %Fe I g n i t i o n Temperature (degrees C) T r a n s i t i o n Temperature (degrees C) In A i r In 0 2 In A i r In 0 2 Brunswick 19.5 550 390 560 400 S u l l i v a n 13.8 590 520 600 530 Galena 0.6 700 610 710 610 DISCUSSION O F E X P E R I M E N T A L R E S U L T S / 66 6.3. EFFECT OF FE + ZN SULPHIDE A lowering of ignition temperature can be seen in particles containing significant quantities of iron and zinc sulphides (Table 5.1). The Brunswick and Sullivan particles ignite at temperatures roughly 160 and 100 °C lower respectively than those for pure galena. Work done by Jorgensen** has shown that pyrite particles ignite at lower temperatures than galena or sphalerite particles and also oxidize to higher temperatures. Jorgensen quoted ignition temperatures, in air, of 600, 700 and 1200 °C for pyrite, galena and sphalerite respectively. Also, as seen in Table 6.2, the enthalpies of reaction for FeS and FeS2 are greater than that of ZnS. Furthermore, the Brunswick and Sullivan concentrates contained significantly larger quantities of iron than zinc (Table 4.1). All this strongly suggests that the presence of iron (rather than zinc) in the Brunswick and Sullivan concentrates is responsible for the lower ignition temperatures observed for these concentrates. As mentioned previously, the rapid loss of mass seen in Figs. 5.5 and 5.6 coincided with the generation of large quantities of pure lead sulphate fume. This indicates that only lead species were volatilized from the particles. Furthermore, it offers some explanation as to why galena particles continued to lose mass at higher furnace temperatures whereas the mass loss for Brunswick and Sullivan particles ceased. The mass loss curves for Brunswick and Sullivan concentrates reacted in air levelled off at close to the theoretical mass of iron oxide plus zinc oxide based on initial assays (as shown in Table 6.3). The theoretical mass values in Table 6.3 were obtained by assuming that the reacted concentrate DISCUSSION OF EXPERIMENTAL RESULTS / T a b l e 6.2 E n t h a l p i e s of O x i d a t i o n F or ZnS, FeS, F e S 2 and PbS C h e m i c a l E q u a t i o n E n t h a l p y A H Z 9 * k j / k g of M e t a l ZnS + 1.5(O a) -> ZnO + S O z -6781 2(FeS) + 2 . 5 ( 0 2 ) -> F e a 0 3 + 2(SO z) -10986 3(FeS) + 5 ( 0 2 ) -> F e 3 0 4 + 3 ( S 0 2 ) -10300 2 ( F e S a ) + 5 . 5 ( 0 2 ) -> F e 2 0 3 + 4 ( S 0 2 ) -14833 3 ( F e S z ) + 8 ( 0 2 ) -> F e 3 0 4 + 6 ( S 0 2 ) -14121 PbS + 2 ( 0 2 ) -> PbSO^ -3966 PbS + 1.5(0 2) -> PbO + S 0 2 -2016 DISCUSSION OF EXPERIMENTAL RESULTS / 68 T a b l e 6.3 Comparison of T h e o r e t i c a l and E x p e r i m e n t a l F i n a l Masses C o n c e n t r a t e %Fe %Zn T h e o r e t i c a l Remaining Mass (%) Mass Curve L e v e l l i n g O f f P o i n t (%) C o n v e r t e d F e z 0 3 + ZnO C o n v e r t e d F ejO^ + ZnO I n A i r I n 0 2 Brunswick 19.5 13.4 44.6 43.6 46.5 24.5 S u l l i v a n 13.8 9.2 31.2 30.5 39.5 30.0 DISCUSSION OF EXPERIMENTAL RESULTS / 69 particles lost all of their lead and sulphur but no iron or zinc. Next, the iron and zinc were converted into either FejO^ or FegO^ and ZnO. These oxides were chosen as final products on the basis of their presence in the reacted particles (the powder diffraction pattern results in Table 5.3) and the predominance area diagrams, Figs. 6.1 and 6.2. These diagrams of Fe-S-0 and Zn-S-0 systems show that, for a typical reactor temperature of 1100 K, the expected products for high oxygen and low SOj pressures are FejO^ and ZnO. Fe^O^ formation is likely at very low oxygen pressures ( < 10 ^ atm). As was stated above, the levelling off values in the mass loss curves for Brunswick and Sullivan concentrates reacted in air (Fig. 5.5) show good agreement with the theoretical masses. However, the Brunswick mass loss curve for the oxygen experiments (Fig. 5.6) levels off at values lower than those expected. This suggests several possibilities. Firstly, there could be experimental error in either the collection of the reacted particles and fume or in their analysis. This is unlikely considering the number of times these experiments were repeated and the consistency of the results. Secondly, the values in the oxygen mass loss curves agree very well with the sum of the initial assays of iron and zinc. This would seem to indicate that none of the iron or zinc oxidized. Again, this possibility is unlikely since the X-ray diffraction patterns showed significant quantities of all three oxides under scrutiny. Thirdly, it is possible that iron and zinc was lost from the particles but did not report to the fume. This could be the case if the particles reacted violently enough to explode and some of the material deposited on the walls of the reaction tube. There are three pieces of evidence which support this hypothesis. First, a visual examination of the quartz reaction tube revealed that, over the course of several experiments, large cn CO UJ o_ UJ o >-X o o -10 - 9 -8 - 7 -6 - 5 - 4 - 3 - 2 -1 LOG OF SULPHUR DIOXIDE PRESSURE Figure 6.1 E q u i l i b r i u m Phase Diagram of the Fe-S-0 System at 1100 Degrees K e l v i n o I—I CO O d CO CO H H O O w X Tl M SO H ?o M co t-1 H CO o to CO L J a. o >-X o o o -1 - 2 -- 3 -- 4 -- 5 -- 6 -- 7 -- 8 --9 -10 ZnO(s) -10 - 9 -8 - 7 -6 - 5 -4 - 3 -2 -1 LOG OF SULPHUR DIOXIDE PRESSURE Figure 6.2 E q u i l i b r i u m Phase Diagram of the Zn-S-0 Syst at 1100 Degrees Ke l v i n CO o C CO co I—I O O *n w ><! M f0 > f"1 to M co a H C O em DISCUSSION OF E X P E R I M E N T A L R E S U L T S / 72 amounts of fume and many particles had indeed deposited on the walls. These particles may represent the unaccounted for mass. Second, the size distributions of the concentrate feeds and reacted particles (Figs. 5.7 and 5.8) show a slight increase in the quantities of small particles (between 30 and 40 microns in diameter) in the products. This occured despite an overall increase in the particle size distribution and supports the theory of exploding particles. Finally, an 17 oxidation study by Kim and Themilis on chalcopyrite particles yielded comparable results. The authors reported the formation of cenospheres similar to those seen in Figs. 5.13 and 5.14. It was also found that larger particles (greater than 50 microns in diameter) tended to explode and create smaller particles approximately 25 microns in size. The fragmentation of the particles was apparently caused by gas formation within the molten core. As can be seen in Fig. 6.3, PbS has a high vapour pressure (greater than 0.1 atm) above its melting point (1392 K) and may be responsible for similar gas formation and fragmentation of Brunswick concentrate. It is likely therefore, that the discrepencies seen in Table 6.3 are due to exploding particles depositing some of their mass onto the reactor wall. Apparently, this was not the case for Brunswick concentrate reacted in air. Although cenospheres were formed, the particles did not seem to explode. A possible explanation for this is that these particles simply did not reach temperatures high enough to generate the quantities of gas required for fragmentation. The slower reaction rate (and therefore heat generation rate) resulting from a lower oxygen partial pressure is likely responsible for this. I I I I I I I I I 2 0 0 4 0 0 6 0 0 8 0 0 1000 1200 1400 1600 1800 2 0 0 0 TEMPERATURE: DEGREES KELVIN Figure 6.3 E q u i l i b r i u m Vapour Pressures of Pb, PbS, and PbO as F u n c t i o n s of Temperature DISCUSSION OF E X P E R I M E N T A L R E S U L T S / 74 6.4. RATE LIMITING PROCESSES The streak photography and mass loss experiments have shown that the feeds being studied oxidized according to the following general pattern: initial reaction rates were very slow. At room temperature, no reaction was observed and very little reaction was observed up until the ignition temperature. Once the particle ignited, oxidation proceeded very rapidly resulting in high particle temperatures and volatilization of lead species. At low temperatures, it is expected that the kinetics of sulphide oxidation are limited by the rate of the chemical reaction which depends exponentially on temperature and has a considerably higher activation energy than other kinetic phenomena such as mass-transfer. As the particle temperature increases, the rate of reaction and the amount of heat liberated by the reaction also increases. But, below the ignition temperature, the oxidation rate is too slow to overcome radiative and convective heat-transfer and the particle temperature remains close to the reactor temperature. At ignition, the oxidation rate is just fast enough so that the heat liberated by the reaction begins to dominate over radiation and convection. As this occurs, the particle temperature begins to rise well above that of its surroundings, the reaction rate further increases and the particle ignites. At some point, the chemical resistance decreases to below the value of the mass-transfer resistance and the rate controlling step changes. Once a particle has ignited and its temperature has increased to well above the reactor temperature, the reaction is likely mass-transfer controlled and is very DISCUSSION OF EXPERIMENTAL RESULTS / 75 rapid. Large quantities of lead species are volatilized from the particle resulting in the steep slopes observed in the mass loss curves. The reaction sequence described above forms the basis of the mathematical model detailed later in the thesis. 6.5. O V E R A L L R E A C T I O N F O R G A L E N A The X-ray diffraction pattern of the fume and reacted galena particles revealed another important characteristic of the oxidation reaction. The only lead species found when the reacted galena particles were examined were PbS and PbSO^ . This indicates that the overall oxidation reaction for lead sulphide under the reactor conditions examined is: PbS(s,l,g) + 20 2 => P b S 0 4 (s) (6.1) This is consistent with Fig. 6.4 which shows a predominance area diagram for the Pb-S-0 system. For typical reaction conditions (T = 900 K; 0 2 pressure = 0.21 atm; S O 2 pressure = low), PbSO^ is the expected product. Figs. 6.5 to 6.8 are predominance area diagrams for the Pb-S-0 system at progressively higher temperatures. As temperature increases to 1700 K, lead sulphate becomes less stable under the reactor conditions and first the oxy-sulphates (Figs. 6.5 and 6.6) and finally PbO (Figs. 6.7 and 6.8) become the expected products. Therefore, below ignition, the particle temperature and oxygen pressure are likely favourable for the reaction given by Eq. (6.1). However, an ignited particle whose reaction is controlled by oxygen transport will have a high temperature and be oxygen starved. Under these conditions gaseous PbS will be generated at the particle LOG OF SULPHUR DIOXIDE PRESSURE 3 Figure 6.4 E q u i l i b r i u m Phase Diagram of the Pb-S-0 System ^ at 900 Degrees Kelvin o> LOG OF SULPHUR DIOXIDE PRESSURE F i g u r e 6.5 E q u i l i b r i u m P h a s e Diagram o f t h e P b - S - O S y s t e m a t 1100 Degrees K e l v i n 3 CO CO UJ ce CL z UJ o >-g u. o o o O-i -1 -2 - 3 - 4 - 5 -6 -7 - 8 "'] -io 4 PbS0 4 (s) PbOPbSO,, (s) 4PbO-PbS0 4(s) PbO(l) Pb(l) •10 - 9 -8 -7 i -6 - 5 r -4 - 3 i —2 PbS(s) -1 -1 0 LOG OF SULPHUR DIOXIDE PRESSURE Figure 6.6 Eq u i l i b r i u m Phase Diagram of the Pb-S-0 System at 1300 Degrees Kelvin CO o a CO CO O O ><; w jo w > f to M co d t-4 H CO 00 LOG OF SULPHUR DIOXIDE PRESSURE Figure 6.7 E q u i l i b r i u m Phase Diagram of the Pb-S-0 System at 1500 Degrees Ke l v i n or Z> CO CO UJ or o_ UJ o >-8 o o -4-r -5H - 9 H -10 - 9 -8 - 7 -6 - 5 - 4 - 3 -2 LOG OF SULPHUR DIOXIDE PRESSURE Figure 6.8 Eq u i l i b r i u m Phase Diagram of the Pb-S-0 System at 1700 Degrees K e l v i n CO O a co CO HH O o tt H S3 w > f SO tt CO d CO CO O DISCUSSION OF E X P E R I M E N T A L R E S U L T S / 81 surface and go shooting out into the oxidizing gas where PbO formation is likely (see Fig. 6.8) until the cooler, more oxygen rich parts of the furnace away from the reacting particles are reached. However, it must be noted that, provided some evidence of reaction was seen, lead sulphate was always found in the product. For particles which did not ignite, this was expected. In the case of ignited particles which were hot enough to vapourize PbS, lead oxide was the expected product. Although some lead oxide was found, lead sulphate was also found in large quantities. The collection of some fume in the product explains this. 6.6. SUMMARY OF THE REACTION SEQUENCE Finally, it remains to account for the formation of cenospheres in the oxidation of concentrate particles as seen in Figs. 5.10 to 5.18. Particles bearing lead, iron and zinc became porous and formed cenospheres which grew despite mass loss (Figs. 5.7 and 5.8) and perhaps even exploded as suggested earlier. Particles of galena melt, form spheres and simply shrink as they lose mass (Fig. 5.9). A possible explanation for this behaviour is illustrated in Figs. 6.9 and 6.10. The galena particle shown in Fig. 6.9[1] falls through the reactor and heats up via convection and radiation. During this time, the reaction is chemically controlled and the rate is very. slow. In Fig. 6.9[2], the particle approaches the ignition temperature and the reaction rate increases causing the particle to heat to just above this critical temperature. DISCUSSION OF EXPERIMENTAL RESULTS / 82 PARTICLE HEATING UP NEGLIGIBLE REACTION 2. - PARTICLE APPROACHES IGNITION TEMPERATURE - REACTION, FUMING BEGIN - ROUNDING OF EDGES OCCURS HEAT OF REACTION CAUSES MELTING REACTION CONTINUES FUME MORE VISIBLE 4 . - MOLTEN PARTICLE SPHERODIZES - REACTION PROCEEDS RAPIDLY - LARGE QUANTITY OF FUME PRODUCED 5. - LOSS OF MATERIAL DUE TO FUMING CAUSES SPHERE TO SHRINK RAPIDLY Figure 6.9 Galena P a r t i c l e Reaction Sequence DISCUSSION OF EXPERIMENTAL RESULTS / 83 1. - PARTICLE HEATING UP - NEGLIGIBLE REACTION 2. - PARTICLE APPROACHES IGNITION TEMPERATURE - REACTION, FUMING BEGIN - ROUNDING OF EDGES OCCURS PbS(g) +Sj(g) OXIDE/SULPHIDE MATRIX OXIDE RICH MATRIX 3. - HIGH TEMPERATURE FROM HEAT GENERATED B7 OXIDATION REACTIONS - VAPORIZATION OF PbS & S T CAUSES FORMATION OF BUBBLES IN HIGH VISCOSITY MATRIX 4. - HIGHER TEMPERATURE, MORE OXIDATION AND VAPORIZATION - SMALL BUBBLES OF PbS & S a FORM SINGLE LARGE BUBBLE IN CENTER OF NEWLY FORMED CENOSPHERE - FRAGMENTATION MAY OCCUR Figure 6 . 1 0 Lead Concentrate (Fe and Zn Bearing) Reaction Sequence DISCUSSION OF E X P E R I M E N T A L R E S U L T S / 84 Up to this point, the only evidence of reaction is the slight rounding of the edges of the particles and a very small amount of fume. At the stage shown in Fig. 6.9[3], the particle begins to react rapidly. Melting begins as the particle temperature increases. The rate controlling step is now most likely mass-transfer of oxygen to the particle. Fume generation is significant as the particle rapidly volatilizes. During the stages seen in Figs. 6.9[3] and 6.9[4], the particle temperature rises well above the melting point causing the particle to form a spherical droplet. This droplet rapidly shrinks as the PbS volatilizes and large quantities of fume are produced. Steps [1] and [2] in Fig. 6.10 for the iron and zinc bearing concentrate are the same as for the galena particle in Fig. 6.9. During step 6.10[3], after ignition, the particle starts to form pockets of gaseous PbS. This may happen because the oxidized iron (and possibly zinc) forms a viscous, semi-fluid matrix which contains PbS. While the particle heats up to temperatures where the vapour pressure of PbS becomes greater than 1.0 atm (> 1609 K -- Fig. 6.3), it does not reach temperatures sufficient to fully liquify the matrix (the melting points of 2 4 PbO, F e 2 O g and ZnO are 1159 K , 1838 K and 2248 K respectively). Gas pockets form and the expanding gas causes the porous particles to swell. Fume is generated by PbSO^ which forms near the surface of the particle and is ejected or by PbO which further reacts away from the particle. In stage 6.3[4], the rising temperature lowers the viscosity and/or the surface DISCUSSION OF EXPERIMENTAL RESULTS / 85 tension to a point where the gas can coalesce into a single bubble. The rising temperature causes further volatilization of PbS and raises the pressure inside the bubble. The pressure finally becomes great enough to deform the particle walls and the cenosphere enlarges (as seen in the size distributions in Figs. 5 .7 and 5.8) or explodes and forms a number of smaller particles. It must be noted that only concentrate particles which contained iron and/or zinc formed cenospheres. Solid spheres containing either iron and sulphur or zinc and sulphur were formed (because of the feed compositions, it was not possible to separate the effects of Fe and Zn although the evidence suggests that Fe is the prime factor). Apparently, the presence of a volatile species and at least one other sulphide was required to form cenospheres. As was previously mentioned in Section 6.3, the formation of cenospheres has been observed and modelled by 1 7 Kim and Themelis who proposed an explanation similar to the one given above. However, in their work, Kim and Themelis were examining chalcopyrite and proposed that dissociated gaseous sulphur caused the observed cenospheres and porous particles. The possibility exists that labile sulphur was also responsible for the cenospheres and porous particles observed in the reacted Brunswick concentrate. Thermodynamic calculations show that, above approximately 900 K, the following reaction produces considerable amounts of gaseous sulphur: 2FeS2 => 2FeS + S2(g) (6.2) These calculations also show that as much as one third of the sulphur present in the Brunswick concentrate could have been liberated by this reaction. In the case of Sullivan concentrate, only about one weight percent of sulphur is DISCUSSION OF EXPERIMENTAL RESULTS / 86 similarly available. Therefore, it is likely that the gaseous species causing cenosphere formation in Sullivan reacted particles is PbS. CHAPTER 7. MATHEMATICAL MODEL AND PREDICTIONS In this chapter, a mathematical model based on the experimental findings reported in the previous chapter together with heat/mass-transfer principles and thermodynamics is developed. The model is capable of predicting the rate of galena oxidation, particle mass and particle temperature as functions of time in a heated oxidizing gas. Model predictions are presented, discussed and compared with the experimental results. 7.1. SYSTEM UNDER CONSIDERATION Owing to the complexities involved with developing a general model for the kinetics of lead concentrate particles of varying size and composition falling through a reactor, an approach involving a single particle of pure galena descending, under gravity, through a stagnant reactor was chosen. This allowed for analysis of galena oxidation kinetics without the influence of particle interactions and the effects of other metallic sulphides, most particularly iron sulphides. At this point, it is useful to summarize what has been discussed in the thesis regarding the oxidation kinetics of galena. 1. The oxidation kinetics at low temperatures appear to be limited by the rate of reaction (as discussed in Section 6.4). 2. Rates of reaction are slow until the particles ignite. Beyond this point, oxidation is rapid and high particle temperatures are attained. Ignition 87 MATHEMATICAL MODEL AND PREDICTIONS / 88 appears to be dictated by the heat balance (Section 6.4). 3. Ignited particles experience significant mass loss and generate large quantities of lead sulphate fume during reaction (Sections 5.2 & 5.3). 4. Evidence from the streak photographs and photomicrographs of fused, reacted particles (Figs. 5.1 to 5.4 and 5.13 to 5.21) suggests that the ignited particles attain very high temperatures - possibly well above 1392 K, the melting point of galena. The large quantities of fume generated also suggests that high temperatures are reached, sufficiently so to volatilize lead species. 5. The overall reaction for galena oxidation, according to fume analysis is given by Eqn.(6.1) (Section 6.5): PbS(s,l,g) + 20 2 => PbS04 (s) (6.1) At higher temperatures however, thermodynamics dictate that PbO formation is likely (Fig. 6.8). Further oxidation to PbSO^ would then have to take place in the cooler regions of the furnace away from the hot particle. 6. The possibility of temperatures above the boiling point of PbS and change in the rate of controlling mechanism must also be considered. It is now possible to conceptualize a model which will incorporate the points made above. The model can be broken down into four stages. Initially, as shown in Fig. 7.1, oxygen is present at the surface of the particle. The particle is at room temperature and the oxidation rate, which is chemically controlled, is negligible. The particle then is dropped into the hot zone of the furnace where it is heated by convection and radiation. As the particle heats up, the reaction rate MATHEMATICAL MODEL AND PREDICTIONS / 89 F i g u r e 7 . 1 : Stage I - C h e mically C o n t r o l l e d Surface Reaction MATHEMATICAL MODEL AND PREDICTIONS / 90 increases to the point where the enthalpy of the reaction (Eqn.(6.1)) ( A H O Q Q = 2 9o -3966 kJ/kg of Pb) elevates the particle temperature to well above that of the furnace and the particle ignites. Rapidly, the particle becomes hot enough to melt or even vaporize PbS and the reaction rate accelerates. However, at some point during this period, two phenomena take hold. First, the chemical resistance which drops exponentially with temperature, diminishes to the extent that mass-transfer of oxygen to the particle becomes rate controlling. Secondly, PbS volatilizes so rapidly that the reaction front lifts off the particle surface into the boundary layer surrounding the particle and fume is produced. This second stage is shown in Fig. 7.2. Transition from the first to the second stage in the model is determined by simultaneously calculating the chemical reaction rate and the _ evaporation rate of PbS. Mass-transfer of PbS and/or 0 2 is likely rate controlling during this stage since the chemical resistance diminishes exponentially with increasing temperature. The second stage of the model holds until the particle begins to boil. At this point, it becomes difficult to consider the presence of a laminar boundary layer surrounding the particle through which mass-transfer takes place. A new stage of the model must be developed to deal with this situation. As the particle continues to react, at the boiling point, heat is generated which can no longer be consumed by raising the temperature of the particle until the remaining PbS is vaporized. A higher rate of PbS vaporization results and this rate is simply dependent on the net rate of heat input to the particle. The particle now consists of a shrinking volume of liquid surrounded by an expanding gas envelope. As the envelope expands, it encounters oxygen and reacts to produce fume and more heat. The heat liberated by the reaction vaporizes more PbS and in this manner the process continues until all the liquid MATHEMATICAL MODEL AND PREDICTIONS / 91 9ft •Ppwe* 0 (Fast chemical r e a c t i o n ) '2 PbS F i g u r e 7.2: Stage I I - Mass-Transfer C o n t r o l l e d Reaction Taking P l a c e at a We l l Defined I n t e r f a c e M A T H E M A T I C A L M O D E L AND PREDICTIONS / 92 PbS is vaporized. This is the third stage of the model and is illustrated in Fig. 7.3. Once all the liquid PbS has been converted to gas, what remains is a small cloud of gaseous PbS surrounded by oxygen which quickly diffuses inward to react with the PbS. This is the fourth stage of the model and is illustrated in Fig. 7.4. The model, therefore, is made up of four distinct stages. The first deals with a surface reaction on the particle which is chemically controlled. The second stage involves a mass-transfer controlled reaction at an interface within the boundary layer. The third stage models the boiling, rapidly expanding particle and the fourth stage, considers oxygen diffusing into the cloud of PbS which remains once boiling is completed. 7.2. MODEL FORMULATION: STAGE I 7.2.1. Assumptions In order to formulate the first stage of the model mathematically, several simplifying assumptions have been made as follows: 1. The particles under consideration are spherical and consist of pure galena. Although galena particles have a cubic structure, the experimental results show that they become progressively more spherical as they react and melt. Consequently, this assumption seems justifiable. MATHEMATICAL MODEL AND PREDICTIONS / 93 l e a v e s a r e a near p a r t i c l e - o x i d i z e s Volume of 0 2 f u r t h e r t o PbS0 4 o u * b y i n c o o l e r r e g i o n expanding of f u r n a c e 9 a s envelope F i g u r e 7.3: Stage I I I - B o i l i n g , Expanding and R e a c t i n g P a r t i c l e MATHEMATICAL MODEL AND PREDICTIONS / PbO Product leaves area near p a r t i c l e - o x i d i z e s f u r t h e r to PbSQi i n c o o l e r region of furnace Boundary l a y e r edge Cloud s u r f a c e / r e a c t i o n i n t e r f a c e P» <s 0 (Fast chemical r e a c t i o n ) F i g u r e 7 .4 : Stage IV - S h r i n k i n g , Reacting PbS Gas Cloud MATHEMATICAL MODEL AND PREDICTIONS / 95 2. Internal resistance to heat conduction is neglected due to the small particle size (Biot numbers of less than 0.1 were calculated; see Appendix II). 3. Furnace wall and gas temperatures are taken to be equal and constant over the length of the furnace hot zone. This assumption was shown to be valid to within 10 K over the first 350 mm of the reactor hot zone as discussed in Section 4.5. 4. The gas in the furnace does not participate in radiation. During the first stage of the reaction, the only species present in the gas are 0 2 and possibly N 2 . Both of these gases are diatomic and therefore transparent to 20 radiation (Gieger and Poirier ). 5. There are two possible oxidation reactions. Galena is either oxidized to lead _ sulphate according to Eq. (6.1) or to lead oxide and S0 2 as follows: PbS(g) + 3/202 => PbO(g) + S 0 2 (7.1) A H 2 9 g = -2016 kJ/kg of Pb From the predominace area diagrams (Figs. 6.4 to 6.8) it can be seen that this second reaction becomes favoured at higher temperatures. Eq.(6.1) will be assumed to describe the oxidation reaction up to 1600K beyond which Eq.(7.1) is applied. This assumption is made based on F*A*C*T thermodynamic calculations which predict that PbO is the favoured oxidation product of PbS reacted in excess oxygen at temperatures above 1600 K. 6. All the heat generated from galena oxidation is absorbed by the particle. This heat then enters into the normal energy exchange between the particle and furnace. 7. The initial stage of the model assumes a surface reaction producing PbSO. , which is consistent with the predominance area diagrams. In this MATHEMATICAL MODEL AND PREDICTIONS / 96 case, it would be expected that, as the reaction proceeded, PbSO^ would slowly build up on the particle and the mass of the particle would increase due to the acquisition of oxygen. However, the mass loss curves (Figs. 5.5 and 5.6) only exhibit decreases in particle mass. A possible explanation for this anomaly rests on the volume change as galena reacts to become lead sulphate. The reaction produces a volume increase of 58% which may be sufficient to cause the product to spall as it forms. Therefore, in order to fit the experimental data, it is assumed that any PbSO^ formed is lost to the furnace. Any error incurred by making this assumption should be small since only about 5% of the particle is reacted during this stage. Further assumptions pertaining to subsequent stages of the model will be detailed later in the thesis. 7.2.2. Heat Balance In developing a model of galena oxidation kinetics, a heat balance must be applied to the particle as it falls through the reactor. Terms must be included for radiation, convection, and the enthalpy of the oxidation reaction: dlmiH - H,,)) . . (7.2) = Aea (r; - T*) + Ah (Ta - T) + RAHre<lct M A T H E M A T I C A L M O D E L A N D P R E D I C T I O N S / 97 Note that A H r e a c t is defined in J/mole of PbS. m d { H d t H " K ( H - H„) (^) = Aea (TF - T*) + Ah(T„ - T)+RAHr,.,l<:t ( 7 > 3 ) Now, Also, let (H - H © ) = H(T). Therefore, the equation becomes: mCp (T) ^ = Ata (TF - T*) + Ah (T„ - T) + RAHreact - (T) (7.4) at at The convective heat-transfer coefficient, h, was calculated from the following correlation 2 0 - 2 1 : MATHEMATICAL MODEL AND PREDICTIONS / 98 for a single sphere of diameter D. Where (7.6) (7.7) The subscript f indicates fluid (gas) properties which were calculated at the mean film temperature. , the terminal velocity of the falling particle relative to the gas, was calculated from Stokes Law: D2g{p- PS) 18f7/ (7.8) 7.2.3. Rate of Reaction As was discussed earlier in this chapter, the first stage of the model involves a particle whose surface is oxygen saturated. The particle temperature is low and the oxidation reaction (Eqn.(6.1)) is chemically rate controlled. Also, some diffusion M A T H E M A T I C A L M O D E L AND PREDICTIONS / 99 of oxygen to the surface of the particle is taking place and this contributes 9 13 16 17 slightly to the overall reaction resistance. Previous modelling work ' ' ' involving similar cases with oxidizing particles showed that the chemical reaction rate could be described be an Arrhenius type equation. Therefore: R = kAe-E"/RT (Jjj^ moles of PbS/s (7.9) Chaubal and Sohn found the oxidation rate of chalcopyrite to be linear with oxygen pressure. Also, a first order reaction allows for simplification in the combining and calculation of resistances as will be seen later in this section. For these reasons, the reaction for galena is assumed to behave similarly. Therefore, Eq.(7.9) becomes: R = kAe-Ea'RT (^f) molesofPbS/s ( 7 < 1 0 ) The diffusion of oxygen to the spherical particle can be described by the equation 22 for diffusion through a surrounding boundary layer (Bird, Stuart and Lightfoot ): nC>5 — —.£/(.»•..i;4Jrr Li — moles s dr (7.11) MATHEMATICAL MODEL AND PREDICTIONS / 100 Integrating Eq.(7.10) yields: />/£ n n ' \ r I ^= In-.' dXo, (7.12) PT pi. (7.13) Rearranging and adding Eqs.(7.10) and (7.13) gives: Pv, ( ? - 1 4 ) D _ i \ - ' — ' _ ' 1'3 kAe-Ea/RT D0,,A7rCT P Now, at steady state and from stoichiometry, R = -(l/2)n Therefore the ° 2 overall reaction rate is given by: 1 Ph Rphs = , . - ~ moles/s (7-15) Eqs. (7.4) and (7.15) describe the initial stage of the model except for the change in particle mass which is assumed to decrease as the PbS reacts. MATHEMATICAL MODEL AND PREDICTIONS / 101 Therefore, the mass loss is given by: fn — - Rph^ Mphs kg/s (7.16) 7.3. MODEL FORMULATION: STAGE n The second stage of the model is more complex. As is seen in Fig. 7.2, the rate of PbS evaporation has increased to a point where PbS is diffusing to and reacting at an interface in the particle boundary layer. 7.3.1. Assumptions Again, several simplifying assumptions are made: Assumptions 1) through 6) from section 7.2.1 are also applied in the second stage of the model with the following qualification. In view of the large quantities of fume generated in this stage of the reaction, assumption 4) may not be valid. Since the fume surrounds the particle, it would interfere with heat-transfer to and from the particle. These effects, however, cannot be quantified readily and are therefore assumed to be small. In addition, the following assumptions are made: 1. Once the vaporization rate of PbS exceeds the rate of reaction dictated by Eq.(7.15), the reaction lifts off the surface and becomes a gas-gas reaction as shown in Fig. 7.2. At this point, the chemical reaction is assumed to be fast. M A T H E M A T I C A L M O D E L A N D P R E D I C T I O N S / 102 2. Mass-transfer takes place through a laminar boundary layer surrounding the particle. 3. A s soon as P b S O ^ is generated in the gas surrounding the particle, it is lost to the furnace atmosphere. Assumption 3) is adopted since no evidence of P b S O ^ , other than a few small particles of fume, was found with the reacted particles. 4. The non-steady state effects associated with the accumulation of PbS in the boundary layer as the reaction moves off the particle surface are negligable. 7.3.2. The Heat Balance A similar analysis to that undertaken in the first stage of the model can be applied to the heat balance in S T A G E II. However, there are two differences. F i rst ly , a term must be included for the enthalpy of PbS vaporization. Secondly, the rate of reaction is strongly effected by the rate of PbS vaporization. The heat balance then becomes: d { m { Hd ; H " ) ] = Aea(TF - T*) + Ak{T„ - T ) + ^ + ^ ) (7.17) Now, following the same series of simplifications as in Section 7.2.2: mCp(T) ^ = Aia (TF - T*) + Ah{Tg - T) + ^  (A#,uW + Atfr,(1,( - H (T)) (7.18) at at MATHEMATICAL MODEL AND PREDICTIONS / 103 Since: dm ~dt dD dm\ dD (pwD2 pnD3 6 A = *D2 m = dt Substitution the above into 7.17 yields: CP(T) dT dt pD H ( T ) ) (7.19) 7.3.3. Rate of Reaction The rate of reaction is now determined by the rate at which PbS and O j arrive at the reaction front in the boundary layer. PbS must first evaporate from the surface and then diffuse to the reaction front. The rate of PbS evaporation was characterized by the Langmuir-Knudsen equation (Rutner, 23 Goldfinger and Hirth ): OC, a surface correcting factor, was assumed to have a value of 1 for these particles. It was assumed that this equation applied to both solid and liquid PbS. As in section 7.2.3, for a spherical particle, the equation for diffusion through a E = Aa (2*MPbSRT)-1/2 [P* - P) moles/s (7.20) MATHEMATICAL MODEL AND PREDICTIONS / 104 surrounding boundary layer (Bird, Stuart, and Lightfoot^ **) is: nphs = -Dpbs.y^r2CTi—-7—^ moles/s (7.21) Note that Dp^g and C ,^^  in this case are evaluated at the particle temperature since the PbS diffuses through the part of the boundary layer which is between the particle and the hot reaction interface. Integrating Eq.(7.21) yields: r * = i X n s (7.22) nPhs = n (P - p/ fcs) moles/s (7.23) Summing Eqs. (7.20) and (7.23) gives: E —2+ P,"™:,, = ( p ' - Pr^) (7.24) I Now, at steady state, -np^g = E. Also, with a fast reaction, P p b g = O. MATHEMATICAL MODEL AND PREDICTIONS / 105 Therefore Eq. (7.24) becomes: -P* - n m p h S = 7 — — - r — moles/s (7.25) Am Dphs .ginC'r 1 The rate of diffusion of 0 2 into the particle can be obtained by applying the same analysis used to generate Eq. (7.23) -D0,,,4nO no, = / i \ v -PL) molt, I a (7.26) Note that in this case, Dp ,g and Cp are evaluated at the mean film temperature. From stoichiometry and the assumption of steady state, - n „ = • j 0.5(n-. ). Also, with a fast reaction, P_ = O. This equation can be used to °2 °2 solve for r^ and then the diffusion/reaction rates back-calculated. -K^UCrPt, _ -P-I— —\ PT [2nMr,sRT)^ i t ~ P r \ r i r° ' Aa DFhs.,j4ntJTi P' + •D/>hs..|2C'ri £),,,.,,2ir<.'T/,,',„ {2nMP,,sRT)l/* D0i.gCTP^ (7.28) AdPT + ?n.s.<|20'Tir/> Therefore, for the second stage of the model, the reaction rate is given by Eq. MATHEMATICAL MODEL AND PREDICTIONS / 106 (7.25) and the mass loss of the particle is determined by n Q V x Mp^g. 7.4. MODEL FORMULATION: STAGE III The third stage of the model, as illustrated in Fig. 7.3, describes the case of a boiling, reacting particle. This "particle" is made up of a shrinking liquid core and an expanding gas envelope which reacts as it grows and comes into contact with 0 2 • 7.4.1. Assumptions The following simplifying assumptions are made: 1. The PbO generated in this stage is generated as fume and is lost to the furnace atmosphere. 2. The rate of expansion of the particle is assumed to be much larger than the diffusion of 0 2 into the particle. Therefore 0 2 diffusion is not considered. 3. The particle exchanges heat with its surroundings by radiation only since a boundary layer required to calculate convection no longer exists. The radius of the particle for these purposes is taken to be the radius of the gas envelope. 4. The rate of reaction depends upon the change in volume of the particle as it expands. The amount of oxygen swept out by the volume expansion is the quantity which reacts (See Fig. 7.3). MATHEMATICAL MODEL AND PREDICTIONS / 107 7.4.2. Heat Balance The heat balance for Stage III is a modified version of the heat balance in Stage I (Eq.(7.4)). On the left-hand side of Eq.(7.4), mCp(T)^I is replaced by dt the product of the rate of vaporization and A H J g ^ . On the right-hand side, Ah(T - T) disappears and the heat balance is: EMr,,s&H„M = Aea {T4F - T4) + — ( A H r c a c t - H (T)) (7.29) dt \ Mpbs J 7.4.3. The Mass Balance The volume of the particle is given by: Therefore, ., nri.s (I) Mri,s nr,,< (g) Mr,,s 3 (7.30) Vr = + m P (--Tl dVr dnphs (/) Mpbs dnp,lS (g) MPh.< 3 , n m\ ~dT = pit + —c^t— m f s ( 7 - 3 1 ) o dn (1) = E is negative and is solved for using Eq.(7.29). rbb o The rate of n_. _(g) generation = -E. n_, _(g) is also consumed at two-thirds rbo rbb M A T H E M A T I C A L M O D E L A N D P R E D I C T I O N S / 108 the rate of oxygen consumption which is given by: Therefore, and fay, dVP _ _ = _ C t moles/s { q ^ dt dt 3 ^ = - ? M r , , C T ^ k g / s l . u (7.34) 7.5. M O D E L F O R M U L A T I O N : S T A G E IV The fourth stage in the model dwells on a "particle" which has finished boiling and is, essentially, a spherical cloud of PbS(g). This cloud rapidly shrinks as Oj diffuses in and reacts with it. 7.5.1. Assumptions M A T H E M A T I C A L M O D E L A N D PREDICTIONS / 109 The simplifying assumptions made in this stage include assumptions 1 and 3 of Stage III as well as the following: 1. The particle cloud holds together and only 0^ diffuses in. 2. The cloud behaves like a solid shrinking particle. 0 2 ^ again assumed to diffuse through a boundary layer and react at the "surface" of the cloud as in Stage I (See Fig. 7.4). 7.5.2. The Heat Balance Since the particle is no longer undergoing a phase change, the temperature may dT rise again and mCp(T) — is .replaced in the left-hand side of the heat balance. dt Therefore the equation is: mCP (T) f = Ma (1* - T 4 ) + Ah[Ta - T) + ^ ( - H (7)) (7.35) 7.5.3. The Mass Balance MATHEMATICAL MODEL AND PREDICTIONS / 110 The rate of oxygen diffusion into the gas cloud is given by Eq.(7.13): . _ D,h,,47rCT ( P;',2 P}; \ PT PT J (7.13) From stoichiometry (Eq.(7.1)) the rate of the PbS reaction can be determined: 2 . Rpus = — 2^o2 moles/s (7.36) = — Mri.s'-no* kg/s (7.37) dt 3 7.6. CALCULATIONS OF GAS AND PARTICLE PROPERTIES 7.6.1. Diffusion Coefficients Values for the diffusion coefficients were calculated from a modification of the MATHEMATICAL MODEL AND PREDICTIONS / 111 Chapman-Enskog equation developed by Fuller (Geiger and Poirier2^): 3.204210" 4 T 1 - 7 j / 1 1 V „ , D * - » = —rr, ^ 2 ( TT ~ XT m ' 3 ( 7 - 3 8 ) PT(v^ + V{j'z) ^ M a and Vg , the diffusion volumes of gas species A and the boundary layer gas were taken from Geiger and Poirier^. For model calculations involving air, the gas species were assumed to be diffusing through nitrogen. For the case of pure oxygen, SO^ was assumed to be the boundary layer gas. 7.6.2. Viscosities The viscosities of air and oxygen were obtained from the Handook of Chemistry and Physics . Variations with temperature were included in the model. 7.6.3. Thermal Conductivity The thermal conductivities of air and oxygen (as functions of temperature) were calculated from Thermal Conductivity vs. Temperature plots in Geiger and Poirier20. 7.6.4. Boundary Layer Thickness The radius of the boundary layer surrounding the particle was calculated using a 22 Sherwood number correlation and the Film Theory (Bird, Stuart and Lightfoot ). MATHEMATICAL MODEL AND PREDICTIONS / 112 For a spherical particle: <1mH. = Sh = 2.0 + 0.60Re1'2Sc1'3 (7.39) D a . b Sc D A.„ (7.40) D_ Sh (7.41) 7.6.5. PbS Thermal Conductivity and Emissivity Unfortunately, values for the thermal conductivity and emissivity of PbS were not available. Therefore, these properties were estimated from values for similar materials listed in Geiger and Poirier^ . For thermal conductivity, values for U 0 2 » Ti02» and Zr0 2 were considered and a value of 3 W/mK assumed. For emissivity, values for carbon, beryllium oxide and Fe _ 0 were considered and a 2 3 value of 0.8 assumed. Since the particles were virtually completely enclosed in a long cylinder, a view factor of 1 was used in the radiation calculations. MATHEMATICAL MODEL AND PREDICTIONS / 113 7.6.6. Thermodynamic Data All of the thermodynamic data used in the thesis were obtained from either Kubaschewski and Evans2^ or the F*A*C*T Data System2**. Solid and liquid densities of the lead compounds were also obtained from F*A*C*T 7.7. NUMERICAL SOLUTION OF THE MODEL The systems of" differential equations presented in this chapter were written into a computer program (Appendix III) and solved using a University of British Columbia Fourth Order Runge-Kutta (with error control) subroutine. 7.8. MODEL FITTING Two parameters in the model, k and Ea, need to be adjusted so that the model predictions fit the experimental results. The magnitude of Ea, the activation energy for the oxidation reaction determines the sensitivity of the reaction rate in the early stages of reaction to temperature. A large value for Ea makes the reaction rate more sensitive and tends to close the separation between the air and oxygen mass-loss curves and the streak test ignition values. The separation of 100 K (Table 5.2) seen in these experimental results for galena in air and oxygen was used to determine Ea. A value of 137 kJ/mole was found to give the best fit to the experimental data. The pre-exponential constant was then used to determine at what temperature transition would occur. When k was adjusted to fit the experimental curves, a value of 6.80 x 10 moles/m s was obtained. M A T H E M A T I C A L M O D E L A N D ' PREDICTIONS / 114 Fig. 7.5 compares the fit of three curves of model predictions to the experimental mass loss results obtained in air. As can be seen in this figure, variations of only 10% in Ea and k from the values listed above have a major effect on model predictions. This is further illustrated in Figs. 7.6 and 7.7 which show the extreme sensitivity of single particle behaviour predicted to Ea and k respectively. 7.9. TYPICAL MODEL PREDICTIONS Figs. 7.8 and 7.9 show typical model predictions of particle behaviour. In Fig 7.8, the temperature vs. time plot for a 100 micron particle reacting in air is shown. The corresponding mass loss curve is given in Fig. 7.9. There are several features of note in these curves. Firstly, the particle is seen to rapidly heat up to the furnace temperature. During this time, very little reaction takes place as , is evidenced by the mass loss curve. Secondly, once the particle ignites, reaction and hence mass loss, is very rapid. Thirdly, the thermal arrests seen in Fig. 7.8 represent the melting and boiling of the particle. Mass loss is seen to be most rapid after the particle has vapourized. Once the particle has ignited, its lifetime (taken to be the time from the onset of fusion to burnout) is of the order of only 100 milliseconds. Roughly 90% of the particle mass is reacted during this time. Finally, the temperature attained by the particle is high, around 2600 K. MATHEMATICAL MODEL AND PREDICTIONS / 11 O CN r g CN SSVN Q33J IVNIOIdO JO % Figure 7.6: E f f e c t of the A c t i v a t i o n Energy on The Mass Loss Curve of a 100 Micron P a r t i c l e Reacting i n A i r . Fnce Te«p • 978 K MATHEMATICAL MODEL AND PREDICTIONS / 117 SSVN IVNIOIdO !N30d3d 3000-1 TIME: SECONDS Pigure 7.B: Temperature vs. Time Curve f o r a 100 Micron P a r t i c l e Reacting i n A i r Fnce Temp - 978 K Figure 7.9: Mass Loss vs. Time Curve f o r a 100 Micron P a r t i c l e Reacting i n A i r Fnce Temp - 978 K M A T H E M A T I C A L M O D E L A N D PREDICTIONS / 120 7.10. THE SENSITIVITY OF THE MODEL TO THE HEAT-TRANSFER COEFFICIENTS The heat-transfer coefficients are important variables since they determine the rate at which the particle exchanges heat with its surroundings. This influences the rate at which the particle heats up and also the point where the oxidation reaction liberates enough heat to cause particle ignition. The convective heat-transfer coefficient is likely to be reasonably accurate since it is calculated using an accepted empirical equation. The radiative heat-transfer coefficient, on the other hand, depends directly on the emissivity of the particle (which has been guessed at). Fortunately, as Fig. 7.10 shows for a 100 micron particle, the convective heat-transfer coefficient is the major contributor to heat exchange between the particle and the furnace. Fig. 7.11 illustrates the effect of 10% variations in the convective heat-transfer coefficient on the mass loss curve of a 100 micron particle. The effect follows the same trend as was observed for Ea in Fig. 7.6. Oddly, decreasing the convective heat-transfer coefficient increases reaction rate despite the fact that the particle would initially heat up more slowly. It is apparently more important that the particle loses heat more slowly once its temperature exceeds that of the furnace. In this manner, ignition is achieved more rapidly. The sensitivity of the model to changes in the convective heat-transfer coefficient appears to be slightly greater than its sensitivity to changes in k. 2000 • 1500-L J 1000 -O O CO z: < 500 0-Legend A CONVECTIVE H.T.C. X RADIATIVE H.T.C. 200 T T 400 600 800 1000 1200 1400 PARTICLE TEMPERATURE: DEGREES K 1600 Figure 7. 10: Comparison of Radiative and Convective Heat Transfer C o e f f i c i e n t s for a 100 Micron P a r t i c l e Reacting in A i r > X > o > tr4 % o a > a T) PO W D I—t O p-3 I—I o co TIME: SECONDS ^ Figure 7 . ll: E f f e c t of the Convective Heat Transfer ^ C o e f f i c i e n t on the Mass Loss Curve of a to 100 Micron P a r t i c l e Reacting i n Air Fnce Temp - 978 K M A T H E M A T I C A L M O D E L A N D PREDICTIONS / 123 7.11. COMPARISON OF MODEL PREDICTIONS AND EXPERIMENTAL RESULTS 7.11.1. Model Predictions vs. Streak Photographs The streaks of reacting particles shown in Figs. 5.3 and 5.4 reveal long distances over which particles are visibly reacting. Examination of Fig. 7.8 will show that particles react very quickly once they reach temperatures high enough for them to emit visible light. The time during which a particle is incandescent therefore is very short as is the distance it travels. For this reason, the long streaks seen in Figs. 5.1 to 5.4 seem somewhat anomalous. Also, the sharp transition observed over a relatively short temperature range (Figs. 5.5 and 5.6) must be predicted by the model. Figs. 7.12 and 7.13 address these specific points. In Fig. 7.12, the effect of furnace temperature on particle temperature vs. position in the reactor is illustrated. Over only a 20 K increase of furnace temperature, a 100 micron particle changes from non-ignition to ignition. Over this temperature range, the ignition point moves from the end of the reactor (350 mm) back to 190 mm. the reactor tube. Thus a sharp transition from no streaks to bright streaks seen further and further up the tube is predicted by the model. Fig. 7.13 shows the temperature vs. distance curves for three particle sizes at a furnace temperature high enough to cause all three to ignite in pure oxygen. A clear distribution of igniting particles is predicted by the model. Small particles ignite near the top of the reactor and larger particles ignite progressively further down. This figure when combined with the size distribution in Table 7.1 definitely predicts the streaks seen in Figs. 5.3 and 5.4. In the 3000 2500-2000 1500 1000-500 L_J t*a»l»*«ly Legend A FURNACE TEMP - 980K x FURNACE TEMP = 970K • FURNACE TEMP = 990K 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 DISTANCE FALLEN THROUGH REACTOR: METERS Figure 7.12: P a r t i c l e Temperature vs. Distance Fallen Through The Reactor For a 100 Micron P a r t i c l e Reacted i n Oxygen 0.40 3000-1 CO 2500 UJ UJ or O UJ Q 2000 T or or UJ Q_ UJ O r— or 51 1500-1000 500-Legend A 100 MICRON PARTICLE X 50 MICRON PARTICLE • 150 MICRON PARTICLE 0.00 0.05 0.10 0.15 0.20 0.25 DISTANCE FALLEN THROUGH REACTOR: METERS 0.30 > H X M > H i—i O > r S o o M > Tl w a O i-9 i—i O 2! CO Figure 7.13: Particle Temperature va. Dutance Fallen Through The Reactor Furnace Aunoaphere = Oxygen, Furnace Temperature = 97a K (S3 Ol MATHEMATICAL MODEL AND PREDICTIONS / 126 Table 7.1: Galena Feed S i z e D i s t r i b u t i o n Mass F r a c t i o n S i z e Range Microns F r a c t i o n of Feed Mass Percent 0 to 40 7.32 40 to 65 10.71 65 to 75 9.67 75 to 85 1 1 .73 85 to 95 14.05 95 to 105 15.61 105 to 115 9.03 -115 to 150 1 1 .68 150 to 185 10.20 MATHEMATICAL MODEL AND PREDICTIONS / 127 case of the particle streaks, agreement between experimental results and model predictions is qualitatively very good. 7.11.2. Model Predictions vs. Mass Loss Curves Fig. 7.14 compares the corrected experimental and predicted mass loss curves based on the size distribution given in Table 7.1. The model predictions are seen to fit the experimental results fairly well. The major regions of difference involve the higher furnace temperature parts of the curves where mass loss begins to slow. In these regions, the model generally predicts more rapid mass loss than the experimental results indicate. A possible explanation for this discrepancy involves the particle interactions which are likely present in the reacting particle stream. The possibility exists that the particles in the stream are close enough to starve each other of oxygen and thereby delay reaction. At higher furnace temperatures, unreacted particles would vaporize PbS faster (resulting in more mass loss). Also, the increased temperature may improve mass-transfer enough to allow more particles to react. In any case, a gradual tailing off of the mass loss curves would be expected in the experimental results. Since the modelling of particle interactions is beyond the scope of this thesis, the predictions of mass loss are determined primarily by whether or not the particle ignites. A second possibility is that the particle size distribution in Table 7.1 is not representative of the galena feed used in the mass loss experiments. As will be MATHEMATICAL MODEL AND PREDICTIONS / 128 SSVkN Q33A IVNIOIdO JO % MATHEMATICAL MODEL AND PREDICTIONS / 129 demonstrated in this chapter, the model is very sensitive to particle size. An incorrect size distribution could certainly be responsible for the differences observed in Fig. 7.14. A third possibility is that lead sulphate fume settled out of the furnace atmosphere onto the collector plate thus adding extra mass as described in Chapter 5. The differences in the low furnace temperature ends of the mass loss curves maj- be due to the assumption that the products of oxidation are immediately lost upon generation. However, since such a small fraction of the overall particle mass is reacted in this part of the curves, the discrepancies seen there are of relatively minor concern. 7.12. MODEL PREDICTIONS 7.1.2.1. The Effect of Particle Size Table 7.1 lists the size distribution, on the basis of mass, for the galena feed determined by Coulter Counter analysis. Although the particle size range is nominally 75 to 150 microns, these results show that the range is in fact much larger. Therefore, the effect of particle size must be examined. Figs. 7.15 to 7.17 show the effect of particle size on the mass loss vs. time and distance and particle temperature vs. time respectively. As can be seen in Fig. 7.15, the 100 micron particle is completely oxidized whereas the 50 and 150 micron particles • 50 MICRON PARTICLE ~i I 1 1 1 I 0 0.5 1 1.5 2 2.5 TIME: SECONDS Figure 7.15: Effe c t of P a r t i c l e S i z e on Mass Loss Curves Reaction in A i r . Fnce Teap - 976K 100 80 CO < < z o or o o 60 40 2 0 -- a Legend A 50 MICRON PARTICLE X 100 MICRON PARTICLE • 150 MICRON PARTICLE PUfttClM H i t c amis j to* O f M I M T I O B n n u i i tat i% COJMHITICMI d f O a i B A T l O a T T T JL 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 DISTANCE FALLEN THROUGH REACTOR: METERS 0.40 > H X w > O > o a w t-1 > z o T) w I—I o H o 2! CO F i g u r e ? . 16: E f f e c t of P a r t i c l e S i z e on Mass Loss Curves Reaction in A i r . Fnce Teap • 978 K 3000-f to 2 5 0 0 - r or O LxJ Q 2000 -or ZD or Ld 1500 r1 r / MITKli or oaii Legend A 50 MICRON PARTICLE X 100 MICRON PARTICLE • 150 MICRON PARTICLE v**ri( t t u rt a tw iM reio* to crioa  i i t * t is» 1000- [fi o r— or 500 t 0.5 1 —T-1.5 2.5 TIME: SECONDS Figure 7.17; E f f e c t of P a r t i c l e S i z e on P a r t i c l e Temperature Reaction i n A i r . Pnce Temp • 91d K > H H HH O > O a t-1 > a 33 tt a HH o HH O •2 CO CO ts3 MATHEMATICAL MODEL AND PREDICTIONS / 133 exit the reactor hot zone (350 mm) before they are totally oxidized. In Figs. 7.16 and 7.17, the 50 micron particle is shown to be the fastest of the three in initial reaction rate and heat up. However, despite a faster reaction rate (relative to the larger particles) the 50 micron particle never ignites. The heat of reaction is not sufficient to overcome convection and radiation. The 100 micron particle obviously does ignite (Fig. 7.17) and the region of rapid mass loss coincides, as expected, with the higher temperatures. The 150 micron particle is the slowest to heat up but also appears about to ignite (Fig. 7.17). This is prevented, however, by the exit of the particle from the reactor (Figs. 7.15 and 7.16). From these three figures, it is possible to understand more fully the effect of particle size on galena oxidation. From Eqs. 7.3 and 7.5 we see that convective heat transfer goes up as D. From Eqn. 7.10 we see that the reaction rate 2 depends on D . As D decreases the reaction rate declines more rapidly than the heat transfer term. Therefore, larger particles ignite at lower temperatures. This effect is countered, however, by the fact that the larger particles have greater terminal velocities (shorter residence times) and longer heat up times (mass to be 3 heated increases with D ). Although the temperature may be high enough to cause ignition, time becomes the limiting factor. For these reasons, the 100 micron particle ignited and oxidized completely whereas the other two particles did not. M A T H E M A T I C A L M O D E L AND PREDICTIONS / 134 7.12.2. The Effect of Oxygen Concentration Figs. 7.18 and 7.19 show the effect of oxygen concentration on the temperature vs. time and distance curves of a 100 micron particle. There are two important effects shown in these figures. Firstly, when pure oxygen is used instead of air, reaction times are shortened significantly and hence the position where the particle ignites is shifted nearer the top of the furnace. Secondly, the final particle temperature is higher. In air, the particle is seen to heat up, ignite, melt and boil and eventually reach a temperature of about 2600 K before it is totally oxidized. In oxygen, the same complete oxidation takes one quarter the time and a temperature of roughly 2800 K is attained. Clearly as expected, increasing oxygen concentration has the effect of accelerating the kinetics of galena oxidation. 7.12.3. The Effect of Furnace Temperature The effect of furnace temperature is illustrated in Fig. 7.13. As has already been discussed, small changes in furnace temperature can be the difference between a particle igniting and oxidizing to completion and virtually no reaction. It is the furnace temperature which determines whether a particle will gain enough heat from its surroundings to reach the ignition temperature. MATHEMATICAL MODEL AND PREDICTIONS / 135 * S33cJ03(] -BcjniVcJBdKGl 310llciVd MATHEMATICAL MODEL AND PREDICTIONS / 136 >i S33U03Q :3dniVd3dW31 310!ldVd CHAPTER 8. CONCLUSIONS 1. Experimental results indicate that the feeds examined all have sharply defined ignition points. These ignition points are dependent primarily on oxygen concentration and composition of the feed (specifically iron content). 2. Large quantities of lead are lost from ignited particles. This lead is volatilized from the particles and oxidized to form lead sulphate fume. 3. The experimental mass loss curves indicate that the kinetics of lead concentrate oxidation are chemically limited at low temperatures and mass-transfer limited at high temperatures. 4. The model predictions show that ignition is determined by the heat balance. Increases in oxygen concentration accelerate the rate of reaction at any given temperature and therefore the rate of heat input to the particle. This has the effect of shifting ignition to a lower furnace temperature. 5. Convection was calculated to be the dominant means for heat-transfer between the particle and its surroundings. 6. Particle size affects ignition temperature in a similar manner as oxygen since increasing the diameter causes a correspondingly larger increase in heat generation than heat loss (per unit mass). This effect can be masked however by the longer heat up times and shorter furnace residence times seen for larger particles. 7. The model predicts that more than 90% of the mass of a galena particle is reacted after that particle ignites. Once a particle has 137 ignited, its lifetime is usually temperatures in excess of 2600 K attained. 8. The model predictions show good results. 8.1. FURTHER WORK Conclusions / 138 less than 100 milliseconds and in air and 2800 K in oxygen are agreement with the experimental Although the kinetics of galena oxidation have been modelled in this study, there still remains the modelling of lead concentrate oxidation kinetics. The thermodynamics and kinetics of mixed concentrate oxidation are more complicated than those of galena. An understanding of these kinetics is essential if modelling of flash smelting of such concentrates is to be successful. REFERENCES 1. H.O. Hofman, Metallurgy of Lead, McGraw-Hill Book Company, Inc, New York, 1918 2. H.O. Hofman, The Metallurgy of Lead and the Desilverization of Base  Bullion, The Scientific Publishing Company, New York, 1899 3. H . F . Collins, The Metallurgy of Lead, Charles Griffin & Company, Ltd., London, 1910 4. Lead Industries Association, Lead in Modern Industry, Lead Industries Association, New York, 1952 5. M.W. lies, Ph.D., Lead-Smelting, John Wiley & Sons, New York, 1902 6. B .H. Morrison, "A Review of New Technologies for Lead and Zinc Reduction Plants", in Canadian Metallurgical Quarterly, 23(4), 1984, pp. 377-381 7. A . K . Biswas and W.G. Davenport, Extractive Metallurgy of Copper, 2nd. Ed. , Pergammon Press, Oxford, 1980 8. Joseph R. Boldt, The Winning of Nickel, Longmans Canada Ltd, Toronto, 1967 9. Y. Fukunaka, T. Monta, Z. Asaki, and Y. Kondo, "Oxidation of Zinc Sulphide in a Fluidized Bed", Met. Trans. B, 7B, 1975, pp. 307-314 10. F .R.A. Jorgensen, "Combustion of Pyrite Concentrate Under Simulated Flash-Smelting Conditions", Trans. Instn. Min. Metall. (Sect. C: Mineral Process. Extr. Metall.), 90, 1981, C l - 9 11. F .R.A. Jorgensen, "On Maximum Temperature Attained During Single-Particle Combustion of Pyrite", Trans. Instn. Min. Metall. (Sect. C: Mineral Process. Extr. Metall.), 90, 1981, C10-16 12. F .R.A. Jorgensen, "Combustion of Chalcopyrite, Pyrite, Galena, and Sphalerite Under Simulated Suspension Smelting Conditions", in Australia/Japan Extractive Metallurgy Symposium 1980, pp 41-51 (The Australasian Institute of Mining and Metallurgy: Melbourne). 13. Z. Asaki, S. Mori, M . Ikeda, and Y. Kondo, "Oxidation of Pyrrhotite Particles Falling Through a Vertical Tube", Met. Trans. B, 16B, 1985, pp 627-638. 14. F .R.A. Jorgensen and E . Ralph Sengit, "Copper Flash Smelting Simulation Experiments", Proc. Australas Inst. Min. Metall., 261, 1977, pp. 39-46. 15. F .R.A. Jorgensen,. "Single-Particle Combustion of Chalcopyrite", Proc. Australas. Inst. Min. Metall., No 288, 1983, pp. 37-46. 139 / 140 16. P.C. Chaubal and H.Y. Sohn, "Intrinsic Kinetics of the Oxidation of Chalcopyrite Particles Under Isothermal and Nonisothermal Conditions", Met. Trans. B, 17B, 1986, pp. 51-60. 17. Y . H . Kim and N.J . Themilis, "Effect of Phase Transformation and Particle Fragmentation on the Flash Reaction of Complex Metal Sulphides", Metallurgical Society/AIME. (Met. A. , 8702-72-0106), 1986, pp. 349-369. 18. Petri Bryk, Rolf Malmstrom, and Erik Nyholm, "Flash Smelting of Lead Concentrates", Journal of Metals, 1966, pp. 1298-1302. 19. B .H. Morrison, "A Review of New Technologies for Lead and Zinc Reduction Plants", Can. Met. Quart., Vol. 23, No. 4, 1984, pp. 377-381 20. G .H. Geiger and D.R. Poirer, Transport Phenomena in Metallurgy, Addisons-Wesley Publishing Co., Reading, 1980, pp. 70, 250, 467. 21. F . Kreith, Principles of Heat Transfer, International Textbook Co., Scranton, 1958, pp. 340-342. 22. R.B. Bird, W . E . Stewart and E . N . Lightfoot, Transport Phenomena. John Wiley and Sons inc., New York, 1960, pp. 644, 646. 23. E . Rutner, P. Goldfinger, and J.P. Hirth, Condensation and  Evaporation of Solids. Gordon and Breach, New York, 1964. pp. 3-5, 149-150. 24. C.R.C. Handbook of Chemistry and Physics, 58th ed., editor, Robert C. Weast, Ph.D., CRC Press Inc., West Palm Beach, 1977. 25. O. Kubaschewski and E . Evans, Metallurgical Thermochemistry , 2nd ed., John Wiley and Sons Inc., New York, 1956. 26. W.T. Thompson, A.D. Pelton, and C W . Bale, F * A * C * T Data System, 1987. APPENDIX I: CALCULATION OF PARTICLE VELOCITIES 141 / 142 C a l c u l a t i o n of Inert P a r t i c l e V e l o c i t i e s i n a Stagnant Gas: Assume that the p a r t i c l e s c o n s i d e r e d have the same d e n s i t y as galena (7500 kg per cubic meter), and that they are i n e r t . A l s o assume that the p a r t i c l e s are s p h e r i c a l . T h e r e f o r e , using Stokes Law and ( i f necessary) the Cunningham C o r r e c t i o n F a c t o r , the f o l l o w i n g r e s u l t s are obt a i n e d : For a 50 micron p a r t i c l e , feed pipe temperature = 300 K rea c t o r temperature = 1000 K Entrance V e l o c i t y = 0.48 m/s V e l o c i t y .05 m down the r e a c t o r = 0.22 m/s E x i t V e l o c i t y = 0.22 m/s Residence Time = 1.54 s For a 100 micron p a r t i c l e , feed pipe temperature = 300 K rea c t o r temperature = 1000 K Entrance V e l o c i t y = 1.25 m/s V e l o c i t y .05 m down the r e a c t o r = 1.12 m/s E x i t V e l o c i t y = 0.91 m/s Residence Time = 0.35 s For a 150 micron p a r t i c l e , feed pipe temperature = 300 K rea c t o r temperature = 1000 K Entrance V e l o c i t y = 1.51 m/s V e l o c i t y .05 m down the r e a c t o r = 1.58 m/s E x i t V e l o c i t y = 1.83 m/s Residence Time - 0.21 s These values above were c a l c u l a t e d from the f o l l o w i n g computer program: / 143 1 C PROGRAM TO CALCULATE VELOCITY AND DISTANCE FALLEN 2 C IN A REACTOR WITH TIME 3 C 4 IMPLICIT REAL*8(A-H,0-Z) 5 DIMENSION Y(5),F(5),T(5),S(5),G(5) 6 EXTERNAL FUNC,FUNC1 7 COMMON /BLK1/GMW,TC,TG,ANUFC,ANUFG,RP,TUBEL,ROP,VF,REP 8 COMMON /BLK2/ROF 9 C 10 C SET VARIABLES WHICH ARE CONSTANT THROUGHOUT THE CALCULATION 1 1 C N: # OF DIFF EQNS GMW: MOLECULAR MASS OF GAS 12 C TC: FEED PIPE TEMP TG: REACTOR TEMP 13 C ANUFC: VISCOSITY OF GAS IN FEED PIPE 14 C RP: PARTICLE RADIUS TUBEL: LENGTH OF FEED PIPE 15 C ROP: PARTICLE DENSITY ROF: FLUID DENSITY IN FEED PIPE 16 C VF: FLUID VELOCITY X: TIME 17 C Y(1),Y(2): INITIAL CONDITIONS - VELOCITY,DISTANCE 1 8 C 1 9 N-2 20 GMW-.02884 21 TC-300 22 TG-1000 23 ANUFC-0.0000107+0.0000000348*TC 24 RP-0.0000750 25 TUBEL-0.15 26 ROP-7500.0 27 ROF-GMW*8.314*TC/101325.0 28 VF--0.0000001 29 X-0.0 30 Y(1)-0.0 31 Y(2)-0.0 32 C 33 C SET VALUES REQUIRED FOR DRKC 34 C 35 Z-0.150 36 H-(Z-X)/64.D0 37 HMIN-.01D0*H 38 E-1.D-5 39 ZH-(Z-X)/50. 40 Z-X 41 C 42 C LOOP TO CALCULATE VELOCITY AT WHICH THE PARTICLE EXITS 43 C THE FEED PIPE AND ENTERS THE HOT REACTOR 44 C 45 DO 30 J-1,75 CALL DRKC (N,X,Z,Y,F,H,HMIN,E,FUNC, G, S , T) 46 47 Z-Z+ZH 48 IF (Y(2).LT.TUBEL) GOTO 30 49 J-76 50 30 CONTINUE 51 C 52 C SET CONDITIONS TO CALCULATE THE VELOCITY PROFILE 53 C ANUFG: REACTOR GAS VISCOSITY 5 4 C 55 Y(2)-0.0 56 X-0.0 57 Z-0.200 58 ANUFG-0.0000107+0.0000000348*TG / 144 59 ROF-GMW*8.314*TG/101325.0 60 H-(Z-X)/64.D0 61 HMIN».01D0*H 62 E-1.D-5 63 ZH«(Z-X)/50. 64 Z-X 65 C 66 C LOOP TO CALCULATE VELOCITY AND DISTANCE WITH TIME 67 C 68 DO 31 J-1,75 69 CALL DRKC (N,X,Z,Y,F,H,HMIN,E,FUNC1,G,S,T) WRITE (6,41 ) X,Y(1) 70 71 41 FORMAT(1X,F6.4,2X,3F10.6) 72 Z-Z+ZH 73 31 CONTINUE 74 STOP 75 END 76 C 77 c SUBROUTINE TO CALCULATE FEED PIPE EXIT VELOCITY 78 c REP: PARTICLE REYNOLDS NUMBER CD: DRAG COEFFICIENT 79 c 80 SUBROUTINE FUNC(X,Y,F) 81 IMPLICIT REAL*8(A-H,0-Z) 82 DIMENSION Y(5),F(5) 83 COMMON /BLK1/GMW,TC,TG,ANUFC,ANUFG,RP,TUBEL,ROP,VP,REP 84 COMMON /BLK2/ROF REP-2.0*ROF*RP*(Y( 1 ) - VF) /ANUFC 85 86 IF(REP.LT.2.0) GOTO 75 87 CD-18.5/(REP**0.6) 88 F(1)«9.81-(CD*1.5*ROF*(Y(1)-VF)**2.0)/(RP*ROP) 89 GOTO 80 90 75 F(1)»9.81-9.0*ANUFC*(Y(1)-VF)/(ROP*RP**2*2.0) 91 80 F(2)-Y(1) 92 RETURN 93 END 94 C 95 C SUBROUTINE TO CALCULATE VELOCITY AND DISTANCE IN 96 C THE REACTOR WITH TIME 97 C 98 SUBROUTINE FUNC1(X,Y,F) 99 IMPLICIT REAL*8(A-H,0-Z) 100 DIMENSION Y(5),F(5) 101 COMMON /BLK1/GMW,TC,TG,ANUFC,ANUFG,RP,TUBEL,ROP,VF,REP 102 COMMON /BLR2/ROF REP«2.0*ROF*RP*(Y(1)-VF)/ANUFG 103 104 IF(REP.LT.2.0) GOTO 120 105 CD-18.5/(REP**0.6) 106 F(1)»9.81-(CD*1.5*ROF*(Y(1)-VF)**2.0)/(RP*ROP) 107 GOTO 250 108 120 F(1)«9.81-9.0*ANUFG*(Y(1)-VF)/(ROP*RP**2*2.0) 109 250 F(2)-Y(1) 1 10 RETURN 1 1 1 END APPENDIX II: BIOT NUMBER CALCULATION 145 / 146 For a sphere of r a d i u s r : Bi = h x r Now, h 2500 W/m K (Fig.7.10) k p 3 W/mK ( S e c t i o n 7.6.5) r = 0.00005 m (a 100 micron p a r t i c l e ) T h e r e f o r e : Bi = (2500 x 0.00005) = 0.0417 3 Since B i < 0.1, i n t e r n a l r e s i s t a n c e to heat conduction can be ignored. A P P E N D I X III: M A T H E M A T I C A L M O D E L - C O M P U T E R P R O G R A M 147 / 148 1 C PROGRAM TO CALCULATE GALENA PARTICLE OXIDATION HISTORY 2 C IN A STAGNANT GAS REACTOR 3 C 4 C 5 C DEFINE VARIABLES AND FUNCTION SUBROUTINES: 6 C 7 C FUNC,FUNC1,FUNC2,FUNC3: SUBROUTINES REQUIRED BY DRKC 8 C TO DEFINE THE SYSTEMS OF DIFFERENTIAL EQUATIONS TO BE 9 C SOLVED IN THE VARIOUS STAGES OF THE MODEL I O C 11 C FUNC: CONTAINS STOKES LAW EQUATIONS FOR CALCULATION OF 12 C PARTICLE VELOCITY AS IT ENTERS THE FURNACE HOT ZONE 13 C 14 C FUNC1: CONTAINS EQUATIONS FOR CALCULATION OF THE 1 5 C FIRST AND SECOND STAGES OF THE MODEL 1 6 C 17 C FUNC2: CONTAINS EQUATIONS FOR CALCULATION OF STAGE 1 8 C THREE OF THE MODEL 1 9 C 20 C FUNC3: CONTAINS EQUATIONS FOR CALCULATION OF STAGE 21 C FOUR OF THE MODEL 22 C 23 IMPLICIT REAL*8(A-H,0-Z) 24 DIMENSION Y(10),F(10),T(10),S(10),G(10) 25 EXTERNAL FUNC,FUNC1,FUNC2,FUNC3 26 REAL*8 X,Z,Y,F,H,HMIN,E,G,S,T,HTC,TF,DHS 27 REAL*8 DHRPBS,AREA,PCNTM,MPBS,MAPBS 28 REAL*8 RP,RB,EPSIG,HTR,DIFRES,RXNRES,RL,MT 29 REAL*8 PEQ,PT,CT,DPBS,D02,CP,HT,RO,PI,LE,P02,XA,ZH1,PEQPBS 30 COMMON /BLK1/Z,H,HMIN,E,G,S,T,HTC,TF,DHS,PCNTM 31 COMMON /BLK2/DHRPBS,AREA 32 COMMON /BLK3/RB,EPSIG,HTR,XA,ZH1,RXNRES 3 3 COMMON /BLK4/PEQ,PT,CT,DPBS,D02,CP,HT,RO,PI,LE,P02,DIFRES,RL 34 COMMON /BLK5/GMW,TC,TG,ANUFC,ANUFG,RP,TUBEL,ROP,VF,REP,ZH 35 COMMON /BLK6/ROF,RPIN,PEQPBS,MPBS,MT,MAPBS 36 COMMON /BLK7/EMOL,RREACT,ZH2 37 C 38 C N = NO. OF DIFFERENTIAL EQNS. TO BE SOLVED BY DRKC 39 C GMW = MOLECULAR WEIGHT OF AIR (kg/mole) 40 C TC = FEED TUBE TEMPERATURE (DEGREES KELVIN) 41 C TF = FURNACE TEMPERATURE (DEGREES KELVIN) 42 C ANUFC = VISCOSITY OF GAS IN FEED TUBE (kg/ms) 43 C RPIN = INITIAL RADIUS OF PARTICLE (m) 44 C TUBEL = FEED PIPE LENGTH (m) 45 C ROP = PARTICLE DENSITY (kg/m**3) 46 C ROF = DENSITY OF GAS IN FEED TUBE (kg/m**3) 47 C VF = FLUID ENTRANCE VELOCITY (m/s) 48 C X = INITIAL TIME (s) 49 C Y(1) = INITIAL PARTICLE VELOCITY (m/s) 50 C Y(2) = INITIAL PARTICLE DISTANCE FALLEN (m) 51 C Z,H,ZH,HMIN,E = VARIABLES REQUIRED BY DRKC - ASSOCIATED 52 C WITH TIME STEPS AND CONVERGENCE 53 C 54 N=2 55 GMW=.02884 56 TC=300 57 TF=1000.0 58 ANUFC=0.0000107+0.0000000348*TC / 149 5 9 R P I N = 0 . 0 0 0 0 5 0 0 6 0 T U B E L = 0 . 1 5 61 R O P = 7 5 0 0 . 0 6 2 R 0 F = G M W * 8 . 3 - 1 4 * T C / 1 01 3 2 5 . 0 6 3 V F = - 0 . 0 0 0 0 0 0 1 6 4 X = 0 . 0 6 5 Y ( 1 ) = 0 . 0 6 6 Y ( 2 ) = 0 . 0 0 6 7 Z = 0 . 1 5 0 6 8 H = ( Z - X ) / 6 4 . D 0 6 9 H M I N = . 0 1 D 0 * H 7 0 E = 1 . D - 5 71 Z H = ( Z - X ) / 5 0 . 7 2 Z = X 7 3 C 7 4 C C A L L F U N C T O C A L C U L A T E E N T R A N C E V E L O C I T Y O F T H E P A R T I C L E 7 5 c I N T O T H E F U R N A C E H O T Z O N E 7 6 c 7 7 DO 3 0 1 = 1 , 7 5 7 8 C A L L D R K C ( N , X , Z , Y , F , H , H M I N , E , F U N C , G , S , T ) 7 9 Z = Z + Z H 8 0 c 81 c C H E C K I F P A R T I C L E E X I T S F E E D T U B E - I F " Y E S " , T H E N B E G I N H E A T I N G A N D R E A C T I O N O F P A R T I C L E ( S T A G E O N E ) 8 2 c 8 3 c 8 4 I F ( Y ( 2 ) , G E . T U B E L ) G O T O 3 3 8 5 3 0 C O N T I N U E 8 6 C 8 7 c B E G I N S T A G E S O N E A N D TWO O F T H E M O D E L 8 8 c 8 9 c Y ( 3 ) = P A R T I C L E V E L O C I T Y ( m / s ) 9 0 c Y ( 4 ) = D I S T A N C E P A R T I C L E H A S F A L L E N T H R O U G H R E A C T O R (i 91 c P 0 2 = O X Y G E N P R E S S U R E I N T H E R E A C T O R ( P a ) 9 2 c M P B S = M O L E C U L A R M A S S O F P b S ( k g / m o l e ) 9 3 c P T = T O T A L S Y S T E M P R E S S U R E ( P a ) 9 4 c RO = D E N S I T Y O F P b S ( k g / m * * 3 ) 9 5 c P I = P I 9 6 c L E = e 9 7 c E P S I G = E M I S S I V I T Y X B O L T Z M A N N ' S C O N S T A N T 9 8 c Y ( 1 ) = P A R T I C L E T E M P E R A T U R E ( I N I T I A L L Y 2 9 8 K ) 9 9 c Y ( 2 ) = M A S S O F P b S W H I C H H A S R E A C T E D ( k g ) 1 0 0 c M A P B S = I N I T I A L P A R T I C L E M A S S ( k g ) 101 c MT = P A R T I C L E M A S S ( k g ) 1 0 2 c R P = P A R T I C L E R A D I U S (m) 1 0 3 c H T C = C O N V E C T I V E H E A T - T R A N S F E R C O E F F I C I E N T 1 0 4 c H T R = R A D I A T I V E H E A T - T R A N S F E R C O E F F I C I E N T 1 0 5 c A N U F G = V I S C O S I T Y O F F U R N A C E G A S ( k g / m s ) 1 0 6 c R O F = D E N S I T Y O F G A S I N F U R N A C E ( k g / m * * 3 ) 1 0 7 c Z H 1 , Z H 2 = V A R I A B L E S U S E D T O R E D U C E T H E T I M E S T E P 1 0 8 c P C N T M = P E R C E N T O F O R I G I N A L P A R T I C L E M A S S R E M A I N I N G 1 0 9 c D R A T E = O X Y G E N D I F F U S I O N R A T E I N T O T H E P A R T I C L E ( m o l e 1 1 0 c 1 1 1 3 3 N = 4 1 1 2 Y ( 3 ) = Y ( 1 ) 1 1 3 Y ( 4 ) = 0 . 0 1 14 P O 2 = 0 . 2 1 * 1 0 1 3 2 5 . 0 1 15 X = 0 . 0 1 16 M P B S = 0 . 2 3 9 2 6 / 150 117 PT=101325 .0 1 1 8 RO=7500.0 119 P I - 3 . 1 4 1 5 9 2 6 120 LE=2 .718282D0 121 E P S I G = 4 . 5 3 6 D - 8 122 Y ( 1 ) = 2 9 8 . 0 123 Y ( 2 ) = 0 . 0 124 M A P B S = 1 . 3 3 3 3 3 * P I * R P I N * * 3 * R O 125 MT=MAPBS 126 R P = ( M T / ( R O * 1 . 3 3 3 3 3 * P I ) ) * * 0 . 3 3 3 3 3 3 127 HTC=( 0 . 0 3 6 6 + 9 . 4 8 D - 5 * ( ( Y ( 1 ) + T F ) / 2 . 0 ) ) / ( R P * 2 . 0 ) 128 H T R = E P S I G * ( T F * * 2 + Y ( 1 ) * * 2 ) * ( T F + Y ( 1 ) ) 129 A N U F G = 0 . 0 0 0 0 1 0 7 + 0 . 0 0 0 0 0 0 0 3 4 8 * T F 130 ROF=GMW*8.3 1 4 * T F / 1 0 1 3 2 5 . 0 131 Z=4.0 132 H = ( Z - X ) / 6 4 . 133 H M I N = 1 0 * * ( - 4 ) * H 134 E - . 0 0 0 0 0 1 135 Z H = ( Z - X ) / 1 2 0 0 . 1 3 6 ZH1=ZH/5 .0 137 ZH2=ZH/50 .0 138 Z=0.000000 139 C 140 C CALL FUNC1 TO BEGIN CALCULATION OF STAGES ONE AND TWO 141 C OF THE MODEL 142 C 143 DO 31 J = 1 , 5 0 0 144 CALL DRKC ( N , X , Z , Y , F , H , H M I N , E , F U N C 1 , G , S , T ) 145 . PCNTM=MT*100.0 /MAPBS 146 D R A T E = A R E A * ( H T R + H T C ) * ( T F - Y { 1 ) ) + 147 & P 0 2 / ( D I F R E S * P T ) * M P B S * ( D H S + D H R P B S - H T ) * ( - 1 . 0 ) 148 W R I T E ( 6 , 4 1 ) X , Y ( 1 ) , P C N T M , Y ( 4 ) 149 41 F O R M A T ( 1 X , F 9 . 7 , 3 X , F 8 . 2 , F 8 . 2 , F 8 . 5 ) 150 Z=Z+ZH 1 51 C 152 C CHECK IF P A R T I C L E E X I T S REACTOR - IF " Y E S " , STOP 1 53 C 154 IF ( Y ( 4 ) . G E . 0 . 3 5 ) GOTO 45 155 C 1 5 6 C CHECK IF P A R T I C L E IS COMPLETELY REACTED IF " Y E S " , STOP 157 C 1 5 8 IF ( M T . L E . 0 . 0 0 0 0 0 0 0 0 0 0 5 ) GOTO 45 159 C 160 C CHECK IF P A R T I C L E HAS REACHED ITS BOIL ING TEMPERATURE 161 C - I F " Y E S " , BEGIN STAGE THREE 162 C 163 I F ( Y ( 1 ) . G E . 1 6 0 9 . 5 ) GOTO 32 164 31 CONTINUE 165 C 166 C STOP TO AVOID E X C E S S I V E NUMBER OF ITERATIONS 167 C 1 6 8 GOTO 45 169 C 170 C BEGIN STAGE THREE OF THE MODEL 171 C 172 C Y ( 1 ) = VOLUME OF THE P A R T I C L E (m) 173 C Y ( 2 ) = NUMBER OF MOLES OF L IQUID PbS 174 C Y ( 3 ) = NUMBER OF MOLES OF GASEOUS PbS / 151 175 C Y ( 4 ) = P A R T I C L E MASS (kg) 176 C Y ( 5 ) = P A R T I C L E V E L O C I T Y IN REACTOR (m/s ) 177 C Y ( 6 ) = DISTANCE F A L L E N THROUGH REACTOR (m) 178 C 179 32 N=6 180 Y ( 5 ) = Y(3 ) 181 Y ( 6 ) = Y(4 ) 182 Y ( 1 ) = 1 .333333*PI*RP**3 1 8 3 Y ( 2 ) = MT/MPBS 184 Y ( 3 ) = 0 .0 185 Y ( 4 ) = MT 186 2=1 .00 187 H = Z / 6 4 . 1 8 8 H M I N = 1 0 * * ( - 4 ) * H 189 E=0.000001 190 ZH=Z/10000 . 191 Z=0.00 1 92 C 1 9 3 C C A L L FUNC2 TO C A L C U L A T E STAGE THREE OF THE MODEL 1 94 C . 195 DO 35 K=1,100 196 C A L L DRKC ( N , X , Z , Y , F , H , H M I N , E , F U N C 2 , G , S , T ) 1 9 7 Z=Z+ZH 198 PCNTM=MT/MAPBS*100.0 199 W R I T E ( 6 , 3 4 ) X , P C N T M 200 34 F O R M A T ( 1X , F 1 0 . 8 , 1 O X , F 1 0 . 4) 201 C 202 C CHECK I F P A R T I C L E E X I T S REACTOR - IF " Y E S " , STOP 203 C 204 IF ( Y ( 4 ) . G E . 0 . 3 5 ) GOTO 45 205 C 206 C CHECK IF P A R T I C L E IS COMPLETELY REACTED I F " Y E S " , STOP 207 C 208 I F ( M T . L E . 0 . 0 0 0 0 0 0 0 0 0 0 5 ) GOTO 45 209 C 210 C CHECK IF P A R T I C L E HAS COMPLETELY VAPOURIZED - IF "YES" 21 1 c BEGIN STAGE FOUR 212 c 213 IF ( Y ( 2 ) . L E . 0 . 0 0 0 0 0 0 0 0 0 0 0 5 ) GOTO 36 214 35 CONTINUE 215 C 216 C BEGIN STAGE FOUR OF THE MODEL 217 C 218 C Y ( 1 ) = P A R T I C L E TEMPERATURE 219 C Y ( 2 ) = P A R T I C L E MASS (kg) 220 c 221 36 N=2 222 Y ( 1 ) = 1609 . 5 223 Y(2)=MT 224 Z=0.01 225 H = Z / 6 4 . 226 E=0.000001 227 C 228 C C A L L FUNC3 TO C A L C U L A T E STAGE FOUR OF THE MODEL 229 C 230 DO 40 L=1,100 231 C A L L DRKC ( N , X , Z , Y , F , H , H M I N , E , F U N C 3 , G , S , T ) 232 Z=Z+ZH / 152 233 C 234 WRITE(6,38)X,Y(1),RP 235 38 FORMAT(1X,F8.6,F10.4,F20.18) 236 C CHECK FOR COMPLETION OF REACTION 237 C 238 IF (MT.LE.0.00000000005) GOTO 45 239 40 CONTINUE 240 45 STOP 241 END 242 C 243 C SUBROUTINE FUNC: CALCULATION OF ENTRANCE VELOCITY OF THE 244 C PARTICLE INTO THE REACTOR 245 C 246 C DEFINE VARIABLES: 247 C REP = PARTICLE REYNOLDS NUMBER 248 C CD = DRAG COEFFICIENT (CUNNINGHAM CORRECTION FACTOR) 249 C F(1) = PARTICLE ACCELERATION (m/s**2) 250 C F(2) = PARTICLE VELOCITY (m/s) 251 C 252 SUBROUTINE FUNC(X,Y,F) 253 IMPLICIT REAL*8(A-H,0-Z) 254 DIMENSION Y(10),F(10),T(10),S(10),G(10) 255 REAL*8 X,Z,Y,F,H,HMIN,E,G,S,T,HTC,TF,DHS 256 REAL*8 DHRPBS,AREA,PCNTM,MPBS,MAPBS 257 REAL*8 RP,RB,EPSIG,HTR,DIFRES,RXNRES,RL,MT 258 REAL*8 PEQ,PT,CT,DPBS,D02,CP,HT,RO,PI,LE,P02,XA,ZH1,PEQPBS 259 COMMON /BLK1/Z,H,HMIN,E,G,S,T,HTC,TF,DHS,PCNTM COMMON /BLK2/DHRPBS,AREA 260 261 COMMON /BLK3/RB,EPSIG,HTR,XA,ZH1,RXNRES 262 COMMON /BLK4/PEQ,PT,CT,DPBS,D02,CP,HT,RO,PI,LE,P02,DIFRES,RL 263 COMMON /BLK5/GMW,TC,TG,ANUFC,ANUFG,RP,TUBEL,ROP,VF,REP,ZH 264 COMMON /BLK6/ROF,RPIN,PEQPBS,MPBS,MT,MAPBS 265 COMMON /BLK7/EMOL,RREACT,ZH2 266 REP=2.0*ROF*RPIN*(Y(1)-VF)/ANUFC 267 C 268 C CHECK IF STOKES LAW HOLDS 269 C 270 IF(REP.LT.2.0) GOTO 75 271 CD=18.5/(REP**0.6) 272 F(1)=9.81-(CD*1.5*ROF*(Y(1)-VF)** 2.0)/(RPIN*ROP) 273 GOTO 80 274 75 F(1)=9.81-9.0*ANUFC*(Y(1)-VF)/(ROP*RPIN**2*2.0) 275 80 F(2)=Y(1) 276 RETURN 277 END 278 C 279 c SUBROUTINE FUNC1: CALCULATION OF MODEL STAGES ONE AND TWO 280 c 281 c DEFINE VARIABLES: 282 c AKF = THERMAL CONDUCTIVITY OF THE FURNACE GAS (W/m**2K) 283 c CPG = HEAT CAPACITY OF THE FURNACE GAS (J/kgK) 284 c ANU' = NUSSELT NUMBER 285 c CT = MOLAR CONCENTRATION OF THE REACTOR GAS AT THE 286 c MEAN FILM TEMPERATURE (moles/m**3) 287 c CT1 = MOLAR CONCENTRATION OF THE REACTOR GAS AT THE 288 c PARTICLE TEMPERATURE (moles/m**3) 289 c DPBS = DIFFUSIVITY OF PbS (m**2/s) 290 c D02 = DIFFUSIVITY OF OXYGEN (m**2/s) / 153 2 9 1 C D H S = E N T H A L P Y O F P b S V A P O U R I Z A T I O N ( J / m o l e ) 2 9 2 C D H R P B S = E N T H A L P Y O F P b S O X I D A T I O N ( J / k g ) 2 9 3 C A R E A = P A R T I C L E S U R F A C E A R E A ( m * * 2 ) 2 9 4 C P E Q P B S = E Q U I L I B R I U M P R E S S U R E O F P b S V A P O U R ( P a ) 2 9 5 C S H P = S H E R W O O D N U M B E R 2 9 6 C R B = B O U N D A R Y L A Y E R R A D I U S ( m ) 2 9 7 C R L = R E A C T I O N I N T E R F A C E R A D I U S ( m ) 2 9 8 C R X N R E S = C H E M I C A L R E S I S T A N C E T O O X I D A T I O N 2 9 9 C D I F R E S , D I F R E 1 = O X Y G E N D I F F U S I O N R E S I S T A N C E T O R E A C T I O N 3 0 0 C I N S T A G E S O N E A N D T W O O F T H E M O D E L R E S P E C T I V E L Y 3 0 1 C F ( 1 ) = d ( P A R T I C L E T E M P E R A T U R E ) / d t 3 0 2 C F ( 2 ) = R A T E O F R E A C T I O N ( m o l e s o f P b S / s ) 3 0 3 C F ( 3 ) = P A R T I C L E A C C E L E R A T I O N ( m / s * * 2 ) 3 0 4 C F ( 4 ) = P A R T I C L E V E L O C I T Y ( m / s ) 3 0 5 C 3 0 6 S U B R O U T I N E F U N C 1 ( X , Y , F ) 3 0 7 I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) 3 0 8 D I M E N S I O N Y ( 1 0 ) , F ( 1 0 ) , T ( 1 0 ) , S ( 1 0 ) , G ( 1 0 ) 3 0 9 R E A L * 8 X , Z , Y , F , H , H M I N , E , G , S , T , H T C , T F , D H S 3 1 0 R E A L * 8 D H R P B S , A R E A , P C N T M , M P B S , M A P B S 3 1 1 R E A L * 8 R P , R B , E P S I G , H T R , D I F R E S , R X N R E S , R L , M T 3 1 2 R E A L * 8 P E Q , P T , C T , D P B S , D 0 2 , C P , H T , R O , P I , L E , P 0 2 , X A , Z H 1 , P E Q P B S 3 1 3 C O M M O N / B L K 1 / Z , H , H M I N , E , G , S , T , H T C , T F , D H S , P C N T M 3 1 4 C O M M O N / B L K 2 / D H R P B S , A R E A 3 1 5 C O M M O N / B L K 3 / R B , E P S I G , H T R , X A , Z H 1 , R X N R E S 3 1 6 C O M M O N / B L K 4 / P E Q , P T , C T , D P B S , D 0 2 , C P , H T , R O , P I , L E , P 0 2 , D I F R E S , R L 3 1 7 C O M M O N / B L K 5 / G M W , T C , T G , A N U F C , A N U F G , R P , T U B E L , R O P , V F , R E P , Z H 3 1 8 C O M M O N / B L K 6 / R O F , R P I N , P E Q P B S , M P B S , M T , M A P B S 3 1 9 C O M M O N / B L K 7 / E M O L , R R E A C T , Z H 2 3 2 0 M T = M A P B S - Y ( 2 ) * 0 . 2 3 9 2 6 3 2 1 R P = ( ( M T / R O ) / ( 1 . 3 3 3 3 3 * P I ) ) * * 0 . 3 3 3 3 3 3 3 2 2 A K F = 0 . 0 1 9 3 + 0 . 0 0 0 0 4 7 4 * ( Y ( 1 ) + T F ) / 2 . 0 3 2 3 C P G = 9 3 6 . 2 5 + 0 . 0 6 5 3 8 * ( Y ( 1 ) + T F ) - 1 3 0 7 5 0 0 . / ( Y ( 1 ) + T F ) * * 2 . 0 3 2 4 R E P = 2 . 0 * R O F * R P * Y ( 3 ) / A N U F G 3 2 5 A N U = 2 . 0 + 0 . 6 * R E P * * 0 . 5 * ( C P G * A N U F G / A K F ) * * 0 . 3 3 3 3 3 2 6 H T C = A N U * A K F / ( 2 . 0 * R P ) 3 2 7 H T R = E P S I G * ( T F * * 2 + Y ( 1 ) * * 2 ) * ( T F + Y ( 1 ) ) 3 2 8 C T = 2 * 1 2 1 8 7 . 3 / ( Y ( 1 ) + T F ) 3 2 9 C T 1 = 1 2 1 8 7 . 3 / Y ( 1 ) 3 3 0 D P B S = 2 . 5 9 3 D - 1 0 * Y ( 1 ) * * 1 . 7 5 3 3 1 D 0 2 = 6 . 0 1 D - 1 0 * ( ( ( Y ( 1 ) + T F ) / 2 . 0 ) * * 1 . 7 5 ) 3 3 2 C P = 1 9 5 . 3 3 + 0 . 0 3 8 4 7 3 * Y ( 1 ) 3 3 3 D H S = 9 7 5 9 8 8 . 9 - 3 9 . 3 4 6 * Y ( 1 ) - 0 . 0 2 3 5 2 * Y ( 1 ) * * 2 3 3 4 D H R P B S = - 3 6 0 3 1 0 2 . 3 + 3 2 3 . 6 6 9 * Y ( 1 ) - 0 . 0 1 0 5 1 4 * Y ( 1 ) * * 2 3 3 5 & - l 3 9 8 9 8 0 . 2 / Y ( 1 ) 3 3 6 A R E A = 4 . 0 * P I * R P * * 2 3 3 7 P E Q P B S = ( L E * * ( 2 0 . 4 7 7 - 2 8 0 9 1 / Y ( 1 ) - 1 . 1 3 2 3 * D L O G ( Y ( 1 ) / 2 9 8 ) 3 3 8 & - 0 . 0 0 l 3 5 4 * Y ( 1 ) ) ) * 1 0 1 3 2 5 . 0 3 3 9 S H P = 2 . 0 + 0 . 6 * R E P * * 0 . 5 * ( A N U F G / ( R O F * D O 2 ) ) * * . 3 3 3 3 3 4 0 R B = R P + 2 . 0 * R P / S H P 3 4 1 R L = ( P E Q P B S + ( D O 2 * 4 . 0 * P I * C T * P O 2 / ( D P B S * 4 . 0 * P I * C T 1 ) ) ) / 3 4 2 & ( D O 2 * 4 . 0 * P I * C T * P O 2 * ( P I * l 6 . 6 2 8 * M P B S * Y ( 1 ) ) * * 0 . 5 / ( A R E A * P T ) + 3 4 3 & ( D O 2 * 4 . 0 * P I * C T * P O 2 / ( D P B S * 4 . 0 * P I * C T 1 * R P ) ) + P E Q P B S / R B ) 3 4 4 R X N R E S * 1 . 0 / ( 6 8 0 0 0 0 0 . 0 * A R E A * L E * * ( - 1 6 5 0 0 . 0 / Y ( 1 ) ) ) 3 4 5 D I F R E S = ( 2 . 0 0 / R L - 2 . 0 0 / R B ) / ( D O 2 * 4 . 0 * P I * C T ) 3 4 6 D I F R E 1 = ( 2 . 0 0 / R P - 2 . 0 0 / R B ) / ( D O 2 * 4 . 0 * P I * C T ) 3 4 7 C 3 4 8 C B E G I N S H O R T E N I N G T H E T I M E S T E P T O A V O I D O V E R S H O O T I N G / 154 349 C THE MELTING TEMPERATURE 350 C 351 IF (Y(1).GE.1260.) GOTO 90 352 GOTO 95 353 90 ZH=ZH2 354 C 355 C CHECK IF MELTING HAS BEGUN - IF "YES", ADJUST THE HEAT 356 C CAPACITY TO INCLUDE THE ENTHALPY OF FUSION OVER A FOUR 357 C DEGREE TEMPERATURE SPAN 358 C 359 IF (Y(1).GE.1390.0) GOTO 100 360 95 HT=-59916.6+195.3 3*Y(1)+0.0192 36*Y(1)**2 361 GOTO 200 362 C 363 C CHECK IF MELTING IS COMPLETED - IF "YES", ADJUST THE 364 C APPROPRIATE VARIABLES TO ACCOUNT FOR LIQUID PbS 365 C 366 100 IF (Y(1).GE.1394.0) GOTO 120 367 ZH=ZH1*2.0 368 CP=38230.7+0.0 38473*Y(1) 369 HT=-77125.5+258.81*Y(1) 370 DHS=874381.0-102.82 5*Y(1)-0.004284*Y(1)**2 371 DHRPBS=-3703383.6+260.189*Y(1)+0.010833*Y(1)**2 372 S.-1398980.2/Y(1) 373 GOTO 200 374 c 375 c BEGIN SHORTENING TIME STEP TO AVOID OVERSHOOTING THE 376 c BOILING TEMPERATURE (ONSET OF STAGE THREE) 377 c 378 1 20 IF (Y(1).GE.1550.0) GOTO 140 379 ZH=ZH1*2.0 380 -GOTO 160 381 1 40 ZH=ZH2*2.0 382 1 60 CP=258.81 383 HT=-77125.5+258.81*Y(1) 384 DHS=874381.0-102.825*Y(1)-0.004284*Y(1)**2 385 DHRPBS=-3703383.6+260.189*Y(1)+0.010833*Y(1j**2 386 &-1398980.2/Y(1) 387 200 EMOL=AREA*PEQPBS/(2.0*PI*MPBS*8.314*Y( 1))**0.5 388 RREACT=1.0/(RXNRES+DIFRE1)*P02/PT 389 C 390 C CHECK FOR TRANSITION TO STAGE TWO 391 C 392 IF (EMOL.GE.RREACT) GOTO 220 393 F(1)=(AREA*(HTR+HTC)*(TF-Y(1))-RREACT*MPBS 394 &*(DHS+DHRPBS-HT))/(MT*(CP+(F(2)*ZH)/MT*700.)) 395 F(2)=RREACT 396 GOTO 240 397 220 F(1)=(AREA*(HTR+HTC)*(TF-Y(1))-P02/(DlFRES*PT)*MPBS 398 &MDHS+DHRPBS-HT) ) / (MT* (CP+(F ( 2 ) *ZH)/MT*700 . ) ) 399 F(2)=(1.0/DIFRES)*P02/PT 400 240 MT=MAPBS-Y(2)*0.23926 401 F(3)=9.8l-9.0 *ANUFG*(Y(3)-VF)/(RO*RP**2*2.0) 402 F(4)=Y(3) 403 260 RETURN 404 END 405 C 406 C SUBROUTINE FUNC2: CALCULATION OF STAGE THREE OF THE MODEL / 155 407 C 408 C DEFINE VARIABLES: 409 C F(1) =d(PARTICLE VOLUME)/dt (m**3/s) 410 C F(2) = RATE OF VAPOURIZATION (moles/s) 41 1 C F(3) = RATE OF LIQUID PbS CONSUMPTION (moles/s) 412 C F(4) = d(PARTICLE MASS)/dt (kg/s) 413 C F(5) = PARTICLE ACCELERATION (m/s**2) 414 C F(6) = PARTICLE VELOCITY (m/s) 415 C 416 SUBROUTINE FUNC2(X,Y,F) 417 IMPLICIT REAL*8(A-H,0-Z) 418 DIMENSION Y(10),F(10),T(10),S(10),G(10) 41.9 REAL*8 X,Z,Y,F,H,HMIN,E,G,S,T,HTC,TF,DHS 420 REAL*8 DHRPBS,AREA,PCNTM,MPBS,MAPBS 421 REAL*8 RP,RB,EPSIG,HTR,DlFRES,RXNRES,RL,MT 422 REAL*8 PEQ,PT,CT,DPBS,D02,CP,HT,RO,PI,LE,P02,XA,ZH1,PEQPBS 423 COMMON /BLK1/Z,H,HMIN,E,G,S,T,HTC,TF,DHS,PCNTM COMMON /BLK2/DHRPBS,AREA 424 425 COMMON /BLK3/RB,EPSIG,HTR,XA,ZH1,RXNRES 426 COMMON /BLK4/PEQ,PT,CT,DPBS,D02,CP,HT,RO,PI,LE,P02,DlFRES,RL 427 COMMON /BLK5/GMW,TC,TG,ANUFC,ANUFG,RP,TUBEL,ROP,VF,REP,ZH 428 COMMON /BLK6/ROF,RPIN,PEQPBS,MPBS,MT,MAPBS 429 COMMON /BLK7/EMOL,RREACT,ZH2 430 RP=(Y(1)/(1. 33 33 3*PI ))**0.333333 431 AREA=4.0*PI*RP**2 432 CT=2.0*12187. 3/(.1609. 5+TF) 433 HTR=EPSIG*(TF**2+1609.5**2)*(TF+1 609.5) 434 HT=-77125.5+258.81*Y(1) 435 DHRPBS=-1596882.-9.613*Y(1)+0.006441*Y(1)**2+2 316008./Y(1) 436 CP=258.81 437 F(1)=F(2)*MPBS/RO-F(3)*MPBS/CT1 438 F(2)=(AREA*(HTR+HTC)*(TF-1609.5)+F(4)*(DHRPBS/MPBS-•HT) 439 &/(MPBS*DHS) 440 F(3)=-1.0*F(2)-0.666667*F(1)*CT 441 F(4)=-0.66667*MPBS*CT*F(1) 442 F(5)=9.81-9.0*ANUFG*(Y(3)-VF)/(RO*RP**2*2.0) 443 F(6)=Y(3) 444 MT=Y(4) 445 RETURN 446 END 447 C 448 C SUBROUTINE FUNC3: CALCULATION OF STAGE FOUR OF THE MODEL 449 C 450 C DEFINE VARIABLES: 451 C F ( 1 ) = d(PARTICLE TEMPERATURE)/dt (kg/s) 452 C F(2) = d(PARTI CLE MASS)/dt (kg/s) 453 C 454 SUBROUTINE FUNC3(X,Y,F) 455 IMPLICIT REAL*8(A-H,0-Z) 456 DIMENSION Y(10),F(10),T(10),S(10),G(10) 457 REAL*8 X,Z,Y,F,H,HMIN,E,G,S,T,HTC,TF,DHS 458 REAL*8 DHRPBS,AREA,PCNTM,MPBS,MAPBS 459 REAL*8 RP,RB,EPSIG,HTR,DlFRES,RXNRES,RL,MT 460 REAL*8 PEQ,PT,CT,DPBS,D02,CP,HT,RO,PI,LE,P02,XA,ZH1 1 ,PEQPBS 461 COMMON /BLK1/Z,H,HMIN,E,G,S,T,HTC,TF,DHS,PCNTM 462 COMMON /BLK2/DHRPBS,AREA 463 COMMON /BLK3/RB,EPSIG,HTR,XA,ZH1,RXNRES 464 COMMON /BLK4/PEQ,PT,CT,DPBS,D02,CP,HT,RO,PI,LE,P02,DlFRES,RL / 156 465 COMMON / B L K 5 / G M W , T C , T G , A N U F C , A N U F G , R P , T U B E L , R O P , V F , R E P , Z H 466 COMMON / B L K 6 / R O F , R P I N , P E Q P B S , M P B S , M T , M A P B S 467 COMMON / B L K 7 / E M O L , R R E A C T , Z H 2 468 C T 1 = 1 2 1 8 7 . 3 / Y ( 1 ) 469 R P = ( M T / ( C T 1 * M P B S * 1 . 3 3 3 3 3 3 * P I ) ) * * 0 . 3 3 3 3 3 3 470 A R E A = 4 . 0 * P I * R P * * 2 471 RB=2.0*RP 472 A K F = 0 . 0 1 9 3 + 0 . 0 0 0 0 4 7 4 * ( Y ( l ) + T F ) / 2 . 0 473 ANU=2.0 474 D 0 2 = 6 . 0 1 D - 1 0 * ( ( Y ( 1 ) + T F ) / 2 . 0 ) * * 1 . 7 5 475 H T C = A N U * A K F / ( 2 . 0 * R P ) 476 H T R = E P S I G * ( T F * * 2 + Y ( 1 ) * * 2 ) * ( T F + Y ( 1 ) ) 477 C P = 1 5 5 . 9 9 - 0 . 0 0 8 5 6 * Y ( 1 ) 478 DHRPBS = - 1 5 9 6 8 8 2 . - 9 . 6 1 3 * Y ( 1 ) + 0 . 0 0 6 4 4 1 * Y ( 1 ) * * 2 + 2 3 1 6 0 0 8 . /Y (1 ) 479 C T = 2 * 1 2 1 8 7 . 3 / ( Y ( 1 ) + T F ) 480 H T = - 4 6 1 0 3 + 1 5 5 . 9 8 5 * Y ( 1 ) - 0 . 0 0 4 2 8 4 * Y ( 1 ) * * 2 481 F ( 1 ) = ( A R E A * ( H T R + H T C ) * ( T F - Y ( 1 ) + F ( 2 ) * ( D H R P B S / M P B S - H T ) ) 482 & / ( Y ( 2 ) * C P ) 483 F ( 2 ) = - 1 . 0 * M P B S * 0 . 6 6 6 6 6 7 * ( D 0 2 * 4 . 0 * P I * C T / ( 1 . 0 / R P - 1 . 0 / R B ) 484 & * ( P 0 2 / P T ) 485 MT=Y(2) 486 RETURN 487 END 

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