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Physical modelling of mass transfer in a Peirce-Smith converter Adjei, Emmanuel 1989

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PHYSICAL MODELLING OF MASS TRANSFER IN A PEIRCE-SMITH CONVERTER By Emmanuel Adjei B. Sc. (Hons.), University of Science and Technology, Kumasi, 1984 A THESIS SUBMITTED IN PARTIAL FULFILLMENT 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 thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1989 © Emmanuel Adjei, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of KA^TIA-L-S" MO b >v \*VT&*U rVUT Q-llO fc^G^U The University of British Columbia Vancouver, Canada D a t e A U C V U S T C U , l ^ g ^ DE-6 (2/88) ii ABSTRACT A 1/4 scale Plexiglas model of a copper converter has been used to measure the fraction of gas absorbed during horizontal gas injection. In this work, sulphur dioxide gas was injected into hydrogen peroxide solution under conditions where mass transfer in the gas phase was rate limited. The S02 absorption rate was measured as a function of the gas flow rate, tuyere submergence, number of tuyeres, and percent filling. The fraction of gas absorbed correlated well with the total trajectory length which included the spout height. It increased with tuyere submergence but decreased with the air flow rate and remained almost constant after a certain flow rate. The mass transfer parameter, kSOi, was computed from the results. It compared favourably with the work of previous investigators but could not be used to explain some of the results, especially, the effect of gas flow rate on the fraction of sulphur dioxide absorbed. Further analysis of the experimental results was based on bubble formation period and rise time. The fraction of gas absorbed increased with total residence time. About 74% of the injected gas was absorbed during the formation period. A mechanism to explain the overall absorption was proposed. An equation relating the absorption efficiency and the residence time was obtained for the physical model. This method was extended to the analysis of industrial data for the comparison of measured and predicted oxygen utilization efficiency. iii TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iii LIST OF SYMBOLS v LIST OF TABLES vii LIST OF FIGURES ix ACKNOWLEDGEMENTS xi 1 INTRODUCTION 1 1.1 Metallurgical uses of submerged gas injection 1 1.2 The copper converting process 1 1.2.1 The slag forming.stage 2 1.2.2 Blister copper forming stage 4 1.3 Problems that occur during copper converting 4 2 LITERATURE REVIEW 7 2.1 Gas injection studies 7 2.1.1 Jetting and bubbling behaviour 7 2 .1.1.1 Static regime N*. < 500 7 2.1.1.2 Dynamic regime 500 <N .^< 2100 8 2.1.1.3 Non-homogeneous jets N^> 2100 8 2.1.1.4Bubble frequency 10 2.1.2 Trajectory equation 11 2.2 Mass transfer models in non-ferrous metallurgy 13 2.2.1 High temperature experiments 13 2.2.2 Physical modelling 14 2.2.3 Mathematical modelling 16 3 OBJECTIVES 23 4 EXPERIMENTAL 24 4.1 Introduction 24 4.2 Gas injection system 24 4.2.1 Tuyeres - 26 4.2.2 Gas delivery system 26 4.2.2.1 Air 26 4 .2.2.2 Sulphur dioxide gas 26 iv 4.3 Method 27 4.4 Conditions for the test 33 4.5 Time exposure photographs 33 4.6 Determination of the fraction of S02 absorbed (X) 33 4.7 Determination of the trajectory length (S0) and the spout height (h) 35 5 EXPERIMENTAL RESULTS 37 5.1 Observations of injection phenomena 37 5.2 Trajectory length, spout height and spout area 43 5.3 Determination of mass transfer parameter, ksoa 49 5.4 Effect of tuyere submergence on the fraction of gas absorbed 51 5.5 Varying the number of tuyeres and tuyere spacing 51 5.6 Effect of gas flow rate 52 6 ANALYSIS AND DISCUSSION 58 6.1 Theoretical prediction of fraction of SOz absorbed (X) as a function of gas residence time 58 6.2 Determination of bubble residence time 62 6.3 Analysis based on the residence time 64 6.3.1 Relationship between the residence time and the fraction of gas absorbed 64 6.3.2 Bubble formation analysis 69 6.4 Effect of injection parameters on the fraction of gas absorbed 73 6.5 Mass transfer parameter, ksoa 74 6.6 Gas residence time and the fraction of gas absorbed 74 6.7 Efficiency analysis on industrial data 75 7 SUMMARY AND CONCLUSIONS 87 REFERENCES 89 APPENDIX A 93 APPENDIX B 98 APPENDIX C 101 V LIST OF SYMBOLS Ak[ Volumetric mass transfer coefficient, c m Y 1 . A Interfacial area of gas bubble, cm2. A 0 Cross-sectional area of tuyere, cm2. C Gas fraction in the jet. [Cso) 0 Concentration of sulphur dioxide gas entering the bath, mol cm"3. [C 5 0l Concentration of sulphur dioxide gas in the bulk of gas phase or gas bubble, mol cm"3. [C s ol Equilibrium concentration of sulphur dioxide gas at the reaction interface, mol cm-3. [CSO)f Concentration of sulphur dioxide gas leaving the bath, mol cm'3. DA-B Diffusion coefficient of substance A in B, c m V 1 . Diameter of tuyere, cm. d Diameter of jet at a horizontal distance x, from the tuyere exit, cm. f Frequency of gas bubble, s1. /. Gas phase mass transfer coefficient for forming bubbles, cm3 s1. 8 Acceleration due to gravity, cm s"2. h Height of spout, cm. K "Enhancement factor" during bubble formation. kx Liquid phase mass transfer coefficient, cm s1. kxa Liquid phase mass transfer parameter per unit length, cm2 s.t. K Gas phase mass transfer coefficient, cm s'1. Ka Gas phase mass transfer parameter per unit length, cm2 s1. kso2a Mass transfer parameter per unit length for sulphur dioxide, cm2 s1 m Slope of the H2S04 concentration-time plots, mol cm"3 s"1. N R . Reynolds number at the tuyere orifice. vi Modified Froude number. Mach number. nSo2 molar flow rate of sulphur dioxide to the bubble interface, mol s'1. p* Partial pressure of a soluble gas in the bulk of gas phase, arm. Partial pressure of sulphur dioxide gas at the tuyere exit, atm. Ps° rso2 Partial pressure of sulphur dioxide gas at the bath surface, atm. Q Gas flow rate, cm3 s1. Qso2 Molar flow rate of sulphur dioxide gas, mol cm'3. R Radius of rising bubbles, cm. R0 Radius of bubble during formation, cm. s Distance along jet trajectory from tuyere orifice, cm. Distance along jet trajectory from tuyere orfice to the quiescent level of the bath, cm. Dimensionless distance along jet trajectory from tuyere orifice. t Time, s. Residence time of bubble, s. k Bubble formation period, s. "o Nominal gas velocity at the tuyere, cm s'1. "* Bubble rise velocity, cm s1. "s Superficial gas velocity, cm s1. vB Volume of bath, cm3. w Mass of gas bubble, g. vD Volume of gas bubble, cm3. X Total fraction of gas absorbed. x< Fraction of gas absorbed during bubble formation. X Horizontal distance of gas jet, cm. XT Dimensionless horizontal distance of a gas jet. y Vertical distance of gas jet or tuyere submergence, cm. yr Dimensionless vertical distance of gas jet. vii £p Specific power density, watts kg"1. Xr A term in Equation 6.8. Xf A term in in Equation 6.12. 6,. Jet cone angle, degrees. Q0 Angle the tuyere makes with the horizontal, degrees. (I Dynamic viscosity, g cmV1. Density of gas, g cm"3. p, Density of liquid, g cm"3, a Surface tension, dynes cm"1. viii LIST OF TABLES Table 2.1 Mass transfer models for gas injection systems 21 Table 4.1 Range of variables used for the experiments 34 Table 5.1 Horizontal distance of jet as determined from photographs and trajectory equation with cone angle of 20° and 30° 42 Table 5.2 Values of the trajectory length, spout height and spout area (25% filling) 44 Table 5.3 Values of the trajectory length, spout height and spout area (30% filling) 45 Table 5.4 Values of the trajectory length, spout height and spout area (35% filling) 46 Table 6.1 Predicted X values using diffusion analysis 67 Table 6.2 Oxygen utilization efficiency data from the Noranda process 78 Table 6.3 Oxygen utilization efficiency data for copper converters from most plants around the world 79 Table 6.4 Oxygen utilization efficiency data from Magma Copper Company 80 ix LIST OF FIGURES Figure 1.1 Schematic of Pekce-Smith copper converter 3 Figure 2.1 Bubbling and jetting behaviour 9 Figure 4.1 Schematic of the physical model 25 Figure 4.2 Interface concentration profile for the SO^H202 system 28 Figure 4.3 Concentration of sulphuric acid in the bath as a function of time at various hydrogen peroxide concentrations 30 Figure 4.4 Concentration of sulphuric acid as a function of time at two sampling positions in the bath 31 Figure 4.5 Plot showing the reproducibility of the experimental results 32 Figure 4.6 Diagram indicating some of the parameters employed in the analyses 36 Figure 5.1 Time-exposure photographs of the bath during gas injection 38 Figure 5.2 Photographs showing liquid circulation and slopping in the bath 39 Figure 5.3 Diagram used in determining X p from the trajectiory equation 41 Figure 5.4 Effect of gas flow rate on spout height at different percent fillings 47 Figure 5.5 Effect of gas flow rate on the vertical cross-sectional area of the spout 48 Figure 5.6 Plot of ln(l-X) against (SQ + h) 50 Figure 5.7 Mass transfer parameter ksoa, as function of tuyere Reynolds number, N R e 53 Figure 5.8 Effect of tuyere submergence on the fraction of sulphur dioxide absorbed 54 Figure 5.9 Effect of number of tuyeres and tuyere spacing on the fraction of sulphur dioxide absorbed 55 Figure 5.10 Effect of gas flow rate on fraction of sulphur dioxide absorbed 56 Figure 5.11 Plot of Gas fraction and the velocity profile along the jet trajectory against the dimensionless horizontal distance 57 Figure 6.1 Gas bubble in contact with the liquid phase 59 Figure 6.2 Effect of sulphur dioxide flow rate on the fraction of gas absorbed 61 Figure 6.3 Effect of gas flow rate on the residence time 63 Figure 6.4 Plot of (1-X) against gas residence time 65 X Figure 6.5 Unsteady state diffusion in a sphere 66 Figure 6.6 Plot of enhancement factor against the specific power density 81 Figure 6.7 Plot showing the proposed absorption phenomena during gas injection for the physical model 82 Figure 6.8 Comparison of the mass transfer parameter, kga, obtained from the present work with that of other investigators 83 Figure 6.9 Prediction of the enhancement factor for the Magma data from the physical model 84 Figure 6.10 Predicted efficiency versus actual efficiency for the Magma data in Table 6.4 (slag blow) 85 Figure 6.11 Predicted efficiency versus actual efficiency for Johnson' s industrial survey presented in Table 6.3 86 Figure Al Air flow rate against differential pressure for the orifice plate with a diameter ratio of 0.4 96 Figure A2 Air flow rate against differential pressure for the orifice plate with a diameter ratio of 0.6 97 Figure BI Schematic of sulphur dioxide calibration set-up 99 Figure B2 Calibration curves for sulphur dioxide gas 100 Figure Cl Schematic of a jet with a rising trajectory 104 xi ACKNOWLEDGEMENTS I would like to thank my supervisor, Dr. G. G. Richards for his helpful supervision, continuous assistance and patience during the course of this research. My sincere thanks to Mr. P. Wenman and my fellow graduate students for their interesting discussions and encouragement. Financial suport provided by the Natural Sciences and Engineering Research Council is very much appreciated. 1 CHAPTER 1 INTRODUCTION 1.1 Metallurgical uses of submerged gas injection. Submerged injection of gas into liquid metals for refining purposes has been in use since the days of the Bessemer. Several metallurgical processes including nickel and copper converting, zinc slag furning, steelmaking and gaseous deoxidation of liquid copper depend on the mass transfer between the injected gas and the melt [1, 2, 3, 4]. Gas injection may be achieved by the use of tuyeres, porous plugs, or a lance, and the gas may be blown from the top, side or bottom of the furnace. The behaviour of the resulting gas bubbles is a subject of considerable interest in the fields of both Chemical and Metallurgical Engineering. In general, high production rates are achieved during gas injection due to the generation of large gas-liquid interfacial areas which result in high mass transfer rates. Gas injection has therefore become the most commonly used technique in metal smelting and refining. 1.2 The copper converting process The most important application of submerged gas injection in the non-ferrous industry is the converting of copper and nickel mattes. Mattes are molten sulphides of heavy metals, often containing oxides. Copper matte, which essentially consists of FeS and CU2S is an intermediate product in the extraction of copper from sulphide ores. These sulphide ores may be chalcopyrite (CuFeS2), bornite (Cu^FeS^ or chalcocite (C^S). LOW 2 concentrations of these minerals occur in an orebody ranging from 1/2 to 2% Cu. Processes involved in extraction are crushing, flotation, roasting, matte smelting and converting. In some cases, hydrometallurgical techniques are used whereby the crushed ore is leached with a suitable solvent followed by precipitation or electrowinning of the copper from solution [5]. The conversion of copper matte to metallic copper is carried out in a Peirce-Smith Converter (Figure 1.1). During converting air is blown through a number of horizontal, side-mounted tuyeres into the molten matte to oxidize iron and sulphur. The process, which takes place in two stages, is exothermic with bath temperatures ranging from 1100 to 1300°C. 1.2.1 The slag forming stage During this stage, iron sulphide is oxidized to ferrous oxide, magnetite and sulphur dioxide according to the following reactions: FeS+3/202 = FeO+S02 (1.1) 3FeS+502 = Fe304 + 3S02 (1.2) The oxides of iron then combine with the silica flux to form slag. As the slag builds up, it is periodically skimmed off and more matte and silica are added. The cyclic operation continues until the charge consists of essentially white metal (Cu2S). 3 Figure 1.1 Schematic of Peirce-Smith copper converter. 4 1.2.2 Blister copper forming stage This is the second stage of blowing and involves the oxidation of the white metal to blister copper according to the reaction below: Ci^S + 02 = 2Cu + S02 (1.3) The product (approximately 99% Cu) is further treated in an anode furnace to remove dissolved oxygen, i.e. 3[0] C u + CH4 = CO+ 2H20 (1.4) The copper is then refined electrolytically. In recent years, considerable work has been done in developing processes where the smelting of the concentrate, the slagging of the iron and the production of blister copper are carried out in a single furnace. Examples include the Worcra, Noranda and Mitsubishi processes [6]. 1.3 Problems that occur during copper converting Several of the problems associated with this process are directly linked to the gas injection process. Examples are accretion growth, tuyere blockage, refractory wear, tuyere erosion, splashing and slopping. In converting operations the tuyeres become plugged periodically with accretions which must be removed by forcing a punching bar through individual tuyeres. The gradual blockage of a given tuyere causes a reduction in the air flow which in turn reduces converter productivity. The exothermic reactions that occur in the furnace, coupled with the high temperatures of operation and the heat and mass flow patterns that exist in the bath, provide harsh conditions in the furnace which lead to tuyere and refractory erosion. The rate of refractory erosion may depend on tuyere punching practice, temperature variation in the converter, matte grade, blowing rate and refractory 5 type. Part of the energy of the injected air results in splashing, a situation where particles of liquid are carried out with the gas above the surface of the bath. Accretions are therefore built at the mouth of the converter which must be removed occasionally. Extensive research work has been pursued at the Department of Metals and Materials Engineering in UBC to investigate the following aspects of converter operations: gas injection into the bath, accretion growth at the tuyere tip, slopping of the bath, kinetics of bath oxidation and heat losses from the interior of the converter during out-of-stack periods. The study of these subjects has involved several types of lab experiments, plant measurements, physical and mathematical modelling [7, 8, 9, 10]. Brimacombe et al. [11] conducted high pressure experiments at Tacoma Smelter of ASARCO to address the problem of accretion growth at the tuyeres and tuyere blockage. At higher pressures, tuyere punching was not necessary but at lower pressures punching was required because of tuyere blockage. The accretion growth was also reduced or eliminated altogether with oxygen enrichment (28% 02). There was a reduction in the build up when air was injected into low-grade mattes because of the increase in the rate of heat evolution. The principal factors which determine the production rates of copper converters are the matte grade and the air blowing rate. In general, high oxygen efficiencies are expected during converting, however, the Johnson survey [12] indicates there are a significant number of operations with relatively low efficiencies. Much attention has not been paid to factors affecting oxygen utilization and its impact on converter productivity. Very few studies have been conducted in this area [12]. For example, it is not clear whether all the reaction between injected gas and the bath takes place below the bath surface or whether other gas-liquid contact areas are important during converting. 6 The processes that occur at the tuyeres including bubble formation and the subsequent reaction may depend on the gas flow rate, the pressure of injection and the heat transfer from the liquid to the bubble. The bubbles formed at the tuyeres tend to expand due to the heat transfer that takes place. Bustos et al. [8] undertook a study on injection phenomena in an industrial converter and measured the pressure pulses at the tuyeres during injection. Periods of constant, low pressure between pressure pulses were observed due to the interaction among adjacent tuyeres. The interaction of adjacent tuyeres led to the coalescence of bubbles which created a horizontal, unstable gas envelope at the tuyere line. Therefore, for part of the time, a given tuyere would feed the envelope with little resistance from surrounding liquid, and the tuyere pressure will remain low until the envelope collapsed. Since there are different regions in the gas-liquid zone of the converter, the extent of reaction in each region is likely to differ. That is, the fraction of gas that reacts during bubble formation may be different from that during bubble rise. If the gas reacted in each region can be estimated quantitatively, then effective control of the various parameters that influence the gas utilization efficiency in the bath can be exercised. Therefore it is necessary to understand the rate controlling process of the reaction during gas injection. A physical model of the converter was used in this work to measure mass transfer rates in injection regimes similar to the industrial converter. Variables such as gas flow rates, tuyere submergence, modified Froude number and gas residence time were employed in this study. 7 CHAPTER 2 LITERATURE REVIEW Since this work involves mass transfer studies in a converter during gas injection, it was necessary to review available literature that provides understanding of the various aspects of the topic. The areas which have been discussed are gas injection studies and mass transfer models in non-ferrous metallurgy. 2.1 Gas injection studies Studies have been conducted by various investigators on injection phenomena in non-ferrous reactors to predict what occurs in the melt. Depending on the type of experiments conducted and the nature of results obtained, various researchers have come out with different models that presumably depict the actual injection process in a converter. Some have observed a jetting behaviour in the bath [13] whilst others have observed that the injected air enters the converter in the form of bubbles [9]. 2.1.1 Jetting and bubbling behaviour. Oryall [7] has reviewed bubble formation in liquids. His findings indicated that three distinct regimes of bubble formation occur during injection. 2.1.1.1 Static regime NRe < 500 At very low flow rates the frequency of bubble formation is proportional to the gas flow rate while the bubble size is almost constant and depends only on the orifice diameter. 8 2.1.1.2 Dynamic regime 500 < N*, < 2100 In this range the bubble volume increases with gas flow rate while the frequency remains almost constant (usually higher than 80 bubbles per second). 2.1.1.3 Non-homogeneous jets NRe > 2100 At higher flow rates (NRe > 2100), a bubble stream is produced where larger bubbles of irregular shapes issuing from the orifice explode into smaller bubbles very close to the tip of the nozzle. McNallan and King [14] reported that at very high flow rates (NRe > 10000, N'Fr > 1000) the bubbling behaviour ceases altogether, and the gas enters the liquid as turbulent jet. In the paper by Hoefele and Brimacombe [9], the conditions which give rise to bubbling and steady jetting were delineated in terms of the modified Froude number, N'Pr and pg/p ; as shown in Figure 2.1. N'Fr is defined by the equation: 2 N>Fr= P ° U ° (2.1) pg is the gas density, p, is the liquid density, u0 is the nominal gas velocity at the tuyere, g is the acceleration due to gravity and d0 is the tuyere diameter. Motion pictures that were taken indicated that at low values of N'Fr, corresponding to blowing conditions in the copper converter (N'Fr = 10 - 14, Pressure = 60 kPa), the gas discharges in the form of bubbles at a frequency of about 10 s"1. At higher flow rates steady jetting predominated in which gas flowed from the tuyeres continuously. The flow regimes and the forward penetration of the gas depended on both the modified Froude number and the gas-liquid density ratio. 9 10 10' -I u. Z 10' 10 0- BOP (cstimottd) •L Transition rtgimt Steady i«t Bubbling I M o m converting 10 .6* Figure 2.1 Bubbling and jetting behaviour as predicted by Hoefele and Brimacombe [9]. 10 Ozawa and Mori [15] in their study on the behaviour of gas jets injected into liquid metal bath (nitrogen in mercury) observed that the injected gas does not form a continuous gas channel even at sonic velocities but breaks up at some distance from the orifice exit. Sano and Mori [16] studied the bubbling and jetting behaviour of gas liquid systems by injecting nitrogen gas (90-7800 cmV1) through an orifice (d0 = 0.2-0.4 cm) into water. In this work, the transition from bubbling to jetting occurred at a NMi (the ratio of gas velocity to the velocity of sound in air) greater than 1.3 -1.4. They observed that the transition occurred more abruptly with increasing ratio of gas to liquid densities. 2.1.1.4 Bubble frequency Irons and Guthrie [17] injected gas (Q = 0.15 - 100 em's'1, nozzle diameter = 0.16 -0.64cm) into molten pig iron and observed a constant bubble frequency of 10 s'1. Andreini et al.[18] reported a bubble frequency of 20 s"1 when liquid copper was deoxidized with a jet of CO (Q = 1 - 8 e m ' s ' 1 and d0 = 0.05 - 0.1 cm). When argon was injected into molten copper and silver at gas flow rates up to 19 cmV1, Fruehan [19] observed a bubble frequency of about 22 s'1. Patankar and Spalding [20] observed that at flow rates between 103 and 104 e m ' s " 1 , a constant frequency regime occurs in which the frequency of bubble formation remains constant at about 10 s"1, and the volume of bubbles formed and released become larger and larger. At very high flow rates (103 to 3»105 cm3 s'1), Hoefele and Brimacombe [9] observed tuyere pressure pulsations of 10 Hz in liquid metals which indicate the formation of large bubbles. These large bubbles or gas envelopes tend to exhibit hydrodynamic instability [21]. Bustos [22] in his experiments in a physical model of the copper converter, observed a dependency of bubble frequency on tuyere diameter, gas flow rate and the modified Froude number. Using five tuyeres with diameters of 1.6 cm and 1.2 cm, bubble 11 frequencies of 13 to 14 s"1 were obtained at low flow rates (5000 to 9000 cmV1 and modified Froude numbers between 0.4 and 2.5). An increase in gas flow rate resulted in an almost linear increase in the pulse frequency. Industrial measurements showed that bubbling frequencies also depended on the state of the refractory. The pulse frequency was about 14 s'1 for the first campaign in a newly relined converter but dropped to about 7 s"1 at the end of the 12th charge [22]. It can be summarized that at lower flow rates bubble frequency is a function of flow rate. At higher flow rates, however, most systems enter a constant frequency regime with a corresponding increase in bubble volume. This relationship can also be expressed as a constant bubble volume regime for very low gas flow rates and a constant frequency regime for higher flow rates. The bubble frequencies reported in the literature show that the formation of bubbles may differ from system to system. In industrial practice, furnace conditions such as the state of the refractory, heat transfer to the bubble and the bath circulation may have a major effect on bubble formation process. The work by Hoefele and Brimacombe [9] has shown that in matte converting and other industrial gas injection systems, the transition from bubbling to jetting regime can be delineated more appropriately by using the modified Froude number (which takes into account the inertia and bouyancy forces that predominates in industrial practice) and the gas-to-liquid density ratio. 2.1.2 Trajectory equation Themelis et al. [13] used momentum and mass balances to derive a dimensionless equation that described the trajectory of a gas injected horizontally into liquid. The jet equation was found to be a function of the modified Froude number. The equation is shown below: 12 j = 4N Fr tan2(9e/2) cos0o 1 + (2.2) yr is the dimensionless vertical distance, xT, the dimensionless horizontal distance, 0C, the jet cone angle, 90, the angle the tuyere makes with the horizontal and C is the gas fraction in the jet. The trajectories of air jets injected into water, as predicted by the above equation were found to correlate well with averaged jet boundaries measured experimentally by a photographic technique. It was concluded that the cone angle is a function of the jet fluid properties and that the jet diameter was a function of the horizontal distance from the orifice. Engh and Bertheussen [23] modified the above model to calculate the trajectories of two-phase submerged jets. In their model, the jet diameter, d, was a function of the distance along the jet axis. The predictions made by this model were in agreement with that predicted by Themelis et al. [13]. The equations used by the two models for the jet diameter are shown below: where s is the distance along the jet trajectory from the tuyere orifice. In computing the trajectory of the air jet, Themelis et al. [13] used a cone angle of 20°. Note that Oryall and Brimacombe [24] found that the apparent cone angle of an air jet injected into a liquid metal (mercury) was 155°. Themelis et al. [13]: d = 2x tan(0e/2) (2.3) Engh and Bertheussen [23]: d=2s tan(9e/2) (2.4) 13 2.2 Mass transfer models in non-ferrous metallurgy Several investigators have used different models to characterize the mass-transfer coefficients, the gas-liquid interfacial areas and the efficiencies of gas-liquid reactions that occur in the industrial practice. These models may be grouped into three categories: high temperature experiments, physical modelling and mathematical modelling. 2.2.1 High temperature experiments The important feature of high temperature experiments is that the thermal conditions in the model are similar to actual industrial practice. Themelis and Schmidt [25] injected a jet of CO vertically upwards into a copper melt at 1170°C. The effects of submersion, orifice diameter, and gas velocity were examined. The rate of deoxidation of liquid copper (0.1 < percent oxygen < 1) was controlled by gaseous mass transfer and the mass transfer parameter (Jc^a, the product of mass transfer coefficient in the gas phase and the interfacial area per unit length of the jet trajectory) varied from about 40 to 240 cm2s"1. The rate of reduction was not affected by the change in melt oxygen content from 1% to less than 0.1%. Nanda and Geiger [26] studied the kinetics of deoxidation of molten copper with CO at 1135°C by monitoring the oxygen content of the bath. The gas, which formed bubbles in the melt, was injected through various-sized orifices. In this analysis, liquid phase control occurred at very low oxygen concentrations in the melt (wt% O < 0.05). It was also observed that kt (mass transfer coefficient in the liquid phase) decreased with flow rate. Fruehan [19] measured the mass transfer rate between submerged bottom-blown and top blown gas jets and liquid melts. At oxygen levels less than 0.025% in copper, the 14 rate was controlled by liquid-phase mass transfer. When the oxygen level was greater than 0.025% the rates were fast and essentially controlled by the rate of gas injection. The mass transfer rates increased with gas flow rate and the depth of jet submersion. This experiment was conducted with orifice diameters of 0.085 and 0.22 cm and gas flow rates varying from 1.8 to 18 cmY1. Though high temperature experiments provide very useful information, it is difficult to gather precise data under such conditions. 2.2.2 Physical modelling Most injection studies are conducted with clear liquids to facilitate optical observations and at ambient temperatures for ease of experimentation. In this type of modelling some principal characteristics of an existing industrial process are studied in a suitable model. The model may not reproduce the entire system but only the characteristic aspect under study based on appropriate similarity criteria. Inada and Watanabe [27] used the NaOH-C02 system to study the efficiency of a bottom blown converter. They measured the rate of absorption of C02 in NaOH by monitoring the pH of the system. The test, which was conducted with pure COz at Reynolds Numbers ranging from 103 to 104, showed mass transport control in the liquid phase. The volumetric mass transfer coefficient, Akx, was a function of the gas flow rate to the 0.65 power and increased with submergence. The nozzle diameter had little effect on the mass transfer parameter. In some of the experiments, the gas was injected horizontally as well as from the top. The highest volumetric mass transfer coefficient was obtained with the bottom injection followed by the horizontal. The efficiency was found to depend on submergence. 15 Taniguchi et al.[28] measured the volumetric mass transfer coefficient, Akx, in the bubble dispersed zone for the absorption of C02 in water in a cylindrical vessel. The gas was bottom injected through a 6 mm diameter nozzle and the flow rate ranged from 16.7 to 167 em's'1. The rate of absorption was controlled by liquid phase transport. The volumetric mass transfer coefficient decreased with increasing bath depth and vessel radius at a constant flow rate. It varied from 0.1 to 3 em's'1. Haida and Brimacombe [29] evaluated the extent of solute inhomogeneity in a bath and the influence of the inhomogeneity on the efficiency of refining reactions. Pure C02 was injected into aqueous bath of 0.1 NaOH solution. The NaOH concentration at different locations in the bath and the efficiency of C02 absorption were measured. The maximum efficiency obtained in the ladle shaped vessel was between 18.1 and 22.8%. Brimacombe et al. [30] injected about 1% S02 gas into a solution of hydrogen peroxide to determine the rate of reaction between a horizontal submerged gas jet and a liquid. The study was undertaken to simulate the rates of reaction in a copper converter. Throughout the experiments mass transfer in the gas phase was found to be rate limiting. The absorption rate of S02 was measured as a function of jet Reynolds number (104 - 4 • 104) and tuyere diameter (0.238 < d„ < 0.476 cm). The mass transfer parameter (kS02a), which was assumed to be independent of the position along the jet trajectory, increased with Reynolds Number and the orifice diameter. Their analysis was based on steady state conditions where the fraction of gas absorbed was a function of the jet trajectory. The fraction of gas absorbed ranged from 0.59 to 0.91. Rocha and Guedes de Carvalho [31] measured the gas-side mass transfer parameter with NH3-HCl system. NH3 was top injected through a submerged lance at various depths. The absorption unit was divided into three sections: bubble formation, bubble rise and surface regions. The fraction of NH3 absorbed was always above 0.92, even at the 16 minimum submersion (3 cm). Steady state conditions were assumed and the gas-side mass transfer parameters were obtained from a plot of the fraction that was not absorbed against the height of submersion by using an equation of the form: ln(l-X) = ln 1-Q (2.5) X is the fraction of gas absorbed and fg is the gas-side mass transfer coefficient for a forming bubble. The correlation which was obtained showed that the mass transfer parameter was a function of the gas flow rate (kga = 0.566(2 °'9). The fraction of gas absorbed during bubble formation was reported as 0.9. The maximum/g value obtained using equation 2.5 and experimental data was 400 em's'1. A SO2/N2 mixture was introduced into an agitated vessel of NaOH solution by Chang and Rochelle [32] to measure the absorption rate of sulphur dioxide under steady state conditions. The rate was calculated from the liquid phase material balance. The mass transfer parameter kga, which was obtained from the rate, was found to be independent of the gas flow rate. 2.2.3 Mathematical modelling In metallurgical processes, high temperatures are encountered and the actual chemical reactions are normally assumed to be fast. The rates may be controlled by transport processes either in the gas phase or the liquid phase and the reactions are said to be mass transfer.limited. Depending on the nature of a process, mathematical expressions are derived to predict the parameters associated with that process. Experiments are usually conducted to confirm these predictions. 17 Below are examples of mathematical models developed by various investigators to predict some of the parameters associated with non-ferrous processes. 1. Mass transfer theory for a submerged jet was presented by Brimacombe et al. [30], In this analysis, a reactant A, in the gas phase reacted at the gas-liquid interface with reactant B in the liquid according to the equation below: aA+bB -»Products (2.6) The equation below was used to predict which phase controls the rate of reaction, based on the assumption that the chemical reaction is instantaneous. aik^CRT N = — — r ^ — (2.7) b(kga)Ps P% is the partial pressure of the soluble gas. If N > 1, gas phase control prevails, while liquid transport is limiting for N < 1; the rate controlling step changes from one phase to the other when N = 1. An experiment was performed by the same authors to determine the mass transfer parameter by injecting 1% S02 into H202 solution. Steady state conditions were assumed and the fraction of gas absorbed was found to be a function of the trajectory length. The mass transfer parameter, kS02a, was calculated from the following relation: Ps° * SO -Q In—- = ksoaS0 (2.8) po r SO. 2 Pso2, PSQ2 are partial pressures of S02 in the gas at the orifice and at the surface of the bath respectively. S0 is the distance along the jet trajectory from orifice to the surface of the bath (calculated from the Themelis et al. [13] model). 18 The left hand-side of the above equation was plotted against S0 and the slope of the straight line yielded kS02a. 2. In the deoxidation of liquid copper by a submerged CO jet Themelis and Schmidt [25] considered a differential element dy, in the vertical direction and equated the rate of decrease of CO concentration (C c o) to the rate of mass transfer across the gas film interface, i.e., -Qd{CC0) = kgady(CC0-Q) (2.9) This equation was used to determine the gas phase mass transfer parameter assuming that kga remains constant with distance from the orifice. The relation was also used to predict the CO utilization efficiency. Most of the models that predict the mass transfer rates with gas phase control are based on the tuyere submergence or the trajectory length of the jet. It is often desirable to understand what happens at the tuyere exit during bubble formation period. 19 3. In the model by Ashman et al. [10], the dynamics of bubble formation in a copper converter were examined. Mathematical equations were derived based on mass, heat and momentum balances on a gas bubble during the formation period. The resulting equations were solved with several assumptions in order to predict the effect of certain factors on bubble growth and detachment. The parameters that were considered are the gas flow rate, bath temperature, and tuyere diameter. The model also incorporates the effects of heat transfer to the bubble, chemical reaction between the bubble and the bath circulation velocity. These parameters were based on a standard set of operating conditions for an industrial practice as shown below: Air flow rate per tuyere 2.5(105) em's1 Inlet air temperature 25°C Oxygen content of air 20.9% Bath temperature 1200°C Tuyere diameter 4 cm Heat-transfer coefficient 0.029 W/cm2K Mass-transfer coefficient 60 cm s1 20 The model was used to predict the concentration of 02, temperature, volume and rise velocity of the bubble as a function of time. The mass transfer coefficient used in this model appeared to be quite arbitrary and it is likely that its value would change with injection conditions such as gas flow rate and tuyere diameter. During the bubble growth (0.1 s), the volume of the bubble increased to 4.4xl04 cm3 and a temperature of 312°C at a flow rate of about 3.33xl05 em's'1. The partial pressure of 02 dropped from 0.21 to 0.13. An increase in air flow rate from 1.67xl05 to 3.33xl05 em's"1 caused the bubble frequency to decrease from 12 to 9 s'1 in contrast to the measurements of Bustos et al. [8]. Increasing the convective heat transfer coefficient caused the frequency to drop, with a corresponding increase in the bubble volume. An increase in the bath velocity from stagnant conditions to 240 cm s'1 increases the bubble frequency from 3 to 15 s"1. Although the model predicts that about 40% of the 02 that enters the bubble is consumed during bubble formation, the model could not predict the total 02 efficiency since the fraction of gas absorbed during bubble rise was not considered. Although these models provide some insight into converter operations, further work needs to be done in order to predict the entire fraction of gas that reacts during injection. Table 2.1 is a summary of most of the mass transfer models dicussed in the literature review. Table 2.1 Mass Transfer Models for Gas Injection Systems. Injection System Type of Injection Parameter! Studied Rate Controlling Step No. of Tuyeres Tuyere Diameter (mm) Gas Flow iUteton'/s) Equations used Purpose Ref l CO-Cu melt wilhO.1-1%0 1 vertically 1 upward! (0.4-2*3°omJ/i) gas phase single lance 1.6-9.5 80-1670 1.4-417 1000-9100 ln(C0) V» y Q %t/£ = 10c(i _/*.*>>*] Deoxidation of liquid copper 25 CO-Cu melt with .05-.005%O j vertically I downwards *. (0.4 cm/!) (max.) liquid phase single orifice in the melt 1.6-4.8 0.4-15 1.5xlO*-0.5 1-90 Kinetics of deoxidation of molten copper 26 COyNaOH vertical (torpedo and ladle ihaped) (0.043*076) liquid phase 4 1.6 and 3.2 168-1325 1.6x10'to 3.12xiaJ 2000-30000 Inhomogeneity of solute in a bath during gas injection 29 COyNaOH bottom fraction absorbed 1 10 400-1300 0.47-5.0 7100-20000 Factors affecting rale of absorption between gas bubble! and liquid 33 COyHjO bottom (O.I-3xI05/i) liquid phase 1 nozzle 6 16.7-166.7 0011-1.1 426-4264 f ««*<C.-C) Mut transfer in gas stirred systems 34 COyNaOH top bottom side 1.3.5 1.3 5.5 1.67-83.3 .09-229 10*10* Efficiency of gas-liquid reactions 27 Air-Oil///,0 1 with thymol 1 top bottom (l-oOcm1/!) liquid phase 1-4 2,4.8 16-167 dC M — »—(C-C) Mass transfer between slag and steel 35 Ar-metal/slag with Si bottom *»(cm/s) (0.01-0.05) Si in the metal phase 1 1 3100 0.54-414 100-10000 Mass transfer between slag and metal 36 Air //,0/bcnzene with iodine top bottom (wp) 4*1=9.40/" (bottom) 4yi = I8.20/)J liquid phase 6 1 250.417. 580 4.02 (avg) K ' ( T ) = V ( C ' ~ C ) Q-BOP 37 Table 2.1 continued. Injection System Type of Injection Parameters Studied Rate Controlling Step No. of Tuyeres Tuyere Diameter (mm) Gas Flow Rate (on'/i) Equations used Purpose Ref Ar-Ag II top OyAg 1 bottom COCu 1 melts I (O-oon'/i) liquid phase 1 0.85.2.2 1.8-18 .002-17.6 3-2500 (7, absorption in Ag. Deoxidation of C« and Ag 19 SOrHfii 1 horizonul W (2O-380cmV») gas phase 1 2.38 3.18 4.76 125 3800 25-4000 lONilO' Copper Convening 30 NHyHCl 1 vertically 1 downwatds *,« «10.4G"D^  (bubble rite) gas phase 5 10 0-600 21.5 (max.) 8800 (mas.) Mass transfer in Absorbers 31 ts) 23 CHAPTER 3 OBJECTIVES It was concluded from the literature review that although mass transfer studies have been undertaken on side injection in non-ferrous systems, no thorough analysis of the copper/nickel converter has been done. It was therefore important to consider the following objectives in this work. 1. To determine the mass transfer rates in a physical model similar to an industrial converter by considering the effect of variables such as bath depth, modified Froude number, gas flow rate and jet residence time. 2. To develop a model that will predict the fraction of gas absorbed and the subsequent efficiency in the system when the injection parameters are known. 3. To relate the findings of this investigation to experimental observations of other physical modelling studies and to copper converter practice. 24 CHAPTER 4 EXPERIMENTAL 4.1 Introduction. Similarity criteria: Since the operating conditions and the physical properties of a model system differ from that of an industrial plant it is difficult to simulate an industrial process with a physical model. Similarity criteria are therefore used in the design of models for the study of an existing or a new process. The problem with physical modelling of a high temperature gas-liquid system lies in the number of physical parameters involved in the process. To attain fluid flow similitude between two gas-liquid systems, four conditions must be satisfied: dynamic, kinematic, geometric and thermal similarity. The use of these similarity criteria in the design and construction of a physical model of a copper converter have been discussed by Bustos [22]. His results have been employed in this work. 4.2 Gas injection system The apparatus employed is illustrated in Figure 4.1. The equipment consisted of a converter shaped vessel which was a sectional, 1/4 scale Plexiglas model of a standard Peirce-Smith copper converter (diameter of 4 m and length 9 m) with five tuyeres. The converter-shaped vessel was constructed from a Plexiglas plate, 0.95 cm thick with internal diameter of 78 cm and length of 27 cm. An 8.5 by 30 cm opening was cut into the upper face of the model, where a fume hood was connected to exhaust excess S02 25 - 6 7. Tuyere Manifold 8. Tuyeres 9. Plexiglas Converter 10. Sampling Tube 11. Fume Hood 2 I E itr 4 1. Compressor 2. Globe Valve 3. Manometer 4. Plate Orifice 5. Sulphur Dioxide Gas 6. Rotameter Figure 4.1 Schematic of the physical model. 26 4.2.1 Tuyeres The assembly consisted of a 5.8 x 7 x 24.8 cm Plexiglas block in which a 5 cm diameter hole was drilled. Into this hole was inserted a 33 cm long Plexiglas bar in which five threaded 7/8" NF holes, 5 cm apart had been drilled. The tuyeres of different diameters were made from 22.2 cm diameter brass rods in which a hole was drilled axially. The external surface was threaded with 7/8" NF thread to allow the tuyeres to be screwed into the tuyere assembly. 4.2.2 Gas delivery system. 4.2.2.1 Air A Sutorbilt 4 MF air compressor, with a capacity of 0.08 mV1 at gauge pressure of 34.5 kPa (5 psig) was used as the air source. The gas flow rate which was measured by an orifice plate was controlled by a globe valve. The calibration of the flow rate is shown in Appendix A. A water manometer was used to record the air flow rate through the orifice. At the exit from the plate orifice the air entered a 51 cm long cylindrical copper manifold (7.5 cm in diameter) for distribution to the tuyeres. 4.2.2.2 Sulphur dioxide gas The solute gas was delivered from a sulphur dioxide cylinder which was connected to a rotameter that measured the flow rate. The S02 flow rate was calibrated by using water displacement method as outlined in Appendix B. The S02 entered the main gas stream well before the manifold in order to ensure effective mixing with the air. 27 4.3 Method SO2/H2O2 system: It has been reported [10] that during converting the reactions that occur according to equations 1.1 and 1.3 are controlled by oxygen transport in the gas phase. A gas/liquid system that will produce an instantaneous, irreversible reaction with gas phase control was therefore required. SO2/H2O2 system, which has been used by Brimacombe et al. [30] for a similar purpose, was employed with an excess concentration of H202. This system has also been used by Chang and Rochelle [32] as a means of removing S02 from waste gases. Jean-Pierre Couilland et al. [38] used an aqueous solution of H202 to absorb S02 gas contained in a waste gas mixture. In this work a total absorption of the S02 gas was observed while the H202 concentration was in excess. Other possible systems that could have been used are SOJNaOH, NH3/H2S04 or NH-JHCl but these require high concentrations of NaOH and acids which would likely have ruined the tuyeres. COJNaOH system could not be used since C02 does not react instantaneously with alkali solution but combines relatively slowly through an intermediate step which would have greatly complicated the analysis. For the system selected, at the reaction plane, the concentration of S02 gas will be zero when the fraction of S02 in air is very low. The percentage of S02 in air varied from 0.03 to 1.0. The rate of reaction will then be equal to the rate at which the various components diffuse to the reaction plane. The actual kinetics of the chemical reaction is immaterial for such reaction [39]. The concentration profile may be represented by Figure 4.2, where the concentration of H202 is in excess. The reaction that occurs in the bath may be represented by the following equation: S02 + H202 = H2S04 (4.1) 28 Figure 4.2 Interface concentration profile for the SO-JH202 system. 29 Procedure: During injection, samples of solution were taken every two minutes. Each experiment lasted twenty minutes. The samples were then titrated against a standard solution of NaOH (l.OxlO'5 to 1.3X10"4 mol cm'3) to determine the concentration of sulphuric acid formed during injection in order to calculate the fraction of S02 gas that reacted. Methyl red was used as the indicator. The titration reaction is represented below: H2S04 + INaOH = Na2S04 + H20 (4.2) Since it was necessary to provide gas phase control in the system, four runs were done with different H202 concentrations as indicated in Figure 4.3. The run with H202 concentration of 9.7x10'5 mol cm'3 was selected for the subsequent experiments. In one run, samples were taken from two different positions in the bath (bottom and central positions). The aim was to find out if the concentration of the acid formed varied from one position to another in the bath. This was also used to examine the nature of mixing in the bulk of the liquid phase. The results, as shown in Figure 4.4 indicate that there was virtually no difference in concentration at the two positions, and that mixing was effective. In all the experiments samples were taken from the central position. It was important to determine how reproducible the experimental results were and therefore three experiments were run under the same conditions but on three different days. The results, as indicated in Figure 4.5 show good agreement. 30 Air flow rate = 15400 cm 3s" 1 Percent filling = 30% 5 A A A A A A Hydrogen Peroxide concentration + o A 9.7x10'5 mol c m 3 5.8x105 mol cm - 3 3.9x10"5 mol cm- 3 1.9x105 mol cm-3 240 480 720 960 Time (sec) 1200 1440 1680 Figure 4.3 Concentration of sulphuric acid in the bath as a function of time at various hydrogen peroxide concentrations. 31 20 0 120 240 360 480 600 720 840 960 1080 1200 Time (sec) Figure 4.4 Concentration of sulphuric acid as a function of time at two sampling positions in the bath. 32 Air flow rate = 15400 cm 3 s "1 Percent filling = 30% i + + • Experiment 1 + Experiment 2 o Experiment 3 — I — 240 —I 1 1 480 600 Time (sec) 120 360 720 840 960 1080 1200 Figure 4.5 Plot showing the reproducibility of the experimental results. 33 4.4 Conditions for the Test The tests were performed over a wide range of conditions to examine the influence of the following parameters on gas phase mass transfer in the system: i. Tuyere diameter ii. Tuyere submergence iii. Gas flow rate iv. Percent filling v. Tuyere spacing The range of variables are shown in Table 4.1. 4.5 Time exposure photographs Time exposure photographs of the spout and the plume were taken under the conditions investigated to delineate the various regions of the gas-liquid zone including jet trajectory and spout height. One side of the vessel was lighted through translucent paper which acted as a diffuser. A Yashica camera set at an f stop of 22 and a shutter speed of 1/2 to 3/2 s was used with 32 ASA black and white film. Some instantaneous pictures were also taken. 4.6 Determination of the fraction of S02 absorbed (X). A mole balance for sulphur dioxide gas was performed over the dispersed region (gas-liquid zone): Rate at which S02 leaves the bath = Moles injected per unit time - Moles reacted per unit time This is represented by the equation below: {Qso)=(lso2-mVB (4.3) Table 4.1. Range of Variables Parameters Range Tuyere diameter, cm 1.6 Tuyere submergence, cm Oto 13.1 (tuyere centre-line = 0) Gas flow rate cm 3 s'1 2xl0 3 to 4.4xl04 Modified Froude number 0.17 to 66 Number of tuyeres 1, 2, 3, 5 Tuyere spacing, cm 5,10,15 Percent filling, % 18.2 to 38.1 S02 flow rate, mol s'1 6.54X10"4 to 27.43X10"4 35 where qSOi: flow rate of S02 gas in moles per second. (Qsojf moles of S02 that leaves the bath per second m : the slope of the H2S04 versus injection time (f) plots (mol cm'V 1). VB : bath volume (cm3) For each experiment, the fraction of S02 absorbed was calculated as the ratio of the moles of S02 gas reacted to that injected, that is: mVB X = (4.4) Qso2 where X is the fraction of gas absorbed. 4.7 Determination of the trajectory length (5„) and the spout height (h) Figure 4.6 is a schematic diagram of the bath during gas injection. The diagram also shows the trajectory length and the spout height. The trajectory length up to the quiescent level was determined by solving equations (2.1) and (2.3). A fourth order Runge Kutta method for solving differential equations with double precision was used in this analysis. The procedure is summarized in Appendix C. The spout height as indicated in Figure 4.6, was determined from the time-averaged photographs. 36 X p : Horizontal distance determined from photographs as well as from the trajectory equation. Figure 4.6 Diagram indicating some of the parameters employed in the analyses. 37 CHAPTER 5 EXPERIMENTAL RESULTS 5.1 Observations of injection phenomena A series of photographs showing the phenomena in the bath during gas injection are shown in Figure 5.1 and Figure 5.2. At lower modified Froude numbers, from visual and photographic examination, the gas entering the bath forms bubbles which break into smaller bubbles at a short distance from the tuyeres. As the modified Froude number is increased the phenomena at the tuyere exit becomes quite complicated. Thus the injection phenomena at the tuyere line is characterized by the generation of an unstable gas-filled packet or envelope which at certain instants may cover several of the tuyeres simultaneously indicating strong interaction between them. This unstable envelope breaks up to form a bubble dispersion zone directly above the tuyeres. In the bulk of liquid phase there was a clockwise recirculating flow which became violent as the gas flow rate increased. This can be observed in Figure 5.2 (b). The time averaged photographs shown in Figure 5.1 indicate a curve along the boundary of trajectory that separates the dispersed region from the rest of the bath. It was observed that at the prevailing modified Froude numbers (N'Fr < 14, da - 1.6 cm), the gas did not penetrate very far into the liquid before moving in the vertical direction. The surface of the bath formed a wave-like curvature with amplitude decreasing as the extreme end of the bath is approached (Figure 5.2(a)). (a) 5 cm Figure 5.1 Time-exposure photographs of the bath during gas injection (a) Q = 5000 (b) Q = 30900 cm3 s1; filling = 38.1%. 40 A spout with varying height was formed at all flow rates direcdy above the tuyeres. A horizontal distance Xf, as indicated in Figure 5.3 was measured from the time-averaged photographs and compared with the calculated values from the trajectory equation (2.2) to establish the validity of the use of the trajectory equation. To obtain Xp from the trajectory equation, the following analyses based on Figure 5.3 were employed. For an incremental value, dx, in the horizontal direction, Xt andXp were calculated from the following equations: X,=|cos(90-e) (5.1) where tan(9) = dyldx Xp=x+Xi (5.2) Y p , which represents the vertical component of the jet trajectory, was used to calculate the lengthy which corresponds toXp, as indicated in equations 5.3 and 5.4. />|cos(9) (5.3) y=Yp-Yi (5.4) The values of Xp and y were computed for each incremental value of x, until a value of y which represents the actual submergence used in the experiment was obtained. This final value of Xp was then compared with that measured from the photographs. The jet cone angle was varied but those that gave comparable results were 20° and 30°. The results, as shown in Table 5.1, indicates that the differences between the values obtained from the two techniques are quite small, especially at higher percent filling. At lower percent filling the values predicted by using a cone angle of 30° seems to be quite good. 41 Figure 5.3 Diagram used in calculating Xp from the trajectory equation. 42 Table 5.1 Horizontal distance of jet as determined from photographs and trajectory equation (2.1) with a cone angle of 20° and 30°. QxlO" 3 35 percent filling 30 percent filling (em's 1) X P (cm) Xp(cm) Photograph Trajectory Photograph Trajectory equation equation 20° 30° 20° 30° ±0.5 ±0.5 15.40 5.7 5.58 5.3 5.5 5.3 5.05 30.90 9.3 9.87 9.1 7.5 9.2 8.64 43.80 12.2 12.85 11.7 10.2 12.05 10.21 43 5.2 Trajectory length, spout height and spout area. The results of these measurements for three different percent filling are shown in Tables 5.2, 5.3 and 5.4. The spout height was found to increase with gas flow rate and tuyere submergence. As shown in Figure 5.4, at the same flow rate, the spout height increased with submergence. The vertical cross-sectional area of the spout, which was determined from the pictures, also increased with gas flow rate. At a flow rate of about 1.5X104 cmV1, the rate of increase decreased (Figure 5.5). Sahajwalla [40] injected air through a nozzle located at the bottom of a physical model of a ladle furnace and measured the spout height and the potential energy in the spout as a function of flow rate. At an air flow rate of about 1350 cm3s \ an abrupt increase in the slope of the curves were observed, which was quite different from the observations in this work. In the present work, at lower flow rates the gas bubbles rise up close to the wall with little penetration. Thus the horizontal component of the gas jet is very small as compared to the vertical component. As the flow rate increases, the momentum of the jet also increases with a correseponding increase in the horizontal component. The horizontal component increases faster than the vertical component. The rate of increase in the vertical component starts to decrease at a flow rate of about 1.5xl04 e m ' s 1 . Thus the spout height tends to collapse as observed in Figure 5.4. In the work of Sahajwalla [40], as the gas flow rate increases, the spout height increases accordingly since there is no horizontal component. The abrupt decrease in the slope of the curve observed in the present work may be due to a "breakthrough effect" or gas channelling (accelerated escape of injected gas from the bath). 44 Table 5.2. (25% filling, tuyere submergence = 5.7 cm) Gas flow rate, Trajectory length, Spout height, h S0 + h Spout area Qxlfj3 (em's1) S0 (cm) (cm) (cm) (cm2) 5.00 5.9 2.2 8.1 27.3 6.80 6.0 2.5 8.5 41.6 8.73 6.2 3.1 9.3 55.9 9.30 6.3 3.4 9.7 62.4 10.92 6.5 4.2 10.7 82.6 15.40 6.9 5.7 12.6 97.5 30.90 8.7 7.6 16.3 116.1 43.80 11.0 9.6 20.6 161.4 45 Table 5.3. (30% filling, tuyere submergence = 8.6 cm) Gas flow rate, Trajectory length, Spout height, h Sa + h Spout area QxW3 (em's 1) 5 0 (cm) (cm) (cm) (cm2) 5.00 8.7 4.9 13.5 63.7 6.80 8.8 6.0 14.8 71.5 8.73 9.4 6.75 16.15 94.9 9.30 9.5 6.8 16.3 103.4 10.92 9.9 7.2 16.8 117.0 15.40 10.2 8.7 18.9 153.4 30.90 12.9 10.3 23.2 202.6 43.80 14.1 14.1 28.2 254.3 46 Table 5.4. (35% filling, tuyere submergence = 11.7 cm) Gas flow rate, Trajectory length, Spout height, h S0 + h Spout area (2x l0 3 (em's 1) S0 (cm) (cm) (cm) (cm2) 5.00 12.5 5.9 18.4 66.3 6.80 12.7 7.0 19.7 85.8 8.73 12.9 8.0 20.9 105.0 9.30 13.1 8.4 21.5 109.8 10.92 13.6 . 8.7 22.3 128.7 15.50 14.5 9.1 23.6 200.0 30.90 16.8 14.9 31.7 308.1 43.80 17.7 17.1 34.8 430.0 47 4 H Tuyere submergence • 35% filling 11.7 cm 30% filling 8.6 cm o 25% filling 5.6 cm ~l I I I I 1 1 1 — 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 Gas flow rate (cnfs1) Figure 5.4 Effect of gas flow rate on spout height at different percent fillings. 48 400 H Tuyere diameter = 1.6 cm • 35% filling + 30% filling O 25% filling co + o I I 1 — 10000 20000 30000 Gas flow rate (crrfsr1) i 40000 Figure 5.5 Effect of gas flow rate on the vertical cross-sectional area of the spout. 49 5.3 Determination of mass transfer parameter, k s o a . In order to determine ksoa, a rate equation for the reaction within the jet and the spout was derived by considering a differential element ds along the jet trajectory and equating the decrease of S02 concentration in the gas flow to the rate of mass transfer across the gas film interface. The equation, which is similar to that used by Themelis and Schmidt [25] and Brimacombe et al. [30] is given below: -Qd[CS0) =ksoads{[CSo) - [ Q o j J (5.5) where [CSOz] = equilibrium concentration at the reaction interface; in this system [Cso] = 0. Rearranging equation (5.5) and integrating with the boundary conditions [Cso) = [Cso)o,s=0and [Cso) = [Cso)f s=S0 + h yields In ^ kSo <* ~ ( 5 8 + /i) (5.6) According to equation (5.6), a plot of the left hand side against S0 + h should yield a straight line of slope -ksoalQ ( assuming that ksoa does not change with position). Since [Cso] /[Cs0] can be represented by 1-X, equation (5.6) can be written as T f T o ln(l-X) = —-g-(S0 + h) (5.7) where X is the fraction of gas absorbed. Figure 5.6 is a plot resulting from equation (5.7) for three different flow rates. Analyses were done based on the trajectory length, S0, alone but the correlation coefficients were not as good as those obtained using (S0 + h). This may indicate that a significant amount of absorption occurs in the spout region. 50 Figure 5.6 Plot of ln(l-X) against (S 0 + h) for different flow rates. 51 The mass transfer parameter, ksoa, determined from the above relationship was plotted against the tuyere Reynolds Number, NRe, as presented in Figure 5.7. Similar plots were obtained by [25] and [30]. As shown in the plot, ksoa values tend to increase with the tuyere Reynolds number. 5.4 Effect of tuyere submergence on the fraction of gas absorbed. At a constant flow rate of 1.54x10* cm3 s'\ the tuyere submergence was varied from the tuyere centre-line (0 cm) to 13.1 cm. The fraction of gas absorbed increased with tuyere submergence as depicted in Figure 5.8. As the submergence is increased, the time spent by the gas in the bath also increases, resulting in an increase in the fraction of gas absorbed. The fraction of gas absorbed was very high at all the submergences studied. Even with the bath surface at the tuyere level the fraction of gas absorbed was quite substantial, about 45%. 5.5 Varying the number of tuyeres and tuyere spacing The number of tuyeres were varied at a constant gas flow rate (9.3xl03 cm3 s"1) and a tuyere diameter of 1.6 cm. This is illustrated in Figure 5.9. There was a slight increase in the fraction of gas absorbed as the number of tuyeres decreased from three to one. With five tuyeres, the fraction of gas absorbed was slightly higher than that with three. The effect of tuyere spacing was observed when three tuyeres were used to conduct two experiments. In the first experiment, the three tuyeres were placed in the central positions of the tuyere assemblage with spacing of 5 cm. The tuyere spacing for the second experiment was 10 cm with two of the tuyeres close to the wall of the vessel. Though Figure 5.9 shows a slight decrease in the fraction of gas absorbed as the tuyere spacing is increased for the same number of tuyeres, the difference is not quite significant. Since 52 two of the tuyeres for the second experiment were close to the wall of the vessel, it is possible that the gas entering the bath through these tuyeres may have a reduced gas-liquid contact area. 5.6 Effect of gas flow rate Detailed studies on the effect of gas flow rate on the fraction of gas absorbed were undertaken with 25%, 30% and 35% fillings. The gas issuing from the tuyeres did not form a continuous jet but a series of gas pulses which penetrated slightly in the horizontal direction before turning upward. The forward penetration increased slightly with the modified Froude number. The plume and the spout formed a dispersed region of gas bubbles and liquid solution which covered the region above the tuyeres. At constant S02 flow rate, increasing the air flow rate resulted in a decrease of the fraction of gas absorbed up to a certain flow rate and stayed almost constant at all higher levels (Figure 5.10). The analyses based on the mass transfer parameter could not explain the observations in Figure 5.10. Equation (5.7) could not be used to explain some of the results that were obtained, and this may be due to the assumption that the mass transfer parameter does not change with position along the 'jet'. This assumption may be valid if the velocity of the jet were to remain constant throughout the system or the structure did not change. When the velocity profile (u^lu0) was calculated using the trajectory equation (2.2), a rapid drop was observed immediately in front of the tuyeres (Figure 5.11). (The constant, 2.8, that forms part of the horizontal axis appears during the derivation of the dimensionless horizontal length and represents the distance the jet cone extends into the tuyere). It was therefore decided to analyze the results in terms of the gas residence time. 53 2800 2400 -4 ^ 2000 H w C M E o "r" 1600 H © I o 1200 800 -\ 400 H 0.0 0.0 1— 10000 i 20000 1 30000 Figure 5.7 Mass transfer parameter kso a, as function of tuyere Reynolds Number NR e. 54 1.0 0.9 • Total gas flow rate = 15400 cm%-1 •a cu \ o C O CO CD •g 'x g T3 JO Q . 3 C O o c o co 0.8 — 0.7 — 0.6-0.5 — 0.4 • 0.3 £ 0.2 0.1 — 0.0 0.0 Tuyere Submergence Percent filling Non-dimensional units a. 13.1 cm 1.000 38.1 b. 11.7cm 0.893 35.0 c. 10.1 cm 0.771 33.2 d. 8.6 cm 0.656 30.0 e. 7.0 cm 0.534 28.2 f. 5.6 cm 0.427 25.0 g. 3.9 cm 0.298 23.0 h. 1.9 cm 0.145 21.0 i. tuyere centre-line 0.000 18.2 0.2 0.4 0.6 0.8 Tuyere submergence in non-dimensional units 1.0 Figure 5.8 Effect of tuyere submergence on the fraction of sulphur dioxide absorbed. 55 1 . 0 0 0 . 9 9 0 . 9 8 -0 . 9 7 -0 . 9 6 -0 . 9 5 0 . 9 4 -0 . 9 3 -0 . 9 2 0 . 9 1 0 . 9 0 0 . 8 9 0 . 8 8 -0 . 8 7 -0 . 8 6 -0 . 8 5 Gas flow rate = 9300 c m V Tuyere spacing Froude number • 1 4 . 9 + 1 0 cm 3 . 7 3 A 5 cm 1 . 6 6 o 1 0 cm 1 . 6 6 X 5 cm 0 . 6 0 (central position) Number of tuyeres Figure 5.9 Effect of number of tuyeres and tuyere spacing on the fraction of sulphur dioxide absorbed. 56 1 . 0 0 O 0 - 7 0 1 0 0 0 0 3 0 0 0 0 4 0 0 0 0 Gas flow rate (cm3 s'1) Figure 5.10 Effect of gas flow rate on fraction of sulphur dioxide absorbed. 57 Note: 2.8 = \/[2\an(Bj2)] Figure 5.11 Plot of gas fraction and the velocity profile along the jet trajectory as a function of dimensionless horizontal distance. 58 CHAPTER 6 ANALYSIS AND DISCUSSION 6.1 Theoretical prediction of fraction of S02 absorbed (X) as a function of gas residence time. This model presents a method of calculating the fraction of S02 absorbed within a bubble as it rises in the bath. The equations are based on the residence time of the bubble. The following assumptions were made: 1. The system is isothermal. 2. There is a constant flow of gas through the tuyeres. 3. The volume of the bubble does not change since the concentration of the reactive gas in the bubble is very small. 4. Reaction is irreversible and instantaneous. 5. The bubbles are spherical in shape and unaffected by adjacent bubbles. 6. The gas inside the bubble is back-mixed. Figure 6.1 shows the gas bubble in contact with the hydrogen peroxide solution. The rate equation for the reaction between S02 in the bubble and the liquid can be derived by considering a differential time element dt, and equating the decrease in S02 concentration in the bubble to the rate of mass transfer to the gas-liquid interface. Rate of S02 transfer to the gas-liquid interface is given by equation (6.1) 59 Figure 6.1 Gas bubble in contact with the liquid phase. 60 nSOl = lcsoA[[Cso)-[Cso)^ (6.1) Rate of decrease of S02 in the bubble is represented by: nso2 = -Vb—^- (6-2) where Vb is the bubble volume. The other variables are as defined before. Equating the two equations above gives _ V b ^ ^ = k s o ^ [ C s o } - [ C s o ) ^ ( 6 .3) Since the reaction is assumed to be instantaneous and irreversible, [Cso) - 0. Equation (6.3) was integrated with the following boundary conditions: [Cso) = [Cso)0, t = 0 and [Cso) = [Cs0)f t = tr This yields In r[CSo)f\ kS0A [CSo)\ Vb r where tT is the residence time of the bubble. This can also be expressed as (6.4) ln(l-X) = —-f-tr (6.5) where X is the fraction of gas absorbed. Knowing the fraction of gas absorbed from experiments and the residence time, it is possible to determine the mass transfer parameter for each flow rate. The above relationship indicates that the fraction of gas absorbed does not depend on the initial concentration of the gas. Some experiments were conducted by varying the flow rate of S02 gas. The results as indicated in Figure 6.2 shows that the fraction of gas absorbed remains almost constant with S02 flow rate. 61 1 . 0 0 0.98 H 0 . 9 6 0.94 H 0.92 H ^ 0 . 9 0 - \ 0.88 0.86 5 0 . 8 4 0.82 Total gas flow rate = 15400 cm%-1 Percent filling = 30 0.80 1 0.0008 1 0.0024 0.0004 0.0012 0.0016 0.002 Flow rate of sulphur dioxide (mol/s) ~i 1 — 0.0028 Figure 6.2 Effect of sulphur dioxide flow rate on the fraction of gas absorbed. 62 6.2 Determination of 'bubble' residence time An equation that expressed the residence time as a function of the horizontal distance of the jet was incorporated into the trajectory equation developed by Themelis et al.[13]. This was solved by using the same procedure for determining the trajectory length as outlined in Appendix C. The velocity of the jet and the gas fraction at any position were also determined. This procedure was used to calculate the residence time of the gas up to the quiescent level. It is necessary to note that residence time at a constant percent fill decreases with flow rate to an asymptote. The residence time in the spout was obtained by dividing the spout height by the calculated velocity of the gas at the quiescent level based on the assumption that the gas travels with a constant velocity in the spout. The sum of the residence time up to the quiescent level and that of the spout gave the total residence time of the gas in the bath. The residence time was plotted against the gas flow rate as shown in Figure 6.3. This parallels the observed absorption of S02 in Figure 5.11. That is, the relationship between the fraction of gas absorbed and the gas flow rate indicated in Figure 5.11 is similar to that between the residence time and gas flow rate. At lower flow rates, the gas trajectory is almost vertical and the residence time tends to be inversely proportional to the flow rate. The residence time is expected to decrease at higher flow rates as well but at higher flow rates the gas penetrates further into the bath leading to a longer trajectory length which requires a longer residence time. The decrease in residence time with flow rate is balanced by a longer residence time required by longer trajectory length which leads to a constant residence time at high flow rates. 63 0.35 0.30 0.25 H Percent filling (%) • 35 + 30 O 25 <| 0.20 - \ OJ O {= 0.15 •g CD CC o.io H 0.05 H <><> o ° 0.00 — I 1 1 1 1— 10000 20000 30000 Gas flow rate (cmV) 40000 Figure 6.3 Effect of gas flow rate on the residence time. 64 6.3 Analysis based on the residence time 6.3.1 Relationship between the residence time and the fraction of gas absorbed The plot in Figure 6.4 shows the fraction of gas that is not absorbed (7-X) as a function of gas residence time for all the gas flow rates and percent fillings investigated in this work at a tuyere diameter of 1.6 cm (30 data points). Following equation 6.5, regression analysis of the plot resulted in an exponential curve that can be represented by the equation below: (1-X) = 0.255e"942'r (6.6) The correlation coefficient of equation 6.6 in logarithmic form is 0.974. This equation shows that the fraction of gas absorbed increases with gas residence time. Figure 5.11 indicates that for each percent filling the fraction of gas absorbed is maximum at lower flow rates which also correspond to maximum residence time. The above relationship can be used to predict the fraction of gas absorbed if the gas residence time is known. Comparing equations (6.5) and (6.6), it can be seen that the pre-exponential constant has changed from 1 to 0.255. The value 0.255 indicates that when the residence time is zero part of the gas has been absorbed already. Further explanation of this result will be given in the next section. The equation obtained from the experimental analysis was found to be similar to that for unsteady-state diffusion in a sphere as shown in Figure 6.5 [2]. As an example, the predictions of this analysis for flow rates ranging from 5000 to 30900 cm3 s'1 at 30 and 35 percent fillings are shown in Table 6.1. 65 Figure 6.4 Plot of (1-X) against gas residence time. 0.1 0.2 0.3 0.4 0.8 0.6 0.7 V Figure 6.5 Unsteady state diffusion in a sphere [2] Table 6.1. Comparison of measured and predicted X values using diffusion analysis [2]. 35% filling 30% filling X (Expt.) X (Predicted) X (Expt.) X (Predicted) 0.982 1.000 0.940 0.970 0.973 1.000 0.929 0.965 0.968 0.990 0.926 0.950 0.964 0.990 0.924 0.945 0.953 0.985 0.923 0.940 0.945 0.980 0.900 0.920 0.941 0.970 0.888 0.890 68 The parameter was evaluated and used to determine X using Figure 6.5. DSOi_Air (0.139 cmV1) is the diffusion coefficient of S02 in air which was calculated from the equation developed by Hirschefelder et al.[41], is the gas rise time and R is the radius of gas bubble during its rise through the bath. R was determined from the correlation obtained by Liebson et al. [42] when they injected air into water at higher Reynolds numbers (NRe > 104, R = 0.335A£05). The average value ofR was 0.225 cm. This approach was able to predict values close to those of the experimental results. The actual equation for the diffusion curve shown in Figure 6.5 for a spherical bubble is given below [2]: X = 1 — ; 1 —e (6.7) The above equation can be simplified to the following form with linle error, i.e, X = l-e~K'r (6.8) where Xr is a term that must be determined in order to represent equation (6.7) by equation (6.8). In this form it can easily be manipulated and compared to the earlier analysis. In order to determine the form of Xr various values of X were selected in Figure 6.5 and the corresponding values of -strlR2 obtained. Different values of DA _B/R2 were chosen arbitrarily in order to calculate tt. The fr values obtained this way were substituted in equation 6.8 and the corresponding Xr were computed. A plot of Xr values against DA_B/R2 yielded an equation of the form 69 \ = 9.%1DA_BIR2 (6.9) with a correlation coefficient of 0.999. The constant 9.87 in equation 6.9 resulted whilst transforming equation 6.7 to equation 6.8. By comparing the equation obtained from experimental analysis (equation 6.6) to that of equation 6.8, the experimental one can be transformed to the following form: where \ would equal 9.42. This equation can be used to predict the gas utilization efficiency of the system which is given by: 6.3.2 Bubble formation analysis From equation (6.10), at *v = 0,1-X = 0.255, which indicates that the fraction of gas absorbed during formation is constant for the physical model at about 0.74. The bubble size and internal mixing tend to increase with gas flow rate. At higher flow rate mixing inside the bubble is vigorous but takes place over a large volume whilst at lower flow rate mixing is less vigorous but takes place over a smaller volume. This may be the reason why there is a constant value for gas absorption during bubble formation at different flow rates. Submergence may not have any effect on this constant value since the bubbles may have fully formed for all the submergences considered during this analysis. When calculating the gas residence time the bubble formation period was not considered. It was assumed that the bubble has already formed and that the time calculated was for bubble rise. During bubble formation, the absorption process is expected to be similar to that of bubble rise, that is, mass transfer inside a gas cavity. Assuming that the forming bubble is spherical, the mass transfer might be described by equation 6.10. A corresponding (1-X) = 0.255e (6.10) (6.11) 70 equation can therefore be written for the reaction that occurs during this period i.e., (1-Xf) = e^< (6.12) where Xt is the fraction of gas absorbed during formation, Xf is a term similar to \ and rf is the bubble formation time. Xf is assumed to be a function of Dso ^R^, where /?av is the time-averaged bubble radius during formation. Rm was calculated from the procedure outlined below: During bubble formation period, the volume of bubble is expected to change with time. Therefore at any time, t, the volume of bubble is given by Vh = Qt (6.13) The instantaneous radius of bubble can be evaluated as R = y 4K The area of bubble can be represented by A =4K y4% j The time-average area is therefore calculated as 1 C'f A„=-\ Adt tfJO The time-average radius, RiV, is determined from Am as 4TC (6.14) (6.15) (6.16) (6.17) 71 The bubble formation time, t{, in equation 6.16 was obtained from the results of Bustos work on bubble frequency analysis for similar range of gas flow rates in the same physical model [22]. Xf was represented by the equation below: X^KD^JRl (6.18) To determine K, Xf was calculated as ^ = --111(0.225) (6.19) h The constant K was calculated for all the measured flow rates and was termed as the "enhancement factor" during bubble formation. During gas injection, power for stirring in the bath and gas phase is supplied by the injected gas. At high velocities the kinetic energy of the injected gas will enhance mixing in the bath and the gas phase. If bubbles are formed during injection, some fraction of kinetic energy will show up inside the bubble as internal mixing and gas circulation. It has been reported by Humphrey et al. [43] that circulation inside a bubble which depends on the momentum of injected gas entering the bubble have an important effect on the rate of mass transfer in bubbles forming from tuyeres. The rate at which the gas jet transfers kinetic energy to the bubbles can be represented by equation 6.20. %=\ P1Q 7 (6-20) ep: Specific power density due to kinetic energy of the gas (watts/kg). pg: Density of gas (g/cm3) A0: Cross sectional area of the tuyere (cm2) W: Mass of the gas bubble (g) 72 The specific power density was computed for each gas flow rate employed in the physical model. Since the kinetic energy contributes to mixing at higher flow rates, the power density may therefore influence the enhancement factor during bubble formation. Figure 6.6 shows a plot of the enhancement factor against the specific power density. Equation 6.21 provides a good fit between the enhancement factor and the power density with a correlation coefficient of 0.99. K = 7le°pw (6.21) As will be seen in section 6.7 it was necessary to use such relationship to predict the enhancement factor for various industrial data. Though equation 6.21 indicates a good fit it predicts low K values for the industrial data. Equation 6.22 is another fit for Figure 6.6. K =742.1 +0.057ep (6.22) with a correlation coefficient of 0.96. Though equation 6.22 has a lower correlation coefficient it is able to predict good K values for the industrial data. The above analyses based on bubble formation and bubble rise can be used to describe the absorption phenomena in the bath during injection. Thus equation 6.12 represents absorption during bubble formation period, whilst equation 6.10 describes that of bubble rise. This phenomena is illustrated in Figure 6.7. The above results indicate that the fraction of gas absorbed during bubble formation forms a higher percent of the total fraction of gas absorbed. Other investigators have also observed similar behaviour. Clift et al. [21] have reported that a very significant fraction of the total mass transfer in industrial extraction or absorption operations occurs during bubble formation. Transfer tends to be particularly favourable because of the exposure of fresh surface with vigorous internal circulation during bubble formation period. Rocha and Carvalho [31] observed that the fraction of gas absorbed during bubble formation in 73 the NHJHCl system when gas phase control prevailed was 90%. The mathematical model presented by Ashman et al. [10] also predicted 40% of absorption during bubble formation in a copper converter. In most absorption analyses based on the jet model a lumped value for the mass transfer parameter, kga has been used to describe absorption from bubble formation to bubble rise. This seems to be inappropriate since it has been realized from the present work that the absorption rate during zthe bubble formation period differ from that during bubble rise where no internal circulation takes place inside the bubble. 6.4 Effect of injection parameters on the fraction of gas absorbed Due to the low modified Froude numbers used, the gas did not penetrate very far into the liquid before turning in the vertical direction. This resulted in the formation of a dispersed region above the tuyeres and close to the wall of the vessel. This dispersed region which may form in an industrial converter is likely to contribute to tuyere erosion and refractory wear. The spout height measured in this work increased with gas flow rate. It formed a significant percentage of the "total trajectory length", and in some of the measurements, the percentage was as high as 50%. When the spout height was included in the total trajectory length, a good correlation was obtained between the fraction of gas absorbed and the total trajectory length, indicating that part of the absorption also occurs in the spout. Previous analyses have neglected the spout height [30, 25]. As the spout height increases, splashing is also expected to increase accordingly. From visual observation, bath slopping, which is an unfavourable phenomena during injection, was also found to increase with spout height. The effect of tuyere submergence on fraction of gas absorbed as shown in Figure 5.8 is quite remarkable. As the submergence is increased from the 74 tuyere centre-line to 13.1 cm (18.2% to 38.1% filling), the fraction of gas absorbed is increased from 0.445 to about 0.99. It is clear that the fraction of gas absorbed when the submergence was at the tuyere center-line (0.445) is different from the extrapolated value during bubble formation (0.74). This is due to the fact that at the tuyere center-line, the bubbles are not fully formed and also a fraction of the injected gas escapes through the surface of the bath without complete reaction. The increase in absorption with tuyere submergence is related to the gas residence time in the bath. There is more contact between the solute gas and the liquid as tuyere submergence is increased. 6.5 Mass transfer parameter, The mass transfer parameter, ksoa, as defined in equation 5.3 and calculated from plots similar to Figure 5.5, varied from 1100 to 2560 cmV1. The purpose was for comparison with the results of previous investigators. Brimacombe et al. [30], using a single tuyere obtained ksoa values that ranged from about 20 to 380 cmV1. In the work of Themelis and Schmidt [25], the mass transfer parameter, kcoa, ranged from 40 to 230 cmV 1 when CO gas was injected into liquid copper. In all the above results, kga increased with the tuyere Reynolds number, NRe. The effect of NRe on kga is presented in Figure 6.8. The results of the present work are much higher than that of the other investigators due to the high flow rates used in this work. 6.6 Gas residence time and the fraction of gas absorbed The analysis which involves the mass transfer parameter indicates that high mass transfer coefficients and interfacial areas are obtained as the gas flow rate is increased. This may therefore lead to higher absorption, contrary to the present results (Figure 5.10). It was observed in Figure 5.10 that as the gas flow rate is increased, the fraction of gas 75 absorbed decreases up to a certain value where it remains almost constant. However, the fraction of gas absorbed increases with the gas residence time as indicated in Figure 6.4. From Figure 6.3, as the flow rate is increased, the gas residence time also decreases and tends to flatten out as higher flow rates are approached. Therefore the fraction of gas absorbed is apparently dependent on the gas residence time only. The lower fraction of gas absorbed at higher flow rates are thus due to the shorter residence time of the gas. Since it was possible to express the results of the experiments in terms of the residence time with one equation, this provides an alternative route to evaluate the fraction of gas absorbed in an industrial converter by using the trajectory equation to determine the residence time as shown in the next section. This method of analysis may be simpler since it does not involve the mass transfer parameter, which is difficult to evaluate for an industrial practice. 6.7 Efficiency analysis on industrial data Data on efficiency and other operating parameters for various copper converters are listed in Table 6.2 to Table 6.4. The bubble rise time for each data point was calculated using the trajectory equations presented in Appendix C. Two different cone angles were used in the analysis. The first one was a cone angle of 155° which has been reported by Oryall and Brimacombe [24]. The rise time obtained varied from 45 to 80 seconds which from the practical point is too long. With this cone angle the jet expands quickly leading to a sudden drop in the gas velocity which results in a longer rise time. The second was an angle of 20° which has been used by Themelis et al. [13]. The values were mostly less than a second which were quite reasonable. Since the values of the spout height for the industrial data have not been measured, they were not included in the determination of 76 the gas rise time. The two equations derived for absorption during bubble formation period and bubble rise time were employed in calculating the oxygen utilization efficiency during converting, that is, Oxygen Utilization Efficiency = 100(i - g~ e^"v') (6.23) where Xr = 9.87D0 _Air/R2, Da _Air is the diffusion coefficient of oxygen in air and R is the radius of bubble during its rise. R was calculated from equation 6.24 developed by Sano and Mori [45] for bubble swarms when nitrogen was injected into mercury. R=0.0455(c/plf5u°M (6.24) where a is surface tension of liquid and ws is the superficial gas velocity. Xf in equation 6.23 is given by the relation below: Xf = KD0i_AiJRl (6.25) The specific power density, ep, was calculated for the industrial data and was used to determine K by employing equation 6.22. Figure 6.9 shows K values of the physical model and that of the Magma data. The bubble frequency for the industrial data was assumed to be 10 s'1 based upon the work of Hoefele and Brimacombe [9]. Figure 6.10 shows a plot of the predicted efficiency against the actual efficiency for the Magma data. The figure indicates a good agreement between the predicted and the actual values. This indicates that the analysis based on residence time is very useful in predicting efficiencies. The procedure based on the physical model where the bubble formation period and the gas rise time are used to determine the efficiencies is strengthened by this results. When the bubble frequency and the tuyere submergence are known, it is possible to calculate the total residence time and the efficiency of gas utilization. The slightly lower predicted values may be due to the fact that the spout 77 height was not included in the analysis. Figure 6.11 is a plot of the predicted efficiency against the actual efficiency for the Johnson's data on most plants around the world [12]. As can be observed in the figure, most of the predicted efficiencies are lower than the reported values. Bustos [22] observed that there was absolutely no correlation between the efficiencies of Johnson's industrial data and other parameters like tuyere diameter, tuyere submergence, bath depth and gas flow rate. It must be emphasized that in predicting the efficiencies, these parameters were employed in the calculation of the gas residence time. 78 Table 6.2. The Noranda process (1-ton pilot plant) [44] (a) Conversion of white metal with slag. Tuyere velocity(c m/s) N'Fr Efficiency (%) Tuyere submergence 7.6 cm Tuyere submergence 15.2cm Tuyere submergence22 .9 cm bath depth 15.4 cm bath depth 22.9 cm bath depth 22.9 cm bath depth 30.5 cm 30000 13880 94.1 51,55,70 77 40000 18510 167.3 45000 20830 211.9 87 53000 24530 293.8 60000 27770 376.6 86 66000 30550 455.8 62 80,88 85000 39340 755.9 88000 40730 810.2 78 94000 43510 924.6 75,82 (b) Conversion of white metal without slag. Tuyere velocity(c m/s) Efficiency (%) Tuyere submergence 7.6 cm Tuyere submergence 15.2 cm Tuyere submergence 25.4 cm bath depth 22.9 cm bath depth 22.9 cm bath depth 33.0 cm 30000 13880 173.9 78 40000 18510 309.2 53, 57 45000 20830 391.7 53000 24530 543.0 72,86 60000 27770 696.1 88 66000 30550 842.5 68 85000 39340 1397.1 71,77 88000 40730 1497.5 94000 43510 1708.9 100000 46280 1933.5 92 126000 58315 3069.8 87,88 142000 65720 3899.0 91 Tuyere diameter, d0 = 0.3175 cm 79 Table 6.3. Copper converter operating practice from most plants around the world [12]. Plant Blowing rate (SCFM) No. Of tuyeres Tuyere diameter (in) Efficiency (%) Mount Isa 29000 54 2 85 Port Kembla 14000 30 2 82 Hoboken 16000 18 1.5 90 Gaspe 22000 50 1.9 80 Hudson Bay 20000 38 2 76 Copper Cliff 18000 48 1.9 93 Noranda 23000 48 1.9 95 Chuquicamata 20000 52 2 98 Caletones 18000 45 2 82 Ronnskar 16000 48 1.9 95 Ghatisla 5000 27 1.6 83 Khetri 5000 28 1.6 90 Kosaka 14000 44 1.6 90 Lamano 22000 50 2 90 Naoshima 21000 46 1.7 98 Saganoseki 19000 48 2 87 Jang Hanj 8000 33 1.5 80 Empress 12000 35 2 70 O'okeip 12000 36 • 2 88 Palabora 18000 46 2 81 Samson 13000 40 1.6 75 Bor 15000 50 1.9 85 San Manuel 22000 52 1.8 86 Ajo 17000 52 1.7 70 Douglas 15000 52 1.7 52 Hildalgo 18000 52 1.9 80 White Pine 16000 42 1.9 43 Ray 20000 42 2 96 Neveda 19000 43 2 95 Chino 19000 48 1.9 78 Utah 18000 46 2 70 Inspiration 15000 44 2 81 Anaconda 22000 48 1.7 88 El Paso 18000 39 1.7 73 Tacoma 19000 44 1.6 63 Copper Hill 13000 40 1.5 71 Plant A 17000 50 2.1 75 80 Table 6.4. Oxygen utilization data from Magma Copper Company (San Manuel) [12] Tuyere air Tuyere Efficiency Blows flow rate (SCFM) velocity (ft/s) (%) 11,858 197 3.22 58.00 12,803 212 3.73 62.80 13,634 226 4.23 61.30 14,494 241 4.81 66.60 15,712 256 5.43 68.50 Slag 16,608 276 6.31 72.80 Blow 17,491 285 6.73 75.10 17,788 290 6.97 77.60 18,721 311 8.02 73.30 18,331 305 7.71 80.10 18,974 315 8.23 77.10 19,566 325 8.76 80.00 20,159 328 8.92 81.40 22,344 363 10.92 83.80 14,837 246 2.84 70.40 17,211 284 3.84 76.60 17,805 295 4.08 77.10 19,343 321 4.83 83.80 Copper 19,948 331 5.14 84.70 Blow 20,178 334 5.23 88.50 20,772 344 5.55 88.50 21,261 352 5.81 93.80 22,552 374 6.56 90.40 24,619 408 7.81 95.70 26,772 448 9.25 96.40 81 Figure 6.6 Plot of enhancement, K, factor against the specific power density (The straight line represents equation 6.22). 82 Figure 6.7 Plot showing the proposed absorption phenomena during gas injection for the physical model. 83 2600 2200 H 1800 co o ^ 1400 -\ 5-• Present work + Brimacombe et al. [30] o Themelis et al. [25] 1000 H 600 H 200 o o t-— I — 10000 20000 N Re + + 30000 40000 Figure 6.8 Comparison of the mass transfer parameter, kga, obtained from the present work with that of other investigators. 84 14000 Specific power density (watts/kg) Figure 6.9 Prediction of the enhancement factor for the Magma data from the physical model. 85 5 0 6 0 7 0 8 0 9 0 1 0 0 Actual efficiency (%) Figure 6.10 Predicted efficiency versus actual efficiency for the Magma data in Table 6.4 (slag blow). 86 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 Actual efficiency (%) Figure 6.11 Predicted efficiency versus actual efficiency for Johnson's industrial survey presented in Table 6.3. 87 CHAPTER 7 SUMMARY AND CONCLUSIONS The fraction of gas absorbed during horizontal gas injection in a system similar to the copper converter has been determined by using S02IH202 system. It was observed that the gas entering the bath forms large bubbles which break into smaller bubbles a short distance from the tuyeres. For a constant S02 flow rate, the fraction of gas absorbed decreased with air flow rate up to a certain value and stayed almost constant at all higher flow rates. The spout height which increased with air flow rate formed a high percentage of the total trajectory length. It was included in the analysis since it is believed that part of the absorption takes place in the spout. The fraction of gas absorbed increased with tuyere submergence and percent filling from 0.445 to about 0.99. A substantial fraction of gas (0.445) was absorbed at zero submergence (tuyere center-line). In analysing the results, the mass transfer parameter, ksoa, was initially evaluated. It was found to increase with the tuyere Reynolds number and gas flow rate. Though the mass transfer parameter increased with gas flow rate, the fraction of gas absorbed did not increase with the gas flow rate. It was therefore not possible to use this parameter to explain the results. With this approach, the different zones that exist in the bath cannot be separated. The second method for analysing the results was based on bubble formation and rise periods. The bubbles are formed at the tuyeres and a high percentage of the gas is 88 absorbed during bubble formation. This was attributed to the vigorous internal mixing and gas circulation inside the bubble. The absorption phenomena during bubble formation and bubble rise was considered to be an "enhanced" diffusion process. The gas rise time was calculated by applying the trajectory equations outlined in Appendix C. This analysis resulted in a single equation which could be used to determine the fraction of gas absorbed for a known gas flow rate and tuyere submergence. It was observed that the fraction of gas absorbed increased with the total gas residence time. Since there was a parallel between the effect of gas flow rate on residence time and effect of gas flow rate on the fraction of gas absorbed, it was possible to explain the effect of gas flow rate on the fraction of gas absorbed with this approach. From this method, it is possible to determine the utilization efficiency during gas injection when the gas residence time and the bubble formation time is known. Thus for a horizontal injection process similar to the converter, a substantial part of the reaction is likely to occur at the tuyere. This method was extended to the analysis of industrial data for the comparison of measured and predicted oxygen utilization efficiencies. The predicted efficiencies for some cases were very good whilst in others, there appeared to be no direct correlation between the gas residence time and efficiency. For an industrial practice, the absorption rate and gas utilization efficiency for gas-metal reactions will depend on the gas flow rate, the bubble formation period, bubble rise time and the tuyere submergence. 89 REFERENCES 1. R. D. Pehlke: Unit Processes of Extractive Metallurgy, American Elsvier Publishing Company Inc., N. Y., 1973. 2. J. Szekely and N. J. Themelis: Rate Phenomena in Process Metallurgy, John Wiley and Sons, Inc., 1971. 3. G. C. McKerrow and D. G. Pannel: Canadian Met. Quarterly, Vol. 11, 1972, pp. 629-633. 4. R. Deilly: J. Of Metals, Vol. 25, No.3, 1973, pp. 33-41. 5. A. K. Biswas and W. G. Davenport: Extractive Metallurgy of Copper, Pergamon Press Ltd., Great Britain., 1980. 6. T. Rosenqvist: Principles of Extractive Metallurgy, McGraw- Hill Book Company, N. Y., 1983. 7. G. N. Oryall: M. A. Sc. Thesis, The University of British Columbia, 1975. 8. A. A. Bustos, G. G. Richards, N. B. Gray and J. K. Brimacombe: Met. Trans. B, Vol. 15B, 1984, pp. 77-89. 9. E. O. Hoefele and J. K. Brimacombe: Met. Trans. B, Vol. 10B, 1979, pp. 631-648. 10. D. W. Ashman, J. W. Mckelliget and J. K. Brimacombe: Canadian Metallurgical Quarterly, Vol. 20, No.4,1981, pp. 387-397. 11. J. K. Brimacombe, S. E. Meredith and R. G. H. Lee: Met. Trans. B, Vol. 15B, 1984, pp. 243-250. 12. R. E. Johnson: Copper and Nickel Converters, The Metallurgical Society of AIME, 1979. 90 13. N. J. Themelis, P. Tarassoff, and J. Szekely: Transactions of ATME Vol. 245, 1969, pp. 2425-2433. 14. M. J. McNallan and T. B. King: Metallurgical Trans. B, Vol. 13B, 1982, pp. 165-173. 15. Y. Ozawa and K. Mori: Trans ISIJ, Vol. 23, 1983, pp. 759-763. 16. M. Sano and K. Mori: Trans ISIJ, Vol. 26,1986, pp. 291-297. 17. G. A. Irons and R. I. L. Guthrie: Met. Trans. B, Vol. 9B, 1978, pp. 101-110. 18. R. J. Andreini, F. S. Foster and R. W. Callen: Met. Trans. 8B, 1977, pp. 625-631. 19. R. J. Fruehan: Metals Technology, March 1980, pp. 95-101. 20. S. V. Patankar and D. B. Spalding: Int. J. of Heat and Mass Transfer, 1972, Vol. 15, pp. 1785-1805. 21. R. Clift, J. R. Grace and M.E. Webber: Bubbles, Drops, and Particles, 1978, Academic Press, N. York. 22. A. A. Bustos: Ph. D. Thesis, The University of British Columbia, 1984. 23. T. A. Engh and H. Bertheussen: Scandinavian Journal of Metallurgy, Vol. 4, 1975, pp. 241-249. 24. G. N. Oryall and J. K. Brimacombe: Met. Trans. B, Vol. 7B, 1976, pp. 391-403. 25. N. J. Themelis and P. R. Schmidt: Transactions of the Metallurgical Society of AIME, Vol. 239,1967, pp. 1313-1318. 26. S. Nanda and G. H. Geiger: Met. Trans. Vol. 2, 1971, pp. 1101-1106. 27. S. Inada and T. Watanabe: Transaction, ISIJ, Vol. 17 1977, pp. 21-27. 28. S. Taniguchi, A. Kikuchi, H. Matsuzaki and Bassho: Transactions, ISIJ, Vol. 28, No.4,1988, pp. 262-270. 29. O. Haida and J. K. Brimacombe: Proc. SCANINJECT, MEFOS/Jernkontoret, Lulea, 1984,5:1-5:17. 91 30. J. K. Brimacombe, E. S. Stratigakos and P. Tarassoff: Met. Trans. B, Vol. 5, 1974, pp. 763-771. 31. F. A. N. Rocha and J. R. F. Guedes de Carvalho: Chemical Eng. Res. Des. Vol. 62, 1984, pp. 303-314. 32. C. H. Chang and G. T. Rochelle: AIChE Journal 27, No.2,1981, pp. 292-298. 33. T. Stapurewicz and N. J. Themelis: Canadian Metallurgical Quarterly, Vol. 26, No.2,1987, pp. 123-128. 34. A. Kikuchi, S. Taniguchi and N. Bessho: Fifth Int. Iron and Steel Congress, Vol. 6, Washington D. C , 1986, pp. 369-375. 35. S. Kim and R. J. Fruehan: Met. Trans. B, Vol. 18B, 1987, pp. 381-390. 36. M. Hirasawa, K. Mori, M. Sano, A. Hatanaka, Y. Shimatani and Y. Okazaki: Transaction, ISIJ, Vol. 27, 1987, pp. 277-282. 37. S. Paul and D. N. Ghosh: Met. Trans B, Vol. 17B, 1986, pp. 461-469. 38. A. Jean-Pierre Couilland and V. Jean Louise: United States Patent No.3733393, 1973. 39. Jean-Claude Charpentier: Advances in Chemical Engineering, Vol. 11 1981, edited by T. B. Drew, G. R. Cokelet, J. W. Hoopes, Jr. and T. Vermeulen, Academic Press, New York. 40. V. Sahajwalla: Masters Thesis, University of British Columbia, 1988. 41. J. O. Hirschfelder, R. B. Bird, and E. L. Spotz: Chem. Revs., 44, 1949, pp. 205-231. 42. I. Liebson, E. G. Holcomb, A. G. Cacoso and J. J. Jamie: A.I.Ch.E. J., Vol. 2, No.3,1956, pp. 296-306. 43. J. A. C. Humphrey, R. L. Hummel and J. W. Smith: Chemical Eng. Sc., Vol. 29, 1974, pp. 1496-1500. 92 44. P. Tarassoff: Process R & D-The Noranda Process given at The 1984 Extractive Metallurgy Lecture, The Metallurgical Society of AIME. 45. M. Sano and K. Mori: Transactions ,ISIJ, Vol. 20,1980, pp. 668-674. 46. A.S.M.E. 'Fluid Meters, their Theory and Application', Report of the A.S.M.E. Research Committee on Fluid Meters, 6th Ed., H. S. Bean, Ed., New York, 1971. 93 APPENDIX A Calibration of Orifice Plate The air flow rate was measured with a thin square-edged orifice plate. The diameter for the plate was made to be equal to that of the nominal internal diameter of the pipe in which it was installed. Two pressure taps were located one inch (flange taps) from the inlet and the oudet faces of the orifice plate. Tubes were connected to a water manometer from the taps which measured the differential and upstream pressures of the air passing through the orifice plate. The recommendations of A.S.M.E. report on fluid meters [46] were considered and related to the design of the orifice plate. The ratio between the orifice diameter and the inside diameter of the pipe, was designated as (3. Two (3 values were used. For relatively higher flow rates, |3 was equal to 0.6 and for lower flow rates, (3 was 0.4. The dimensions of the device are listed in Table Al . Table Al Inside diameter of pipe, Dp (cm) 5.04 Orifice diameter ((3 = 0.4), D0 (cm) 2.02 Orifice diameter (|3 = 0.6), DQ (cm) 3.02 Plate thickness (cm) 0.30 The downstream face of the plate orifice was bevelled at 45° as recommended by the A.S.M.E. report [46]. The theoretical rate of flow of a compressible fluid is given by the equation below: 94 Mideal=A-1/2 (A l ) 1 ( Y - l ) ( l - r ^ p 4 ) M i d e a l Ideal mass flow rate of gas r Static pressure ratio (PQ/P,) Y Ratio of specific heats (Cp/Q,) A0 Cross-sectional area of the orifice p g Density of the fluid upstream of the orifice Px Absolute static pressure upstream the orifice P0 Absolute static pressure downstream the orifice P Diameter ratio (PJDp) The actual rate of flow is given by MaemtU = CDM^ (A2) The values for the discharge coefficient, C D , were obtained from the relationship given below, that is CD = Co + AC[l04/NRf (A3) where C 0 and AC represent the discharge coefficient when the throat Reynolds number tends to infinity and the increase in the discharge for an arbitrary Reynolds number change from 10 4 to infinity respectively. The exponent a = 1 for flange taps. For (3 = 0.4, C0 = 0.60139 and A C = 0.01809 For p = 0.6, C0 = 0.60799 and A C = 0.03920 [22] Equation A l was combined with the experimental measurements of the upstream and downstream preasures to determine the f low rate. The relationship between the air 95 flow rate Q, and the pressure drop across the orifice plate are shown in Figure A l and Figure A2. It was possible to express the air flow rate as a function of the pressure drop in the form of exponential relation: p = 0.4, Q = 2 7 8 0 # 0 4 8 with a correlation coeffient of 0.998. p = 0.6, Q = 6670// 0 4 8 with a correlation coefficient of 0.999. These equations were then used to calculate the air flow rate at any known pressure drop. H. Figure Al Air flow rate against differential pressure for the orifice plate with a diameter ratio of 0.4. 97 45000 0 20 40 60 Differential pressure across the orfice plate (cm) Figure A2 Air flow rate against differential pressure for the orifice plate with a diameter ratio of 0.6. 98 APPENDIX B Calibration of Rotameter The rotameter used in the experiments was No. 603 (Model 7600) by Matheson. The tube consisted of a cylindrical glass graduated from 0 to 150 mm and contained two floats - a stainless steel and a pyrex.The calibration was done with the two balls simultaneously by using a water displacement method. An inverted, graduated cylinderical container was filled with water which was covered with a little oil. The rotameter was then connected to the inverted cylinder as shown in Fig. BI. The S02 was supplied from a gas cylinder at a pressure of 65.5 kPa (9.5 psig). The positions of the balls as well as the volume of water displaced were recorded when the gas was turned on for a certain time interval. The flow rate of the gas at each position of the balls was calculated by dividing the volume of water displaced by the the time of injection. The calculated flow rates were plotted against the various positions of the balls (Fig B2). When the S02 flow rates were correlated with the height of the balls the folowing equations resulted: Steel ball qSOi = 7.8 x 10"5//°7 5 with a correlation coefficient of 0.999. Pyrex ball qSo2 = 4.5 x 10"5//°"73 with a correlation coefficient of 0.994 Since the correlations obtained were very good, the above equations were used to determine the S02 flow rates in the experiments. 99 - _ Inverted Cylidrical Tube Rotameter Water Sulphur Dioxide Gas Figure B1 Schematic of sulphur dioxide calibration set-up. 100 150 O 0.001 0.002 0.003 0.004 Sulphur dioxide flow rate (mol/s) Figure B2 Calibration curves for sulphur dioxide gas. 101 APPENDIX C Determination of jet trajectory length (S0) and residence time (t). The jet trajectory length was calculated by using the theoretical expression derived by Themelis et al. [13] and Engh et al. [23]. The general equation relating the vertical and the horizotal components of the jet trajectory as indicated in Figure Cl is given below: tan2(9e/2) cos 9„ 1 + KdxrJ x2C (Cl) xr = xld0: dimensionless horizontal distance of jet. yr = yld0: dimensionless vertical distance of jet. C : gas fraction in the jet. NFr: modified Froude number. For an incremental distance, dx, in the horizontal direction, the incremental distance, ds, along the axis of the jet trajectory is given by ds=stcQdx (C2) Since sec 9 = [1 + (dy/dx)2]m, the equation for the tajectory length can be written as 1/2 ds_ dx 1 + 'dy}2 ydXjj (C3) In dimensionless form, equation C3 can be represented by C4. d^ dXr 1 + KdxrJ 1/2 (C4) where Sr = sld0 is the dimensionless distance along the axis of the jet trajectory. 102 The time it takes for a bubble to travel an incremental length ds, along the axis of the jet trajectory is given by equation C5. ds dt=- (C5) where is the velocity of the gas along the jet trajectory for that short distance. Equation C5 can be written as ds ux Since dtlds = (dtldx).(dxlds), it implies that, dt__dt_ ds_ dx ds' dx Substituting C3 and C6 into C7 gives equation C8. (C6) (C7) dt__\_ dx ux 1 + KdXj 1/2 (C8) If t{ is the time the gas travels the initial distance dJ2 tan(8c/2) before it enters the bath with the nominal tuyere velocity u0, and t is the time it takes to travel a distance s, along the jet trajectory, then t can be written in dimensionless form as tr = t/ti. Therefore dt = ttdtr In dimensionless form equation C8 can be written as dtL_d£_ dxr tiUx 1 + KdxrJ 1/2 (C9) The velocity along the jet trajectory is represented by equation CIO. 103 u0dl (CIO) where d is the diameter of the jet trajectory at any point as indicated in Figure C l . d = 2d0S,tan(9c/2) (Cl l ) The gas fraction in the jet is given by C12. 8S2tan2(8c/2) Equations C l , C4 and C9 were solved numerically using a fourth order Runge-Kutta method to obtain the trajectory coordinates, the trajectory length and the residence time. 11/2 C=~ 2 / / L — (C12) do/2tanpc/2) X Figure CI Schematic of a jet with a rising trajectory. 

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