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Fluid flow in horizontal injection regimes Shook, Andrew A. 1986

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FLUID FLOW IN HORIZONTAL INJECTION REGIMES By ANDREW A. SHOOK B.E., The University of Saskatchewan, 1983 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 Metallurgical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1986 (c) Andrew Shook, 1986 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 The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) i i A B S T R A C T P h y s i c a l and mathematical modelling studies have been performed to inve s t i g a t e l i q u i d flow driven by a h o r i z o n t a l l y i n j e c t e d gas. The experimental work consisted of water v e l o c i t y measurements made at 100 l o c a t i o n s w i t h i n a plexiglass tank. A i r was introduced into the tank through a s e r i e s of side-mounted tuyeres, and the e f f e c t of a i r flowrate on water r e c i r c u l a t i o n v e l o c i t y was observed. The r e s u l t s of the experiments i n d i c a t e that the maximum water v e l o c i t y occurs at the water surface. The e f f e c t of bubbles coalescing from adjacent tuyeres was observed with increasing a i r flowrate, and was found to diminish the water r e c i r c u l a t i o n r a t e . The mathematical model employed a variant of the Marker and C e l l (MAC) technique to compute f l u i d flow with a free surface. The model predictions i n d i c a t e that the flow i n the experimental tank i s l a r g e l y driven by water flowing across the free surface. Based on t h i s knowledge, q u a l i t a t i v e p r e d i c t i o n s of the flow regimes i n a Peirce-Smith copper converter and a zinc slag fuming furnace were made. i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS • i i i LIST OF TABLES v i LIST OF FIGURES v i i NOMENCLATURE x i ACKNOWLEDGEMENTS x i v 1. INTRODUCTION 1 1.1. Copper Smelting 1 1.2. The Copper Converter 4 1.2.1. H i s t o r y and Development 4 1.2.2. D i f f i c u l t i e s 4 1.3. The Zinc Slag Fuming Furnace 5 1.4. O b j e c t i v e s of the Work 6 2. LITERATURE REVIEW 7 2.1. Flow Regimes i n Non-Ferrous M e t a l l u r g i c a l Reactors 7 2.2. Bubble Formation I n v e s t i g a t i o n s ..... 10 2.3. I n v e s t i g a t i o n s of Gas-Driven Flow . 12 2.3.1. Experimental Studies 12 2.3.2. Numerical Studies 15 2.4. Conclusions 19 3. EXPERIMENTAL 20 3.1. O b j e c t i v e s . 20 3.2. Apparatus 20 3.3. Procedure 26 3.4. Experimental R e s u l t s 28 i v TABLE OF CONTENTS (cont'd) Page 3.4.1. I n t r o d u c t i o n 28 3.4.2. Check of the Two D i m e n s i o n a l i t y of the Flow 30 3.4.3. V e l o c i t y Patterns 34 3.4.4. Surface Shape 36 3.4.5. Bath S t i r r i n g 37 4. THE MATHEMATICAL MODEL 43 4.1. General Statement of Problem 44 4.1.1. Governing Equations 44 4.1.2. Boundary Conditions 46 4.2. S e l e c t i o n of S o l u t i o n Algorithm 48 4.3. The SOLASMAC Algorithm 53 4.4. Treatment of Boundary Conditions 56 4.4.1. Locating and Moving the Free Surface , 56 4.4.2. Free Surface V e l o c i t i e s and Pressures 58 4.4.2.1. Normal Stress C o n d i t i o n 61 4.4.2.2. Tangential S t r e s s C o n d i t i o n 61 4.4.3. Rectangular Wall Boundaries 63 4.4.4. Round Bottom Boundary 64 4.4.5. Gas-Liquid Boundary 65 5. CALCULATIONS - COMPARISON WITH EXPERIMENTAL RESULTS 67 5.1. Constant V e r t i c a l V e l o c i t y 70 5.2. V a r i a b l e Density 73 5.3. Pulsed Boundary 77 5.4. C o l l a p s i n g Surface 79 5.5. D i s c u s s i o n 83 6. INDUSTRIAL CALCULATIONS 86 6.1. F l u i d Flow i n a Copper Converter 86 6.1.1. Assumptions 86 6.1.2. Mathematical D e s c r i p t i o n of Gas-Liquid I n t e r f a c e ... 87 6.1.3. Results 88 V TABLE OF CONTENTS (cont'd) Page 6.2. F l u i d Flow i n a Zinc Slag Fuming Furnace 90 6.2.1. Assumptions 91 6.2.2. Modifications to Program 92 6.2.3. Mathematical D e s c r i p t i o n of Bubble Column 93 6.2.4. Results 95 7. CONCLUSIONS 97 7.1. Experimental 97 7.2. Calculations - Agreement with Experimental Results 98 7.3. I n d u s t r i a l Calculations 99 7.3.1. Flow i n a Copper Converter 99 7.3.2. Flow i n a Zinc Slag Fuming Furnace 99 7.4. Recommendations for Further Work 100 REFERENCES 101 APPENDIX 1 107 TABLES 110 FIGURES 123 vi LIST OF TABLES Page Chapter 3 Table 3.1. Experimental Air Injection Rates I l l Table 3.2. Values of in Experimental Tank, Experiment 1 112 Table 3.3. Minimum Mean Values of of Experimental Runs 113 Chapter 4 Table 4.1. Approximations Used for Rectangular Wall Boundaries 114 Chapter 5 Table 5.1. Effective Viscosity Values Predicted by Model of Sahai and Guthrie (44) 115 Table 5.2. Comparison of Results Calculated by Constant Velocity Condition with Experiments 116 Table 5.3. Estimated Experimental Bubble Column Porosity Values.... 117 Table 5.4. Comparison of Results Calculated by Variable Density Condition with Experiments 118 Table 5.5. Comparison of Results Calculated by Pulsed Boundary Condition with Experiments 119 Table 5.6. Comparison of Results Calculated by Collapsing Surface Condition with Experiments 120 Chapter 6 Table 6.1. Data Used to Model Flow in Peirce-Smith Copper Converter 121 Table 6.2. Data Used to Model Flow in Zinc Slag Fuming Furnace 122 v i i LIST OP FIGURES Page Chapter 1 Figure 1.1. Schematic Diagram of Peirce-Smith Copper Converter 124 Figure 1.2. Schematic Diagram of Zinc Slag Fuming Furnace 125 Chapter 2 Figure 2.1. Estimate of Flow i n a Copper Converter, from Themelis et a l . (3) 126 Figure 2.2. P r e d i c t e d Flow P r o f i l e i n Two-Phase Region of Copper Converter, from Nakanishi and Szekely (4) 127 Figure 2.3. J e t Behaviour Diagram, from Hoefele and Brimacombe (7) 128 Chapter 3 Figure 3.1. Schematic Diagram of Experimental Apparatus 129 Figure 3.2. Dimensions and Construction of Experimental Tank 130 Figure 3.3. Diagram of Laser-Doppler System . 131 Figure 3.4. V e l o c i t y Measurement Locations W i t h i n Experimental Tank 132 Figure 3.5. H o r i z o n t a l V e l o c i t y V a r i a t i o n Through Depth of Tank 133 Figure 3.6. V e r t i c a l V e l o c i t y V a r i a t i o n Through Depth of Tank.... 134 Figure 3.7. Experimental V e l o c i t y Vector and Surface P l o t -Experiment 1 135 Figure 3.8. Experimental V e l o c i t y Vector and Surface P l o t -Experiment 2 136 Figure 3.9. Experimental V e l o c i t y Vector and Surface P l o t -Experiment 3 137 Figure 3.10. Experimental V e l o c i t y Vector and Surface P l o t -Experiment 4 138 Figure 3.11. Experimental V e l o c i t y Vector and Surface P l o t -Experiment 5 139 v i i i Page Figure 3.12. Experimental Velocity Vector and Surface Plot -Experiment 6 140 Figure 3.13. Experimental Velocity Vector and Surface Plot -Experiment 7 141 Figure 3.14. Experimental Velocity Vector and Surface Plot -Experiment 8 142 Figure 3.15. Experimental Velocity Vector and Surface Plot -Experiment 9 143 Figure 3.16. Experimental Velocity Vector and Surface Plot -Experiment 10 144 Figure 3.17. Variation of Air Holdup in Experimental Tank with Air Flowrate 145 Figure 3.18. Variation of Air Holdup in Experimental Tank and Zinc Slag Fuming Furnace with Modified Froude Number 146 Figure 3.19. Mean Cell Kinetic Energy of Experimental Measurements as a Function of Air Flowrate 147 Figure 3.20. Mean Cell Kinetic Energy of Non-Surface Cells in Experimental Measurements as a function of Air Flowrate 148 Figure 3.21. Variation of Experimental Mean Cell Kinetic Energy with Air Input Energy 149 Figure 3.22. Bubble Formation at Tuyeres, N F r, = 0.4 150 Figure 3.23. Bubble Formation at Tuyeres, N F r, = 2.3 151 Figure 3.24. Bubble Formation at Tuyeres, N p r, = 15.6 152 Chapter 4 Figure 4.1. Schematic Description of Boundary Conditions Necessary to Describe the Flow in the Experimental Tank 153 Figure 4.2. Velocity Fluctuations in a Finite-Difference Cell Computed by SOLASMAC Method 154 i x Page F i g u r e 4.3. F i n i t e D i f f e r e n c e G r i d Used to Compute Flow i n Experimental Tank 155 Fig u r e 4.4. Flowchart of the SSMCR Program 156 Figure 4.5. Diagram of Free Surface L o c a t i o n and Movement Technique as used i n SSMCR 157 Fig u r e 4.6. Recognition of Surface O r i e n t a t i o n f o r T a n g e n t i a l S t r e s s C o n d i t i o n 158 Chapter 5 F i g u r e 5.1. Diagram of Problem used to Test SSMCR 159 Fig u r e 5.2. Agreement Between P r e d i c t i o n s Made by SSMCR and SOLASMAC f o r Flow i n a Square C a v i t y 160 Fig u r e 5.3. Erroneous P r e d i c t i o n of Square C a v i t y Flow 161 Fig u r e 5.4. Correct P r e d i c t i o n of Square C a v i t y Flow 162 Fig u r e 5.5. P r e d i c t i o n of Experimental Flow Regime Made by Constant V e l o c i t y C o n d i t i o n , = 10 g/cm»s 163 Fi g u r e 5.6. P r e d i c t i o n of Experimental Flow Regime Made by Constant V e l o c i t y C o n d i t i o n , = 40 g/cm«s 164 Fi g u r e 5.7. P r e d i c t i o n of Experimental Flow Regime Made by Constant V e l o c i t y C o n d i t i o n , \x^ff = 400 g/cm»s 165 Fig u r e 5.8. P r e d i c t i o n of Experimental Flow Regime Made by V a r i a b l e Density C o n d i t i o n 166 Fig u r e 5.9. P r e d i c t i o n of Experimental Flow Regime Made by V a r i a b l e Density C o n d i t i o n w i t h L e f t W a l l V e l o c i t y 167 Fig u r e 5.10. P r e d i c t i o n of Experimental Flow Regime Made by Pulsed Boundary C o n d i t i o n 168 Figure 5.11. P r e d i c t i o n of Experimental Flow Regime Made by C o l l a p s i n g Surface C o n d i t i o n , T = 0.5 s 169 Fi g u r e 5.12. P r e d i c t i o n of Experimental Flow Regime Made by C o l l a p s i n g Surface C o n d i t i o n , T = 0.7 s 170 X Page Chapter 6 Figure 6.1. Prediction of Flow Regime in a Copper Converter Made by Constant Velocity Boundary Condition 171 Figure 6.2. Prediction of Flow Regime in a Copper Converter Made by Collapsing Surface Condition 172 Figure 6.3. Prediction of Flow Regime in a Zinc Slag Fuming Furnace Made by Collapsing Surface Condition 173 Figure 6.4. Prediction of Flow Regime in a Zinc Slag Fuming Furnace Made by Constant Velocity Condition 174 Figure 6.5. Prediction of Flow Regime in a Zinc Slag Fuming Furnace Made by Variable Density Condition 175 NOMENCLATURE Area of bubble column (m ) = depth of column x bubble forward p e n e t r a t i o n 2 Area of tank face (m ) Mean c e l l d i s t r i b u t i o n d e v i a t i o n (-) Mean c e l l angular d e v i a t i o n (degrees) divergence (s ^) Buoyant energy input r a t e (watts) Mean c e l l k i n e t i c energy T o t a l energy input r a t e (watts) K i n e t i c energy input r a t e (watts) 2 A c c e l e r a t i o n due to g r a v i t y (9.8 m/s ) Depth of tank (m) Height of f r e e surface above a r b i t r a r y datum (m) Column number (-) Row number (-) Number of experimental c e l l s c o n t a i n i n g data (-) Number of columns (-) Number of rows (-) Pressure (Pa) 3 A i r f l o w r a t e (m/s) Equivalent radius of tank (m) time (s) u V V max w x y z Greek Symbols a Y e n l p T 4> Subscripts c e e f f f 1 g t x i i Horizontal f l u i d v e l o c i t y (m/s) V e r t i c a l f l u i d v e l o c i t y (m/s) Maximum v e l o c i t y (m/s) Transverse f l u i d v e l o c i t y (m/s) Horizontal axis (-) V e r t i c a l axis (-) Transverse axis (-) Gas f r a c t i o n (-) F r a c t i o n a l upwind d i f f e r e n c e parameter (-) Surface i n c l i n a t i o n (degrees) V i s c o s i t y (g/cm«s) V o r t i c i t y ( s " 1 ) 3 Density (kg/m ) Stress (Pa) Tuyere Diameter (m) calculated experimental e f f e c t i v e f l u i d laminar gas turbulent Superscripts t+At t next time step time dependent Dimensionless Groups 2 N Froude Number, u /g»<b) Fr 8 N_ , Mo d i f i e d Froude Number, Fr 2 u p g m x S4> P f " N Nozzle Reynolds Number, Re Cb O U V Hg g ^g xiv ACKNOWLEDGEMENTS The author wishes to express his sincere appreciation to: Dr. G.G. Richards for his helpful supervision and great patience. Mrs. J. Richards for her instructions on the operation of the LDV. Mr. P. Wenman for drawing many of the graphs and figures. Mrs. M. Jansepar for typing, correcting and helping to prepare this thesis. And to my fellow graduate students and friends for their interesting discussions, support and encouragement. Financial support for this work was obtained from the Natural Sciences and Engineering Research Council. 1 1. INTRODUCTION The injection of air into a vessel through a number of horizontal side-mounted tuyeres i s a common practice In the non-ferrous metallurgical industry. In particular, side-blowing of air is carried out in the Peirce-Smith copper converter, and the zinc slag fuming process (1). This i s contrasted with the top and bottom blown vessels currently used in the ferrous industry (2). Unlike the case of top and bottom blowing, relatively l i t t l e research has been performed on the description of the f l u i d flow regimes within side-blown metallurgical vessels (3,4,5). The major objective of this work was therefore to develop a mathematical model capable of predicting the flow regimes within side-blown furnaces. Due to i t s industrial importance, particular emphasis was placed on the description of the fl u i d flow profile in a Peirce-Smith copper converter. In addition however, predictions were also made for the slag flow in the zinc slag fuming furnace. 1.1. Copper Smelting The production of copper i s of great importance to the industrialized world. Used mainly in electrical components, annual world copper production exceeds 8 million metric tons, with proven reserves of 550 million metric tons (1982 Figures) (6). In Canada, copper production is a b i l l i o n dollar industry: approximately 650,000 metric tons are produced annually, and accounts for 2.4% of a l l exports (6). 2 Throughout the world, most copper Is found In the form of sulphide o r e s , e i t h e r c h a l c o c l t e ( C ^ S ) , c h a l c o p y r i t e (CuFeS2) or c o v e l l i t e (CuS). Mineable copper ores are t y p i c a l l y low-grade - 3 wt.% or lower, depending upon current p r i c e s . Since the e a r l y 20th century, copper has been produced from these ores by the f o l l o w i n g procedure: 1. The ore i s crushed and concentrated by f l o t a t i o n . This concentrate i s u s u a l l y between 15 and 35% copper, the remainder being mainly i r o n , sulphur and gangue. 2. The concentrate i s roasted e l i m i n a t i n g some of the sulphur as SO^-3. The c a l c i n e i s melted i n a reverberatory furnace. Slag ( c o n t a i n i n g s i l i c a t e s , FeO and some copper and sulphur) i s removed, and the matte (a mixture of C ^ S and FeS) i s tapped o f f . 4. The matte i s placed i n a copper converter. S i l i c a f l u x i s al s o added to the matte at t h i s time. The conversion of the matte to m e t a l l i c copper i n t h i s furnace i s c a r r i e d out by blowing a i r through the matte i n two stages: a) slagataklng: The i r o n sulphide i n the matte p r e f e r e n t i a l l y o x i d i z e s to FeO and sulphur d i o x i d e : FeS + 3/2 0. + FeO + S0„ 2 +• L (1-1) 3 This o x i d a t i o n r e l e a s e s a great amount of heat, and proceeds autogenously. The sulphur d i o x i d e , n i t r o g e n , and unreacted oxygen r i s e through the melt and are drawn out of the converter mouth. The FeO combines w i t h the s i l i c a f l u x t o form a s l a g on the s u r f a c e of the b a t h . P e r i o d i c a l l y , the s l a g i s poured out of the con v e r t e r , more s i l i c a and matte are added, and the process i s continued. b) b l i s t e r making: Once a l l of the i r o n has been removed from the melt, the molten copper sulphides are o x i d i z e d to copper by f u r t h e r blowing of a i r : Cu 2S + 0 2 > Cu + S0 2 (1.2) This produces " b l i s t e r copper" which i s approximately 99% pure with the remainder l a r g e l y oxygen and precious metals. 5. The b l i s t e r copper i s normally reduced In an anode furnace and then cast i n t o e l e c t r o d e s and r e f i n e d e l e c t r o l y t i c a l l y . T his y i e l d s 99.98% pure copper, and "anode s l i m e s " from which any precious metals can be e x t r a c t e d . Over the years, many improvements have been made i n t h i s smelting process. The r o a s t i n g and reverberatory steps have been elimina t e d i n the new " f l a s h - s m e l t i n g " techniques. These methods u t i l i z e the heating value of the sulphur i n the ore to accomplish m e l t i n g . They are therefore f a r more 4 energy efficient than earlier methods. Despite such general improvements however, the vast majority of the copper produced in the world today s t i l l involves the oxidation of sulphide matte in a copper converter. 1.2. The Copper Converter 1.2.1. History and Development By far the most common type of copper converter in use today is the Peirce-Smith side-blown converter (Figure 1.1). The design of this vessel arose from research in the mid-19th century. Bottom blown Bessemer-type converters were found to be unsuccessful due to the blockage of the tuyeres by frozen copper matte. As well, the refractory was found to be strongly attacked by the s i l i c a slag. In 1905 Messrs. Peirce and Smith conducted successful tests on a side-blown cylindrical vessel with a basic refractory lin i n g . Within a few years, the Peirce-Smith converter was in use world-wide. 1.2.2. Difficulties Despite i t s success and wide use, the design of the Peirce-Smith converter has several serios problems: 1. Because the air is blown into the converter at low pressure and consequently in the form of bubbles, the bath is able to wash up against the tuyeres (7). Cooled by the incoming a i r , matte in this region s o l i d i f i e s to form accretions which eventually completely block the 5 tuyeres (7). In order to maintain blowing r a t e s , the plug of frozen matte must be "punched" p e r i o d i c a l l y . In a d d i t i o n to i t s o v e r a l l i n e f f i c i e n c y , t h i s punching of the tuyeres i s thought to add g r e a t l y to r e f r a c t o r y wear. 2. At c e r t a i n a i r f l o w r a t e s and bath depths, the matte i n the converter may " s l o p " u n c o n t r o l l a b l y . This may r e s u l t i n the e j e c t i o n of molten matte from the top of the converter and cause great danger to operating personnel i n the immediate v i c i n i t y . Dust l o s s e s and the b u i l d up of m a t e r i a l at the converter mouth may a l s o become excessive under these c o n d i t i o n s ( 8 ) . 1.3. The Zinc Slag Fuming Furnace A diagram of the z i n c s l a g fuming furnace i s shown i n Figure 1.2. Zinc s l a g fuming i s t o p o l o g i c a l l y s i m i l a r to the copper converting process i n that i t i n v o l v e s gas i n j e c t i o n from m u l t i p l e , h o r i z o n t a l l y o r i e n t e d tuyeres i n t o a molten bath. Zinc s l a g fuming i s a process whereby molten s l a g from lead smelting ( c o n t a i n i n g up to 20% ZnO) i s reduced by the a c t i o n of c o a l i n j e c t e d i n t o the furnace. The reduced z i n c leaves the bath as a vapour, where i t i s r e - o x i d i z e d , and removed by the furnace o f f - g a s e s . The k i n e t i c s of the s l a g fuming furnace have been e x t e n s i v e l y stu d i e d (9,10). However, the general l a c k of accurate information on the bath c i r c u l a t i o n v e l o c i t y has l i m i t e d the a p p l i c a b i l i t y of mathematical models d e s c r i b i n g t h i s process (9,10). 6 1.4. Objectives of the Work The primary objective of the research performed i n t h i s work was to i develop a mathematical model that was capable of describing the f l u i d flow regime i n h o r i z o n t a l l y i n j e c t e d m e t a l l u r g i c a l processes. For the case of the copper converter, such a model has several obvious a p p l i c a t i o n s . 1. To y i e l d i n s i g h t s of the general flow patterns and s t i r r i n g within conventional copper converters. 2. To provide input data for sophisticated heat and mass transf e r models of the converter. To date, the development of such models has been hampered by a lack of information on the bath c i r c u l a t i o n v e l o c i t y . In turn, such models can provide information to optimize the production of current copper converters, and aid i n the design of more e f f i c i e n t v e s s e l s . 3. To study and possibly suggest ways of preventing bath slopping. S i m i l a r l y , the a p p l i c a t i o n of such a model to the zinc slag fuming furnace w i l l provide information, previously unavailable, on the bath r e c i r c u l a t i o n v e l o c i t i e s . When considered along with k i n e t i c models (9,10) t h i s has a p p l i c a t i o n to the o p t i m i z a t i o n of the f u r n a c e o p e r a t i n g conditions. 7 2. LITERATURE REVIEW As described in Chapter 1, a major objective of this study was to provide a mathematical description of the fluid flow regimes in the Peirce-Smith copper matte converter and the Zinc Slag-Fuming furnace. As gas Injection into a liquid bath is the central feature of each of these processes, the scientific literature was searched in the following fields: 1. Studies of the flow regimes in non-ferrous reactors. 2. Experimental studies of simple bubble formation in water and liquid metals. 3. Investigations of the flow regimes in other gas-driven flow systems. These studies are largely concerned with inert-gas injected ladles and gas l i f t systems. 4. Numerical methods capable of predicting the velocity and pressure regimes within a copper converter and a zinc slag fuming furnace. This review will be discussed in detail in Chapter 4. 2.1. Flow Regimes i n Non-Ferrous Metallurgical Reactors Among the first to consider the fluid flow fields in copper matte converting was Themelis et al. (3). Based on air-water jet injection studies, Themelis et al. estimated the fluid flow regime within a copper converter to be as shown in Figure 2.1. 8 Based on the assumptions of Theraelis et a l . , Nakanishi and Szekely (4) formulated a f l u i d flow model of domain "A" i n Figure 2.1. The r e s u l t s of t h e i r c a l c u l a t i o n s are shown i n Figure 2.2, and show the maximum c a l c u l a t e d v e l o c i t i e s to be about 5.0 m/s. However, the s t u d i e s of O r y a l l (11) and O r y a l l and Brimacombe (12) showed that the model of Themelis et a l . was unable to p r e d i c t the behaviour of an a i r j e t i n l i q u i d metal. The j e t cone angle measured by Themelis et a l . i n water (20 degrees) was found to be 155 degrees i n mercury. In a d d i t i o n , bubbles i n mercury were found to r i s e almost v e r t i c a l l y , w i t h very l i t t l e forward p e n e t r a t i o n . These f i n d i n g s threw doubt upon the accuracy of the d e s c r i p t i o n shown i n Figure 2.2. I n d u s t r i a l measurements performed by Hoefele and Brimacombe (7) on a N i c k e l converter showed that the a i r enters t h i s v e s s e l i n the form of b u b b l e s - at a formation ra t e of 10 s ^ - and not as a j e t as was thought by The r a e l i s et a l . ( 3 ) . The bubble formation frequency of 10 s * agreed w e l l with that found a c c o u s t i c a l l y by Irons and Guthrie (13) i n a bath of l i q u i d i r o n . However, t h i s low frequency i n d i c a t e d l a r g e bubbles (estimated diameters of 0.40 to 0.68 m) and hence the p o s s i b i l i t y of bubble coalescence o c c u r r i n g . The l a b o r a t o r y measurements of Hoefele and Brimacombe (7) of gas i n j e c t i o n i n t o water, mercury and Zinc ( I I ) C h l o r i d e s o l u t i o n i n d i c a t e d that the t r a n s i t i o n between bubbling and j e t t i n g phenomena could be r e l a t e d to the modified Froude number and the r a t i o between the gas and l i q u i d d e n s i t i e s . This c r i t e r i o n ( s i m i l a r to the i n j e c t i o n number c o n d i t i o n employed by Wraith and C h a l k l e y (14)) i s i l l u s t r a t e d i n Figure 2.3. Bubbling behaviour i s 9 c l e a r l y p r e d i c t e d f o r matte c o n v e r t i n g . P l o t t i n g r e l e v a n t o p e r a t i o n a l data f o r the s l a g fuming furnace on Figure 2.3 i n d i c a t e s that t h i s too i s operating i n a bubbling mode. A mathematical model formulated by Ashman et a l . (15) was used to study the r e a c t i o n k i n e t i c s and oxygen u t i l i z a t i o n e f f i c i e n c y i n a copper c o n v e r t e r . A modified v e r s i o n of the bubble formation model proposed by Davidson et a l . (16) was employed. Data was not a v a i l a b l e f o r the bath r e c i r c u l a t i o n v e l o c i t y (upon which bubble formation depended s t r o n g l y ) . Therefore, t h i s parameter was v a r i e d independently u n t i l the p r e d i c t e d bubble formation frequency was approximately equal to that measured by Hoefele and Brimacorabe ( 7 ) . Using t h i s method, the v e r t i c a l v e l o c i t y of the l i q u i d matte i n the region of the tuyeres was c a l c u l a t e d to be between 1.2 and 2.6 ra/9. I n d u s t r i a l t r i a l s c a r r i e d out by Bustos et a l . (17) on an operating copper converter and a z i n c s l a g fuming furnace showed that the bubbling behaviour i n these v e s s e l s i s more complex than was p r e v i o u s l y b e l i e v e d . A n a l y s i s of tuyere pressure' measurements i n d i c a t e d that the copper converter behaves d i f f e r e n t l y depending upon the s t a t e of i t s r e f r a c t o r y l i n i n g . The bubbles forming at the tuyeres of a newly-lined copper converter were shown to r i s e independently, and not to i n t e r f e r e s i g n i f i c a n t l y with one another. However, a f t e r s e v e r a l heats (and corresponding r e f r a c t o r y wear) the bubbles a r i s i n g from adjacent tuyeres were found to i n t e r a c t with one another, forming a l a r g e , unstable gas envelope i n the region of the tuyeres. The bubbles i n the s l a g fuming furnace were found to act independently under a l l t e s t i n g c o n d i t i o n s at a frequency of 5-6 s ^. 10 U n f o r t u n a t e l y , no s t u d i e s were found that p r e d i c t e d the complete f l u i d flow regimes i n e i t h e r the copper matte converter or the z i n c s l a g fuming furnace as has been done f o r a r g o n - s t i r r e d l a d l e s . 2.2. Bubble Formation Investigations The behaviour of gas bubbles i n j e c t e d i n t o water has been e x t e n s i v e l y s t u d i e d . The e a r l i e r l i t e r a t u r e concerning bubble formation i n aqueous systems have been thoroughly reviewed by Kumar and Kuloor (18), Themelis, et a l . (3) and C l i f t , Grace and Weber ( 1 9 ) . Among the more s i g n i f i c a n t s t u d i e s were those performed by Davidson and co-workers (16) and Wraith and C h a l k l e y ( 14). The behaviour of a i r bubbles i n j e c t e d i n t o a l i q u i d (from a s i n g l e tuyere) has been summarized by O r y a l l (11) as f o l l o w s : 1. At v e r y low a i r f l o w r a t e s ( N R g < 500) the bubble volume i s r e l a t i v e l y i n v a r i a n t w i t h gas flow. Therefore, the bubble frequency of formation v a r i e s w i t h gas f l o w r a t e i n t h i s low-flow c o n d i t i o n . 2. At h i g h e r a i r f l o w s (500 < N R g < 2100), the bubble formation frequency ^ remains almost constant (at approximately 8-10 s ^ ) , hence bubble volume incre a s e s w i t h i n c r e a s i n g a i r flowrate i n t h i s regime. Simple models (15) have been formulated (based only upon a balance of buoyancy force and i n e r t i a ) y i e l d i n g formulae f o r bubble volume that agree w e l l with experiment, though nozzle diameter e f f e c t s have been seen to be s i g n i f i c a n t . 11 3. Increasing air flowrates further causes significant interaction between rising bubbles. Wraith and Chalkely (14) and Nilmani and Robertson (20,21) have investigated the effects of bubble-bubble interaction. Two different effects, termed "binary coalescence" and "stem coalescence" have been found to occur at low and high air flowrates respectively. The transition between these two types of behaviour is determined by gas momentum. Stem coalescence occurs when the bubbles elongate and form columns of air that reach the surface and break up unpredictably. 4. At higher flowrates s t i l l , (N„ > 10,000, N_ > 1000) the bubbling Re r r behaviour ceases altogether, and the gas enters the liquid as a turbulent jet. It is generally agreed that the transition between these types of behaviour are dependent upon increasing gas momentum. The nozzle Reynolds number (11) was i n i t i a l l y used to predict the transition between "bubbling" and jetting phenomena. As mentioned above, Hoefele and Brimacombe (7) used the modified Froude Number as an indicator of this gas momentum effect, and successfully predicted the transition between "bubbling" and "jetting" regimes. The injection number has been used similarly by Wraith and Chalkely (14) to correlate the effect of gas momentum for air injection into both water and liquid metal. 12 2.3. Investigations of Gas-Driven Fluid Flow 2.3.1. Experimental Studies One of the earliest investigations of the velocity fields set up by air injected into water was that performed by Bulson (22). The conclusions formed from this large-scale study may be summarized as follows: 1. A stable recirculation pattern was clearly observed at a l l air flowrates tested. 2. The maximum water velocities were found to occur at the surface. 3. This maximum water velocity was found to be directly related to the cube root of the air injection rate. Unfortunately, Bulson did not provide sufficient information to allow later calculation of the modified Froude Number of the injected gas. A large number of studies on gas-lift systems have been reported in the Chemical Engineering literature. For example, Kumar et a l . (23) correlated the liquid recirculation rate with the dimensionless gas flowrate. In addition, the gas holdup in the column was shown to be directly related to the injected gas flowrate. Some of the earliest experimental measurements that directly applied to metallurgical operations were performed by Szekely and Asai (24). They studied the general recirculation pattern of water in an air-injected physical model of a steel ladle using photography. This was largely a 13 qualitative study, used in conjunction with a mathematical model (25). Szekely, Wang and Riser (26) measured the velocity field in a water-filled plexiglass model of a ladle, by a combination of hot-wire anemometry and photographic techniques. Air was introduced through a vertically-oriented tuyere at the bottom of the tank at a flowrate of .2 1/s (a modified Froude Number of 1.3). Water velocities were measured at 24 different points in the model, and the general recirculatory nature of the flow was clearly discerned. Salcudean and Guthrie (27,28,29) employed two physical models in their study of the fluid flow generated while tapping ladles. Although primarily a liquid injection study, the effect of air entrainment on the fluid velocity distribution was observed. Szekely, Dilawari and Metz (30) also used a physical model of a ladle, but employed a motor-driven belt, rather than gas bubbles, as the means of generating the recirculatory flow regime. Laser-doppler measurements of the flow pattern were made. This investigation was primarily for the purpose of validating a mathematical model, and was not intended as a rigorous physical description of gas-liquid interaction. In a following paper, Szekely, El-Kaddah and Grevet (31) carried out laser-doppler measurements of the flow in an air-injected water model of a 6 ton ladle. Two air flowrates were considered, 12.3 Nl/min and 25 Nl/min, corresponding to (modified) Froude Numbers of .021 and .080 respectively. 14 At each flowrate, the water velocity in the plexiglas model was measured at 26 different locations. Hsiao, Chang and Lehner (32) measured gas plume velocities in a 1/7 scale water model of a ladle. The maximum water velocity was found to occur at the f l u i d surface and was found to be related linearly to the cube root of the air injection rate. A comprehensive analysis of gas-stirring was performed by Sahai and Guthrie (33,34) who carried out experiments in a .17 scale model of a 150 ton ladle. The gas was introduced into the tank through an immersed, downward directed lance, at a Modified Froude Number of about 650. The velocity profile within the tank was measured by a photographic technique: the motion of small seed particles within the tank was recorded at different times, and their velocity was assumed to be the same as the water velocity. These measurements were carried out in 106 different locations in the tank. Oesters, Dromer, and Kepura (35) employed hot wire anemometry to measure the velocity f i e l d in a water model of a ladle. Both bottom blowing (through a perforated plate) and top blowing (through a lance) were studied. An equation based on the studies of Hsiao et a l . (21) was formulated and shown to give a good estimate of the velocity in the gas plume. Two injection rates were studied: 3 1/s and 1.2 1/s, corresponding to modified Froude numbers of 2 and 32 respectively. In addition, the "mixing time" in the ladle as a function of air flowrate was shown to decrease with air flowrate asymtotically to a minimum. 15 Mazumdar and Guthrie (36) studied the flow in a .3 scale model of a 150 ton ladle. Photographic techniques and the motion of small threads immersed in the flow were used to measure the velocity profiles. Injection was carried out at a Froude number of approximately 2. An investigation by Haida and Brimacombe (37) confirmed the existence of a maximum in the stirring efficiency of an air-driven water model. Electrical probes were used to study the shear stress of the liquid at the wall (which is directly related to the velocity of the fluid) at various air injection rates. At a modified Froude Number of approximately 50, this shear stress was seen to reach a maximum. This was explained as being due to the onset of gas "channelling" through the liquid, and a means of predicting this behaviour was derived. 2.3.2. Numerical Studies As with the experimental work, virtually a l l effort in this field has been concerned with calculating the fluid flow regime in argon-stirred ladles. Drawing on earlier work by Gosman et al. (38), Szekely and Asai (24,25) formulated a mathematical model of the turbulent recirculating flow fields in an inert-gas stirred ladle. Two dimensional flow was assumed and the Kolmogorov-Prandtl single equation model was used to calculate the turbulent viscosity. The solution was performed in stream function and vorticity. The free surface was assumed to be horizontal, and to be of zero 16 vorticity. Reasonable qualitative agreement was found with visual experimental observations, although quantitative comparisons of the calculated flow field with experimental results were not made. In the mathematical segment of the work performed by Szekely et al. (26), the k-W technique proposed by Spalding (39) was used to model the turbulent viscosity in a two dimensional mathematical model of an experimental ladle. They neglected free surface effects, and considered the bubble column to apply a constant upward velocity to adjacent fluid. Again, fair qualitative agreement was found with experiment, but actual velocities were found to be up to an order of magnitude different from computed velocities. Deb Roy, Majumadar and Spalding (40) also used a two-dimensional mathematical model in an attempt to predict the flow regime in an air-stirred water model (31) of a ladle. An algebraic model, proposed by Pun and Spalding (41), was used to calculate the turbulent viscosity. The bubble-water boundary condition was modelled by varying the density in this boundary region according to a "void fraction" - the amount of space occupied by the bubbles. Two separate methods of calculating this void fraction were then tested. 1. No slip. The air moves with the same velocity as the liquid. 2. Slip. The bubbles move through the liquid at their terminal velocity. 17 In a l l cases, the free surface was considered to be a horizontal, non-retarding wall. These calculations were found to yield better agreement with experiment than those published by Szekely et a l . (26). Szekely, Dilawari and Metz (30) used a model similar to Szekely et a l . (26), but applied wall functions in the region of the walls to better approximate these boundaries. Good agreement was found between the model predictions and experiments in which the recirculation of the bath was caused by a motor-driven belt, rather than by bubbles. This indicated that the main reason for the differences between previous calculations and experiments was due to inaccurate approximations of the bubble-liquid interface. When this boundary was simplified to a form that was easily approximated mathematically (a belt-driven system), good agreement was observed between calculations and experiments. Again neglecting deformation of the free surface, Szekely, El Kaddah and Grevet (31,42,43) employed the improved approximation of the bubble boundary suggested by Deb Roy et al. (40) and calculated the flow regime in a gas-agitated system. Velocity fields were calculated based on both the k-W model and the algebraic model of Pun and Spalding (41). The algebraic model was found to yield acceptable results, and was much less expensive in computer time. However, considerable differences between experiment and calculations were s t i l l observed. Sahai and Guthrie (44) proposed a new algebraic relation for a constant effective viscosity, deliberately designed for bubble-driven systems 18 (the v i s c o s i t y model of Pun and Spalding (41) was based on the a c t i o n of tu r b u l e n t j e t s ) . In two succeeding papers (33,34) they examined the mathematical d e s c r i p t i o n of the bubbling mechanism and developed a r e l a t i o n f o r mean bath r e c i r c u l a t i o n v e l o c i t y and bubble plume v e l o c i t y . I n a d d i t i o n , they formulated a turbu l e n t model, based on Patanker and Spalding's (45) SIMPLE a l g o r i t h m . This model was found to give good q u a l i t a t i v e agreement with experiments. The f i r s t to examine the e f f e c t s of the f r e e surface were Salcudean, Low, Hurda and Guthrie ( 4 6). A three dimensional model was w r i t t e n that used a Marker and C e l l (MAC) type method to des c r i b e the deformation of the free s u r f a c e . The d e s c r i p t i o n of the b u b b l e - l i q u i d boundary suggested by Deb Roy et a l . (40) was employed as was the v i s c o s i t y model of Pun and Spalding (41). C a l c u l a t i o n s were performed f o r symmetrical and o f f - c e n t r e d gas i n j e c t i o n . Reasonable agreement was found both with two dimensional c a l c u l a t i o n s (40) and w i t h experimental r e s u l t s . A three dimensional, steady s t a t e , turbulent model formulated by Salcudean and Wong (47) was a l s o compared to these c a l c u l a t i o n s . Salcudean, L a i and Guthrie (48) and Mazuradar and Guthrie (49) extended these c a l c u l a t i o n s to compare the accuracy of the turbulent v i s c o s i t y models proposed by Sahai and Guthrie (44) and Deb Roy et a l . (40) w i t h the k-e model of Jones and Launder (50). A l l models were compared with the experimental r e s u l t s of Oeters et a l . (35). S u r p r i s i n g l y , the f l u i d p r o f i l e s p r e d i c t e d by the a l g e b r a i c models were c l o s e r to experimental values than were the r e s u l t s c a l c u l a t e d from the k-e model. This was thought to be 19 due to the influence of buoyancy and curvature effects, as well as Inaccuracies introduced by upwind differencing. 2.4* Conclusions From the literature mentioned above, a number of conclusions can be made about the prediction of the flow regimes in the copper matte converter and the zinc slag fuming furnace: 1. The air enters these vessels in the form of bubbles. Therefore the mathematical method used to model the flow in these vessels should be one that is capable of modelling flow induced by bubble injection. 2. The study of Salcudean et a l . (48) has shown that for the computation of flow in a ladle, the complex k-e differential model of turbulence does not yield as accurate results as the simple algebraic viscosity model of Sahai and Guthrie (44). 3. A number of mathematical approximations have been used to describe the stirring effect of bubbles in liquid ladles. The most successful of these to date has been the variable density or void fraction model of Deb Roy et al . (40). 4. Independent air-injection studies by Hsiao et al. (32) and Bulson (22), under very different conditions, have shown that the maximum liquid velocity frequently occurs at the surface of the vessel. This velocity may be related to the cube root of the air injection rate. 20 3. EXPERIMENTAL 3.1. Objectives The two major objectives of the experimental portion of this work were as follows: 1. To provide a quantitative description of the stirring of a liquid by air bubbles, at Injection conditions similar to the industrial processes of copper converting and zinc slag fuming. 2. To verify a mathematical model written to describe both the experimental and the industrial systems. 3.2. Apparatus The experimental apparatus used in this investigation is shown schematically in Figure 3.1. The two major components of this system were a water-filled plexiglass tank and a laser-doppler velocimeter. The design of the water tank assembly was determined by the following criteria: 1. Rectangular in cross section. 2. Air injected from a series of horizontal side-mounted tuyeres. 3. Open at the top. 21 4. Adjacent tuyeres do not Interact significantly at low air flowrates. The tank was chosen to be rectangular in cross section to simplify both velocity measurements and later mathematical modelling. A rectangular computatational domain allows finite-difference calculations (which are carried out in a rectangular grid) to be performed easily - therefore a rectangular tank is much simpler to treat in this manner than tanks of other shapes. Further, i t is difficult to obtain laser-doppler velocity measurements in a tank with surfaces not normal to the incident laser beam, due to refraction of the laser light. For example, i t is hard to obtain laser-doppler measurements at a l l positions within a cylindrical tank. Therefore a rectangular tank was constructed to allow measurements to be made easily at a l l positions. Side-blowing of air from more than one tuyere was specified for two reasons: a) The metallurgical processes of interest in this study (copper converting and zinc slag fuming) are side-Injected. b) There already exist many excellent studies of stirring by a single, vertically oriented tuyere. The surface of the bath was unconstrained to achieve similarity with the metallurgical processes of interest. 22 The spacing between the tuyeres was given much design consideration as i t has been shown by Bustos et a l . (17) that the interaction of bubbles forming at adjacent tuyeres is a significant effect in nickel matte converting and copper converting. To investigate this effect experimentally, a tuyere spacing was sought in the experimental tank that would prevent or permit the coalescence of bubbles at adjacent tuyeres depending on air flowrate. Preliminary experiments were performed to determine the appropriate minimum tuyere spacing to achieve interaction at Froude Numbers approximating those used in copper converting. This distance was found to be approximately 2.5 cm. Provision was made to blow air from two sides of the tank i f so desired. Two-sided gas injection was the source of qualitative data only, and is discussed in Chapter 6. Note that exact dimensional similarity with a zinc slag fuming furnace or a copper converter was not a design criterion. The tank constructed for the purposes of this study was not, nor was i t intended to be, a physical model of any metallurgical process in particular. In fact, the aforementioned design criteria effectively rule out any possible direct correspondence between the dimensions of the experimental tank and either of the two metallurgical process vessels of interest to this study. Rather, the purposes of the velocity measurements carried out in this tank were both to verify, and to provide input data for, a mathematical model of the flow regimes within a copper converter and a zinc slag fuming furnace. In particular, a means of characterizing the interaction of air bubbles with a 23 s u r r o u n d i n g l i q u i d w a s o f p r i m a r y i m p o r t a n c e . T h e r e f o r e , t h e a c t u a l p h y s i c a l d i m e n s i o n s o f t h e e x p e r i m e n t a l t a n k w e r e n o t i m p o r t a n t , p r o v i d e d t h a t t h e y f a c i l i t a t e d v e l o c i t y m e a s u r e m e n t s a n d s u b s e q u e n t m a t h e m a t i c a l m o d e l l i n g . H o w e v e r , s i m p l y b e c a u s e r i g i d p h y s i c a l s i m i l a r i t y d o e s n o t e x i s t b e t w e e n t h e e x p e r i m e n t a l t a n k a n d t h e two i n d u s t r i a l p r o c e s s e s o f i n t e r e s t , t h i s d o e s n o t n e c e s s a r i l y i m p l y t h a t t h e m e a s u r e d v e l o c i t y p r o f i l e s w e r e c o m p l e t e l y i n a p p l i c a b l e t o t h e s e p r o c e s s e s . D e s p i t e a l l d i f f e r e n c e s , s i m i l a r i t y o f t h e m o d i f i e d F r o u d e N u m b e r w a s m a i n t a i n e d t h r o u g h o u t t h e s e e x p e r i m e n t s , i n d i c a t i n g t h a t t h e b u b b l e b e h a v i o u r t h a t o c c u r r e d i n t h e e x p e r i m e n t a l t a n k was s i m i l a r t o t h a t o c c u r r i n g i n t h e i n d u s t r i a l c a s e s . I t I s p r i m a r i l y t h e i n t e r a c t i o n o f t h e s e b u b b l e s w i t h t h e s u r r o u n d i n g l i q u i d w i t h w h i c h t h i s s t u d y i s c o n c e r n e d . I t i s n o t c l a i m e d t h a t t h e v e l o c i t y m e a s u r e m e n t s m a d e w i t h i n t h i s t a n k a r e i d e n t i c a l t o t h o s e o c c u r r i n g i n d u s t r i a l l y , b u t r a t h e r t h a t t h e r e e x i s t s ( i f o n l y q u a l i t a t i v e l y ) s i m i l a r l i t y o f t h e i n t e r a c t i o n m e c h a n i s m b e t w e e n t h e g a s b u b b l e s a n d t h e s u r r o u n d i n g f l u i d . D e s p i t e s u c h o b v i o u s c o m p l i c a t i o n s a s t h e n o n - w e t t i n g c h a r a c t e r i s t i c s o f l i q u i d m e t a l s , a n d t h e d i f f e r e n c e b e t w e e n l i q u i d i n t e r a c t i o n w i t h r e a c t i n g a n d n o n - r e a c t i n g g a s e s , t h e r e s u l t s o f t h i s e x p e r i m e n t a l s t u d y w i l l s t i l l b e g e n e r a l l y a p p l i c a b l e t o z i n c s l a g f u m i n g a n d c o p p e r c o n v e r t i n g f o r two r e a s o n s : 1. I n a l l o f t h e s e p r o c e s s e s , a l i q u i d b a t h i s s t i r r e d w h o l l y b y i n c o m i n g g a s t h a t e n t e r s t h e v e s s e l i n t h e f o r m o f b u b b l e s . T h u s e v e n t h o u g h 24 various liquid characteristics vary among the three processes, the basic method of stirring remains the same: lifting and shearing of the surrounding liquid by air bubbles. 2. A mathematical model that is capable of describing the flow regime within the experimental tank should therefore be capable of extrapolating these results to industrial processes, with greater confidence than calculations based simply on assumptions about the bubble-stirring mechanism. With these conclusions in mind, the internal dimensions of the plexiglass tank were designed to be 30 cm x 20 cm. The tank was made 60 cm ta l l to prevent escape of liquid by splashing. Five holes, each 1.0 cm in diameter, were drilled along two sides of the tank at 2.5 cm intervals, and located 3.0 cm above the tank bottom. Water-tight fittings were placed in each of these holes through which 0.6 cm diameter nylon tubing was inserted, forming tuyeres (Figure 3.2). Air was introduced into these tuyeres through the manifolds shown in Figure 3.2. These manifolds were constructed of 3/8 in. PVC tubing and hoseclamps. Great care was taken In their construction to ensure that the air flow out of each tuyere was identical (this was verified by successive rotameter measurements of each tuyere). The air flow into the tank was measured by a calibrated rotameter attached to a manometer (Figure 3.1). 25 The laser-doppler velocimeter (LDV) consisted of a 15 mW Spectra-Physics Helium-Neon laser connected to optics and data-collection electronics manufactured by TSI Inc. The detailed operation of an LDV has been outlined elsewhere (65,66) and w i l l not be discussed here. The velocity of the water in the tank was measured within a region of space 1.8 cm long, defined by the intersection of two laser beams (Figure 3.3). The value recorded as the water velocity at a particular point in the tank was actually an average of 512 separate velocity readings taken from anywhere within this 1.8 cm measuring length. The LDV was only capable of measuring one component of the f l u i d velocity at any one time, (either horizontal or vertical) but the orientation of the refracting optics could be altered to allow later measurement of the other component. The e n t i r e l a s e r - d o p p l e r apparatus was mounted on a computer-controlled traversing table. This table was constructed so that i t could position the measuring volume of the LDV at any position in the cross-section of the tank with an error of less than 1.0 mm. Data collection was also performed by computer. At each point within the tank, the LDV sent 512 independent water velocity measurements to an Apple He microcomputer. The computer then averaged these measurements, and printed the result (along with the standard deviation of the points) on the screen and saved these values on disk. x 26 3.3. Procedure The laser-doppler measurements were carried out in the central ve r t i c a l plane of the experimental tank, in a 10 x 10 grid with points spaced 3.0 cm apart (Figure 3.4). Each separate square region within this grid is termed a " c e l l " , due to the similarity between this experimental mesh, and a f i n i t e difference grid. Velocity measurements were carried out in the centre of each of these c e l l s , as shown. The numbered ce l l s correspond to the measurement locations mentioned in Figures 3.5 and 3.6. Velocity measurements were carried out at ten separate air flowrates, ranging from 38 to 220 std 1/min. (Table 3.1). Due to the inability of the LDV to measure both horizontal and vertical f l u i d velocity components simultaneously, a complete experimental run involved two complete sweeps over this mesh of 100 c e l l s . On the f i r s t sweep the horizontal component was measured, and the optics were then adjusted to measure the vertical component. The second sweep was then performed, measuring the vertical water velocity in each of the 100 positions in the tank. The experimental procedure was therefore very straightforward: 1. The tank was f i l l e d to i t s desired operating level (always 30.0 cm) with d i s t i l l e d water. 27 2. Seed particles (small plastic spheres measuring 2.0 x 10 m in diameter) of neutral buoyancy were added to the water in the tank. 3. The tank was positioned perpendicular to the laser, and i t s exact position relative to the LDV was recorded. 4. The air supply was switched on and set to the desired flowrate by observing the rotameter/manometer assembly. 5. The traversing table, computer, and LDV were switched on. 6. The data collection and traversing program was run on the computer. After these preliminary steps had been performed, the traversing-table LDV system performed the following tasks, with l i t t l e or no intervention: 7. The traversing table positioned the laser at a point in the 100 point scan. 8. The LDV made 512 separate measurements of the (horizontal or vertical) flu i d velocity at this new location. 9. These values were transferred to the computer, which averaged them and stored the mean and standard deviation on a floppy diskette. The computer then calculated the next location for the traversing table, and gave these instructions to the computer controlling the table. Steps 7 to 9 were repeated as necessary un t i l velocity measurements had been taken in each of the 100 locations. 28 Once both velocity traverses had been completed, the shape of the water surface was traced onto a piece of clear plastic. This surface shape was later digitized, and stored on a mainframe computer along with the velocity data, allowing later analysis and plotting. The only serious d i f f i c u l t y encountered in implementing this procedure arose from the presence of smaller air bubbles remaining in the water near the surface and left wall. These bubbles randomly blocked and scattered the light from the laser beam, sometimes causing a very long period of time to elapse before 512 water velocity measurements could be taken. Further, this scattering of the laser could cause spurious velocity measurements i f the LDV happened to measure the velocity of a passing bubble, rather than the surrounding water. This problem was overcome by introducing very small and precisely sized seed particles into the water tank. This then enhanced signal quality, and permitted the LDV to distinguish between particles and bubbles. 3.4. Experimental Results 3.4.1. Introduction In addition to the water velocity and surface profile measurements, a number of general observations have been made on the experimental system as a whole: 1. The air was seen to enter the tank in the form of bubbles at a l l flowrates tested. This was expected since even the maximum experimental modified^ F^yide number used (15.5) was well within the bubbling regime 29 A s d e s i g n e d , a d j a c e n t t u y e r e s d i d n o t i n t e r a c t s i g n i f i c a n t l y a t l o w a i r f l o w r a t e s . H o w e v e r , b o t h p h o t o g r a p h s a n d v i s u a l o b s e r v a t i o n s I n d i c a t e t h a t a t a i r f l o w r a t e s g r e a t e r t h a n a p p r o x i m a t e l y 90 s t a n d a r d l i t r e s p e r m i n u t e , b u b b l e s r i s i n g f r o m a d j a c e n t t u y e r e s i n t e r f e r e d w i t h o n e a n o t h e r , a n d d i d n o t r i s e i n s e p a r a t e , d i s c r e t e c o l u m n s . T h i s c o r r e s p o n d e d t o a m o d i f i e d F r o u d e n u m b e r o f 2 . 3 2 . T h e b u b b l e c o l u m n was s e e n t o b e c o n f i n e d t o t h e r e g i o n a r o u n d t h e l e f t w a l l , o n l y p e n e t r a t i n g 2 t o 3 cm i n t o t h e b a t h . T h e t a n k w a s o b s e r v e d t o o s c i l l a t e s l i g h t l y w h i l e v e l o c i t y m e a s u r e m e n t s w e r e b e i n g t a k e n . T h i s o c c u r r e d b e c a u s e t h e t a n k w a s n o t s e c u r e l y a n c h o r e d , b u t was s i m p l y p l a c e d o n a s u p p o r t i n g t a b l e . V a r i o u s a t t e m p t s w e r e made t o p r e v e n t t h i s a c t i o n , b u t n o n e w a s c o m p l e t e l y s u c c e s s f u l . T h i s v i b r a t i o n was n o t s e v e r e ( a d e f l e c t i o n o f a p p r o x i m a t e l y ± 2 mm a t i t s w o r s t ) a n d i t i s u n l i k e l y t h a t i t a f f e c t e d t h e v e l o c i t y m e a s u r e m e n t s s i g n i f i c a n t l y f o r t h e f o l l o w i n g r e a s o n s : a ) T h e r e w e r e n o s i g n i f i c a n t o s c i l l a t i o n s a b o u t t h e y - a x i s ( F i g u r e 3 . 1 ) . b ) O s c i l l a t i o n s a b o u t t h e x - a x i s h a v e a v e r y s m a l l e f f e c t s i n c e t h e v a r i a t i o n o f t h e f l o w t h r o u g h t h e d e p t h o f t h e t a n k i s s m a l l ( s e e b e l o w ) . F u r t h e r , v e l o c i t y m e a s u r e m e n t s w e r e made a l o n g a 1.8 cm r e g i o n d e f i n e d b y t h e i n t e r s e c t i o n o f two l a s e r b e a m s ( s e e A p p a r a t u s s e c t i o n i n t h i s c h a p t e r ) . H a r m o n i c m o t i o n a b o u t t h e x - a x i s w o u l d m e r e l y s e r v e t o e x t e n d t h i s l e n g t h s l i g h t l y . c ) O n l y v i b r a t i o n a l m o t i o n a b o u t t h e z - a x i s w o u l d h a v e a n e f f e c t , h o w e v e r d u e t o t h e a v e r a g i n g o f v e l o c i t y m e a s u r e m e n t d a t a , e v e n t h i s e f f e c t w o u l d b e s m a l l . A s s u m i n g t h a t t h e s e o s c i l l a t i o n s o c c u r s y m m e t r i c a l l y 30 about the z-axis, the v e l o c i t y measurements would be evenly distributed in the region within 2 millimetres to either side of the desired position. Since a velocity measurement at a particular point was used to represent the average velocity of the fl u i d for 15 mm in each direction, i t is unlikely that this effect was significant, compared to the coarseness of the experimental data grid. 3*4.2. Check of the Two Dimensionality of the Flow Before carrying out any of the complete velocity scans, several checks were made on the variation on flow through the width of the tank. From the point of view of subsequent computer modelling, i t would be ideal i f the horizontal and vertical velocity components at each of the 100 measurement locations were constant at any point in the tank width. This would be the case of truly two-dimensional behaviour. However, there were two reasons for suspecting that this type of behaviour would not be observed: 1. The effect of the walls of the tank would be to retard the nearby f l u i d . Thus, one would expect the water velocity to be lower in the region of the walls than in the centre of the f l u i d . 2. The sti r r i n g of the tank was caused by bubbles injected from a number of adjacent tuyeres. Any significant variation in the air flowrates among these tuyeres could cause a corresponding variation in the local fluid velocity. 31 To quantify these effects, the variation of the horizontal and vertical components with tank width was measured at four separate locations in the tank (Figure 3.4). When plotted (Figures 3.5 and 3.6) they show that there i s , in fact, some variation of flow with width. The wall effect is noticeable and as expected the velocities are higher i n the centre of the tank than at the walls. However, the flatness of the profiles (a maximum velocity variation of 10%) indicates that this effect i s slight. More importantly, these profiles are relatively symmetrical - indicating that the air flowrates through the tuyeres were equal. Because of this slight velocity variation, a l l velocity scans were performed in the central vertical plane of the tank, where the wall effect was least significant. The location of the measuring volume that was used in these scans is clearly seen in Figures 3.5 and 3.6 (the dashed l i n e ) . Another, more rigorous, method of checking the two dimensionality of the flow is to apply the principle of conservation of mass to the measured velocity profiles. That i s , once a l l vertical and horizontal velocity measurements had been made at a given air flowrate, the two dimensional continuity equation was applied to each experimental c e l l . If the horizontal and vertical mass flows into each c e l l were equal to the horizontal and vertical flows out, then there would be no reason to suspect any transverse flow. Stated mathematically, for an incompressible f l u i d , the three dimensional continuity equation i s : 32 | E + SS. + J £ « o (3.1) 5x dy dz Applying a backward fi n i t e difference approximation, this becomes (for an experimental c e l l ) : u i j k " U i - l , j k + v i j k " V i j - l , k + w i j k " W i j , k - 1 m Ax Ay Az (3.2) Therefore i f the summation: ~ U ± - 1 ' ^ + ^ " ^ J - 1 (3.3) Ax AY i s everywhere equal to zero, then —— = 0 and the flow is confined to the QZ vertical plane. Applying Equation (3.3) to several cells in the velocity vector plot of Figure 3.7 yields the values shown in Table 3.1. It is immediately apparent that Equation (3.3) does not, In fact, equal zero at a l l locations. However, even with truly two-dimensional experimental flow, there are several reasons why one might expect Equation (3.3) to be non-zero. 1. The equation assumes an incompressible flu i d of constant overall density. Both visual observation and photographs have confirmed that there are several regions of the tank (such as near the surface or the bubble column) where the air is entrained in the liquid phase. In these regions, the local density could vary significantly, invalidating Equation (3.3). This effect is shown by considering the variable density form of the 33 continuity equation: dp. + o ( p u ) + 5 ( p v ) 0 0 / 3 4 ) at ax ay Equation (3.3) assumes p • constant. If this is not the case, then the value calculated by Equation (3.3) is erroneous. Unfortunately, without knowing the values of •^ D-, i t i s impossible to determine the "true" two a t ' ax' ay dimensional divergence for these c e l l s . Therefore any subsequent consideration of the two dimensionality of the flow in the experimental tank w i l l be confined to regions uncontaminated by air bubbles. 2. A further reason why Equation (3.3) might be non-zero arises from the coarseness of the experimental mesh. Equation (3.3) is a finite-difference approximation to the true relationship: + ?• - 0 (3.5) ax ay and this approximation rapidly loses accuracy as mesh size is increased. Accuracy w i l l be most adversely affected in regions where the flow is highly rotational, such as in the central vortex region of Figure 3.7. In this case, the f l u i d velocity changes strongly with position and there are significant changes in the orientation of the flow vectors from one c e l l to the next. The velocity at one position may be tremendously different from that measured only a few centimetres away. Thus the continuity equation that applies for the infinitesimal region around this point may bear no resemblance whatever to that for the c e l l as a whole, and therefore the fact 34 that Equation (3.3) does not equal zero does not necessarily imply that — is dz finite. The simplest solution to this difficulty would be to reduce the size of the experimental mesh. Unfortunately, time constraints eliminated this as an option. Therefore further analysis of the two dimensionality of the flow in the tank will be performed on regions in which the velocity vectors are relatively parallel, and do not change greatly with position. Selecting regions that do not violate the two criteria mentioned dw above reduces the mean values of — . These are shown for each set of dz experimental velocity measurements in Table 3.2. It is clear that the overall flow in the experimental tank was not perfectly two-dimensional. While this is only to be expected, i t is important to note for subsequent comparisons between experimental and calculated results that the mathematical model developed in the following chapter was In fact truly two dimensional, and was incapable of considering any transverse flow. 3.4.3. V e l o c i t y P a t t e r n s The series of laser-doppler measurements carried out at ten air flowrates between 38 and 220 standard l/min yielded the velocity vector plots shown in Figures 3.7 to 3.16. Each of these plots has been scaled so that the longest velocity vector corresponds to the maximum measured velocity (shown at the bottom of the plot). The border around each plot represents the position of the inside walls of the tank. Cells that do not contain 35 arrows indicate that it was not possible to obtain velocity measurements at that position (usually due to bubbles blocking the laser beam). Such cells are common at the higher air flowrates and near the surface of the tank. Despite the presence of such "empty" cells, the general recirculatory motion of the flow is clearly observed. The left-most column of each plot is the region occupied by the bubble column. It was not possible to obtain accurate water velocity measurements in this region, due to the large numbers of bubbles in this region, which block the laser beam. The expected recirculation pattern is clearly evident in each plot, with the fluid rotating around a quiescent central region. For each velocity vector plot, the highest velocities appear to occur at, or close to, the surface of the bath. Low water velocities occur in the centre of the vortex and in the vicinity of the walls. A close inspection of the velocity plots reveals notable differencs between those taken at widely different air flowrates. For example, the centre of the vortex in Figure 3.7 (corresponding to a blowing rate of 38 l/min) is in the approximate centre of the tank, while that of Figure 3.16 (at an air flowrate of 216 l/min) is in the upper left corner. Further, such prominent features as the strongly downward-directed velocity vectors in the upper right quarter of Figure 3.16 are not present at a l l in Figure 3.7: the vectors in this region of Figure 3.7 are horizontal. This shift in the location of the vortex centre and the orientation of the velocity vectors 36 indicates that there was a s i g n i f i c a n t change i n the flow conditions between the lowest and the highest a i r flowrate. This hypothesis i s supported by observing the figures l y i n g between these two extremes. They show a st e a d i l y - i n c r e a s i n g downward motion of surface v e l o c i t y vectors, with a corresponding s h i f t i n the l o c a t i o n of the vortex. 3.4.4. Surface Shape In a d d i t i o n to v e l o c i t y vectors, Figures 3.7 to 3.16 also contain the steady state f l u i d surface p r o f i l e s . These p r o f i l e s were obtained by d i g i t i z i n g the traces of the experimental surfaces. As with the vector p l o t s , the surface p r o f i l e s also show considerable change with a i r blowing r a t e . This v a r i a t i o n i s emphasized by i n t e g r a t i o n of these surface p r o f i l e s . By conservation of mass, the volumes defined by each of these curves should a l l be equal to the volume of the undisturbed bath (18 l i t r e s ) . Performing t h i s i n t e g r a t i o n by the trapezoidal method, (and assuming only a two-dimensional surface v a r i a t i o n ) reveals that t h i s volume i s i n a l l cases larger than the quiescent volume. This "extra" volume must therefore be due to the presence of a i r bubbles. C a l c u l a t i n g a percentage gas holdup defined by: „ ,. Actual Bath Volume - Quiescent Bath Volume mnv / o 0 Gas Holdup = tr—: — .j. -, x 100% (3.6) r Quiescent Bath Volume 37 and plotting against injection air flowrate gives Figure 3.17. There is clearly a direct relationship between the gas fraction and the flowrate of the injected air. More importantly however, the gas fraction at the higher flowrates tested Is as high at 14%. This has serious ramifications for the mathematical model discussed in the following chapter, since fluid properties such as viscosity and density will be altered significantly by this amount of gas. In addition, any calculations performed to model an industrial operation (such as zinc slag fuming) must take this effect into account. Plotting gas holdup against modified Froude number In Figure 3.18 -for both the experimental results and data obtained on the slag fuming furnace - shows a very poor correlation. The gas fraction of the slag fuming furnace is much higher than any experimentally measured value, and yet the modified Froude Number of the furnace injection is lower than many experimental runs. Therefore, the gas fraction variation is not due to the inertial variation of the incoming air, and is not a bubbling-jetting phenomenon. 3 . 4 . 5 . Bath S t i r r i n g A quantitative indication that some type of flow transition occurs between the lowest and the highest air flowrates tested, is given by the bath kinetic energy (or stirring energy). The (experimental) mean cell kinetic energy can be calculated by: 38 E c e l l * N where N i s equal to the number of c e l l s which c o n t a i n v e l o c i t y data. C e l l s f o r which i t was not p o s s i b l e to c o l l e c t v e l o c i t y measurements ( u s u a l l y those near the surface) are excluded from t h i s c a l c u l a t i o n . The e f f e c t t h i s may have on the summation i s discussed below. E q u a t i o n ( 3.7) has been a p p l i e d t o each s e t o f v e l o c i t y measurements. When p l o t t e d against a i r f l o w r a t e , one ob t a i n s the curve shown i n Figure 3.19. This shows a steady increase i n c e l l energy w i t h a i r flow up to a maximum at an a i r flo w r a t e of about 90 1/min and corresponding to a modified Froude number of 3. At t h i s p o i n t , a sharp t r a n s i t i o n occurs, and the mean c e l l energy a c t u a l l y decreases with i n c r e a s i n g blowing r a t e ( t h i s i s most unexpected, as the i n j e c t e d a i r i s the source of a l l k i n e t i c energy i n the bath). The mean c e l l energy continues to drop f o r a l l remaining a i r f l o w r a t e s . U n f o r t u n a t e l y , the e x c l u s i o n of c e l l s l a c k i n g v e l o c i t y measurements may have a f f e c t e d t h i s summation somewhat, e s p e c i a l l y s i n c e the water v e l o c i t i e s near the surface were the l a r g e s t , and i t was p a r t i c u l a r l y d i f f i c u l t to o b t a i n surface v e l o c i t y measurements at higher a i r f l o w r a t e s . This e f f e c t can be i n v e s t i g a t e d by c a l c u l a t i n g the mean c e l l k i n e t i c energy of non-surface c e l l s and p l o t t i n g t h i s as a f u n c t i o n of a i r f l o w r a t e , as shown i n Figure 3.20. This curve f o l l o w s the same o v e r a l l trend as Figure 3.19, i n d i c a t i n g that t h i s observed t r a n s i t i o n i s not due to a lack of surface v e l o c i t y measurements. n n J 1 2 2 Z Z j (pAxAyz)(u . + v ) (3.7) i - l j - l L - 1 J 1 J 39 As a l l other important factors such as depth of f i l l , tuyere diameter and vessel dimensions were constant throughout each run, i t can only be concluded that this effect is due to a change in the mechanism of air-water energy transfer with air flowrate. The interaction of the incoming air with the water must have undergone a sharp change at an air blowing rate of about 90 l/min, otherwise the cell kinetic energy would continue to increase with increasing air flowrate. A plot of the mean cell energy as a function of air input energy (Figure 3.21) illustrates this further. The energy input rate is defined as being the sum. of the air buoyant and kinetic energy: • • • . E. . = E, + E. . . (3.8) input buoyant kinetic - 2 <P„ " p J Q h + 1 / 2 P - Q ( Q / A ) 2 < 3- 9) Figure 3.21 shows that, after a certain point, increasing the energy Input rate actually causes a decrease in the kinetic energy possessed by the bath. Obviously therefore, the mechanism of energy transfer changes with air flowrate. To characterize this transition, a series of photographs were taken of the air bubbles in the tank (Figures 3.22 to 3.24). Figure 3.22 shows bubble formation at a relatively low air flow of 38 l/min N , = 0.4, r r corresponding to the velocity regime of Figure 3.7. It can be seen that the air enters as distinct bubbles that do not interact to any significant degree. The photograph taken at the transition air flowrate of 90 l/min AO N„ , « 2.3) shows that the bubbles tend to Interfere with one another, or Fr' . . ' coalesce. Finally, at an air flowrate of 256.8 1/min (corresponding to the plot of Figure 3.16) Figure 3.2A shows the bubbles coalescing continuously, and the air "channels" i t s way to the surface. It i s this coalescence or channeling that i s probably responsible for the reduced efficiency of energy transfer. The maximum amount of fl u i d entrained by a bubbled gas seems to occur when the bubbles do not interfere with one another. If the gas "channels" to the surface in the form of a continuous column of a i r , very l i t t l e f l u i d is entrained. In such a case, the surrounding f l u i d is unable to penetrate the gas column, and much of the gas rushes to the surface without encountering any liquid at a l l , and l i t t l e energy transfer can occur. The transition between Figures 3.7 and 3.16, and the shape of Figure 3.21 can then be explained by the following hypothesis: 1. For air flows of less than about 90 1/min (N , = 2.3) the air enters the r r bath in the form of discrete, non-interacting bubbles, which entrain liquid well. Increasing air flowrate merely increases energy transfer. This corresponds to the increasing section of the curve in Figure 3.21 and the profiles of Figure 3.7 to 3.10. 2. At an air flowrate of about 90 1/min (the maximum of Figure 3.19 and corresponding to a modified Froude Number of 2.3) the bubbles start to interact significantly both longitudinally, and with bubbles rising from adjacent tuyeres. 4 1 3 . A i r f l o w r a t e s g r e a t e r t h a n t h i s c a u s e g r e a t e r c o a l e s c e n c e , a n d i n c r e a s i n g l y p o o r g a s - l i q u i d c o n t a c t i n g . T h e f l u i d f i n d s i t m o r e a n d m o r e d i f f i c u l t t o p e n e t r a t e t h e g a s c o l u m n . T h i s c o r r e s p o n d s t o t h e d e c r e a s i n g s e c t i o n o f F i g u r e 3 . 1 9 , a n d c o v e r s F i g u r e s 3 . 1 1 t o 3 . 1 4 . 4 . A t a n a i r f l o w r a t e o f 1 5 7 . 8 l / m i n ( t h e m i n i m u m o f F i g u r e 3 . 7 a n d c o r r e s p o n d i n g t o a m o d i f i e d F r o u d e n u m b e r o f 1 2 ) c o m p l e t e c h a n n e l l i n g o c c u r s . T h i s d r a s t i c a l l y d i f f e r e n t g a s - l i q u i d c o n t a c t i n g m e c h a n i s m i s r e s p o n s i b l e f o r p r o d u c i n g t h e v e l o c i t y f i e l d o f F i g u r e s 3 . 1 5 a n d 3 . 1 6 . C h a n n e l i n g b e h a v i o u r h a s b e e n o b s e r v e d p r e v i o u s l y b y s e v e r a l p r e v i o u s s t u d i e s ( 3 7 , 1 4 ) . H a i d a a n d B r i m a c o m b e ( 3 7 ) o b s e r v e d t h e o n s e t o f c h a n n e l i n g t o o c c u r a t a m o d i f i e d F r o u d e n u m b e r o f 2 3 , q u i t e d i f f e r e n t f r o m t h e v a l u e o f 2 . 3 d e t e r m i n e d b y t h i s s t u d y . H o w e v e r , m u c h o f t h e c h a n n e l i n g b e h a v i o u r i n t h i s s t u d y i s d u e t o t h e i n t e r a c t i o n o f b u b b l e s o r i g i n a t i n g f r o m a d j a c e n t t u y e r e s , a n d n o t t o t h e c o a l e s c e n c e o f b u b b l e s r i s i n g f r o m a s i n g l e t u y e r e , a s d e f i n e d b y H a i d a a n d B r i m a c o m b e . T h e r e f o r e , i t i s r e a s o n a b l e t h a t t h i s i n t e r a c t i o n w o u l d o c c u r a t l o w e r i n j e c t i o n r a t e s i n t h i s e x p e r i m e n t a l s y s t e m . T h i s i n t e r a c t i o n o f b u b b l e s r i s i n g f r o m a d j a c e n t t u y e r e s h a s a l s o b e e n s h o w n t o b e s i g n i f i c a n t i n t h e n i c k e l m a t t e c o n v e r t i n g p r o c e s s (17), a t F r o u d e n u m b e r s a s l o w a s 1 2 . U n f o r t u n a t e l y , t h e p r e s e n c e o f t h i s s i g n i f i c a n t c h a n g e i n t h e g a s - l i q u i d i n t e r a c t i o n i n t h i s s y s t e m i n v a l i d a t e s t h e a p p l i c a t i o n o f a l g e b r a i c m o d e l s o f b a t h r e c i r c u l a t o r y m o t i o n b a s e d u p o n s i m p l e e n e r g y b a l a n c e s ( 3 3 , 3 4 ) s i n c e a n i n c r e a s i n g a m o u n t o f i n p u t b u o y a n t e n e r g y i s n o t 42 t r a n s f e r r e d t o t h e b a t h , b u t e s c a p e s t h e s y s t e m e n t i r e l y . I t i s t h i s " l o s t " e n e r g y t h a t i s n o t a c c o u n t e d f o r I n s i m p l y e q u a t i n g t h e k i n e t i c e n e r g y o f t h e b a t h w i t h t h e e n e r g y o f t h e i n p u t a i r . A n y p r e d i c t i o n s f o r t h e v e l o c i t y f i e l d s i n m u l t i - t u y e r e i n d u s t r i a l s y s t e m s a t s i m i l a r m o d i f i e d F r o u d e n u m b e r s m u s t t a k e t h i s c o a l e s c e n c e e f f e c t i n t o a c c o u n t i f a c c u r a t e c a l c u l a t i o n s a r e t o b e m a d e . 43 4. THE MATHEMATICAL MODEL The objectives of the mathematical model written for this work were twofold: 1) To provide a means of analyzing the experimental velocity and surface shape measurements. 2) To provide predictions for the fluid flow regimes in the industrial processes of lead slag fuming and copper matte converting. The overall similarity between the experiments and the industrial processes allowed the same computer model to predict values for both of these cases with l i t t l e modification. Thus the experimental data serves both to verify the mathematical model, and to provide an accurate and detailed description of the gas-liquid stirring mechanism common to each of these systems. The formulation of the mathematical model developed In this work proceeded in four distinct stages: 1) The general statement of the problem, where the governing equations and boundary conditions were identified. Within this general description, assumptions and approximations had to be introduced, to allow solution. 2) A numerical technique was chosen to solve the simplified system. 3) A computer progam employing this technique was run for many different sets of input conditions in an attempt to model the experimental data collected previously. 44 4 ) F i n a l l y , c a l c u l a t i o n s w e r e m a d e t o d e s c r i b e t h e f l u i d f l o w p a t t e r n i n a Z i n c S l a g F u m i n g F u r n a c e , a n d a P e i r c e - S m i t h C o p p e r C o n v e r t e r . 4.1. General Statement of Problem 4.1.1. Governing Equations T h e g e n e r a l p r i n c i p l e s t h a t a l l o w c a l c u l a t i o n o f t h e m o t i o n o f a n i s o t h e r m a l f l u i d a r e t h e l a w s o f c o n s e r v a t i o n o f m a s s a n d c o n s e r v a t i o n o f m o m e n t u m . T h e s e l a w s a r e s t a t e d m a t h e m a t i c a l l y b y t h e t h r e e d i m e n s i o n a l N a v i e r - S t o k e s e q u a t i o n s : % ^ + l r (P" 2> + f- ( P « v ) + I" (P"w) at ax K ay H az k 5 Tyx d Tzx ay ) az ( 4 . 1 ) |- (puv) + |- (pv 2) + |- (pvw) at ax y ay p az K ah _ ( a V ay ax at yy oy ax zy ) az ( 4 . 2 ) a( pw) + a( puw) + a(pvw) + a(pw ) at ax ay az „ ah , 5 T Z X ~ p g az" " ( — 3x zy ax. zz. ay az (4.3) ap_ + a(pu) + a( Pv) + a( Pz) = 0 ( 4 4 at ax ay az H e r e , E q u a t i o n s ( 4 . 1 ) t o ( 4 . 3 ) r e p r e s e n t t h e c o n s e r v a t i o n o f x , y a n d z f l u i d m o m e n t u m , w h i l e E q u a t i o n ( 4 . 4 ) s t a t e s t h e l a w o f c o n s e r v a t i o n o f m a s s : a n d i s t e r m e d t h e " c o n t i n u i t y e q u a t i o n " . W h e n s u p p l i e d w i t h t h e c o r r e c t i n i t i a l 45 a n d b o u n d a r y c o n d i t i o n s , t h e s e e q u a t i o n s c a n b e u s e d t o c a l c u l a t e t h e f l o w o f a f l u i d u n d e r a n y c o n d i t i o n s . T o a c h i e v e a s o l u t i o n t o t h e s e e q u a t i o n s f o r t h e e x p e r i m e n t a l s y s t e m d i s c u s s e d i n S e c t i o n 3 , t h e f o l l o w i n g a p p r o x i m a t i o n s w e r e m a d e : 1) T h e f l o w o f t h e l i q u i d i n t h e t a n k w a s a s s u m e d t o b e p e r f e c t l y t w o - d i m e n s i o n a l . T h e l a s e r - d o p p l e r m e a s u r e m e n t s i n d i c a t e t h a t t h i s i s a f a i r a s s u m p t i o n , a n d i t g r e a t l y s i m p l i f i e d f u r t h e r c a l c u l a t i o n s . 2) T h e t u r b u l e n t a c t i o n o f t h e e x p e r i m e n t a l b a t h w a s d e s c r i b e d b y t i m e - a v e r a g i n g t h e s e e q u a t i o n s , a n d e m p l o y i n g t h e t e c h n i q u e o f a t u r b u l e n t v i s c o s i t y t o d e s c r i b e t h e v i s c o u s a n d t u r b u l e n t s h e a r s t r e s s e s ; T h i s a s s u m p t i o n i n t r o d u c e d l i t t l e e r r o r b y i t s e l f , a n d w a s n e c e s s a r y i n o r d e r t o c a l c u l a t e t h e t u r b u l e n t f l o w f i e l d . T h e a c t u a l d e t e r m i n a t i o n o f t h i s t u r b u l e n t v i s c o s i t y h o w e v e r i n t r o d u c e d c o n s i d e r a b l e a p p r o x i m a t i o n . 3 ) O n e p h a s e o n l y ( l i q u i d ) w a s c o n s i d e r e d : t h e d o m a i n o f c a l c u l a t i o n w a s a s s u m e d t o e n d a t t h e g a s - l i q u i d i n t e r f a c e . T h u s a l l s u b s e q u e n t c a l c u l a t i o n s w e r e o n l y c o n c e r n e d w i t h t h e f l o w o f t h e w a t e r i n t h e t a n k . T h i s a p p r o x i m a t i o n a l s o a l l o w e d c o n s t a n t f l u i d d e n s i t y t o b e a s s u m e d . T h i s i s a v e r y i m p o r t a n t a s s u m p t i o n . I t g r e a t l y r e d u c e s t h e m a t h e m a t i c a l e f f o r t n e c e s s a r y t o s o l v e t h e p r o b l e m , b u t c o n s e q u e n t l y c o m p l i c a t e s t h e b o u n d a r y c o n d i t i o n s , a n d l i m i t s t h e a p p l i c a b i l i t y o f t h e m o d e l . U n f o r t u n a t e l y , t h e r e w a s v e r y l i t t l e c h o i c e i n t h i s m a t t e r a s t h e c o m p l e x i t y o f a t w o - p h a s e m o d e l w a s b e y o n d t h e s c o p e o f t h i s i n v e s t i g a t i o n . T h e a d e q u a c y o f a o n e p h a s e m o d e l i n d e s c r i b i n g t h e e x p e r i m e n t a l r e s u l t s i s d i s c u s s e d i n S e c t i o n 5 . 46 4 ) C o n v e c t i v e s t i r r i n g e f f e c t s d u e t o t e m p e r a t u r e g r a d i e n t s w e r e n e g l e c t e d . T h i s i s a g o o d a s s u m p t i o n f o r t h e e x p e r i m e n t s p e r f o r m e d i n t h i s w o r k ( w h i c h w e r e e s s e n t i a l l y i s o t h e r m a l ) . F o r t h e c a s e o f t h e two i n d u s t r i a l p r o c e s s e s , S a l c u d e a n e t a l . ( 3 9 ) h a v e s h o w n t h a t t h e m a g n i t u d e o f t h e A r c h i m e d e s N u m b e r ( G r / R e ) i s o f s i g n i f i c a n c e w h e n d e t e r m i n i n g t h e s i g n i f i c a n c e o f t h e s t i r r i n g d u e t o t e m p e r a t u r e g r a d i e n t s . F o r t h e c a s e o f s l a g f u m i n g a n d c o p p e r m a t t e c o n v e r t i n g , t h e A r c h i m e d e s n u m b e r i s v e r y h a r d t o e s t i m a t e a c c u r a t e l y , d u e t o t h e l a c k o f a d e q u a t e i n f o r m a t i o n o n t e m p e r a t u r e g r a d i e n t s , a n d t h e r e f o r e t h e e f f e c t o f t h i s a s s u m p t i o n i s d i f f i c u l t t o q u a n t i f y . T h e a c t u a l e q u a t i o n s e m p l o y e d b y t h e m a t h e m a t i c a l m o d e l a r e t h e r e f o r e : S + £ + " * v t (J + ZH) + g x ( 4 . 5 ) + 8V - .0 ( 4 . 7 ) ax dy T h i s i s t h e c o n s e r v a t i v e f o r m o f t h e t i m e - a v e r a g e d two d i m e n s i o n a l t u r b u l e n t N a v i e r - S t o k e s e q u a t i o n s , w r i t t e n f o r a n i n c o m p r e s s i b l e , i s o t h e r m a l f l u i d . T h e t e r m y i s a n e f f e c t i v e t u r b u l e n t v i s c o s i t y , d e f i n e d a c c o r d i n g t o t h e B o u s s i n e s q a p p r o x i m a t i o n . 4.1.2. Boundary Conditions A s c h e m a t i c d e s c r i p t i o n o f t h e b o u n d a r y c o n d i t i o n s n e c e s s a r y t o q u a n t i f y t h e e x p e r i m e n t a l r e s u l t s i s s h o w n i n F i g u r e 4 . 1 . T h e ( t w o -47 d i m e n s i o n a l , s i n g l e - p h a s e ) l i q u i d o f t h e e x p e r i m e n t s i s a s s u m e d t o b e b o u n d e d b y t h e f o l l o w i n g : 1) T h e w a l l s o f t h e p l e x i g l a s s t a n k . 2) T h e s u r f a c e o f t h e l i q u i d . 3) T h e b u b b l e c o l u m n . E a c h o f t h e s e t h r e e c o n d i t i o n s m u s t b e c h a r a c t e r i z e d m a t h e m a t i c a l l y i f a q u a n t i t a t i v e d e s c r i p t i o n o f t h e f l u i d f l o w w i t h i n t h e e x p e r i m e n t a l t a n k i s t o b e a c h i e v e d . T h i s i s b y n o m e a n s a t r i v i a l e x e r c i s e : t w o o f t h e t h r e e b o u n d a r y c o n d i t i o n s a r e e x t r e m e l y d i f f i c u l t t o d e s c r i b e w i t h i n t h e c o n f i n e s o f t h i s s i n g l e - p h a s e m o d e l . F o r e x a m p l e , w e r e a two p h a s e m o d e l e m p l o y e d , t h e b u b b l e c o l u m n w o u l d s i m p l y b e d e s c r i b e d a s a c o n s t a n t i n f l o w o f g a s , a n d t h e f r e e s u r f a c e a s a g a s - l i q u i d i n t e r f a c e . B u t t h e o n e - p h a s e m o d e l r e q u i r e s t h a t t h e h o r i z o n t a l a n d v e r t i c a l v e l o c i t y c o m p o n e n t s a n d t h e l i q u i d p r e s s u r e b e s p e c i f i e d a l o n g t h e e n t i r e l e n g t h o f e a c h b o u n d a r y . I n t h e c a s e o f t h e b u b b l e c o l u m n , t h i s i s a l m o s t i m p o s s i b l e t o a c h i e v e , w h i l e d e r i v i n g t h e s e v a l u e s f o r t h e f r e e s u r f a c e r e q u i r e s c o n s i d e r a b l e m a t h e m a t i c a l e f f o r t . F o r t u n a t e l y , t h e d r a g o f t h e w a l l s o n t h e l i q u i d i s r e l a t i v e l y s i m p l e t o c h a r a c t e r i z e , a s t h e f l u i d v e l o c i t y i n t h e r e g i o n o f t h e w a l l s i s z e r o . 48 4.2. Selection of Solution Algorithm As is usually the case, the selection of the numerical method for this particular problem was heavily influenced by the boundary conditions of the problem. An algorithm was sought that could adequately describe the pertinent boundary conditions, while s t i l l maintaining accuracy and computational efficiency. In this case, the combination of awkward boundary conditions and the type of flow made the selection of the solution algorithm more difficult: 1) The rectangular shape of the experimental tank meant that finite difference schemes could be applied easily to this simple system. However, this makes the description of flow In more complex (eg. round) geometries more difficult. This was unfortunate, since a partial objective of this work was to calculate the fluid flow regime in a (round) copper converter. Finite element methods are frequently used instead to describe the flow in such a domain. 2) The free surface at the top of the bath greatly complicated matters. Finite element methods have only been applied very sparingly to problems containing a free surface boundary. The nature of the finite element method makes i t very difficult for this technique to calculate the fluid velocity and position at a free surface. Some fin i t e element calculations have been made for such cases (51,52), but these require considerable approximation and are generally computationally inefficient, especially when compared to equivalent finite-difference techniques. Further the flow in the experimental tank has been shown to be highly rotational: 49 This i n v a l i d a t e s (53) to describe p o t e n t i a l flow (£ such techniques the p o s i t i o n of = 0 ) . ay as the method of the free surface, (4.8) free streamlines as t h i s requires The only t r u l y s a t i s f a c t o r y numerical method capable of describing the free surface i s the Marker and C e l l (MAC) technique (54). The MAC technique and i t s variants allow c a l c u l a t i o n of the p o s i t i o n of the free surface, as well as the v e l o c i t y and pressure f i e l d s i n t h i s region. Again, several factors prevent the immediate s e l e c t i o n of t h i s type of technique: a) The MAC methods are generally only used f o r laminar flows, as they are r e l a t i v e l y complex numerical techniques; the addition of turbulent v i s c o s i t y c a l c u l a t i o n s can cause these methods to be p r o h i b i t i v e l y expensive i n computer time. b) MAC methods are transient techniques - steady state can only be attained by solving the system at successively greater points i n time and determining when these solutions cease to change appreciably. Since we are only interested i n the steady state v e l o c i t y values, t h i s i s a very i n e f f i c i e n t method of c a l c u l a t i o n . c) MAC methods are generally very computationally expensive. They frequently require manipulation of vast amounts of data, and also employ many thousands of i t e r a t i o n s to solve the Navier-Stokes equations. Even r e l a t i v e l y "small" MAC programs often comprise several thousand executable statements. 50 There are large numbers of numerical methods that are capable of directly calculating a steady state velocity regime under turbulent conditions. However few, i f any, of these can describe the position of a free surface adequately, since most of these rely upon the derived values of vortic i t y and stream function. (Vorticity and stream function simplify solution of the turbulent Navier-Stokes equations, but are of l i t t l e use in the location of a free surface, as this relies upon the continuity equation, which vanishes). Because the motion of the free surface was considered to be of significance in determining the overall velocity f i e l d , these two functions could not be employed - solution had to be made in terms of the primitive variables (velocities and pressure). Therefore, the ideal numerical technique for this problem would possess the following characteristics: 1) Capable of describing the free surface, as in the MAC methods. 2) Able to deal with turbulence adequately. 3) Direct calculations of the steady state velocity regime. 4) Capable of dealing with non-rectangular domains of calculation. Unfortunately an efficient numerical method with a l l of these capabilities does not exist. Because of i t s a b i l i t y to describe the free surface, a MAC method was selected and modified in an attempt to meet the other c r i t e r i a : 51 1) Turbulence was to be modelled by employing a constant turbulent viscosity, calculated by the algebraic technique proposed by Sahai and Guthrie (44) and employed by Salcudean et al. (48). 2) Circular domains were to be dealt with by employing the techniques by Viecelli (55,56). 3) Two attempts were made to improve the transient nature of the MAC methods calculations of steady state. a) Direct Steady-State Calculation: A means was sought whereby the steady state velocities and surface profile could be calculated directly by a MAC type method. The MAC equations that describe the free surface were modified to apply to steady flow situation, and the steady Navier-Stokes equations were employed. A program was written based on this method: - the i n i t i a l conditions were set into the velocity and pressure arrays. - the steady Navier-Stokes equations were solved to produce new values for velocity and pressure. - the surface was moved in accordance with these new values. - iteration proceeded until a steady state was attained. Unfortunately, despite a l l attempts, this program failed to converge. Eventually, i t was determined that this failure to converge was due to the movement of the free surface. In effect, every time that the 52 surface was moved, a new computational problem was created. Since the surface was moved with each i t e r a t i o n , convergence was impossible. When the surface movement was deleted, t h i s program r a p i d l y converged. b) Accelerated Steady State Solution When the previous attempt f a i l e d , i t was decided to revert to the MAC technique to advancing the free surface along d i s c r e t e time steps. However, an attempt was made to predict the steady state s o l u t i o n from a very few transient s o l u t i o n s . A second computer program was written, t h i s time employing the transient Navier Stokes equations. The program was run (with boundary conditions approximating those of the experimental apparatus) and allowed to ca l c u l a t e the v e l o c i t y p r o f i l e s and surface shape at several d i f f e r e n t time increments. These c a l c u l a t e d values were s t i l l very d i f f e r e n t f o r those at steady state. The manner i n which the cal c u l a t e d v e l o c i t y i n each f i n i t e d ifference c e l l changed with time was then used i n an attempt to predict the steady state. This predicted steady state was then compared with the "true" calculated value. Unfortunately, t h i s attempt was also unsuccessful. The reason f o r t h i s i s revealed by a time plot of the calculated v e l o c i t i e s of a f i n i t e d i f f e r e n c e c e l l (Figure 4.2). The o s c i l l a t o r y nature of the convergence i s Immediately obvious: the v e l o c i t y values swing quite unpredictably at each time step, though they do approach a steady state. Again, t h i s o s c i l l a t i o n of the v e l o c i t i e s i s due to the unstable influence of the free surface. Moving the free surface at the beginning of each time step introduces a 53 complete change in the geometry of the computational mesh, which affects the calculated velocity values somewhat. When the free surface was not allowed to move, the program immediately predicted the expected velocity field for flow In a square cavity, from a very few time steps. The velocity values of this predicted steady state were within .1% of those calculated for the true steady state, but used less than 10% of the computer time. After these unsuccessful attempts to reduce the cost of employing a MAC method, i t was decided that the only other alternative was to employ an efficient MAC algorithm, and to use a l l available computational techniques to increase efficiency and reduce cost. 4.3. The SOLASMAC Algo r i t h m The computational algorithm selected for these calculations was based on the SOLASMAC method developed by H i l l (57), which simplified and combined the SOLA progra proposed by Hirt (58) and the SMAC method of Amsden and Harlow (59). For the purposes of this study, the SOLASMAC method was altered slightly to yield the SSMCR (SOLASMAC - ROUND DOMAIN) program. As with SOLASMAC the SSMCR program method uses an explicit transient finite-difference scheme to solve the viscous incompressible Navier-Stokes equations. Calculations are performed in primitive variables, allowing the position of the free surface to be determined easily. To increase stability and accuracy, upwind differencing is used to a slight extent. 54 A s i n a l l f i n i t e d i f f e r e n c e s c h e m e s , t h e s o l u t i o n d o m a i n i s d i v i d e d i n t o a n u m b e r o f s m a l l s q u a r e s , o r " c e l l s " . F i g u r e 4 . 3 s h o w s a s k e t c h o f t h e f i n i t e d i f f e r e n c e g r i d u s e d t o m o d e l t h e l a s e r - d o p p l e r m e a s u r e m e n t s . B o u n d a r y c o n d i t i o n s ( e x c e p t f o r t h e f r e e s u r f a c e ) a r e a p p l i e d i n t h e f i c t i t i o u s c e l l s o n t h e e d g e s o f t h e d o m a i n . T h e v e l o c i t y a n d p r e s s u r e v a l u e s f o r a c e l l a r e c o n s i d e r e d t o a p p l y a t t h e c e l l c e n t r e . T h e c e n t r a l f i n i t e d i f f e r e n c e s c h e m e s u s e d t o a p p r o x i m a t e t h e t e r m s o f E q u a t i o n s 4 . 5 t o 4 . 8 a r e c o n t a i n e d i n A p p e n d i x 1 . A f l o w c h a r t o f t h e c o m p u t e r p r o g r a m i s s h o w n i n F i g u r e 4 . 4 . A f t e r t h e v e l o c i t i e s a n d p r e s s u r e s h a v e b e e n l o a d e d i n t o t h e b o u n d a r y c e l l s , t h e f l u i d v e l o c i t y w i t h i n a l l o t h e r c e l l s i s s e t t o z e r o , a n d t h e p r e s s u r e d i s t r i b u t i o n i s s e t t o h y d r o s t a t i c . T h i s r e p r e s e n t s t h e i n i t i a l c o n d i t i o n o f t h e e x p e r i m e n t a l t a n k , b e f o r e t h e a i r h a s b e e n i n t r o d u c e d . H o r i z o n t a l a n d v e r t i c a l momentum e q u a t i o n s a r e t h e n s o l v e d e x p l i c i t l y f o r e a c h n o n - b o u n d a r y c e l l c o n t a i n i n g f l u i d . T h i s p r o c e d u r e s t a r t s w i t h t h e c e l l a t t h e b o t t o m l e f t o f t h e f i n i t e - d i f f e r e n c e m e s h , a n d c o n t i n u e s u p t h e f i r s t c o l u m n . O n c e t h e t o p o f a c o l u m n o f c e l l s i s r e a c h e d , t h e p r o c e d u r e s t a r t s a g a i n a t t h e b o t t o m o f t h e n e x t c o l u m n , a n d c o n t i n u e s b o t t o m t o t o p , l e f t t o r i g h t . T h e s o l u t i o n o f t h e momentum e q u a t i o n s y i e l d s a n e s t i m a t e o f l i q u i d v e l o c i t i e s f o r t h e n e x t t i m e s t e p . T h e c o n t i n u i t y e q u a t i o n i s t h e n a p p l i e d t o e a c h c e l l : d i v , " i J " U 1 - 1 , J + V * J " V 1 » J - 1 i j Ax Ay ( 4 . 9 ) 55 The value of this sum i s termed the "divergence" and is used to adjust the pressure and velocity fields using the following scheme: t\P±i = Xp d i v ± J (4.10) p i j = P i j + ^ i j At AP., J u,, = u,., + ^- (4.12) i j i j pAx AtAP, , u ', = u, . , (4.13) 1-1,J i - l , j pAx At AP. . v. . = v., + ^ (4.14) i j i j pAY At AP, , = v.., - i i (4.15) 'ij-1 ' i j - 1 pAy where \ is an input parameter to the program, usually set by Ax AY Two constraints are placed on the time step, At: X = ^ 1 (4.16) . 2At(-^2 + -^y) At < minimum (-^*-, -^ -) (4.17) u v At < § , (4.18) ^ ( A X 2 + Ay2) This is carried out until the divergence of each c e l l f a l l s below a specified error limit. When this occurs, these new c e l l velocities are used to calculate the new position of the free surface, and the entire procedure is repeated for the next time step. This method of solving the transient Navier-Stokes equations i s very similar to many other numerical techniques. 56 H o w e v e r , t h e SOLASAMC m e t h o d a s u s e d b y SSMCR d i f f e r s g r e a t l y b y i t s t r e a t m e n t o f t h e b o u n d a r y c o n d i t i o n s , m o s t n o t a b l y t h e f r e e s u r f a c e . 4.4. Treatment of Boundary Conditions 4.4.1. Locating and Moving the Free Surface T h e r e a r e two m a i n d i f f i c u l t i e s a s s o c i a t e d w i t h c a l c u l a t i o n s i n v o l v i n g a f r e e s u r f a c e b o u n d a r y : l o c a t i n g i t s p o s i t i o n , a n d s p e c i f y i n g t h e v e l o c i t i e s a n d p r e s s u r e s a t t h e s u r f a c e . O f t h e t w o , t h e f o r m e r i s s i m p l e r . U s i n g m a s s b a l a n c e s , i t i s r e l a t i v e l y s t r a i g h t f o r w a r d t o l o c a t e t h e p o s i t i o n o f t h e s u r f a c e w i t h i n t h e f i n i t e d i f f e r e n c e m e s h , a n d t o m o v e i t i f a p p r o p r i a t e . P r e v i o u s MAC m e t h o d s ( 5 4 , 6 0 ) u s e d m a r k e r p a r t i c l e s t o l o c a t e t h e s u r f a c e p o s i t i o n . T h e s e w e r e f i c t i o u s p a r t i c l e s o f n e u t r a l b u o y a n c y t h a t w e r e c o n s i d e r e d t o m o v e w i t h t h e f l o w o f t h e l i q u i d . A t t h e e n d o f a t i m e i t e r a t i o n , t h e p o s i t i o n s o f t h e m a r k e r p a r t i c l e s l o c a t e d t h e p o s i t i o n o f t h e f r e e s u r f a c e . H o w e v e r , t o k e e p t r a c k o f t h e s e p a r t i c l e s r e q u i r e d c o n s i d e r a b l e a m o u n t s o f c o m p u t e r t i m e a n d m e m o r y . T h e S O L A S M A C m e t h o d , f o l l o w i n g t h e s u g g e s t i o n s o f H i r t , e l i m i n a t e s t h e n e e d o f t h e s e p a r t i c l e s a l t o g e t h e r . I n s t e a d t h e s u r f a c e i s l o c a t e d b y a v e c t o r d e s c r i b i n g t h e h e i g h t o f t h e s u r f a c e a b o v e a n a r b i t r a r y d a t u r a a t a n y p o s i t i o n w i t h i n t h e f i n i t e d i f f e r e n c e m e s h ( F i g u r e 4 . 5 ) . T h e s e " h e i g h t s " a r e t h e n m o v e d a c c o r d i n g t o a m a s s b a l a n c e . T h e c o m p l e t e d e r i v a t i o n h a s b e e n p e r f o r m e d b y H i l l , b u t a s h o r t e r d e s c r i p t i o n l s g i v e n b e l o w . 57 Considering an element of the free surface: Mass input rate: = u(h 2 - h x) + VAX (4.17) Mass output rate: = <u + £ ^< h2 + ^ Ax - h x) + [v + -g (h 2 + | i M _ h i ) ] ^ ( 4 . 1 8 ) Rate of Accumulation: ft a h2 A X 2 Applying the continuity equation, letting AX •+ 0, and observing = v one obtains (in finite difference form): at h t _ h t h ± t + A t = h* + At ( v t j - u i j ( 1 ) ) (4.20) Equation (4.20) is then used to advance the position of the free surface. In practice, the x and y fluid velocity components employed in Equation (4.20) are averages of surface cell velocities and those in adjacent cells. It has been found by H i l l and others, that this increases stability and accuracy. The actual averaging scheme used depends upon the position of the surface in relation to the finite difference mesh. For example, i f the free surface 58 (from the previous time step) lay below the centre of the surface cell, the averaged velocities would be: S " i ¥ I ( u i - 2 , j + u i j + 2 u i - 1 , j ) + T ^ i i J - i + 2 u i - i i J . i > ( 4' 2 1> * = ^ ( V i j + V I . H + v i j + v i ) H ) + I ( V i ( j - i + V i j - 2 ( j - i . 5 ) A y - \ where f = Ay Additional velocity averaging methods are used i f the surface is above the cell centre, or i f there is a boundary cell nearby. These averaging schemes are largely a result of trial and error investigation by many different researchers. They are completely empirical, but have been shown to yield accurate results. 4.4.2. Free Surface Velocities and Pressures Unlike locating and moving the free surface, specifying the values of the surface cell velocities and pressures is truly complex, since these values depend inherently on surface orientation. The equations that describe these values arise from the vanishing of the normal and tangential stresses on the fluid at the free surface. The normal stress condition is usually employed to estimate the pressure of a surface cel l , while the tangential stress is used with the continuity equation to determine the vertical and ( 4 . 2 2 ) ( 4 . 2 3 ) 59 horizontal fluid velocity components. Again, the f u l l derivation has been performed by H i l l , and only an abridged version is given below. The only significant departure from Hill's derivation is the use of time averaged velocities to describe turbulence. For a two-dimensional Newtonian fluid under laminar flow conditions, the velocity gradient descriptions of the normal and tangential stresses at a surface are: Normal Stresses: ^X T = P„ - 2 U (—^) (4.24) xy v % K Sx v ' x = p - 2u(-r- JS:) (4.25) yy *x *ay Tangential Stresses: ou? av x = T u (—^ + —-) (4.26) xy yx p ay 3x If the flow of the fluid is turbulent, the x and y velocities and the isentropic pressure can be represented as being the sum of an average component and a fluctuating component: u = u + u' (4.27) v = v + v' (4.28) Pt = P + P' (4-29) 60 Substituting these values into the normal and tangential stress conditions and time-averaging, one obtains: Tangential Stresses: T - P - 2n (-^-) yy ay Normal Stresses: T = -n" (£• + £L) (4.31) xy K v5x ay7 v ' Note that the laminar viscosity, rather than the apparent turbulent viscosity, is used in the turbulent stress conditions. Therefore, the derivation of the normal and tangential stress conditions for a turbulent fluid surface proceeds similarly to that for a laminar surface, i f time-averaged values are used. H i l l has shown that for a fluid surface inclined at an angle 9 to the horizontal, a two-dimensional force balance on a unit length of surfaces gives: Tangential Stress Condition: -2 tan e ( j g - + (tan 2 0 - 1 ) < J £ + - g - ) = 0 (4.32) o x oy oy ox Normal Stress Condition: . _ .. C 2 tan e vay dx 2 _ (1 + tan e) ,au . av P H 9 for, n + "57> (A-33) Since the sum of the forces on a fluid surface are zero. 6 1 T h u s t o d e s c r i b e t h e f l u i d v e l o c i t y a n d p r e s s u r e a t t h e s u r f a c e , o n e i s f o r c e d t o s a t i s f y a f u r t h e r t w o p a r t i a l d i f f e r e n t i a l e q u a t i o n s , w h o s e f o r m d e p e n d s u p o n s u r f a c e o r i e n t a t i o n . F o r t u n a t e l y , a p p r o x i m a t i o n s c a n b e m a d e . 4.A.2.1. Normal Stress Condition T h e S O L A S M A C m e t h o d e m p l o y s t h e f u l l n o r m a l s t r e s s c o n d i t i o n t o s p e c i f y t h e p r e s s u r e i n t h e f i n i t e d i f f e r e n c e c e l l s a t t h e f l u i d s u r f a c e . H o w e v e r , S O L A S M A C w a s w r i t t e n t o s t u d y f l o w s a t R e y n o l d s n u m b e r s , w h e r e t h e v i s c o u s p r e s s u r e c o r r e c t i o n t e r m i n t h e n o r m a l s t r e s s e q u a t i o n b e c o m e s s i g n i f i c a n t . H i r t a n d S h a n n o n ( 6 0 ) h a v e s h o w n t h a t t h e f u l l n o r m a l s t r e s s c o n d i t i o n n e e d o n l y b e u s e d a t R e y n o l d s n u m b e r s b e l o w 1 0 . T h e r e f o r e , t h e p r e s s u r e o f t h e s u r f a c e c e l l s w a s c o n s i d e r e d t o b e a t m o s p h e r i c . T h i s a p p r o x i m a t i o n i s common i n M A C - t y p e c o d e s ( i t w a s u s e d i n t h e o r i g i n a l MAC p r o g r a m ) a s i t r e d u c e s c o m p l e x i t y a n d i n c r e a s e s s t a b i l i t y a t l i t t l e c o s t i n a c c u r a c y . A.A.2.2. Tangential Stress Condition T h e s u r f a c e v e l o c i t i e s s e t b y t h e t a n g e n t i a l s t r e s s c o n d i t i o n a l s o d e p e n d u p o n s u r f a c e i n c l i n a t i o n . T h e S O L A S M A C m e t h o d c o n s i d e r s s u r f a c e o r i e n t a t i o n i n 15 d e g r e e I n c r e m e n t s . T h a t i s , f o r s u r f a c e s o r i e n t e d b e t w e e n t h e h o r i z o n t a l a n d +15 d e g r e e s , o n e e q u a t i o n i s u s e d . F o r t h o s e i n c l i n e d b e t w e e n +16 a n d +30 d e g r e e s , a d i f f e r e n t e q u a t i o n i s e m p l o y e d a n d t h i s c o n t i n u e s i n 15 d e g r e e i n c r e m e n t s . 62 H i l l chose to be this accurate because he was solving problems at low velocities, where differences of a few millimeters per second were significant. This type of accuracy is not just i f i e d for the purposes of this investigation. Therefore, in the SSMCR program, the tangential stress condition was approximated much more roughly. As in the original MAC program, surface slopes are considered to be either horizontal, vertical or at 45 degrees. Figure 4.6 shows how these slopes are recognized by the computer program: the three cells surrounding a surface c e l l (the central c e l l in Figure 4.6) are examined. If only one i s empty of f l u i d , the surface i s either horizontal or v e r t i c a l . If two are empty, the surface i s at 45 degrees. For a surface c e l l that contains a section of surface considered to be horizontal, the horizontal velocity component is calculated by the x-wise momentum equation, while the vertical component is defined by the continuity equation: \l - "1,3-1 + f <»1J " V l , J > <4-24> The horizontal tangential stress condition is applied by adjusting the horizontal velocity in cells outside the free surface: - i > J + i - " i j - 5 ( T u " * i - i t J > ( 4 - 2 5 > This velocity is then used to solve the x-wise momentum equation at the next time step. 63 V e r t i c a l c e l l s a r e t r e a t e d s i m i l a r l y . C e l l s l o c a t e d o n a 45 d e g r e e s u r f a c e h a v e t h e i r v e l o c i t i e s s p e c i f i e d b y s e t t i n g : U i j " V i . j ( A- 2 6> v i j - v i , j - l < 4 ' 2 7 > T h u s s i m u l t a n e o u s l y s a t i s f y i n g b o t h t h e t a n g e n t i a l s t r e s s c o n d i t i o n a n d t h e c o n t i n u i t y e q u a t i o n . 4 . 4 . 3 . Rectangular Wall Boundaries T h e b o u n d a r y c o n d i t i o n s u s e d t o d e s c r i b e t h e i n f l u e n c e o f t h e w a l l s o f t h e e x p e r i m e n t a l t a n k w e r e , i n c o m p a r i s o n t o t h e o t h e r s , e x t r e m e l y s i m p l e t o a p p l y . A s s t a t e d e a r l i e r , t h e s e c o n d i t i o n s w e r e a p p l i e d i n t h e f i c t i t i o u s c e l l s s u r r o u n d i n g t h e c o m p u t i n g r e g i o n . I n t h e s e r e g i o n s , t h e v e l o c i t y c o m p o n e n t p e r p e n d i c u l a r t o t h e w a l l w a s s e t t o b e z e r o . T h e v e l o c i t y c o m p o n e n t t a n g e n t i a l t o t h e w a l l was e i t h e r z e r o ( n o s l i p ) o r u n a f f e c t e d b y t h e w a l l ( f r e e - s l i p ) , a s s e e n i n T a b l e 4 . 1 . T h e c h o i c e b e t w e e n t h e n o s l i p a n d f r e e s l i p c o n d i t i o n s i s m a d e d e p e n d i n g o n t h e r e l a t i v e s i z e s o f t h e b o u n d a r y l a y e r a n d t h e f i n i t e - d i f f e r e n c e m e s h . F o r l o w R e y n o l d s n u m b e r f l o w s , w h e r e t h e b o u n d a r y l a y e r i s l a r g e c o m p a r e d t o t h e c e l l s i z e , t h e n o - s l i p a p p r o x i m a t i o n s h o u l d b e u s e d . A t h i g h e r R e y n o l d s n u m b e r s , t h e b o u n d a r y l a y e r i s s m a l l , a n d t h e r e t a r d i n g e f f e c t o f t h e w a l l i s l i m i t e d t o a r e g i o n s m a l l e r t h a n a f i n i t e d i f f e r e n c e c e l l . F o r t h i s c a s e , f r e e - s l i p s h o u l d b e u s e d . I n a c t u a l f a c t , t h e c h o i c e o f t h e s e b o u n d a r y c o n d i t i o n s h a d l i t t l e e f f e c t o n t h e c a l c u l a t i o n s . B o t h t y p e s o f a p p r o x i m a t i o n w e r e t e s t e d , a n d s i m i l a r r e s u l t s w e r e o b t a i n e d w i t h e a c h . 64 4.4.4. Curvilinear Boundary T o p r e d i c t t h e f l o w r e g i m e i n a c o p p e r c o n v e r t e r , i t i s n e c e s s a r y t h a t t h e m a t h e m a t i c a l m o d e l b e c a p a b l e o f d e a l i n g w i t h a c u r v e d o r n o n - r e c t a n g u l a r b o u n d a r y . I n i t i a l l y , t h e t e c h n i q u e o f V i e c e l l i w a s e m p l o y e d , w h e r e b y t h e p r e s s u r e o f b o u n d a r y c e l l s w a s a d j u s t e d t o c o n s t r a i n t h e f l u i d t o f l o w p a r a l l e l t o a n a r b i t r a r y b o u n d a r y . H o w e v e r , t h i s p r e s s u r e a d j u s t m e n t w a s f o u n d t o b e i n s u f f i c i e n t : t h e f l u i d w a s n o t s i g n i f i c a n t l y a f f e c t e d b y t h e p r e s s u r e i n t h e b o u n d a r y c e l l s t o c a u s e i t t o f l o w p a r a l l e l t o a c u r v e d w a l l . T h i s was d u e t o t h e r e l a t i v e l y s m a l l s i z e o f t h e t e r m s -7*^ a n d i n t h e m o m e n t u m e q u a t i o n s , 3 dx o y a 2 v 5 2 u c o m p a r e d t o s u c h v i s c o s i t y - d e p e n d e n t t e r m s a s —2" o r 2 ~ s :'- n c e t n e d y dx e f f e c t i v e v i s c o s i t y u s e d i n t h i s s t u d y was m a n y o r d e r s o f m a g n i t u d e h i g h e r t h a n t h e l a m i n a r v i s c o s i t y e m p l o y e d b y V i e c e l l i . T o f o r c e t h e v e l o c i t i e s n e a r t h e w a l l t o f l o w t a n g e n t i a l l y , i t was t h e r e f o r e n e c e s s a r y t o a d j u s t t h e b o u n d a r y v a l u e v e l o c i t i e s , n o t t h e p r e s s u r e s . T h e f o l l o w i n g t e c h n i q u e was t h e n e m p l o y e d : 1) T h e c u r v i l i n e a r b o u n d a r y was i m p o s e d o v e r a f i n i t e d i f f e r e n c e g r i d . C e l l s t h a t i n t e r s e c t e d t h e b o t t o m b o u n d a r y w e r e t e r m e d b o u n d a r y c e l l s . 2) F o r e a c h t i m e s t e p , t h e f l u i d i n t h e s e b o u n d a r y c e l l s w a s c o n s t r a i n e d t o f l o w t a n g e t i a l l y t o t h e w a l l . T h e r e was no v e l o c i t y c o m p o n e n t n o r m a l t o t h e w a l l . T h e m a g n i t u d e o f t h e t a n g e n t i a l v e l o c i t y w a s c a l c u l a t e d f r o m t h e n e t m a s s f l o w i n t o t h e c e l l ( i . e . b y t h e c o n t i n u i t y e q u a t i o n ) . 65 3) The pressure i n these boundary c e l l s was adjusted i n accordance with a B e r n o u l l i - t y p e equat ion: the pressure change being ca l cu la ted by the change i n f l u i d momentum. The i n t e r s e c t i o n of a surface c e l l and a w a l l c e l l was treated by s e t t i n g the pressure of the c e l l equal to zero , and determining the v e l o c i t i e s by the c o n t i n u i t y equat ion. When t h i s (admittedly crude) boundary approximation were employed, the f l u i d was found to flow t a n g e n t i a l l y to the round w a l l . 4.4.5. Gas-Liquid Boundary The f i n a l boundary cond i t ion r e q u i r i n g treatment i s the gas l i q u i d i n t e r f a c e - the bubble column. This i s by far the hardest to charac ter i ze mathematical ly due to i t s inherent complexity and the lack of experimental d a t a . The i n j e c t i o n of a gas into a quiescent bath causes the l i q u i d to r i s e by a number of i n t e r - r e l a t e d e f f e c t s . The f i r s t of these i s f l u i d displacement or entrainment. A r i s i n g gas bubble pushes and shears l i q u i d as i t moves upward, causing the f l u i d to r i s e . Sahai and Guthr ie (33) have argued that t h i s e f f ec t occurs with the a c t i o n of l a r g e , s t a b l e , gas bubbles . Another mechanism for l i q u i d movement i s buoyancy. The presence of a bubble lowers the dens i ty i n i t s v i c i n i t y . Th i s dens i ty d i f f erence causes the f l u i d beneath the bubble to r i s e , fo l lowing the bubble to the sur face . 66 Despite these general guidelines, and the existence of some experimental data, the mathematical description of the gas-liquid interface i s by no means complete. For example, simple models such as that proposed by Sahai and Guthrie (34) make no mention of the effect of bubble coalescence on the liquid rise velocity - a serious (though understandable) omission. Mathematical models based on these descriptions have had some qualitative success, but there remains considerable disparity between experiments and calculations. Therefore before reliable calculations can be made, an adequate description of this boundary is necessary, which requires the water velocity and pressure distribution along the length of the rising gas column as a function of Injected gas flowrate. Unfortunately, this data i s not yet available. Therefore, a l l descriptions of the gas-liquid boundary condition used by the mathematical model in this work w i l l be, at best, semi-empirical. Various theoretical velocity and pressure distributions w i l l be applied to the model, and the results studied. Those predicting velocity and surface values that conform closest to the experimental data w i l l be accepted as valid descriptions. That i s , the mathematical description of the gas-liquid boundary condition w i l l be inferred by the agreement of calculations with experiments. 67 5. CALCULATIONS - COMPARISON WITH EXPERIMENTAL RESULTS Before calculations of flow regimes were performed, the computer program used in this study was subjected to several tests to ensure that i t was error-free. Foremost among these was the calculation of flow in a square cavity: the SSMCR program was set the task of computing the flow regime of a fictious fluid (having a viscosity of 0.4 g/cm s) in a square cavity measuring 1 cm x 1 cm (Figure 5.1). This problem is suitable for test conditions as it is relatively simple (involving no free-surface conditions or turbulence) and has been extensively studied (61-62). The output of the SSMCR program was compared to that published by H i l l (57). This comparison is shown in Figure 5.2 and clearly there is excellent quantitative agreement between the two sets of data. Hill's program was, in turn, verified by comparison to an analytical solution. Unfortunately, the velocity vector plots so commonly used to illustrate vortex flow patterns are of l i t t l e use in quantitative comparisons of different velocity regimes. For example, during the development and "debugging" of the SSMCR program, an error was discovered in which the viscous drag term of the Navier-Stokes equations was incorrectly described. This had a considerable effect on the magnitude of the calculated velocities, but as Figures 5.3 and 5.4 show, the vector plots of the velocities calculated by the erroneous (Figure 5.3) and correct (Figure 5.4) conditions are very similar. In fact, i t was not until actual numerical values were examined that this error was discovered. Therefore, these plots by themselves are not suitable for comparison between experimental and 68 c a l c u l a t e d v a l u e s , s i n c e a c o n s i d e r a b l e d i s p a r i t y b e t w e e n v e l o c i t y p a t t e r n s c o u l d p a s s u n d e t e c t e d . I n a n a t t e m p t t o q u a n t i f y t h e c o m p a r i s o n b e t w e e n t h e m o d e l r e s u l t s a n d t h e e x p e r i m e n t s , t w o v a l u e s w e r e d e f i n e d : t h e m e a n c e l l a n g u l a r d e v i a t i o n , a n d t h e m e a n c e l l m a g n i t u d e d e v i a t i o n . T h e s e a r e c a l c u l a t e d a s f o l l o w s : n . n D f t = -±— z E t a n " 1 ( - ± ^ ) - t a n _ i ( - i ^ - ) ( 5 . 1 ) 9 n . n . . - . , v . . v . . v ' I j i = l j = l i j , e i j . c •i Ui S~i*7*Z + v . , 2 - / ^ T T 2 + v , ,2 ~ 1 1 i j »e i j »e i j , c i j , c , _ D = E £ =!—J J ( 5 . 2 ) ">s Vj I=I 1=1 ^ V , e + V , = T h e a n g u l a r d e v i a t i o n g i v e s a n i n d i c a t i o n o f h o w w e l l t h e s h a p e s o f t h e c o m p u t e d a n d m e a s u r e d v o r t i c e s a g r e e , b y c o m p a r i n g t h e o r i e n t a t i o n o f t h e v e l o c i t y v e c t o r s . T h e m a g n i t u d e d e v i a t i o n i n d i c a t e s h o w w e l l t h e p r e d i c t e d v e l o c i t y d i s t r i b u t i o n a g r e e s w i t h t h e e x p e r i m e n t a l m e a s u r e m e n t s . H o w e v e r , a n o n - z e r o m a g n i t u d e d e v i a t i o n c a n i n d i c a t e t w o p o s s i b i l i t i e s : 1 . T h e o v e r a l l p r e d i c t e d b u l k m o t i o n o f t h e f l u i d i s t o o h i g h o r l o w - t h a t i s , t h e l e f t h a n d b o u n d a r y c o n d i t i o n u s e d b y t h e p r o g r a m i s i m p a r t i n g t o o g r e a t o r t o o s m a l l a v e l o c i t y t o t h e f l u i d . T h i s w o u l d b e t h e c a s e i n w h i c h t h e h i g h e s t c a l c u l a t e d v e l o c i t i e s a r e l o c a t e d i n t h e s a m e r e g i o n a s t h e h i g h e s t m e a s u r e d v e l o c i t i e s , b u t t h e r e l a t i v e m a g n i t u d e s o f t h e s e v a l u e s d i f f e r . 69 2. The overall distribution of the predicted velocity field is incorrect. This would be the case in which the highest predicted velocities are not located in the same region as the highest measured velocities. To identify which of these effects is significant, an additional factor, the velocity distribution deviation defined by: dist n i n j ± m l j = 1 i / o u 4 1 V 2 v . . V 2 1/2 , .1/2 _ u ij,c max,e. . lj . c max,e. -, ^ i j ,e i j ,e' Lv v ) T K v ) J J max,c max,c  < Uij 2> e + V,e> 1 / 2 ( 5 .3 ) is calculated. If this value is low relative to the magnitude deviation, i t can be concluded that the first of these effects is predominant, and the predicted fluid velocity distribution is essentially correct, but the bulk of the fluid is moving too slowly or too fast. If the velocity distribution deviation is high relative to the magnitude deviation, the second effect is predominant, and the general velocity distribution is incorrect. This relatively complex method of comparison between the predicted and measured velocity distributions allows precise, quantitative, conclusions to be drawn as to the success or failure of both the model as a whole and its various boundary approximations. This in turn allows the model to be "tuned" or fitted to the experimental data. When considered along with the vector plots, the three deviation values give a good indication of the accuracy of a given flow calculation. 7 0 O n c e t h e p r o g r a m h a d b e e n v e r i f i e d I n t h e m a n n e r m e n t i o n e d p r e v i o u s l y , c a l c u l a t i o n s w e r e p e r f o r m e d i n a t t e m p t s t o c o m p u t e t h e e x p e r i m e n t a l f l o w p a t t e r n s . A v a r i e t y o f a p p r o x i m a t i o n s ( s h o w n b e l o w ) w e r e u s e d f o r t h e g a s - l i q u i d b o u n d a r y . A s w e l l , t h e v a l u e f o r t h e t u r b u l e n t v i s c o s i t y u s e d b y t h e m o d e l w a s v a r i e d f r o m r u n t o r u n i n a n a t t e m p t t o o b t a i n g o o d a g r e e m e n t w i t h t h e e x p e r i m e n t s . T h e a l g e b r a i c e f f e c t i v e v i s c o s i t y m o d e l o f S a h a i a n d G u t h r i e ( 4 4 ) w a s t e s t e d , a s w e r e c o m p l e t e l y e m p i r i c a l v i s c o s i t y v a l u e s . T h e e f f e c t i v e v i s c o s i t y v a l u e s p r e d i c t e d b y t h e m o d e l o f S a h a i a n d G u t h r i e ( 4 4 ) a r e s h o w n I n T a b l e 5 . 1 . 5.1. Constant Vertical Velocity T h e f i r s t a p p r o x i m a t i o n t o t h e t w o - p h a s e r e g i o n u s e d b y t h e m a t h e m a t i c a l m o d e l w a s t h a t o f c o n s t a n t u p w a r d v e l o c i t y . T h a t i s , t h e b u b b l e s i n t h e e x p e r i m e n t a l t a n k w e r e c o n s i d e r e d t o i m p a r t a c o n s t a n t u p w a r d v e l o c i t y t o t h e f l u i d i n t h i s r e g i o n . T h i s s h e a r i n g o f t h e l i q u i d w a s t h e n t r a n s m i t t e d i n t o t h e b u l k o f t h e f l u i d b y v i s c o u s f o r c e s . T h i s a s s u m p t i o n w a s i n c o r p o r a t e d i n t o t h e m a t h e m a t i c a l m o d e l b y a p p l y i n g a c o n s t a n t v e r t i c a l v e l o c i t y t o t h e c e l l s i n t h e l e f t w a l l b o u n d a r y r e g i o n , a n d s e t t i n g t h e h o r i z o n t a l v e l o c i t y o f t h e s e c e l l s t o b e z e r o . T h e i n i t i a l p r e s s u r e d i s t r i b u t i o n w a s a s s u m e d t o b e h y d r o s t a t i c . C a l c u l a t i o n s w e r e p e r f o r m e d b y t h e p r o g r a m u n t i l s t e a d y s t a t e h a d b e e n a t t a i n e d . T h e o n s e t o f s t e a d y s t a t e w a s a r b i t r a r i l y d e t e r m i n e d t o o c c u r w h e n t h e v e l o c i t i e s o f e a c h c e l l c h a n g e d b y l e s s t h a n 5% o v e r 20 t i m e s t e p s . A n a d d i t i o n a l c h e c k o n t h i s c o n d i t i o n w a s m a d e b y c o m p a r i n g a 71 " s t e a d y s t a t e " v e l o c i t y p r o f i l e w i t h p r o f i l e s c a l c u l a t e d a t s e v e r a l p r e v i o u s t i m e s , a n d e n s u r i n g t h a t t h e d i f f e r e n c e w a s b o t h s m a l l a n d d i m i n i s h i n g w i t h t i m e . T h e r e s u l t s o f m a n y o f t h e c a l c u l a t i o n s p e r f o r m e d u s i n g t h i s b o u n d a r y c o n d i t i o n a r e s h o w n i n T a b l e 5 . 2 . M a n y o t h e r c a l c u l a t i o n s w e r e p e r f o r m e d b e s i d e s t h o s e s h o w n i n T a b l e 5 . 2 , b u t t h e s e w e r e g e n e r a l l y o f a p r e l i m i n a r y n a t u r e , a n d s e r v e d m a i n l y t o d e t e r m i n e t h e o p t i m u m v a l u e s o f t h e t w o i n p u t p a r a m e t e r s ( v i s c o s i t y a n d l e f t w a l l v e l o c i t y ) . B y e x a m i n i n g T a b l e 5 . 2 , a n u m b e r o f i n f e r e n c e s may b e d r a w n : 1 . I t i s p o s s i b l e t o o b t a i n v e r y l o w v a l u e s o f b o t h t h e m e a n c e l l v e l o c i t y d e v i a t i o n a n d t h e v e l o c i t y d i s t r i b u t i o n d e v i a t i o n (< 10%) u s i n g t h i s b o u n d a r y c o n d i t i o n . 2 . G e n e r a l l y s p e a k i n g , t h e t h r e e d e v i a t i o n v a l u e s a r e r e l a t i v e l y c o n s t a n t f o r a l l e x p e r i m e n t s . T h a t i s , t h e c a l c u l a t i o n s p e r f o r m e d a t c e r t a i n v a l u e s o f v i s c o s i t y a n d v e l o c i t y s e e m t o a p p l y e q u a l l y w e l l ( o r p o o r l y ) t o a l l o f t h e e x p e r i m e n t a l m e a s u r e m e n t s . 3 . I n a l l c a s e s c a l c u l a t e d w i t h t h i s b o u n d a r y c o n d i t i o n , t h e m e a n c e l l d i r e c t i o n a l d e v i a t i o n i s v e r y h i g h : b e t w e e n 30 a n d 60 d e g r e e s . H o w e v e r , t h i s d i s a g r e e m e n t a p p e a r s t o b e m o r e p r o n o u n c e d w i t h t h e e x p e r i m e n t s p e r f o r m e d a t l o w e r a i r b l o w i n g r a t e s . 4 . I n c r e a s i n g t h e b o u n d a r y v e l o c i t y d o e s n o t s e e m t o l o w e r t h e d i r e c t i o n a l d e v i a t i o n a t a l l . B y i n c r e a s i n g t h e b o u n d a r y v e l o c i t y i t i s p o s s i b l e t o l o w e r t h e * m e a n c e l l v e l o c i t y d e v i a t i o n t o b e o n l y 1 0 % , b u t t h e d i r e c t i o n a l d e v i a t i o n r e m a i n s h i g h a t 3 0 t o 40 d e g r e e s . 72 5 . T h e e f f e c t o f i n c r e a s i n g t h e v i s c o s i t y a l s o a p p e a r s t o b e s l i g h t . I n c r e a s i n g t h e v i s c o s i t y f r o m a b o u t 1 0 g / ( c m » s ) t o 4 0 g / ( c m » s ) a n d t o 4 0 0 g / ( c m s ) l o w e r s t h e m e a n c e l l v e l o c i t y d e v i a t i o n b u t d o e s n o t c h a n g e t h e v a l u e s o f t h e d i r e c t i o n a l d e v i a t i o n a p p r e c i a b l y . O b s e r v a t i o n s 4 a n d 5 a b o v e i n d i c a t e t h a t t h e o v e r a l l s h a p e o f t h e v e l o c i t y p r o f i l e i s r e l a t i v e l y i n d e p e n d e n t o f e i t h e r t h e v i s c o s i t y o r t h e b o u n d a r y v e l o c i t y , a n d i s i n s t e a d d e t e r m i n e d m a i n l y b y t h e n a t u r e o f t h i s b o u n d a r y c o n d i t i o n i t s e l f . T h i s c o n c l u s i o n i s s u p p o r t e d b y c o m p a r i n g F i g u r e s 5 . 5 , 5 . 6 a n d 5 . 7 w i t h t h e e x p e r i m e n t a l p l o t s o f C h a p t e r 3 ( t h e v e l o c i t y a n d v i s c o s i t y v a l u e s u s e d t o g e n e r a t e t h e c a l c u l a t e d f i g u r e s a r e s h o w n o n t h e p l o t s t h e m s e l v e s ) . I t i s o b v i o u s t h a t t h e e f f e c t o f i n c r e a s i n g t h e v i s c o s i t y ( f r o m 10 t o 4 0 t o 4 0 0 g / ( c m » s ) ) i s t o i n c r e a s e t h e v e r t i c a l v e l o c i t y o f t h e c e l l s t o t h e r i g h t o f t h e b o u n d a r y , a n d t o t r a n s m i t t h i s i n c r e a s e d v e l o c i t y f u r t h e r i n t o t h e f l u i d ( t h i s i s r e a s o n a b l e , a s v i s c o s i t y i s t h e m a i n m e a n s o f e n e r g y t r a n s f e r f r o m t h e b o u n d a r y t o t h e l i q u i d ) . T h i s i s t u r n h a s t w o e f f e c t s : f i r s t l y , t h e m e a n c e l l m a g n i t u d e d e v i a t i o n i s l o w e r e d d u e t o t h e i n c r e a s e d o v e r a l l v e l o c i t y o f t h e f l o w . S e c o n d l y , t h e v e l o c i t y d i s t r i b u t i o n d e v i a t i o n i s l o w e r e d , a s t h e i n t e r i o r a n d s u r f a c e c e l l s i n c r e a s e i n v e l o c i t y w i t h r e s p e c t t o t h e l e f t w a l l c e l l s ( F i g u r e s 5 . 6 a n d 5 . 7 ) ; t h e b u l k o f t h e f l u i d m o t i o n i s n o l o n g e r c o m p l e t e l y c o n f i n e d t o a s m a l l r e g i o n i n t h e l e f t s i d e o f t h e c o m p u t a t i o n a l r e g i m e ( a s I n F i g u r e 5 . 5 ) . T h i s i s c l o s e r t o t h e e x p e r i m e n t a l p l o t s o f F i g u r e s 3 . 7 t o 3 . 1 6 , w h e r e t h e l a r g e s t f l u i d v e l o c i t i e s o c c u r i n t h e r e g i o n o f t h e s u r f a c e , a n d n o t a t t h e l e f t s i d e o f t h e t a n k . T h e v e l o c i t y 73 distribution deviation cannot be reduced to zero however, as the maximum calculated velocities s t i l l occur at the left wall. However, the overall directions of the calculated velocity vectors do not change significantly from Figure 5.5 to 5.7: those near the left wall possess the highest velocities, and are largely vertical. The surface velocities remain generally low, and largely horizontal. Both of these predictions are strongly contradicted by the experimental vector plots. Therefore, because of the inability of the mathematical model to agree with the direction of the experimental velocity vectors, one is forced to conclude that this description of the gas-liquid boundary is not accurate. Thus the stirring of the water in the experimental tank was not significantly due to the shearing action of the bubbles on the liquid; i f i t were, this boundary condition would have been able to f i t the experimental data. An entirely different mechanism must therefore be responsible for stirring the liquid in the tank. 5.2. V a r i a b l e Density Apart from shear due to bubble rise velocity, another mechanism that may be significant in causing the experimental flow regimes is the lowering of the density of the liquid in the region of the bubbles. An hypothesis for such a mechanism is as follows: 74 1 . T h e e f f e c t o f t h e p r e s e n c e o f a b u b b l e i n t h e l i q u i d i s t o l o w e r t h e d e n s i t y ( a n d t h e r e f o r e t h e h y d r o s t a t i c p r e s s u r e ) i n t h e r e g i o n o f t h e b u b b l e . 2 . T h i s l o w p r e s s u r e r e g i o n t h e n c a u s e s a n i n f l u x o f t h e h i g h e r - d e n s i t y a d j a c e n t f l u i d . 3 . A s t h i s b u b b l e r i s e s u n d e r t h e i n f l u e n c e o f b u o y a n t a n d i n e r t i a l f o r c e s , t h i s l o w - d e n s i t y ( a n d t h e r e f o r e l o w p r e s s u r e ) r e g i o n m o v e s u p w a r d w i t h I t . T h i s a p p r o x i m a t i o n h a s b e e n u s e d i n a g r e a t m a n y m a t h e m a t i c a l s t u d i e s o f g a s i n j e c t i o n i n t o l i q u i d l a d l e s ( 3 1 , 4 0 , 4 2 , 4 3 ) . T o a p p l y t h i s m e t h o d t o a m a t h e m a t i c a l m o d e l , t h e a v e r a g e d e n s i t y o f f i n i t e d i f f e r e n c e c e l l s i n t h e l e f t h a n d b o u n d a r y i s l o w e r e d b y e m p l o y i n g a v o i d f r a c t i o n : p i j = a p g + ( 1 " a ) p f ( 5 , 4 ) t o s i m u l a t e t h e p r e s e n c e o f g a s b u b b l e s . T h e l i q u i d i n t h e b o u n d a r y c e l l s w i l l r i s e d u e t o t h e v e r t i c a l p r e s s u r e d i f f e r e n c e t e r m i n t h e N a v i e r - S t o k e s e q u a t i o n s . M o d i f i c a t i o n s h a v e t o b e m a d e t o t h e c o n t i n u i t y e q u a t i o n t o a l l o w f o r t h i s r e d u c t i o n o f f l u i d d e n s i t y : P i j v i j " P i j - l V l j - l + P i j u i j " P i - l , j U i + j A y Ax = 0 ( 5 . 5 ) 75 Frequently, the shearing e f f e c t mentioned above i s ap p l i e d along w i t h t h i s approximation. In t h i s case, a v e r t i c a l v e l o c i t y i s applied to these boundary c e l l s , along w i t h a reduced d e n s i t y . Both methods were tested i n t h i s i n v e s t i g a t i o n . To apply the v a r i a b l e - d e n s i t y method to the SSMCR program, i t i s necessary to o b t a i n values f o r the void f r a c t i o n i n the bubble column. U n f o r t u n a t e l y v o i d f r a c t i o n measurements of t h i s type were not made d i r e c t l y . An estimate of t h i s value can be derived from the bulk, voidage measurements shown i n Figure 3.19. D i r e c t observation i n d i c a t e s that v i r t u a l l y a l l of the bubbles are confined to the region 6 cm from the l e f t w a l l of the tank. Assuming that these bubbles alone are re s p o n s i b l e f o r the increase i n the o v e r a l l tank volume, the bubble column gas holdup can be c a l c u l a t e d as f o l l o w s : a , • a, ,, • ( i ) (5.6) column ^ j u l k A . column The gas concentrations In the bubble column c a l c u l a t e d by t h i s means are shown In Table 5.3. These voidage values are at best only a crude estimate s i n c e they assume a constant gas co n c e n t r a t i o n throughout the length of t h e two-phase r e g i o n , w h i l e o b s e r v a t i o n s i n d i c a t e t h a t the g a s co n c e n t r a t i o n i s higher at the free surface than i n the bubble column. Applying these values to the SSMCR program y i e l d s the v e l o c i t y v e c t o r p l o t s such as those shown i n Figures 5.8 and 5.9. The values f o r the mean c e l l v e l o c i t y , angle and d i s t r i b u t i o n d e v i a t i o n are shown i n Table 5.4 76 f o r the best run w i t h t h i s boundary c o n d i t i o n . I t i s immediately c l e a r from Table 5.4 (and from the vector p l o t s of Figures 5.8 and 5.9) that t h i s boundary c o n d i t i o n gives very d i f f e r e n t r e s u l t s from the constant v e l o c i t y c o n d i t i o n . For example, the comparison of run VK.4 w i t h the experimental measurements shows that f o r a l l cases, the mean angular d e v i a t i o n i s very low, with a maximum (absolute) value of -12.4 degrees and a minimum value of 0.40 degrees. This c e r t a i n l y very good agreement, and i s f a r b e t t e r than that a t t a i n e d by the previous c o n d i t i o n . However, the v e l o c i t y d e v i a t i o n s I n d i c a t e that t h i s boundary approximation i s not p e r f e c t . For run VK.4, the mean v e l o c i t y d e v i a t i o n i s very high at 99.8% f o r a l l experiments. This i n d i c a t e s that the c a l c u l a t e d v e l o c i t i e s were only about 0.2% of the measured v e l o c i t i e s which i s extremely poor agreement. More important however, the v e l o c i t y d i s t r i b u t i o n d e v i a t i o n i s q u i t e h i g h , at a value of 10 to 30%. An examination of Figure 5.8 e x p l a i n s t h i s : the maximum v e l o c i t i e s s t i l l occur at the l e f t side of the tank and not at the surface as i s shown by the experiments. This i n d i c a t e s that the main e f f e c t of lowering the den s i t y i n the boundary c e l l s i s to impart a v e r t i c a l v e l o c i t y to them, as i n the previous boundary c o n d i t i o n . However, t h i s c o n d i t i o n has been shown to be more accurate than the constant upward v e l o c i t y c o n d i t i o n . Therefore, t h i s increased agreement with experiments (the lowering of the angular d e v i a t i o n ) must be due to the e f f e c t of the term. This terra was i n i t i a l l y zero i n OX the constant upward v e l o c i t y c o n d i t i o n , but was non-zero (and negative) i n \ 77 t h i s c a s e . A s s h o w n b y t h e x - w i s e m o m e n t u m e q u a t i o n t h i s t e r m d i r e c t l y a f f e c t s t h e x - w i s e v e l o c i t y . T h u s , t h e m a j o r e f f e c t o f t h i s t e r m i s t o c a u s e a n e g a t i v e h o r i z o n t a l v e l o c i t y i n c e l l s a d j a c e n t t o t h e g a s b o u n d a r y . I m p a r t i n g a p o s i t i v e v e r t i c a l v e l o c i t y t o t h e b o u n d a r y c e l l s ( a l o n g w i t h t h e l o w e r e d d e n s i t y ) r e s u l t s i n v e l o c i t y p l o t s s u c h a s t h a t s h o w n i n F i g u r e 5.9. T h e d e v i a t i o n v a l u e s s h o w n i n T a b l e 5.4 I n d i c a t e t h a t t h e i n c r e a s e d b o u n d a r y v e l o c i t y l o w e r s t h e m e a n v e l o c i t y d e v i a t i o n , b u t c o r r e s p o n d i n g l y i n c r e a s e s t h e m e a n a n g u l a r d e v i a t i o n . I n e f f e c t , t h i s c a s e i s a r e v e r s i o n t o t h e c o n s t a n t u p w a r d v e l o c i t y b o u n d a r y c o n d i t i o n . T h e e f f e c t o f t h e t e r m o n s h a p i n g t h e f l o w i s r e d u c e d b y t h e g r e a t l y i n c r e a s e d v e r t i c a l b o u n d a r y v e l o c i t i e s . F r o m t h e r e s u l t s o f t h e c a l c u l a t i o n s m a d e w i t h t h i s c o n d i t i o n , i t c a n b e c o n c l u d e d t h a t t h e r e d u c e d d e n s i t y b o u n d a r y c o n d i t i o n a c c u r a t e l y p r e d i c t s t h e o v e r a l l s h a p e o f t h e e x p e r i m e n t a l v e l o c i t y p r o f i l e s . T h i s i n d i c a t e s t h a t t h e r e i s , i n f a c t , a s m a l l n e t h o r i z o n t a l f l o w o f f l u i d i n t o t h e b u b b l e c o l u m n . U n f o r t u n a t e l y , t h i s m e t h o d w a s u n a b l e t o p r e d i c t t h e m a g n i t u d e o r t h e d i s t r i b u t i o n o f t h e e x p e r i m e n t a l v e l o c i t i e s a c c u r a t e l y . T h u s i t c a n b e c o n c l u d e d t h a t t h i s a p p r o x i m a t i o n , b y i t s e l f , i t I s n o t a n a c c u r a t e d e s c r i p t i o n o f t h e b u b b l e c o l u m n . 5.3 . Pulsed Boundary T h e t h i r d a t t e m p t t o c h a r a c t e r i z e t h e b u b b l e c o l u m n m a t h e m a t i c a l l y i n v o l v e d " p u l s i n g " t h e v e l o c i t y i n t h e b o u n d a r y c e l l s . T h e b u b b l e s w e r e 78 once more considered to give a vertical velocity to the surrounding liquid, but the effect of the bubbles was crudely simulated by the following mechanism: 1. Bubbles were considered to impart a constant upward velocity to boundary cells in which they reside. 2. Boundary cells were considered to contain a bubble, or be ful l of fluid. The size of a bubble was one boundary c e l l . 3. Bubbles were moved upward to the next c e l l at time intervals corresponding to their rise velocity. This was initiall y assumed to be 40 cm/s, based on the observations of Davenport (65), but was freely varied. 4. The bubble frequency was assumed to be 10 Hz. This was based on the observations of Hoefele and Brimacombe (7). 5. Bubbles that reached the surface of the bath were allowed to escape; new bubbles were introduced at the bottom of the bath. Thus as the imaginary (two-dimensional) bubbles move through the left hand boundary cells, they Impart a vertical velocity to these cells and shear the surrounding liquid. Once the "bubbles" leave the surface of the liquid, the fluid is allowed to f a l l under gravitational forces. The values used for the bubble rise velocity and formation frequency were freely varied in an attempt to obtain good agreement with experimental data. These two parameters were found to have l i t t l e influence on the overall results. 79 The best results of this approximation are shown in Figure 5.10. Qualitatively, this method appears to yield better results than the simpler constant velocity condition. The surface velocities are larger in comparison to the cells at the left side of the tank than those produced by the previous condition. This observation is supported by the relatively low values of the velocity distribution deviation generated by this boundary condition (Table 5.5). The collapse of the surface after the passage of a bubble is responsible for these higher surface velocities. 5.4. C o l l a p s i n g Surface The previous three boundary approximations have been shown to predict that the maximum fluid velocities would occur at the bubble-liquid interface, which differs from the experimental results. It is evident from Figures 3.7 to 3.16 (particularly 3.14 through 3.16) that the flow of fluid down from the free surface is of great importance in determining the experimental velocity regime, and therefore a means of incorporating this phenomenon into the computer model was sought. The "collapsing surface" condition described below was an attempt at describing this effect. Using this condition, the free surface was set at an i n i t i a l position and allowed to collapse under gravity, causing the fluid under It to move. The collapsing surface approximation was applied to the SSMCR program in the following manner: 80 The surface of the fluid in the finite difference grid was assumed to have an i n i t i a l , non-horizontal, orientation. Several different ways of determining this i n i t i a l surface position were utilized: 1. Calculated surface profiles were tested (for example, that occurring in Figure 5.9 was used as an i n i t i a l surface position to generate Figure 5.11). 2. Experimental surface measurements. 3. Arbitrary surface profiles were also used to see the effect of changing the surface shape. The i n i t i a l velocity of the fluid was set to be zero. With these i n i t i a l conditions, the computer program was run, and allowed to calculate the fluid velocity at subsequent time intervals. As calculation progressed, the fluid surface was allowed to collapse under gravity. The effect of bubbles was taken Into account by raising the surface to its i n i t i a l position at time intervals corresponding to an input bubble frequency (usually set to be 8-10 s Thus, at computational times corresponding to every l/10th of a second, the surface collapse was stopped, and the free surface was set to its i n i t i a l position. Calculation then proceeded from this new position, with the surface allowed. to collapse as before. Detection of the onset of steady-state was somewhat more complex with this boundary condition as compared to the previous four conditions. 8 1 T h i s i s b e c a u s e t h e f l u i d v e l o c i t y c h a n g e s c o n t i n u o u s l y t h r o u g h o u t t h e t i m e t h a t t h e s u r f a c e i s c o l l a p s i n g . T h e r e f o r e , t o d e t e c t s t e a d y - s t a t e , i t w a s n e c e s s a r y t o c o m p a r e v e l o c i t y p r o f i l e s t a k e n a t i d e n t i c a l t i m e i n t e r v a l s i n t h e s u r f a c e m o v e m e n t c y c l e . F o r e x a m p l e , i f t h e s u r f a c e w a s r e s e t e v e r y 0 . 1 s e c o n d s , a n d t h e s y s t e m t i m e s t e p w a s 0 . 0 0 5 s e c o n d s , t h e n t w o v e l o c i t y p r o f i l e s c a l c u l a t e d a t 0 . 1 1 0 s a n d 0 . 2 1 0 s c o u l d b e c o m p a r e d ; two p r o f i l e s c a l c u l a t e d a t 0 . 1 1 0 s a n d 0 . 2 1 5 s s h o u l d n o t b e c o m p a r e d f o r t h i s p u r p o s e h o w e v e r . F i g u r e s 5 . 1 1 a n d 5 . 1 2 s h o w s t e a d y - s t a t e v e l o c i t y v e c t f c p l o t s g e n e r a t e d b y t h i s c o n d i t i o n a t t w o p o i n t s i n t h e s u r f a c e c o l l a p s e c y c l e . I t i s o b v i o u s t h a t t h e r e i s , i n f a c t , l i t t l e d i f f e r e n c e b e t w e e n t h e s e t h r e e p l o t s . T h e i n i t i a l s u r f a c e s h a p e u s e d w a s t h a t o f F i g u r e 5 . 9 . T h e m e a n c e l l d e v i a t i o n v a l u e s f o r c a l c u l a t i o n s p e r f o r m e d b y t h i s c o n d i t i o n a r e s h o w n i n T a b l e 5 . 6 . B y e x a m i n i n g T a b l e 5 . 6 , i t i s o b v i o u s t h a t t h i s b o u n d a r y c o n d i t i o n g i v e s v e r y l o w v a l u e s f o r t h e m e a n c e l l a n g u l a r d e v i a t i o n ( a m i n i m u m v a l u e o f 0 . 8 d e g r e e s a n d a m a x i m u m v a l u e o f 12 d e g r e e s ) . I n a d d i t i o n , t h i s a p p r o x i m a t i o n a l s o y i e l d s v e r y l o w v e l o c i t y d i s t r i b u t i o n d e v i a t i o n v a l u e s ( 8 t o 1 5 % ) . O n e c a n c o n c l u d e b y t h e s e f a c t s t h a t t h e c o m p u t e r m o d e l a c c u r a t e l y p r e d i c t s b o t h t h e d i r e c t i o n a n d t h e r e l a t i v e m a g n i t u d e s o f t h e e x p e r i m e n t a l v e l o c i t y v e c t o r s . T h i s b o u n d a r y a p p r o x i m a t i o n i s e v e n m o r e s u c c e s s f u l t h a n t h e r e d u c e d d e n s i t y m e t h o d d e s c r i b e d i n S e c t i o n 5 . 2 . 82 However, the mean cell magnitude deviations of the plots generated by this boundary condition are a l l very high, (greater than 99%), indicating that the predicted velocities are much lower than those occurring in the experimental tank. This is reasonable, as i t is unlikely that the collapse of a surface only a few centimetres would be capable of producing water velocities as high as those found in the experiments. In an attempt to increase the velocities predicted by this condition, a positive vertical velocity was applied to the left wall boundary cells. The results of this approxiation are also shown in Table 5 . 6 , and i t is clear that this attempt is only partially successful: the mean cell magnitude deviation values are lowered, but there is a corresponding increase in the directional and distribution deviations. This indicates that as the left wall velocity is increased, this boundary condition tends to the constant upward velocity condition, with a corresponding decrease in accuracy. Therefore, i t can be concluded that the collapsing surface condition accurately predicts both the orientation and the velocity distribution of the experimental flow patterns, though is unable to predict the absolute magnitude of the experimental velocities. This indicates that, i f each velocity value predicted by this boundary condition were multiplied by the ratio of the maximum measured velocity to the maximum predicted velocity, one would obtain excellent agreement between the calculated results and the experiments. 83 5.5 . Discussion Based upon the results discussed above, i t can be concluded that the stirring of the water in the experimental tank must be largely driven from the surface. The first three boundary approximations - a l l of which assumed that the flow was driven from the left boundary - were a l l unsuccessful in predicting the experimental velocity profiles. However, when a boundary condition was used in the mathematical model that assumed a l l of the liquid motion was due to the action of a collapsing surface, much better agreement with the experimental results was achieved, even though the magnitudes of the predicted velocities are too low. It would not be correct to assume that the rather simplistic "collapsing surface" model is necessarily a true description of what is actually occurring in the experiments. Rather, this model indicates that the free surface is of paramount importance in determining the experimental velocity regimes. In fact, the bulk of the stirring in the tank appears to be driven from the surface, and not from the left side as has been assumed previously. Figures 3.7 to 3.16 and the calculations discussed above allow an hypothesis of the stirring mechanism within the water tank to be proposed: 1. The effect of the bubble column is to cause the water in this region of the tank to rise, largely due to density differences. 84 2. The air-water column rises along the left side of the tank, but does not shear the adjacent fluid significantly. 3. At the surface, the trapped air escapes. The water then falls under the influence of gravity, but is displaced outward by more fluid rising from below. 4. This water then travels downward and outward across the free surface at high speed. It is this fluid which is largely responsible for the stirring of the liquid in the tank. In this case, the "collapsing surface" condition would only be an approximation to the actual case - even though this approximation agrees with the experiments. It is proposed that the high experimental surface velocities are due to fluid being continuously introduced across the entire length of the surface, and not due to a "collapse" of the surface Itself. Therefore, even though the collapsing surface boundary condition yields relatively accurate results, i t would not necessarily be a physically accurate description of the experimental process of stirring. A more physically correct boundary condition, might be to constantly introduce downward-directed fluid along the length of the mathematical free-surface, while maintaining a constant surface shape. Unfortunately, a l l attempts made to apply this type of condition failed due to computational instability. 85 T h e r e f o r e , a c o m p l e t e l y a c c u r a t e d e s c r i p t i o n o f t h e b u b b l e - l i q u i d b o u n d a r y a p p e a r s o n l y t o b e p o s s i b l e w i t h a m a t h e m a t i c a l m o d e l c a p a b l e o f p e r f o r m i n g c a l c u l a t i o n s o n t w o p h a s e s ( a i r a n d w a t e r ) s i m u l t a n e o u s l y . A s i n g l e - p h a s e m o d e l ( s u c h a s S S M C R ) i s c l e a r l y l i m i t e d i n i t s a b i l i t y t o d e s c r i b e t h e I n t r i c a c i e s o f t h e g a s - l i q u i d b o u n d a r y . 86 6. INDUSTRIAL CALCULATIONS 6.1. Fluid Flov i n a Copper Converter 6.1.1. Assumptions The dimensions and p h y s i c a l constants that were used to model the flow i n a Peirce-Smith converter are presented i n Table 6.1. This data was obtained mainly from Johnson (63), as w e l l as from Bustos et a l . (17) and Hoefele and Brimacombe ( 7 ) . A number of s i m p l i f y i n g assumptions were made about the flow regime i n t h i s i n d u s t r i a l v e s s e l to a l l o w c a l c u l a t i o n to proceed: 1. A l l of the flow i s two dimensional and due only to the i n f l u e n c e of bubbles. Temperature gradients have no e f f e c t . 2. The e f f e c t of s l a g on the surface of the copper matte was neglected. 3. The bath was assumed to be completely i s o t h e r m a l , and Incompressible. 4. The r e a c t i o n of a i r with the matte and the generation of sulphur d i o x i d e was neg l e c t e d . 5. C a l c u l a t i o n s were only performed on the bulk of the f l u i d . The bubble column i t s e l f , and a l l f l u i d between t h i s r e g i o n and the adjacent w a l l was neglected from c a l c u l a t i o n s . The value f o r the e f f e c t i v e v i s c o s i t y was determined from the c a l c u l a t i o n s presented i n the previous chapter: the v i s c o s i t y values which were the most s u c c e s s f u l i n p r e d i c t i n g the experimental data (400 g/cm s) were 87 "scaled up" to apply to the industrial system. This was done be defining a new constant to apply to the effective viscosity model of Sahai and Guthrie (44). The effective viscosity predicted by the model of Sahai and Guthrie is given by: /n % n 1/3 H e f f " KHpt<^=fM] (6.1) -3 where K Is taken to be 5.5 x 10 Setting the effecting viscosity value of 400 g/cm s equal to the left hand side of Equation (6.1), a new value for K can be determined that applies to the side-blown, multi-tuyere systems of interest: ^eff K = — T-pr = 6.28 (6.2) u ,(l - a ) g Q- ' which is three orders of magnitude ldrger than that previously used. Employing this new value of K to the copper converter yields an effective viscosity value of approximately 5000 g/cm s. 6.1.2. Mathematical Description of Gas-Liquid Interface Based on the calculations presented in the previous chapter, i t was concluded that the most accurate description of the bubble column in the 89 distribution of the velocities, are probably predicted accurately. 2. The highest fluid velocities are predicted to lie in the region of the free surface. 3. The kinetic energy of the fluid bath appears to be poorly distributed compared to that observed in the experimental tank: there are large regions of the copper converter that have very low fluid velocities relative to the surface. This may be due to two effects: a) The copper converter (when 35% full) is bounded by a much larger area of wall (40% larger) than does a square tank of equivalent volume. This indicates that the wall will retard the fluid to a much greater extent in the copper converter than in the experimental tank. b) The bubble column in the copper converter does not enter near the bottom of the vessel (as in the experimental tank). Instead, the air column enters through tuyeres that are elevated 0.5 m off the bottom of the converter (Figure 2.1). This has the effect of reducing the buoyant input power of the injected air relative to bottom-injection. As well, operating with the tuyeres close to the surface reduces the amount of fluid that is able to come in contact with the air column at any given time. The closer the tuyeres are placed to the surface, the shorter the length of the bubble column becomes, and therefore the volume available for stirring and reaction is reduced. It is reasonable to conclude therefore that the stirring efficiency in the copper converter could be increased by the following factors: 90 a) If the converter were arranged as a vertical cylinder instead of a horizontal cylinder, the amount of wall that contacts the fluid would be reduced by as much as 40%. This would have the effect of reducing the wall drag effect, and increasing the circulation velocities. b) Blowing from the bottom of the vessel, rather than from the sides would increase the buoyant input power to the bath, and consequently (in the absence of channelling) increase the recirculation velocity. As has been discussed by Bustos et a l . (17) an unstable gas envelope forms at the mouths of the tuyeres in the copper converter. Based on the experimental investigation in which this effect was observed, i t can be concluded that the stirring efficiency in the Peirce-Smith converter is less than optimal when operating in this manner. An increased tuyere spacing or a reduction in air flowrate could conceivably increase this efficiency. 6.2. Fluid Flow in a Zinc Slag Fuming Furnace The following set of predictions for the fluid flow regimes in a zinc slag-fuming involve significantly greater approximation than those made for the Peirce-Smith copper converter. This is for two reasons: 1. The liquid in the fuming furnace has an extremely high concentration of entrapped gases. Studies by Richards (9,10) indicate that the porosity of the furnace approaches 40%. Therefore, the single-phase SSMCR model is greatly limited in its ability to model this system. A two phase model is needed i f accurate calculations are to be performed. 9 1 2 . U n l i k e t h e c a s e o f t h e c o p p e r c o n v e r t e r , s o l i d s ( p o w d e r e d c o a l ) a r e i n j e c t e d i n t o t h e s l a g f u m i n g f u r n a c e a l o n g w i t h g a s . I t i s n o t k n o w n h o w t h i s w i l l a f f e c t t h e b u b b l e f o r m a t i o n a n d s t i r r i n g e f f e c t s w i t h i n t h e f u r n a c e . H o w e v e r , a s n o e x p e r i m e n t a l o r m a t h e m a t i c a l s t u d i e s o f t h e f l o w i n a s l a g f u m i n g f u r n a c e h a v e b e e n p u b l i s h e d t o d a t e , t h e m a t h e m a t i c a l m o d e l p r e d i c t i o n s a r e p r e s e n t e d b e l o w a s a g e n e r a l q u a l i t a t i v e d e s c r i p t i o n o f t h e f l o w i n t h i s v e s s e l . 6.2.1. Assumptions A s w i t h t h e p r e v i o u s c a l c u l a t i o n s i t w a s n e c e s s a r y t o m a k e s e v e r a l s i m p l i f y i n g a s s u m p t i o n s t o a l l o w c o m p u t a t i o n o f f l u i d f l o w p r o f i l e s i n t h i s p r o c e s s v e s s e l . A l l o f t h e a s s u m p t i o n s o u t l i n e d i n S e c t i o n 6 . 1 . 1 w e r e u s e d , w i t h t h e f o l l o w i n g a d d i t i o n s : 1 . T h e v e r y h i g h c o n c e n t r a t i o n o f e n t r a i n e d g a s e s i n t h e l i q u i d p h a s e o f t h e s l a g f u m i n g f u r n a c e was t a k e n i n t o a c c o u n t b y l o w e r i n g t h e d e n s i t y o f t h e f u r n a c e s l a g . T h u s t h e a c t u a l s l a g d e n s i t y was a s s u m e d t o b e : ° = ° s l a g ( 1 _ o c ) + a Pg ( 6 * 3 ) w h e r e a i s t h e f r a c t i o n o f g a s i n t h e s l a g . T h i s v a l u e w a s s e t t o b e 30% ( 9 , 1 0 ) . 92 2. The effective viscosity of the furnace slag was calculated as for the copper converter. 3. The effects of the particle injection were neglected, along with the heat and gas evolution from coal oxidation. The data used to model the furnace are presented in Table 6.2 and are taken from Richards (9,10). 6.2.2. M o d i f i c a t i o n s t o Program The nature of the boundary conditions of the slag fuming furnace necessitated modifications to the SSMCR program. While both the experimental tank and the Peirce-Smith copper converter have flow regimes that are driven from only one side or from the surface, the slag-fuming furnace is driven from both sides. The finite difference code for the SSMCR program was written in central and backward differences, and calculations were repeatedly performed over the computational mesh from left to right (Section A.3). This has the effect of "left-biasing" the flow - right side boundary conditions do not have as much effect as left side conditions. A simple way of overcoming this difficulty is to rotate the mesh through 90 degrees. Unfortunately in this case, the influence of the free surface would be reduced and complicated by such a method. In an attempt to reduce this effect while maintaining the free-surface conditions, calculation over the finite difference mesh was 93 altered to sweep from left to right, and then from right to left. The continuity equation was re-written in both forward and reverse finite difference approximations. However, the momentum equations and the free surface conditions were not altered and remained in their "left-justified" code. Calculations performed with this new solution procedure were seen to have a diminished (but slightly noticeable) "left-biasing" effect - i.e. fluid flow profiles with identical left and right boundary conditions were not found to be perfectly symmetrical. However, due to the extreme difficulty in making additional adjustments to the SSMCR program, and the amount of approximation already introduced into these calculations, further refinements were not made to the program. All calculations shown below contain some of the "left-biasing" effect. 6.2.3. Mathematical Description of Bubble Column Describing the bubble column in the slag fuming furnace was not as straightforward as that in the copper converter. Initially, the collapsing surface condition was applied. This produced plots such as Figure 6.3. It can be seen that this condition predicts the surface at the centre of the bath to rise as the left and right sides collapse. Therefore, i f this condition were true, the slag bath would be constantly oscillating, with the centre velocities switching from positive to negative as the surface rose and f e l l . 94 Qualitative observations of the experimental tank when injected with air from two sides indicate that this is not what is occurring at a l l . The surface of the experimental tank was seen to be highly stable,-and not to fluctuate to any significant extent. In addition, two stable and distinct vortices were observed to exist in the separate halves of the tank. The reasons for the failure of the collapsing surface condition in this case have been discussed in the previous chapter, where i t was noted that, although this boundary condition can result in accurate predictions, i t is not necessarily an accurate description of what is occurring In the tank. With one-sided injection this was acceptable - the approximation of a collapsing surface successfully modelled the constant influx of high-velocity fluid falling away from the bubble column without creating obviously Incorrect results. However, in the case of two-sided gas injection, the shortcomings of this approximation are more obvious, as it is evident from observations that the surface cannot possibly be continuously collapsing and reforming. The collapsing surface condition fails because i t is forced to predict - by conservation of mass - a corresponding rise in the surface at the centre of the tank which is a physically unrealistic result. This predicted surface rise does exist in the calculations made for the flow in the copper converter, as can be seen in Figure 6.2. However, since the surface is distorted less, and is collapsing from one side only, the effect Is much less obvious. 95 Therefore, even though this condition can predict the expected high surface velocities in the slag fuming furnace, it should not be used, as the orientation of the predicted velocities must certainly be incorrect. Instead, the variable density and the constant velocity boundary conditions were employed to model the bubble column, in the hope of obtaining at least a correct qualitative description of the flow regime in the slag-fuming furnace. The variable density condition has been shown to be the next most accurate boundary condition, after the collapsing surface approximation. The value used for the porosity of the bubble column was set to 80% to give the predictions of Figure 6.5. The wall velocity used to generate Figure 6.4 was 1.0 m/s. 6.2.4. R e s u l t s The results of the calculations performed on the zinc slag fuming furnace are shown in Figures 6.4 and 6.5. The general recirculatory nature of the flow is clearly discerned, with the fluid rotating in two distinct convective cells as mentioned earlier. The "left-biasing" is noticeable in the asymmetry of the flow pattern, particularly In the region of the right wall. It must be stressed that the accuracy of these flow predictions is quite poor, even when compared to the predictions made for the flow in the copper converter. 96 However, even such inaccurate flow predictions as these indicate that the slag fuming furnace appears to be more highly stirred than the copper converter, with the f l u i d maintaining relatively high velocity throughout the vessel. As well, the slag in the centre of the furnace appears to acquire a very high downward velocity due to the meeting of the two opposing vortices. However, this seems to have the effect of leaving a relatively stagnant zone at the bottom centre of the tank, where the two vortices diverge. 97 7. CONCLUSIONS 7*1* Experimental In the experimental section of this research, air was injected through side-mounted tuyeres into a water-filled plexiglas tank at modified Froude numbers varying from 0.4 to 15.6 and the resulting fluid velocity and surface profiles were recorded. Under a l l conditions, the water was found to move in a recirculating vortex with the highest velocities in the region of the free surface. Little variation in flow was found through the width of the tank. At gas injection rates greater than approximately 90 1/min. (corresponding to a modified Froude number of 2.3) the mean kinetic energy of the fluid was found to diminish with increasing air flowrate. Photographic evidence suggests that this effect is largely due to interaction of bubbles originating from adjacent tuyeres. Integration of the steady-state surface profiles has shown that the experimental tank contains a significant amount of entrapped air. The air holdup in the tank has been seen to vary linearly with air flowrate. 98 7.2. Calculations - Agreement with Experimental Results I n a n a t t e m p t t o p r e d i c t t h e e x p e r i m e n t a l ( a n d i n d u s t r i a l ) f l o w r e g i m e s , a m a t h e m a t i c a l m o d e l c a p a b l e o f p r e d i c t i n g b o t h f l u i d v e l o c i t y a n d s u r f a c e p r o f i l e s i n b o t h r e c t a n g u l a r a n d c i r c u l a r g e o m e t r i e s h a s b e e n d e v e l o p e d , b a s e d o n t h e S O L A S M A C m e t h o d o f H i l l ( 4 8 ) . T u r b u l e n c e h a s b e e n m o d e l e d i n t h e p r o g r a m b y a p p l y i n g t h e B o u s i n e s q a p p r o x i m a t i o n , a n d a s s u m i n g a c o n s t a n t e f f e c t i v e v i s c o s i t y . T h e a l g e b r a i c v i s c o s i t y m o d e l o f S a h a i a n d G u t h r i e ( 3 5 ) w a s t e s t e d , a s w e r e c o m p l e t e l y e m p i r i c a l v i s c o s i t y v a l u e s , a n d t h e i r r e s p e c t i v e p r e d i c t i o n s w e r e c o m p a r e d w i t h t h e e x p e r i m e n t a l d a t a . O v e r a l l , t h e m o d e l o f S a h a i a n d G u t h r i e w a s f o u n d t o u n d e r - p r e d i c t t h e e f f e c t i v e v i s c o s i t y I n t h e e x p e r i m e n t a l s y s t e m . A n u m b e r o f d i f f e r e n t b o u n d a r y c o n d i t i o n s w e r e a p p l i e d t o t h e m a t h e m a t i c a l m o d e l i n a n a t t e m p t t o p r e d i c t t h e e x p e r i m e n t a l f l o w p a t t e r n s . T h e b o u n d a r y a p p r o x i m a t i o n t h a t was f o u n d t o y i e l d t h e m o s t a c c u r a t e r e s u l t s w a s t h e " c o l l a p s i n g s u r f a c e " c o n d i t i o n , w h e r e i n t h e r e c i r c u l a t i o n o f t h e f l u i d w a s d r i v e n f r o m t h e f r e e s u r f a c e . T h i s w a s t h e o n l y b o u n d a r y a p p r o x i m a t i o n t e s t e d t h a t p r e d i c t e d t h e m a x i m u m f l u i d v e l o c i t i e s t o o c c u r I n t h e r e g i o n o f t h e f r e e s u r f a c e . B a s e d u p o n t h i s o b s e r v a t i o n , i t w a s s u g g e s t e d t h a t t h e s t i r r i n g i n t h e e x p e r i m e n t a l t a n k w a s " d r i v e n " f r o m t h e s u r f a c e , a n d l a r g e l y d u e t o f l u i d f a l l i n g a w a y f r o m t h e t o p o f t h e b u b b l e c o l u m n a c r o s s t h e f r e e s u r f a c e . 99 7.3. Industrial Calculations 7.3.1. Flow In a Copper Converter U t i l i z i n g the "collapsing surface" boundary approximation, predictions have been made for the fluid flow regime in a Peirce-Smith copper converter. The effective viscosity was determined by redefining the constant K in the model of Sahai and Guthrie. The resulting fluid flow profiles indicate that the stirring in the copper converter is very poor. The bath recirculates very slowly, with velocities in the bulk of the bath an order of magnitude lower than those occurring at the surface near the bubble column. It has been suggested that this is due to both the large amount of wall exposed to the bath, and the relatively short length of the bubble column. It has also been proposed that the stirring efficiency would be much higher in a vertically-oriented cylindrical vessel, with air injected at the bottom. 7.3.2. Flow i n the Zinc Slag-Fuming Furnace With significant approximation, a qualitative prediction of the flow in a slag-fuming furnace has been made. Employing the constant upward velocity boundary condition, this prediction indicates that the fluid in the furnace recirculates in two counter-rotating vortices, with high velocities located in the centre of the vessel. 1 0 0 7.4 . Recommendations for Further Work I t h a s b e e n o b s e r v e d d u r i n g t h e c o u r s e o f t h i s i n v e s t i g a t i o n t h a t t h e f l u i d w i t h i n t h e m a t h e m a t i c a l m o d e l o f t h e c o p p e r c o n v e r t e r c o u l d b e i n d u c e d t o " s l o p " b y v a r y i n g t h e l e f t s i d e b o u n d a r y c o n d i t i o n . T h a t i s , u n d e r t h e r i g h t c o n d i t i o n s , t h e p r o g r a m p r e d i c t s a c o l l a p s e o f t h e f l u i d s u r f a c e a t t h e l e f t s i d e o f t h e c o n v e r t e r , a n d a c o r r e s p o n d i n g r i s e i n t h e r i g h t s i d e f l u i d . I t i s s u g g e s t e d t h a t t h i s " s l o p p i n g " b e h a v i o u r i n t h e m a t h e m a t i c a l m o d e l b e i n v e s t i g a t e d m o r e f u l l y , t o p o s s i b l y y i e l d i n f o r m a t i o n o n t h e p r e v e n t i o n o f t h i s p h e n o m e n o n i n i n d u s t r i a l p r a c t i c e . 101 REFERENCES 1. T. Rosenqvist, Principles of Extractive Metallurgy, McGraw-Hill, 1974. 2. R.D. Pehlke, Unit Processes of Extractive Metallurgy, Elsevier North Holland, 1973. 3. N.J. Themelis, P. Tarassoff and J. Szekely, Trans. AIME 245, p. 2425, 1969. 4. K. Nakanishi and J. Szekely, Unpublished Research, from J. 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S c a n i n j e c t I I , 2 n d I n t e r n a t i o n a l C o n f e r e n c e o n I n j e c t i o n M e t a l l u r g y , L u l e a , S w e d e n , J u n e 1 2 - 1 3 , 1 9 8 0 . 6 9 . S c a n i n j e c t I V , 4 t h I n t e r n a t i o n a l C o n f e r e n c e o n I n j e c t i o n M e t a l l u r g y , L u l e a , S w e d e n , J u n e 1 1 - 1 3 , 1 9 8 6 . 7 0 . G a s I n j e c t i o n i n t o L i q u i d M e t a l s , U n i v e r s i t y o f N e w c a s t l e u p o n T y n e , 1 9 7 9 . 7 1 . T . D e b R o y , A . K . M a j u m d a r , J . M e t a l s , N o v e m b e r 1 9 8 1 . p p . 4 2 - 4 7 . 7 2 . N . T a k e m i t s u , J . C o m p u t . P h y s . , 3 6 , p p . 2 3 6 - 2 4 8 , 1 9 8 0 . 7 3 . G . d e V a h l D a v i s a n d G . D . M a l l i n s o n , C o m p u t e r s a n d F l u i d s , V o l . 4 , p p . 2 9 - 4 3 , 1 9 7 6 . 7 4 . W . S . H w a n g a n d R . A . S t o e h r , J . M e t a l s , O c t o b e r 1 9 8 3 , p p . 2 2 - 2 8 . 7 5 . T . M i y a u c h i a n d H . O y a , A i C h E , J o u r n a l , V o l . 1 1 , 3 , p p . 3 9 5 - 4 0 2 , 1 9 6 5 . 7 6 . J . L . L . B a k e r a n d B . T . C h a o , A i C h E J o u r n a l , V o l . 1 1 , 3 , p p . 2 6 8 - 2 7 3 , 1 9 6 5 . 106 77. R.D. Mills, J. Roy. Aero. Soc, 69, Feb. 1965, pp. 116-120. 78. B.J. Daly, J. Comput. Phys., 4, pp. 97-117, 1969. 79. E.O. Hoefele, M.A.Sc. Thesis, The University of British Columbia, 1978. 80. CO. Bennett and J.E. Myers, Momentum, Heat and Mass Transfer, 2nd ed., McGraw-Hill, New York, 1962. 81. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, John Wiley and Sons, New York, 1960. 82. Injection Phenomena in Extraction and Refining, University of Newcastle, April 1982. 107 APPENDIX 1 Finite-Difference Approximations used by SSMCR 1. Continuity Equation: ax ay where au = u i , j " u i - i , j ax AX 8JL = V i , j " V i , j - 1 ay Ay 2. x-Wise Momentum Equation: 2 2 2 au i ap a(uv) a(u ) + ueff . a u a u, at p ax g x ay ax P &x2 ay2 where: t+At au _ u - u at At i ap „ P i , j " P i + l , j p ax pAx 2 2 / 2 N (u. , + n.,, ,) - (u. , . + u. .) a(u ) = i , j i + i . y 1-1,J 1>3 a x A A x y | u i , j + u i + i „ i I ( u i , j " u i + i , j ) ~ Y 4 Ax u. . . + u. . I (u. n . - u. . N i - 1 , J i»3 1 1-1,3 i . j ) 108 5(uv) = ( u i , j + U l , J + l ) ( V i , 1 + - < UJ,j + U J , j - l ) ( v i , j - l + vl+1.1-l> ay A Ay . , j + V l - r l , j I ( " l . j - "I.j-H) - T V l , j - 1 + V J + l , j - l ' ( U l , j - l - " l t j > 4 Ay (£± + all* - i^l r u i + i , j " 2 u i , j + u i - i , j . u i , j + i ~ Z u i , j - 2u. , + . i , j - l ] p ax 2 ay 2 Ax Ay 3 . y-Wise Momentum Equation: ay, = _ i a p _ _ a(uv) _ at p ay y ax ay a(y 2) . j±ff r ^ v J \ l 9 9 i P "ax2 ay where: t+At av _ v - v at At _ I ap = p i , j " p i , j + i P ay pAy 2 2 ~ / 2N (v. . + v. ....) - (v. . - + v. .) a(v ) m 1,3 i,3+l i , j - l 1,3 ay 4 Ay , j + v i , j + l 1 ( v i , j - v i t 3 + i > ' ? | v i , 3 - l + v i , 3 ( v i , j - l " v i , j ) 4 AY / x (u, . + u. . ,,)(v. . + v.,, .) - (u. , . + u. T ..,)(v, , + V. .) a(uv) = v i , j i , J + l / v i , J i+1,3 1-1,3 i - l , j + l / v i - l > j i,3 ax 4 Ax T I + u i .j+i ( v i , j ' ' w , ] ' - y 1 V i , ] + ( V i , . i - v i . . i ? 4 AX 109 ^ e f f , o 2 v ^ e f f rV i + l . i ~ 2 V 1 „ 1 + V 1 - 1 J 5x ay Ax ^, -i - 2 v . . + v. . , i , j + l i , j i , 3 - l 1 . 2 Ay I n t h e s e e q u a t i o n s , y i s u s e d t o v a r y t h e a m o u n t o f u p w i n d d i f f e r e n c i n g . 110 TABLES I l l Table 3.1. Experimental Air Injection Rates Run Number Air Flowrate (std 1/min) N F r, 1 38 0.43 2 68 1.40 3 78 1.80 4 88 2.30 5 120 4.40 6 142 - 6.30 7 154 7.40 8 166 8.70 9 178 10.20 10 216 15.60 Table 3.2. Values of -A£ at Six Locations l n Experimental Tank, Experiment Number 1 Cell Location i i 3 3 3.4 4 4 3.8 5 5 1.5 6 6 -1.4 7 7 -3.1 8 8 -0.4 113 Table 3.3. Minimum Mean Values of -Ajr of Experimental Runs 114 Table 4.1. Approximations Used for Rectangular Wall Boundaries Type of Wall Vertical Horizontal No Slip U i . J - ° vi,j= " v i - l , j ui,j= - u i , j + l v i , r 0 Free Slip u i , r 0  v i , r v i - i , j u i , r u i , j + i v i , r 0 Table 5.1. Effective Viscosity Values Predicted by the Model of Sahai and Guthrie (44) Experiment Meff^cms) Number 1 2.9 2 3.4 3 3.5 4 3.6 5 3.8 6 3.8 7 3.8 8 3.8 9 3.8 10 3.5 Table 5.2. Comparison of Results Calculated by Constant Velocity Condition with Experiments Experiment Number Run Name u e f f(g/cras) Left Wall Velocity(m/s) Deviation Values 1 2 3 4 5 6 7 8 9 10 C«.ag 87.8 86.9 88.9 89.0 86.9 87.9 87.8 86.2 86.4 84.5 VC8.6 3 1.0 ce -40.1 -32.3 -34.3 -35.4 -33.2 -35.1 -26.8 -33.2 -20.7 -27.0 Bdi„t 47.7 44.3 40.1 41.9 39.1 35.1 45.3 49.6 52.9 45.7 "mag 67.6 65.4 69.3 70.5 66.6 64.8 67.2 65.5 62.6 65.6 VC6.6 10 0.80 5 9 -62.6 -54.0 -57.5 -52.1 -39.0 -51.6 -47.8 -48.5 -39.6 -47.2 "dist 44.3 40.9 33.4 37.6 37.8 24.2 40.9 49.4 48.1 42.4 Bmag 51.1 46 53.1 55.4 48.9 44.1 50.9 46.6 45.1 49.2 VC.4 40 0.80 B 9 -81.5 -70.6 -76.6 -71.1 -58.0 -64.1 -62.4 -67.5 -57.4 -60.2 "dist 39.4 33.5 26.5 31.8 31.6 13.1 36.2 43.5 45.1 38.7 "mag 15.8 9.8 19.9 25.4 10.9 16.3 18.4 5.5 13.1 9.1 VC2.8 400 0.80 C 9 -50.6 -41.2 -46.3 -41.0 -31.0 -37.1 -30.8 -39.7 -25.5 -31.5 "dist 24.2 19.4 8.9 17.4 13.3 5.7 23.0 27.5 36.9 20.4 "mag -0.6 6.7 3.2 10.3 -6.5 0.6 1.3 -12.4 -5.8 -17.0 VC7.4 400 1.0 V -63.3 -54.8 -59.1 -53.8 -41.1 -46.8 -43.2 -50.2 -36.8 -17.8 C d l s t 22.0 17.8 5.1 14.3 10.7 3.4 19.7 25.7 33.8 8.2 Table 5 . 3 . Estimated Experimental Bubble Column Porosity Values Experiment Number Bubble Column P o r o s i t y 1 8.7 2 19.6 3 24.1 4 26.7 5 34.9 6 43.6 7 51.9 8 56.2 9 57.2 10 70.6 Table 5.4. Comparison of Results Calculated by Variable Density Condition with Experiments Experiment Number Run Name H e f f(g/cins) Porosity(Z) Left Wall VelocltyO/s) Deviation Values 1 2 3 4 5 6 7 8 9 10 "mag 99.8 99.8 99.8 99.8 99.8 99.8 99.8 99.8 99.8 99.8 VK.4 400 30.0 0.0 B 9 -12.1 -12.4 2.1 -2.6 -8.9 -8.1 0.3 -3.6 4.3 -3.9 B d i s t 23.7 19.0 7.1 15.1 16.5 -10.7 21.3 30.5 30.1 22.2 "mag 57.2 59.8 65.6 65.0 67.3 61.6 58.9 63.8 59.0 60.9 VL.4 400 30.0 0.4 ° 9 -35.3 ,-25.7 -31.1 -28.7 -25.9 -21.6 -24.8 -21.9 -27.0 -17.8 5 d i a t 10.8 16.7 9.4 18.2 16.0 13.4 -7.3 20.9 27.0 20.5 Table 5.5. Comparison of Results Calculated by Pulsed Boundary Condition with Experiments Experiment Number Run No. Bubble Frequency Rise Velocity t'eff Deviation Values 1 2 3 4 5 6 7 8 9 10 "mag 29.4 22.4 30.3 34.8 24.9 17.6 26.6 21.2 20.7 19.5 BB2.4 20 1.0 400 B9 -76 -66.4 -72.3 -66.9 -54.3 -60 -56.6 -63.3 -51.4 -54.3 B d i s t 21.1 13.8 1.52 10.2 9.2 -15.5 13.9 24.8 28.4 12.3 "mag 30.7 25.1 33.7 38.1 27.5 27.6 32.8 24.5 26.1 28.2 BB2.4 20 0.8 400 B 8 -67.9 -59.2 -62.5 -57.2 -44.6 -50.3 -47.6 -53.6 -42.4 -49.1 "disc 26.8 21.4 11.5 19.6 17.2 4.3 25.6 32.0 37.1 26.2 Bmag 68.3 64.6 69.9 71.2 66.5 64.5 68.5 65.5 64.3 67.3 BB3.8 10 1.0 40 B9 -76.7 -67.7 -70.9 -65.4 -52.7 -58.4 -57.1 -61.8 -53.6 -60.6 5 d i s t 43.9 37.8 32.7 37.3 36.0 -21.3 40.8 48.0 49.0 43.7 "mag 20.4 12.3 21.9 26.8 16.5 5.5 18.2 12.6 11.8 11.6 BB4.7 10 1.0 400 B9 -78.7 -68.5 -74.4 -69.0 -56.3 -62.1 -59.1 -65.4 -54.1 -57.1 B d i s t 21.5 14.1 2.8 11.0 11.0 -16.7 15.5 26.5 29.9 15.2 Table 5.6. Comparison of Result* Calculated by Collapsing Surface Condition with Experiments Experiment Number Run Name r eff(g/cms) Left Wall Velocity(m/B) Deviation Values 1 2 3 4 5 6 7 8 9 10 "mag 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 99.9 S.C0L.4 AOO 0.0 c e 0.9 2.9 5.4 6.5 10.9 5.1 0.8 -18.0 27.0 14.1 B d l 8 t 1A.5 -8.1 -3.0 7.8 12.4 3.6 -5.5 27.7 15.6 12.2 Cmag 24.7 19.1 29.0 33.3 21.1 21.9 27.0 16.1 22.2 19.8 VG.6 AOO 70.0 B e -40.1 -30.6 -35.4 -30.3 -23.9 -30.7 -27.8 -33.2 -14.6 -23.7 C d l s t 24.1 18.9 9.6 17.3 14.0 1.4 22.8 27.9 36.7 21.3 121 Table 6.1. Data Used to Model Flow in Peirce-Smith Copper Converter Quantity Value Bath Density 4600 kg/m 3 Bath V i s c o s i t y 0.1 g/cms @ 1200°C Furnace Diameter 3.85 m Tuyere Submergence 0.35 m Table 6.2. Data Used to Model Flow ln Zinc Slag Fuming Furnace Quantity Value Bath Density 3500 kg/m 3 Bath V i s c o s i t y 0.2 g/cms Furnace Width 2.9 m Tuyere Submergence 0.68 m Slag P o r o s i t y 0.30 123 FIGURES 124 F i g u r e 1.1. Schematic Diagram of Peirce-Smith Copper Converter. 125 Primary air 05-33Nm3/s Coal: I-15 kg/5 Secondary air 3 3-75 Nm3/s Punching Valve & Tuyere C,CO—C0 2 Zn. +1/2 0, —ZnO.. (g) 2 (s) Tertiary air (unregulated) Slag Fuming Process Water-jacketed Walls Figure 1.2. Schematic Diagram of Zinc Slag Fuming Furnace. 126 Figure 2 . 1 . Estimate of Flow in a Copper Converter, from Themelis et a l . (3 ) . 127 2.2. Predicted Flow Profile in Two-Phase Region of Copper Converter, from Nakanishi and Szekely (4). 128 F i g u r e 2.3. J e t Behaviour Diagram, from Hoefele and Brimacombe ( 7 ) . 1 2 9 Figure 3.1. Schematic Diagram of Experimental Apparatus. \* 20 cm *| 130 60 cm Front •30 cm o o o o o Side 5 holes (j) = I cm Balanced manifold Top view F i g u r e 3.2. Dimensions and C o n s t r u c t i o n of Experimental Tank. 131 Photodetector 1.8 cm X.Y.Z traversing table Tank' «—Manifold Stand Figure 3.3. Diagram of Laser-Doppler System. 132 Water sur face 4 2 fjm VM 3 1 Two - phase region F i g u r e 3 .4 . V e l o c i t y Measurement Locations W i t h i n Experimental Tank. 133 F i g u r e 3.5. H o r i z o n t a l V e l o c i t y V a r i a t i o n Through Depth of Tank. 134 Fract ional width F i g u r e 3 . 6 . V e r t i c a l V e l o c i t y V a r i a t i o n Through Depth of Tank. 135 * V t \ t 4 \ \ \ \ \ • 1 ft. / \ \ \ \ t l 1 \ \ \ / \ \ / / 1 / i ^ t MAXIMUM VELOCITY (M/S) = 0.455 F i g u r e 3.7. Experimental V e l o c i t y Vector and Surface P l o t - Experiment 1. 136 \ \ f \ - \ \ • \ \ \ \ \ \ \ I / J / i / ' / . te * MAXIMUM VELOCITY (M/S) = 0.452 F i g u r e 3.8. Experimental V e l o c i t y Vector and Surface P l o t - Experiment 2. 137 MAXIMUM VELOCITY (M/S) = 0.575 Figure 3 .9 . Experimental Velocity Vector and Surface Plot - Experiment 3 . 138 1 \ \ \ t 4 \ \ \ \ \ 1 % »» ' f \ \ 1 \ \ / / I I \ \ 0 / 4 I \ 4 I * MAXIMUM VELOCITY (M/S) = 0.560 Figure 3.10. Experimental Velocity Vector and Surface Plot - Experiment 4. 139 \ \ * \ \ I * 1 V \ I i 1 \ 0 / I 1 0 >- 0 — <• *• MAXIMUM VELOCITY (M/S) = 0.492 Figure 3.11. Experimental Velocity Vector and Surface Plot - Experiment 5. 140 e B B — — i ) w \ \ \ A \ \ \ \ \ \ 1 I \ \ \ / 1 V \ V If 4 V * t # f MAXIMUM VELOCITY (M/S) = 0.570 Figure 3.12. Experimental Velocity Vector and Surface Plot - Experiment 6. 141 - B -f 4 \ \ \ ? \ ' • * V \ \ * \ \ V 1 \ \ \ \ / / \ \ / i \ \ — i i V — — - 4 ^ ^ 0 MAXIMUM VELOCITY (M/S) = 0.477 3.13. Experimental Velocity Vector and Surface Plot - Experiment 7 142 M A X I M U M VELOCITY (M/S) = 0.388 Figure 3.14. Experimental Velocity Vector and Surface Plot - Experiment 8. 143 t • • >•> \ \ X - * \ \ ^ - ' J I > J \ \ ^ ^ / / I I \ N — - — • / 1 j N. N ^ ^ ^ ^ / * MAXIMUM VELOCITY (M/S) = 0.367 Figure 3.15. Experimental Velocity Vector and Surface Plot - Experiment 9. 144 f—1 n n i — =*——•— LJ 4 LJ l 4 P •> \ \ 9 \ 0 \ \ \ T \ \ / \ \ 1 \ \ / / 1 \ \ — s / — 0' 0 ^—. ~- — — — 4t- 0 MAXIMUM VELOCITY (M/S) = 0.443 Figure 3.16. Experimental Velocity Vector and Surface Plot - Experiment 10. 145 20 S 15 o 0> o. a. I 10 o l l 1 1 1 1 — o — o ° o — o o o ° o o . 1 1 1 1 1 1 40 80 120 160 200 240 Air flowrate (1/min) Figure 3.17. Variation of Air Holdup in Experimental Tank with Air Flowrate. 146 50 40 30r-* o w a 20 z. i o 1 1 1 1 1 1 1 ~ T — Slag fuming operation — a D ° ° • • » 1 I I 1 1 1 1 0 2 4 6 8 10 12 14 16 Modified Froude number Figure 3.18. Variation of Air Holdup in Experimental Tank and Zinc Slag Fuming Furnace with Modified Froude Number. 147 5.0 o X o» c U c 4.0 - 3.0 a> u c o 0) 2.0 .1 1 1 1 1 1 o o — o o — o o o o o 1 1 1 1 1 1 40 70 100 130 160 Air flowrate (l/mlh) 190 F i g u r e 3.19. Mean C e l l K i n e t i c Energy of Experimental Measurements as a Function of A i r Flowrate. 148 O >» o> •_ 0) c u «-0) o 0) 3.5 3.0 2.5 2.0 1 1 1 1 1 1 o O O o o o — — ; o ° o 1 1 1 1 1 1 4 0 7 0 100 130 160 Air f lowrate ( l / m i n ) 190 Figure 3.20. Mean Cell Kinetic Energy of Non-Surface Cells in Experimental Measurements as a function of Air Flowrate. 149 c 5.0 i — r — i — i — i — i — i — i r X ? 4.0 o> c Q> = 3.0| o c o tt) 2 2.01 J L _ J [ L 1 J I I L 7 II 15 Energy input rote (J/s) 19 23 Figure 3.21. Variation of Experimental Mean Cell Kinetic Energy with Air Input Energy. 150 Figure 3.22. Bubble Formation at Tuyeres, N p r , =0 .4 . Figure 3.23. Bubble Formation at Tuyeres Figure 3.24. Bubble Formation at Tuye 153 B u b b l e column Figure 4.1. Schematic D e s c r i p t i o n of Boundary Conditions Necessary to Describe the Flow i n the Experimental Tank. 154 4.2. Velocity Fluctuations in a Finite-Difference Cell Computed by SOLASMAC Method. 1 5 5 'fy • fy 'A A A, 'A A //// F i g u r e 4 . 3 . F i n i t e D i f f e r e n c e G r i d Used to Compute Flow i n Experimental Tank. 156 ( BEGIN ) INITIALIZE VARIABLES SET PRESSURE DISTRIBUTION TO HYDROSTATIC APPLY BOUNDARY CONDITIONS I SOLVE MOMENTUM EQUATIONS READ IN PREVIOUS RESULTS CALCULATE DIVERGENCE FOR EACH FULL CELL UPDATE PRESSUREy AND VELOCITIES MOVE FREE SURFACE INCREMENT TIME WRITE OUT RESULTS 0 <D c STOP Figure 4 . 4 . Flowchart of the SSMCR Program. 156 l a l t l o l i f * Variable* bat! In D i s t r i b u t i o n to B>4rotL«tir ( s. Apply BouDdary Cosditiooa Solve Equations 9 Calculate Divergence for Each F u l l C e l l Update Freeaure* V e l o c i t i e s Figure A.A. Flowchart of the SSMCR Program. 157 F i g u r e 4.5. Diagram of Free Surface L o c a t i o n and Movement Technique as used i n SSMCR. 1 5 8 E E E S * --^„. F F F Horizontal >- v S v E F \ \ \ \ E F • > F E * empty cell F« full cell S * surface cell Vertical \ E E \ F ' N N E F F S 4 5 degrees Figure 4.6. Recognition of Surface Orientation for Tangential Stress Condition. 159 1.0 cm / s I cm '///////////////////// I cm Figure 5 . 1 . Diagram of Problem used to Test SSMCR. 160 1 1 ' ' • • • • • 1 1 0.0 0.2 0.4 0.6 0.8 1.0 H, Vertical Distance (cm) 5.2. Agreement Between P r e d i c t i o n s Made by SSMCR and SOLASMAC Flow i n a Square C a v i t y . B B B B B B B B B \ \ \ * ' 4 + * * * t \ \ \ v * • - ' , i I \ A \ V - - * • • / H \ V ^ -~ — ^ 0- t MAXIMUM VELOCITY (M/S) = 0.600 Figure 5.3. Erroneous P r e d i c t i o n of Square C a v i t y Flow. If - B B - B B B B B B B \ \ \ h ' 4 * * + \ \ \ V - - 0 i i H \ \ ^ ^ — r *• / MAXIMUM VELOCITY (M/S) = 0.600 Figure 5 .4 . Correct Prediction of Square Cavity Flow. 163 / 1 -\ * e B B B B P-MAXIMUM VELOCITY (M/S) = 0.265 Figure 5 . 5 . Prediction of Experimental Flow Regime Made by Constant Velocity Condition, Mgff = 1 ° g/cm«s. 164 F i g u r e 5.6. P r e d i c t i o n of Experimental Flow Regime Made by Constant V e l o c i t y C o n d i t i o n , u g f f = 40 g/cm«s. 165 / / > ^ - « -i - i n 1 f / / j r « * « . K Lf— LJ 1 r > / t t / r »• * ? r * \ t * * r r \ * •* * * \ <+ 0 t \ +r Jf f 0 •» \ ^ *^ *r -* *r -* \ -\ - + * + MAXIMUM VELOCITY (M/S) 0.586 F i g u r e 5.7. P r e d i c t i o n of Experimental Flow Regime Made by Constant V e l o c i t y C o n d i t i o n , u f f • 400 g/cm»s. 166 -e-i I t \ \ / 4 * v ^ / MAXIMUM VELOCITY (M / S ) = 0.001 F i g u r e 5 .8 . P r e d i c t i o n of Experimental Flow Regime Made by V a r i a b l e Density C o n d i t i o n . 167 f \ \ \ ~B —B B B B 11 w •r *• MAXIMUM VELOCITY (M/S) = 0.218 F i g u r e 5.9. P r e d i c t i o n of Experimental Flow Regime Made by V a r i a b l e Density C o n d i t i o n w i t h L e f t Wall V e l o c i t y . 168 F i g u r e 5.10. P r e d i c t i o n of Experimental Flow Regime Made by Pulsed Boundary C o n d i t i o n . 169 Figure 5.11. Prediction of Experimental Flow Regime Made by Collapsing Surface Condition, T = 0.5 s. 170 \ \ t / t \ \ \ \ \ > r \ i \ \ \ \ / I i \ \ ^ / / i v if t V t F 1 Figure 5.12. Prediction of Experimental Flow Regime Made by Collapsing Surface Condition, T • 0 . 7 s . 171 Figure 6.1. Prediction of Flow Regime in a Copper Converter Made by Constant Velocity Boundary Condition. 172 F i g u r e 6 . 2 . P r e d i c t i o n o f F l o w R e g i m e i n a C o p p e r C o n v e r t e r M a d e b y C o l l a p s i n g S u r f a c e C o n d i t i o n . 173 MAXIMUM VELOCITY (M/S) = 0.001 Figure 6.3. P r e d i c t i o n of Flow Regime i n a Zinc Slag Fuming Furnace Made by Collapsing Surface Condition. 174 MAXIMUM VELOCITY (M/S) = 0.625 Figure 6.4. Prediction of Flow Regime in a Zinc Slag Fuming Furnace Made by Constant Velocity Condition. 175 -e B B - - B B -' - » - B B B B -MAXIMUM VELOCITY (M/S) = 0.062 Figure 6.5. P r e d i c t i o n of Flow Regime i n a Zinc Slag Fuming Furnace Made by Variable Density Condition. 

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