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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 t h i s thesis as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA October, 1986 (c) Andrew Shook, 1986  In  presenting  degree  at  this  the  thesis in  University of  partial  fulfilment  of  British Columbia, I agree  freely available for reference and study. I further copying  of  department publication  this or of  thesis for by  his  or  her  representatives.  requirements that the  for  an advanced  Library shall make it  agree that permission for extensive  scholarly purposes may be It  is  granted  by the  understood  that  this thesis for financial gain shall not be allowed without  permission.  Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  DE-6(3/81)  the  head of copying  my or  my written  ii  ABSTRACT  Physical  and mathematical  investigate liquid  The at  100  e x p e r i m e n t a l work c o n s i s t e d of water v e l o c i t y measurements made  a series  o f side-mounted  recirculation  indicate effect  f l o w d r i v e n by a h o r i z o n t a l l y i n j e c t e d g a s .  l o c a t i o n s w i t h i n a p l e x i g l a s s tank.  through water  m o d e l l i n g s t u d i e s have been performed t o  that of  increasing  velocity  t u y e r e s , and t h e e f f e c t  was observed.  t h e maximum water v e l o c i t y  bubbles  A i r was i n t r o d u c e d i n t o the tank  coalescing  from  The r e s u l t s  occurs  adjacent  a i r f l o w r a t e , and was found  o f a i r f l o w r a t e on o f the  experiments  a t t h e water s u r f a c e . tuyeres  was  observed  t o d i m i n i s h t h e water  The with  recirculation  rate.  The (MAC)  mathematical  technique  model employed  t o compute  fluid  flow  a variant with  a  o f the Marker  free  p r e d i c t i o n s i n d i c a t e t h a t the flow i n the e x p e r i m e n t a l by  water  qualitative  flowing  across  predictions  the f r e e of  the  surface.  flow  regimes  surface.  and C e l l The model  tank i s l a r g e l y d r i v e n  Based  on  in a  Peirce-Smith  c o n v e r t e r and a z i n c s l a g fuming furnace were made.  this  knowledge, copper  iii  TABLE OF CONTENTS Page ABSTRACT  i i  TABLE OF CONTENTS  •  i i i  LIST OF TABLES  vi  LIST OF FIGURES  v i i  NOMENCLATURE  xi  ACKNOWLEDGEMENTS  1.  2.  xiv  INTRODUCTION  1  1.1. Copper S m e l t i n g  1  1.2. The Copper C o n v e r t e r 1.2.1. H i s t o r y and Development 1.2.2. D i f f i c u l t i e s  4 4 4  1.3. The Z i n c S l a g Fuming Furnace  5  1.4. O b j e c t i v e s of t h e Work  6  LITERATURE REVIEW  7  2.1. Flow Regimes i n Non-Ferrous M e t a l l u r g i c a l R e a c t o r s  7  2.2. Bubble F o r m a t i o n I n v e s t i g a t i o n s  .....  10  2.3.  .  12 12  I n v e s t i g a t i o n s of Gas-Driven Flow 2.3.1. E x p e r i m e n t a l S t u d i e s 2.3.2. N u m e r i c a l S t u d i e s  3.  15  2.4. C o n c l u s i o n s  19  EXPERIMENTAL  20  3.1. O b j e c t i v e s  .  20  3.2. Apparatus  20  3.3. P r o c e d u r e  26  3.4. E x p e r i m e n t a l R e s u l t s  28  iv  TABLE OF CONTENTS (cont'd)  Page 3.4.1. 3.4.2. 3.4.3. 3.4.4. 3.4.5. 4.  Introduction Check o f the Two D i m e n s i o n a l i t y o f the F l o w V e l o c i t y Patterns S u r f a c e Shape Bath S t i r r i n g  THE MATHEMATICAL MODEL  43  4.1. G e n e r a l Statement of Problem 4.1.1. G o v e r n i n g E q u a t i o n s 4.1.2. Boundary C o n d i t i o n s  44 44 46  4.2. S e l e c t i o n of S o l u t i o n A l g o r i t h m  48  4.3. The SOLASMAC A l g o r i t h m  53  4.4. Treatment of Boundary C o n d i t i o n s  56  4.4.1. L o c a t i n g and Moving the F r e e S u r f a c e 4.4.2. Free S u r f a c e V e l o c i t i e s and P r e s s u r e s 4.4.2.1. Normal S t r e s s C o n d i t i o n 4.4.2.2. 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 4.4.3. R e c t a n g u l a r W a l l B o u n d a r i e s 4.4.4. Round Bottom Boundary 4.4.5. G a s - L i q u i d Boundary 5.  6.  28 30 34 36 37  ,  56 58 61 61 63 64 65  CALCULATIONS - COMPARISON WITH EXPERIMENTAL RESULTS  67  5.1. Constant  70  Vertical Velocity  5.2. V a r i a b l e D e n s i t y  73  5.3. P u l s e d Boundary  77  5.4. C o l l a p s i n g S u r f a c e  79  5.5. D i s c u s s i o n  83  INDUSTRIAL CALCULATIONS  86  6.1. F l u i d Flow i n a Copper C o n v e r t e r  86  6.1.1. Assumptions 6.1.2. M a t h e m a t i c a l 6.1.3. R e s u l t s  D e s c r i p t i o n o f G a s - L i q u i d I n t e r f a c e ...  86 87 88  V  TABLE OF CONTENTS (cont'd) Page 6.2.  7.  F l u i d Flow i n a Z i n c S l a g Fuming Furnace 6.2.1. Assumptions 6.2.2. M o d i f i c a t i o n s to Program 6.2.3. Mathematical D e s c r i p t i o n of Bubble Column 6.2.4. R e s u l t s  90 91 92 93 95  CONCLUSIONS  97  7.1.  Experimental  97  7.2.  C a l c u l a t i o n s - Agreement w i t h E x p e r i m e n t a l R e s u l t s  98  7.3.  Industrial Calculations  99  7.3.1. Flow i n a Copper C o n v e r t e r  99  7.3.2. Flow i n a Z i n c S l a g Fuming Furnace  99  7.4.  Recommendations f o r F u r t h e r 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  Table 3.2.  Values of Experiment  Table 3.3.  Ill  in Experimental Tank, 1  Minimum Mean Values of  112 of Experimental Runs  113  Chapter 4 Table 4.1.  Approximations Used for Rectangular Wall Boundaries  114  Effective Viscosity Values Predicted by Model of Sahai and Guthrie (44)  115  Chapter 5 Table 5.1. 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 Comparison of Results Calculated by Pulsed Boundary Condition with Experiments  119  Comparison of Results Calculated by Collapsing Surface Condition with Experiments  120  Data Used to Model Flow in Peirce-Smith Copper Converter  121  Data Used to Model Flow in Zinc Slag Fuming Furnace  122  Table 5.5. Table 5.6.  118  Chapter 6 Table Table  6.1. 6.2.  vii  LIST OP FIGURES Page Chapter 1 F i g u r e 1.1. F i g u r e 1.2.  Schematic Diagram of P e i r c e - S m i t h Copper Converter  124  Schematic  125  Diagram of Z i n c S l a g Fuming F u r n a c e  Chapter 2 F i g u r e 2.1.  F i g u r e 2.2.  E s t i m a t e of Flow i n a Copper C o n v e r t e r , T h e m e l i s e t a l . (3)  from 126  P r e d i c t e d Flow P r o f i l e i n Two-Phase R e g i o n of Copper C o n v e r t e r , from N a k a n i s h i and S z e k e l y ( 4 )  127  J e t B e h a v i o u r Diagram, from H o e f e l e and Brimacombe (7)  128  F i g u r e 3.1.  Schematic  129  F i g u r e 3.2.  Dimensions and C o n s t r u c t i o n of E x p e r i m e n t a l Tank  130  F i g u r e 3.3.  Diagram of L a s e r - D o p p l e r  131  F i g u r e 3.4.  V e l o c i t y Measurement L o c a t i o n s W i t h i n E x p e r i m e n t a l Tank  F i g u r e 2.3.  Chapter 3  F i g u r e 3.5.  Diagram of E x p e r i m e n t a l A p p a r a t u s  System  .  132  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  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....  134  F i g u r e 3.7.  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 and S u r f a c e P l o t Experiment 1 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 and S u r f a c e P l o t Experiment 2  136  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 and S u r f a c e P l o t Experiment 3  137  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 and S u r f a c e P l o t Experiment 4  138  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 and S u r f a c e P l o t Experiment 5  139  F i g u r e 3.8. F i g u r e 3.9.  F i g u r e 3.10.  F i g u r e 3.11.  135  viii 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 i n 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 , = 0.4  150  Figure 3.23. Bubble Formation at Tuyeres, N , = 2.3  151  Figure 3.24. Bubble Formation at Tuyeres, N , = 15.6  152  Fr  Fr  pr  Chapter 4  Figure 4.1.  Figure 4.2.  Schematic Description of Boundary Conditions Necessary to Describe the Flow i n the Experimental Tank  153  Velocity Fluctuations in a Finite-Difference Cell Computed by SOLASMAC Method  154  ix  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 t o Compute Flow i n Experimental  Tank  155  F i g u r e 4.4.  Flowchart  F i g u r e 4.5.  Diagram o f Free S u r f a c e L o c a t i o n and Movement Technique as used i n SSMCR R e c o g n i t i o n of S u r f a c e O r i e n t a t i o n f o r T a n g e n t i a l  157  Stress Condition  158  F i g u r e 5.1.  Diagram o f Problem used t o Test SSMCR  159  F i g u r e 5.2.  Agreement Between P r e d i c t i o n s Made by SSMCR and  F i g u r e 4.6.  of the SSMCR Program  156  Chapter 5  SOLASMAC f o r Flow i n a Square C a v i t y  160  F i g u r e 5.3.  E r r o n e o u s P r e d i c t i o n o f Square C a v i t y F l o w  161  F i g u r e 5.4.  C o r r e c t P r e d i c t i o n o f Square C a v i t y F l o w  162  F i g u r e 5.5.  P r e d i c t i o n o f E x p e r i m e n t a l 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 P r e d i c t i o n of E x p e r i m e n t a l F l o w 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 P r e d i c t i o n o f E x p e r i m e n t a l Flow Regime Made by  F i g u r e 5.6. F i g u r e 5.7.  C o n s t a n t V e l o c i t y C o n d i t i o n , \x^ff F i g u r e 5.8. F i g u r e 5.9.  163 164  = 400 g/cm»s  165  P r e d i c t i o n of E x p e r i m e n t a l Flow Regime Made by V a r i a b l e Density Condition  166  P r e d i c t i o n of E x p e r i m e n t a l F l o w Regime Made by V a r i a b l e Density Condition with L e f t Wall Velocity  167  F i g u r e 5.10. P r e d i c t i o n of E x p e r i m e n t a l P u l s e d Boundary C o n d i t i o n  F l o w Regime Made by 168  F i g u r e 5.11. P r e d i c t i o n of E x p e r i m e n t a l Flow Regime Made by 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 , T = 0.5 s  169  F i g u r e 5.12. P r e d i c t i o n of E x p e r i m e n t a l 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. Figure 6.2. Figure 6.3. Figure 6.4. Figure 6.5.  Prediction of Flow Regime in a Copper Converter Made by Constant Velocity Boundary Condition  171  Prediction of Flow Regime in a Copper Converter Made by Collapsing Surface Condition  172  Prediction of Flow Regime in a Zinc Slag Fuming Furnace Made by Collapsing Surface Condition  173  Prediction of Flow Regime in a Zinc Slag Fuming Furnace Made by Constant Velocity Condition  174  Prediction of Flow Regime in a Zinc Slag Fuming Furnace Made by Variable Density Condition  175  NOMENCLATURE  A r e a o f b u b b l e column (m ) = depth o f column x bubble forward  penetration  2 A r e a o f tank f a c e (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 a n g u l a r divergence  d e v i a t i o n (degrees)  ( s ^)  Buoyant energy i n p u t r a t e  (watts)  Mean c e l l k i n e t i c energy T o t a l energy i n p u t r a t e  (watts)  K i n e t i c energy i n p u t r a t e  (watts)  2 A c c e l e r a t i o n due t o g r a v i t y (9.8 m/s ) Depth o f tank (m) H e i g h t o f f r e e s u r f a c e above a r b i t r a r y datum (m) Column number (-) Row number (-) Number o f e x p e r i m e n t a l Number o f columns  c e l l s c o n t a i n i n g data  (-)  Number o f rows (-) Pressure (Pa)  3 A i r flowrate (m/s) Equivalent time ( s )  r a d i u s o f tank (m)  (-)  xii  u  Horizontal  V  Vertical fluid  V  Maximum v e l o c i t y  max  fluid  velocity  velocity  (m/s)  (m/s)  Transverse f l u i d  velocity  x  Horizontal axis  (-)  y  V e r t i c a l axis  z  Transverse axis  w  (m/s)  (m/s)  (-) (-)  Greek Symbols a  Gas f r a c t i o n (-)  Y  F r a c t i o n a l upwind d i f f e r e n c e parameter  e  Surface i n c l i n a t i o n ( d e g r e e s )  n  Viscosity  l  Vorticity (s" )  p  (g/cm«s) 1  3 D e n s i t y (kg/m  )  T  Stress  (Pa)  4>  Tuyere Diameter  Subscripts c  calculated  e  experimental  eff  effective  f  fluid  1  laminar  g  gas  t  turbulent  (m)  (-)  Superscripts t+At  next time s t e p  t  time dependent  Dimensionless Groups  2 N  Froude Number, u  Fr N_ , Fr  M o d i f i e d Froude Number,  2  u g Re  p x  m  S4>  N  /g»<b) 8  P  f  "  N o z z l e Reynolds Number, Cb V  H  O U  g 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  Sciences and Engineering Research Council.  from the Natural  1  1.  The side-mounted industry.  INTRODUCTION  i n j e c t i o n of a i r into a vessel through tuyeres  a number of h o r i z o n t a l  i s a common practice In the non-ferrous  In p a r t i c u l a r ,  side-blowing  metallurgical  of a i r i s c a r r i e d  out i n the  Peirce-Smith copper converter, and the zinc slag fuming process (1). contrasted  with  the top and bottom  blown vessels  This i s  currently used  in  the  ferrous industry (2).  Unlike  the case  research has been performed  of top and bottom  blowing,  relatively  on the description of the f l u i d  within side-blown metallurgical vessels (3,4,5).  flow  little regimes  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 i n d u s t r i a l  importance,  p a r t i c u l a r emphasis was placed on the description of the f l u i d flow p r o f i l e i n a Peirce-Smith  copper converter.  In addition however, predictions were  also made for the slag flow i n the zinc slag fuming furnace.  1.1.  Copper S m e l t i n g  The industrialized  production world.  of  copper  Used mainly  i s of great  in electrical  importance  to the  components, annual  world  copper production exceeds 8 m i l l i o n metric tons, with proven reserves of 550 m i l l i o n metric tons billion  dollar  (1982 Figures) (6).  industry:  approximately  In Canada, copper production i s a 650,000 metric  annually, and accounts for 2.4% of a l l exports (6).  tons  are produced  2  Throughout t h e w o r l d , most copper I s found ores, either Mineable  chalcoclte (C^S),  copper  ores  The o r e i s crushed usually  low-grade - 3 wt.% o r lower,  (CuS).  depending  S i n c e t h e e a r l y 20th c e n t u r y , copper has been produced  from these o r e s by t h e f o l l o w i n g  1.  c h a l c o p y r i t e (CuFeS2) o r c o v e l l i t e  are t y p i c a l l y  upon c u r r e n t p r i c e s .  I n t h e form o f s u l p h i d e  between  procedure:  and c o n c e n t r a t e d  15 and 35% copper,  by f l o t a t i o n .  This concentrate i s  the remainder  being  mainly  iron,  s u l p h u r and gangue. 2.  The c o n c e n t r a t e i s r o a s t e d e l i m i n a t i n g some o f t h e s u l p h u r as SO^-  3.  The c a l c i n e  i s melted  in a  reverberatory  furnace.  Slag  (containing  s i l i c a t e s , FeO and some copper and s u l p h u r ) i s removed, and t h e matte ( a m i x t u r e o f C ^ S and FeS) i s tapped o f f . 4.  The matte i s p l a c e d i n a copper c o n v e r t e r . the matte a t t h i s in  this  furnace  time.  Silica  f l u x i s a l s o added t o  The c o n v e r s i o n o f the matte t o m e t a l l i c  i s carried  out by b l o w i n g  a i r through  copper  the matte i n two  stages:  a) slagataklng:  The  i r o n s u l p h i d e i n t h e matte p r e f e r e n t i a l l y  o x i d i z e s t o FeO and  sulphur d i o x i d e :  FeS  +  3/2 0. 2  + FeO +•  +  S0„  L  (1-1)  3  This  oxidation  autogenously. through with  The  t h e melt  releases  sulphur  a  great  amount  dioxide, nitrogen,  of heat,  and u n r e a c t e d  and a r e drawn out o f t h e c o n v e r t e r mouth.  the s i l i c a  flux  t o form  a  slag  on  and  the surface  proceeds  oxygen  rise  The FeO combines of  the  bath.  P e r i o d i c a l l y , t h e s l a g i s poured out o f t h e c o n v e r t e r , more s i l i c a and matte a r e added, and t h e p r o c e s s i s c o n t i n u e d .  b) b l i s t e r making:  Once a l l o f t h e i r o n has been removed from t h e m e l t ,  the molten  copper s u l p h i d e s a r e o x i d i z e d t o copper by f u r t h e r b l o w i n g o f a i r :  Cu S 2  This  produces  "blister  +  copper"  0  2  >  which  Cu  +  S0  (1.2)  2  i s approximately  99% pure  w i t h the  remainder l a r g e l y oxygen and p r e c i o u s m e t a l s .  5.  The b l i s t e r copper i s n o r m a l l y reduced into  e l e c t r o d e s and r e f i n e d  copper,  and  "anode  slimes"  I n an anode f u r n a c e and then c a s t  electrolytically. from  which  any  This  yields  precious  99.98% pure  metals  can be  Over t h e y e a r s , many improvements have been made i n t h i s  smelting  extracted.  process.  The r o a s t i n g and r e v e r b e r a t o r y s t e p s have been e l i m i n a t e d i n the  new " f l a s h - s m e l t i n g " t e c h n i q u e s . the  sulphur  These methods u t i l i z e t h e h e a t i n g v a l u e o f  i n the ore to accomplish  melting.  They a r e t h e r e f o r e f a r more  4  energy e f f i c i e n t  than  earlier  methods.  Despite  such  general  improvements  however, the vast majority of the copper produced i n the world today  still  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 i n use today i s the Peirce-Smith arose  from  side-blown research  in the mid-19th  converters were found by  converter (Figure 1.1).  to be unsuccessful due  frozen copper matte.  attacked  by  the  silica  Within  a  design of this vessel  Bottom blown Bessemer-type  to the blockage of the tuyeres  As well, the refractory was slag.  In  successful tests on a side-blown lining.  century.  The  few  1905  found  Messrs. Peirce and  to be strongly Smith  conducted  c y l i n d r i c a l vessel with a basic refractory  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  consequently  i n the form of bubbles,  the  (7).  tuyeres  solidifies  to  form  Cooled  by  the  converter  at  low  pressure  and  the bath i s able to wash up against incoming  accretions which  a i r , matte  eventually  i n this  completely  region  block  the  5  tuyeres matte  (7). must  In order be  inefficiency,  to maintain  "punched" this  blowing  periodically.  punching  rates,  In  the  plug  a d d i t i o n to  of the t u y e r e s i s thought  of  its  frozen overall  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 "slop" uncontrollably. from  the  top  personnel material  of  the  i n the at  the  bath d e p t h s ,  T h i s may  result  converter  and  mouth may  may  i n the e j e c t i o n of molten matte  cause  immediate v i c i n i t y . converter  the matte i n the c o n v e r t e r  great  danger  Dust l o s s e s and  to  operating  the b u i l d  up  a l s o become e x c e s s i v e under  of  these  conditions (8).  1.3.  The  Zinc Slag Fuming Furnace  A diagram of the z i n c  slag  fuming f u r n a c e  i s shown i n F i g u r e  1.2.  Z i n c 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 c o n v e r t i n g process i n that  i t i n v o l v e s gas  injection  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 t u y e r e s  i n t o a molten bath.  Zinc  slag  fuming  is  a  s m e l t i n g ( c o n t a i n i n g up t o 20% ZnO) i n t o the f u r n a c e .  The  reduced  process  whereby  i s reduced  molten  slag  from  lead  by the a c t i o n of c o a l i n j e c t e d  z i n c l e a v e s the b a t h as a vapour, where i t i s  r e - o x i d i z e d , and removed by the f u r n a c e o f f - g a s e s .  The studied bath  kinetics  (9,10).  circulation  of  the  However, t h e velocity  has  models d e s c r i b i n g t h i s p r o c e s s  slag  fuming  general limited (9,10).  furnace  have  l a c k of a c c u r a t e the  been e x t e n s i v e l y  i n f o r m a t i o n on  applicability  of  the  mathematical  6  1.4.  Objectives of the Work  The develop  primary  o b j e c t i v e o f the r e s e a r c h performed i n t h i s work was t o i  a mathematical model  t h a t was  capable  o f d e s c r i b i n g 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 p r o c e s s e s .  For  the case  of the copper  converter,  such  a model has s e v e r a l  obvious a p p l i c a t i o n s .  1.  To  yield  insights  c o n v e n t i o n a l copper 2.  To p r o v i d e  of  flow  patterns  and  stirring  f o r s o p h i s t i c a t e d heat  and mass t r a n s f e r models o f  To date, the development o f such models has been hampered  by a l a c k o f i n f o r m a t i o n on the bath c i r c u l a t i o n v e l o c i t y . models  within  converters.  i n p u t data  the c o n v e r t e r .  the g e n e r a l  can p r o v i d e  information  to optimize  I n t u r n , such  the p r o d u c t i o n  of current  copper c o n v e r t e r s , and a i d i n the d e s i g n o f more e f f i c i e n t v e s s e l s . 3.  To study and p o s s i b l y suggest  Similarly, furnace  will  recirculation this  has  conditions.  ways o f p r e v e n t i n g bath  slopping.  the a p p l i c a t i o n of such a model t o t h e z i n c s l a g  provide  information,  velocities.  application  previously  When c o n s i d e r e d to  the  along  optimization  unavailable,  on  with  models  of  kinetic  the  furnace  fuming  the  bath  (9,10)  operating  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  Peirce-Smith copper matte converter and the Zinc Slag-Fuming furnace. Injection  into  a liquid  bath  i s the central  in the As gas  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  i n water and liquid  metals. 3.  Investigations of the flow regimes i n 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  w i l l be discussed in detail in Chapter 4.  2.1.  Flow Regimes i n Non-Ferrous M e t a l l u r g i c a l Reactors  Among the f i r s t to consider the fluid flow fields in copper matte converting  was Themelis  studies, Themelis  et a l . (3).  et a l . estimated  Based  the fluid  converter to be as shown in Figure 2.1.  on air-water  jet injection  flow regime within a copper  8  Based on the assumptions (4) formulated a f l u i d of  their  of Theraelis et a l . , N a k a n i s h i and S z e k e l y  f l o w model of domain "A"  calculations  are  shown  in  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  Figure  i n F i g u r e 2.1. 2.2,  and  show  The  results  the  maximum  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 showed t h a t the model of Themelis  et a l . was  o f an a i r j e t i n l i q u i d m e t a l .  The  al.  found  i n water  (20  a d d i t i o n , bubbles little  degrees)  was  forward p e n e t r a t i o n .  j e t cone a n g l e measured by Themelis to  bubbles  c o n v e r t e r showed  t h a t found  iron.  155  degrees  i n mercury.  t h a t the  by H o e f e l e and Brimacombe (7) on  a i r enters t h i s  vessel  i n the form  occurring.  The  The  bubble  thought  f o r m a t i o n f r e q u e n c y of 10 s * agreed  a c c o u s t i c a l l y by I r o n s and G u t h r i e (13) i n a bath of  However,  d i a m e t e r s of 0.40  In  2.2.  - at a f o r m a t i o n r a t e of 10 s ^ - and not as a j e t as was  Theraelis et a l . ( 3 ) . with  be  et  These f i n d i n g s threw doubt upon the a c c u r a c y o f  I n d u s t r i a l measurements performed Nickel  unable t o p r e d i c t the behaviour  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 v e r y  the d e s c r i p t i o n shown i n F i g u r e  a  (12)  this t o 0.68  low m)  frequency  indicated  large  of by  well  liquid  bubbles  (estimated  and hence the p o s s i b i l i t y of bubble  coalescence  l a b o r a t o r y measurements of H o e f e l e and Brimacombe (7) of gas  i n j e c t i o n i n t o w a t e r , mercury and Z i n c ( 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 t h a t the t r a n s i t i o n between b u b b l i n g and j e t t i n g phenomena c o u l d be r e l a t e d to the modified  Froude number and  the r a t i o  between the gas  and  liquid  densities.  T h i s 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 W r a i t h and  Chalkley  (14))  is illustrated  i n F i g u r e 2.3.  Bubbling  behaviour  is  9  clearly for  p r e d i c t e d f o r matte c o n v e r t i n g .  the  slag  fuming  furnace  on  Plotting  Figure  2.3  r e l e v a n t o p e r a t i o n a l data  indicates  that  this  too  is  o p e r a t i n g i n a b u b b l i n g mode.  A mathematical study  the  reaction  kinetics  converter.  A  Davidson  a l . (16)  et  recirculation  model f o r m u l a t e d by Ashman et a l . (15) was  modified  oxygen u t i l i z a t i o n  v e r s i o n of  was  velocity  and  the bubble  employed.  (upon  which  in a  copper  f o r m a t i o n model proposed  Data was bubble  efficiency  used to  not  available  formation  f o r the  depended  by bath  strongly).  T h e r e f o r e , t h i s parameter was v a r i e d i n d e p e n d e n t l y u n t i l the p r e d i c t e d bubble formation frequency Brimacorabe  (7).  was  a p p r o x i m a t e l y e q u a l to t h a t measured by H o e f e l e  and  U s i n g t h i s method, the v e r t i c a l v e l o c i t y o f the l i q u i d matte  i n the r e g i o n o f the t u y e r e s 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 o p e r a t i n g copper  converter  behaviour  and  i n these  a  zinc  slag  fuming  furnace  v e s s e l s i s more complex  than  A n a l y s i s of t u y e r e pressure' measurements i n d i c a t e d behaves d i f f e r e n t l y  showed was  that  forming  t h a t the copper  at the t u y e r e s of a n e w l y - l i n e d copper  to r i s e  i n d e p e n d e n t l y , and  not  However, a f t e r s e v e r a l heats adjacent  to i n t e r f e r e  arising  from  tuyeres  forming  a l a r g e , u n s t a b l e gas  bubbles  i n the s l a g fuming  were  found  envelope  s ^.  lining.  w i t h one  r e f r a c t o r y wear) the interact  i n the r e g i o n of  f u r n a c e were found  t e s t i n g c o n d i t i o n s at a frequency of 5-6  to  converter The  c o n v e r t e r were shown  significantly  (and c o r r e s p o n d i n g  bubbling  previously believed.  depending upon the s t a t e of i t s r e f r a c t o r y  bubbles  the  with  one  another. bubbles another,  the t u y e r e s .  The  to a c t i n d e p e n d e n t l y under a l l  10  Unfortunately, fluid  flow  regimes  no  studies  in either  were  found  that  the copper  matte  converter or  fuming f u r n a c e as has been done f o r a r g o n - s t i r r e d  2.2.  Bubble Formation  The extensively aqueous  systems  Themelis,  et  significant  of  The  have  the  complete  the z i n c  slag  ladles.  Investigations  behaviour  studied.  predicted  bubbles  earlier  been  a l . (3)  gas  and  injected  into  water  has  been  l i t e r a t u r e c o n c e r n i n g bubble f o r m a t i o n i n  thoroughly reviewed Clift,  Grace  and  by  Kumar  Weber  and  (19).  Kuloor (18),  Among  the  more  s t u d i e s were those performed by D a v i d s o n and co-workers (16) and  W r a i t h and C h a l k l e y ( 1 4 ) .  The b e h a v i o u r of a i r bubbles i n j e c t e d t u y e r e ) has been summarized  1.  w i t h gas  (from a s i n g l e  by O r y a l l (11) as f o l l o w s :  At v e r y low a i r f l o w r a t e s invariant  into a liquid  flow.  (N  R g  < 500)  t h e b u b b l e volume i s r e l a t i v e l y  T h e r e f o r e , the bubble  f r e q u e n c y of f o r m a t i o n  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 l o w - f l o w 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  ^  remains increases  Rg  < 2100), the b u b b l e f o r m a t i o n frequency  a l m o s t c o n s t a n t ( a t a p p r o x i m a t e l y 8-10 with  increasing  a i r flowrate  i n this  s ^ ) , hence bubble volume regime.  Simple  models  (15) have been f o r m u l a t e d (based o n l y upon a b a l a n c e of buoyancy and  inertia)  experiment, significant.  yielding though  formulae  nozzle  f o r bubble  diameter  volume  effects  that  have  force  agree w e l l  been  seen  to  with be  11  3.  Increasing a i r flowrates further causes rising  bubbles.  Wraith  and  significant  Chalkely (14)  and  i n t e r a c t i o n between  Nilmani  (20,21) have investigated the e f f e c t s of bubble-bubble different  effects,  termed  "binary coalescence" and  and  Robertson  interaction.  Two  "stem coalescence"  have been found to occur at low and high a i r flowrates respectively.  The  transition  between these  gas  momentum.  Stem coalescence occurs when the bubbles  two  types of behaviour  i s determined  by  elongate and  form  columns of a i r that reach the surface and break up unpredictably. 4.  At  higher flowrates s t i l l ,  (N„ > 10,000, N_ > 1000) Re rr  the b u b b l i n g  behaviour ceases altogether, and the gas enters the l i q u i d as a turbulent jet. I t i s generally agreed that the t r a n s i t i o n between these types of behaviour are dependent upon increasing gas momentum. number (11) was  The  nozzle Reynolds  i n i t i a l l y used to predict the t r a n s i t i o n between "bubbling"  and j e t t i n g phenomena.  As mentioned  above, Hoefele and Brimacombe (7) used  the modified Froude Number as an indicator of this gas momentum e f f e c t , and successfully regimes. (14)  predicted  the  transition  between  "bubbling"  and  "jetting"  The i n j e c t i o n number has been used s i m i l a r l y by Wraith and Chalkely  to correlate  the effect  water and l i q u i d metal.  of gas momentum for a i r i n j e c t i o n  into  both  12  2.3.  Investigations of Gas-Driven F l u i d 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  al.  (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 water-filled anemometry  plexiglass model of a ladle, and  photographic  techniques.  by  field  a combination  Air was  of  introduced  in a  hot-wire 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 their  and Guthrie (27,28,29) employed two  study of the fluid  primarily a liquid  flow generated  physical models in  while tapping ladles.  injection study, the effect of air entrainment  fluid velocity distribution was  Although on the  observed.  Szekely, Dilawari and Metz (30) also used a physical model of a ladle, but employed a motor-driven means  of  generating  the  belt, rather than gas bubbles, as 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. corresponding  Two air flowrates were considered, 12.3 Nl/min and 25 Nl/min, to (modified) Froude Numbers of .021 and  .080  respectively.  14  At each flowrate, the water v e l o c i t y i n the plexiglas model was measured at 26 d i f f e r e n t l o c a t i o n s .  Hsiao, Chang and Lehner (32) measured gas plume v e l o c i t i e s i n a 1/7 scale water model of a l a d l e .  The maximum water v e l o c i t y was  found to occur  at the f l u i d surface and was found to be related l i n e a r l y to the cube root of the a i r i n j e c t i o n r a t e .  A comprehensive analysis of g a s - s t i r r i n g was performed Guthrie (33,34) who ladle.  The  directed  carried out experiments  gas was  lance, at  by Sahai and  i n a .17 scale model of a 150 ton  introduced into the tank through an immersed, downward a Modified Froude Number of  about  650.  The  p r o f i l e within the tank was measured by a photographic technique: of  small seed p a r t i c l e s within the tank was  their  velocity  was  assumed  to be  the  velocity  the motion  recorded at d i f f e r e n t times, and  same as the water v e l o c i t y .  These  measurements were c a r r i e d out i n 106 d i f f e r e n t locations i n the tank.  Oesters, Dromer, and measure the v e l o c i t y f i e l d  Kepura (35) employed hot wire anemometry to  i n a water model of a l a d l e .  (through a perforated plate) and top blowing An  equation based  shown  to  give  a  on good  the studies of Hsiao estimate  i n j e c t i o n rates were studied:  of  the  3 1/s and  Froude numbers of 2 and 32 respectively. the  ladle  as  a  (through a lance) were studied. et a l . (21) was  velocity 1.2  i n the  formulated  gas  plume.  and Two  1/s, corresponding to modified  In addition, the "mixing time" i n  function of a i r flowrate was  flowrate asymtotically to a minimum.  Both bottom blowing  shown to decrease  with a i r  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 i s directly related various air injection rates. 50, this shear stress was  to the velocity of the fluid) at  At a modified Froude Number of approximately  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 a l . (38), Szekely and Asai (24,25) formulated a mathematical model of the turbulent recirculating flow fields in an inert-gas stirred ladle. the Kolmogorov-Prandtl turbulent viscosity. vorticity.  single The  Two dimensional flow was assumed and  equation model was  solution was  performed  used  to calculate  in stream  the  function and  The free surface was assumed to be horizontal, and to be of zero  16  vorticity.  Reasonable  qualitative  experimental  observations,  although  agreement  was  quantitative  found  with  visual  comparisons  of  the  calculated flow field with experimental results were not made.  In the mathematical segment of the work performed by Szekely et a l . (26), the k-W turbulent  technique proposed  viscosity  experimental ladle.  i n a two  by Spalding (39) was used  to model the  dimensional  model  mathematical  of an  They neglected free surface effects, and considered the  bubble column to apply a constant upward velocity to adjacent f l u i d .  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. Spalding  (41), was  used  to  An algebraic model, proposed calculate  the  turbulent  by Pun and  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 s l i p . 2.  Slip.  The air moves with the same velocity as the liquid.  The bubbles move through the liquid at their terminal velocity.  17  In  a l l cases,  the  non-retarding wall.  free  surface was  considered  to  be  a  horizontal,  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 Kaddah and  neglecting  Grevet  deformation  (31,42,43) employed  of  the free  the improved  surface,  Szekely, E l  approximation of the  bubble boundary suggested by Deb Roy et a l . (40) and calculated the flow regime in a gas-agitated system. both the k-W  model and the algebraic model of Pun and Spalding (41).  algebraic model was expensive  Velocity fields were calculated based on  found to yield  in computer  time.  The  acceptable results, and was much less  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 turbulent  model  jets).  mathematical  o f Pun and S p a l d i n g I n two  description  succeeding  ( 4 1 ) was based  papers  (33,34)  on t h e a c t i o n o f  they  o f t h e b u b b l i n g mechanism and developed  f o r mean b a t h 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 . they  formulated  a t u r b u l e n t model, based  SIMPLE a l g o r i t h m . with  examined t h e  T h i s model was found  on P a t a n k e r  a relation In addition,  and S p a l d i n g ' s (45)  t o g i v e good q u a l i t a t i v e  agreement  experiments.  The Salcudean, written  first  to  Low, Hurda  that  deformation  used  examine  and G u t h r i e  a Marker  of the free  boundary suggested  the e f f e c t s (46).  and C e l l  surface.  of  the free  A three  (MAC) t y p e  surface  dimensional method  The d e s c r i p t i o n  were  model was  t o d e s c r i b e the  o f the b u b b l e - l i q u i d  by Deb Roy et a l . (40) was employed as was t h e v i s c o s i t y  model o f Pun and S p a l d i n g ( 4 1 ) . C a l c u l a t i o n s were performed f o r s y m m e t r i c a l and  o f f - c e n t r e d gas i n j e c t i o n .  dimensional dimensional,  calculations steady  Reasonable agreement was found both w i t h two  ( 4 0 ) and w i t h  s t a t e , t u r b u l e n t model f o r m u l a t e d  (47) was a l s o compared t o these  Salcudean, extended  these  experimental  by Salcudean  A  three  and Wong  calculations.  L a i and G u t h r i e  calculations  results.  to  ( 4 8 ) and Mazuradar and G u t h r i e (49)  compare  v i s c o s i t y models proposed by S a h a i  the accuracy  of  the t u r b u l e n t  and G u t h r i e ( 4 4 ) and Deb Roy et a l . (40)  w i t h t h e k-e model of Jones and Launder ( 5 0 ) . A l l models were compared w i t h the  experimental  results  o f Oeters  et a l . ( 3 5 ) .  Surprisingly,  the f l u i d  p r o f i l e s p r e d i c t e d by t h e a l g e b r a i c models were c l o s e r t o e x p e r i m e n t a l than were t h e r e s u l t s c a l c u l a t e d  from t h e k-e model.  T h i s was thought  values t o 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 a i r 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 stirring  effect of bubbles  i n liquid  have been used to describe the ladles.  The most successful of  these to date has been the variable density or void fraction model of Deb Roy et a l . (40). 4. Independent air-injection studies by Hsiao et a l . (32) and Bulson (22), under very different  conditions, have shown that the maximum liquid  velocity frequently occurs at the surface of the vessel. may be related to the cube root of the air injection rate.  This velocity  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  schematically in Figure 3.1.  investigation is shown  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  following c r i t e r i a :  1.  Rectangular in cross section.  2.  Air injected from a series of horizontal side-mounted tuyeres.  3.  Open at the top.  by the  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. computatational  domain  allows  finite-difference  A rectangular  calculations (which are  carried out in a rectangular grid) to be performed easily - therefore a rectangular tank i s 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. laser-doppler  measurements  For example, i t i s hard to obtain  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 with the metallurgical processes of interest.  to achieve  similarity  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 permit  the  flowrate.  sought in the experimental tank that would prevent or  coalescence  of bubbles  Preliminary  at adjacent  experiments  were  tuyeres depending on air  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 i s discussed in Chapter 6.  Note that exact dimensional furnace  or  a  copper  converter was  similarity with a zinc slag fuming 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. the aforementioned  In fact,  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  regimes within a copper  converter and  a zinc  slag  model of the flow  fuming furnace.  In  particular, a means of characterizing the interaction of air bubbles with a  23  surrounding  liquid  dimensions  of  facilitated  was  the  velocity  this  the  does  not  of  the  which  exists between  this  1.  (if  only  the  gas  slag  gas  a l l  of  the  of  that  and  these enters  was  behaviour in  bubbles  exist  interest,  profiles  were  differences,  throughout  these  occurred  industrial the  they  not  of  a l l  that  the  with  does  velocity  maintained  that  modelling.  processes  Despite  occurring  these  that  those  bubbles  this  fuming  that  the  and  in  the  cases.  surrounding  the  the  study  converting  vessel  a in  with  for  liquid the  the  of  reacting  will  s t i l l two  bath form  but  be  made  It  liquid  within  rather  Despite liquid and  there  mechanism  such  obvious  metals,  and  non-reacting  generally  this  that  interaction  f l u i d .  characteristics  experimental  processes,  of  surrounding  interaction  copper  measurements  industrially,  similarlity  non-wetting  liquid  velocity  occurring  qualitatively)  as  measured  physical  concerned.  claimed to  between  results  In  not  identical  difference  zinc  of  bubble  provided  similarity  industrial  Number  actual  mathematical  physical  the  the  important,  processes.  the  interaction  not  two  that  Froude  to  is  the  these  that  Therefore,  subsequent  rigid  and  similar  study  is  complications  the  was  the  It are  to  were  and  imply  modified  tank  primarily  tank  tank  indicating  experimental  tank  because  necessarily  experiments,  with  simply  inapplicable  similarity  importance.  measurements  experimental  completely  Is  primary  experimental  However, between  of  the  gases,  applicable  to  reasons:  is of  stirred bubbles.  wholly Thus  by  incoming  even  though  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  calculations  based  processes,  simply  on  with  assumptions  greater about  confidence  the  than  bubble-stirring  mechanism.  With these conclusions in mind, the internal dimensions plexiglass tank were designed to be 30 cm x 20 cm.  of the  The tank was made 60 cm  t a 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 in Figure 3.2. hoseclamps.  introduced into these tuyeres through the manifolds shown  These manifolds were constructed of 3/8 i n . PVC tubing and  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  Spectra-Physics  Helium-Neon  velocimeter laser  electronics manufactured by TSI  (LDV)  connected  Inc.  The  to  consisted optics  and  of  a  15  mW  data-collection  detailed operation of an LDV  has  been outlined elsewhere (65,66) and w i l l not be discussed here.  The  v e l o c i t y of the water i n the tank was  of space 1.8 3.3).  The  tank was  cm long, defined  by the i n t e r s e c t i o n of two  laser beams (Figure  value recorded as the water v e l o c i t y at a p a r t i c u l a r point i n the actually  anywhere within measuring  measured within a region  one  an  average of  t h i s 1.8 component  512  separate v e l o c i t y readings taken from  cm measuring length. of  the  fluid  The  velocity  LDV  at  was  any  only capable of  one  time,  (either  horizontal or v e r t i c a l ) but the orientation of the r e f r a c t i n g optics could  be  altered to allow l a t e r measurement of the other component.  The  entire  laser-doppler  apparatus  computer-controlled traversing table.  This table was  could  of  position  the  measuring  volume  the  LDV  was  c o l l e c t i o n was  within the tank, the LDV an Apple H e and  printed  sent 512  microcomputer. the result (along  the screen and  also  The  performed  by  on  a  constructed so that i t  at  cross-section of the tank with an error of less than 1.0  Data  mounted  any  position  in  the  mm.  computer.  At  each  point  independent water v e l o c i t y measurements to computer then averaged these measurements,  with the  standard deviation of the points) on  saved these values on disk.  x  26  3.3.  Procedure  The  laser-doppler  measurements were  carried  out i n the central  v e r t i c a l plane of the experimental tank, i n a 10 x 10 grid with points spaced 3.0 cm apart (Figure 3.4).  Each separate square region within this grid i s  termed a " c e l l " , due to the s i m i l a r i t y between this experimental mesh, and a f i n i t e difference g r i d . of  each  Velocity measurements were c a r r i e d out i n the centre  of these c e l l s ,  as shown.  The numbered  cells  correspond to the  measurement locations mentioned i n Figures 3.5 and 3.6.  Velocity  measurements  were  carried  out  at  ten  separate a i r  flowrates, ranging from 38 to 220 std 1/min. (Table 3.1).  Due vertical  fluid  to the i n a b i l i t y velocity  run involved two complete sweep  the horizontal  adjusted  to measure  of the LDV  to measure both horizontal and  components simultaneously, a complete  experimental  sweeps over this mesh of 100 c e l l s .  On the f i r s t  component  was  the v e r t i c a l  measured, and  component.  The  the optics second  were  then  sweep was  then  performed, measuring the v e r t i c a l water v e l o c i t y i n each of the 100 positions i n the tank.  The experimental procedure was therefore very straightforward:  1.  The tank was f i l l e d distilled  water.  to i t s desired operating l e v e l (always 30.0 cm) with  27  2.  Seed p a r t i c l e s (small p l a s t i c spheres measuring 2.0 x 10  m i n diameter)  of neutral buoyancy were added to the water i n the tank. 3.  The tank  was  positioned  perpendicular  to the l a s e r ,  and i t s exact  position r e l a t i v e to the LDV was recorded. 4.  The a i r 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 c o l l e c t i o n 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  i n the 100 point  scan. 8.  The LDV made 512 separate measurements of the ( h o r i z o n t a l or v e r t i c a l ) f l u i d v e l o c i t y at t h i s new l o c a t i o n .  9.  These values were transferred stored  the mean  and standard  to the computer, which averaged them and deviation  on a floppy  diskette.  The  computer then calculated the next location f o r the traversing table, and gave these instructions to the computer c o n t r o l l i n g the table.  Steps 7 to 9 were repeated as necessary u n t i l v e l o c i t y measurements had been taken i n each of the 100 locations.  28  Once both v e l o c i t y traverses had been completed, the shape of the water surface was traced onto a piece of clear p l a s t i c . was  later  digitized,  and stored  on a mainframe  This surface shape  computer  along  with the  v e l o c i t y data, allowing l a t e r analysis and p l o t t i n g .  The  only  procedure arose  serious  from  difficulty  encountered  the presence of smaller  water near the surface  and l e f t  wall.  i n implementing  a i r bubbles remaining  this i n the  These bubbles randomly blocked and  scattered the l i g h t from the laser beam, sometimes causing a very long period of  time to elapse  Further,  this  before  scattering  512 water v e l o c i t y measurements could of  the laser  could  cause  be taken.  spurious  velocity  measurements i f the LDV happened to measure the v e l o c i t y of a passing bubble, rather than the surrounding  water.  This problem was overcome by introducing  very small and p r e c i s e l y sized seed p a r t i c l e s into the water tank. enhanced  signal  quality,  and permitted  the LDV  This then  to d i s t i n g u i s h between  p a r t i c l e s and bubbles.  3.4.  Experimental  3.4.1.  Results  Introduction  In addition to the water v e l o c i t y and surface p r o f i l e measurements, a number of general observations have been made on the experimental  system as  a whole:  1.  The a i r was  seen  flowrates tested. modified^ F^yide  to enter  the tank  i n the form  of bubbles  This was expected since even the maximum  at  all  experimental  number used (15.5) was well within the bubbling  regime  29  As  designed,  adjacent  flowrates.  However,  that  flowrates  at  air  minute, and  bubbles  did  not  modified  The  wall,  Froude  The  tank  were  being  taken.  were  made  This  vibration  its  to  worst)  and  it for  a)  There  no  b)  Oscillations  were  variation  merely c)  Only due  the  in  this  serve  to  would  be  placed  into  90  standard  interfered  the  low  with  This  air  Indicate  litres  one  per  another,  corresponded  to  a  the  that  it  x-axis  have  a  the  the  left  measurements  was  table.  not  securely  Various  completely  the  velocity  the  y-axis  attempts  successful.  approximately  very  depth  of  Harmonic length  about  Assuming  tank  of  about  measurements  chapter).  of  velocity  was  affected  intersection  motion  none  oscillations  this  around  ±  2  mm  at  measurements  reasons:  velocity  extend  the  supporting but  region  while  deflection  through  the  a  the  bath.  because  action, (a  to  slightly  on  severe  averaging  small.  cm  occurred  flow  by  at  observations  columns.  confined  oscillate  significant  defined  the  to  be  following  about  visual  tuyeres  discrete  unlikely  the  vibrational to  3  not  Further,  section  2  this  is  of  below).  to  This  was  and  significantly  approximately  adjacent  seen  simply  interact  2.32.  to  prevent  significantly  region  of  was  was  than  separate,  number  not  photographs  from  observed  but  did  greater  penetrating  was  anchored,  in  column  only  both  rising  rise  bubble  tuyeres  the  velocity that  two  of  small the  were  motion  effect tank  made  laser  (Figure  about  since  is  small  along  beams  a  (see  the  3.1). the (see  1.8  cm  Apparatus  x-axis  would  slightly. z-axis  would  measurement  these  have  an  data,  oscillations  effect,  even  occur  this  however effect  symmetrically  30  about  the  z-axis,  distributed  velocity  measurements  would  be  evenly  i n the region within 2 millimetres to either side of the  desired p o s i t i o n . was  the  Since a v e l o c i t y measurement at a p a r t i c u l a r point  used to represent the average v e l o c i t y of the f l u i d for 15 mm i n  each  direction,  i t i s unlikely  that  this  effect  was  significant,  compared to the coarseness of the experimental data g r i d .  3*4.2.  Check of the Two Dimensionality of the Flow  Before  carrying  out  any  of the complete  velocity  checks were made on the v a r i a t i o n on flow through  scans, several  the width of the tank.  From the point of view of subsequent computer modelling, i t would be i d e a l i f the  horizontal  and  vertical  velocity  components  measurement locations were constant at any point  at  each  of  the  i n the tank width.  100 This  would be the case of t r u l y two-dimensional behaviour.  However, there were two reasons for suspecting that this type of behaviour would not be observed:  1.  The e f f e c t of the walls of the tank would be to retard the nearby f l u i d . Thus, one would expect the water v e l o c i t y to be lower i n the region of the walls than i n the centre of the f l u i d .  2.  The s t i r r i n g of the tank was caused by bubbles injected from a number of adjacent  tuyeres. Any  s i g n i f i c a n t v a r i a t i o n i n the a i r flowrates among  these tuyeres could cause a corresponding v a r i a t i o n i n the l o c a l velocity.  fluid  31  To  quantify these  effects,  the  v e r t i c a l components with tank width was i n the tank (Figure 3.4). there i s , i n f a c t , noticeable tank  and  the  of  horizontal  and  measured at four separate locations  When plotted (Figures 3.5  as expected  variation  of  and 3.6)  some v a r i a t i o n of flow with width.  than at the walls.  velocity  variation  The  they show that wall effect i s  the v e l o c i t i e s are higher i n the centre of the However, the flatness of the p r o f i l e s  10%)  indicates  that  this  effect  (a maximum  is slight.  More  importantly, these p r o f i l e s are r e l a t i v e l y symmetrical - i n d i c a t i n g that the a i r flowrates through the tuyeres were equal.  Because of this s l i g h t v e l o c i t y v a r i a t i o n , a l l v e l o c i t y scans were performed was  i n the c e n t r a l v e r t i c a l plane of the tank, where the wall effect  least s i g n i f i c a n t .  The location of the measuring volume that was used i n  these scans i s c l e a r l y seen i n Figures 3.5 and 3.6 (the dashed l i n e ) .  Another,  more rigorous, method of checking the two dimensionality  of the flow i s to apply the p r i n c i p l e of conservation of mass to the measured velocity  profiles.  measurements had  That  i s , once  been made at  a l l vertical  and  horizontal  a given a i r flowrate, the  continuity equation was applied to each experimental c e l l . and  vertical  mass flows  into  each c e l l  were equal  two  velocity  dimensional  I f the horizontal  to the  horizontal  and  v e r t i c a l flows out, then there would be no reason to suspect any transverse flow.  Stated  mathematically,  for  dimensional continuity equation i s :  an  incompressible  fluid,  the  three  32  |E  SS.  +  5x  Applying  a backward  experimental  u  finite  +  J£ dz  dy  difference  o  «  (3.1)  approximation,  t h i s becomes (for an  cell):  ijk" i-l,jk Ax U  v +  ijk" ij-l,k Ay V  w +  i j k " ij,k-1 Az W  (3.2)  m  Therefore i f the summation:  ~  U  ±  Ax  - '^  +  1  ^  " ^ J -  (3.3)  1  AY  i s everywhere e q u a l to zero, then —— = 0 and  the flow i s confined to the  QZ  v e r t i c a l plane.  Applying  Equation  plot of Figure 3.7 apparent  (3.3)  to several c e l l s  i n the v e l o c i t y vector  y i e l d s the values shown i n Table 3.1.  I t i s immediately  that Equation (3.3) does not, In f a c t , equal zero at a l l locations.  However, even with t r u l y two-dimensional  experimental flow, there are several  reasons why one might expect Equation (3.3) to be  1.  The  o v e r a l l density.  equation  assumes  an  non-zero.  incompressible  fluid  of  constant  Both v i s u a l observation and photographs have confirmed that  there are several regions of the tank (such as near the surface or the bubble column) where the a i r i s entrained i n the l i q u i d phase. the This  local  density could vary  effect  is  shown  by  significantly,  considering the  invalidating  variable  In these regions, Equation  density  form  (3.3). of  the  33  continuity equation:  dp.  o( pu)  +  at  by Equation  the v a l u e s of  0  /  0  3  4  )  ay  Equation (3.3) assumes p • constant. calculated  5(pv)  +  ax  I f this i s not the case, then the value  (3.3) i s erroneous.  Unfortunately, without knowing  •^-, i t i s i m p o s s i b l e to determine  the " t r u e " two  D  a t ' ax' ay dimensional  divergence  for  these  cells.  Therefore  any  subsequent  consideration of the two dimensionality of the flow i n the experimental tank w i l l be confined to regions uncontaminated by a i r bubbles. 2. from  the  A further reason why  coarseness  of  the  Equation  experimental  (3.3) might be non-zero arises mesh.  Equation  (3.3)  is  a  f i n i t e - d i f f e r e n c e approximation to the true r e l a t i o n s h i p :  +  ax and  this  approximation  rapidly  ?•  -  0  (3.5)  ay loses accuracy  as mesh  size  i s increased.  Accuracy w i l l be most adversely affected i n regions where the flow i s highly rotational, case,  the  such  as  fluid  i n the central vortex region of Figure 3.7.  velocity  changes  strongly with  position  and  In this there  are  s i g n i f i c a n t changes i n the orientation of the flow vectors from one c e l l to the next.  The  v e l o c i t y at one position may  that measured only a few centimetres away. applies  for  the  infinitesimal  region  be tremendously  different  from  Thus the continuity equation that 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. this as an option.  Unfortunately, time constraints eliminated  Therefore further analysis of the two dimensionality of  the flow in the tank w i l l 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 above reduces experimental overall  dw the mean values of — . dz  velocity  These are shown for each set of  measurements in Table 3.2.  flow in the experimental  tank was  It i s clear  not perfectly  that the  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.  Velocity Patterns  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 (shown at the bottom of the plot).  to the maximum measured velocity  The border around each plot represents  the position of the inside walls of the tank.  Cells that do not contain  35  arrows indicate that i t 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" c e l l s , the general recirculatory motion of the flow is clearly observed.  The bubble  left-most column of each plot is the region occupied by the  column.  It was  not  possible to obtain accurate  measurements in this region, due  water velocity  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  centre of the vortex in Figure 3.7  air flowrates.  (corresponding  For example, the  to a blowing rate of 38  l/min) is in the approximate centre of the tank, while that of Figure (at an air flowrate of 216 l/min) is in the upper left corner.  3.16  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: vectors in this region of Figure 3.7 location of the vortex centre and  are horizontal.  the  This shift in the  the orientation of the velocity vectors  36  indicates that the  lowest  t h e r e was a s i g n i f i c a n t  and t h e h i g h e s t  observing  the  figures  steadily-increasing corresponding  3.4.4.  shift  steady  lying  downward  between motion  This  these of  hypothesis  two  surface  i s supported by  extremes. velocity  They  show  a  vectors,  with  a  i n t h e l o c a t i o n o f the v o r t e x .  addition to velocity vectors, state  digitizing plots,  a i r flowrate.  c o n d i t i o n s between  S u r f a c e Shape  In the  change i n t h e f l o w  fluid  the traces  the s u r f a c e  surface  profiles.  These  of the experimental  profiles  also  3.7  Figures  t o 3.16  profiles  surfaces.  show c o n s i d e r a b l e  also  contain  were o b t a i n e d  As w i t h  change w i t h  by  the v e c t o r a i r blowing  rate.  This profiles.  variation  By c o n s e r v a t i o n  c u r v e s should Performing  a l l be equal  this  of mass,  by  integration  t h e volumes  defined  t o the volume of the u n d i s t u r b e d  i n t e g r a t i o n by the t r a p e z o i d a l method,  two-dimensional s u r f a c e larger  i s emphasized  variation) reveals  than t h e q u i e s c e n t  volume.  This  that  this  of  these  by each  surface of these  bath (18 l i t r e s ) .  (and assuming o n l y a  volume i s i n a l l cases  " e x t r a " volume must t h e r e f o r e be due  t o t h e presence of a i r b u b b l e s .  C a l c u l a t i n g a percentage gas holdup d e f i n e d by:  Gas  „ ,. Holdup r  =  A c t u a l Bath Volume - Quiescent Bath Volume tr—: — .j. -, Quiescent Bath Volume  mnv x 100%  /o (3.6) 0  37  and  plotting against injection a i r flowrate gives Figure 3.17.  There i s  clearly a direct relationship between the gas fraction and the flowrate of the injected a i r . 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 w i l l 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  furnace - shows a very poor correlation.  on the slag fuming  The gas fraction of the slag fuming  furnace i s much higher than any experimentally measured value, and yet the modified  Froude  Number  of the furnace  injection  i s lower  than  many  experimental runs.  Therefore, the gas fraction variation i s not due to the  inertial  of the incoming  variation  air, and i s 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, i s given by the bath kinetic energy (or stirring energy). energy can be calculated by:  The (experimental) mean c e l l kinetic  38  n  n  1 2 Z j (pAxAyz)(u . + v j - l -  2  J  E  cell  where N i s e q u a l for  *  Z i-l  N  L  )  (3.7)  1 J  t o t h e number o f c e l l s which c o n t a i n v e l o c i t y d a t a .  Cells  which i t was not p o s s i b l e t o 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 s u r f a c e )  a r e excluded  from t h i s  have on t h e summation i s d i s c u s s e d  Equation measurements.  ( 3 . 7 ) has  calculation.  The e f f e c t  t h i s may  set of  velocity  below.  been  applied  to  each  When p l o t t e d a g a i n s t a i r f l o w r a t e , one o b t a i n s t h e c u r v e  i n F i g u r e 3.19. T h i s shows a steady to  1J  i n c r e a s e i n c e l l energy w i t h a i r f l o w up  a maximum a t an a i r f l o w r a t e o f about 90 1/min and c o r r e s p o n d i n g  modified  Froude number o f 3.  most u n e x p e c t e d , as t h e i n j e c t e d a i r i s the source bath).  The mean c e l l  to a  At t h i s p o i n t , a sharp t r a n s i t i o n o c c u r s , and  the mean c e l l energy a c t u a l l y d e c r e a s e s w i t h i n c r e a s i n g b l o w i n g  the  shown  energy  continues  rate (this i s  o f a l l k i n e t i c energy i n  t o drop  f o r a l l remaining a i r  flowrates.  U n f o r t u n a t e l y , t h e e x c l u s i o n o f c e l l s l a c k i n g v e l o c i t y measurements may  have  affected  velocities difficult This of  near  summation  the surface  to obtain  effect  this  surface  were  somewhat,  t h e l a r g e s t , and  velocity  measurements  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  non-surface  cells  shown  i n Figure  3.19,  indicating  3.20. that  and p l o t t i n g  this  This  curve  this  observed  s u r f a c e v e l o c i t y measurements.  e s p e c i a l l y since i t was  at higher  the water particularly  a i r flowrates.  the mean c e l l k i n e t i c  energy  as a f u n c t i o n o f a i r f l o w r a t e , as  f o l l o w s t h e same o v e r a l l t r e n d as F i g u r e transition  i s not due t o a l a c k o f  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  i s 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 c e l l  kinetic  energy would continue to  increase with increasing air flowrate.  A plot of the mean c e l l 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  •  . . input  -  Figure  3.21  2  =  •  E, buoyant  <P„ " p J  Q h  +  1  +  /  shows that, after  2  .  E. . . kinetic  P- Q(Q/ ) A  (3.8)  <-)  2  3  9  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 r e l a t i v e l y low a i r flow of 38 l/min N  , = 0.4, rr  corresponding to the velocity regime of Figure 3.7. air  enters as distinct  degree.  bubbles that  It can be seen that the  do not interact  to any  significant  The photograph taken at the transition air flowrate of 90 l/min  AO  N„ , « 2.3) shows that the bubbles Fr' . coalesce.  tend to Interfere with one another, or . '  F i n a l l y , at an a i r flowrate of 256.8 1/min (corresponding to the  plot of Figure 3.16) Figure 3.2A shows the bubbles  coalescing continuously,  and the a i r "channels" i t s way to the surface.  It for  i s t h i s coalescence or channeling that i s probably responsible  the reduced  e f f i c i e n c y of energy  entrained by a bubbled with one another. continuous  transfer.  The maximum amount of f l u i d  gas seems to occur when the bubbles do not i n t e r f e r e  I f the gas "channels" to the surface i n the form of a  column of a i r , very l i t t l e  fluid  i s entrained.  In such a case,  the surrounding f l u i d i s unable to penetrate the gas column, and much of the gas rushes to the surface without encountering any l i q u i d 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 a i r flows of less than about 90 1/min (N , = 2.3) the a i r enters the rr bath  i n the form  liquid well.  of d i s c r e t e ,  non-interacting bubbles,  Increasing a i r flowrate merely  This corresponds  to the increasing  which  increases energy  entrain  transfer.  section of the curve i n Figure 3.21  and the p r o f i l e s of Figure 3.7 to 3.10. 2.  At an a i r flowrate of about 90 1/min (the maximum corresponding  of Figure 3.19 and  to a modified Froude Number of 2.3) the bubbles  start to  interact s i g n i f i c a n t l y both longitudinally, and with bubbles r i s i n g adjacent tuyeres.  from  41  3.  A i r  flowrates  increasingly more  At  poor  difficult  decreasing 4.  greater  an  to  occurs.  to  This  responsible  a  for  studies  channeling the  to  value  of  2.3  behaviour  in  this  adjacent tuyere, this  tuyeres, as  interaction  system. been  defined  This  shown  Froude  to  numbers  of  modified  at  Haida  a  study and by  is  to  Haida  would  as  of  as  algebraic balances  interaction models  (33,34)  of since  field  number  coalescence  bubbles the  of  of  rising  rates  from  matte  it  and  by  the  several onset  adjacent converting  of from  channeling  from  a  from single  reasonable  this  is  3.16.  originating  is  in  and  channelling  the  rising  Therefore,  3.7  different  of  bubbles  bubbles  injection  nickel  observed quite  the  mechanism  3.15  much  to  Figure  previously  23,  more  3.14.  complete  (37) of  to  of  Figures  However,  Brimacombe.  in  of  interaction  and  it  contacting  observed  study.  lower  12)  finds  3.11  minimum of  and  corresponds  Figures  Brimacombe  the  at  This  c o a l e s c e n c e ,  that  experimental  tuyeres  has  process  also  (17),  at  12.  Unfortunately, gas-liquid  and  significant low  to  fluid  gas-liquid  Froude  this  the  occur  interaction be  by  due  not  number  been  and  modified  determined  (the  velocity  has  The  covers  l/min  different the  greater  column.  and  Froude  behaviour  (37,14).  gas  3.19,  157.8  producing  occur  the  Figure  drastically  Channeling previous  of  cause  contacting.  penetrate  flowrate  corresponding  this  gas-liquid  section  air  than  the in  bath an  presence this  of  system  recirculatory increasing  this  significant  invalidates motion  amount  of  the  based input  upon  buoyant  change  in  application simple energy  the of  energy is  not  42  transferred  to  the  bath,  energy  is  not  accounted  bath  with  fields must to  be  that  in  take  the  energy  multi-tuyere this  made.  of  but  escapes for  the  effect  system  simply  input  industrial  coalescence  In  the  air.  into  equating Any  systems  entirely.  at  the  account  kinetic  predictions  similar if  It  for  modified  accurate  is  this  "lost"  energy the Froude  of  the  velocity numbers  calculations  are  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. verify  the mathematical  Thus the experimental data serves both to  model, and  description of the gas-liquid  to provide an  accurate and  detailed  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  sets of input conditions in an attempt collected previously.  run for many different  to model the experimental data  44  4)  Finally, Zinc  4.1.  calculations  Slag  Fuming  Furnace,  made  to  describe  and a P e i r c e - S m i t h  the  fluid  Copper  flow  pattern  in  a  Converter.  General Statement of Problem  4.1.1.  Governing Equations  The isothermal momentum.  general  fluid  +  are  These  Navier-Stokes  %^  were  lr  principles the  laws  that  laws  are  of  allow  calculation  conservation  stated  of  of  mass  mathematically  by  of  an  and conservation  of  the  the  motion  three  dimensional  equations:  f-  (P«v) ay  yx ay  at  at  |- (puv) + | - ( p v ) + | - (pvw) ax ay az  2  +  K  +  az  H  k  dT  (4.1)  ah _ ay  2  y  zx ) az  5T  I" (P"w)  (P" > ax  p  K  a (  at  V ax  ax  yy oy  zy ) az  (4.2)  a( pw) at  +  a( puw) ax  +  a(pvw) ay  +  „ ah ~ az" "  a(pw ) az  p g  , ZX — 5 T  3x  ax. zz. az  zy ay  (  (4.3)  ap_ at  Here,  Equations  momentum,  while  is  the  termed  (4.1)  +  a(pu) ax  +  a( v) ay P  to  (4.3)  represent  Equation  (4.4)  states  "continuity  equation".  a( z) az P  +  the  the When  =  0  conservation  law o f  (  of  with  the  4  x , y and z  conservation  supplied  4  of  mass:  correct  fluid and  i n i t i a l  45  and a  boundary  fluid  conditions,  under  To system  1)  any  The  flow  in  of  a  the  assumption,  The  turbulent  This  this 3)  One  turbulent phase  it  effort  a  very  at  only  Unfortunately,  important  the  for  experimental  flow  of  of  a  and was  very  adequacy is  It  problem,  limits  two-phase  The  results  the  the  but  l i t t l e model of  a  discussed  in  of  the  density reduces  in  beyond phase  Section  5.  is  a  the  model  a  stresses;  necessary  calculation a l l  in of  to the  was  subsequent in  the  be  tank.  assumed.  mathematical  complicates the  the  model.  matter scope  in  of  approximation.  of  this  by  determination  water  applicability  was one  of  consequently  choice  this  described  was  Thus  fluid  greatly  the  and  domain  flow  that  shear  actual  interface.  constant  assumption.  turbulent  The  the  perfectly  technique  considerable  considered:  be  was  the  itself,  f i e l d .  to  made:  calculations.  bath  and  by  were  indicate  further  introduced  with  the  assumed  employing  error flow  allowed  solve  was  viscous  gas-liquid  also  there  investigation. experimental  calculate  approximations  measurements  and  l i t t l e  was  equations  tank  the  concerned  conditions,  complexity  to  experimental  however  the  to  the  turbulent  (liquid)  necessary  boundary  used  simplified  describe  viscosity  were  the  equations,  to  approximation is  of  the  end  calculations  be  these  following  greatly  introduced  only to  can  laser-doppler  these  calculate  assumed  This  in  viscosity  to  This  liquid  action  assumption  order  the  and  time-averaging  to  3,  The  fair  turbulent  solution  Section  two-dimensional.  2)  equations  conditions.  achieve  discussed  these  as of  describing  the this the  46  4)  Convective  stirring  This  good  is  (which  a were  processes,  hard  of  d i f f i c u l t  actual  et  (39) is  and  +  the  employed  +  4.1.2.  is  term  that  the  the  the  two  this  when  of  For  adequate  effect  of  the  determining  this  the  the  case  is  very  A r c h i m e d e s number of  work  industrial  magnitude  gradients.  the  lack  of  in  neglected.  information  on  assumption  is  by  the  mathematical  the  conservative  y  is  equations, an  effective  *  v (J  8V  -  t  model  are  ZH)  +  +  therefore:  gx  (4  .0  .  5)  ( 4  .  7 )  dy  form  of  the  time-averaged  written  for  an  turbulent  two  dimensional  incompressible,  v i s c o s i t y ,  defined  isothermal  according  to  approximation.  Boundary Conditions  A  quantify  shown  the  therefore  "  Navier-Stokes  Boussinesq  case  temperature  to  ax  The  the  converting,  +  f l u i d .  performed  significance  to  were  quantify.  S £  turbulent  For  of  due  gradients  experiments  have  due  accurately,  equations  This  temperature  the  copper matte  gradients,  to  a l .  stirring  and  to  isothermal).  (Gr/Re)  the  estimate  temperature  The  for  Number  fuming  to  assumption  Salcudean  significance slag  due  essentially  Archimedes  of  effects  the  schematic  description  experimental  results  of  the  is  boundary  shown  in  conditions  Figure  4.1.  necessary  The  to  (two-  47  dimensional, by  the  single-phase)  a  is  to  1)  The  walls  2)  The  surface  3)  The  bubble  be  of  this  would a  specified  along  column, for  the  zero.  to  three  is  assumed  to  be  bounded  tank.  conditions  description This  is  are  of  by  no  fluid  means  extremely  were  a  a  be  characterized  flow  within  t r i v i a l  difficult  as  two a  interface. vertical the  is  characterize,  But  the as  the  to  the  describe  mathematically  experimental  exercise:  model inflow  two  within  of  employed, of  each  impossible  boundary.  to  achieve,  considerable  drag  the  the  fluid  the  walls  velocity  on in  the  and  model  and  requires  of  gas,  one-phase  components  length  almost  surface  phase  constant  velocity  entire  this free  the  must  of  tank  the  the  three  confines  model.  Fortunately, simple  experiments  liquid.  described  gas-liquid and  values  the  example, be  plexiglass  these  single-phase  horizontal  bubble  the  column.  conditions  simply  the of  achieved.  For  as  of  quantitative  boundary of  of  following:  Each if  liquid  bubble  the  requires  the  surface  that  the  liquid  pressure  In  case  the  while  be the  these  effort.  liquid region  of  deriving  mathematical  the  free  column  is of  the  relatively walls  is  48  4.2.  Selection of Solution Algorithm  As i s usually the case, the selection of the numerical method for this particular problem was heavily influenced by the boundary conditions of the problem. pertinent  An algorithm was  boundary  sought  conditions,  computational efficiency.  while  that could adequately describe the still  maintaining  accuracy  and  In this case, the combination of awkward boundary  conditions and the type of flow made the selection of the solution algorithm more d i f f i c u l t :  1)  The  rectangular  difference  shape  schemes  of  could  the be  experimental  applied easily  tank to  meant this  that  simple  finite system.  However, this makes the description of flow In more complex (eg. round) geometries  more d i f f i c u l t .  This was  objective of this work was (round) copper  converter.  unfortunate,  to calculate the fluid  since a  partial  flow regime in a  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 d i f f i c u l t for this technique to calculate the fluid velocity  and  position  at  a  free  surface.  Some f i n 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  (4.8)  ay  This  invalidates  such  (53)  to describe  the  as  techniques position  the method  the  of  free  of  free  surface,  as  streamlines  this  requires  p o t e n t i a l f l o w (£ = 0 ) .  The the  free  only truly  surface  technique  s a t i s f a c t o r y n u m e r i c a l method capable o f d e s c r i b i n g  i s the Marker  and C e l l  and i t s v a r i a n t s a l l o w  (MAC) t e c h n i q u e  calculation  of the p o s i t i o n  s u r f a c e , as w e l l as the v e l o c i t y and p r e s s u r e f i e l d s s e v e r a l f a c t o r s prevent  a)  The MAC  complex  viscosity expensive b)  numerical  calculations i n computer  can  solving  determining are  the  system  when these  only interested  o f the f r e e  i n this region.  Again,  f o r laminar  techniques; cause  these  the  f l o w s , as they a r e  addition  methods  t o be  of  turbulent  prohibitively  time.  MAC methods a r e t r a n s i e n t by  The MAC  the immediate s e l e c t i o n o f t h i s type o f t e c h n i q u e :  methods a r e g e n e r a l l y o n l y used  relatively  (54).  techniques at  - steady  s t a t e can o n l y be a t t a i n e d  successively greater  solutions  cease  i n the steady  points  in  time  t o change a p p r e c i a b l y .  state v e l o c i t y  values, this  and  S i n c e we i s a very  i n e f f i c i e n t method o f c a l c u l a t i o n . c)  MAC  methods  are  generally  very  computationally  expensive.  They  f r e q u e n t l y r e q u i r e m a n i p u l a t i o n o f v a s t amounts of d a t a , and a l s o many thousands of i t e r a t i o n s relatively executable  "small"  MAC  statements.  t o s o l v e the N a v i e r - S t o k e s  programs  often  comprise  employ  equations. several  Even  thousand  50  There directly  are large numbers of numerical methods that are capable of  calculating  conditions.  a  steady  However few,  i f any,  state  velocity  regime  under  turbulent  of these can describe the position of a  free surface adequately, since most of these r e l y upon the derived values of vorticity  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 i n the location of a free surface, as this r e l i e s upon the continuity equation, which vanishes). Because the motion of the free surface was considered to be of  significance  in  determining  the  overall  functions could not be employed - solution had  velocity  field,  these  two  to be made i n terms of the  primitive variables ( v e l o c i t i e s and pressure).  Therefore,  the  ideal  numerical  technique  f o r this  problem  would  possess the following c h a r a c t e r i s t i c s :  1) Capable of describing the free surface, as i n the MAC  methods.  2) Able to deal with turbulence adequately. 3) Direct calculations of the steady state v e l o c i t y  regime.  4) Capable of dealing with non-rectangular domains of c a l c u l a t i o n .  Unfortunately capabilities  does not  surface, a MAC other c r i t e r i a :  an  efficient  exist.  method was  numerical  method  Because of i t s a b i l i t y  with  a l l of  these  to describe the free  selected and modified i n an attempt  to meet the  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 a l . (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, converge.  despite  a l l attempts,  this  program  failed  to  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  s u r f a c e was moved w i t h each i t e r a t i o n ,  problem  was  convergence  created.  Since the  was i m p o s s i b l e .  s u r f a c e movement was d e l e t e d , t h i s program r a p i d l y  When the  converged.  b) Accelerated Steady State Solution  When t h e p r e v i o u s attempt MAC  technique  t o advancing  failed,  the f r e e  i t was d e c i d e d t o r e v e r t  surface  along  discrete  time  t o the steps.  However, an attempt was made t o p r e d i c t t h e steady s t a t e s o l u t i o n from a v e r y few  transient  employing  the t r a n s i e n t  boundary  to c a l c u l a t e  different  a t steady  finite  predict  computer  N a v i e r Stokes  this  time  The program was run ( w i t h  o f t h e e x p e r i m e n t a l apparatus) and  profiles  and s u r f a c e  shape  at  several  These c a l c u l a t e d v a l u e s were s t i l l v e r y d i f f e r e n t  state.  difference c e l l  the steady s t a t e .  program was w r i t t e n ,  equations.  those  the v e l o c i t y  time i n c r e m e n t s .  those  each  A second  c o n d i t i o n s approximating  allowed  for  solutions.  The manner i n which t h e c a l c u l a t e d changed w i t h time was then used  This predicted  velocity i n  i n an attempt t o  steady s t a t e was then compared w i t h  the " t r u e " c a l c u l a t e d v a l u e .  U n f o r t u n a t e l y , t h i s attempt this  i s r e v e a l e d by a time  difference Immediately time of  cell  (Figure 4.2).  obvious:  s t e p , though  the free  o f the c a l c u l a t e d  The o s c i l l a t o r y  the v e l o c i t y  they do approach  the v e l o c i t i e s  Moving  plot  was a l s o u n s u c c e s s f u l .  v a l u e s swing a steady  i s due t o t h e u n s t a b l e  surface at the beginning  The reason f o r  velocities  of a  finite  n a t u r e o f the convergence i s quite  state.  Again, t h i s  influence o f each  u n p r e d i c t a b l y a t each oscillation  of the free  time  step  surface.  introduces a  53  complete change in the geometry of the computational mesh, which affects the calculated velocity values somewhat.  When the immediately  free  predicted  surface was  the expected  cavity, from a very few time steps.  not  allowed  velocity  to move, the program  field  for flow In a square  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 A l g o 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. with  SOLASMAC  the  SSMCR program  method  uses  an  explicit  As  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  As into  a  in  number  finite  a l l  of  small  difference  Boundary  cells  values  for  a  finite  difference  are  shown into  the  c e l l  the  i n i t i a l  the  are  or  the  of  considered  in  to  Appendix  1.  After  4.4. c e l l s ,  pressure  the  of  solution  surface) The  approximate  the  terms  flowchart  of  velocities  and  fluid  velocity  within  is  set  to  experimental  applied  of  tank,  The  other  before  central to  4.5  been  cells This  the  the  program  have  hydrostatic.  the  pressure  Equations  pressures  of  in  and  computer  a l l  divided  measurements.  centre.  the  is  sketch  velocity  c e l l  the  the  are  the  A  a  laser-doppler  domain.  apply  domain  shows  4.3  at  distribution  condition  the  free  the to  the  Figure  model  for  used  schemes,  "cells".  to  edges  schemes  boundary  and  used  (except  on  Figure  the  zero,  grid  contained  in  difference  squares,  conditions  fictitious  4.8  finite  is  loaded  is  set  to  represents  air  has  been  introduced.  Horizontal explicitly starts  with  continues the  for  to  top,  The velocities to  each  for  at  first  again to  solution the  the  Once  at  momentum c e l l  bottom  column.  starts left  vertical  non-boundary  c e l l  the  procedure  bottom  each  the  up  and  the  containing  left the  equations  of  the  top  bottom  of  of  are  then  f l u i d .  This  procedure  finite-difference a  column  the  next  of  cells  column,  solved  mesh, is  and  reached,  and  continues  right.  of  next  the time  momentum step.  equations  The  yields  continuity  an  estimate  equation  is  of  then  liquid applied  c e l l :  d  i  ,  v  ij  " i J  "  U  Ax  1-1,J  V +  * J  "  V  1 » J - 1  Ay  (4.9)  55 The value of this  sum i s termed the "divergence" and i s used to  adjust the pressure and v e l o c i t y f i e l d s using the following scheme: t\P  p  ij  u,, ij  u  =  ±i  =  P  =  =  'ij-1  =  i j  (4.10)  ± J  ^ i j  +  u,., + i j  At AP., J ^pAx  AtAP, , u, . , i - l , j pAx  ', = 1-1,J  v. . ij  Xp d i v  v., + i j  At AP. . ^ pAY  At AP, , v . . , ' i j - 1 Ay- i i  (4.12)  (4.13)  (4.14)  (4.15)  p  where \ i s an input parameter to the program, usually set by X  = .  ^ 1 2At(-^2 + -^y) Ax AY  (4.16)  Two constraints are placed on the time step, At: At  At  <  minimum (-^*-, -^-) u v  < § ^  ( A X  2  , + Ay )  (4.17)  (4.18)  2  This i s carried out u n t i l the divergence of each c e l l f a l l s below a s p e c i f i e d error l i m i t .  When this occurs, these new c e l l v e l o c i t i e s are used  to  calculate the new position of the free surface, and the entire  is  repeated  Navier-Stokes  for the next equations  time step.  procedure  This method of solving the transient  i s very similar to many other numerical techniques.  56  However,  the  treatment  of  4.4.  SOLASAMC the  boundary  There  are  a  surface  velocities  free and  Of relatively finite  conditions,  the  two,  by  most  SSMCR  notably  iteration,  the  considerable following  differs the  greatly  free  by  its  surface.  to  of  mesh  above (Figure The  description  to  complete ls  given  and  calculations specifying  the  used  is  the  mass the  balances,  surface  track  located datura  of  memory.  by at  a  has  are  been  the the  it  is  within  the  locate  the  need  that  end  time  of  a  position  these  describing within  moved  performed  of  method, particles  the  height  the  finite  according by  the  required  SOLASMAC  of  position then  buoyancy  particles The  vector  any  "heights"  At  these  the  to  neutral  located  of  and  particles  l i q u i d .  particles  derivation below.  marker  eliminates  These  of  particles  time  Hirt,  Using  appropriate.  of  keep  arbitrary  4.5).  with  position,  position  if  marker  surface  an  it  flow  computer  the  the  fictious  the  of  its  simpler.  (54,60)  the  However,  suggestions  surface  move  were  of  is  locate  to  with  associated  surface.  methods  move  Instead  balance.  shorter  to  locating  former  These  amounts  the  altogether.  difference  the  and  positions  surface.  the  at  the  MAC  d i f f i c u l t i e s  boundary:  mesh,  position.  considered  free  main  straightforward  difference  surface  two  pressures  Previous  mass  used  Locating and Moving the Free Surface  involving  of  as  Treatment of Boundary Conditions  4.4.1.  were  method  H i l l ,  to  but  a a  57  Considering an element of the free surface:  Mass input rate:  =  u(h - h ) + VAX 2  (4.17)  x  Mass output rate:  =  <  u  +  £  ^< 2 h  + ^  Ax - h ) x  +  [v + -g ( h  | i M  2 +  _  h  i  )  ]  ^  (  4  .  1  8  )  Rate of Accumulation:  ft  a h  2  AX  2  Applying the continuity equation, letting AX •+ 0, and observing at  = v one obtains (in finite difference form):  h  h  t + A t ±  =  h* + At ( v  t j  - u  t  _ t h  1 i j (  ))  (4.20)  Equation (4.20) i s 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 c e l l 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 c e l l , the averaged velocities would be:  "  S  *  =  ^  i  ¥  (  I  (  u  V  i - 2 , j  i  j  +  V  +  I  u  .  i  +  2  u  j  H  v +  (j  where  i -  i j  ,  1  +  v  )  +  j  i  T ^ i  )  +  )  H  - i.5)Ay -  i  I  J  - i  (  V i  +  2 u  (  j  i-i  - i  +  i J  .i>  ( 4  ' > 2 1  V i j - 2  (  4  .  2  2  )  (  4  .  2  3  )  \  f = Ay  Additional velocity averaging methods are used i f the surface i s above the c e l l  centre, or i f there  i s a boundary c e l l  nearby.  These  averaging schemes are largely a result of t r i a l 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 c e l l velocities and pressures i s 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 i s usually  employed to estimate the pressure of a surface c e l l , while the tangential stress i s used with the continuity equation to determine the vertical and  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 i s given below. significant  departure  from H i l l ' s  The only  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: T  =  xy  x = yy  v  ^X P„ - 2 (—^) % Sx K  p  -  *x  v  U  (4.24) '  (4.25)  2u(-r- ) JS:  *ay  Tangential Stresses: ou u (—^ ay  ?  x xy  =  T yx  p  +  av —-) 3x  (4.26)  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)  =  P + P'  (4-29)  P  t  60  Substituting these values conditions and time-averaging,  into the normal and  tangential stress  one obtains:  Tangential Stresses:  -  T  P - 2n  (-^-)  yy  ay  Normal Stresses: T xy  =  -n"5x (£• + ay£L) K  v  v  7  (4.31) '  Note that the laminar viscosity, rather than the apparent turbulent viscosity,  is used  in the  turbulent  derivation of the normal and fluid  surface  proceeds  stress conditions.  Therefore,  the  tangential stress conditions for a turbulent  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  (jg o  -  +  (tan  2  oy  x  0 - 1)<J£ + oy  -g- ) ox  =  0  (4.32)  Normal Stress Condition:  . __ P  2  .. (1C + t a n e) ,au . av H  29 tan for, en  v  ay  +  "dx 57>  Since the sum of the forces on a fluid surface are zero.  (A-33)  61  Thus one  is  form  to  forced  to  depends  describe satisfy  upon  the  a  fluid  further  surface  velocity  two  and  partial  orientation.  pressure  at  differential  Fortunately,  the  surface,  equations,  whose  approximations  can  be  condition  to  made.  4.A.2.1.  Normal Stress Condition  The specify  the  However,  SOLASMAC pressure  SOLASMAC  viscous  significant. condition  be  Therefore,  the  This  original  l i t t l e  cost  orientation the  continues  +16 in  the have  Reynolds  pressure  as  flows  in  (60)  approximation  surface surface  in  horizontal  between  the  at  it  of is  normal  difference  study  term  full  at  shown  reduces  in  the  the  below  10.  cells  complexity  and  surface. where  equation f u l l  was  MAC-type  fluid  numbers,  stress  that  surface  common  at  Reynolds  normal  numbers  the  cells  stress  becomes  normal  stress  considered  codes  (it  increases  the  was  to  be  used  stability  in at  Tangential Stress Condition  The upon  used  the  accuracy.  A.A.2.2.  depend  to  Shannon  MAC p r o g r a m ) in  f i n i t e  written  and  only  atmospheric.  the  employs  correction  Hirt  need  in  was  pressure  method  15  15  set  inclination.  degree  and  and  velocities  +15  +30  degree  degrees,  one a  increments.  the  The  Increments. degrees,  by  tangential  SOLASMAC  That  i s ,  equation  different  for is  stress  method  considers  surfaces used.  equation  For is  condition  oriented those  employed  also  surface between inclined and  this  62  H i l l chose to be t h i s accurate because he was low  velocities,  significant.  where  differences of  a  few  solving problems at  millimeters  per  second  were  This type of accuracy i s not j u s t i f i e d for the purposes of this  investigation.  Therefore, was  i n the  approximated much more roughly.  slopes  are  considered  Figure  4.6  shows how  to be these  the three c e l l s surrounding are  examined.  If  only  h o r i z o n t a l or v e r t i c a l . a  SSMCR program, the  surface  horizontal,  cell the  that  tangential stress condition  As i n the o r i g i n a l MAC  program, surface  e i t h e r h o r i z o n t a l , v e r t i c a l or at 45 slopes  are  recognized  a surface c e l l one  is  empty  by  degrees.  the computer program:  (the c e n t r a l c e l l i n Figure  of  fluid,  the  surface  4.6)  i s either  I f two are empty, the surface i s at 45 degrees. contains  a  horizontal v e l o c i t y  section  of  component  surface  considered  i s calculated by  For  to  be  the x-wise  momentum equation, while the v e r t i c a l component i s defined by the continuity equation:  -  "1,3-1  f  tangential  stress  condition  \l  The  horizontal  +  <»1J  " Vl,J>  is  applied  <- > 4  by  24  adjusting  the  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 outside the free surface:  -i  > J +  i  -  "ij - 5  ( T  u  " *i-i > t J  ( 4  -  2 5 >  This v e l o c i t y i s then used to solve the x-wise momentum equation at the next time step.  63  Vertical surface  have  cells  their  are  treated  velocities  specified  U  v  Thus  simultaneously  continuity  4.4.3.  the  to  apply.  boundary  experimental As  stated  wall  the  on  a  45  degree  setting:  V i . j  ( A  i , j - l  v  the  on  large At  - > 2 6  < ' 4  tangential  retarding  effect  stress  difference  c e l l .  choice  calculations. obtained  of  Both with  was  either  in  Table  the  the  condition  and  2 7  >  the  the  is  case,  boundary  types  of  slip  applied  to  zero  and  of  size,  numbers,  others,  these  set  (no  of  the  walls  extremely in  the  simple  fictitious  regions,  the  velocity  zero.  The  velocity  be  slip)  the  flows,  no-slip  boundary to  free-slip  a  or  unaffected  conditions  should  were  had  be  layer where  by  is  l i t t l e  tested,  and  made  and  the  boundary should  small,  smaller used.  is  the  approximation  layer  region  c o n d i t i o n s  approximation  slip  boundary  number  the  limited  free  the  Reynolds  c e l l  were  influence  4.1.  sizes low  wall  this  no  the  the  In  wall  to  each.  region.  the seen  to  conditions  was  For  these  comparison  wall  Reynolds  For  describe  the  mesh.  of  to  to  r e l a t i v e  compared  higher  in these  between  the  finite-difference is  as  choice  used  computing  to  (free-slip),  depending  were  earlier,  tangential  The  the  -  both  were,  perpendicular  component  used.  i j  conditions  tank  surrounding  component  layer  "  located  Rectangular Wall Boundaries  of  the  satisfying  ij  by  Cells  equation.  The  cells  similarly.  than In  and a  similar  the  finite  actual  effect  be  fact,  on  the  results  64  4.4.4.  C u r v i l i n e a r Boundary  To that  the  predict  flow  the  to  an  cells  boundary  this  was  not  to  cause  it  r e l a t i v e l y  of  be  small  s i z e  the  of  converter,  dealing  with  adjusted  to  constrain  affected  the  to  was by  a  found  the  curved  -7*^ a n d  terms  dx  to  v i s c o s i t y - d e p e n d e n t  viscosity  than  the  laminar  near  the  wall  boundary then  1)  value  in  this  viscosity flow  to  employed  not  it  the  was  by  insufficient:  in  the  This  in  momentum  the  was  boundary  due  to  the  equations,  oy  terms  study  tangentially,  velocities,  fluid  5 u  as  2  2 ~ '-  o r  s:  —2"  many  was  To  therefore  pressures.  The  t n e  dx  orders  V i e c e l l i .  nce  of  magnitude  force  the  necessary following  to  higher  velocities adjust  the  technique  was  employed:  The  curvilinear  Cells 2)  to  used  or  employed,  the  wall.  dy effective  curved  was  be  pressure  2  such  necessary  a  V i e c e l l i  a v to  is  of  was  3  compared  it  technique  adjustment  parallel  of  copper  boundary.  pressure  flow  a  capable  cells  significantly to  in  I n i t i a l l y ,  arbitrary  However, fluid  regime  model  boundary.  pressure  parallel  the  flow  mathematical  non-rectangular whereby  the  For flow  that  each  boundary  intersected  time  step,  tangetially  the  wall.  the  net  The  mass  to  the  fluid  the  wall.  into  the  imposed  bottom  the  magnitude  flow  was  of  in  boundary these  There the  c e l l  over  were  boundary  was  no  tangential (i.e.  by  a  the  finite termed  boundary  cells  velocity velocity  difference  was  continuity  cells.  constrained  component was  grid.  normal  calculated equation).  to to  from  65  3)  The p r e s s u r e  in  Bernoulli-type change  boundary c e l l s  equation:  the  was  pressure  adjusted  change  i n accordance with  being  calculated  by  a  the  i n f l u i d momentum.  The setting  these  the  intersection  of  pressure  the  of  a surface cell  cell  equal  and a w a l l c e l l was t r e a t e d to  zero,  v e l o c i t i e s by the c o n t i n u i t y e q u a t i o n .  When t h i s  a p p r o x i m a t i o n were employed,  was  the  fluid  and  by  determining  the  ( a d m i t t e d l y crude) boundary  found to  flow  tangentially  to  the  round w a l l .  4.4.5.  G a s - L i q u i d Boundary  The interface  -  final  the  mathematically data.  bubble due  to  column. its  inter-related  The f i r s t  of  these  bubble pushes and shears  rise.  S a h a i and G u t h r i e (33)  a c t i o n of  large,  stable,  This  inherent  The i n j e c t i o n of a gas  by a number of  gas  boundary c o n d i t i o n r e q u i r i n g treatment  gas  is  by f a r  complexity  into a quiescent  the  the  f l u i d beneath  density  and  hardest the  lack  the  gas  liquid  to  characterize  of  experimental  bath causes the l i q u i d to  rise  effects.  is  fluid  liquid  displacement  as  it  or e n t r a i n m e n t .  A rising  moves upward, c a u s i n g the  have argued that  this  effect  fluid  occurs with  to  the  bubbles.  Another mechanism f o r l i q u i d movement a bubble lowers  the  is  in its  vicinity.  the bubble to r i s e ,  following  is  buoyancy.  The presence  This density difference the bubble to the  of  causes  surface.  66  Despite experimental  these  general  the existence  of some  data, the mathematical description of the gas-liquid  interface  i s by no means complete.  guidelines,  and  For example, simple models such as that proposed by  Sahai and Guthrie (34) make no mention of the e f f e c t of bubble coalescence on the  liquid  rise  Mathematical success,  velocity  models based  but there  calculations.  -  a  serious  on these  (though  before  omission.  descriptions have had some qualitative  remains considerable d i s p a r i t y  Therefore  understandable)  reliable  between experiments and  calculations  can be made, an  adequate description of this boundary i s necessary, which requires the water v e l o c i t y and pressure d i s t r i b u t i o n along the length of the r i s i n g gas column as a function of Injected gas flowrate.  Unfortunately, t h i s data i s not yet  available.  Therefore,  a l l descriptions of the g a s - l i q u i d  boundary condition  used by the mathematical model i n t h i s work w i l l be, at best, semi-empirical. Various  t h e o r e t i c a l v e l o c i t y and pressure d i s t r i b u t i o n s w i l l  the model, and the results studied. values  that  conform  closest  be applied to  Those predicting v e l o c i t y and surface  to the experimental  data w i l l  be accepted as  v a l i d d e s c r i p t i o n s . That i s , the mathematical description of the gas-liquid boundary condition w i l l experiments.  be inferred  by the agreement of calculations  with  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: fictious  the SSMCR program was set the task of computing the flow regime of a fluid  (having  a viscosity  of 0.4 g/cm s) i n a square cavity  measuring 1 cm x 1 cm (Figure 5.1).  This problem i s suitable for test  conditions as i t i s 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 i n Figure 5.2 and clearly there i s excellent quantitative agreement between the two sets of data.  H i l l ' s program was, i n 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 i n quantitative comparisons of different  velocity regimes.  For example, during  the development and  "debugging" of the SSMCR program, an error was discovered viscous drag term of the Navier-Stokes  in which the  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. examined themselves  that  this  In fact, i t was not until actual numerical values were error  was discovered.  are not suitable  for comparison  Therefore, between  these  plots by  experimental  and  68  calculated  values,  could  undetected.  pass  In and  the  an  since  attempt  experiments,  deviation,  and  a  the  considerable  disparity  between  velocity  the  to  quantify  the  comparison  between  two  values  were  defined:  the  mean  c e l l  magnitude  deviation.  model  results  c e l l  angular  mean  These  are  patterns  calculated  as  follows:  D  9  -±— n.n. I j  •i  ~ D  U  1  =  E  Vj  ">s  The the  =  f t  computed  and  vectors.  velocity  distribution  non-zero  magnitude  1.  The  overall  i s ,  the  left  great  or  which  the  the  too  values  E . , j=l  i  1  predicted  a  »e  gives  can  bulk  -  v..  + v.,2  t a n  an  _  ( - i ^ - ) v. . i j . c  i  to  velocities,  but  J  ,c  comparing  the  orientation  indicates  used  velocities  ij  well  the  the  + v, ,2  how  two  of  ,c  of  experimental  condition  velocity  ij  ,_  (5.2)  well  the  shapes  the  measurements.  of  of the  predicted However,  a  p o s s i b i l i t i e s :  fluid by  are  is  the  f l u i d .  the  how  (5.1) '  v  V,=  +  by  indicate  motion  /^TT2  indication  agree,  the  -  ij »e =!—J  deviation  with  calculated  measured  ( - ± ^ )  ^V,e  boundary  small  differ.  ij  magnitude agrees  1  ij,e  vortices  deviation  hand  tan"  S~i*7*Z  deviation  The  highest  highest  z . i=l  1=1  measured  velocity  n  £  I=I  angular  n.  too  high  program  This located  relative  is  would in  or  the  low  -  that  imparting  be  the  same  magnitudes  too  case  in  region  as  of  these  69  2.  The overall distribution of the predicted velocity field i s 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 i s significant, an additional factor, the velocity distribution deviation defined by:  dist  , ^ i j ,e J  i/o  .1/2 _  i j ,e'  u  u  ±  2  e +  m  l  2  v  U  can  V  4 1  i n j  i j , c max,e. Lv ) max,c  < ij >  is calculated.  n  V,e>  j = 1  v .  T  .  V  2 1/2  . l j . c max,e. -, K ) J max,c v  (5.3)  1 / 2  If this value i s low relative to the magnitude deviation, i t  be concluded that the f i r s t of these effects  i s predominant, and the  predicted fluid velocity distribution i s essentially correct, but the bulk of the  fluid i s moving too slowly or too fast.  If the velocity distribution  deviation i s high relative to the magnitude deviation, the second effect i s 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 i t s various boundary approximations. or fitted  This in turn allows the model to be "tuned"  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.  70  Once  the  program  previously,  calculations  experimental  flow  used  for  the  viscosity obtain  patterns.  good  by  agreement  with  model  empirical  viscosity  model  of  Sahai  of  and  variety  boundary.  model  Sahai  was  values. Guthrie  (44)  the  from  run  (44)  effective shown  manner to  approximations  experiments.  are  the  attempts  well,  varied  Guthrie  The  In  in  of  As  the  and  verified  performed  A  the  viscosity  been  were  gas-liquid  used  had  to  was  Table  the  below)  were  the  in  an  turbulent attempt  algebraic  tested,  viscosity In  for  run  The  compute  (shown  value  mentioned  as  effective  were  values  to  completely  predicted  by  the  5.1.  5.1. Constant Vertical Velocity  The  first  mathematical bubbles velocity  in  model the  to  transmitted was  to  horizontal  was  the into  the  was  this  the  bulk  of  into  the  cells  attained.  when  the  An  The  velocities  a d d i t i o n a l  were  onset of  check  the  the  to  of  each on  be  the  upward  This  fluid  left  by  to  viscous by  impart  be  of  used That  a  the  a  region,  zero.  by  the  i s ,  the  constant liquid  forces.  applying  boundary  to  region  velocity.  shearing  model  wall  cells  two-phase  considered  region.  these  assumed  were  mathematical  in  of  to  constant  tank  in  Calculations been  of  fluid  velocity  distribution  that  experimental  incorporated  velocity  approximation  This  was  then  assumption  constant and  upward  vertical  setting  The  i n i t i a l  until  steady  the  pressure  hydrostatic.  performed steady c e l l t h i s  by  the  program  state  was  arbitrarily  changed  by  less  c o n d i t i o n  than  was  determined  5%  over  made  by  20  state to  time  had occur  steps.  comparing  a  71  "steady times,  state" and  velocity  ensuring  profile  that  the  with  profiles  difference  was  calculated both  small  at  several  and  previous  diminishing  with  time.  The boundary  are  besides  those  preliminary input  nature,  It  is  deviation boundary 2.  a l l  the  of  the  mainly and  but  wall  performed  other  using  this  calculations  were  these  were  the  optimum  determine  number low  Many  5.2,  to  left  a  calculations 5.2.  Table  very  generally values  of of  a the  velocity).  of  inferences  values  of  distribution  deviation  lower  the at  the  directional  and  both  may  the  deviation  is  be  drawn:  mean (<  c e l l  10%)  velocity  using  this  *mean  By c e l l  deviation  the  values  are  calculations  seem  to  apply  relatively performed  equally  well  constant  at  certain  (or  poorly)  measurements.  very to  blowing  boundary  a l l .  i s ,  with  appears air  deviation  velocity  calculated  disagreement  Increasing  That  deviation  at  three  experimental  cases  performed  lower  the  experiments.  to  this  in  velocity  viscosity  a l l  Table  5.2,  obtain  speaking,  directional  4.  to  Table  of  In  shown served  values a l l  in  the  condition.  Generally for  3.  and  of  (viscosity  examining  possible  many  shown  and  parameters  By 1.  of  condition  performed  two  results  this  high: be  between more  condition,  30  and  pronounced  60  the  mean  degrees.  c e l l  However,  with  the  experiments  lower  the  directional  rates.  velocity  increasing velocity remains  boundary  high  does the  not  boundary  deviation at  seem  30  to  to 40  to  velocity be  it  only  degrees.  is  possible  10%,  but  to the  72  5.  The  effect  of  Increasing 400  g/(cm  the  values  increasing  the s)  viscosity  lowers  of  the  profile  boundary  velocity,  boundary  condition  This with  the  used  to  It  is  400 of  from the  the mean  lowered,  as  no  the  left  longer  the  plots region  of of  the  10  also  appears  g/(cm»s)  velocity  deviation  to  to  40  deviation  be  slight.  g/(cm»s)  but  does  and  not  to  change  appreciably.  indicate  independent  that  of  determined  supported of  effect  increase  and  to  to  the  flow.  cells  completely regime  Figures  c e l l  above  the  overall  either  the  mainly  3.7  the  surface,  this  liquid). deviation  by  shape  viscosity  the  In  and  5.6 to  Figure  3.16,  where  not  at  shown  is  the  the  is  nature  of  the  or  the  of  and a  the the  5.7);  the  this  largest left  in  of  the  to  the  velocities tank.  to  right  into  the  transfer f i r s t l y , overall  deviation with  fluid  left  40  the  increased  the  the  to  effects:  of  5.7  values  to  energy  velocity  closer  fluid  side  of  distribution  bulk  10  further  the  in  and  themselves).  (from cells  two  to  region is  the  has  due  This  plots  means  increase  small  5.5).  main  5.6  viscosity  velocity  velocity  cells  the  of  5.5,  and  viscosity  turn  lowered  the  on  velocity  is  Figures  velocity  increased  This  surface  confined  to  vertical  Secondly,  (as  are  viscosity  (Figures  (the  increasing  transmit  and  comparing  3  figures  of  as  by  Chapter  calculated  interior  wall  computational  is  plots  magnitude  the  about  instead  reasonable,  c e l l of  is  the  to  boundary  velocity  to  is  5  viscosity  i t s e l f .  that  boundary, (this  and  and  the  is  mean  relatively  experimental  g/(cm»s)) the  4  conclusion  obvious  fluid  is  generate  from  directional  Observations velocity  the  the  is  respect  motion  side  of  is the  experimental occur The  in  the  velocity  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  vertical.  the highest velocities,  velocities  and  remain generally low,  and  are  largely  largely horizontal.  The  surface  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 i s forced to conclude that this description of the gas-liquid boundary i s 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.  Variable Density  Apart that may  be  from shear due  significant  to bubble  rise velocity, another mechanism  in causing the experimental flow regimes is the  lowering of the density of the liquid in the region of the bubbles. hypothesis for such a mechanism is as follows:  An  74  1.  The  effect  density  of  (and  the  presence  therefore  of  the  a  bubble  in  hydrostatic  the  liquid  pressure)  in  is  to  lower  the  the  region  of  the  the  higher-density  bubble. 2.  This  low  pressure  adjacent 3.  As  this  this  region  then  causes  an  influx  of  f l u i d . bubble  rises  low-density  under  (and  the  influence  therefore  low  of  buoyant  pressure)  and  region  inertial  forces,  moves  upward  with  many  mathematical  It.  This studies  of  gas  To of a  f i n i t e void  approximation injection  apply  this  difference  into  cells  in  rise  this  the  due  equations. for  to  ladles  a  the  used  in  a  (31,40,42,43).  mathematical  left  great  hand  model,  boundary  the  is  average  lowered  by  density  employing  fraction:  simulate  w i l l  been  liquid  method  p  to  has  presence  to  the  P i j  v  of  i j  =  of  have  fluid  "  P i j - l Ay  a  gas  vertical  Modifications  reduction  i j  p  g  +  1  "  bubbles.  pressure to  (  be  a  )  p  The  f  (  liquid  difference  made  to  the  i j  "  in  term  the  in  boundary  the  continuity  5  ,  4  )  cells  Navier-Stokes  equation  to  allow  density:  V  l j - l  +  P i j  u  P i - l , j Ax  U  i + j  =  0  (5.5)  75  Frequently, with  this  the shearing  approximation.  these boundary c e l l s ,  effect  In this  case,  mentioned  above  a vertical  a l o n g w i t h a reduced d e n s i t y .  i s applied  velocity  along  i s applied to  B o t h methods were t e s t e d  in this investigation.  To necessary  apply  t h e v a r i a b l e - d e n s i t y method t o t h e SSMCR program, i t i s  to obtain  values  f o r the void  fraction  i n t h e 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 o f 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 d e r i v e d  from t h e bulk, v o i d a g e measurements  shown i n F i g u r e 3.19. D i r e c t o b s e r v a t i o n i n d i c a t e s t h a t v i r t u a l l y a l l o f t h e bubbles Assuming overall  are confined that tank  these  to the region bubbles  volume,  alone  6 cm from  the l e f t  are responsible  t h e bubble  column  a  a, ,, • ( i ) ^julk A . column  w a l l o f t h e tank.  f o r the increase  gas holdup  i n the  c a n be c a l c u l a t e d as  follows:  The  , column  gas c o n c e n t r a t i o n s  means a r e shown I n T a b l e 5.3. estimate of  •  I n t h e bubble  region,  column  c a l c u l a t e d by  this  These v o i d a g e v a l u e s a r e a t best o n l y a crude  s i n c e they assume a c o n s t a n t  t h e two-phase  (5.6)  while  gas c o n c e n t r a t i o n throughout t h e l e n g t h observations  indicate  that  the gas  c o n c e n t r a t i o n i s h i g h e r a t t h e f r e e s u r f a c e than i n t h e bubble column.  Applying  these  values  t o t h e SSMCR program  yields  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 F i g u r e s 5.8 and 5.9. The v a l u e s f o r t h e  mean c e l l v e l o c i t y , a n g l e  and d i s t r i b u t i o n d e v i a t i o n a r e shown i n T a b l e 5.4  76  f o r t h e b e s t r u n w i t h t h i s boundary c o n d i t i o n .  It of  Figures  i s immediately  clear  5.8 and 5.9) t h a t  from T a b l e  this  5.4 (and from the v e c t o r p l o t s  boundary c o n d i t i o n g i v e s v e r y  different  r e s u l t s from t h e c o n s t a n t v e l o c i t y c o n d i t i o n . F o r example, t h e comparison o f run  VK.4 w i t h  the experimental  measurements shows t h a t  f o r a l l cases, the  mean a n g u l a r d e v i a t i o n i s v e r y l o w , w i t h a maximum ( a b s o l u t e ) v a l u e o f -12.4 degrees  and a minimum  value  o f 0.40 d e g r e e s .  This  certainly  very  good  agreement, and i s f a r b e t t e r than t h a t a t t a i n e d by t h e p r e v i o u s c o n d i t i o n .  However, approximation  the v e l o c i t y  i s not p e r f e c t .  deviations  Indicate  that  this  boundary  F o r r u n VK.4, t h e mean v e l o c i t y d e v i a t i o n i s  v e r y h i g h a t 99.8% f o r a l l e x p e r i m e n t s .  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 o n l y about 0.2% o f t h e measured v e l o c i t i e s which i s e x t r e m e l y poor agreement. is  quite  explains  high, this:  More important at a value  however, 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  o f 10 t o 30%.  t h e maximum v e l o c i t i e s  still  An e x a m i n a t i o n occur  of F i g u r e  at the l e f t  5.8  s i d e o f the  tank and not a t t h e s u r f a c e as i s shown by t h e e x p e r i m e n t s .  T h i s i n d i c a t e s t h a t t h e main e f f e c t o f l o w e r i n g the d e n s i t y i n the boundary c e l l s  i s t o impart  boundary  condition.  accurate  than  increased  a v e r t i c a l v e l o c i t y t o them, as i n t h e p r e v i o u s  However,  the constant  this  upward  agreement w i t h experiments  must be due t o t h e e f f e c t o f t h e  c o n d i t i o n has been velocity  condition.  shown  t o be more  Therefore,  this  ( t h e lowering of the angular d e v i a t i o n ) term.  T h i s terra was i n i t i a l l y z e r o 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 n e g a t i v e ) i n  \  this  case.  affects a  As  the  negative  shown  x-wise  horizontal  the  Figure  lowered  increased  a  reversion  e f f e c t  of  increased  From can  be  concluded  predicts  the  indicates the  bubble  magnitude Thus  it  accurate  5.3.  or can  the be  vertical  velocity  to  the  boundary  results  mean  on  in  the  angular upward  plots  mean  flow  cause  that  In  (along  shown that  effect,  this  but case  condition.  reduced  by  the  in the  deviation,  boundary is  to  cells  Indicate  velocity  velocity  the  as  5.4  deviation.  is  boundary.  such  Table  directly  The  g r e a t l y  velocities.  the shape in  the  calculations  reduced of  the  fact,  a  Unfortunately, distribution  of  shown  shaping  of  concluded  velocity  lowers  the  i s ,  in  values  constant  that  description  term  positive  results  column.  this gas  boundary  there  of the  overall  that  effect  term  to  term  the  major  this  adjacent  the  the  the  equation  c e l l s  velocity  to  momentum  in  increases  vertical  Thus,  deviation  boundary  correspondingly is  a  x-wise  velocity  density)  The  5.9.  the  velocity.  Imparting with  by  77  that  the  small  the  this  with  boundary  experimental  this  of  bubble  density  made  net  velocity  was  experimental  approximation,  condition,  condition  horizontal  method  this  accurately  profiles. flow  unable  of to  by  i t s e l f ,  bubble  column  This  fluid predict  velocities  it  into the  accurately. it  Is  not  an  column.  P u l s e d Boundary  The involved  third  "pulsing"  attempt the  to  characterize  velocity  in  the  the  boundary  c e l l s .  The  mathematically bubbles  were  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 f u l l of f l u i d . The size of a bubble was one boundary c e l l .  3.  Bubbles  were moved  upward  to  the  corresponding to their rise velocity. 40 cm/s,  based  next  cell  at  time  intervals  This was i n i t i a l l y assumed to be  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 i s allowed to f a l l under gravitational  The  values  used  frequency were freely varied experimental data.  for  the  bubble  in an attempt  rise  forces.  velocity  and  formation  to obtain good agreement with  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 Qualitatively, this method appears to yield better results than the constant velocity condition. to  the  cells  previous values  at  the  condition. of  left  5.10. simpler  The surface velocities are larger in comparison side of  the  This observation  tank than those produced by  is supported by  the relatively  the velocity distribution deviation generated by  the low  this boundary  The collapse of the surface after the passage of a  condition (Table 5.5).  bubble is responsible for these higher surface velocities.  5.4.  Collapsing 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. Figures 3.7 down from  to 3.16 the  free  (particularly 3.14 surface  i s of  experimental velocity regime, and  through 3.16) great  It is evident from  that the flow of fluid  importance  in determining  the  therefore a means of incorporating this  phenomenon into the computer model was sought.  The  "collapsing surface" condition described below was  at describing this effect.  an attempt  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  program in the following manner:  approximation was  applied  to  the SSMCR  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 allowed  to calculate the fluid velocity at subsequent time intervals.  calculation gravity. to  progressed,  the  fluid  surface was  allowed  As  to collapse under  The effect of bubbles was taken Into account by raising the surface  i t s i n i t i a l position at time intervals corresponding  frequency  ( u s u a l l y set to be 8-10  corresponding and  run, and  s  to an input bubble  Thus, at computational  times  to every l/10th of a second, the surface collapse was stopped,  the free surface was  proceeded from this new  set to i t s i n i t i a l  position.  Calculation then  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.  81  This  is  because  that  the  surface  necessary the  the  to  surface  seconds,  is  velocity  movement the  cycle.  system  calculated  calculated  at  velocity  changes  collapsing.  compare  and  profiles  fluid  at  0.110  s  Therefore,  profiles For  time  0.110 and  continuously  taken  example, step  s  and  0.215  if  was  s  to  detect  at  identical  the  0.005  0.210  s  should  throughout  not  steady-state,  surface  time was  seconds,  could  be  be  the  compared;  compared  for  it  was  intervals  reset  then  time  in  every  two  0.1  velocity  two  profiles  this  purpose  however.  Figures generated is  by  obvious  plots.  deviation Table  this that  The  values  0.8  and  5.12  condition there  i n i t i a l  at  i s ,  for  two  in  surface  show  points  fact,  shape  steady-state  l i t t l e  used  calculations  in  was  performed  surface  difference  that  of  vectfc  collapse between  Figure  5.9.  this  condition  obvious  that  this  plots  cycle. these  The  by  very  examining  low  degrees  values and  a  Table for  5.6,  the  it  mean  maximum  is  c e l l  value  angular  of  12  approximation  also  yields  very  low  velocity  to  by  these  facts  15%).  One  can  conclude  predicts  both  the  direction  velocity  vectors.  the  the  velocity  It three  mean  are  c e l l  shown  in  5.6.  By gives  5.11  reduced  density  This  and  boundary  method  the  degrees). distribution  that  relative  in  the  Section  is  (a In  even  5.2.  of more  condition  minimum  value  addition,  deviation  computer  magnitudes  approximation  described  deviation  boundary  model the  values  of this (8  accurately experimental  successful  than  82  However, the mean c e l l 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 i n the  experimental tank.  This i s reasonable, as i t i s 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. is  The results of this approxiation are also shown in Table 5 . 6 , and i t  clear that this attempt i s only partially  successful:  the mean c e l l  magnitude deviation values are lowered, but there is a corresponding increase in the directional and distribution deviations. left  wall  constant  velocity i s increased, upward  velocity  this  condition,  This indicates that as the  boundary condition  with  a  corresponding  tends  to the  decrease in  accuracy.  Therefore, condition  accurately  i t can be concluded predicts  both  that  the collapsing  the orientation  and  surface  the velocity  distribution of the experimental flow patterns, though i s 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. assumed  that  The f i r s t three boundary approximations - a l l of which  the flow was driven  from  the left  boundary  unsuccessful i n predicting the experimental velocity profiles.  - were  all  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 "collapsing  would  not be correct  to assume that  the rather  simplistic  surface" model i s necessarily a true description of what i s  actually occurring in the experiments. Rather, this model indicates that the free  surface  i s of paramount importance  velocity regimes.  in determining the experimental  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 i s 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 i s displaced outward by more fluid rising from below. 4. This water then travels downward and outward across the free surface at high  speed.  It i s this  fluid  which i s 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 i s 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 constantly mathematical  more  physically  introduce  correct  downward-directed  free-surface, while  boundary fluid  maintaining  condition, along  a  might  the length  constant  surface  be to of the shape.  Unfortunately, a l l attempts made to apply this type of condition failed due to computational  instability.  85  Therefore, boundary  appears  performing single-phase describe  the  a  only  completely to  calculations model  be on  (such  Intricacies  possible two  as of  accurate  the  with  phases  SSMCR)  description a  (air  is  gas-liquid  of  the  mathematical and  clearly  water) limited  boundary.  bubble-liquid  model  capable  simultaneously. in  its  a b i l i t y  of A to  86  6.  6.1.  INDUSTRIAL CALCULATIONS  F l u i d Flov i n a Copper Converter  6.1.1.  Assumptions  The dimensions  and p h y s i c a l c o n s t a n t s t h a t were used t o model t h e  f l o w i n a P e i r c e - S m i t h converter are presented  i n T a b l e 6.1. T h i s d a t a was  o b t a i n e d m a i n l y from Johnson ( 6 3 ) , as w e l l as from Bustos e t a l . (17) and H o e f e l e and Brimacombe ( 7 ) .  A number o f s i m p l i f y i n g assumptions were made about t h e f l o w 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 t o proceed:  1.  A l l of the flow bubbles.  i s two d i m e n s i o n a l  and due o n l y  to the i n f l u e n c e o f  Temperature g r a d i e n t s have no e f f e c t .  2.  The e f f e c t o f s l a g on t h e s u r f a c e o f t h e copper matte was n e g l e c t e d .  3.  The b a t h was assumed t o be c o m p l e t e l y i s o t h e r m a l , and I n c o m p r e s s i b l e .  4.  The r e a c t i o n o f a i r w i t h t h e matte and t h e g e n e r a t i o n o f s u l p h u r d i o x i d e was n e g l e c t e d .  5.  C a l c u l a t i o n s were o n l y performed column i t s e l f ,  and a l l f l u i d  on t h e b u l k o f t h e f l u i d .  between t h i s  The bubble  r e g i o n and t h e a d j a c e n t  wall  was n e g l e c t e d from c a l c u l a t i o n s .  The calculations  value presented  f o r the e f f e c t i v e i n the previous  viscosity chapter:  was determined the v i s c o s i t y  from t h e  values  which  were t h e most s u c c e s s f u l i n p r e d i c t i n g t h e e x p e r i m e n t a l d a t a (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  H  n  KHpt<^=fM]  "  eff  1/3  %  (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:  K  ^eff —  = u  ,(l-a)gQ-  T-pr  =  6.28  (6.2)  '  which i s 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 l i e in the region of the free surface.  3.  The kinetic energy of the fluid bath appears to be poorly distributed compared to that observed i n the experimental regions  of the copper converter  relative to the surface.  tank:  that have very  there are large  low fluid  velocities  This may be due to two effects:  a) The copper converter (when 35% f u l l ) i s bounded by a much larger area of wall (40% larger) than does a square tank of equivalent volume. This indicates that the wall w i l l 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 i s 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 i s 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 i n the copper converter.  Based  on the experimental investigation i n which this effect was observed, i t can be concluded  that the stirring efficiency i n the Peirce-Smith converter i s  less than optimal when operating i n 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 i n a zinc slag-fuming involve significantly greater approximation than those made for the Peirce-Smith copper converter.  1.  This is for two reasons:  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 i t s a b i l i t y to model this system.  model i s needed i f accurate calculations are to be performed.  A two phase  91  2.  Unlike  the  injected how  this  case  into  of  the  w i l l  the  slag  affect  copper fuming  the  converter,  furnace  bubble  solids  along  formation  with  and  (powdered gas.  stirring  It  coal)  is  effects  not  are known  within  the  furnace.  However, a  slag  fuming  predictions flow  in  this  6.2.1.  simplifying  with  1.  furnace  The slag  experimental  have  presented  with  the  below  very  previous  assumptions A l l  following  high  fuming  furnace  of  a  (9,10).  mathematical  published as  a  to  general  studies  date,  the  qualitative  of  the  flow  mathematical  in  model  description  of  the  is  the  calculations  allow  the  it  computation  assumptions  was of  necessary  fluid  outlined  in  flow  to  make  profiles  Section  6.1.1  several in  this  were  used,  additions:  concentration  furnace  slag.  to  Thus  fraction  of  was  taken  the  actual  °  where  been  or  vessel.  vessel.  the  no  Assumptions  As  process  are  as  of  =  gas  entrained into  in  account  slag  °slag  gases  (  1  the  density  _  o  c  )  +  slag.  a  by  in  the  liquid  lowering  was  assumed  the to  phase  of  the  density  of  the  be:  Pg  This  (  value  was  set  to  6  *  be  3  )  30%  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 i s 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 i s 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 free-surface  an attempt  to reduce  this  conditions, calculation over  effect  while  maintaining the  the finite difference mesh was  93  altered  to sweep from left  to right, and then from right  continuity  equation was re-written  difference  approximations.  in both  forward  to l e f t .  and reverse  The finite  However, the momentum equations and the free  surface conditions were not altered and remained i n 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.  difficulty i n making additional adjustments  However, due to the extreme to the SSMCR program, and the  amount of approximation already introduced into these calculations, further refinements were not made to the program.  A l l 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. surface condition was applied.  I n i t i a l l y , the collapsing  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  rose and f e l l .  switching from  positive  to negative  as the surface  94  Qualitative observations of the experimental  tank when injected  with a i r from two sides indicate that this i s not what i s 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 i n 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 i s 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  Incorrect results. shortcomings  from  the bubble  column  without  However, in the case of two-sided  of this approximation  creating obviously gas injection, the  are more obvious, as i t i s evident from  observations that the surface cannot possibly be continuously collapsing and reforming.  The collapsing surface condition f a i l s because i t i s forced to  predict - by conservation of mass - a corresponding rise in the surface at the centre of the tank which i s a physically unrealistic result.  This predicted surface rise does exist in the calculations made for the flow i n the copper converter, as can be seen in Figure 6.2.  However,  since the surface i s distorted less, and i s 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, i t 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. 1.0  The wall velocity used to generate Figure 6.4  was  m/s.  6.2.4.  Results  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 convective cells as mentioned earlier.  distinct  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 that  the slag  copper  fuming  converter,  throughout  furnace  with  the v e s s e l .  the  inaccurate flow predictions as these appears fluid  As w e l l ,  to be more highly s t i r r e d  maintaining the slag  relatively  i n the centre  high  indicate than the velocity  of the furnace  appears to acquire a very high downward v e l o c i t y due to the meeting of the two opposing relatively  vortices.  stagnant  v o r t i c e s diverge.  However, this seems to have the e f f e c t of leaving a  zone at the bottom centre of the tank, where the two  97  7.  7*1*  CONCLUSIONS  Experimental  In  the experimental  section of this research, a i r 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.  L i t t l e 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 a i r flowrate.  Photographic  evidence suggests that this effect i s 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 a i r . The a i r holdup in the tank has been seen to vary linearly with air flowrate.  98  7.2.  Calculations - Agreement with Experimental Results  In  an  attempt  regimes,  a  surface  profiles  mathematical  developed,  based  modeled  the  a  in  constant  were  on  and  the  was  of  boundary  in  an  "collapsing driven tested  the  the  of  Based  to  H i l l  industrial)  fluid  flow  velocity  geometries  (48).  Bousinesq  of  and  has  Turbulence  approximation,  Sahai  and  Guthrie  values,  experimental  under-predict  from  boundary  attempt that  the  that  free  upon  experimental away  of  both  circular  viscosity the  surface"  from  approximation  falling  the  different  approximation  the  with  found  The  region  applying  predicting and  method  empirical  compared  number  was  rectangular  v i s c o s i t y model  model  fluid  of  (and  been  has  and  been  assuming  viscosity.  mathematical  the  experimental  and  data.  the  (35)  was  their  Overall,  effective  tested,  respective the  model  viscosity  In  of the  system.  A  was  the  capable  SOLASMAC by  algebraic  Guthrie  experimental  model  program  were  predict  both  completely  predictions Sahai  in  effective  The as  to  to was  predict found  predicted  the  to  condition, free  conditions  the  wherein  flow  the  most  the  recirculation  This  maximum  applied  experimental  yield  surface.  were  fluid  was  to  patterns.  accurate  the  only  velocities  to  the  results of  the  boundary occur  In  stirring  in  surface.  this  observation,  tank  was  "driven"  the  top  of  the  from  bubble  it  was  the  suggested  surface,  column  across  and the  that  the  largely free  due  to  surface.  fluid  99  7.3.  I n d u s t r i a l Calculations  7.3.1.  Flow In a Copper Converter  Utilizing  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 i n the copper  converter i s 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 i s 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  flow i n a slag-fuming furnace has been made.  prediction of the  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.  100  7.4.  Recommendations f o r Further Work  It the  fluid  induced under  right  to  right  at  the  side  be  prevention  observed  the  "slop"  mathematical  by  varying  side  of  the the  the  model  the  conditions, left  during  left  course of  side  program  of  the  copper  boundary  predicts  converter,  this  and  a  a  investigation converter  could  condition. collapse  That  of  corresponding  that  the rise  be i s ,  fluid in  the  f l u i d .  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July  Scaninject Lulea,  International  into  International 12-13,  Liquid  on  Injection  Metallurgy,  Conference  on  Injection  Metallurgy,  1980.  International 11-13,  Conference  1986. Metals,  University  of  Newcastle  upon  Tyne,  1979. 71.  T.  Deb  Roy,  72.  N.  Takemitsu,  73.  G.  de  Vahl  29-43, 74.  W.S.  75.  T.  76.  J . L . L .  Majumdar,  J .  Comput.  Davis  and  J .  Metals,  Phys.,  G.D.  36,  November  pp.  Mallinson,  1981.  236-248,  Computers  pp.  42-47.  1980. and  Fluids,  V o l .  4,  pp.  1976.  Hwang  Miyauchi  1965.  A.K.  and  R.A.  and  Baker  H.  and  Stoehr, Oya, B.T.  J .  AiChE, Chao,  Metals, Journal, AiChE  October V o l .  Journal,  1983,  11,  3,  Vol.  pp.  22-28.  pp.  395-402,  11,  3,  pp.  1965. 268-273,  106  77.  R.D. Mills, J . Roy. Aero. S o c , 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 F i n i t e - D i f f e r e n c e Approximations used by SSMCR  1.  Continuity Equation:  ax  ay  where au ax  u =  "  i-i,j  u  AX  8JL ay  2.  i,j  =  V  i,j  " i,j-1 Ay V  x-Wise Momentum Equation:  2  au  i ap  at  p  ax  g  x  2  a(uv)  a(u )  ay  ax  u +  eff . a u P  &x  2  2  a u, ay  2  where:  i  p / 2 a(u )  N =  | i,j u  +  u  i+i„i  _  ap ax  „  t+At u -u At  P  i,j  " i+l,j pAx P  2 2 ( u . , + n.,, ,) - ( u . , . + u. .) i,j i+i.y 1-1,J 1>3 A Ax  ax  y  au at  I  (  u  i , j "  u  i+i,j  )  ~  Y  4 Ax  u. . . + u. . I ( u . . - u. . i-1, J i»3 1 1-1,3 i . j ) n  N  108  5(uv)  ( =  u  i , j + l,J+l U  i,1 +  ) ( V  - < J,j + U  U  J,j-l  )  (  v  i,j-l  +  v  l+1.1-l>  A Ay  ay  I ( " l . j - "I.j-H) - T  .,j + l - r l , j V  V  l,j-1  +  V  J+l,j-l  '  (  U  l,j-l  - "l  t j  >  4 Ay  £± + all* - i ^ l  (  ax  p  3.  ay  2  u r  i+i,j "  2  i , j  u  +  u  i-i,j  .  i,j+i ~  u  Z  u  2u. i , j  Ax  2  , +  . i,j-l]  Ay  y-Wise Momentum E q u a t i o n :  ay, at  =  _ i a p _ _ p ay y  a(uv) _  a(y ) 2  . j±ff  ay  ax  l  P  J \  ^ v  r  9  "ax  ay  2  9 i  where: t+At av at  _ I  ap  P ay  ~/ 2 a(v ) N  m  _  v  - v At  p =  i , j" i , j + i p  pAy  2 2 ( v . . + v. ....) - ( v . . - + v. .) 1,3 i,3+l i , j - l 1,3  4 Ay  ay  , j  / x a(uv)  v =  +  v  i , j + l  1  (  i,j  v  - i v  t 3  +i>  I  4 AY  ( u , . + u. . , , ) ( v . . + v.,, .) i , j i,J+l i,J i+1,3 / v  4 Ax  ax  T  ' ?  +  u  i.j+i  ( v  i , j ' 'w,]'  | i,3-l v  +  i,3  (  v  i,j-l  - ( u . , . + u. T . . , ) ( v , , 1-1,3 i-l,j+l i-l>j  - y 1V i , ] +  4 AX  v  / v  (  "  v  i , j  )  + V. .) i,3  Vi,.i - i..i v  ?  109  ^eff  2  5x  In  ^eff  ,o v  these  V r  i  +  l . i  ~  2  1 „ 1  +  V  ^, -i -  i,j+l  1 - 1 J  Ax  ay  equations,  V  y  is  used  to  vary  the  .  2v. i2 ,j  . + v. . ,  Ay  amount  of  upwind  differencing.  i , 3 - l  1  110  TABLES  Ill  Table 3.1. Experimental A i r Injection Rates  Run Number  1 2 3 4 5 6 7 8 9 10  A i r Flowrate (std 1/min)  38 68 78 88 120 142 154 166 178 216  N , Fr  0.43 1.40 1.80 2.30 4.40 - 6.30 7.40 8.70 10.20 15.60  Table 3.2.  Values of -A£ at Six Locations l n Experimental Tank, Experiment Number 1  Cell Location i i 3 4 5 6 7 8  3 4 5 6 7 8  3.4 3.8 1.5 -1.4 -3.1 -0.4  113  Table 3.3. Minimum Mean Values of -Ajr of Experimental Runs  114  Table 4.1.  Approximations Used f o r Rectangular Wall Boundaries  Type of Wall Vertical  Horizontal  Free Slip  No Slip  U  i.J-  u  °  i,r  0  v  i,j= " i - l , j  v  i,r  v  i-i,j  u  i,j= - i,j+l  u  i,r  u  i,j+i  v  v  u  i,r  0  v  i,r  0  Table 5.1. Effective Viscosity Values Predicted by the Model of Sahai and Guthrie (44)  Experiment Number 1 2 3 4 5 6 7 8 9 10  Meff^cms)  2.9 3.4 3.5 3.6 3.8 3.8 3.8 3.8 3.8 3.5  Table 5.2.  Comparison of Results Calculated by Constant V e l o c i t y Condition with  Experiment Run Name  u (g/cras) eff  Left Wall Velocity(m/s)  Deviation Values «.ag  C  VC8.6  3  1.0  e  c  10  0.80  VC7.4  400  0.80  1.0  87.8  86.2  86.4  84.5  -35.1  -26.8  -33.2  -20.7  -27.0  86.9  88.9  89.0  -32.3  -34.3  -35.4  -33.2 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  -62.6  -54.0  -57.5  -52.1  -39.0  -47.8  -48.5  -39.6  -47.2  9  5  B  9  "mag 400  87.9  87.8  41.9  "dist  VC2.8  10  40.1  mag  0.80  9  86.9  3  44.3  B  40  8  6  2  47.7  "dist  VC.4  7  5  1  -40.1  Number  di„t  B  VC6.6  4  Experiments  C  9  -51.6  44.3  40.9  33.4  37.6  37.8  24.2  40.9  49.4  48.1  42.4  51.1  46  53.1  55.4  48.9  44.1  50.9  46.6  45.1  49.2  -58.0  -64.1  -62.4  -67.5  13.1  36.2  43.5  -70.6  -76.6  39.4  33.5  26.5  31.8  31.6  15.8  9.8  19.9  25.4  10.9  -41.2  -46.3  -41.0  -31.0  -81.5  -50.6  -71.1  -57.4  -60.2  45.1  38.7 9.1  16.3  18.4  5.5  13.1  -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  "mag  -0.6  6.7  3.2  10.3  -6.5  0.6  1.3  -12.4  -5.8  -17.0  -46.8  -43.2  -50.2  -36.8  -17.8  3.4  19.7  25.7  33.8  8.2  V C  dlst  -63.3 22.0  -54.8 17.8  -59.1 5.1  -53.8 14.3  -41.1 10.7  20.4  Table 5 . 3 .  Estimated Experimental Bubble Column Porosity Values  Experiment Number  B u b b l e Column Porosity  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 Run Name  VK.4  H (g/cins) eff  400  Porosity(Z)  30.0  Left Wall VelocltyO/s)  0.0  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  -12.1  -12.4  2.1  -2.6  -8.9  -8.1  23.7  19.0  7.1  15.1  16.5  -10.7  21.3  30.5  30.1  22.2  57.2  59.8  65.6  65.0  67.3  61.6  58.9  63.8  59.0  60.9  -31.1  -28.7  -25.9  -21.6  -24.8  -21.9  -27.0  -17.8  18.2  16.0  13.4  -7.3  20.9  27.0  20.5  B  B  9 dist  "mag VL.4  400  30.0  0.4  Number  °9 5  diat  -35.3 10.8  ,-25.7 16.7  9.4  0.3  -3.6  4.3  -3.9  Table 5.5.  Comparison of Results Calculated by Pulsed Boundary Condition with Experiments  Experiment Number Run No.  BB2.4  Bubble Frequency  20  Rise Velocity  1.0  t'eff  400  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  -66.4  -72.3  -66.9  -54.3  -56.6  -63.3  -51.4  -54.3  B  9  -76  B  dist  21.1  13.8  1.52  10.2  9.2  -15.5  13.9  24.8  28.4  12.3  30.7  25.1  33.7  38.1  27.5  27.6  32.8  24.5  26.1  28.2  -67.9  -59.2  -62.5  -57.2  -44.6  -50.3  -47.6  -53.6  -42.4  -49.1  26.8  21.4  11.5  19.6  17.2  25.6  32.0  37.1  26.2  68.3  64.6  69.9  71.2  66.5  64.5  68.5  65.5  64.3  67.3  -76.7  -67.7  -70.9  -65.4  -52.7  -58.4  -57.1  -61.8  -53.6  -60.6  43.9  37.8  32.7  37.3  36.0  -21.3  40.8  48.0  49.0  43.7  20.4  12.3  21.9  26.8  16.5  18.2  12.6  11.8  11.6  -78.7  -68.5  -74.4  -69.0  -56.3  -62.1  -59.1  -65.4  -54.1  -57.1  21.5  14.1  11.0  11.0  -16.7  15.5  26.5  29.9  15.2  "mag BB2.4  20  0.8  400  B  8  "disc B  BB3.8  10  1.0  40  mag  B  9  5  dist  "mag BB4.7  10  1.0  400  B  9  B  dist  2.8  -60  4.3  5.5  Table 5.6.  Comparison of Result* Calculated by Collapsing Surface Condition with Experiments  Experiment Number Run Name  re  ff(g/cms)  Left Wall Velocity(m/B)  Deviation Values "mag  S.C0L.4  VG.6  AOO  AOO  0.0  70.0  c  e  1  2  3  4  5  6  7  8  9  10  99.9  99.9  99.9  99.9  99.9  99.9  99.9  99.9  99.9  99.9  0.9  2.9  5.4  6.5  10.9  5.1  0.8  -18.0  27.0  14.1  B  dl8t  1A.5  -8.1  -3.0  7.8  12.4  3.6  -5.5  27.7  15.6  12.2  C  mag  24.7  19.1  29.0  33.3  21.1  21.9  27.0  16.1  22.2  19.8  -40.1  -30.6  -35.4  -30.3  -23.9  -30.7  -27.8  -33.2  -14.6  -23.7  24.1  18.9  9.6  17.3  14.0  1.4  22.8  27.9  36.7  21.3  B  C  e  dlst  121  Table 6.1. Data Used to Model Flow in Peirce-Smith Copper Converter  Quantity  Value  Bath D e n s i t y  4600 kg/m  Bath V i s c o s i t y @ 1200°C  0.1 g/cms  Furnace  3.85 m  Diameter  Tuyere Submergence  0.35 m  3  Table 6.2. Data Used to Model Flow ln Zinc Slag Fuming Furnace  Quantity  Value  Bath  3500 kg/m  Density  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 Porosity  0.30  3  123  FIGURES  124  F i g u r e 1.1.  Schematic Diagram o f P e i r c e - S m i t h Copper C o n v e r t e r .  125  C,CO—C0 Zn. +1/2 0, —ZnO.. (g) 2 (s) 2  Tertiary air (unregulated)  Primary air 05-33Nm /s Coal: I-15 kg/5 3  Water-jacketed Walls  Secondary air 3 3-75 Nm/s 3  Punching Valve & Tuyere Slag Fuming  Figure 1.2.  Process  Schematic Diagram  of Z i n c S l a g Fuming F u r n a c e .  126  Figure 2 . 1 .  Estimate of Flow i n 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 B e h a v i o u r Diagram, from H o e f e l e a n d Brimacombe ( 7 ) .  129  F i g u r e 3.1.  Schematic Diagram o f E x p e r i m e n t a l A p p a r a t u s .  130  \*  20 cm  *|  60 cm  o o o o o  5 holes (j) = I cm  Side  Front •30 cm  Balanced manifold  Top F i g u r e 3.2.  view  Dimensions and C o n s t r u c t i o n o f E x p e r i m e n t a l  Tank.  131  Photodetector  1.8 cm  X.Y.Z  traversing table  Tank'  «—Manifold  Stand  F i g u r e 3.3.  Diagram o f L a s e r - D o p p l e r  System.  132  Water  4  2  3  1  surface  fjm VM  Two - p h a s e  Figure 3.4.  region  V e l o c i t y Measurement L o c a t i o n s W i t h i n E x p e r i m e n t a l  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 o f Tank.  134  Fractional  Figure 3.6.  width  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 o f Tank.  135  V  *  \  t  t  4  \  \  •  \  \  \  \  1  ft.  /  t  l  /  \  \ \ \ 1 \ \  \  /  \  /  1  /  i  ^ MAXIMUM VELOCITY (M/S)  F i g u r e 3.7.  t  =  0.455  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 and S u r f a c e P l o t - E x p e r i m e n t  1.  136  \  •  f  -  \  \  \ \ \ \  \  \ \  /  F i g u r e 3.8.  =  I i  / .  te  MAXIMUM VELOCITY (M/S)  \  J /  /'  \  *  0.452  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 and S u r f a c e P l o t - Experiment 2.  137  MAXIMUM VELOCITY  Figure 3 . 9 .  (M/S)  =  0.575  Experimental Velocity Vector and Surface Plot - Experiment  3.  138  1  \  \  \  \  \  \  \  f  \  \  1  t  4  \  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  \  I  1  *  \ \  0  /  \  0  0  >-  <•  —  MAXIMUM VELOCITY (M/S)  =  *•  0.492  Figure 3.11. Experimental Velocity Vector and Surface Plot - Experiment 5.  140  e  w  \  B  B  \  \  A  \  \  \  \  1  I  \  1  V  \  \  \  \  /  \  V  If  ——i)  4  *  V  t #  MAXIMUM  VELOCITY (M/S)  =  f  0.570  Figure 3.12. Experimental Velocity Vector and Surface Plot - Experiment 6.  141  -B-  f  4  \  ' •  \  \  V  \  \  \  \  \  *  V  \  \  1  \  \  /  /  \  /  i  \  i  i  \ \ V  — —  —  ^  ^  MAXIMUM VELOCITY  3.13.  ?  -  *  4 0  (M/S)  =  0.477  Experimental V e l o c i t y Vector and Surface Plot - Experiment  7  142  M A X I M U M V E L O C I T Y (M/S)  Figure 3.14.  =  0.388  Experimental Velocity Vector and Surface Plot - Experiment 8.  143  t  • • >•>  \  \  ^  \  \  \  N  N.  N  -  \  \  -  *  '  J  I  >  J  ^  ^  /  /  I  I  —  -  —  •  /  1  j  ^  ^  ^  ^  /  *  (M/S)  =  MAXIMUM VELOCITY  Figure 3.15.  X  0.367  Experimental Velocity Vector and Surface Plot - Experiment  9.  144  nL J  nLJ  f—1  =*——•—  —  4  il  4  P  •>  \  \  \  0  \  \  \  /  \  \  /  \  —  9  \  \  T  \  \  /  1  \  s  /  1  0'  —  0  ^—.  ~-  —  —  —  4t-  0  MAXIMUM VELOCITY (M/S) = 0.443 Figure 3.16.  Experimental Velocity Vector and Surface Plot - Experiment 10.  145  20  S o  15  l  l  1  1  1  1  —  —  o  0>  o. a.  Io  10  o  o  o ° —  o o  o°  o .  1  1  40  80  1  120  1  160  1  200  1  240  Air flowrate (1/min)  Figure 3.17. Variation of A i r Holdup i n Experimental Tank with A i r Flowrate.  146  50  1  1  1  1  1  1  1  ~T  40  —  Slag fuming operation  o  30r-*  w  a  20 — a  z.  io  •  D  °  °  •  » 0  1  I I  2  4  6  Modified  Figure 3.18.  1  1  8  10  Froude  V a r i a t i o n of A i r Holdup  1  12  1  14  16  number  i n Experimental Tank and Zinc  Fuming Furnace with Modified Froude Number.  Slag  147  5.0  .1  1  1  1  1  1  o X  o»  4.0  c  —  o  o  o  o o  —  o  U  c -  a>  o  3.0  o o  u c o  0)  2.0  1  1  1  40  70  100 Air  F i g u r e 3.19.  Mean  Cell  Kinetic  1 130  1 160  1 190  flowrate (l/mlh)  Energy  Function of A i r Flowrate.  of Experimental  Measurements  as a  148  1  1  1  1  1  1  O O 3.5  O  >» o •_>  o  —  o o  o  0)  c  3.0  —  o  ;  °  u  o  «0)  2.5  o 0)  2.0  1  1  40  70  1  1  1  1  100  130  160  190  Air  Figure 3.20. Mean C e l l K i n e t i c  flowrate  (l/min)  Energy of Non-Surface C e l l s i n Experimental  Measurements as a function  of A i r Flowrate.  149  r  i—r—i—i—i—i—i—i  5.0  X ? o> c  4.0  Q> c  =  3.0|  o  c o  tt)  2  2.0  1  JL_J  [  7  L  II  1  J  I  15  I  19  L  23  Energy input rote ( J / s )  Figure 3.21.  Variation  of Experimental Mean C e l l  Input Energy.  Kinetic  Energy with A i r  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  Bubble  F i g u r e 4.1.  Schematic  column  Description  of  Boundary  D e s c r i b e the Flow i n the E x p e r i m e n t a l  Conditions Tank.  Necessary  to  154  4.2.  Velocity Fluctuations in a Finite-Difference Cell Computed by SOLASMAC Method.  155  'fy •  fy  'A  A Figure 4 . 3 .  Finite Tank.  A,  'A A  Difference Grid  Used  //// t o Compute F l o w i n E x p e r i m e n t a l  156  BEGIN  (  )  INITIALIZE VARIABLES  READ IN PREVIOUS RESULTS  SET PRESSURE DISTRIBUTION TO HYDROSTATIC  APPLY BOUNDARY CONDITIONS  I  SOLVE MOMENTUM EQUATIONS  CALCULATE DIVERGENCE FOR EACH FULL CELL  UPDATE PRESSUREy AND VELOCITIES  0 MOVE FREE SURFACE  INCREMENT TIME  <D WRITE OUT RESULTS  c Figure 4 . 4 .  STOP  Flowchart of the SSMCR Program.  156  laltlolif* Variable*  bat!  In  Distribution to B>4rotL«tir  s.  (  Apply BouDdary Cosditiooa  Solve Equations  9  Calculate Divergence f o r Each F u l l C e l l  Update Freeaure* Velocities  F i g u r e A.A.  F l o w c h a r t o f t h e SSMCR Program.  157  F i g u r e 4.5.  Diagram of F r e e S u r f a c e L o c a t i o n and Movement Technique as used i n SSMCR.  158  E  E  E  S *  --^„.  F  F  F  Horizontal  >-  v S  E  v  E * empty cell F« full cell S * surface cell  \ \  F  E  \ \  •  F  >  F Vertical  E  E  \ \  F  'N  E N  F  F 45  Figure 4.6.  Recognition Condition.  S  degrees  of Surface  Orientation  for Tangential  Stress  159  1.0 cm / s  I cm  '///////////////////// I cm  Figure 5 . 1 .  Diagram o f Problem used t o T e s t SSMCR.  160  1  1  0.0  '  0.2  '  •  0.4  •  •  0.6  •  •  0.8  1  1  1.0  H, Vertical Distance (cm)  5.2.  Agreement Between P r e d i c t i o n s Flow i n a Square C a v i t y .  Made by SSMCR and SOLASMAC  B  B  B  \  \  \  t  \  \  \ H  B  *  B  B  B  B  '  4  +  \  v  *  • -  A  \  V  -  -  * •  •  \  V  ^  -~  —  ^  0-  MAXIMUM VELOCITY (M/S)  F i g u r e 5.3.  B  '  *  *  *  ,  i  I  =  /  t  0.600  Erroneous P r e d i c t i o n of Square C a v i t y F l o w .  If-  B  B  \  -B  \  B  \  h  B  B  '  B  4  B  *  B  *  +  \  \  \  V  -  -  0  i  i  H  \  \  ^  ^  —  r  *•  /  MAXIMUM  Figure 5 . 4 .  VELOCITY  (M/S) =  0.600  Correct Prediction of Square Cavity Flow.  163  e  B  B  B  B  P-  /  1 \  *  MAXIMUM VELOCITY (M/S) = 0.265  Figure 5 . 5 .  Prediction  of  Experimental  Velocity Condition, Mgff  =  Flow  1 ° g/cm«s.  Regime  Made  by  Constant  164  F i g u r e 5.6.  Prediction  of  Experimental  Velocity Condition, u  g f f  Flow  Regime  = 40 g/cm«s.  Made  by  Constant  165  /  >^-«-  /  f  /  /  jr  «*  «.  /t  t  /  r  »•  *  ?  \  t  *  *  r  r  \  *  •*  *  *  \  <+  0  t  \  +r  Jf  f  ^  \  n LJ  i-i Lf— K  *^  1  1  r  >  r  *  0  •»  *r  -*  *r  -*  \ \  -  MAXIMUM  F i g u r e 5.7.  Prediction  +  *  +  V E L O C I T Y (M/S)  of  Experimental  Velocity Condition, u  f f  Flow  0.586  Regime  • 400 g/cm»s.  Made  by  Constant  166  -e-  \ \  v  MAXIMUM  Figure 5.8.  ^  i  I  t  /  4  *  (M/S)  =  /  VELOCITY  0.001  P r e d i c t i o n of E x p e r i m e n t a l Flow Regime Made by V a r i a b l e D e n s i t y Condition.  167  ~B —B  B  B w  B •r  11  *•  f \  \ \ MAXIMUM  F i g u r e 5.9.  VELOCITY  P r e d i c t i o n of E x p e r i m e n t a l Condition with Left Wall  (M/S) =  0.218  F l o w Regime Made by V a r i a b l e D e n s i t y Velocity.  168  F i g u r e 5.10.  P r e d i c t i o n of E x p e r i m e n t a l F l o w Regime Made by P u l s e d Boundary Condition.  169  Figure 5.11.  Prediction of  Experimental  Surface Condition, T = 0.5  s.  Flow Regime Made by Collapsing  170  \ \ \  \ \ \ \ \ \ i \ / I i  \ \  /  t  /  t  \ \  v  >  r  ^  /  i  if  t  V  F t  Figure 5.12.  Prediction  of  Experimental  Surface Condition, T • 0 . 7 s .  Flow  1  Regime  Made  by  Collapsing  171  Figure 6.1.  Prediction of Flow Regime in a Copper Converter Made by Constant Velocity Boundary Condition.  172  Figure  6.2.  Prediction Collapsing  of  Flow  Surface  Regime  Condition.  in  a  Copper  Converter  Made  by  173  MAXIMUM VELOCITY (M/S) =  F i g u r e 6.3.  0.001  P r e d i c t i o n of Flow Regime i n a Z i n c S l a g Fuming Furnace Made by C o l l a p s i n g Surface  Condition.  174  MAXIMUM VELOCITY (M/S) =  Figure 6 . 4 .  0.625  Prediction of Flow Regime i n a Zinc Slag Fuming Furnace Made by Constant V e l o c i t y Condition.  175  -e  B  B-  -B B  -  MAXIMUM VELOCITY (M/S) =  F i g u r e 6.5.  -B  '  B  -  B  »  B  -  0.062  P r e d i c t i o n o f Flow Regime i n a Z i n c S l a g Fuming Furnace Made by Variable Density  Condition.  

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