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Holdup studies in three-phase fluidized beds and related systems Bhatia, Vinay Kuma 1972

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HOLDUP STUDIES IN THREE-PHASE FLUIDIZED BEDS AND RELATED SYSTEMS by VINAY KUMAR BHATIA B. Sc. (Chem. Eng.), Agra University, India, 1963 M. Tech. (Chem. Eng.), I.I.T., Kharagpur, India, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of CHEMICAL ENGINEERING We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1972 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t p u b l i c a t i o n , i n p a r t or i n whole, or the copying of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed w i t h -out my w r i t t e n p e r m i s s i o n . VINAY K. BHATIA Department of Chemical E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT Previous research i n three-phase f l u i d i z a t i o n has provided some experimental data on in d i v i d u a l phase hold-ups, but no un i f i e d bed model for predicting these holdups under a variety of circumstances. A step i n t h i s d i r e c t i o n i s made here with the development of a "general-ized wake model", which builds upon the e a r l i e r analyses of 0 s t e r g a a r d , of Efremov and Vakhrushev, and of Rigby and Capes. The present analysis takes into account 1. the e f f e c t of siz e and p a r t i c l e content of the bubble wakes, 2. the c i r c u l a t i o n of sol i d s i n i t i a t e d by p a r t i c l e entrainment i n the bubble wakes, and 3. the r e l a t i v e motion between the continuous phase and the dispersed phases. I t does not take into account any surface e f f e c t s , e s p e c i a l l y the s o l i d s w e t t a b i l i t y . The expression used for estimating the wake volume f r a c t i o n was necessarily a r b i t r a r y , due to the paucity of relevant information on bubble wakes, es p e c i a l l y i n the presence of s o l i d s . Comparison of the generalized wake model with previous analyses indicates that the e a r l i e r models are sp e c i a l cases of the generalized Wake model. Where the wake volume fr a c t i o n can be neglected, the generalized model reduces i i i to the "gas-free model," which f o l l o w s from the mechanism proposed by V o l k . T h i s s i m p l i f i e d model g i v e s good p r e d i c t i o n of s o l i d s holdup f o r p r e v i o u s experimental data oh three-phase f l u i d i z a t i o n of heavy and/or l a r g e p a r t i c l e s , i n which the p a r a d o x i c a l bed c o n t r a c t i o n on i n t r o d u c t i o n of gas i s no longer observed. Experiments to t e s t the g e n e r a l i z e d wake model were c a r r i e d out over a p a r t i c l e diameter range o f 1/4 - 3 mm 3 and a p a r t i c l e d e n s i t y range of 2.5 - 11.1 gm/cm , u s i n g water (1 c . p . ) , aqueous g l y c e r o l (2.1 c.p.) and aqueous p o l y e t h y l e n e g l y c o l (63 c . p . ) , c o v e r i n g the p a r t i c l e Reynolds number range of 0.36 - 1800. The experiments were performed i n 20 mm and 2 i n c h diameter t r a n s p a r e n t columns. The l i q u i d s u p e r f i c i a l v e l o c i t y was v a r i e d from 0.4 to 39 cm/sec and the gas ( a i r ) s u p e r f i c i a l v e l o c i t y from 0.2 to 21.0 cm/sec. Holdups i n the three-phase f l u i d i z e d bed, as w e l l as i n the g a s - l i q u i d r e g i o n s above and below the bed, were measured by the p r e s s u r e drop g r a d i e n t method and by the v a l v e s h u t - o f f t e c h n i q u e . Attempts were made to a n a l y z e , as w e l l as t o modify, the methods used f o r measuring h o l d u p s . Thus, whereas the expanded bed h e i g h t i n the 2 0 mm g l a s s column was obtained from somewhat a r b i -t r a r y v i s u a l o b s e r v a t i o n s , the expanded bed h e i g h t i n the 2 i n c h perspex column was o b t a i n e d by the i n t e r s e c t i o n of two s t r a i g h t l i n e s , one o f p o s i t i v e (three-phase region) and one of n e g a t i v e (two-phase region) s l o p e , r e s u l t i n g i v from a p l o t o f the p r e s s u r e drop p r o f i l e i n the a x i a l d i r e c t i o n . S i m i l a r l y , attempts to improve upon the gas holdup measurement techniques produced an e l e c t r o - r e s i s t i v i t y probe w i t h a s l i g h t v a r i a t i o n i n d e s i g n from t h a t employed by Nassos and B a n k o f f , w e l l s u i t e d f o r measurements i n a i r -water f l o w . The subsequent use of the probe f o r measuring gas holdups i n three-phase f l u i d i z e d beds was not as s u c c e s s f u l . For the beds of small l i g h t p a r t i c l e s / the knowledge of wake c h a r a c t e r i s t i c s — t h e s i z e as w e l l as the p a r t i c l e content of the bubble wakes—was shown to be of c r i t i c a l importance f o r s o l i d s holdup p r e d i c t i o n s and of l i t t l e consequence f o r gas holdup p r e d i c t i o n s . On the other hand, f o r the l a r g e or heavy p a r t i c l e s , the g a s - f r e e model again s u f f i c e d f o r p r e d i c t i o n s of s o l i d s holdup, thereby suggest-i n g the i n s i g n i f i c a n t r o l e of bubble wakes i n these systems. The o p e r a t i o n of the three-phase f l u i d i z e d beds i n the Stokes regime, though s u b j e c t t o p a r t i c l e e l u t r i a t i o n , s i m i l a r l y showed no apparent e f f e c t of bubble wakes. The g e n e r a l i z e d wake model was thus v i n d i c a t e d by the experiments, as was a proposed c o r r e l a t i o n f o r bubble r i s e v e l o c i t y i n the absence or presence of s o l i d s , which was i n c o r p o r a t e d i n t o the model. V ACKNOWLEDGEMENTS I g r a t e f u l l y acknowledge the meaningful a s s i s t a n c e p r o v i d e d by Dr. Norman E p s t e i n , under whose quidance t h i s i n v e s t i g a t i o n was conducted. I am indebted t o him f o r r e -viewing the progress of t h i s work both p a t i e n t l y and c r i t i c a l l y . Thanks are due to Dr. S.D. Cavers f o r h e l p i n g w i t h the wake s i z e data i n l i q u i d - l i q u i d systems, to D r s . R.M.R. Branion and G.H. Neale f o r h e l p i n g w i t h some t h e o r e t i c a l a n a l y s e s , and to Dr. K.B. Mathur f o r h e l p f u l d i s c u s s i o n s a t v a r i o u s stages of p r e p a r a t i o n of t h i s t h e s i s . I acknowledge w i t h g r a t i t u d e the help rendered by the Workshop S t a f f i n c o n s t r u c t i o n of experimental f a c i l i t i e s a t v a r i o u s stages of the work and i n b u i l d i n g p a r t s of the apparatus. The f i n a n c i a l a s s i s t a n c e p r o v i d e d by N a t i o n a l Research C o u n c i l of Canada and the U n i v e r s i t y o f B r i t i s h Columbia are g r a t e f u l l y and most c o r d i a l l y acknowledged. My ve r y s p e c i a l thanks are due to Mr. P.M.G. Rao f o r t i m e l y a s s i s t a n c e i n p r e p a r a t i o n o f a l l the diagrams i n t h i s d i s s e r t a t i o n . L a s t but not the l e a s t , thanks are due to my c o l l e a g u e s and f r i e n d s f o r c o n t r i b u t i n g to my "knowledge" over the y e a r s . :And to my f a m i l y , I owe i t a l l . v i TABLE OF CONTENTS PAGE ABSTRACT i i ACKNOWLEDGEMENTS V LIST OF FIGURES x i i LIST OF TABLES xix NOTATION xxiv CHAPTER 1. INTRODUCTION- 1 1.1 Three-phase f l u i d i z e d beds 1 1.2 Gas holdup and expansion c h a r a c t e r i s t i c s of three-phase f l u i d i z e d beds 5 1.3 Wake model for three-phase f l u i d i z e d beds . . 12 1.4 Importance of turbulence phenomena i n three-phase f l u i d i z e d beds 23 1.5 Scope of research 26 2. THEORY 30 2.1 Holdup i n gas-liquid flow 30 2.1.1 Holdup studies 31 2.1.1.1 Bubble dynamics 32 2.1.1.2 Bubble column 43 2.1.1.3 V e r t i c a l cocurrent flow 46 2.1.2 Models for gas holdup predictions . . 49 2.1.3 C i r c u l a t i o n and turbulence i n two-phase gas- l i q u i d flow 58 v i i CHAPTER PAGE 2.2 Voidage i n l i q u i d - s o l i d f l u i d i z e d beds . . . . 59 2.2.1 E f f e c t o f tu r b u l e n c e on voidage i n l i q u i d - s o l i d f l u i d i z e d beds 66 2.3 Holdup i n g a s - l i q u i d - s o l i d f l u i d i z e d beds . . 69 2.3.1 Models f o r three-phase f l u i d i z e d beds 7 0 (A) The g a s - f r e e model . 7 0 (B) The wake model 76 (C) The c e l l model 100 2.3.2 Gas holdup i n three-phase f l u i d i z e d beds 102 2.3.3 Voidage i n three-phase f l u i d i z e d beds 107 3 . EXPERIMENTAL 110 3.1 Apparatus I l l 3.1.1 The 2 0 mm bench top g l a s s column . . . 112 3.1.2 The 2 i n c h perspex column . . . . . . . . 117 3.1.2.1 L i q u i d c y c l e and t e s t s e c t i o n . . . 117 3.1.2.2 Gas c y c l e and bubble n o z z l e . . . . 122 3.1.3 E l e c t r o - r e s i s t i v i t y probe 126 3.1.4 D e s c r i p t i o n of a u x i l i a r y c i r c u i t s f o r measurement of l o c a l gas holdup and . bubble frequency 129 Q u a n t i t i e s measured 12 9 (a) l o c a l gas f r a c t i o n 12 9 (b) bubble frequency 131 A n a l y s i s of the probe s i g n a l 132 3.2 Range o f v a r i a b l e s s t u d i e d 134 (A) Gas holdup i n two-phase g a s - l i q u i d f l o w . . 135 (B) S o l i d s and gas holdup i n a three-phase f l u i d i z e d bed 136 v i i i CHAPTER PAGE 3.3 Experimental procedure 138 3.3.1 P h y s i c a l p r o p e r t i e s of the l i q u i d s used 138 3.3.2 P h y s i c a l p r o p e r t i e s of the s o l i d s used . . 140 3.3.3 Measurement of gas holdup i n gas-l i q u i d flow 143 3.3.4 Holdup s t u d i e s i n three-phase f l u i d i z e d beds 146 3.4 Data p r o c e s s i n g 148 3.4.1 Expanded bed h e i g h t and s o l i d s holdup 148 3.4.2 Gas holdup 156 4. RESULTS AND DISCUSSION 160 4.1 Comparison of proposed mathematical models w i t h p r e v i o u s work 160 4.1.1 Gas holdup i n g a s - l i q u i d flow . . . . 160 4.1.2 Gas holdup i n three-phase f l u i d i z e d beds 167 4.1.3 Voidage i n three-phase f l u i d i z e d beds 18 2 Beds of 6 mm p a r t i c l e s (Figures 4.2a, b and Table 4.7) 189 Beds o f 3 mm p a r t i c l e s ( F i g u r e s 4.3a, b) 194 Beds o f 1 mm p a r t i c l e s (Table 4.8 and F i g u r e 4.4) 197 4.2 D i s c u s s i o n o f experimental r e s u l t s and comparison w i t h t h e o r i t i c a l p r e d i c t i o n s . . . 2 08 4.2.1 E v a l u a t i o n o f experimental techniques 2 08 i x CHAPTER PAGE 4.2.2 Gas holdup r e s u l t s 2 09 4.2.2.1 Gas holdup i n ga s - l i q u i d flow . . . 209 (A) 20 mm glass column 210 (B) 2 inch perspex column 211 (i) Bubble column 211 ( i i ) Cocurrent flow 218 E l e c t r o - r e s i s t i v i t y probe measurements 2 36 (i) Air-water flow 2 36 ( i i ) Air-PEG solution flow . . 241 4.2.2.2 Gas holdup i n three-phase f l u i d i z e d beds 249 (A) 20 mm glass column 249 (B) 2 inch perspex column 254 (i) Air-water-1/4 mm glass beads 254 (i i ) Air-water-1/2 mm glass beads . 262 ( i i i ) Air-water-1 mm glass beads 27 3 (iv) Air-water-lead shot 279 (v) Air-PEG solution-1 mm glass beads 286 (vi) Air-PEG s o l u t i o n - s t e e l shot 290 4.2.3 Voidage r e s u l t s 295 4.2.3.1 Voidage i n l i q u i d - s o l i d f l u i d i z e d beds 295 (A) 20 mm glass column 296 (B) 2 inch perspex column 298 glass beads-water 302 lead shot-water 305 s o l i d particles-PEG s o l u t i o n . . 307 4.2.3.2 Voidage i n three-phase f l u i d i z e d beds 308 (A) 20 mm glass column 308 (i) Air-water-1/2 mm sand . . . . 308 (ii ) Air-water-1 mm glass beads. . 311 ( i i i ) Air-aqueous glycerol-1 mm glass beads 314 (B) 2 inch perspex column 318 X CHAPTER PAGE (i) Air-water-1/4 mm glass beads 318 ( i i ) Air-water-1/2 mm glass beads 327 ( i i i ) Air-water-1 mm glass beads 338 (iv) Air-water-lead shot 343 (v) Air-PEG solution-1 mm glass beads 351 (vi) Air-PEG s o l u t i o n - s t e e l shot 354 5. CONCLUSIONS 358 6. NOMENCLATURE 365 7. LITERATURE CITED 370 8. APPENDICES A l 8.1 A c e l l model for three-phase f l u i d i z a t i o n . . A l 1. The case of a stationary s o l i d p a r t i c l e with v e r t i c a l l y upwards l i q u i d flow . . . A4 2. The case of a r i s i n g bubble i n a co-current l i q u i d flow A10 Results A17 8.2 Measurement techniques for gas holdup (and bed height) i n three-phase f l u i d i z e d beds and related systems A2 3 1. Modified pressure drop gradient method. . A24 2. Direct volumetric measurements using quick c l o s i n g valves A33 8.3 Estimation of bubble size from l o c a l bubble frequency measurements A^3 8.4 Ca l i b r a t i o n of l i q u i d flow meters A 4 6 8.5 C a l i b r a t i o n of rotameters A 4 9 8.6 Physical properties of materials used . . . A^2 x i CHAPTER PAGE (A) S o l i d densities and p a r t i c l e sizes A 5 2 (B) Densities and v i s c o s i t i e s of l i q u i d s . . . A53 8.7 Processed data and re s u l t s A58 8.8 Publications A70 8.9 Error Estimation A71 8.10 Calculations A74 x i i LIST OF FIGURES FIGURE PAGE 2.1 Gas holdup data from l i t e r a t u r e f o r a i r -water system i n bubble columns of 1-42 i n c h diameter 44 2.2 Schematic r e p r e s e n t a t i o n of the g a s - f r e e model 72 2.3 Schematic r e p r e s e n t a t i o n o f the wake model 7 8 2.4 Schematic of wake s t r u c t u r e suggested by de Nevers and Wu [61] 96 3.1 Schematic diagram o f 2 0 mm g l a s s column apparatus 113 3.2 The 20 mm g l a s s column 116 3.3 Schematic diagram o f 2 i n c h perspex column apparatus . 118 3.4 P r e s s u r e drop apparatus f o r measuring l o n g i t u d i n a l p r e s s u r e drop p r o f i l e i n the experimental s e c t i o n 121 3.5 L o c a t i o n of pressure taps and b a l l v a l v e s f o r gas holdup measurements 12 3 3.6 Design of gas i n l e t and d i s t r i b u t o r 125 3.7 C i r c u i t diagram f o r e l e c t r o - r e s i s t i v i t y probe 127 3.8 E l e c t r o - r e s i s t i v i t y probe and mount f o r t r a v e r s i n g mechanism 130 3.9 Schematic diagram of the a n a l o g u e - l o g i c c i r c u i t f o r measuring l o c a l bubble-frequency and gas f r a c t i o n 133 3.10 T y p i c a l p ressure drop p r o f i l e i n l i q u i d -s o l i d f l u i d i z a t i o n 150 x i i i FIGURE PAGE 3.11 T y p i c a l p r e s s u r e drop p r o f i l e i n t h r e e -phase f l u i d i z a t i o n 153 4.1 Proposed models f o r d r i f t v e l o c i t y o f bubble swarms 163 4.2a L i q u i d f r a c t i o n data o f Michelsen and 0 s t e r g a a r d [14] f o r 6 mm g l a s s beads 190 4.2b Bed voidage data o f Michelsen and 0s t e r g a a r d [14] f o r 6 mm g l a s s beads 191 4.3a L i q u i d f r a c t i o n data o f M i c h e l s e n and 0s t e r g a a r d [14] f o r 3 mm g l a s s beads 195 4.3b Bed voidage data o f Michelsen and 0 s t e r g a a r d [14] f o r 3 mm g l a s s beads 196 4.4 L i q u i d f r a c t i o n data o f M i c h e l s e n and 0 s t e r g a a r d [14] f o r 1 mm g l a s s beads 199 4.5a Bed voidage data o f 0 s t e r g a a r d and The i s e n [18] f o r 2 mm g l a s s beads 2 0 2 4.5b Comparison of eV' by g e n e r a l i z e d wake model wit h e£ by Efremov-Vakhrushev equations f o r 2 mm g l a s s beads a t <j1> = 11.0 cm/sec 203 4.6 Comparison of average bubble r i s e v e l o c i t i e s p r e d i c t e d by equation 4.13 w i t h experimental data i n 20 mm g l a s s column 212 4.7 A x i a l v a r i a t i o n of gas holdup i n bubble column (<j1> = 0.0) 214 4.8a Gas holdup i n bubble column 2IV 4.8b Comparison of measured and p r e d i c t e d gas holdups i n 2 i n c h bubble columns 219 4.9 A x i a l d i s t r i b u t i o n o f gas holdup i n a i r -water flow a t <J1> = 1.25 cm/sec 2 23 4.10 A x i a l d i s t r i b u t i o n of gas holdup i n a i r -water flow at <j-,> = 1.87 cm/sec 2 24 x i v FIGURE PAGE 4.11 A x i a l d i s t r i b u t i o n of gas holdup i n a i r -water flow a t <j , 2> = 4.55 cm/sec 22 6 4.12 A x i a l d i s t r i b u t i o n o f gas holdup i n a i r -PEG s o l u t i o n flow 227 4.13 Comparison of equation 4.18 w i t h e x p e r i -mental data i n 2 i n c h perspex column 2 33 4.14 Gas holdup f o r c o c u r r e n t a i r - w a t e r flow i n 2 i n c h perspex column 23 4 4.15 Gas holdup f o r c o c u r r e n t air-PEG s o l u t i o n flow i n 2 i n c h perspex column 235 4.16 Comparison of r a d i a l gas holdup p r o f i l e s computed by equation 2.27b w i t h e x p e r i -mental data 238 4.17 R a d i a l gas holdup p r o f i l e s i n air-PEG s o l u t i o n flow 24 3 4.18 Gas holdup i n 2 0 mm g l a s s column f o r three-phase f l u i d i z a t i o n by a i r and water of 1 mm g l a s s beads 250 4.19 Gas holdup i n 20 mm g l a s s column f o r three-phase f l u i d i z a t i o n by a i r and aqueous g l y c e r o l of 1 mm g l a s s beads 251 4.20 Gas holdup i n 20 mm g l a s s column f o r three-phase f l u i d i z a t i o n by a i r and water of 1/2 mm sand p a r t i c l e s 2 5 2 4.21 Gas holdup i n three-phase beds of 1/4 mm g l a s s beads f l u i d i z e d . b y a i r and water . . . . 255 4.22 Gas holdup i n three-phase beds of 1/2 mm g l a s s beads f l u i d i z e d by a i r and water . . . . 264 4.23 Comparison o f gas holdup p r o f i l e s computed by^equation 2.27b w i t h experimental d a t a . . . 270 4.24 Gas holdup i n three-phase beds of 1 mm g l a s s beads f l u i d i z e d by a i r and water . . . . 274 4.2 5 Gas holdup i n three-phase beds of 2 mm l e a d shot f l u i d i z e d by a i r and water 28 0 XV FIGURE PAGE 4.2 6 Gas holdup i n three-phase beds of 1 mm g l a s s beads f l u i d i z e d by a i r and PEG s o l u t i o n 287 4.27 Gas holdup p r o f i l e s i n three-phase f l u i d i z e d bed 2 8 9 4.28 Gas holdup i n three-phase beds of 3 mm s t e e l shot f l u i d i z e d by a i r and PEG s o l u t i o n 291 4.2 9 The expansion c h a r a c t e r i s t i c s o f 1/2 mm sand p a r t i c l e s f l u i d i z e d by water 299 4.30 E s t i m a t i o n of expanded bed h e i g h t and s o l i d s holdup d i s t r i b u t i o n from l o n g i t u d -i n a l p r essure drop p r o f i l e i n bed of 1 mm g l a s s beads f l u i d i z e d by water 303 4.31 E s t i m a t i o n of expanded bed h e i g h t and s o l i d s holdup d i s t r i b u t i o n from l o n g i t u d -i n a l p r e s s u r e drop p r o f i l e i n bed of 1/4 mm g l a s s beads f l u i d i z e d by water 304 4.32 E s t i m a t i o n of expanded bed h e i g h t from l o n g i t u d i n a l p ressure drop p r o f i l e i n bed of 2 mm l e a d shot f l u i d i z e d by water . . . . . 306 4.3 3 Bed voidage i n three-phase beds of 1/2 mm sand p a r t i c l e s f l u i d i z e d by a i r and water i n 20 mm g l a s s column 310 4.34 Bed voidage i n three-phase beds of 1 mm g l a s s beads f l u i d i z e d by a i r and water i n 20 mm g l a s s column 312 4.35 Bed voidage i n three-phase beds of 1 mm g l a s s beads f l u i d i z e d by a i r and aqueous g l y c e r o l i n 20 mm g l a s s columns 316 4.36 Comparison of measured (a) and p r e d i c t e d (b) v a l u e s of bed voidages i n three-phase beds of 1 mm g l a s s beads f l u i d i z e d by a i r and water w i t h those f l u i d i z e d by a i r and aqueous g l y c e r o l . 317 7 X V I FIGURE PAGE 4.37 Variation i n observed pressure drop for e l u t r i a t i n g three-phase bed of 1/4 mm glass beads (dp = 0.323 mm) f l u i d i z e d by a i r (<j 5> = 18.0 cm/sec) and water < j 1 > = 3.18 cm7sec . . . . . . 319 4.38 Bed voidage i n three-phase beds of 1/4 mm glass beads f l u i d i z e d by a i r and water . . . 321 4.39 Comparison of e , calculated from equations 1.3, 2.91, 2.94 and 2.106 with x^ = 0.2, using the measured values of gas and s o l i d s holdup i n three-phase f l u i d i z e d beds of 1/4 mm glass beads, with predicted values ( ) from equation 4.7 with p = 3 . . . . 327 4.40 Bed voidage i n three-phase beds of 1/2 mm glass beads f l u i d i z e d by a i r and water . . . 329 4.41 Comparison of e , calculated from equations 1.3, 2.91, 2.94 and 2.106 with x^O, using the measured values of gas and s o l i d s hold-up i n three-phase f l u i d i z e d beds of 1/2 mm glass beads, with predicted values ( ) from equation 4.7 with. p=3 335 4.42 Comparison of bed voidage data for 1/2 mm sand p a r t i c l e s i n 20 mm glass column with those for 1/2 mm glass beads i n 2 inch perspex column 336 4.43 Bed voidage i n three-phase beds of 1 mm glass beads f l u i d i z e d by a i r and water . . . 338 4.44 Comparison of calculated from equations 1.3, 2.91, 2.94 and 2.106, using the measured values of gas and sol i d s holdup i n three-phase f l u i d i z e d beds of 1 mm glass beads, with predicted values ( ) from equation 4.7 with p=3 342 4.45 Comparison of bed voidage data for 1 mm glass beads i n 20 mm glass column with those i n 2 inch perspex column 344 4.46 Bed voidage i n three-phase beds of 2 mm lead shot f l u i d i z e d by a i r and water 346 x v i i FIGURE 4 .47 4 .48 8.1.1 8.1.2 8.1.3 8.2.1 8.2.2 8.2.3 8.2.4 8.3.1 8.4.1 8.4.2 8.5.1 PAGE Bed voidage i n three-phase beds of 1 mm g l a s s beads f l u i d i z e d by a i r and PEG s o l u t i o n 353 Bed voidage i n three-phase beds of 3 mm s t e e l shot f l u i d i z e d by a i r and PEG s o l u t i o n 357 Schematic of c e l l model f o r three-phase f l u i d i z a t i o n A3 S o l u t i o n of c e l l model f o r three-phase f l u i d i z a t i o n under a r b i t r a r i l y chosen c o n d i t i o n s A19 Expansion behaviour of a three-phase f l u i d i z e d bed as p r e d i c t e d by the c e l l model A20 Schematic of s t a t i c p r essure drop g r a d i e n t along the t e s t s e c t i o n A29 P l o t of manometric e q u a t i o n s : t h r e e -phase f l u i d i z e d bed; — - — l i q u i d - s o l i d f l u i d i z e d bed; — - - two-phase gas-l i q u i d flow A31 Diagramatic r e p r e s e n t a t i o n of the e x p e r i -mental column (a) under running c o n d i t i o n , and (b) w i t h v a l v e s c l o s e d A35 L o n g i t u d i n a l d i s t r i b u t i o n o f a b s o l u t e pressure i n the experimental column w i t h v a l v e s open . A38 A p l a n view of bubble t r a v e r s e over the probe A44 C a l i b r a t i o n curves f o r water flow meters i n 2 i n c h c i r c u l a t i o n loop A47 C a l i b r a t i o n c u rves f o r PEG s o l u t i o n flow meters i n 2 i n c h c i r c u l a t i o n loop A48 C a l i b r a t i o n curve f o r a i r rotameter i n 20 mm g l a s s column setup A50 x v i i i FIGURE PAGE 8.5.2 C a l i b r a t i o n curve for water rotameter i n 20 mm glass column setup A^0 8.5.3 C a l i b r a t i o n curves for a i r rotameter i n 2 inch c i r c u l a t i o n loop A51 8.6.1 Dynamic v i s c o s i t y of polyethylene g l y c o l solutions A^7 x i x LIST OF TABLES TABLE PAGE 2.1 R a t i o of wake to bubble volume f o r v a r i o u s v a l u e s of d i s p e r s e d phase holdup i n two-phase f l u i d systems 98 3.1 Experimental c o n d i t i o n s f o r two-phase gas-l i q u i d flow i n the 20 mm g l a s s column 137 3.2 Experimental c o n d i t i o n s f o r two-phase gas-l i q u i d flow i n the 2 i n c h perspex column 137 3.3 Experimental c o n d i t i o n s f o r three-phase f l u i d i z a t i o n i n 20 mm g l a s s column 139 3.4 Experimental c o n d i t i o n s f o r three-phase f l u i d i z a t i o n i n 2 i n c h perspex comumn 139 4.1 R a t i o of wake to bubble volume i n t h r e e -phase f l u i d i z e d bed (D_>4 inches) 169 4.2 R a t i o o f wake t o bubble volume i n t h r e e -phase f l u i d i z e d bed (D=2 inch) . . 17 0 4.3-A R a t i o of gas holdup i n three-phase f l u i d i z e d bed to gas holdup i n two-phase g a s - l i q u i d flow (D>_4 inches) 174 4.3- B R i s e v e l o c i t y of bubble swarms i n t h r e e -phase f l u i d i z e d bed (D>4 inches) 175 4.4- A R a t i o of gas holdup i n three-phase f l u i d -i z e d bed t o gas holdup i n two-phase gas-l i q u i d flow (D=2 inches) 17 9 4.4-B R i s e v e l o c i t y of bubble swarms i n t h r e e -phase f l u i d i z e d bed (D=2 inches) 180 4.5 Degree of agreement between equations 4.7 and 2.128 f o r p r e d i c t i n g r a t i o o f wake f r a c t i o n to gas f r a c t i o n i n a three-phase f l u i d i z e d bed 185 4.6 Comparison of measured and p r e d i c t e d bed voidages f o r l i q u i d s o l i d f l u i d i z a t i o n . . . . 187 XX TABLE PAGE 4.7 Comparison of measured and p r e d i c t e d gas and l i q u i d f r a c t i o n s i n three-phase f l u i d -i z e d beds 192 4.8 Comparison o f measured and p r e d i c t e d gas and l i q u i d f r a c t i o n s i n three-phase f l u i d i z e d beds 198 4.9 Comparison of measured and p r e d i c t e d gas and l i q u i d f r a c t i o n i n three-phase f l u i d -i z e d beds 204 4.10 T r a n s i t i o n from bubbly t o s l u g flow i n a i r -water flow 220 4.11 Comparison of gas holdups i n v a r i o u s s e c t i o n s of the column f o r a i r - w a t e r flow 229 4.12 E s t i m a t i o n of d i s t r i b u t i o n parameter, CQ, from gas holdup measurements 231 4.13 Gas holdup by e l e c t r o - r e s i s t i v i t y probe f o r a i r - w a t e r flow 239 4.14 Gas holdup by e l e c t r o - r e s i s t i v i t y probe f o r a i r - PEG s o l u t i o n flow 244 4.15 L i q u i d f i l m t h i c k n e s s , 6 * , from e l e c t r o -r e s i s t i v i t y probe measurements i n a i r -PEG s o l u t i o n flow 246 4.16 Average bubble s i z e , r , from e l e c t r o -r e s i s t i v i t y probe measurements i n a i r -PEG s o l u t i o n flow 248 4.17 Comparison o f gas holdup i n three-phase f l u i d i z e d bed t o t h a t i n two-phase r e g i o n s of the column 257 4.18 Comparison of gas holdup i n three-phase f l u i d i z e d bed t o t h a t i n g a s - l i q u i d flow . . . 260 4.19 Comparison of gas holdup i n three-phase f l u i d i z e d bed t o t h a t i n two-phase r e g i o n s of the column 2 65 xx i TABLE PAGE 4.2 0 Comparison o f gas holdup i n three-phase f l u i d i z e d bed to t h a t i n g a s - l i q u i d flow . . . 268 4.21 Gas holdup by e l e c t r o - r e s i s t i v i t y probe i n three-phase f l u i d i z e d bed 271 4.22 Comparison o f gas holdup i n three-phase f l u i d i z e d bed to t h a t i n two-phase r e g i o n s of the column 275 4.23 Comparison o f gas holdup i n three-phase f l u i d i z e d bed to t h a t i n g a s - l i q u i d flow . . . 277 4.24 Comparison of gas holdup i n three-phase f l u i d i z e d bed to t h a t i n two-phase r e g i o n s of the column 281 4.25 Comparison o f gas holdup i n three-phase f l u i d i z e d bed to t h a t i n g a s - l i q u i d flow . . . 283 4.26 C h a r a c t e r i s t i c measurements of gas bubble phase i n three-phase f l u i d i z e d by e l e c t r o -r e s i s t i v i t y probe 288 4.27 Comparison of gas holdup i n three-phase f l u i d i z e d bed to t h a t i n two-phase r e g i o n s of the column 292 4.28 Comparison of average bubble s i z e i n t h r e e -phase f l u i d i z e d bed to t h a t i n g a s - l i q u i d flow 294 4.29 Expansion r e s u l t s f o r l i q u i d - s o l i d f l u i d -i z a t i o n i n 20 mm column 297 4.30 Expansion r e s u l t s f o r l i q u i d - s o l i d f l u i d -i z a t i o n i n 2 i n c h column 300 4.31 S e n s i t i v i t y o f bed v o i d a g e , e, p r e d i c t e d from g e n e r a l i z e d wake model w i t h xk=0.2, to wake volume f r a c t i o n , e^, i n t h r e e -phase beds o f 1/4 mm g l a s s beads at <j - j _> = 1.87 cm/sec 323 4.32 Estimated v a l u e s o f wake volume f r a c t i o n , ek, from experimental data i n three-phase beds of 1/4 mm g l a s s beads f l u i d i z e d by a i r and water x x i i TABLE PAGE 4.33 S e n s i t i v i t y of bed v o i d a g e , e, p r e d i c t e d from g e n e r a l i z e d wake model wit h x^=0.0 to wake volume f r a c t i o n , ej^, i n t h r e e -phase beds o f 1/2 mm g l a s s beads 332 4.34 Estimated v a l u e s of wake volume f r a c t i o n , e^, from experimental data i n three-phase beds o f 1/2 mm g l a s s beads f l u i d i z e d by a i r and water. 333 4.35 Estimated v a l u e s of wake volume f r a c t i o n , ek, from experimental data i n three-phase beds of 1 mm g l a s s beads f l u i d i z e d by a i r and water 341 4.3 6 Comparison o f measured bed voidages i n three-phase bed of 2 mm l e a d shot f l u i d i z e d by a i r and water wi t h the va l u e s p r e d i c t e d by the g a s - f r e e model 350 4.37 Comparison of measured bed voidages i n three-phase bed o f 1 mm g l a s s beads f l u i d -i z e d by a i r and PEG s o l u t i o n w i t h the va l u e s p r e d i c t e d by the g a s - f r e e model 355 8.6.1 S o l i d d e n s i t i e s and p a r t i c l e s i z e s A 52 8.6.2 D e n s i t y and v i s c o s i t y o f w a t e r - g l y c e r o l s o l u t i o n A5 3 8.6.3 C a l i b r a t i o n of Canon viscometer (300-H-304) A54 8.6.4 D e n s i t y and v i s c o s i t y o f p o l y e t h y l e n e g l y c o l - w a t e r s o l u t i o n (I) A^5 8.6.5 D e n s i t y and v i s c o s i t y of p o l y e t h y l e n e g l y c o l - w a t e r s o l u t i o n (II) A56 8.7.1 Gas holdup data f o r a i r - w a t e r flow i n 20 mm g l a s s column A5 9 8.7.2 Gas holdup data i n 2 i n c h perspex column f o r (A) a i r - w a t e r flow A6 0 (B) air-PEG s o l u t i o n flow A 6 0 8.7.3 S o l i d s holdup data f o r l i q u i d - s o l i d f l u i d i z a t i o n i n 20 mm g l a s s column A61 x x i i i TABLE PAGE 8.7.4 Solids holdup data for l i q u i d - s o l i d f l u i d i z a t i o n i n 2 inch perspex column . . . . A62 8.7.5 Gas and s o l i d s holdup data i n 20 mm glass column for three-phase beds of (A) air-water-1/2 mm sand A6 3 (B) air-water-1 mm glass beads A64 (C) air-aqueous glycerol-1 mm glass beads . . A63 8.7.6 Gas and sol i d s holdup data i n 2 inch perspex column for three-phase beds of (A) air-water-l/4 mm glass beads A65 (B) air-water-1/2 mm glass beads A66 (C) air-water-1 mm glass beads A67 (D) air-water-2 mm lead shot A68 (E) air-PEG solution-1 mm glass beads . . . . A69 (F) air-PEG s o l u t i o n - s t e e l shot A69 8.9.1 Estimated errors i n experimental r e s u l t s . . A7 3 NOTATION The a n a l y s i s o f multiphase f l o w , i n which s e v e r a l phases flow s i m u l t a n e o u s l y , has drawn c o n s i d e r a b l e a t t e n t i o n i n r e c e n t y e a r s . For two-phase f l o w , the s i m p l e s t case of multiphase f l o w , the i n f o r m a t i o n a v a i l a b l e i n the l i t e r a t u r e , though o f t e n c o n t r a d i c t o r y , i s s t a g g e r i n g . S i n c e no standard d e f i n i t i o n s of the terms used f o r d e s c r i b i n g the multiphase flow phenomenon e x i s t , the terminology adopted by v a r i o u s r e s e a r c h e r s i s o f t e n vague and sometimes v e r y confus-i n g . Zuber and F i n d l a y [39] t r i e d to develop a r i g o r o u s terminology which g i v e s a p h y s i c a l meaning to each o f the terms. L a t e r the same terminology was used e x t e n s i v e l y by W a l l i s [ 27]. The terminology employed i n t h i s t h e s i s f o l l o w s c l o s e l y t h a t adopted by W a l l i s , and r e c e n t l y extended by -Bhaga [ 1 ] . Although the nomenclature i s l i s t e d a t the end, some e x p l a n a t i o n o f the more common terms i s g i v e n here i n order to p r o v i d e some f a m i l i a r i t y w i t h the simple r e l a t i o n -s h i p s among these terms. The three phases i n v o l v e d i n three-phase f l u i d i z a t i o n are, i d e n t i f i e d by s u b s c r i p t s 1, 2 and 3 i n g e n e r a l . S u b s c r i p t d e s c r i b e s the continuous l i q u i d phase, w h i l e s u b s c r i p t s 2 and 3 r e p r e s e n t the d i s p e r s e d gas and s o l i d phases r e s p e c t i v e l y . For the three-phase system c o n s i d e r e d i n t h i s XXV study the gas and l i q u i d phases flow cocurrently upwards i n a v e r t i c a l c y l i n d r i c a l pipe. A c y l i n d r i c a l coordinate system i s used i n which Z represents the distance measured v e r t i c a l l y upwards and r denotes the r a d i a l distance from the pipe axis. The volumetric flow rate i s represented by the symbol Q. The t o t a l volumetric flow rate i s then the sum of the i n d i v i d u a l component flows: Q = Q-L + Q 2 + Q 3 (i) Since i n t h i s investigation flow of s o l i d p a r t i c l e s i s not considered, i s zero and equation (i) reduces to Q = Q1 + Q 2 ( i i ) In a three-phase f l u i d i z e d bed, every part of the three-phase region w i l l be occupied by one phase or another at any instant of time. I f we consider an element of volume which i s very much smaller than the volume of either a gas bubble or a s o l i d p a r t i c l e , then a ( t ) , representing the f r a c t i o n of elemental volume occupied by one phase, can p r a c t i c a l l y only be either 0 or 1. The temporal average of t h i s event (occupation by one phase) occurring over a long period w i l l represent the s t a t i s t i c a l average volume f r a c t i o n of the given phase for that s p a t i a l l y fixed elemental volume: xxv i a = ^ / T a (t) d t ( i i i ) 1 0 For homogeneous d i s t r i b u t i o n o f the g i v e n phase, <a> = a. However, i f the three-phase r e g i o n i s not of homogeneous d i s t r i b u t i o n , i t becomes n e c e s s a r y , i n order to o b t a i n a t r u l y s i g n i f i c a n t v a l u e of the average v o l u m e t r i c f r a c t i o n , t h a t the averaging process be c a r r i e d out both i n space and time. Then - / / q ( rft ) r d r d t . < a > ~ / r d r /dt U v ) Thus i n order to av o i d any a m b i g u i t i e s about the measurement of average v o l u m e t r i c f r a c t i o n o f a phase, i t i s e s s e n t i a l to d e f i n e e x p l i c i t l y how the averaging process was c a r r i e d o u t . For many purposes the d i s t r i b u t i o n o f phases i n the e n t i r e f i e l d i s not r e q u i r e d ; t h e r e f o r e the averaging can be done over a much l a r g e r volume of the three phase r e g i o n , and a then r e p r e s e n t s the average v o l u m e t r i c f r a c t i o n (hold-up) o f one phase. In t h i s case a i s measured over the e n t i r e c r o s s - s e c t i o n of the c o n d u i t and over s u f f i c i e n t l e n g t h to e l i m i n a t e any spurious l o n g i t u d i n a l v a r i a t i o n s . Thus, i f a batch o f s o l i d p a r t i c l e s of mass, W, i s expanded to a h e i g h t , L ^ , i n a pipe o f c r o s s - s e c t i o n a l a r e a , A, the average v o l u m e t r i c f r a c t i o n o f s o l i d s ( s o l i d s holdup) i n the flow r e g i o n w i l l be xx v i i <a_,> = W/p 3 A L b (v) Sim i l a r l y i f we i s o l a t e the three-phase flow region by cutting o f f the gas and the l i q u i d flow rates simultaneously, then the average volumetric f r a c t i o n of gas (gas holdup) can be obtained by measuring the ultimate volume of gas c o l l e c t e d , 0, - , at the top: flux or volumetric flow rate per unit area of conduit, and i s equivalent to what has been commonly c a l l e d " s u p e r f i c i a l " v e l o c i t y i n the l i t e r a t u r e . Flux i s indeed a vector quantity, but i n this study j w i l l be used to represent the scalar component i n the d i r e c t i o n along the pipe. Then by d e f i n i t i o n , = n 2 / A L b (vi) The symbol j i s used to represent the volumetric QL/A (vii) Q2/A ( v i i i ) Q3/A = 0 (ix) and the t o t a l average volumetric f l u x i s X X V I 1 1 The l o c a l volumetric flux i s related to l o c a l phase volumetric f r a c t i o n and v e l o c i t y as follows: h = a ± v i <xi> J 2 = a2 v2 (xii) j 3 = a 3 v 3 ( x i i i ) and the t o t a l l o c a l volumetric f l u x i s j a s - 3 i + J 2 + 3 3 (xiv) The r e l a t i v e v e l o c i t y between the f l u i d phases i s defined as V21 = v2 " V l = _ V12 ( X V ) D r i f t v e l o c i t i e s are defined as the difference between the phase v e l o c i t i e s and the t o t a l volumetric f l u x . Thus v x j = v x - j (xvi) V 2 j = v2 " ^ (xvii) v 3 j = v 3 - j ( x v i i i ) The d r i f t f l u x of phase i represents the volumetric flux of that phase based on d r i f t v e l o c i t y : j i j = a i V i i = a i < v i - 3 ) < x i x ) xx ix The symbol Ap i s used to describe the pressure drop i n a pipe. If dp/dZ represents the rate at which pressure increases with distance i n the Z d i r e c t i o n , then pressure drop over a length of pipe L w i l l be - Ap = - / L (dp/dZ) dZ 0 (xx) XXX The following brief citation on Indian Philosophy is presented to illuminate "thought": • • • One of the most important t o p i c s of orthodox Indian p h i l o s o p h y was the q u e s t i o n of pramana — "means of r e l i a b l e knowledge." Four means o f r e l i a b l e knowledge were r e c o g n i z e d : p e r c e p t i o n {pratyaksa) , i n f e r e n c e (anumana) , i n f e r e n c e by analogy or comparison (upmana) , and "word" (sabda) , the pronouncement of r e l i a b l e a u t h o r i t y , such as the Vedas. The m a t e r i a l i s t s allowed only p e r c e p t i o n , but I n d i a developed her own system of l o g i c i n the study o f the process o f i n f e r e n c e . A c o r r e c t i n f e r e n c e was e s t a b l i s h e d by s y l l o g i s m , of which the Indian form (pancavayava) comprises f i v e members: p r o p o s i t i o n {pratijna) , reason (hetu), example (udaharana) , a p p l i c a t i o n (upnaya), and c o n c l u s i o n (nigamana). The c l a s s i c a l example of Indian s y l l o g i s m may be paraphrased as f o l l o w s : 1. There i s f i r e on the mountain, 2. because there i s smoke above i t , 3. and where th e r e i s smoke the r e i s f i r e , a s , f o r i n s t a n c e , i n a k i t c h e n ; 4. such i s the case w i t h the mountain, 5. and t h e r e f o r e there i s f i r e on i t . The I n d i a n s y l l o g i s m r e v e r s e d the order o f t h a t o f c l a s s i c a l l o g i c ( A r i s t o t e l i a n : major premiss ( 3 ) , minor premiss ( 2 ) , and c o n c l u s i o n ( 1 ) ) , the argument being s t a t e d i n the f i r s t and second c l a u s e s , e s t a b l i s h e d by the g e n e r a l r u l e and example i n the t h i r d , and f i n a l l y c l i n c h e d by v i r t u a l r e p e t i t i o n of the f i r s t two c l a u s e s . The "example" ( i n the above s y l l o g i s m the kitchen) was g e n e r a l l y looked on as an e s s e n t i a l p a r t of the argument, and helped to strengthen i t s xxxi r h e t o r i c a l force. The basis of generalization (for example "where there i s smoke there i s f i r e " ) on which every i n f e r -ence rests was believed to be the q u a l i t y of universal concomitance (vyapti) . . . . But the world i s more complex and subtle than we think i t , and that what i s true of a thing i n one of i t s aspects may at the same time be f a l s e i n another. Basham, A.L., "The wonder that was India," Grove Press, Inc., New York (1959). 1 CHAPTER 1 INTRODUCTION 1.1 Three-phase f l u i d i z e d beds A l a r g e number of chemical and process e n g i n e e r i n g systems r e q u i r e b r i n g i n g two or more phases (gas, l i q u i d and s o l i d ) i n t o i n t i m a t e c o n t a c t w i t h each o t h e r . The chemical c h a r a c t e r i s t i c s o f the phases i n v o l v e d are determined by the requirements of the process i t s e l f , b u t the dynamic behaviour o f these phases i s p r i n c i p a l l y governed by the r e l a t i v e motion between the l i g h t e r and the h e a v i e r phases. F l u i d i z a t i o n i s a phenomenon i n v o l v i n g r e l a t i v e motion between a f l u i d phase and a s o l i d phase i n such a manner th a t s o l i d p a r t i c l e s are supported and maintained i n a suspended s t a t e by upward f l o w i n g f l u i d . Under these c o n d i t i o n s g r a v i t y f o r c e s on the p a r t i c l e s , m o d i f i e d by buoy-ancy, are balanced by drag f o r c e s a r i s i n g from the r e l a t i v e motion between the f l u i d and the s o l i d p a r t i c l e s . The g e o m e t r i c a l s t r u c t u r e o f a f l u i d i z e d bed i s very much dependent on the nature o f the f l u i d i z i n g medium. I f the f l u i d used i s a l i q u i d , a f l u i d i z e d bed o f s i n g l e s i z e d s p h e r i c a l p a r t i c l e s w i l l g e n e r a l l y appear to be homogeneously d i s t r i b u t e d . There are then no i n t e r - p a r t i c l e c o l l i s i o n s , each p a r t i c l e appears to have i t s i n d i v i d u a l i d e n t i t y and t h e r e f o r e such a l i q u i d - s o l i d f l u i d i z e d bed i s d e s c r i b e d as 2 " p a r t i c u l a t e l y f l u i d i z e d " . The gross expansion characteris-t i c s of a p a r t i c u l a t e l y f l u i d i z e d bed under these i d e a l conditions can be predicted t h e o r e t i c a l l y by various c e l l models (Happel/ Kuwabara), but i s best represented by an empirical r e l a t i o n developed by Richardson and Zaki [2]. However, i f the f l u i d used i s a gas, large or small s o l i d -free aggregates usually r i s e up through the bed, giving i t the appearance of a b o i l i n g l i q u i d , and therefore such a gas-solid f l u i d i z e d bed i s described as "aggregatively f l u i d -i z e d " . The s o l i d - f r e e aggregates or gas bubbles give r i s e to heterogeneous p a r t i c l e d i s t r i b u t i o n s i n gas-solid f l u i d i z e d beds, and therefore the description of bed expansion becomes more d i f f i c u l t and complex. In three-phase f l u i d i z a t i o n , a mixture of gas and l i q u i d i s used as the f l u i d i z i n g medium to maintain s o l i d p a r t i c l e s i n the suspended state. Three phases (gas, l i q u i d and solid) are thereby brought into contact simultaneously. The term 'three-phase f l u i d i z a t i o n 1 has been used rather loosely to describe the process of bringing gas, l i q u i d and suspended s o l i d p a r t i c l e s into contact, where the gas and the l i q u i d may be flowing either cocurrently or counter-currently. As defined i n this thesis, however, the term i s r e s t r i c t e d to the f l u i d i z a t i o n of s o l i d p a r t i c l e s by a mixture of gas and l i q u i d flowing cocurrently and v e r t i c a l l y upwards i The term gas-liquid* f l u i d i z a t i o n has also been used to describe t h i s mode of contacting. The pred i c t i o n of holdup 3 for each phase of a three-phase f l u i d i z e d bed requires a knowledge of equilibrium conditions between the three phases, which i n turn i s governed by the l o c a l r e l a t i v e v e l o c i t i e s between the phases. The geometrical structure and physical appearance of the bed w i l l depend on the l o c a l holdup of each phase i n the bed. The bed w i l l appear to be p a r t i c u l a t e l y f l u i d i z e d at low gas-to-liquid r a t i o s [7,8] and aggregatively f l u i d i z e d at large gas-to-liquid r a t i o s , as has been v i s u a l l y observed by various investigators [13]. Jackson [3] and Bhatia [4] observed q u a l i t a t i v e l y that for suspension of granular p a r t i c l e s i n a pool of l i q u i d by gas i n j e c t i o n , the r e s u l t i n g bed consists of s o l i d p a r t i c l e s dispersed i n a continuous l i q u i d phase, with the gas r i s i n g through the medium as discrete bubbles, tran s f e r r i n g i t s momentum to the l i q u i d phase during the ascent. Thus i t i s envisaged that i n a three phase f l u i d i z e d bed the s o l i d p a r t i c l e s are supported e n t i r e l y by the l i q u i d phase alone, which suggests that the c h a r a c t e r i s t i c properties of a three-phase f l u i d i z e d bed can be synthesized on the basis of the properties of two simpler two-phase systems, namely, cocurrent g a s - l i q u i d flow and l i q u i d - s o l i d f l u i d i z a t i o n . The analysis of the complex behaviour of three-phase f l u i d i z e d beds becomes somewhat s i m p l i f i e d i n view of the above observations, since the holdup of l i q u i d - s o l i d f l u i d i z e d beds i s very well described by the empirical r e l a t i o n of Richardson and Zaki, as well as by other s i m i l a r c o r r e l a t i o n s . 4 However, knowledge of l o c a l holdup d i s t r i b u t i o n s of the gas and the l i q u i d phases, and th e i r i n t e r a c t i o n , i s also very important i n order to analyse the phenomenon completely. The study of the d i s t r i b u t i o n of gas and l i q u i d phases i n three-phase f l u i d i z e d beds requires knowledge of two-phase gas - l i q u i d flow and the e f f e c t of s o l i d p a r t i c l e s i n perturbing the d i s t r i b u t i o n of phases i n the gas - l i q u i d flow. The f i e l d of gas-liquid flow has been investigated by a large number of researchers, but s t i l l the understanding of the e f f e c t of i n t e r a c t i o n of the two phases on t h e i r respective flow and holdup p r o f i l e s i s far from complete [39]. This incomplete understanding of the two phase gas- l i q u i d flow phenomenon puts attempts to analyse the behaviour of three-phase f l u i d i z e d beds at some disadvantage. Nevertheless, the three-phase f l u i d i z a t i o n technique has already found i t s way into c e r t a i n process industries involving gas-liquid reaction i n the presence of a suspended s o l i d c a t a l y s t . In such reaction systems, where interphase_ mass transfer i s not the co n t r o l l i n g " r e s i s t a n c e , cocurrent gas-liquid flow i s employed since much higher throughputs can be achieved without flooding than for countercurrent. flow, while the higher mass transfer d r i v i n g force obtainable from the l a t t e r i s not required. The a p p l i c a b i l i t y of the three-phase f l u i d i z a t i o n technique of bringing three phases into contact simultaneously has not been completely investigated [54] but three-phase 5 f l u i d i z e d beds have the i n d u s t r i a l p o t e n t i a l of being employed i n any of the following combinations: 1. gas reacting with l i q u i d i n the presence of s o l i d s as the c a t a l y s t , v i z . hydrogenation of vegetable o i l i n the presence of nickel or other powdered ca t a l y s t s . 2. gas and l i q u i d reacting with s o l i d p a r t i c l e s , e.g. i n the production of calcium b i - s u l f i t e cooking liquor from s o l i d limestone p a r t i c l e s i n the presence of water and sulf u r dioxide [55]. 3. gas reacting with s o l i d p a r t i c l e s i n the presence of l i q u i d c a t a l y s t ; c e r t a i n e s t e r i f i c a t i o n reactions could be c l a s s i f i e d as s p e c i f i c examples. 4. gas reacting with l i q u i d to form the s o l i d phase which i s kept i n suspension [3]. Besides i t s applications i n chemical reaction systems, three-phase f l u i d i z a t i o n can also be employed for physical operations. Thus a three-phase c r y s t a l l i z e r has been designed to regulate the c r y s t a l growth of ammonium-sulfate. [5]. 1.2 Gas holdup and expansion c h a r a c t e r i s t i c s of three-phase  f l u i d i z e d beds The actual and pote n t i a l uses of three-phase f l u i d i z -ation i n various i n d u s t r i a l processes have led various investigators to study i t experimentally. Thus, Turner [6] 6 suggested the p o s s i b i l i t y of using the three-phase f l u i d i z -ation technique i n d e s u l f u r i z i n g higher molecular weight residual petroleum feedstocks. He therefore car r i e d out preliminary hydrodynamic experiments on the air-water-sand system and thus demonstrated f o r the f i r s t time the peculiar behaviour of three-phase f l u i d i z e d beds. Turner observed that introducing a small flow rate of gas to a l i q u i d - s o l i d f l u i d i z e d bed, keeping the l i q u i d v e l o c i t y constant, resulted i n contraction of the bed. Stewart and Davidson [7], 0stergaard [8] , and Adlington and Thompson [9] then investigated t h i s unexpected behaviour of three-phase f l u i d i z e d beds, using a i r and water as gas and l i q u i d respectively, with spherical, single-sized s o l i d p a r t i c l e s of various d e n s i t i e s . They too observed the contraction of a l i q u i d - s o l i d f l u i d i z e d bed on introduction of gas at constant l i q u i d flow rate i n a l l the systems which they investigated. The degree of contraction observed was, however, found to depend on the i n i t i a l degree of expansion of the l i q u i d - s o l i d f l u i d i z e d bed. A s i m i l a r investigation had been c a r r i e d out e a r l i e r by Volk [10], using nitrogen as gas, heptane as l i q u i d and extruded c y l i n d r i c a l c a t a l y s t p e l l e t s as s o l i d s i n d i f f e r e n t columns varying from 0.625 inch to 6 inches i n diameter. Volk, however, observed that under a l l conditions studied the l i q u i d - s o l i d f l u i d i z e d bed expanded further on introduction of gas at constant l i q u i d flow rate. This contradictory behaviour of three-phase f l u i d i z e d beds can perhaps be ascribed to the difference i n physical properties and espec i a l l y surface tension of the l i q u i d phase used. The e f f e c t of l i q u i d properties on the behaviour of three-phase f l u i d i z e d has not been investigated systematic-a l l y , but recently Dakshinamurty et a l . [11], using kerosene as the l i q u i d phase, a i r as the gas phase and sin g l e - s i z e d glass beads as the s o l i d phase, observed that the smoothly f l u i d i z e d l i q u i d - s o l i d f l u i d i z e d bed expanded further on introduction of gas at constant l i q u i d v e l o c i t y . Contrary behaviour was observed with water as l i q u i d . I t should be noted that the organic l i q u i d s used by Volk and by Dakshinamurty et a l . have surface tensions about one-third that of water, and that these investigators paid no attention to the possible presence of trace impurities i n t h e i r technical grade l i q u i d s . I t i s known that minute organic impurities could change the surface c h a r a c t e r i s t i c s of the s o l i d p a r t i c l e s , such as rendering them non-wettable by the l i q u i d phase. The importance of s o l i d s w e t t a b i l i t y was pointed out e a r l i e r by Guha et a l . [13] i n th e i r study of the suspension of s o l i d p a r t i c l e s i n a l i q u i d medium by means of a gas flow. In c a r e f u l l y planned experiments to show the e f f e c t of w e t t a b i l i t y of p a r t i c l e s i n three-phase f l u i d i z a t i o n , Evans [12] observed that a' bed of l i q u i d and wettable p a r t i c l e s contracted, while the same bed with the 8 p a r t i c l e s rendered non-wettable expanded further, on i n t r o -duction of gas at a fixed l i q u i d rate. I t i s , then, believed that the physical properties of the l i q u i d , e s p e c i a l l y surface tension, and the nature of the p a r t i c l e surface are probably i n t e r r e l a t e d and play an important r o l e i n determining the behaviour of three-phase f l u i d i z e d beds [83]. However, since no systematic i n v e s t i g a t i o n has yet been car r i e d out. to elucidate the e f f e c t of l i q u i d and s o l i d surface properties, no generalizations can be made with any confidence. Almost a l l studies, including the present one, have been carried out using s o l i d p a r t i c l e s which were f u l l y wettable by the l i q u i d phase. Michelsen and 0 s t e r g a a r d [14], i n conducting detailed experiments on 3 mm and 6 mm glass beads, found that water f l u i d i z e d beds consisting of such r e l a t i v e l y large p a r t i c l e s did not show any contraction when a i r was introduced as observed for smaller p a r t i c l e s . Almost a l l of the e a r l i e r works were car r i e d out using only very small gas flow rates, since i t was d i f f i c u l t to discern the expanded bed height at high gas rates. Michelsen and 0 s t e r g a a r d extended the range of gas v e l o c i t i e s studied, though they did not indicate how the height of expanded bed was measured. They observed that a l i q u i d - s o l i d f l u i d i z e d bed of 1 mm p a r t i c l e s con-tracted on introducing the gas at small flow rates. On further increasing the gas v e l o c i t y at a fixed l i q u i d v e l o c i t y , the bed height tended to reach a d e f i n i t e minimum 9 and then increase with increase of gas v e l o c i t y . Another important aspect of three-phase f l u i d i z e d bed operation i s the behaviour of the gas phase inside the bed. Numerous attempts have been made to study th i s aspect, but i t i s not yet f u l l y understood. Massimilla et a l . [15] studied the r i s e v e l o c i t y of a single gas bubble i n a l i q u i d -s o l i d f l u i d i z e d bed, using water as the l i q u i d . They observed that the r i s e v e l o c i t y was a function of the bubble diameter, but that the functional dependence for small bubbles (bubble diameter up to 8mm), was r a d i c a l l y d i f f e r e n t than i n pure water. However, for larger bubbles the r i s e v e l o c i t y i n the l i q u i d - s o l i d f l u i d i z e d beds approached the r i s e v e l o c i t y i n pure water. Another important observation to be made from th e i r measurements i s that, although the r i s e v e l o c i t y of bubbles i n pure water i s p r a c t i c a l l y constant f o r bubble diameters i n the range 3 mm - 20 mm, the r i s e v e l o c i t y of bubbles i n a l i q u i d - s o l i d f l u i d i z e d bed for the same range of bubble diameters increases monotonically. This observation then leads to the p o s s i b i l i t y of v e r t i c a l bubble coalescence i n a l i q u i d - s o l i d f l u i d i z e d bed, i f the bed contains bubbles of various s i z e s . Thus large bubbles, with t h e i r character-i s t i c high v e l o c i t y , would be prevalent i f coalescence were the predominant phenomenon inside the bed, while small bubbles, with their c h a r a c t e r i s t i c low v e l o c i t y , would p r e v a i l i f bubble break-up were the predominant phenomenon. The rol e of the s o l i d p a r t i c l e s can then be characterized by noticing 10 how the gas holdup, defined as the volumetric f r a c t i o n of the bed occupied by the gas bubbles, i s affected. To compare the gas holdup i n a three-phase f l u i d i z e d bed with that i n two-phase ga s - l i q u i d flow i t i s necessary that they both be referred^ to a common s o l i d s free basis. The gas holdup i n a three-phase f l u i d i z e d bed on a s o l i d s -free basis i s given by M l e 2 = e 2 / ( l - e 3 ) (1.1) The gas holdup from equation 1.1 can then be compared to the II gas holdup i n two-phase gas-liquid flow, e 2 , t o elucidate the e f f e c t of the s o l i d p a r t i c l e s . Most investigators [14,16,17] have, however, compared the d i r e c t l y measured absolute gas holdup i n a three-phase f l u i d i z e d bed, e 2, with that i n two-phase g a s - l i q u i d flow, at the same flow rates of both the gas and the l i q u i d phases. Thus from such a comparison Michelsen and 0stergaard [14], who obtained the gas holdup inside the bed from the s t a t i c pressure drop measurements i n a 6 inch diameter column, concluded that i n beds of 3 mm and 6 mm glass beads break-up of bubbles takes place i n the lower portion of the bed, whereas i n beds of 1 mm glass beads coalescence takes place i n the same region. No mechanistic c o r r e l a t i o n was, however, obtained between the gas holdup i n a three-phase f l u i d i z e d bed and the gas holdup i n two-phase gas-liquid flow. 11 Efremov and Vakhrushev [16] used the same p r i n c i p l e of s t a t i c pressure drop measurement to obtain the gas holdup inside the bed i n a 10 cm diameter column, although the accuracy of th e i r s t a t i c pressure drop measurement technique, using a s t a t i c tube immersed i n the bulk of the f l u i d , i s questionable [82]. They employed narrow fracti o n s of glass beads with mean diameters ranging between 0.32 mm and 2.15 mm and they observed that for a l l sizes of p a r t i c l e s studied, the gas holdup i n a three-phase f l u i d i s e d bed was considerably smaller than i n two-phase gas- l i q u i d flow under i d e n t i c a l gas and l i q u i d flow rates, respectively. They were further-more able to. empirically correlate the gas holdup inside the three-phase f l u i d i z e d bed with the gas holdup i n two phase gas-l i q u i d flow. V a i l et a l . [17] used the method of i s o l a t i n g the t e s t section by simultaneously cutting o f f the gas and the l i q u i d flows and then recording the amount of gas c o l l e c t e d at the top of the bed. As w i l l be discussed under "Experimental Technique," t h i s method has inherent errors, which were not f u l l y corrected for by these investigators. They studied 0.73 mm glass beads and two d i f f e r e n t c a t a l y s t powders of the same s i z e , the measurements being c a r r i e d out i n a 14.6 cm diameter column. They found that for a l l the three s o l i d p a r t i c l e s studied, the gas holdup i n three-phase f l u i d i z a t i o n was always smaller than the gas holdup i n two-phase gas-l i q u i d flow under i d e n t i c a l flow conditions. They attributed 12 thi s r e s u l t to the f a c t that the s o l i d phase displaces part of the l i q u i d , while the gas bubbles can r i s e and e x i s t only within the l i q u i d phase. Based on t h i s reasoning,an empirical c o r r e l a t i o n between the gas holdup i n a three-phase f l u i d i z e d bed and i n two-phase gas-liquid flow was presented. In view of the reported works [14,15,79] i t would seem to be correct that i f the p a r t i c l e size i s small as compared to the bubble siz e , bubble coalescence w i l l r e s u l t , whereas i f the p a r t i c l e size i s comparable or bigger than the bubble s i z e , bubble break-up w i l l occur, i n the bed. 0 s t e r g a a r d and Theisen [18] observed that t h i s does not seem to a f f e c t the contraction or expansion behaviour of the bed, since p a r t i c l e size does not seem to influence the d i s t r i b u t i o n of l i q u i d between the l i q u i d - s o l i d f l u i d i z e d (particulate) phase and the gas-liquid (bubble) phase. However, since no det a i l e d study has been done of flow d i s t r i b u t i o n or so l i d s c i r c u l a t i o n i n three-phase f l u i d i z e d beds, i t i s rather important to understand the c h a r a c t e r i s t i c behaviour of each phase and the mutual i n t e r a c t i o n between phases, at l e a s t qua1i ta t i v e l y . 1.3 Wake model for three-phase f l u i d i z e d beds The q u a l i t a t i v e d e s c r i p t i o n of the behaviour of i n -d i v i d u a l phases can provide the basis for a mathematical model describing the gross behaviour of three-phase f l u i d i z e d beds. However, to keep the mathematical model r e a l i s t i c , assumptions 13 have to be made about those aspects of three-phase fluidized beds which are not well understood. One aspect of three-phase fluidized beds which, although i t plays an important role in determining the behaviour of such beds, has not been sufficiently investigated, is the phenomenon of wake formation behind the dispersed gas phase. It i s very well recognized now that a dispersed phase, when moving through a continuous medium, carries along with i t some continuous phase as i t s wake. The average amount of continuous phase carried as the wake of the dispersed gas phase w i l l depend upon the size and shape of the wake [8] and on the mode, frequency and rate of wake shedding [80]. Thus, in a three phase fluidized bed, gas bubbles rising through the continuous liquid medium w i l l carry part of the liquid phase in their wake, making that part of the liquid phase unavailable for support of the solid particles. On the basis of this wake phenomenon, Stewart and Davidson [7] were able to explain the observed contraction in three-phase fluidized beds. Using a two-dimensional liquid-fluidized bed contained between perspex plates spaced 0.25 inch apart, Stewart and Davidson observed photographically that when an air bubble rises through the bed, some liquid follows the bubble as i t s wake. They also observed that the liquid wake was practically free of solid particles. The resulting combined gas-liquid bubble rises through the bed at a much greater velocity than the 14 average v e l o c i t y of l i q u i d through the i n t e r s t i c e s of the bed, thus removing some l i q u i d from the continuous phase and, according to the reasoning of these investigators, thereby reducing the o v e r a l l f l u i d i z i n g force. Because of t h i s r e -duction, the bed s e t t l e s to a lower depth. No phenomenological c o r r e l a t i o n was attempted to represent the observed bed behaviour. The model subsequently proposed by 0stergaard [8] was in i t s main features the same as that suggested by Stewart and Davidson, but i t was based on representing a three-phase f l u i d i z e d bed as consisting of a l i q u i d - f l u i d i z e d p a r t i c u l a t e phase, a gas bubble phase and a wake phase. The wake phase was assumed to follow the bubble phase at the bubble v e l o c i t y and have a porosity i d e n t i c a l to that of the pa r t i c u l a t e phase. This l a s t assumption, based on photographic observation of a bubble emerging from a l i q u i d - s o l i d f l u i d i z e d bed being followed by a long t r a i l containing s o l i d s , contradicts the observation of Stewart and Davidson that a bubble i s followed by a wake of l i q u i d devoid of p a r t i c l e s . There i s , unfortunately, no conclusive evidence available yet to support or repudiate either claim. The model proposed by 0stergaard i s presented here i n i t s en t i r e t y , since i t i s f e l t that t h i s model attempts to consider the fundamental d i s t r i b u t i o n of the l i q u i d between the pa r t i c u l a t e phase and the wake phase. Further, i t i s hoped that d e f i c i e n c i e s of the model can be i d e n t i f i e d so that modifications can be incorporated to 15 explain three-phase bed behaviour more r e a l i s t i c a l l y . Let us consider a bed of p a r t i c l e s of weight, W, expanded to a height, L^, i n a column of cross-sectional area, A, under the influence of gas and l i q u i d volumetric fluxes < j - j _ > and < J 2 > respectively. Then the average volumetric solids f r a c t i o n or s o l i d s holdup w i l l be e , = — (1.2) 3 P s ^ b The bed porosity, e, defined as the f r a c t i o n of the bed volume occupied by gas and l i q u i d , w i l l be e = e- + E, = (1-e-.) (1.3) If i t i s assumed that the porosity of the wake phase i s i d e n t i c a l to the porosity of the pa r t i c u l a t e ( i . e . the l i q u i d -s o l i d f l u i d i z e d ) phase, i t follows that £ = £ 1(l-£ 2) + £ 2 (1.4) where i s the volume f r a c t i o n of l i q u i d i n the p a r t i c u l a t e phase and i s related to the volumetric f l u x through t h i s n region, j ^ , by the well known Richardson - Zaki [2] equation, i . e . , " ... \ 1/n *1 = <3i/Vj ( 1 . 5 ) 16 where n = f (Re , d /D) p' p (1.6) Let us assume that the volume f r a c t i o n of the three-phase bed occupied by the wake phase i s e^. If we consider the three-phase f l u i d i z e d bed to be macroscopically homogeneous, then the f r a c t i o n of any cross-sectional area occupied by the gas bubble phase and the wake phase w i l l be e 2 a n <^ eyi respectively. Therefore on the basis of the physical picture assumed by 0stergaard for a three-phase f l u i d i z e d bed, the area occupied by the pa r t i c u l a t e or l i q u i d - s o l i d f l u i d i z e d phase i n the plane perpendicular to the p r i n c i p l e flow axis w i l l be The d i s t r i b u t i o n of l i q u i d phase between the p a r t i c u l a t e phase and the wake phase i s obtained by carrying out a material balance across any cross-section inside the three-phase bed: ALS = ( 1 - £ 2 - £ k ) A (1.7) Total volumetric flow rate of l i q u i d through the column Volumetric flow rate of l i q u i d through the pa r t i c u l a t e phase ) * (Volumetric flow rate of l i q u i d through the wake phase 17 or Q l = Q l f + Q l k ( 1 ' 8 ) If i t i s assumed that the wake phase tra v e l s with the gas bubble phase at the bubble v e l o c i t y , a good assumption i f one disregards the wake shedding phenomenon, then equation 1.8, with the aid of equation 1.7, may be rewritten as II _ II <j 1> A = J 1 ( l - e 2 - e k ) A + v 2 e k A e 1 (1.9) II Solving equation 1.9 for gives the volumetric f l u x through the p a r t i c u l a t e phase as _ II 3 ± = — (1.10) 1 _ e 2 - e k In order to predict the behaviour of a three-phase f l u i d i z e d bed from the above set of equations, one requires independent knowledge of e 2 and e^. The f r a c t i o n of the bed volume occupied by the gas phase i s related to the average r i s e v e l o c i t y of the bubbles by = < J o > / v 9 (1.11) As discussed i n the preceding section, the r i s e v e l o c i t y of a single bubble in a l i q u i d - s o l i d f l u i d i z e d bed was 18 studied by Massimilla [15]. However, no data on the r i s e v e l o c i t y of a swarm of bubbles i n a l i q u i d - s o l i d f l u i d i z e d bed are a v a i l a b l e . 0 s t e r g a a r d therefore obtained an estimate of bubble r i s e v e l o c i t y i n a three-phase f l u i d i z e d bed from the data of N i c k l i n [19] for two-phase gas- l i q u i d flow, which 0 s t e r g a a r d correlated by the empirical equation, v 2 = 21.7 - 4.6 In <j 2> + <j x> (1.12) However, since equation 1.12 i s based on two-phase data, i t could hardly be expected to accurately represent the gas bubble phase i n a three-phase bed. The phenomenon of wake formation behind a gas bubble has not been studied extensively. The importance of the wake phenomenon i n c o n t r o l l i n g transport processes was f i r s t r e a l i z e d i n l i q u i d - l i q u i d operations [36]. Subsequently, therefore, attempts were made to obtain information regarding the shape and size of wakes behind l i q u i d drops [84], while study of the mode, frequency and rate of wake shedding i n l i q u i d - l i q u i d systems i s currently i n progress [85]. Generalizable quantitative information regarding the wake c h a r a c t e r i s t i c s i n l i q u i d - l i q u i d systems, however, i s rather l i m i t e d , whereas even s p e c i f i c information on the wake c h a r a c t e r i s t i c s i n gas l i q u i d systems i s minimal and i n three-phase systems almost non-existent. Therefore i n 19 order to obtain an estimate for the volume f r a c t i o n occupied by the wakes, e^ ., 0 s t e r g a a r d [8] postulated, q u a l i t a t i v e l y that 1. increases with increasing z^i the rate of increase i n e^ . being slower at large values of 2. increases with increasing l i q u i d flow rate, that is,with increasing bed f l u i d i t y . These two assumptions were then incorporated by t r i a l and error into the following equation i n such a way as to s a t i s f y the data for bed contraction: e k = 0.14 e 2 0 - 5 (<j]_> - <j 1> m f) (1.13) where < j ] _ > m f 1 S the minimum volumetric l i q u i d f l u x required to i n i t i a t e f l u i d i z a t i o n of the s o l i d p a r t i c l e s . 0 s t e r g a a r d thus presented the set of equations 1.2 -1.10, which when used i n conjunction with equations 1.11 -1.13, s a t i s f i e d a limited quantity of experimental data on bed contraction. But l a t e r , i n a more detailed i n v e s t i g a t i o n , 0 s t e r g a a r d and Theisen [18] reported that the model proposed by 0 s t e r g a a r d could not s a t i s f a c t o r i l y predict the observed contraction of three-phase f l u i d i z e d beds over a wider range of operating v a r i a b l e s . Furthermore, the model f a i l s to describe f u l l y the observed bed behaviour of three-phase 20 f l u i d i z e d beds as outlined i n the preceding section, v i z . that the bed height drops on introduction of gas at a con-stant l i q u i d flow rate, then reaches a d e f i n i t e minimum on increasing the gas flow rate, and f i n a l l y slowly expands again as the gas flow rate i s further increased. Only the i n i t i a l bed contraction i s predicted by the model. Neverthe-les s 0 s t e r g a a r d ' s model for a three-phase f l u i d i z e d bed does have the v i r t u e of describing, a l b e i t approximately, the d i s t r i b u t i o n of l i q u i d between the p a r t i c u l a t e phase and the wake phase. The equations used for estimating the volumetric gas f r a c t i o n and the volumetric wake f r a c t i o n cannot be expected to accurately predict these quantities i n a three-phase f l u i d i z e d bed, since they were obtained from a very limited range of data. Even i n two-phase gas-l i q u i d flow the information on gas holdup i s non-conclusive and ambiguous, while the information on volumetric wake f r a c -t i o n and i t s r o l e i n determining v e l o c i t y p r o f i l e s i n gas-l i q u i d flow i s almost non-existent. I t i s useful at t h i s point to digress from 0 s t e r g a a r d ' s model and consider the simple model proposed by Davidson [20] to explain the bubble phenomenon i n gas s o l i d f l u i d i z a t i o n , and the modification introduced to t h i s simple model by Kunii and Levenspiel [21] to explain bubble behaviour i n gas-solid f l u i d i z e d beds more r e a l i s t i c a l l y . Davidson postulated that a bubbling gas-solid f l u i d i z e d bed can be considered as being constituted of a gas bubble phase and a 21 gas-solid emulsion phase. Davidson also assumed the bubbles i n the bubble-phase to be spherical and the flow around the spherical c a v i t i e s to be i r r o t a t i o n a l and incompressible. The flow pattern of gas and s o l i d and the pressure d i s t r i -bution i n the v i c i n i t y of the r i s i n g bubble predicted by this model have been shown to be e s s e n t i a l l y correct [21]. However, Rowe and Partridge [22] observed that the r i s i n g bubbles each carry a wake behind them containing s o l i d p a r t i c l e s . The reason given for the presence of thi s wake was that the pressure i n the lower part of the bubble i s less than i n the nearby emulsion phase, a reason predictable by Davidson's model. Gas i s thereby drawn into the bubble, r e s u l t i n g i n an i n s t a b i l i t y , p a r t i a l collapse of the bubble, and turbulent mixing behind i t . This turbulence r e s u l t s i n s o l i d s being drawn up behind the bubble and forming a wake. The wake of the bubble exchanges s o l i d material continually during i t s r i s e , depending on the mode, frequency and rate of wake shedding, but ultimately the s o l i d p a r t i c l e s c a r r i e d i n the wake of the bubble are deposited at the bed surface when the bubble emerges from the bed, thus giving r i s e to downward s o l i d movement i n the emulsion phase. Kunii and Levenspiel [21] modified Davidson's simple model by incorporating the wake phenomenon. They considered the bubbling gas-solid f l u i d i z e d bed as consisting of a gas bubble phase, a gas-solid emulsion phase, and a wake phase. This model i s analogous to that proposed by 0stergaard for a 22 three-phase f l u i d i z e d bed. Kunii and Levenspiel pointed out that the s o l i d p a r t i c l e s i n the emulsion phase develop a c i r c u l a t o r y motion promoted by the r i s i n g bubble wakes, containing s o l i d p a r t i c l e s . However, they also showed that so l i d s movement did not a f f e c t the behaviour of the bubble phase markedly, but that the movement of the enti r e emulsion phase could be reversed due to the motion of the p a r t i c l e s . In a three-phase f l u i d i z e d bed as pictured i n 0 s t e r g a a r d ' s model, i t i s the r e l a t i v e v e l o c i t y between the l i q u i d and the s o l i d p a r t i c l e s i n the p a r t i c u l a t e phase that would control the expansion or contraction of the bed. 0 s t e r g a a r d considered the d i s t r i b u t i o n of l i q u i d between the p a r t i c u l a t e phase and the wake phase, but the c i r c u l a t o r y motion of the solids i n the p a r t i c u l a t e phase as induced by the gas bubbles, was not considered. By analogy with gas-solid f l u d i z a t i o n , a c i r c u l a t o r y motion of s o l i d p a r t i c l e s would also e x i s t i n three-phase f l u i d i z e d beds, i f i t i s assumed that the wake accompanying a gas bubble contains s o l i d p a r t i c l e s . Thus the main drawbacks i n the simple but elegant model proposed by 0 s t e r g a a r d seem to be: (i) The assumption that the porosity of the wake phase is.equal to that of the p a r t i c u l a t e phase, ( i i ) The neglect of so l i d s c i r c u l a t i o n induced by the motion of gas bubbles carrying wakes containing s o l i d p a r t i c l e s . ( i i i ) The quantitative representation of wake volume f r a c t i o n by equation 1.13 and of bubble r i s e v e l o c i t y by equation 1.12. 1.4 Importance of turbulence phenomena i n three-phase  f l u i d i z e d beds Turbulence i s known to exert s i g n i f i c a n t influence on momentum transfer (and other transfer processes) from a p a r t i c l e immersed i n a f l u i d by a l t e r i n g the flow f i e l d around the p a r t i c l e . However, very l i t t l e e f f o r t has gone i n to qu a n t i t a t i v e l y c o r r e l a t i n g these e f f e c t s with the measureable fundamental properties of a turbulent f i e l d , v i z . i n t e n s i t y and scale of turbulence. The importance of turbulence i n a f l u i d i z e d bed can be appreciated i f we consider that the bed consists of s o l i d p a r t i c l e s of f i n i t e s i z e . Then a flow f i e l d i s developed around each p a r t i c l e due to the r e l a t i v e motion between the f l u i d and the p a r t i c l e and the no s l i p condition to be s a t i s f i e d at the p a r t i c l e surface. The l a t t e r causes generation of v o r t i c i t y at the p a r t i c l e surface, the growth and decay of which determines whether the flow f i e l d near the p a r t i c l e i s either turbulent or non-turbulent. At high r e l a t i v e v e l o c i t y , the v o r t i c i t y i s convected downstream with the flow and i s concentrated at the rear of the p a r t i c l e , causing a backward flow to be induced near the surface. This backward flow counters the forward moving f l u i d and d e f l e c t s i t away from the rear, 24 strengthening the r o t a t i o n a l motion i n the standing eddy. The term wake i s commonly applied to t h i s whole region of non-zero v o r t i c i t y on the downstream side of the p a r t i c l e . At s t i l l higher r e l a t i v e v e l o c i t i e s the wake no longer remains permanently attached to the p a r t i c l e s but i s shed at regular i n t e r v a l s i n an otherwise uniform stream of f l u i d . Thus formation of a s u f f i c i e n t l y high v o r t i c i t y wake behind a p a r t i c l e can be considered as the onset of a turbulence f i e l d i n the f l u i d medium around the p a r t i c l e . The turbulence f i e l d i n a f l u i d i z e d bed, which i s constituted of an assemblage of p a r t i c l e s , can then be considered as the composite e f f e c t of the wakes of i n d i v i d u a l p a r t i c l e s [87]. However, i n three-phase f l u i d i z e d beds as i n two-phase gas-liquid systems, i t i s the gas bubbles which play the dominant r o l e i n creating turbulence i n the f l u i d phase. The mechanism for generation of turbulence i s p r i n c i p a l l y the same as described above, v i z . the formation of wakes behind the bubbles and consequent wake shedding at higher r e l a t i v e v e l o c i t i e s . Thus a cocurrent gas - l i q u i d flow under non-laminar flow conditions may be considered as a system which generates f l u i d phase turbulence through the presence of randomly moving bubbles. The presence of s o l i d p a r t i c l e s i n such a turbulence generating system may suppress the l i q u i d phase turbulence i n varying degrees, depending on the r e l a t i v e i n e r t i a of the p a r t i c l e s and the i n t e n s i t y of t u r b u l e n c e . A l a r g e d e n s i t y d i f f e r e n c e between the p a r t i c l e and the l i q u i d w i l l tend to damp out t u r b u l e n c e [88] , so t h a t the l i q u i d - p h a s e i n t e n s i t y of tu r b u l e n c e i n a three-phase f l u i d i z e d bed may be much s m a l l e r than i n the c o r r e s p o n d i n g c o c u r r e n t g a s - l i q u i d f l o w , f o r equal f l u i d phase v e l o c i t i e s . A systematic i n v e s t i g a t i o n o f the expansion c h a r a c t e r i s t i c s o f a f l u i d i z e d bed should then i n v o l v e a study of t u r b u l e n c e g e n e r a t i o n i n the bed and the i n f l u e n c e o f fundamental t u r b u l e n c e p r o p e r t i e s on the drag c o e f f i c i e n t of an i n d i v i d u a l p a r t i c l e . The knowledge o f wake fo r m a t i o n behind the bubble and the mode, frequency and r a t e of wake shedding has not been examined to any e x t e n t . Stewart and Davidson [ 7 ] , i n t h e i r i n v e s t i g a t i o n o f a three-phase f l u i d i z e d bed i n a two dimensional column, observed the wake shedding phenomenon p h o t o g r a p h i c a l l y . Rigby and Capes [8 0],. f o l l o w i n g up t h e i r i n v e s t i g a t i o n of a two-dimensional three-phase f l u i d i z e d bed, s u c c e s s f u l l y e x p l a i n e d the observed c o n t r a c t i o n phenomenon on the b a s i s o f a few measurements o f wake shedding and the consequent r i s e o f shed v o r t i c e s through the bed. T h e i r work thus i n d i r e c t l y demonstrates the p o s s i b l e r e l e v a n c e o f t u r b u l e n c e , generated by shedding o f wakes from the r i s i n g b u b b l e s , i n the study o f three-phase f l u i d i z e d beds. S t u d i e s [14,17] on the s t a t e o f mixing i n three-phase f l u i d i z e d beds have a l s o demonstrated i n d i r e c t l y the dominant r o l e o f turbulence i n multiphase flow. However, attempts to gain better i n s i g h t into the mechanisms c o n t r o l l i n g other trans-port phenomena cannot be e n t i r e l y s a t i s f a c t o r y u n t i l the flow f i e l d s around both the dispersed phases are f u l l y understood. Thus we see that turbulence probably plays a r o l e i n defining the behaviour of three-phase f l u i d i z e d beds. Given the present state of knowledge of three-phase systems t however, i t appears s u f f i c i e n t to know the size and shape of a wake behind an i s o l a t e d bubble, the influence of other bubbles and p a r t i c l e s on the s i z e and shape of the wake, and the mode, frequency and rate of wake shedding—in order to quantify the contraction phenomenon. Nevertheless, i n order to gain complete knowledge of the f l u i d dynamics of three-phase f l u i d i z e d beds, i t i s important that the turbulence phenomenon be systematically examined. The present state of knowledge of turbulence can be used to describe single phase flow, but i t has not advanced enough to predict fundamental quantities for multiphase flow. 1.5 Scope of research The information available on three-phase f l u i d i z e d beds i s rather scanty. Therefore the primary aim of t h i s research was to devise and carry out an experimental programme for c o l l e c t i n g r e l i a b l e and accurate data on the holdup of so l i d s and gas i n a three-phase f l u i d i z e d bed under a wide range of flow conditions. As has been suggested e a r l i e r [3,4], a three-phase f l u i d i z e d bed can be considered as a complex system, the properties of which are a composite of two simpler systems: gas-liquid cocurrent flow and a l i q u i d - s o l i d f l u i d i z e d bed. The parameters to be studied were selected on the basis of information about the two-phase systems. Thus, since i t had been established i n various studies on two-phase gas- l i q u i d flow that v a r i a t i o n i n the properties of the gas phase does not play an important r o l e under normal atmospheric conditions, i t was decided to use atmospheric a i r as the gas phase for the entire programme. Water was chosen as the l i q u i d phase for most of the study for s i m p l i c i t y and for purposes of comparing the data c o l l e c t e d i n th i s study with the data available i n the l i t e r a t u r e from the work of various investigators, most of whom.used water as the l i q u i d and a i r as the gas. In order to study three-phase f l u i d i z e d beds under conditions where turbulence i n the continuous phase i s i n s i g n i f i c a n t , water was replaced by a high v i s c o s i t y l i q u i d . A 30% (by weight) solution of polyethylene g l y c o l i n water was selected because i t gives high v i s c o s i t y (-60 c.p.) without a f f e c t i n g density and surface tension of the solut i o n markedly. Surface tension of the l i q u i d phase, though an important parameter i n g a s - l i q u i d flow systems and found to be even more important for three-phase systems [11], was not d e l i b e r a t e l y varied because of the experimental d i f f i c u l t y of keeping traces of 28 impurities from entering the system and thus r a d i c a l l y chang-ing the s t a t i c equilibrium (contact angle) between the phases. So l i d p a r t i c l e s selected for the study had to be non-reacting with the l i q u i d s chosen and of well defined shape. The behaviour of i r r e g u l a r l y shaped p a r t i c l e s i n l i q u i d -s o l i d f l u i d i z e d beds i s not e n t i r e l y understood and therefore c l o s e l y sized spherical p a r t i c l e s of various densities and sizes were chosen. Also the behaviour of p a r t i c u l a t e l y f l u i d i z e d beds with spherical p a r t i c l e s lends i t s e l f to sa t i s f a c t o r y explanation by simple mathematical models. The secondary aim of t h i s research was to derive a mathematical model which, when coupled with some empirical information, would y i e l d better understanding of the behaviour of a three-phase f l u i d i z e d bed. For t h i s purpose the wake model proposed by 0 s t e r g a a r d was chosen as a s t a r t i n g point, since i t describes the d i s t r i b u t i o n of l i q u i d between the wake phase and the p a r t i c u l a t e phase. However, as pointed out e a r l i e r , i t was expedient to incorporate into the model the r e c i r c u l a t i o n of solids induced by movement of gas bubbles i n order to explain not only the observed contraction, but also the subsequent expansion, of three-phase f l u i d i z e d beds. To develop a model for three-phase f l u i d i z e d beds, knowledge of gas holdup i n cocurrent gas - l i q u i d flow i s e s s e n t i a l . Since the models available i n the l i t e r a t u r e for predicting gas holdup i n gas-liquid flow are mostly empirical i n nature, there i s quite a v a r i a b i l i t y and ambiguity i n t h e i r range of a p p l i c a b i l i t y . I t therefore became necessary to study two-phase gas- l i q u i d cocurrent flow from the point of view of developing a l o g i c a l and reasonable physical model for gas holdup i n two-phase gas- l i q u i d flow. Attempts were also made to develop a model for three-phase f l u i d i z e d beds under conditions i n which the turbulence phenomenon can be neglected. Experimental measurements carried out with small p a r t i c l e s i n a highly viscous l i q u i d to support such a model were not e n t i r e l y successful, but the model .-is. presented herein for possible future investigations. 30 CHAPTER 2 THEORY Th i s chapter i s d i v i d e d i n t o three s e c t i o n s — d e a l i n g w i t h holdup i n g a s - l i q u i d f l o w , i n l i q u i d - s o l i d f l u i d i z e d beds and i n three-phase f l u i d i z e d beds, r e s p e c t i v e l y . These d i s p e r s e d phase o p e r a t i o n s have been i n v e s t i g a t e d i n the p a s t to v a r y i n g e x t e n t s . Mathematical models p u r p o r t i n g to p r e d i c t the d e s i r e d q u a n t i t i e s have been p r e s e n t e d , o f t e n without any understanding of the phenomena i n v o l v e d . There-f o r e , an attempt i s made to o u t l i n e the mechanisms of these phenomena from the knowledge a v a i l a b l e i n the l i t e r a t u r e and then, on the b a s i s of these mechanisms, to propose e i t h e r new models or m o d i f i c a t i o n s to e x i s t i n g models, i n or d e r to d e s c r i b e the c h a r a c t e r i s t i c s of the d i s p e r s e d phase o p e r a t i o n s more s a t i s f a c t o r i l y . 2.1 Holdup i n g a s - l i q u i d flow Two-phase g a s - l i q u i d flow has been s t u d i e d f o r many y e a r s , so t h a t the amount o f i n f o r m a t i o n c u r r e n t l y a v a i l a b l e i n the l i t e r a t u r e , though o f t e n i n c o n c l u s i v e , i s s t a g g e r i n g . E x c e l l e n t t r e a t i s e s [23,24,25] have been w r i t t e n on the s u b j e c t and an index of over 5000 a r t i c l e s , r e p o r t s and books on two-phase g a s - l i q u i d flow has been prepared by Gouse [26]. 31 Recently, the book by Wallis [27] has put the subject into some perspective. Due to the complex nature of two-phase flow phenomena, i t has become necessary to r e l y heavily on experimental data, since a r e a l i s t i c analysis has been lacking. For example, the presence of wakes behind a r i s i n g bubble swarm has been recognized but as yet no analysis has adequately taken the wake phenomenon into consideration. I t i s , there-fore, necessary that a l l the basic information available be c a r e f u l l y studied, and that the s a l i e n t features found to be c o n t r o l l i n g the behaviour of ga s - l i q u i d flow i n a given regime be i d e n t i f i e d . A r e a l i s t i c model can then be developed based on those parameters found to be relevant i n the physical observations. To t e s t the a p p l i c a b i l i t y and l i m i t a t i o n s of the model so developed, c a r e f u l l y planned and s t a t i s t i c a l l y designed experimental data w i l l be required. Only an approach which i s appropriately balanced between th e o r e t i c a l and experimental e f f o r t s w i l l lead to better understanding of two-phase ga s - l i q u i d flow. 2.1.1 Holdup studies When we are considering a dispersed two-phase system i n which the gas i s uniformly d i s t r i b u t e d i n a l i q u i d medium as discrete bubbles, the r i s e v e l o c i t y of the swarm of bubbles i s subject to two influences, one a r i s i n g from the motion of the bubbles and the other from th e i r presence. The 32 rela t i o n s h i p between the average r i s e v e l o c i t y of the bubble swarm, v 2 , and the average volume f r a c t i o n of gas i n the swarm, <a2> / i s simply v 9 = <j 9> / <a„> (2.1) where <J2> i s the average volumetric f l u x of the gas through the system. Thus i n order to predict the r i s e v e l o c i t y of the bubble swarm, and consequently the gas holdup, i t i s necessary to understand the motion of a bubble and how i t i s affected by the proximity of other bubbles. 2.1.1.1 Bubble dynamics The r i s e v e l o c i t y of a single bubble i n an i n f i n i t e medium has been studied extensively. Haberman and Morton [29] presented a comprehensive review of bubble motion studies up to 1956. Although the work of Haberman and Morton [28] and of Peebles and Garber [30] elucidates the importance of the physical properties of the l i q u i d (the properties of the gas phase are found not to be important under.normal pressure) on the r i s e v e l o c i t y of a bubble, most of the experimental data available for the r i s e v e l o c i t y of single bubbles i s for bubble motion i n water. In the following description the data obtained for bubble motion i n water [31] are used to i l l u s t r a t e the important aspects of the r i s e of a gas bubble through a pool of stagnant l i q u i d . I t has been observed that small gas bubbles (r < 0.4 mm) which are almost perfect spheres because of the dominant surface tension f o r c e s , behave very much l i k e small s o l i d p a r t i c l e s . However, the Stokes solution for the terminal r i s e v e l o c i t y of a bubble, V , can be used only for s t i l l smaller bubbles (r < 0.2 mm, Re^ < 2 ) : V „ = g ( P l - p 2 ) / 1 8 u 1 (2.2) Equation 2.2 assumes that the l i q u i d v e l o c i t y at the bubble surface r e l a t i v e to the bubble i s zero, an assumption which, however, breaks down for bubbles with i n t e r n a l c i r c u l a t i o n . Hadamard [32] and Rybczynski [33] modified the above equation for perfect f l u i d spheres with complete transference of shear stress at the bubble-liquid i n t e r f a c e , and obtained db g ^ i - p2) 3^ i + 3 l i 2 „ Voo = * (2-3) 18y J L 2 y 1 + 2u2 which for y^>> u2 reduces to V o = d* g ( p1- p 2 ) / 1 2 u 1 (2.4) Equation 2.4 applies to small bubble sizes (Re^<2), but only i n the complete absence of surface i m p u r i t i e s . At the other extreme, when the bubbles are very large (r > 9.0 mm, Re, > 5000) and show a spherical cap shape, the Reynolds and Weber numbers are known to c h a r a c t e r i z e the motion of such b u b b l e s . For p r e d i c t i n g the shape of the bubble i n t h i s regime, the s u r f a c e t e n s i o n and v i s c o u s f o r c e s are normally c o n s i d e r e d to be n e g l i g i b l e as compared to the g r a v i t y and i n e r t i a f o r c e s . Based on these assumptions, Davies and T a y l o r [34] c o n s i d e r e d the motion around the f r o n t s t a g n a t i o n p o i n t to be i r r o t a t i o n a l and o b t a i n e d where R i s the r a d i u s of c u r v a t u r e o f the bubble a t the s f r o n t s t a g n a t i o n p o i n t . I t i s important to note here t h a t although Davies and T a y l o r observed a s i z a b l e wake r e g i o n behind s p h e r i c a l cap b u b b l e s , the i r r o t a t i o n a l flow model was a p p l i e d to the r e g i o n around the f r o n t s t a g n a t i o n p o i n t o n l y and no attempt was made to c o n s i d e r the d e t a i l e d s t r u c t u r e o f the wake i t s e l f . The a b i l i t y o f e q u a t i o n 2.5 to p r e d i c t the t e r m i n a l r i s e v e l o c i t y o f l a r g e bubbles (r > 9.0 mm) stems from the f a c t t h a t the bubble shape, the bubble v e l o c i t y and the r a t e o f energy d i s s i p a t i o n are i n t e r - r e l a t e d . For the i n t e r m e d i a t e bubble s i z e s ( 0 . 6 < rg < 9.0 mm), where the s u r f a c e t e n s i o n and v i s c o u s f o r c e s are comparable to the g r a v i t y and i n e r t i a f o r c e s , the bubble shape and bubble v e l o c i t y are d i f f i c u l t to model. In t h i s range the gas 35 bubbles are neither spherical nor do they r i s e r e c t i l i n e a r l y . Although no study has considered the wake structure behind a r i s i n g bubble i n t h i s range, the studies on r i s e or f a l l of a l i q u i d drop through a l i q u i d medium with which i t i s immiscible by Edge and Grant [35], Letan and Kehat [36], and Magarvey and Bishop [37] a l l suggest that wake a c t i v i t y i n t h i s region of Reynolds number i s quite predominant. The bubbles with r g > 0.8 mm are quite noticeably deformed and th e i r path of ascent i s h e l i c a l . Because of the h e l i c a l path, the r i s e v e l o c i t y of these bubbles i n the v e r t i c a l d i r e c t i o n decreases slowly as the bubble size increases u n t i l r g - 2.4 mm, which corresponds to the minimum i n the r i s e v e l o c i t y vs. bubble si z e r e l a t i o n s h i p . Peebles and Garber [30], from t h e i r extensive experimental data, observed that the average bubble r i s e v e l o c i t y for 1.0 < r g < 2.4 mm i s best represented by the equation = 1.35(o-/ P lr e) 0* 5 (2.6) For bubbles with r g > 2.4 mm, the bubble shape i s not regular but pulsates around an oblate spheroid. The path of r i s e of such bubbles becomes less h e l i c a l and therefore the v e r t i c a l r i s e v e l o c i t y of the bubble once again begins to slowly increase with increasing bubble s i z e . Nevertheless Peebles and Garber [30] , Haberman and Morton [28] , and Levich [38] termed the bubble movement to be turbulent i n t h i s range and found t h a t the average v e r t i c a l r i s e v e l o c i t y i s governed mainly by the p h y s i c a l p r o p e r t i e s of the gas-l i q u i d system and i s t h e r e f o r e approximately a c o n s t a n t f o r a g i v e n system. The e m p i r i c a l e quation suggested by Haberman and Morton, which has been v i n d i c a t e d [ 3 9 ] f o r gas-l i q u i d systems w i t h 2.4 < rQ < 4.0 mm i s ag 0.25 = 1.53 (—) (2.7) Hl For bubbles w i t h r > 4 mm, the t r a n s i t i o n from an o b l a t e e s p h e r o i d to a s p h e r i c a l bubble cap becomes n o t i c e a b l e and i s manifested by an almost r e c t i l i n e a r path o f r i s e . Bubbles w i t h r > 9.0 mm r i s e r e c t i l i n e a r l y and have a w e l l d e f i n e d e J s p h e r i c a l cap w i t h a f l a t u n d u l a t i n g t a i l . T h e i r r i s e v e l o c i t y i s c o n t r o l l e d not by the volume of the bubble but by the c u r v a t u r e of the l e a d i n g edge of the s p h e r i c a l c a p . Rigorous t h e o r e t i c a l a n a l y s e s o f bubble motion i n t h i s i n t e r m e d i a t e bubble s i z e range (0.4 < rg < 9.0 mm) have met w i t h l i t t l e success because o f the d i f f i c u l t y i n d e f i n i n g (a) the shape o f the bubble and (b) the flow f i e l d around the b u b b l e . L e v i c h [38] p o s t u l a t e d t h a t i n low v i s c o s i t y systems the energy i s d i s s i p a t e d i n t o a t h i n boundary l a y e r around the bubble and the flow f i e l d o u t s i d e the t h i n boundary l a y e r i s e s s e n t i a l l y u n a f f e c t e d by the b u b b l e . Moore [40] presented a model based on t h i s p o s t u l a t e and found i t to p r e d i c t the r i s e v e l o c i t y s a t i s f a c t o r i l y f o r s u i t a b l y s i z e d bubbles w i t h f o r e and a f t symmetry i n a h i g h 37 surface tension low v i s c o s i t y l i q u i d . Because.of the d i f f i c u l t y of defining the bubble boundary, the a p p l i c a b i l i t y of t h i s model i s lim i t e d to bubble sizes of r < 2.0 mm. e — Most of the bubble sizes encountered i n two-phase gas- l i q u i d flow are generally greater than 2.0 mm. Therefore we cannot expect Moore's th e o r e t i c a l model to be of much p r a c t i c a l use. large bubbles (r > 1.5 mm) as being analogous to i n t e r f a c i a l disturbances whose dynamic motion i s assumed to be sim i l a r to those of waves on an i d e a l l i q u i d , because of the i n v i s c i d nature of the motion of large bubbles [38]. For waves of small wave length, X, compared to the depth of the l i q u i d , the wave v e l o c i t y , C^, i s given by Lamb [42] as Mendelson replaced the wave length i n equation 2.8 by the circumference of the equivalent bubble defined by the r e l a t i o n Mendelson [41], on the other hand, considered the C 2ira g_X Xp^ 2TT (2.8) CO X 2 TO: e (2.9) and obtained the bubble r i s e v e l o c i t y , V (-C ) as • * ' CO CO V a (2.10) 00 + g r e 38 On comparing the r i s e v e l o c i t y predicted by equation 2.10 with experimental data [28,30], Mendelson observed that t h i s s i m p l i s t i c model predicts the r i s e v e l o c i t y of a bubble i n an i n f i n i t e medium quite s a t i s f a c t o r i l y for r g > 1.5 mm. Therefore i t i s recommended that, i n order to predict the r i s e v e l o c i t y of bubbles i n low v i s c o s i t y l i q u i d s , the following r e l a t i o n s be used : Theoretical solutions of Moore [40], r < 1.5 mm and Re,< 800. e fc> V = (gr + a/p,r ) 0 * 5 r > 1.5 mm and Re, > 800 (2.10) co .»= Q < e e b The r i s e v e l o c i t y of large gas bubbles i n narrow ducts i s yet another i n t e r e s t i n g aspect of two-phase gas- l i q u i d flow, as i t forms a flow regime quite d i s t i n c t from the discrete bubble flow regime, and has been studied by Dumi-trescu [43] and by Davies and Taylor [34]. These bubbles occupy almost the entire cross-section of the duct and are c a l l e d slugs. The r i s e v e l o c i t y of an i n d i v i d u a l slug i s given by Dumitrescu's equation, V = k, (2.11) oo 1 gD where k-^  i s i n general a complex function of v i s c o s i t y and surface tension, but for a low v i s c o s i t y l i q u i d i s well approximated by a constant value of 0.35 [44]. The r i s e v e l o c i t y of a p a r t i c u l a r bubble i n a bubble swarm, with respect to the column boundaries, i s influenced by the walls of the containing vessel as well as by the bubble around i t . The r i s e of a gas bubble i n a confined l i q u i d medium i s somewhat analogous to the corresponding sedimen-t a t i o n of a s o l i d ' p a r t i c l e ; but t h i s analogy has been mis-interpreted i n the past by various authors who were therefore obliged either to set narrow operating l i m i t s on the v a l i d i t y of t h e i r equations [4 5] or to correct them by means of empiric factors [46]. These authors used the basic equation desired for the sedimentation of a s o l i d p a r t i c l e i n a confined l i q u i d medium on the assumption that, due to the return flow caused by the displacement of the l i q u i d by the f a l l i n g p a r t i c l e , additional resistance would be encountered by the p a r t i c l e and thus i t would s e t t l e at a lower v e l o c i t y . Therefore V/V^ OC ( l - d p / D ) k (2.12) For dp/D 1, i t can be seen from equation 2.12 that V -> zero. However, for large bubbles r i s i n g i n a narrow conduit (d^D) i t i s well known that the bubble r i s e v e l o c i t y i s not zero but i s given by equation 2.11. Thus i t i s not surprising that an empirical c o r r e l a t i o n of the basic form of equation 2.12 does not s a t i s f a c t o r i l y predict the r i s e v e l o c i t y of a gas bubble i n a r e s t r i c t e d l i q u i d medium, despite the fa c t that the above form has been successfully used to predict the wall e f f e c t i n l i q u i d - s o l i d f l u i d i z a t i o n [47]. Mendelson and Maneri [48] , therefore, questioned the t h e o r e t i c a l basis for such formulations, and suggested again that a bubble can be considered as an i n t e r f a c i a l disturbance whose dynamic behaviour i s analogous to the motion of surface waves on an i d e a l l i q u i d . The extension of this analogy to account for wall proximity ef f e c t s was obtained by arguing that a dynamic s i m i l a r i t y should e x i s t between the propagation of waves over shallow water and the r i s e v e l o c i t y of a bubble i n a r e s t r i c t e d medium. In shallow l i q u i d s , the wave v e l o c i t y i s given by where h i s the depth of the undisturbed l i q u i d . Substituting for the wave length i n equation 2.13, as before, the circum-ference of the equivalent bubble defined by equation 2.9, Mendelson and Maneri[4 8] obtained the bubble r i s e v e l o c i t y V(=C) i n conjunction with equation 2.10 as [42] (2.13) (2.14) The parameter h, which by analogy should be re l a t e d to some e f f e c t i v e l i q u i d depth, was assumed by Mendelson and Maneri to be d i r e c t l y proportional, to the tube radius. Then 41 tanh C1(R/re) (2.15) The value of the constant was then obtained from the known r i s e v e l o c i t y of slugs, assuming that large bubbles with v a l i d by Dumitreseu [43] and many other investigators. For comparatively large tubes, at large N E q, containing low v i s c o s i t y and high surface tension l i q u i d , the constant was found to be 0.25. Thus the r i s e v e l o c i t y of a single bubble i n a confined medium i s given by The e f f e c t of the presence of other bubbles on the motion of a bubble has not been investigated systematically. A t h e o r e t i c a l analysis has not been possible since the flow f i e l d around the conglomeration of bubbles can not be defined except for very small spherical bubbles. However, the motion of s o l i d p a r t i c l e s s i n g u l a r l y and i n groups undergoing laminar flow within a confined space has been extensively studied by Happel, Brenner and co-workers [49] u t i l i z i n g c e l l model techniques. This involves the concept that an assemblage of s o l i d p a r t i c l e s can be divided into a number of i d e n t i c a l c e l l s , each of which contains a p a r t i c l e surrounded by a f l u i d envelope containing a volume of f l u i d s u f f i c i e n t to r e = R behave l i k e slugs, an assumption already shown to be (2.16) make the f r a c t i o n a l void volume i n the c e l l i d e n t i c a l to that i n the entire assemblage. The c e l l model technique was found to apply with greatest success for concentrated assemblages where the p a r t i c l e s i n the assemblage are d i s t r i -buted more or less randomly and the e f f e c t of the container walls i s not important. Happel [4 9] assumed a t y p i c a l c e l l envelope to be sphe r i c a l . Then for a spherical p a r t i c l e , E3 " ( V r c e l l > 3 " <^»3 <2-17> Happel and Ast [50] , however, considered the t y p i c a l c e l l envelope to be c y l i n d r i c a l . To characterize the i n d i v i d u a l c e l l completely, both the c e l l radius and the c e l l length are then required. They s t i l l assumed the a p p l i c a b i l i t y of equation 2.17, so that the length of t h e i r c y l i n d r i c a l c e l l would have to be 4/3 ( r c e - ^ ) . They found that the predicted values of s e t t l i n g v e l o c i t i e s for t h e i r c y l i n d r i c a l c e l l model agreed reasonably well with the values predicted by the concentric sphere c e l l model up to a s o l i d s holdup of e3 = 0.216 (corresponding to r g / r c e l l - 0.6). Thus Happel concluded that the shape of the f l u i d envelope to be used i n such c e l l u l a r representations of assemblages i s of no s i g n i f i c a n t importance up to substantial s o l i d s holdups. A similar c e l l model representation i s subsequently applied In the present, work, to swarms of bubbles. 43 2.1.1.2 Bubble column If a two-phase gas- l i q u i d operation i s c a r r i e d out i n a v e r t i c a l column under conditions of zero net l i q u i d flow rate through the column, then the contacting device i s c a l l e d a "Bubble Column". The c o n t r o l l i n g parameters for the operating c h a r a c t e r i s t i c s of a bubble column are the residence time of the gas phase (determined by the r i s e v e l o c i t y of the swarm) , the i n t e r f a c i a l area (determined by the size of the bubbles i n the swarm) and the mixing c h a r a c t e r i s t i c s (determined by the wake phenomenon and by the geometric structure of the containing v e s s e l ) . Freedman and Davidson [51] presented a summary of data obtained i n bubble columns of diameters ranging from 1 inch to 42 inch which i s shown i n Figure 2.1. As can be seen, a wide spread i n gas holdup exists but the data can mainly be divided into two d i s t i n c t regions, that i s , a region i n which l i q u i d c i r c u l a t i o n i s prevalent (column diameter >^  4 inch) and a region where l i q u i d c i r c u l a t i o n i s not important (column diameter <_ 2 inch) . The studies of Hughmark [52] and of Shulman and Molstad [53] for small diameter columns (1 inch to 4 inch) have shown that the gas holdup i s primarily a function of the volumetric flux of gas, < J 2 > / through the column and can be predicted from a knowledge of bubble size i n the swarm. However, the information concerning bubble sizes i s not r e a d i l y available except at small gas flow rates. At higher gas flow rates, dispersion of the gaseous phase 0.30 0.26 » 0.22 Q . 3 0.18 Q o 0.14 V § 0.10 0.06 0.02 V 3 4 5 6 <j 2>, cm/sec FIGURE 2.1 GAS HOLDUP DATA FROM LITERATURE FOR AIR-WATER SYSTEM IN BUBBLE COLUMNS OF 1-42 INCH DIAMETER (curves are numbered i n accordance with references) into the l i q u i d phase i s brought about by induced turbulence, and the breakup and coalescence of the dispersed phase occurs continuously. The size of bubbles i n the swarm under these turbulent conditions had not been studied conclusively but i t i s believed that i n columns of diameter up to 4 inch, the bubble size increases with gas flow rate [53] u n t i l the bubbles are large enough (r >_ 9.0 mm) to become sp h e r i c a l l y capped [58] . Then, i f the l i q u i d pool i s deep enough, these s p h e r i c a l l y capped bubbles coalesce to form slugs which occupy almost the entire cross-section of the column. Towell et a l . [59] studied the bubble size and gas holdup i n a 16 inch diameter column. With the help of high speed cine photography, they observed that l i q u i d c i r c u l a t i o n i n such large diameter columns i s very s i g n i f i c a n t and i s responsible for a high degree of mixing and a lowering of gas holdup r e l a t i v e to smaller columns. Small diameter columns (up to 4 inch i n diameter), on the contrary, were shown [60] not to have s i g n i f i c a n t l i q u i d c i r c u l a t i o n and were, therefore, found to exhi b i t very i n s i g n i f i c a n t a x i a l mixing and much larger gas holdups. Various explanations have been provided for the presence of l i q u i d c i r c u l a t i o n i n large diameter columns. Freedman and Davidson believe that i t i s caused by the maldistribution of the gas at the bottom of the column. De Nevers [61] suggests that density differences between those parts of the column which are r i c h and those which are poor i n dispersed phase cause the l i q u i d c i r c u l a t i o n to set i n . Yoshitome and S h i r a i [62] measured the i n t e n s i t y of c i r c u l a t i o n i n a 15 cm diameter column and found a strong upward flow of continuous phase i n the central core region and a downward flow near the walls of the column. As observed by these authors [59-62], the turbulent a c t i v i t y i n large columns i s quite s i g n i f i c a n t and under such conditions the size of the bubbles i n the swarm i s controlled by the energy d i s s i p a t i o n i n the two-phase system [63]. Calderblank [64], i n studying the dispersion of gas i n a mechanically s t i r r e d tank, u t i l i z e d this concept to obtain the average bubble si z e i n the tank by balancing the surface tension forces with the turbulent energy d i s s i p a t i o n . Towell et a l . recommended that Calderblank 1s c o r r e l a t i o n be extended to predict average bubble size for bubble columns by assuming that the power dissipated per u n i t volume i n a bubble column can be taken as Power input per u n i t volume = < J 2 > (2.18) They then substituted equation 2.18 into Calderblank 1s c o r r e l a t i o n for s t i r r e d tanks and obtained the following re-lationship to predict the average bubble size i n a swarm: d b = 0.25 [ ( < j 2 > ) " 0 , 4 ( a / p ) 0 * 6 ] e 2 0 * 5 + 0 .09 (2.19) Good agreement was reported between equation 2.19 and the li m i t e d amount of data obtained by Towell et a l . photographic-a l l y for large (> 4 inch) diameter columns. 2.1.1.3 V e r t i c a l cocurrent flow The flow of the gas and the l i q u i d phases cocurrently i n a v e r t i c a l conduit has been studied and a number of methods have been suggested to predict the gas holdup. I t i s important to note that the bubble dynamics as observed i n bubble columns i s not r a d i c a l l y changed due to the flow of the l i q u i d phase. Baker and Chao [65] observed that the r i s e v e l o c i t y of a bubble i n a v e r t i c a l l y moving l i q u i d stream i s not affected by the v e l o c i t y of the l i q u i d stream inasmuch as the r e l a t i v e v e l o c i t y of the bubble i s found to be the same as the bubble v e l o c i t y i n the quiescent l i q u i d stream. I t has also been observed [66] that bubble formation from an o r i f i c e i n a v e r t i c a l l y moving l i q u i d stream i s unaffected by the v e l o c i t y of the l i q u i d stream. Thus the phenomenon of r e l a t i v e v e l o c i t y could be used to generalize the behaviour of two-phase flow, as suggested by Lapidus and E l g i n [67]. The e f f e c t of column diameter on the mixing character-i s t i c s and gas holdup i n cocurrent gas-liquid flow was investigated by Reith et a l [68] i n 5, 14 and 29 cm columns. They found that a 5 cm column displays very l i t t l e a x i a l mixing and high gas holdup, as was observed e a r l i e r for bubble columns. But for larger vessels, although no systematic 48 c i r c u l a t i o n of the l i q u i d stream was observed, the rates of a x i a l mixing were found to be much higher, and the gas holdup much lower, than for the 5 cm column. Therefore they suggested that a x i a l mixing i n large columns i s caused by the generation of large scale eddies i n the l i q u i d phase due to passage of the bubbles. The average bubble size i n cocurrent gas - l i q u i d flow has not been investigated systematically. However, Patrick [69], based on a limited amount of data obtained i n a 5 cm column, reported that the average bubble size i n two-phase gas-l i q u i d flow i s a function of the average l i n e a r l i q u i d v e l o c i t y , the following r e l a t i o n s h i p representing the data for the cocurrent bubble flow regime: The t r a n s i t i o n point from the bubble flow regime to the slug flow regime has been studied only q u a l i t a t i v e l y . Reith et a l . suggested that for a 5 cm column, slug flow occurred at gas v e l o c i t i e s above < J2> = 5 cm/sec. E l l i s and Jones [60] observed the t r a n s i t i o n point v i s u a l l y and found the following r e l a t i o n s h i p to describe the t r a n s i t i o n approximately: 0.666 d b = 5.074/v 15 < v 1 < 150 (2.20) > = 0.2 <j.> + 3 .05 (2.21) Although no slugs were act u a l l y observed i n pipes greater than 10 cm i n diameter up to <j 9> = 45 cm/sec [68], E l l i s 49 and Jones c l a s s i f i e d the flow to be i n the slug flow regime i f the average r e l a t i v e v e l o c i t y of the gas phase was found to be greater than the v e l o c i t y of r i s e of a single slug as given by Dumitrescu's r e l a t i o n , equation 2.11, for the given column diameter. Several methods have been suggested to predict the gas holdup i n cocurrent gas-liquid flow. Of a l l the empirical correlations available, that of Lockhart and M a r t i n e l l i [70], o r i g i n a l l y developed for horizontal flow but subsequently applied also to v e r t i c a l flow, i s s t i l l the most convenient to use when information concerning the detailed flow structure i s either not available or not desired [71], Duckler et a l . [72] compiled a l l the available data and checked the v a l i d i t y of various empirical correlations that have been recommended i n the l i t e r a t u r e . They found that Hughmark's [73] c o r r e l a t i o n represents most of the data reported over a wide range of operating l i m i t s very s a t i s f a c t o r i l y . Therefore, Hughmark's co r r e l a t i o n w i l l be used to check the v a l i d i t y of the model proposed herein, i n conjunction with equations 2.20 and 2.21. 2.1.2 Models for gas holdup predictions The r e l a t i v e v e l o c i t y between the dispersed and the continuous phases has been suggested by Lapidus and E l g i n [67] as the single parameter required to completely describe the behaviour of an ideal dispersed phase system. They did 50 not provide a d e f i n i t i o n of an id e a l system, but i t can be inferred from their work that i f the dispersed phase i s uniformly d i s t r i b u t e d throughout the continuous phase i n such a manner that each p a r t i c l e has an i n d i v i d u a l i d e n t i t y , yet i t s behaviour i s i d e n t i c a l to that of a l l the other p a r t i c l e s , then the dispersed phase system i s i d e a l . The l o c a l r e l a t i v e v e l o c i t y i s then uniform and i d e n t i c a l to the average r e l a t i v e v e l o c i t y of the whole assemblage. In a non-ideal system the concept of r e l a t i v e v e l o c i t y can only be employed l o c a l l y . Since the data on l o c a l r e l a t i v e v e l o c i t y are not e a s i l y available, Zuber and Findlay [39] recommended that the concept of l o c a l d r i f t v e l o c i t y be used instead. The property of constancy of d r i f t v e l o c i t y for c e r t a i n s p e c i f i c regimes can then be u t i l i z e d , the information on r e l a t i v e v e l o c i t y i n various flow regimes being often vague and incomplete. The l o c a l d r i f t v e l o c i t y , as defined e a r l i e r , repre-sents the l o c a l v e l o c i t y of the bubble with respect to the l o c a l volumetric flux of the mixture: V 2 j = v2 " j (2.22) In a two-phase system, data on the average values are more re a d i l y available; thus the average volumetric flux of the gas phase, <j 9>, which i s r e a d i l y measurable, i s 5 1 <j 2> = <a 2v 2> = Q 2/A ( 2 . 2 3 ) The weighted mean v e l o c i t y of the gas phase, v 2 , which i s obtained by integration across the cross-section, i s given by <j > < a 9v 2> v 2 = — — = — = - = - ( 2 . 2 4 ) <c*2> < A2> In view of equation 2 . 2 2 , the weighted mean v e l o c i t y of the gas phase can also be expressed as <a93> <a0v0-> v 2 = — - — + —£_j£2_ ( 2 . 2 5 ) < a 2 > < A2> Equation 2 . 2 5 can be put i n several alternate forms which are most useful for analyzing experimental data and for determining the average volumetric gas f r a c t i o n , <a2>. Thus, multiplying and d i v i d i n g the f i r s t term on the r i g h t hand side by <j>, we obtain <a 2v 2,> v2 = C 0 < j > + ( 2 ' 2 6 ) < a 2 > where CQ i s the d i s t r i b u t i o n parameter [ 3 9 ] and i s given by < a 9 > <j> C0 = — ( 2 . 2 7 ) The d i s t r i b u t i o n parameter takes into account that both the volumetric f l u x of the mixture and the gas holdup are not uniform over the cross-section. However, i f either the volumetric flux of the mixture or the gas holdup i s uniform over the cross-section, i t can be e a s i l y seen from equation 2.27 that CQ w i l l be unity. Thus where e2 = < a 2 > = a2 (2.28) Equation 2.26 can be s i m p l i f i e d to v 2 = <j> + <v 2j> = <j x> + <j 2> + <v 2j> (2.29) Combination of equation 2.29 with equation 2.24 y i e l d s e 9 ( < j 1 > + <v 9^>) <j 2> = — ±3 (2.30) 1 - e 2 For i d e a l bubbly flow, a l l the averaging brackets on the v e l o c i t i e s can also be dropped. I f , however, neither the volumetric flux of the mixture nor the volumetric gas f r a c t i o n i s uniform over the cross-section, the d i s t r i b u t i o n parameter for a x i a l l y symmetric flow through a c i r c u l a r duct, assuming flow and gas holdup p r o f i l e s i n the r a d i a l d i r e c t i o n to be j * M — = 1 - (R ) (2.27a) =>c 53 and —^— = 1 - (R ) (2.27b) a2C respectively, i s given by [39] „ M + M1 +4 / 0 0_ x C Q = (2.27c) M + M* + 2 For p o s i t i v e values of (M + M ' ) , CQ i s obviously greater than unity. Zuber and Findlay have pointed out that the main problem i n two-phase flow i s determining the correct equation for the d r i f t v e l o c i t y , v 9 ., with p a r t i c u l a r regard to the flow regime. In general, the l o c a l d r i f t v e l o c i t y i s found to be affected by the bubble spacing or the gas holdup. Thus for the bubble flow regime, Zuber and Hench [74] reported that v 2 j = V w (1 - a 2 ) m (2.31) where m was found to vary between 0 and 3 depending on the bubble s i z e . As a basis for considering the more complicated case of three-phase flow, Bhaga [1] developed a model for two-phase gas- l i q u i d flow by considering the r e l a t i v e v e l o c i t y between the phases. The l o c a l r e l a t i v e v e l o c i t y i s related to the l o c a l d r i f t v e l o c i t y by the equation 54 (1 - a 2) v 21 (2.32) Thus, combining equations 2.31 and 2.32, we get v 2 1 = V„ (1 - a 2) m-1 (2.33) which i s the re l a t i o n s h i p used by Bhaga [1] who expected the exponent (m-1) to vary between -1 and 2. For the lim i t e d data tested, however, Bhaga found that a value of m = 2 best represented his re s u l t s for both cocurrent and countercurrent air-water flow i n a v e r t i c a l conduit. Nevertheless, the model proposed by Bhaga provides no advantage over the o r i g i n a l model proposed by Zuber and Findlay, at l e a s t for two-phase flow. pation i n i r r o t a t i o n a l flow [38,40] i n conjunction with the c e l l u l a r representation of a bubble swarm suggested by Happel [49] to predict the r i s e v e l o c i t y of a bubble swarm. His method, applicable i n the bubble flow regime, assumes that the v o r t i c i t y generated i n the wake of a bubble i s not transferred far enough downstream to a f f e c t the motion of any downstream bubbles. Based on these assumptions he found that the energy-destroying v e l o c i t y (due to buoyancy alone), or the d r i f t v e l o c i t y , i s given by Marrucci [7 5] u t i l i z e d the concept of energy d i s s i -55 V „ (1 - a2)2/ d - a 2 5 / 3 ) (2.34) He noted q u a l i t a t i v e l y t h a t the sparse data o f N i c k l i n [19] and h i s own da t a i n d i c a t e d a r e l a t i v e l y s m a l l decrease i n r i s e v e l o c i t y w i t h i n c r e a s i n g a2, i n co n f o r m i t y w i t h equation 2.34. Although no e x p l i c i t mention i s made of the r a d i u s o f bubbles which make up the swarm, M a r r u c c i ' s model does i m p l i c i t l y i n d i c a t e the e f f e c t o f bubble r a d i u s on the r i s e v e l o c i t y of the swarm through the term V . [7 6] f o r p r e d i c t i n g the d r i f t v e l o c i t y i n the bubble flow regime, based on the method o f Maneri and Mendelson [48] f o r p r e d i c t i n g the r i s e v e l o c i t y o f l a r g e bubbles i n c o n f i n e d 2 media. The l a t t e r authors showed t h a t f o r l a r g e N£ o (=gp^R / a ) , t h a t i s , f o r l a r g e tube diameters such t h a t 1/N^ ,Q << 1, the r i s e v e l o c i t y of a s i n g l e bubble i s g i v e n by Equation 2.35 was found by Maneri and Mendelson to c o r r e l a t e experimental data w e l l f o r 1/y = R/re between 1 and 10, the lower l i m i t o f which corresponds to a s l u g . T h i s e quation g i v e s the d r i f t v e l o c i t y of a s i n g l e s p h e r i c a l bubble i n a c y l i n d r i c a l tube. Happel and A s t [50] suggested a sphere-i n - c y l i n d e r - t y p e c e l l model to r e p r e s e n t an assemblage of s o l i d Another model has been proposed by the pr e s e n t author V/V = / ' 00 tanh [0.25 (Vr )] (2.35) p a r t i c l e s . Thus, i f we use the approach of Happel and Ast, we have the necessary r e l a t i o n s h i p between y and the volumetr gas f r a c t i o n , a_, as follows: 3 (2.36) a 2 Y Now, combining equations 2.35 and 2.36, we get (2.37) Thus equation 2.37 would predict the d r i f t v e l o c i t y [or the energy-destroying v e l o c i t y as defined by N i c k l i n [19]] of a bubble swarm i n tubes of large diameters for low v i s c o s i t y systems e.g. water. I t should, however, be pointed out that equation 2.37 w i l l not give the d r i f t v e l o c i t y for slugs, but equation 2.35 with r Q = R does represent the r i s e v e l o c i t y of slugs i n a quiescent medium, for which i t then s i m p l i f i e s [48] i n combination with equation 2.10 for large N„ to v 2j g r e (tanh 0.25) (2.38) or v 2j = 0.35 /~gD (2.39) which i s the Dumitrescu equation for d r i f t v e l o c i t y i n the slug flow regime recommended by several investigators [39,27] Equation 2.26, derived o r i g i n a l l y by Zuber and Findlay, i s quite general and applicable to a l l ga s - l i q u i d flow regimes i f the d i s t r i b u t i o n parameter, and the weighted mean d r i f t v e l o c i t y can be obtained independently. This requires simultaneous measurements of v e l o c i t y and gas holdup p r o f i l e s , which have not generally been made, and thus l o c a l d r i f t v e l o c i t y p r o f i l e s are not av a i l a b l e . The lack of experimental measurements of l o c a l properties thus necessitates suitable assumptions for the d i s t r i b u t i o n parameter and the weighted mean d r i f t v e l o c i t y i n order to advance a meaningful empirical or semi-empirical model for two-phase gas- l i q u i d flow. Zuber and Findlay assumed smooth and symmetric p r o f i l e s for v e l o c i t y and gas holdup and thus found the d i s t r i b u t i o n parameter to be between 1.0 and 1.5 (see equation 2.27c). They did not appreciate the p o s s i b i l i t y of systematic c i r c u l a t i o n , which p e r s i s t s i n bubble columns [59] and which may render these p r o f i l e s to be neither smooth nor symmetric. Thus, the models above cannot be used successfully to predict the gas volume f r a c t i o n i n columns where c i r c u l a t i o n e x i s t s . At the present time not enough information i s available to predict the rate of c i r c u l a t i o n , but a possible mechanism i s suggested i n the next section. 58 2.1.3 C i r c u l a t i o n and turbulence i n two-phase g a s - l i q u i d flow I t has been shown above that for a bubble column greater than 10 cm i n diameter, a systematic c i r c u l a t i o n develops i n the l i q u i d phase with an upward l i q u i d flow i n the central core of the column and a downward l i q u i d flow near the walls. From a l l the evidence presented, the following description of c i r c u l a t i o n can be given: At very small gas flow rates and low Reynolds number (Re^ < 10) the bubbles are uniformly d i s t r i b u t e d and r i s e i n distinguishable bubble chains [77]. Crabtree and Bridgwater [78] showed that the r i s i n g bubble chains drag l i q u i d by viscous shear and thus induce c i r c u l a t i o n i n the l i q u i d phase. This c i r c u l a t i o n i s further enhanced by the presence of both low density (high gas fraction) and high density (low or zero gas fraction) regions. At higher Reynolds number (Re^ - 400), a distinguishable wake behind the bubbles appears and the l i q u i d i n the attached wake travels at the bubble v e l o c i t y . The l i q u i d i n the wake i s deposited near the surface of the main l i q u i d when the bubble reaches t h i s surface and breaks up. Because of the r i s e of l i q u i d i n the wake of the bubble, the c i r c u l a t i o n becomes quite intense, a f f e c t i n g the r a d i a l bubble d i s t r i b u t i o n and giving r i s e to more regions of high and low density, which help to sustain the c i r c u l a t i o n . At s t i l l higher Reynolds number, i t i s believed that the wake no longer remains attached to the bubble but i s shed at regular i n t e r v a l s [79] , as observed by Letan and Kehat [^36] for drops i n l i q u i d - l i q u i d systems. Under these conditions the l i q u i d c i r c u l a t i o n appears to be chaotic as i t i s superimposed on random eddying [59] , but i n f a c t a systematic c i r c u l a t i o n i s maintained i n which the l i q u i d moves upwards i n the central region and downwards near the w a l l . The bubbles mainly r i s e i n the high upward v e l o c i t y region, but the c i r c u l a t i o n i s so intense that a bubble could be trapped i n the downward flow region or even swept back with the downward flow, as observed by Freedman and Davidson [51]. A similar physical model can be advanced for cocurrent gas- l i q u i d flow. Baker and Chao [65] have shown that the dynamics of an i n d i v i d u a l bubble i s not affected by the up-wards l i q u i d v e l o c i t y . I t i s therefore to be expected that a s i m i l a r general model should be applicable to cocurrent flow, where the c i r c u l a t i o n and the wake shedding patterns are superimposed on a net l i q u i d flow i n the v e r t i c a l d i r e c t i o n . However, Reith et a l . [68] observed no systematic c i r c u l a t i o n of l i q u i d i n cocurrent flow; nevertheless, a x i a l mixing was found to occur and was ^apparently .caused by the wake shedding phenomenon. The same e f f e c t has also been observed by Letan and Kehat [36] for the study of a x i a l mixing i n l i q u i d - l i q u i d extraction (spray) columns. I t i s therefore probable that i n cocurrent flow the c i r c u l a t i o n i s contained i n c e l l u l a r regions. 60 I t has been observed for the r i s e or f a l l of a drop through a l i q u i d medium that the wake behind the drop i s shed at regular i n t e r v a l s [36]. Though vortex shedding from behind a bubble has not been studied, i t i s believed to occur on very similar patterns as behind a drop. The shed vo r t i c e s w i l l i n i t i a l l y r i s e i n the l i q u i d at the bubble v e l o c i t y , but w i l l subsequently be dissipated by viscous stresses [79]. The i n t e r a c t i o n of the mean flow with these ejected vortices would create intense and chaotic v e l o c i t y fluctuations i n the mean flow. These fluctuations are believed to be respon-s i b l e for the generation and maintenance of turbulence i n the l i q u i d phase at the expense of the energy of the mean flow. Delhaye [81] had reported some measurements of i n t e n s i t y of turbulence induced i n a l i q u i d due to passage of gas bubbles. He observed that even at small gas flow rates, the turbulence generated i n the l i q u i d phase by the gas bubbles i s quite s i g n i f i c a n t . 2.2 Voidage i n L i q u i d - s o l i d f l u i d i z e d beds The hydrodynamics of f l u i d flow through a bed of granular material has been researched quite successfully i n the past. B a s i c a l l y , three d i s t i n c t approaches have been developed and used to study the hydrodynamics of f l u i d i z e d beds: (i) U sing the r e l a t i o n s h i p between p r e s s u r e drop and f l u i d flow r a t e o f a f i x e d bed a t the f l u d i z a t i o n t r a n s i t i o n p o i n t to study i n c i p i e n t f l u i d i z a t i o n and subsequent bed expansion [ 5 6 ]. ( i i ) Using the concept suggested by Lapidus and E l g i n [67] t h a t the r e l a t i v e v e l o c i t y between the l i q u i d and the s o l i d p a r t i c l e s alone can p r e d i c t the voidage i n l i q u i d - s o l i d f l u i d i z e d beds. The a c t u a l form o f the r e l a t i o n s h i p between the r e l a t i v e v e l o c i t y and the bed voidage has, however, to be determined experimen-t a l l y . ( i i i ) Using a c e l l model to r e p r e s e n t an assemblage o f par-t i c l e s as suggested by Happel and Brenner [49]; then s o l v i n g the f u l l Navier-Stokes equations w i t h i n the c e l l a n a l y t i c a l l y f o r low Reynolds numbers (Re < 1 ) , P when the i n e r t i a l terms are not im p o r t a n t , or numeri-c a l l y f o r moderately h i g h Reynolds numbers [65,66] . A l l t hree techniques have been found to p r e d i c t the r e l a t i o n s h i p between voidage and v o l u m e t r i c l i q u i d f l u x q u i t e s a t i s f a c t o r i l y over a wide range o f system v a r i a b l e s . Since a l l these techniques have been t r e a t e d q u i t e e l o q u e n t l y by v a r i o u s i n v e s t i g a t o r s [56,2,49], no attempt i s made here to pr e s e n t them i n d e t a i l . However, a summary of these techniques i s g i v e n below. Andersson [56] s t u d i e d the a p p l i c a b i l i t y o f the p r e s -sure drop e q u a t i o n f o r f i x e d beds to o b t a i n a r e l a t i o n s h i p between the l i q u i d v e l o c i t y and the bed voidage for l i q u i d -s o l i d f l u i d i z e d beds. The a p p l i c a b i l i t y of t h i s technique mainly depends on the experimentally observed f a c t that the pressure drop i n f l u i d i z e d beds i s always equal to the buoyed weight of the bed per uni t area, i . e . , -Ap = e 3 ( p 3 - p 1 ) L b g (2.40) Ergun [57] successfully represented the pressure drop through a fixed bed of spherical p a r t i c l e s by an equation which/at the t r a n s i t i o n point to f l u i d i z a t i o n , i s Ap d e? u ( 2 — j ) 9 ( — ) = 1 5 0 ( h L f + 1 ' 7 5 L P l ^ ^ £3 mf ^ l ^ l S (2.41) Since the pressure drop through a fixed bed of p a r t i c l e s must equal the buoyed weight of the bed per u n i t area i n order to i n i t i a t e f l u i d i z a t i o n , combination of equations 2.40 and 2.41 provides the desired r e l a t i o n s h i p between the void f r a c t i o n and the volumetric l i q u i d f l u x . However, the knowl-edge of bed voidage at i n c i p i e n t f l u i d i z a t i o n , ( e]_) mf/ 1 S s t i l l required. A value of 0.4 i s recommended fo r spherical p a r t i c l e s [21,49] . Neuzil and Hrdina [47] observed that the voidage at minimum f l u i d i z a t i o n , ( £ i ) m f ^ s affected by the d. ./D r a t i o and recommended the following r e l a t i o n s h i p for J? spherical p a r t i c l e s : 63 ^l^mf = ° ' 4 0 4 + ° - 4 2 9 ( d p/°) (2.42) The second approach u t i l i z e s the concept of r e l a t i v e v e l o c i t y between the l i q u i d and the s o l i d p a r t i c l e s to describe the re l a t i o n s h i p between the voidage and the volu-metric l i q u i d f l u x . Richardson and Zaki [2], based on a dimensional analysis of the relevant variables involved i n f l u i d i z a t i o n , demonstrated that v i 3 v r v 3 V V OO 00 = f, (Re , e,, d /D) (2.43) 1 p 1 p Since i n batch f l u i d i z a t i o n the net v e l o c i t y of the s o l i d p a r t i c l e s i s zero, the r e l a t i v e v e l o c i t y between the l i q u i d and the s o l i d p a r t i c l e s i s simply v 1 3 = v x = <j1>/e1 (2.44) Now,combining equations 2.43 and 2.44, the functional r e l a t i o n -ship can be written as = f 2 (Re p, e±f dp/D) (2.45) On the basis of a large amount of data obtained for sedimen-ta t i o n and f l u i d i z a t i o n of p a r t i c l e s for a wide range of system variables, Richardson and Zaki [2] found that the following simple r e l a t i o n s h i p represented the r e s u l t s i n the best manner: <J!>/v; = ej (2.46) where V* = V and exponent n was found to be a function of 00 CO ^ p a r t i c l e Reynolds number (R©p) alone, only i f the r a t i o dp/D was s u f f i c i e n t l y small. The following values of n were reported by Richardson and Zaki over a wide range of Re , i n the absence of wall e f f e c t : n=4.65 R e < 0 . 2 (2.47) P ' n = 4.35 Re"* 0 3 0.2<Re < 1 (2.48) P P n = 4.45 R e " 0 , 1 KRe < 500 (2.49) P P n = 2.39 500 < Re (2.50) P Richardson and Zaki [2] also observed that the r a t i o dp/D affected the f l u i d i z a t i o n quite markedly. However, Neuzil and Hrdina [47] found that the correction factors reported by Richardson and Zaki to take wall e f f e c t s into account were not adequate. They, therefore, questioned the basis on which Richardson and Zaki had formulated t h e i r wall e f f e c t correction f a c t o r s . Neuzil and Hrdina, following an approach similar to that suggested by Steinour [93] , obtained the following semi-empirical r e l a t i o n s h i p to describe the bed expansion: < J l > / v = o = ° - 6 7 R e p ' ° 3 [1-1.27 ( d / D ) 1 * 1 5 ] el' 1 (2.51) which i s based on a large quantity of experimental data i n the range 0.0454 < dp/D < 0.3 and 75.5 < Re p < 1795. The c o r r e l a t i o n for minimum f l u i d i z a t i o n of spherical p a r t i c l e s was obtained by combining equations 2.42 and 2.51, the r e s u l t being J ± m f = [1-1.27 (d / D ) 1 , 1 5 ] [0.348+0.370 (d /D) ] 2 * 7 Re° * 0 3 V M p' P - P (2.52) However when the wall e f f e c t s are not important, e.g. for dp/D = 0.1 the r e l a t i v e error i n the l i q u i d phase volume f r a c t i o n i s only 3.5% [47]. Equation 2.46 proposed by Richardson and Zaki [2], with the value of the exponent n obtained from equations 2.4 7 -2.50, should be used because i t has been checked for a wider var i e t y of data and i s generally more acceptable. Thus the equation proposed by Neuzil and Hrdina, when the wall e f f e c t s are important, and the equations proposed by Richardson and Zaki, when the wall e f f e c t s are n e g l i g i b l e , w i l l be used i n t h i s work to predict the r e l a t i o n s h i p between 66 the voidage and the volumetric l i q u i d flux for the l i q u i d -s o l i d systems investigated. The t h i r d method for l i q u i d - s o l i d f l u i d i z a t i o n i s e s s e n t i a l l y a n a l y t i c a l i n i t s approach, and depends on the a b i l i t y of representing an assemblage of p a r t i c l e s by a unit c e l l consisting of one spherical p a r t i c l e surrounded by a concentric spherical envelope, which contains a volume of l i q u i d such that the l i q u i d phase volume f r a c t i o n i n the c e l l i s the same as that for the entir e assemblage [49]. This approach then requires solving the f u l l Navier-Stokes equa-tions of motion within the c e l l f or appropriate boundary conditions. The Navier Stokes equations, with the required boundary conditions, can be solved a n a l y t i c a l l y i f the i n e r t i a l terms i n the equations can be neglected (Re < 0.2). P However, when the i n e r t i a l terms are important, solution of the complete Navier-Stokes equations can only be obtained by numerical techniques. Such techniques have been developed and used by Masliyah and Epstein [82,] and by L e c l a i r and Hamielec [90] to obtain solutions for an assemblage of spher-i c a l or spheroidal p a r t i c l e s up to quite large Reynolds numbers (Re - 100) . P. 2.2.1 E f f e c t of turbulence on voidage i n l i q u i d - s o l i d f l u i d i z e d beds Free-stream turbulence has been shown to have a s i g -n i f i c a n t e f f e c t on the drag c o e f f i c i e n t of a single p a r t i c l e [94-96] . Changes i n the drag c o e f f i c i e n t of a p a r t i c l e were found to be influenced by the free-stream Reynolds number, based on the r e l a t i v e velocity.,of free stream with respect to the p a r t i c l e , and the c h a r a c t e r i s t i c s of the free-stream turbulence, v i z . , the i n t e n s i t y of turbulence [94,95] and the scale of turbulence r e l a t i v e to the p a r t i c l e size [96]. Torobin and Gauvin [95] measured drag c o e f f i c i e n t s of aero-dynamically smooth spheres by studying t h e i r v e l o c i t y history i n a wind tunnel, i n which turbulence was generated by screen g r i d s . They observed that, at constant Reynolds number, an increase i n the i n t e n s i t y of turbulence at f i r s t produced a moderate increase and then a sharp decrease i n the drag c o e f f i c i e n t of the p a r t i c l e . The increase was assumed to be caused by the disruption of the p a r t i c l e wake, whereas the sharp decrease was believed to be caused by the premature t r a n s i t i o n of the laminar boundary layer into a turbulent boundary layer and i t s consequent reattachment to the p a r t i c l e , thereby reducing the form drag, which con-s t i t u t e s the main portion of the t o t a l drag force under these conditions. The value of the c r i t i c a l Reynolds number, based on r e l a t i v e v e l o c i t y between the a i r stream and the p a r t i c l e , at which t h i s t r a n s i t i o n took place varied with the i n t e n s i t y of free-stream turbulence, 1^ .. Torobin and Gauvin found that the c r i t e r i o n for t r a n s i t i o n was adequately described by the r e l a t i o n s h i p 68 Re = 45 (2.53) where Re c If the free-stream Reynolds number was now increased beyond Re c, for a fixed i n t e n s i t y of free-stream turbulence, the drag c o e f f i c i e n t of 'the p a r t i c l e decreased to a d e f i n i t e minimum after which i t increased again quite noticeably [94]. In f l u i d i z a t i o n , the p a r t i c l e s are maintained i n a suspended state because t h e i r weight, modified by buoyancy, i s balanced exactly by the drag forces due to the r e l a t i v e motion of the l i q u i d with respect to the p a r t i c l e s . Should the f l u i d i z i n g stream be turbulent and the c r i t e r i o n estab-lis h e d by equation 2.53 be s a t i s f i e d on increasing the flow rate, the drag c o e f f i c i e n t as well as the t o t a l drag force experienced by the p a r t i c l e s would then be reduced. This would r e s u l t i n contraction of the bed, that i s , reduction of bed voidage. For a l l other ranges of Reynolds number, any increase of turbulence i n the f l u i d i z i n g l i q u i d stream on increasing the flow rate would give r i s e to an increase i n the drag c o e f f i c i e n t over and above that caused by the increased flow i t s e l f , thereby causing further expansion of the l i q u i d - s o l i d f l u i d i z e d bed. No detailed study has been conducted to elucidate the e f f e c t s of turbulence on the expansion c h a r a c t e r i s t i c s of a l i q u i d - s o l i d f l u i d i z e d bed. However, a l i m i t e d study involving water f l u i d i z a t i o n of very large and very heavy spheres, carried out by Trupp [87], revealed that such f l u i d i z e d beds expanded more than predictable by the Richardson - Zaki c o r r e l a t i o n s . I t i s known that a f l u i d i z e d bed of randomly spaced and randomly moving s o l i d p a r t i c l e s generates fluid-phase turbulence [97]. The quantitative c h a r a c t e r i s t i c s of t h i s induced turbulence, v i z . , the i n t e n s i t y and the scale, have not, however, been measured. Therefore at t h i s stage i t can only be surmised that the turbulence of the f l u i d i z i n g l i q u i d stream s i g n i f i c a n t l y influences the behaviour of a l i q u i d - s o l i d f l u i d i z e d bed. But for a better understanding of the importance of the turbulence phenomenon i n f l u i d i z a t i o n , i t i s necessary to qu a n t i t a t i v e l y study the c h a r a c t e r i s t i c s of the turbulence and t h e i r influence on bed voidage of l i q u i d - s o l i d f l u i d i z e d beds. 2.3 Holdup i n g a s - l i q u i d - s o l i d f l u i d i z e d beds The active academic i n t e r e s t i n the study of three-phase f l u i d i z e d beds originated from the preliminary i n v e s t i -gations undertaken by Turner [6] and Adlington and Thompson [9], who reported that on slow addition of gas to a l i q u i d -s o l i d f l u i d i z e d bed, the bed contracted. Volk [10], however, i n an e a r l i e r and detailed study of three-phase f l u i d i z a t i o n , had observed that the l i q u i d - s o l i d f l u i d i z e d bed expanded smoothly on introduction of the gas phase. Based on these contrary observations, i t was postulated [7,8,10] that i n a three-phase f l u i d i z e d bed the s o l i d p a r t i c l e s were e n t i r e l y supported by the l i q u i d whereas the gas t r a v e l l e d through the bed as d i s c r e t e p a r t i c l e - f r e e bubbles. This postulate forms the basis of various mathematical models formulated to describe the expansion behaviour of three-phase f l u i d i z e d beds, three of which are considered i n the following section. 2.3.1 Models for three-phase f l u i d i z e d beds The three mathematical models which have been or w i l l be proposed to predict the volumetric f r a c t i o n of the i n d i v i d -ual phases i n a three-phase f l u i d i z e d bed are: (A) the gas-free model, (B) the wake model and i t s modifications, and (C) the c e l l model. A fourth model, whereby the gas and l i q u i d are treated as a homogeneous f l u i d i z i n g medium with appropriately averaged f l u i d properties and v e l o c i t y , was considered by Volk [10]. This homogeneous model i s rejected here on the grounds that i t i s unsuitable even for two-phase gas- l i q u i d flow i n most instances, e s p e c i a l l y for the slug flow regime. (A) The g^as-f'ree model Volk [10] , who made an extensive study of the behaviour of three-phase f l u i d i z e d beds using nitrogen as gas, heptane as l i q u i d and porous c y l i n d r i c a l c a t a l y s t p e l l e t s as s o l i d p a r t i c l e s , reported that the bed expanded smoothly on increasing the gas flow rate at a fixed l i q u i d flow rate. He considered the three-phase f l u i d i z e d bed to consist of a bubbling gas phase and a l i q u i d - s o l i d f l u i d i z e d (particulate) phase, the l i q u i d flow rate through which i s modified due to the presence of the bubbles. The observed bed expansion was considered due to (i) the increase i n bed volume caused by the presence of the gas bubbles, and ( i i ) the increase i n i n t e r s t i t i a l l i q u i d v e l o c i t y caused by the reduction i n area available for l i q u i d flow. These two factors are inter-dependent. The data processing scheme used by Volk was empirical i n i t s nature and grossly i n e r r o r. Therefore, by taking the two factors mentioned above into account, a simple mathematical model, along the l i n e s suggested by Volk, i s developed here. Figure 2.2 demonstrates the model schematically. Let A be the cross-sectional area of the column and be the average volumetric f r a c t i o n occupied by the i phase. Then 3 Z e i = 1 (2.54) i=l 72 <i,> <J 2> FIGURE 2.2 SCHEMATIC REPRESENTATION OF THE GAS-FREE MODEL 73 Let us assume that each phase i s homogeneously d i s t r i b u t e d so that the area occupied by each phase i s e.A. Then the Area occupied by the gas phase = £ 2A (2.55) and the Area occupied by the l i q u i d - s o l i d f l u i d i z e d phase = A ( l - e 2 ) (2.56) Since a l l the l i q u i d i s assumed to flow through the l i q u i d -s o l i d f l u i d i z e d bed region, a material balance for the l i q u i d gives S u p e r f i c i a l l i q u i d v e l o c i t y through gas-free region = A <j 1> <j x> A ( l - e 2 ) ( 1" e2 ) (2.57) S i m i l a r l y a material balance of gas through the bed gives <j 2> = v 2 e 2 (2.58) where v 2 i s the average r i s e v e l o c i t y of bubbles through the bed. Lt has been assumed. [8,16,79] that bed expansion of the l i q u i d - s o l i d f l u i d i z e d phase can be well represented by the Richardson - Zaki c o r r e l a t i o n , using the modified super-f i c i a l l i q u i d v e l o c i t y through t h i s region. Then the volume f r a c t i o n of l i q u i d i n the p a r t i c u l a t e phase, e|'^  i s given by 1/n (2.59) 'If e2) J where the exponent n i s a function of p a r t i c l e Reynolds number, Re , while E " i s related to the o v e r a l l volumetric p If f r a c t i o n of l i q u i d i n the three-phase f l u i d i z e d bed, e^, by 'If z^/ ( l - e 2 ) (2.60) Combining equations 2.59 and 2.60, E ^ i s given by £ 1 = <3i> V (1 - £„) ( l - e 2 ) (2.61) and the t o t a l bed voidage, e, i s given by e = ( l - e 3 ) £1 + e2 = <3 ^> V (l-£ 0) co 2 1/n ( l - e 2 ) + e 2 (2.62) By rearranging equation 2.62, the s o l i d s holdup i n the three-phase f l u i d i z e d bed, e^, can be written as 1/n , e 3 = ( l - e 2 ) 1 -<3 ^> LV (1-£0)J L. 00 v 2 (2.63) Thus equation 2.62, i n combination with equation 2.58, provides a simple scheme for describing the voidage and bubble v e l o c i t y i n a three-phase f l u i d i z e d bed. With the help of t h i s model and a measurement or estimate of the volumetric f r a c t i o n of one of the three phases at d i f f e r e n t gas and l i q u i d flow rates, necessary information about the average volumetric fractions.of the other i n d i v i d u a l phases i n a three-phase f l u i d i z e d bed can be obtained. The determination of s o l i d s holdup, by measuring the bed height, L^, i s probably the simplest procedure, as the s o l i d s holdup, e^, i s related to the bed height, L, , by the following r e l a t i o n s h i p : e3 = (2.64) The average volumetric f r a c t i o n of gas, e 2/ can then be obtained, by t r i a l and error, from equation 2.63, and the average volumetric f r a c t i o n of l i q u i d , e^, from equation 2.54. Thus a simple mathematical model can be obtained on the basis of the description of a three-phase f l u i d i z e d bed given by Volk. However, i t i s important to note that t h i s model has the inherent assumption that no l i q u i d i s associated with the gas phase. Such l i q u i d , by d e f i n i t i o n , would not provide any e f f e c t i v e force for f l u i d i z a t i o n of the s o l i d p a r t i c l e s and hence would not contribute to the expansion of the bed. I t i s therefore necessary to understand the d i s t r i b u t i o n of the l i q u i d phase between the gas phase and the p a r t i c u l a t e phase. The wake model derived o r i g i n a l l y by pstergaard [8], as described i n section 1.3, considers 76 thi s d i s t r i b u t i o n by assuming that the l i q u i d associated with the gas phase i s represented by the wake of the bubbles. (B) The wake model Stewart and Davidson [7] and 0 s t e r g a a r d [8] each pro-posed a mechanism for the observed bed contraction i n three-phase, f l u i d i z a t i o n [6,9] by describing a three-phase f l u i d i z e d bed as consisting of (i) a gas phase, (2), ( i i ) a wake phase, (k), and ( i i i ) a l i q u i d - s o l i d f l u i d i z e d (or particulate) phase, ( f ) . The mechanisms proposed were s i m i l a r i n most respects; however, one difference was reported i n the respective observations of these early investigators. Stewart and Davidson observed photographically that the bubble wake i n a two-dimensional bed was e s s e n t i a l l y free of s o l i d p a r t i c l e s , whereas 0 s t e r g a a r d ' s photographic observations of bubbles emerging from a three-phase f l u i d i z e d bed showed them to be followed by a long t r a i l containing s o l i d p a r t i c l e s . 0 s t e r g a a r d , i n proposing the mathematical model presented i n section 1.3, therefore assumed that the volume f r a c t i o n of so l i d s i n the wake was the same as i n the p a r t i c u l a t e phase. The main weaknesses i n the model proposed by 0 s t e r g a a r d , as stated i n section 1.3, are (i) The assumption that the porosity of the wake phase i s equal to that of the p a r t i c u l a t e phase, ( i i ) The neglect of solids c i r c u l a t i o n induced by the motion of gas bubbles carrying wakes containing s o l i d p a r t i c l e s . ( i i i ) The quantitative representation of bubble r i s e v e l o c i t y by equation 1.12 and of wake volume f r a c t i o n by equation 1.13. Modifications to remedy these shortcomings are proposed below. In view of the controversy about the sol i d s content of the wake, the wake model i s rederived here for a wake soli d s content ranging from zero to the value p r e v a i l i n g i n the p a r t i c u l a t e phase, i n order to e s t a b l i s h i t s e f f e c t on the bed c h a r a c t e r i s t i c s . Figure 2 .3.. demonstrates the model schematically. Consider the d i s t r i b u t i o n of l i q u i d between the wake phase, k, and the par t i c u l a t e phase, f . Let us assume that Volume of l i q u i d i n wake phase = (2.65) Volume of l i q u i d i n p a r t i c u l a t e phase = (2.66) so that the t o t a l volume of l i q u i d i n the three-phase f l u i d i z e d bed i s given by ^1 = filk + R l f (2.67) and the average volumetric f r a c t i o n of l i q u i d i n the three-phase f l u i d i z e d bed i s 78 V If — I l l / * s 1 1 © 1 1 1 SOLID 1 .LIQUID 1 € 3 f | *1f ® (D WAKE GAS <* I I I I I I •— i l € 3 k V 1f _ III Vo FIGURE 2.3 SCHEMATIC REPRESENTATION OF THE WAKE MODEL 79 £1 = ^ l / ^ b (2.68) Let us now define the average volumetric f r a c t i o n of wake l i q u i d i n the bed as e,, = -±± (2.69) AL. b and the average volumetric f r a c t i o n of p a r t i c u l a t e phase l i q u i d as " i f e 1 f = — (2.70) AL, b Then i t can e a s i l y be seen that £1 - £ l k + £ l f ( 2 ' 7 1 ) S i m i l a r l y l e t us consider the d i s t r i b u t i o n of so l i d s between the wake phase, k, and the pa r t i c u l a t e phase, f . Let us assume.that W .^ i s the weight of s o l i d p a r t i c l e s i n the wake phase and the weight of s o l i d p a r t i c l e s i n the part i c u l a t e phase. Then the t o t a l weight of s o l i d p a r t i c l e s i s given by W = W f + w k (2.72) 80 and the average'volumetric f r a c t i o n of s o l i d s i n the three-phase f l u i d i z e d bed i s e 3 = W/p3 AL b (2.64) Let us define the average volumetric f r a c t i o n of wake sol i d s i n the bed as W k £3k = — I T " ( 2'7 3 ) p 3 AL b and the average volumetric f r a c t i o n of p a r t i c u l a t e phase solids as W f e 3 f = (2.74) p^ AL, H3 rtJ-<b I t can e a s i l y be seen that £3 = £3f + £3k ( 2'7 5 ) Since a l l the gas passes through the bed as discrete bubbles, the average volumetric f r a c t i o n of gas i n the three-phase f l u i d i z e d bed, z^, i s i n d i v i s i b l e . Then by material balance, £1 + £2 + £3 ~ 1 (2.76) 81 Combining equations 2.71, 2.75 and 2.76 we get £ l k + e l f + £2 + £3k + £ 3 f = 1 ( 2 ' 7 7 ) Now the t o t a l volume occupied by the wake phase = volume of s o l i d s i n wake phase + volume of l i g u i d i n wake phase W k = + °lk = ^ b ( e3k + £ l k ) and the average volumetric f r a c t i o n of the wake phase i n the three- phase f l u i d i z e d bed, (= volume of wakes/bed volume), i s therefore given by ek = £ l k + £3k ( 2 ' 7 8 ) Combining equations 2.77 and 2.78, and rearranging, we get £ l f + £ 3 f = 1 ' £2 " £k ( 2 ' 7 9 ) or -Al— + Z 3 f = i (2.80) 1 _ £ 2 - £ k 1 - £ 2 - £ k In order to esta b l i s h a rel a t i o n s h i p between the par t i c u l a t e (or l i q u i d - s o l i d f l u i d i z e d bed) region and the l i q u i d - s o l i d wake region, l e t us f i r s t consider the p a r t i c u -82 l a t e region. Let us define the average volumetric f r a c t i o n of s o l i d s i n t h i s region, e^'f* as volume of s o l i d s i n pa r t i c u l a t e phase/volume of p a r t i c u l a t e phase or e" = V p3 " 3 f Wf/p3 + 0 l f Dividing both numerator and denominator by the volume of the three-phase f l u i d i z e d bed we get (W-/p-AL. ) e e — 1 J a = J I (2.81) 3 f (N f/p 3AL b) + (fi l f/AL b) e 3 f + e l f substituting equation 2.79.into, equation 2.81 y i e l d s " £ 3 f = — (2.82) ( l - £ 2 - E k ) S i m i l a r l y i t can be shown that the average volumetric f r a c t i o n of l i q u i d i n the p a r t i c u l a t e phase i s - " e l f e l f = — (2.83) ( 1 - £ 2 - £ k ) Then from equations 2.80, 2.8 2 and 2.83, we can obtain that II n £ l f + £ 3 f 1 (2.84) 83 which also follows from the d e f i n i t i o n s of and z'^f For the l i q u i d - s o l i d wake region l e t us define the average volumetric f r a c t i o n of so l i d s i n the wake as volume of s o l i d s i n wake/total volume of wake or V p 3 £3k wk/p3 + n l k Dividing both numerator and denominator by the volume of the three-phase f l u i d i z e d bed we get y p s ^ b = £3k ( V ^ ^ V + ( "lk/ A L b ) £lk + £3k e3'k = ^ — ^ — - = — (2.85) Substituting equation 2.78 into equation 2.8 5 y i e l d s £3k " i f <2-86> Sim i l a r l y i t can be shown that the average volumetric f r a c t i o n of l i q u i d i n the wake i s II e. k Then from equations 2.78, 2.86 and 2.87, we can obtain that II II £ l k + £3k = 1 (2.88) 84 II II which also follows from the d e f i n i t i o n s of £^ k and £ 3 k . Now since the amount of sol i d s i n the wake of a bubble i s not known a p r i o r i , i t has been expedient to assume that e 3 k i s related to v i a e3k = X k £ 3 f ( 2 * 8 9 ) where x ^ i s a c o e f f i c i e n t of pro p o r t i o n a l i t y which can take any po s i t i v e value between zero and unity. Then the case of Xj^ . = 0 s i g n i f i e s that there are no s o l i d p a r t i c l e s i n the wake (as observed by Stewart and Davidson), whereas the case of Xj^  =1 s i g n i f i e s that the s o l i d s f r a c t i o n i n the wake i s the same as that i n the p a r t i c u l a t e phase (as postulated by ^stergaard). I t can e a s i l y be seen that e l k - 1 - x k e 3 f = ^ W l f (2-90) and therefore the t o t a l bed voidage i n a three-phase f l u i d i z e d bed, from equations 2.71, '2..'83 r "2% 8'7 "and 2r. 90, i s £ = = ( l - e 3 ) = e2 + £ k ^ 1 - x k ^ + e l f ( x k e k + 1 ~ e2 ~ ek^ (2.91) In order to f i n d the v e l o c i t y of l i q u i d through the pa r t i c u l a t e phase, l e t us assume that the t o t a l volumetric flow rate of l i q u i d through the column i s Q, and the t o t a l 85 volumetric flow rate of gas through the column i s Q 2. Then the average l i q u i d f l u x through the column i s <Ji> = Q-./A (2.92) and the average gas flux through the column i s <Jo> = Qo/A (2.93) Now, since a l l the gas through the column travels as discrete bubbles, a simple material balance over any cross-section perpendicular to the flow path w i l l give <j 2> = e 2 v 2 (2.94) Let us now consider a sim i l a r material balance for the l i q u i d over a cross-section perpendicular to the flow path. Then volumetric flow rate of l i q u i d through the column / l i q u i d flux\ | through the J \ particulate J x Vphase / (cross^sectional area occupied by the pa r t i c u l a t e phase (l i q u i d f l u x \ through the j wake phase / x cross-sectional I area occupied by \ the wake phase or <j 1> A = A ( l - e 2 - e k ) ] + [ ( v 2 e l k). (A efc)] (2.95) 86 Substituting equation 2.90 into equation 2.95 and rearranging gives <j x> - v 2 (1-x k+x k^ f) ek j-^ - (2.96) ( l - e 2 - e k ) The l i q u i d f l u x through the pa r t i c u l a t e phase can be related to the average volumetric f r a c t i o n of l i q u i d i n the par t i c u l a t e phase, by the Richardson - Zaki c o r r e l a t i o n , II 1/n e l f = ( V V J (2-97) where n = f (Re )' and the values of n for various Re are P P given by equations 2.47 to 2.50. Thus for a d i s t r i b u t i o n of s o l i d p a r t i c l e s between the wake phase and the pa r t i c u l a t e phase as given by equation 2.89, equations 1.4 and 1.10 derived e a r l i e r for J^stergaard 1 s model have been modified to equations 2.91 and 2.96, respectively. Another deficiency of the wake model, as postulated by 0stergaard [8] and used subsequently by Efremov and Vakhrushev [16] as well as Rigby and Capes [80] , has been the neglect of so l i d s c i r c u l a t i o n i n the pa r t i c u l a t e phase. A l -though the existence of so l i d s c i r c u l a t i o n was not reported by these authors, the bubbles r i s i n g i n a three-phase f l u i d i z e d bed have been observed to carry .sizeable wakes- containing some p a r t i c l e s . A simple consideration of continuity would then suggest a downward motion of s o l i d p a r t i c l e s i n the 87 pa r t i c u l a t e phase as a r e s u l t of the i r upward motion i n the wakes of the bubbles. Based on t h i s simple mechanism for so l i d s c i r c u l a t i o n i n a three-phase f l u i d i z e d bed, a model w i l l now be developed for i t . Let us assume that each bubble i n a three-phase f l u i d i z e d bed c a r r i e s with i t an attached wake containing s o l i d p a r t i c l e s . Then the rate at which s o l i d s reach the surface of the three-phase f l u i d i z e d bed Solids f l u x through .wake x [ cross-sectional area occupied by wake phase or 'volumetric flow rate of s o l i d s to bed surface ] = [*2 £3k] x [ A £ k ] <2'98> After a bubble emerges from the bed, s o l i d s carried i n i t s wake w i l l be washed out of i t by continuous exchange with the surrounding l i q u i d [80] . The distance for which the s o l i d s w i l l be carr i e d above the bed would then depend on the exchange rate between the surrounding l i q u i d phase and the wake phase. The s o l i d p a r t i c l e s , a f t e r leaving the wake, would move downwards to the bed and downwards i n the bed to compensate for the upward movement i n the wake. There-fore the downward flow rate of s o l i d p a r t i c l e s i n the par-88 t i c u l a t e phase Solids flux through pa r t i c u l a t e phase x cross-sectional area occupied by pa r t i c u l a t e phase = [- v. '3f " x [A(l-e 2-e k) ] (2.99) Since there i s no net flow of s o l i d p a r t i c l e s through the bed, the upward flow of so l i d s i n the wake must exactly equal the downward flow of sol i d s i n the pa r t i c u l a t e phase. Hence, equating equations 2.98 and 2.99 and rearranging, the li n e a r v e l o c i t y of s o l i d p a r t i c l e s i n the pa r t i c u l a t e phase i s given by .. v „ e,_ e. v 3 = ^—-——— (2.100) e 3 f ( 1 " e 2 - £ k ) and i n combination with equation 2.89, _„ = _ V2 £k X k V3 " ( l - e 2 - e k ) ( 2 * 1 0 1 ) From equation 2.101 i t can be seen that i f there are no s o l i d p a r t i c l e s i n the wake (*k= 0), the c i r c u l a t i n g v e l o c i t y of s o l i d p a r t i c l e s i n the f l u i d i z e d bed i s zero. Thus no c i r c u l a t i o n would e x i s t by the mechanism postulated here. 89 The presence of other more complex modes of c i r c u l a t i o n i s recommended for further investigations. For the present i t w i l l be assumed that equation 2.101 describes the s o l i d s c i r c u l a t i o n phenomenon adequately for x^ > 0. As has been stated e a r l i e r i n section 2.2, the r e l a t i v e v e l o c i t y between the l i q u i d and the s o l i d p a r t i c l e s controls the bed expansion [67]. The Richardson - Zaki c o r r e l a t i o n , equation 2.97, can be modified and represented i n terms of the r e l a t i v e v e l o c i t y : _ II „ v, , l / ( r i - l ) e l f = ( ) (2.102) CO The l i q u i d f l u x through the p a r t i c u l a t e phase i s given by equation 2.96. Therefore the l i n e a r v e l o c i t y of l i q u i d through the p a r t i c u l a t e phase w i l l be *1 < ^ l > - V 2 { 1 - x k + £ l f x k ) £ k v± = = — n '• (2.103) £ l f £ l f ( 1 ~ £ 2 - £ k ) and the r e l a t i v e v e l o c i t y between the l i q u i d and the s o l i d p a r t i c l e s i n the p a r t i c u l a t e phase w i l l be _.. ... < i 1 > - v 2 ( l - V £ l f x k ) e k ^ V 2 £ k . x k V13 = v l ~ v3 = • + £ l f ( 1 - £ 2 - £ k } ( 1 - £ 2 - £ k } (2.104) 90 By simplifying equation 2.10 4 we get v 13 : j l > " v 2 ( 1 _ x k ) e } £ l f ^ 1 - £2~ ek^ (2.105) Then the volume f r a c t i o n of l i q u i d i n the p a r t i c u l a t e phase, II e l f ' w ^ k e given by the modified Richardson - Zaki c o r r e l -ation, equation 2.102, which when combined with equation 2.105 reduces to " = r ! M r n _ 1 ) <j 1> - v 2 ( l - x k ) £ ] ( l - e 0 - e , ) V 2 k 0 0 1/n (2.106) Thus, i n the presence of s o l i d s c i r c u l a t i o n i n a three-phase f l u i d i z e d bed, equations.2.97 and 2.96 are modified and replaced by equation 2.106 to predict the average volumetric II f r a c t i o n of l i q u i d i n the p a r t i c u l a t e phase, ^-^f ^ ne average volumetric f r a c t i o n of gas, £ 2, and the t o t a l voidage, e, i n the bed are s t i l l given by equations 2.94 and 2.91 re s p e c t i v e l y . In order to complete the model, independent r e l a t i o n -ships for the r i s e v e l o c i t y of a bubble swarm and the volume f r a c t i o n occupied by the wakes i n a three-phase f l u i d i z e d 91 bed have to be developed. For dispersed phase operations, Lapidus and E l g i n [67] have demonstrated that the r e l a t i v e v e l o c i t y between the phases determines the holdup of either phase. Thus i f we postulate that the r e l a t i v e v e l o c i t y between the gas and the l i q u i d i n a three-phase f l u i d i z e d bed can be predicted by the same correlations as for two-phase gas- l i q u i d flow, the problem then reduces to c o r r e c t l y formulating the r e l a t i v e v e l o c i t y i n two-phase gas-liquid flow, several competing models having been proposed (section 2.1.2). Towell et a l . [59] have suggested rather a r b i t r a r i l y that the r e l a t i v e v e l o c i t y between the gas and the l i q u i d i n large diameter (> 4 inch) columns-, where system-a t i c c i r c u l a t i o n of l i q u i d i s predominant, i s given by This empirical c o r r e l a t i o n was tested and confirmed by Reith et a l . [68] for large diameter columns. However, for small diameter columns (< 4 i n ) , i n the absence of systematic l i q u i d c i r c u l a t i o n , the r e l a t i v e v e l o c i t y for the bubble flow II v 21 = v + 2 <j~> (2.107) regime can be obtained by combining equations 2.32 and 2.3 7 to give v n (2.108) 21 II 92 I t has been shown [76] that the model presented i n section 2.1.2 s a t i s f a c t o r i l y represents a wide v a r i e t y of data for bubble.columns and for cocurrent gas-liquid flow. Equation 2.108, which i s based on t h i s model, i s therefore recommended for predicting the r e l a t i v e v e l o c i t y between gas and l i q u i d i n small columns (< 4 inch) . In the absence of adequate under-standing of the c i r c u l a t i o n phenomenon p r e v a i l i n g i n larger columns, the description of bubble motion i n such columns i s incomplete and therefore modelling attempts have not been successful. Nevertheless, the empirical c o r r e l a t i o n developed by Towell et a l . [59], equation 2.107, has been confirmed for large columns [68] and i s therefore recommended. The r e l a t i v e v e l o c i t y i n three-phase f l u i d i z a t i o n i s defined as the difference i n l i n e a r v e l o c i t i e s between the gas and the l i q u i d and i s given by -"» _ - - < j 2 > j l V21 " v2 " V l "~ (2.109) £2 £ l f II Substituting the value of from equation 2.96 into equation 2.109 we get < h > v 2 1 = = n—— (2.110) £2 £ l f ( 1 " e 2 - £ k ) which a f t e r substitutions and rearrangements s i m p l i f i e s to 93 v 2 1 = n (2.111) £2 £ l f ( 1 - £ 2 - £ k ) Substituting the value of from equation 2.76 into equation 2.111 and rearranging gives II < j 2 > < J l > +  <Jo > ET f ( 1 " £ 2 " £ k ) -•'« = _± £_ + _ i i f—JL_ v (2.112) £ 2 ( l - e 3 ) ( l - e 3 ) 1 Bhaga [1], by a method very si m i l a r to the one employed by Zuber and Findlay [39] for gas-liquid flow, obtained for one-dimensional simultaneous flow of gas, l i q u i d and s o l i d i n a v e r t i c a l conduit <(1-a,)j 9> „, <a,a9 v 9,> " — = Co < 3 1 + J 2 > + — — — — (2.113) <a2> < a 2 > in where CQ i s the d i s t r i b u t i o n parameter for cocurrent gas-l i q u i d - sol i d flow and i s given by <a 2(j,+j 2)> C Q = 1 Z (2.27d) <a 2><j 1 +j 2> Comparing equation 2.113 with equation 2.26 i t can be seen in that the d i s t r i b u t i o n parameter, C Q, represents the e f f e c t of r a d i a l d i s t r i b u t i o n of both volumetric flux of the mixture and i n s i t u volume f r a c t i o n of the gas ( i . e . gas holdup) on the r i s e v e l o c i t y of a bubble swarm. As was observed for g a s - l i q u i d flow, a value of C Q greater than unity indicates that neither the gas holdup nor the volumetric flux of the gas-liquid stream are constant over the cross-section. Therefore, for a three-phase f l u i d i z e d bed, i f both the gas holdup and the volumetric flux of the ga s - l i q u i d stream vary r a d i a l l y at a given v e r t i c a l l e v e l i n the bed, then equation 2.112 should be modified to include the d i s t r i b u t i o n MI parameter, C Q. Thus <J2> = V ^ l * + ^ 2 > } , £ l f ( 1- £2" £k } -»' (2.114) e2 " (1-63) ( l - e 3 ) ^ Since i t has been assumed that the r e l a t i v e v e l o c i t y i n three-_ 11 j phase f l u i d i z a t i o n , v 2^, can be represented by the c o r r e l -•M ations developed for two-phase gas- l i q u i d flow, v 2 ^ can be obtained from equation 2.107 for large diameter columns (_> 4 inch) and from equation 2.108 for small diameter columns (< 4 inch). Both these correlations require a knowledge of V^, which can be estimated from the average diameter of bubbles i n the swarm. Since the structure and size of wakes behind gas bubbles has not.been investigated systematically i n either two-phase gas-l i q u i d flow or three-phase f l u i d i z a t i o n , i t i s d i f f i c u l t to develop a r e a l i s t i c model for the volume f r a c t i o n of wakes i n a three-phase f l u i d i z e d bed. Letan and Kehat [36] have presented a limited set of data for the volume f r a c t i o n of wake at various values of the dispersed phase holdup i n a l i q u i d - l i q u i d system. In the absence of gas - l i q u i d data, the Letan - Kehat r e s u l t s w i l l be used i n t h i s thesis on the assumption that the wake behind a gas bubble has sim i l a r c h a r a c t e r i s t i c s to the wake behind a l i q u i d drop. To j u s t i f y t h i s assumption, a tentative model i s suggested based on the wake structure postulated by de Nevers and Wu [61]. From the v i s u a l observation of large a i r bubbles (d g =• 1-2 cm), which were e s s e n t i a l l y hemispherical i n shape, r i s i n g i n either glycerine or water, these investigators inferred that the bubbles were each followed by a conical wake whose influence extended to a distance 1, behind the bubble. k Thus, i f the t r a i l i n g bubble followed at a distance greater than 1^ . it.would not be affected by the wake of the leading bubble. On the basis of such a structure for the bubble and the wake, de Nevers and Wu found that a dimensionless distance, l^/Rg/ of seven s a t i s f i e d t h e i r data for bubbles r i s i n g and coalescencing i n both the air-water and a i r -glycerine systems. If we v i s u a l i z e a similar structure for the r i s e of a non-coalescing swarm of bubbles, we can therefore estimate that the v e r t i c a l distance between any two bubbles i n the swarm should be at l e a s t (1, + R ). Then the volume of the FIGURE 2.4 SCHEMATIC OF WAKE STRUCTURE SUGGESTED dE NEVERS AND WU [61] 97 wake as compared to the volume of the bubble i s given by k B V ^ k 2/3TTRJ 2R s (2.115) For a l i q u i d - l i q u i d system, both Letan and Kehat [36] and Hendrix et a l . [99] have shown that the wake size i s markedly affected by the presence of other drops. The e f f e c t of gas volume f r a c t i o n on the wake siz e can be estimated i f i t i s assumed for computational purposes that: (a) the bubbles have an orderly d i s t r i b u t i o n i n the swarm, and (b) the wake length i s equal to the v e r t i c a l distance between the bubbles, so that the wake of the leading bubble would j u s t f a l l short of a f f e c t i n g the r i s e v e l o c i t y of the t r a i l i n g bubble. Now, i f we assume a simple cubic d i s t r i b u t i o n for the bubbles i n -the swarm, with (1, + R ) as the c h a r a c t e r i s t i c length of s the c e l l , i t can e a s i l y be shown that the r a t i o of wake size to bubble size i s given by 1/3 - 1] (2.116) In Table 2.1 are presented values of wake to bubble volume r a t i o for a cubic and two other .sys'tematic bubble d i s t r i -butions, as well as the values recommended by Letan and 98 TABLE 2.1 RATIO OF WAKE TO BUBBLE VOLUME FOR VARIOUS VALUES OF DISPERSED PHASE HOLDUP IN TWO-PHASE FLUID SYSTEMS Dispersed phase holdup, e2 V"B ( 1 ) V"B ( 2 ) V ° B ( 3 > (*) v v 0.1 0.878 0.946 1.047 0.93 0.15 0.704 0.763 0.852 0.83 0.2 0.594 0.648 0 .728 0.83 0.25 0.516 0.565 0.640 0.83 0.3 0.456 0.503 0 .573 0.83 0.35 0.408 0.452 0.519 0.83 0.4 0.368 [ 0.411 0.475 0.83 0.45 0.335 0.376 0.437 0.83 0.5 0.306 0 .346 0.405 0.77 0.6 0.259 0.296 0.351 0.61 (1) 27T 1 / 3 * 'Cubic C e l l - n./nn = 1/2 [(=-?—) - II K a 2 1/3 ( 2 ) Orthorhombic C e l l - fl./fiB = 1/2 [ (— ) - 1] K B 3/3~e2 ( 3 ) Rhombohedral C e l l - fi. /n_ = 1/2 [ ( ^ I L ! ^ ) 1 ^ 3 - i ] K B _ 3£2 From Figure 6 of Letan and Kehat [36] Continuous phase: d i s t i l l e d water Dispersed phase: kerosene Kehat [36] for wakes behind l i q u i d drops. As can be seen from Table 2.1, the model improvised here for estimation of the volume f r a c t i o n of wakes i n ga s - l i q u i d systems, though undoubtedly oversimplified, are matched by the trend :.of the l i q u i d - l i q u i d data"closely enough to^ j u s t i f y the use of these data for g a s - l i q u i d systems as a f i r s t approximation. A more complicated model based on the exponential wake structure recommended by de Nevers and Wu [61] and by Crabtree and Bridgwater [78] can be developed s i m i l a r l y , but lack of detailed data for wake sizes i n gas-l i q u i d systems does not warrant such complicated models at this stage. The presence of s o l i d p a r t i c l e s i n three-phase f l u i d i z -ation would a f f e c t the wake si z e , as inferred by 0stergaard [8] and c l e a r l y demonstrated by Rigby and Capes [80] and by Efremov and Vakhrushev [16]. Thus, following the l a t t e r ' s recommendations, i t i s postulated that the volume f r a c t i o n of wakes i n three-phase f l u i d i z a t i o n w i l l be represented by ( P k / P B ) " ' = ( n k / n B ) n f(e) (2.117) The exact form of the function f can only be developed from experimentally observed values of wake sizes i n two-phase gas-liquid flow and i n three-phase f l u i d i z a t i o n . Thus the model proposed here i s si m i l a r i n p r i n c i p l e to the wake model proposed by 0stergaard, the main difference 100 being the in c l u s i o n of sol i d s c i r c u l a t i o n and i t s e f f e c t on bed expansion i n the former. A possible mechanism for sol i d s c i r c u l a t i o n , based on the r i s e of s o l i d p a r t i c l e s i n the wake of r i s i n g bubbles, i s suggested, along with relationships for both the r i s e v e l o c i t y of a bubble swarm and the volume f r a c t i o n of wakes i n a three-phase f l u i d i z e d bed. (C) The c e l l model As stated e a r l i e r , the c e l l model technique has been successfully used to describe various dispersed phase opera-tions [49,90,86], This technique consists of representing an assemblage of p a r t i c l e s by a spherical (or sometimes a c y l i n d r i c a l ) c e l l containing a single p a r t i c l e and l i q u i d i n such a proportion that the voidage i n the c e l l i s equal to the voidage of the entire assemblage. The Navier-Stokes equations of motion are then solved for the closed c e l l , with adequate and consistent boundary conditions, to obtain the r e l a t i v e v e l o c i t y of the p a r t i c l e with respect to the l i q u i d i n the c e l l . The proposed c e l l model for three-phase f l u i d i z a t i o n consists of representing a bubble and a s o l i d p a r t i c l e i n d i v i d u a l l y by two separate spherical c e l l s . Smith [91] used such a model to analyse the rate of sedimentation of p a r t i c l e s of two d i f f e r e n t species. In order to form a consistent set, Smith matched the two spherical c e l l s each 101 representing an i n d i v i d u a l species, by using the boundary conditions of (a) equal tangential v e l o c i t i e s at the equator . of both envelopes and (b) equal pressure gradients at a l l . s u r f a c e s of both envelopes. The proposed model, which i s presented i n i t s enti r e t y i n Appendix 8.1, uses si m i l a r but s l i g h t l y modified boundary conditions to match the two spherical c e l l s v i z . (a) equality of tangential v e l o c i t i e s at the equatorial surface of both envelopes and (b) equality of drag per unit volume i n each c e l l such that the o v e r a l l pressure gradient (-Ap/L) i s the same for both the c e l l s and throughout the bed. The solution obtained for the c e l l model, using the creeping flow equations with the above boundary conditions to match the two spherical c e l l s , and Happel 1s [49] boundary conditions for each i n d i v i d u a l c e l l ( i t i s r e a l i z e d that these may not be the unique set of boundary conditions) shows the l i q u i d - s o l i d f l u i d i z e d bed to expand smoothly on introduction of gas. Since the wake behind a gas bubble i s non-existent i n creeping flow, th i s r e s u l t i s not surprising because, as explained by 0stergaard [8] and by Stewart and Davidson [ 7 ] , i t i s the wake behind the gas bubble that i s responsible for the observed contraction i n three-phase f l u i d i z a t i o n . I t i s therefore recommended that the f u l l Navier-Stokes equations be solved by ex i s t i n g numerical techniques [ 8 2 , , .90] , f or a c r i t i c a l evaluation of the proposed c e l l model. 102 2.3.2 Gas holdup i n three-phase f l u i d i z e d beds The behaviour of bubbles i n a three-phase f l u i d i z e d bed has been the subject of li m i t e d study [15,79], although many investigators [14,18,100] have observed that i n well expanded beds of small s o l i d p a r t i c l e s bubble coalescence predominates, whereas i n s l i g h t l y expanded (close to packed bed voidage) beds of large p a r t i c l e s bubble breakup generally occurs. A number of attempts [14,16,17,79] have been made to study the gas holdup i n three-phase f l u i d i z a t i o n , but as yet no r e a l i s t i c model has been formulated to incorporate the q u a l i t a t i v e observations and to point out the areas r e -quired for further study, i n order to complete the under-standing of gas bubble behaviour. Nevertheless, empirical correlations have been suggested by various investigators. Thus V a i l et a l . [17], who measured the gas holdup i n a 146 mm diameter column by quickly shutting o f f the gas and l i q u i d flow rates simultaneously, thus i s o l a t i n g the experi-mental section, recommended the following c o r r e l a t i o n for a bed of 0.77 mm glass beads f l u i d i z e d by a i r and water: e'" = 0.1026 ( l - e . ) 2 , 0 9 ( < j o > / < j 1 > ) 0 , 7 8 (2.118)* for .1.5 <j 1< 9.0, 4.0 <j 2< 20 * The exponents 2.09 and 0.78 were erroneously inter-changed i n the o r i g i n a l paper [17]. 103 They further suggested that when £3=0, the above c o r r e l a t i o n s a t i s f a c t o r i l y predicted the gas holdup i n two-phase gas-l i q u i d flow. Therefore ,, 0.78 e 2 = 0.1026 (<j2>/<j1>) (2.119) and combining the above two equations we see that .., .. 2.09 e2 = e2 ( 1 " e 3 ) (2.120) Equation 2.120 thus shows the importance of bed expansion, which had been implied by other investigators but not formulated l o g i c a l l y . However, equation 2.120 i s only an empirical c o r r e l a t i o n and cannot be extrapolated beyond the range of i t s supporting data without the r i s k of serious error. Michelsen and 0 s t e r g a a r d [14] measured gas holdups by measuring the s t a t i c pressure drop across the length of the bed and also by employing a tracer i n j e c t i o n technique. They proposed the following c o r r e l a t i o n for a bed of 1 mm glass beads f l u i d i z e d by a i r and water i n a 216 mm diameter column: ,v nn x°-37^ . .0.78, e2 = 0.011 <D-j. ^2 for 2 <J1-< 7.5 and 0.35 <J 2 < 2.2 (2.121) 104 and for two-phase ga s - l i q u i d flow of the air-water system, e2' = 0 .0394 <j 1>" 0' 1 6 <j 2> 1 , 0 5 (2.122). for 0.7 < 17.0 and 0.35 < J2 < 2.2 The format of equation 2.121 contributes l i t t l e to the under-standing of bubble behaviour i n three-phase f l u i d i z e d beds, but the equation i t s e l f i s found to be i n quantitative agree-ment with equation 2.118 within the l i m i t s of a p p l i c a b i l i t y of equation 2.121. Capes et a l . [7 9] studied the l o c a l properties of gas bubbles i n three-phase (air-water-glass beads) f l u i d i z e d beds by locating two e l e c t r o - r e s i s t i v i t y probes, separated v e r t i c a l l y by a short distance, inside the bed. The measured bubble r i s e v e l o c i t i e s were correlated with the measured voidage i n the bed and the average length,!, of bubbles i n the swarm, by means of the following r e l a t i o n s h i p : v 2 - C<j1+j2>) = 32.5 l 1 ' 5 3 ( ^ ) 2 (2.123) for 0.03 < j x < 2.61 and 0.5 < J 2 < 2.0 They also suggested that i f a spherical cap bubble with a f l a t base i s assumed to have an included wake angle of 135°, then the average length and the equivalent radius of the 105 bubble can be correlated by 1 = 1.14 r e (2.124) I t has been found, however, by the present author that for an included wake angle of 135° the above re l a t i o n s h i p i s incorrect and should read 1 = 1.02 r e (2.125) while only for an included wake angle of 158° does equation 2.124 give the correct r e l a t i o n s h i p between 1 and r . Equation 2.123 i n combination with equation 2.125 shows the influence of bubble diameter and the important e f f e c t of bed voidage on bubble v e l o c i t y . Equation 2.123 i s found to be i n quantitative agreement with equation 2.118 on assum-ing a r e a l i s t i c bubble diameter, no measurements of bubble diameter having been reported by V a i l et a l . Thus a l l these empirical correlations are quantit a t i v e l y compatible and demonstrate the importance of bed voidage and average bubble diameter i n determining the r i s e v e l o c i t y of a bubble swarm, and thereby also the gas holdup, i n three-phase f l u i d i z a t i o n . From the generalized wake model for a three-phase f l u i d i z e d bed, presented i n section 2.3.1, the r i s e v e l o c i t y of bubbles i s given by where 106 IM II v 2 = -°- ± — f _ + - J i _ £_JL_ v 2 1 (2.114) _ lit v 2 1 = ( V J B + 2 <j 2> (2.107) for D > 4 inch and j "» 1/3 ( V J B "tanh [0.25 ( l / e 2 ) " L / J] _ v 2 j (2.108) v 2 1 = — • m - ~Tfr~ for D < 4 inch, where (V o) B i s the rise' v e l o c i t y of a single bubble i n a three-phase f l u i d i z e d bed and depends mainly on the average bubble diameter. The gas holdup i s then obtained from e2 =  < ^ 2 > ^ 2 (2.94) Thus equation 2.114 i n combination with equation 2.108 (or equation 2.107) and equation 2.94 provides a general model for describing the gas bubble behaviour i n three-phase f l u i d i z e d beds. Q u a l i t a t i v e l y t h i s model i s better than any of the empirical c o r r e l a t i o n s , as i t not only takes into account the e f f e c t of bed voidage and bubble diameter on the r i s e v e l o c i t y of bubbles, but also i l l u s t r a t e s the necessity of investigating the r a d i a l p r o f i l e s of gas holdup and gas-l i q u i d f l u x through the bed (to determine the d i s t r i b u t i o n 107 in parameter C Q) and the phenomenon of wake formation (to determine £jj , i n order to better the understanding of gas bubble behaviour i n three-phase f l u i d i z a t i o n . 2.3.3 Voidage i n three-phase f l u i d i z e d beds The number of independent models postulated for over-a l l voidage (l-e^) i n three-phase f l u i d i z e d beds i s - l i m i t e d . 0stergaard [8] derived a wake model, which has subsequently been used and modified by various investigators without signif-i c a n t l y enhancing the understanding of bed expansion behaviour i n three-phase f l u i d i z a t i o n . Thus Efremov and Vakhrushev [16], who used the wake model on the assumption that no p a r t i c l e s are present i n the wake [7], postulated that the bed voidage i s given by <i x> - ( y n B ) <j 2> £ = : < 1- e2~ ek ) + £2 + ek 00 (2.126) where £k = e 2 . f e < 2 ' 1 2 7 ) B and n 1 (=Re^*1/4.4 5) i s the improvised Richardson - Zaki exponent evaluated by using the Reynolds number, Re, based on the l i q u i d flux through the pa r t i c u l a t e phase instead of the free s e t t l i n g Reynolds number, Re p/ as suggested by Richardson and Zaki [2]. The r a t i o of wake volume to bubble 108 volume, fij,/flB, as determined from the measured values of e and e2 using equations 2.126 and 2.127, was then empirically correlated.to f i t the observed bed voidage data by the following r e l a t i o n : = 5.Me, r , 0 D [ l - tanh {40 -3- (e.) - '3 .2 ( e , ) • ^ O] K B X <J 2 ^  J- J-(2.128) II where i s the voidage i n the l i q u i d - s o l i d f l u i d i z e d bed before the introduction of the gas. The other noteworthy attempt to improve the wake model i s due to Rigby and Capes [80]. They tested t h i s model to find out the e f f e c t of assumed p a r t i c l e content of the wake by considering the two extremes suggested by Stewart and Davidson [7] and 0stergaard [8]. They concluded that the presence of p a r t i c l e s i n the wake had a marked influence on the wake volume, which was also found to be affected by the bed voidage.and to a les s e r degree by the p a r t i c l e s i z e . No phenomenological equation was presented to correlate the wake volume with various bed parameters. Other empirical correlations for the voidage i n three-phase f l u i d i z e d beds are either unnecessarily complicated [10] or o u t r i g h t l y misleading [11], and w i l l therefore not be included i n further discussions. From the generalized wake model for three-phase fluidized-beds presented i n section 2.3.1, the bed voidage i s given by 109 e = e 2 + £ k ( l - x k ) + e[f (x ke k+ 1 - e 2 - ek> (2.91) II where e^. i s given by equation 2.106. The terms * k and e k are the two quantities for which estimates have to be obtained experimentally, since no r e l i a b l e information con-cerning them e x i s t i n the three-phase f l u i d i z a t i o n l i t e r a t u r e . As has already been pointed out i n section 2.3.1, the improvised wake model takes into consideration not only wake formation behind the bubbles and the p a r t i c l e content of the wake, but also p a r t i c l e c i r c u l a t i o n and i t s e f f e c t on the contraction-expansion c h a r a c t e r i s t i c s of a three-phase f l u i d i z e d bed. 110 CHAPTER 3 EXPERIMENTAL The main aim of t h i s work was to est a b l i s h the e f f e c t of l i q u i d and s o l i d phase properties on holdups of gas, l i q u i d and s o l i d i n a three-phase f l u i d i z e d bed. The sol i d s holdup, or the volume f r a c t i o n of so l i d s inside the bed, e 3, can be d i r e c t l y calculated i f the weight of s o l i d p a r t i c l e s present i n the bed, W, i s known and i f the expanded bed height, L^, can be measured. Thus e 3 = W/p3ALb (3.1) However, the expanded bed height, L^, cannot always be measured d i r e c t l y . - At low gas flow rates, e s p e c i a l l y i n a bed of large or heavy p a r t i c l e s , the upper boundary of the bed i s very well defined and can be measured e a s i l y by v i s u a l observation through transparent column walls. But at large gas flow rates, the upper bed l e v e l i s not so c l e a r l y delineated and i t becomes d i f f i c u l t to e s t a b l i s h the expanded bed height v i s u a l l y . I t i s therefore necessary to develop a c r i t e r i o n to define the expanded bed height consistently under a l l conditions of gas and l i q u i d flow rates. Such a c r i t e r i o n was developed and i s discussed i n d e t a i l i n I l l Appendix 8.2. Thus with the knowledge of W and L^, can be calculated from.equation 3.1. Since £1 + £2 + £3 = 1 * ° (3.2) becomes known i f the gas holdup inside the.bed, e 2' could be measured. Of the various techniques available for measure-ment of gas holdup, the following two were chosen for th i s work: 1. d i r e c t volumetric measurements using quick clo s i n g valves, 2. the measurement of s t a t i c pressure drop gradient. A t h i r d technique, that of measuring the l o c a l gas f r a c t i o n by an e l e c t r o - r e s i s t i v i t y probe, was l a t e r developed and used i n part of the work. The experimental equipment used was designed to i n -corporate these techniques for measuring the expanded bed height and gas holdup into t h i s study. 3.1 Apparatus The three-phase f l u i d i z a t i o n studies were carr i e d out i n two columns: (i) a 20 mm i . d . glass column and ( i i ) a 2 inch i . d . perspex column. The two experimental set ups are discussed separately i n the following sections. 112 3.1.1 The 20 mm bench top g l a s s column The design of the 20 mm g l a s s column was based on the apparatus used f o r l i q u i d - s o l i d f l u i d i z a t i o n s t u d i e s by Andersson [56] , with s u i t a b l e m o d i f i c a t i o n s f o r three-phase f l u i d i z e d bed o p e r a t i o n . The o v e r a l l l a y o u t of the equip-ment i s shown s c h e m a t i c a l l y i n F i g u r e 3.1. The main experimental column c o n s i s t e d of a 660 mm l o n g , 2 0 mm i-.d. g l a s s column w i t h a s t r a i g h t 168 mm long entrance s e c t i o n . A 60 mesh copper s c r e e n , S, was p u s h - f i t t e d t o separate the calming s e c t i o n D from the experimental s e c t i o n E and a l s o to a c t as a bed s u p p o r t . Two pressure t a p s , 4 9.3 cm a p a r t , were p r o v i d e d w i t h a 6 mm U-tube manometer, , to measure the s t a t i c p r e s s u r e drop a c r o s s the bed. The t e s t l i q u i d from the feed tank was c i r c u l a t e d by a c e n t r i f u g a l pump d r i v e n by a 1/15 horsepower .motor. A bypass was p r o v i d e d to r e g u l a t e the flow and a needle v a l v e to c o n t r o l the flow r a t e to the experimental column. The l i q u i d flow r a t e was measured by a c a l i b r a t e d rotameter, R-^ . The l i q u i d from the experimental column was allowed to o v e r f l o w i n t o an e x i t s e c t i o n , X, from which i t was r e t u r n e d to the feed tank, thus completing the l i q u i d c y c l e . The temperature of the l i q u i d was measured by a thermometer, T, and kept c l o s e to the room temperature by adding some f r e s h tap water o c c a s i o n a l l y . However, when the water g l y c e r o l s o l u t i o n was used as the t e s t l i q u i d , an immersion c o o l e r was used to keep the l i q u i d temperature w i t h i n ± 1°F of the FIGURE 3.1 SCHEMATIC DIAGRAM OF 20 MM GLASS COLUMN APPARATUS LEGEND FOR FIGURE 3.1 a i r source (85 psig) buffer b o t t l e 1 mm glass c a p i l l a r y gas d i s t r i b u t o r calming section experimental section triple-necked 2 l i t e r f l a s k three-way glass stop-cocks ground perspex b a l l carbon tetrachloride U-tube manometer open mercury manometer pressure regulator rotameters 60 mesh copper screen bed support thermometer l i q u i d reservoir e x i t section 115 room temperature. The three-way stopcock was used to turn o f f the l i q u i d flow. The a i r supply was obtained from the laboratory out-l e t through a f i l t e r - r e d u c e r valve assembly, P. The a i r flow rate to the column was c a r e f u l l y controlled by a needle value and measured by the cal i b r a t e d rotameter, R 2. The a i r entered at the base of the glass column through a 5 cm long, 1 mm glass c a p i l l a r y , held i n a v e r t i c a l p o s i t i o n close to the column axis by a spacer fixed to the column wa l l . A 5 cm long, 1/2 mm glass c a p i l l a r y was also used e s p e c i a l l y for the low a i r flow rates studied. To damp out the fluctuations i n the a i r l i n e a damper b o t t l e , B, was used. The pressure at which the a i r was supplied to the column was measured by the open mercury manometer, M2. The three-way stop-cock, G 2, was used to i n s t a n t l y shut o f f the gas flow to the column. A c a r e f u l l y ground perspex b a l l , L, which f i t t e d quite snugly into a ground glass j o i n t , was used as a stop-valve to i s o l a t e the experimental section, once the gas and the l i q u i d flows had been cut o f f . The gas i n section D col l e c t e d below the.screen S and the gas i n the experimental section c o l l e c t e d near the top of the glass column. Only the l a t t e r reading was recorded. The d e t a i l s of the 20 mm glass column are given i n Figure 3.2. C O 7F co CO T LO CO ~7V 06 2-0 C D 06 C\J 116 ,60 MESH COPPER SCREEN 2-5cm GROUND PERSPEX BALL FIGURE 3.2- THE 20 MM GLASS COLUMN 117 3.1.2 The 2 inch perspex column The main bulk of the three-phase f l u i d i z a t i o n study was carr i e d out i n a 2 inch perspex column, the schematic drawing of which i s given i n Figure 3.3. 3.1.2.1 Liquid cycle and tes t section The d e t a i l s of the bulk of the equipment have been given by LeGlair [101], who designed and used most of the same apparatus for an e a r l i e r study. Therefore only the main features of the equipment are discussed here. The l i q u i d c i r c u l a t i o n loop i s made from seamless copper tubing. A 152 inch long s t r a i g h t run of the 2 inch copper tubing pre-ceding the experimental section acts as the calming section. The t e s t l i q u i d i s c i r c u l a t e d by a cen t r i f u g a l pump driven by a 3 horse-power motor. A bypass i s provided to regulate the pressure at which the l i q u i d i s pumped. With water as the tes t l i q u i d , the setting of the bypass valve i s not important, but when polyethylene-glycol solution i s used, the setting i s found to be c r i t i c a l since an excessive c i r c u l a t i o n of the l i q u i d i n the bypass loop makes i t quite frothy. Therefore the opening of the bypass valve was so regulated as to maintain a pump del i v e r y pressure of over 4 0 psig when using polyethylene-glycol solution as the tes t l i q u i d . The l i q u i d from the pump flows through a heat exchanger where i t i s cooled to maintain a steady temperature X 118 B, L, N IV s t^ Xj— T 0 0, 0, feed tank heat exchanger pump FIGURE 3.3 -SCHEMATIC DIAGRAM OF 2 INCH PERSPEX COLUMN APPARATUS 119 LEGEND FOR FIGURE 3.3 A - A i r source (35 psig) B^,B2 - 2 inch f u l l - b o r e b a l l valves C - C a p i l l a r y flow meter E - Experimental perspex t e s t section G - Pressure gauge I - Glass tube l e v e l indicator N - A i r i n l e t cone O^/OpyO^ - O r i f i c e meters P - A i r f i l t e r and pressure regulator R1' R2 ~ Rotameters S - Entry section T - Thermometer X - E x i t section L j - Lever arm po s i t i o n when b a l l valves are f u l l y open L - Lever arm pos i t i o n when b a l l valves are f u l l y c closed 120 i n the l i q u i d c y c l e . The temperature of the l i q u i d i s measured at a location downstream from the measuring st a t i o n by a thermometer, T. The l i q u i d then enters the main experimental column through an annular entry section, S. The l i q u i d from the experimental column overflows into the e x i t section, X, whence i t i s returned to the feed tank, thus completing the l i q u i d c ycle. The l i q u i d flow rate to the column i s measured at the measuring station, either by one of the three o r i f i c e -meters O^, C>2 or 0^ / or by the capillary-tube meter, C. The c a l i b r a t i o n curves for these flow meters are given i n Appendix 8.4. The t e s t section consists of a 5 f t long 2 i n . i n s i d e diameter perspex tube. A l l along the t e s t section c a r e f u l l y d r i l l e d pressure taps are provided, each of which houses a c a r e f u l l y shaped 1/4 inch copper tube with a 1/16 inch opening into the column, to f i t f l u s h with the inside of the t e s t section. The pressure taps are connected to a 100 cm long, 8 mm i . d . U-tube manometer through a pressure manifold system (Figure 3.4) which permits the pressure drop to be measured between any two taps. Carbon tetrachloride dyed with a c r y s t a l of potassium permanganate was used as the manometric f l u i d for most of the study, while t e t r a -bromo-ethane was used for the r e s t . An open mercury manometer was also located i n the t e s t section to measure the absolute pressure i n the column, s A screen trap i s clamped on to the opening of the l i q u i d return l i n e into the feed tank, to catch any p a r t i c l e s e l u t r i a t e d out of the experimental column. 1 p MANOMETER HEADERS 0 DISTANCE IN INCHES FROM DATUM OPEN MERCURY MANOMETER --DATUM FIGURE 3.4 PRESSURE DROP APPARATUS FOR MEASURING LONGITUDINAL PRESSURE DROP PROFILE IN THE EXPERIMENTAL SECTION 122 The test section i s separated from the calming section by a 60 mesh copper screen held i n the recess of a rubber gasket, which i n turn i s held between two flanges. Since th i s screen also acted as the bed support, for f l u i d i z i n g 0.25 mm glass beads another f i n e r screen (100 mesh) was used on top of the 60 mesh screen to prevent the small p a r t i c l e s from -falling through. Two 2 inch f u l l bore b a l l valves were used to trap the flowing mixture i n the t e s t section by clo s i n g them simultaneously. One valve was located 5 feet, below the t e s t section and the other at the top of the t e s t section. The two valves were connected through lever arms by a l i n k rod and could be shut completely and simultaneously by quickly rotating the b a l l s through 90° v i a the l i n k rod. Ca r e f u l l y d r i l l e d taps were provided i n the section below the t e s t section for l i q u i d l e v e l i n d i c a t i o n , and both below and above the t e s t section for s t a t i c pressure drop measure-ments. Figure 3.5 shows the l o c a t i o n of these pressure taps. Again carbon t e t r a - c h l o r i d e dyed with potassium per-manganate was used as the manometric f l u i d . 3.1.2.2 Gas cycle and bubble nozzle A i r was taken from the laboratory supply at 35 psig through a 1/2 inch copper l i n e and reduced to a pressure of 12-16 psig by a pressure regulator and f i l t e r assembly, P, which maintained the reduced supply pressure constant at any desired l e v e l . This pressure was read downstream from the CO -ixi- 4*. IO ro in ALL DIMENSIONS IN CM. Hr ±1 123 FIGURE 3.5 -LOCATION OF PRESSURE TAPS AND BALL VALVES FOR GAS HOLDUP MEASUREMENTS 12 4 measuring st a t i o n on a Bourdon tube type pressure gauge, G. A i r was then brought to the bottom of the main experimental column through a 1/2 inch copper l i n e and admitted through the gas d i s t r i b u t o r , N. The a i r leaving the experimental column at the top was vented to the atmosphere. The a i r flow rate to the experimental column was measured by either of the two c a l i b r a t e d rotameters, and R 2. Yet another rotameter was used for part of t h i s study to measure very small gas flow rates. The c a l i b r a t i o n curves for these rotameters are given i n Appendix 8.5. The gas entered the column through a gas d i s t r i b u t o r , the d e t a i l s of which are given i n Figure 3.6. The main bubble nozzle, N, was turned from a brass block to house various gas d i s t r i b u t o r s that can be screwed on to i t . In order to d i s t r i b u t e the gas uniformly, a perforated 1/4 inch thick perspex plate d i s t r i b u t o r with 1 hole per square cm [4] was designed. Similar perforated plate d i s t r i -butors with fewer holes were also designed i n order to check any e f f e c t of the gas d i s t r i b u t o r design on the gas;,:holdup i n two-phase gas-liquid flow. Preliminary investigations revealed l i t t l e or no e f f e c t of the gas d i s t r i b u t o r geometry, and therefore a perforated 1/4 inch thick perspex plate d i s t r i b u t o r with four 1/16 inch holes was used i n most of the studies. The gas d i s t r i b u t o r was located 12 feet below the bed support'screen with-the. hopeirithat the flow "and gas d i s t r i b u t i o n 125 FIGURE 3.6 DESIGN OF GAS INLET AND DISTRIBUTOR 126 p r o f i l e s would be f u l l y developed i n the te s t section. V i s u a l observation of the te s t section, however, showed bubble coalescence occurring at d i f f e r e n t gas flow rates. I t was therefore considered doubtful that the gas d i s t r i -bution p r o f i l e s were f u l l y developed. Nevertheless the gas d i s t r i b u t o r was l e f t at the foot of the column through-out the entire study since i t provided a two-phase gas-l i q u i d region preceding the three-phase f l u i d i z e d bed region; since a gas - l i q u i d zone also followed the f l u i d i z e d bed, the e f f e c t of the presence of s o l i d p a r t i c l e s i n the tes t section on the gas holdup i n the two-phase region above i t could therefore be determined. 3.1.3 E l e c t r o - r e s i s t i v i t y probe An e l e c t r o - r e s i s t i v i t y probe was o r i g i n a l l y developed by Neal and Bankoff [103] for measuring the l o c a l volumetric gas f r a c t i o n i n mercury-nitrogen flow. The sensing element of t h e i r probe consisted of an insulated sewing needle with i t s exposed t i p pointing into the flow. The probe was supplied with a D.C. pote n t i a l and grounded through the continuous phase to complete the c i r c u i t , which i s shown schematically i n Figure 3.7. When an i n d i v i d u a l bubble passed over the probe, i t served to open the c i r c u i t , which resulted i n a nearly square wave output. Nassos and Bankoff [104] tested the a p p l i c a b i l i t y of the same probe i n a i r -127 5 V o-2 0 0 K 4 0 0 K P r o b e O / P FIGURE 3.7 CIRCUIT DIAGRAM FOR ELECTRO-RESISTIVITY PROBE 128 water flow and found that, due to d e f l e c t i o n of bubbles away from the probe, the average gas f r a c t i o n obtained by integrating the l o c a l gas f r a c t i o n p r o f i l e was smaller than the value obtained by s t a t i c pressure drop measurements. Some modifications were suggested to improve the agreement [104], but since the d e f l e c t i o n of a bubble from a pointed sensing element remained as a basic problem, i t was decided to change the design of the probe s l i g h t l y for the present study. The e l e c t r o - r e s i s t i v i t y probe used i n thi s study consisted of two electrodes held at a small but fixed distance apart. The a r r i v a l of an i n d i v i d u a l bubble i s sensed by the passage of the bubble through the gap. Although the probe supports could d e f l e c t a bubble into or away from the gap, i t was nevertheless believed that the p r o b a b i l i t y of re g i s t e r i n g an impinging bubble would be increased over that of the o r i g i n a l needle probe. Since the diameter of the bubbles encountered i n the gas-liquid flow study was always larger than the o v e r a l l probe dimension (1.7 mm), the s p a t i a l r e s o lution of the probe could be considered good. However, i t i s believed that a quick penetration of the bubble on impingement remains a problem and would become a major source of error when the probe i s used i n more viscous l i q u i d s . The probe used was o r i g i n a l l y a miniature hot-film probe (1270-20W-6) supplied by Thermo Systems Inc., from which the hot-film filament was c a r e f u l l y cut o f f so as to expose the two electrodes, leaving a gap of 1 mm. The de-t a i l s of the probe are given i n Figure 3.8. The support needles are epoxy coated to insulate them from the continuous phase. The probe was mounted i n the experimental section E through a traversing mechanism to allow for a r a d i a l traverse of the column to positions very close to the column walls. The d e t a i l s of the mounting mechanism are shown i n Figure 3.8. One of the electrodes was maintained at a constant D.C. p o t e n t i a l with respect to the other electrode, which was grounded through a 5 meter coaxial cable. The p o t e n t i a l applied (2-3 volts) was so adjusted as to produce pulses of an amplitude of about 0.22 v o l t s across a 100,000 ohm r e s i s t o r connected i n s e r i e s . The e l e c t r o n i c c i r c u i t used to analyse the probe signal i s described i n the next section. 3.1.4 Description of a u x i l i a r y c i r c u i t s for measurement  of l o c a l gas holdup and bubble frequency Before discussing the c i r c u i t s used, i t i s important to c l e a r l y define the variables being measured. Quantities measured (a) l o c a l gas f r a c t i o n The l o c a l volumetric gas f r a c t i o n i s defined r - - ' as the p r o b a b i l i t y that gas w i l l e x i s t at a point under con-sid e r a t i o n . For flow with stationary time-averaged properties (quasi-steady flow) t h i s p r o b a b i l i t y i s the f r a c t i o n of time the gas e x i s t s at that point [103] . Thus DETAIL OF A (SUPPORT N E E D L E S ) 1< 6 - 3 >| 152 Mi ~7 7 .25 l TVVV 19 < > A L L DIMENSIONS IN mm FIGURE 3.8 ELECTRO-RESISTIVITY PROBE AND MOUNT FOR TRAVERSING MECHANISM 131 a 2 r = fc2/T ( 3 , 3 ) where t 2 i s the time the probe i s exposed to the gas phase and T i s the t o t a l sample i n t e r v a l . In order to obtain a true s t a t i s t i c a l average, the sample i n t e r v a l must be large compared to the time scale of flow o s c i l l a t i o n s , 1/n 1, where n 1 i s the l o c a l bubble frequency. Thus,for a quasi-steady flow, the l o c a l gas f r a c t i o n can be expressed as 1 N a2 r = | I t ± (3.4) i = l In order that the gas f r a c t i o n measured l o c a l l y by t h i s technique could be compared with the o v e r a l l gas f r a c t i o n measured by s t a t i c pressure drop gradient, a traverse of the probe was made to obtain a r a d i a l p r o f i l e of the l o c a l gas f r a c t i o n . These p r o f i l e s were then integrated over the cross-section to provide the o v e r a l l average gas f r a c t i o n , as given by 1 * * <a2> = e2 = 2 y a 2 r R dR (3.5) where R i s the dimensionless distance from the center of the pipe. (b) bubble frequency The bubble frequency at a point, n^ ,, i s defined as the number of bubbles passing through that point per unit time: 132 n r N/T (3.6) where N i s the t o t a l number of bubbles that pass through the point i n time T. The time T must be long enough to obtain a representative sample, which implies that N>>1. Normally 100 - 1000 bubbles, depending on the r a d i a l l o c a t i o n of the probe, were counted i n order to obtain the bubble frequency. Analysis of the probe signal A simple ele c t r o n i c analogue l o g i c c i r c u i t was designed to obtain these quantities from the probe signal and i s shown schematically i n Figure 3.9. The p r i n c i p a l component of the c i r c u i t was the l o g i c d i f f e r e n t i a l comparator, which was used to trigger pulses of width equal to the residence time of an in d i v i d u a l bubble, u t i l i z i n g the following character-i s t i c of the comparator: Y • X • NON-INVERTING INVERTING I/P I / P • Z IF X + Y > 0 , Z = I IF X +Y < 0 , Z = 0 P85AU P35AU IOK 20K 100 K HOOK :IOOK 10 K •MWV 1 20K —wvw-Darcy Frequency f—| Meier P35A iVWSV 1 -I5V P45ALU |sP6o6 P35A — Philbrick solid state amplifier P35AU - Philbrick solid state amplifier P85AU -• Philbrick solid state amplifier P45A - Philbrick solid state amplifier P45ALU - Philbrick solid state amplifier SP656 — Philbrick photochopper stabilized solid state amplifier ^XA7I0 — Fairchild logic differential comparator ^ — To high quality ground -=• — To power common FIGURE 3.9 SCHEMATIC DIAGRAM OF THE ANALOGUE-LOGIC CIRCUIT FOR MEASURING LOCAL BUBBLE-FREQUENCY AND GAS FRACTION co co The p u l s e s o f uniform amplitude thus t r i g g e r e d by the com-p a r a t o r were then i n t e g r a t e d to o b t a i n the t o t a l time the probe i s exposed to the gas phase/ from which the l o c a l gas f r a c t i o n was c a l c u l a t e d by means o f eq u a t i o n 3.4. The c i r c u i t p r e c e d i n g the comparator was designed to a m p l i f y the probe s i g n a l , but most i m p o r t a n t l y to i s o l a t e the measuring c i r c u i t from the probe, so as not to c r e a t e any feedbacks [105] . The bubble frequency was o b t a i n e d by c o u n t i n g the number of p u l s e s t r i g g e r e d by the comparator on a Darcy frequency counter f o r a f i x e d time o f 10 seconds. The t o t a l number o f p u l s e s counted were then read from the e l e c t r o n i c d i s p l a y of the c o u n t e r . A l t e r n a t i v e l y , a s t r i p c h a r t r e c o r d e r was used to r e c o r d the comparator o u t p u t . The number of p u l s e s were then counted from the r e c o r d i n g o f over a minute. E i t h e r method o f o b t a i n i n g the bubble frequency was found to be s a t i s f a c t o r y and used i n t e r -changeably, depending on the a v a i l a b i l i t y o f the equipment,. The bubble frequency measurements were used to o b t a i n an estimate o f average bubble s i z e i n the t e s t s e c t i o n by the method presented i n Appendix 8.3. 3.2 Range of v a r i a b l e s s t u d i e d The experimental programme f o r c o l l e c t i n g the d a t a was d i v i d e d i n t o two p a r t s : 135 (A) The study of gas holdup i n two-phase ga s - l i q u i d flow, and (B) The study of so l i d s and gas holdup i n a three-phase f l u i d i z e d bed. The main experimental programme was c a r r i e d out i n the 2 inch i . d . perspex column located i n a 2 inch diameter forced c i r c u l a t i o n loop. However, the 20 mm i . d . glass column was used to carry out.a preliminary study to e s t a b l i s h the relevance of various parameters involved. Although only a li m i t e d amount of data was obtained i n the l a t t e r , an appreciable range was investigated and therefore the r e s u l t s obtained are included. (A) Gas holdup i n two-phase gas-liquid flow The need to study two-phase ga s - l i q u i d flow arose from the lack of established and r e l i a b l e methods to predict the gas holdup for such flow. The purpose of t h i s study was two-fold: (i) to check the a p p l i c a b i l i t y of the mathematical model proposed i n section 2.1.2, and ( i i ) to obtain data that could be used l a t e r for comparing with the data on gas holdup i n three-phase f l u i d i z a t i o n , i n order to e s t a b l i s h the r o l e of s o l i d p a r t i c l e s i n promoting either the coalescence or breakup of bubbles i n three-phase f l u i d i z e d beds. 136 Therefore the scope of t h i s study was l i m i t e d , Tables 3.1 and 3.2 summarizing the range of variables studied. (B) Solids and gas holdup i n a three-phase f l u i d i z e d bed As has been outlined above, the main aim of t h i s work was to e s t a b l i s h the e f f e c t of l i q u i d - and solid-phase properties on the i n d i v i d u a l gas, l i q u i d and s o l i d holdups i n a three-phase f l u i d i z e d bed, for a wide range of conditions. The choice of f l u i d s selected for t h i s study was guided by the findings i n corresponding two-phase g a s - l i q u i d studies. Thus, a i r was conveniently chosen as the gas phase and used throughout the study, since i t has been shown [108] that the properties of the gas phase had l i t t l e or no e f f e c t under normal atmospheric conditions. Ordinary tap water was used as the l i q u i d phase for the most part, so that the data co l l e c t e d i n t h i s study could be compared with e a r l i e r i nvestigations. In the l a t e r part of the work, an aqueous polyethylene-glycol solution was used to investigate the e f f e c t of l i q u i d v i s c o s i t y . The polyethylene-glycol solution was chosen because i t i s a very viscous l i q u i d , the Newtonian behaviour of which has been v e r i f i e d [101] , and because i t s density and surface tension are only a l i t t l e d i f f e r e n t from that of water. For the s o l i d phase, equi-sized spherical glass beads, lead shot and s t e e l b a l l bearings were chosen to give a broad range of p a r t i c l e s i z e and density. 137 TABLE 3.1 EXPERIMENTAL CONDITIONS FOR TWO-PHASE GAS-LIQUID FLOW IN 20 MM GLASS COLUMN Liquid V e l o c i t y , (cm/sec) Gas Veloc i t y , j2 (cm/sec) Liquid V i s c o s i t y , y, (cp) 1 Gas Holdup, £~ (-)  Z 0.0 - 18.0 5.0 - 18 .0 1.0 0.20 - 0.39 TABLE 3.2 EXPERIMENTAL CONDITIONS FOR TWO-PHASE GAS-LIQUID FLOW IN 2 INCH PERSPEX COLUMN Liquid V e l o c i t y , j . (cm/sec) Gas Veloc i t y , J2 (cm/sec) Liquid V i s c o s i t y , y, (c ) 1 P Gas Holdup,£-(-)  1 Flow Regime 0.0 - 19.0 1.5 - 13.0 1.0 & 69.0 0.05 - 0.28 bubble-slug 138 Tables 3.3 and 3.4 l i s t the range of variables studied i n both the 20 mm glass column and the 2 inch perspex column. 3.3 Experimental procedure The experimental procedure used to obtain data i n the 20 mm glass column and the 2 inch perspex column were e s s e n t i a l l y s i m i l a r . The s a l i e n t features of the procedure adopted are described i n the following sections. 3.3.1 Physical properties of the l i q u i d s used For the major part of t h i s work water was used as the tes t l i q u i d . Ordinary tap water containing 0.2% by weight sodium dichromate and 0.05% by weight sodium hydroxide as corrosion i n h i b i t o r s [101] was t r i e d for the early runs i n the 2 inch perspex column. Since the additives did not i n h i b i t corrosion as e f f e c t i v e l y as had been hoped f o r , ordinary tap water without additives was used thereafter. This required frequent cleaning of the mercury manometer traps and the copper bed support screen. For the studies i n the 20 mm glass column ordinary tap water was used without any problems. The density of the water was checked occasion-a l l y , but i n the f i n a l processing of the data c o l l e c t e d , both the density and v i s c o s i t y of water were obtained from Perry [106] . Surface tension too was measured for the early runs and found to remain e s s e n t i a l l y unchanged. TABLE 3.3 EXPERIMENTAL CONDITIONS FOR THREE-PHASE FLUIDIZATION IN 20 mm GLASS COLUMN Liquid V e l o c i t y , (cm/sec) Gas V e l o c i t y , J 2 (cm/sec) Liquid V i s c o s i t y , y 1 (cp) 1 P a r t i c l e Diameter,d (mm) P Solids Density, (gm/cc) Solids Holdup, £_ (-) 3 Gas Holdup, e 0 •(-) 1.7 - 8.1 0.2 - 8.2 1.0 & 2.1 0.5 - 1.0 2.5 - 3.0 0.5 - 0.2 0.05 - 0.15 TABLE 3.4 EXPERIMENTAL CONDITIONS FOR THREE-PHASE FLUIDIZATION IN 2 INCH PERSPEX COLUMN Liquid V e l o c i t y , j . (cm/sec) Gas V e l o c i t y , j 2 (cm/sec) Liquid V i s c o s i t y , u (cp) 1 P a r t i c l e Diameter,d (mm) p Solids Density,p_ (gm/cc) Solids Holdup, (-) 3 Gas Holdup, £„ (-) 2 Flow Regime 0.4 - 39.0 0.4 - 21.0 1.0 & 63.3 0.25 - 3.2 2.9 - 11.1 0.5 - 0.1 0.05 - 0.25 bubble-slug H 1 CO VD 140 The v i s c o s i t y of the polyethylene g l y c o l solution was measured by a Cannon Viscometer (H-304) which was cali b r a t e d with ASTM Standard O i l No. S-20 and No. S-60, according to the procedure recommended i n the ASTM manual (D445-53T). The v i s c o s i t y of the solution was then measured by the cal i b r a t e d viscometer, following the procedure recommended, and these measurements too are reported i n Appendix 8.6. A pl o t of dynamic v i s c o s i t y against the inverse of the absolute temperature i s presented as Figure 8.6.1 of Appendix 8.6. This p l o t was used to obtain the v i s c o s i t y of the solution at the measured temperature i n the f i n a l analysis of the data. The surface tension of polyethylene g l y c o l solution was also checked and was found to be 63 dynes/em. Since i t i s not very d i f f e r e n t from that of pure water (70 dynes/cm)* no further measurements of the surface tension were made or reported. 3.3.2 Physical properties of the sol i d s used Glass beads of three d i f f e r e n t s i z e s , 0.25, 0.5 and 1.0 mm, lead shot, and s t e e l b a l l bearings were used for studies i n the.2 inch perspex column; washed granular sand and 1.0 mm glass beads were used for studies i n the 20 mm glass column. For a l l the glass beads, lead shot and washed sand, a c a r e f u l l y screened cut was selected from the screen analysis and the average p a r t i c l e s i z e was taken as the arithmetic mean of the two consecutive sieve s i z e s . The diameter of lead shot was also checked by measuring the diameter of some 50 randomly chosen p a r t i c l e s by a micro-meter. The chrome-plated s t e e l b a l l bearings were of pr e c i s e l y ground grade; therefore the quoted diameter was taken as the size of the p a r t i c l e s . A random check on the diameter of a few s t e e l b a l l s with a micrometer showed no difference i n size from the quoted diameter. The density of glass beads and sand was measured by the s p e c i f i c gravity b o t t l e method. Ten to f i f t e e n grams of p a r t i c l e s were placed i n a 10 ml s p e c i f i c gravity b o t t l e and weighed c a r e f u l l y on a balance. The bo t t l e was then c a r e f u l l y f i l l e d with d i s t i l l e d water to the mark and weighed again. The density of the d i s t i l l e d water was measured separately i n another 10 ml, s p e c i f i c gravity b o t t l e . The density of the p a r t i c l e s was then calculated from these measurements and i s reported i n Appendix 8.6. The density of the lead shot and the steel b a l l s was measured by weighing some 50 randomly selected p a r t i c l e s both i n d i v i d u a l l y and c o l l e c t i v e l y on a c a r e f u l l y adjusted balance, and i n the case of the lead shot by measuring the p a r t i c l e s i z e i n two perpendicular d i r e c t i o n s with a micro-meter. The densities calculated from these measurements are reported i n Appendix 8.6. 142 The density of the 25-75 glycerol-water solu t i o n used i n the 20 mm glass column was measured both before and after each run. Since the measured densities agreed with the published values, the density of the solution used was subsequently obtained from Perry [106] . The v i s c o s i t y of the 25-75 gl y c e r o l water solution was taken from Mathur [107] and i s reported i n Appendix 8.6. A 33% by weight solution of polyethylene g l y c o l i n water was used for the measurements i n the 2 inch perspex column. The solution was found to be quite a c i d i c and i t corroded the mechanical seals of the ce n t r i f u g a l pump. I t was then decided to neutralize the solution with a d i l u t e solution of sodium hydroxide; 0.2% by weight of sodium dichromate was also added to i n h i b i t corrosion. I t was also found that for r e s t a r t i n g the pump after a long shut-down, the mechanical seals should be thoroughly washed with fresh water so as to remove from them any deposits of s o l i d i f i e d poly-ethylene g l y c o l . The clea r orange-yellow solution turned dark brown with usage and was replaced with fresh s o l u t i o n . I t was found that the deterioration i n colour of the clear solution was due to the suspended corrosion products. If the solution was allowed to stand undisturbed, i t became clear once again as the corrosion products se t t l e d out. The density of the polyethylene g l y c o l solution was measured by a s p e c i f i c gravity b o t t l e , and the measurements are reported i n Appendix 8.6. 143 3.3.3 Measurement of gas holdup i n gas - l i q u i d flow For s t a r t i n g a run, the l i q u i d was c i r c u l a t e d i n the column u n t i l a constant temperature was achieved and noted. A l l the runs were conducted at about the room temperature. A l l the manometer taps i n the experimental section, and above and below the experimental section, were c a r e f u l l y flushed to remove any a i r bubbles i n the connecting l i n e s . The l i q u i d flow rate was then adjusted to obtain the desired v e l o c i t y through the column. When polyethylene g l y c o l solution was used as the t e s t l i q u i d , s t a t i c pressure drop readings were taken on a l l the manometers i n order to determine the f r i c t i o n a l pressure drop i n single phase flow. The a i r was then introduced by pressurizing the a i r l i n e , and the back-pressure was so adjusted that no f l u c t u a -tions i n the rotameter reading were observable. This back-, pressure i n the a i r l i n e was recorded. The l i q u i d flow rate was once again adjusted to the desired flow rate and the s t a t i c pressure drop measurements along the experimental section, as well as above and below i t , were recorded. The absolute pressure near the top of the experimental section was recorded with the help of the open mercury manometer. The v i s u a l observations of bubble si z e d i s t r i b u t i o n and flow regime encountered were also recorded. The two b a l l valves were then shut o f f by manually actuating the l i n k rod connecting them. The gas flow was 144 cut o f f by venting the a i r to the atmosphere and the l i q u i d flow by switching o f f the motor. The se t t l e d l i q u i d height i n the experimental section was measured d i r e c t l y , that i n the section below i t by noting the l i q u i d l e v e l i n the glass tube indicator (see Figure 3.3) and that above the experimental section by a d i r e c t dip-rod measurement. The absolute pressure near the top of the experimental section was once again recorded with the help of the open mercury manometer. The b a l l valve at the top of experimental section was then opened and the s e t t l e d l i q u i d height below the experimental section was checked again to ensure that the copper screen allowed no l i q u i d to leak through. From these measurements the gas holdup was calculated as described subsequently under Data Processing. During the l a t e r part of the work an e l e c t r o - r e s i s t i v -i t y probe was developed mainly to study the gas holdup i n the three-phase f l u i d i z e d bed region. However, a few runs were conducted to check the a p p l i c a b i l i t y of the probe for measurements of l o c a l gas f r a c t i o n s i n air-water and a i r -polyethylene gl y c o l solution flow. In order to use the el e c t r o n i c c i r c u i t described above, the amplifiers were warmed for 20 minutes under zero load conditions and then checked for any o f f s e t by the c i r c u i t described i n the manual [105]. The probe was located i n the experimental section so that the gap between the electrodes was nearly horizontal and perpendicular to the 145 flow d i r e c t i o n . This adjustment was not found to be c r i t i c a l i n these measurements. The probe was then supplied a D.C. poten t i a l from a constant D.C. source, through a p o t e n t i a l d i v i d e r . The applied voltage was so regulated as to produce pulses of approximately 0.22 v o l t s amplitude across a 100,000 ohm r e s i s t o r i n s e r i e s . As has been stated e a r l i e r , the l o g i c d i f f e r e n t i a l comparator was the central component of the measuring c i r c u i t . The output of the probe was c a r e f u l l y amplified to produce pulses of approximately 3.2 v o l t s amplitude, which were then fed as the non-inverting input to the comparator. The inverting input to the comparator was a reference D.C. po t e n t i a l of approximately -3.0 v o l t s amplitude, taken from a constant D.C. source through a potential d i v i d e r . The reference voltage was adjusted i n such a manner that the cutting o f f l e v e l of the pulses was at approximately 0.2 v o l t s above the datum. Both the input and the output of the comparator were monitored continuously on a dual beam oscilloscope to ensure that the cut-off l e v e l i n the comparator was such that no pulses were triggered from the input signal corresponding to the l i q u i d phase. The comparator output was integrated by the integrat-ing c i r c u i t with a time constant of nearly one second. The time required to integrate the comparator output to 8 v o l t s was noted. The gain of the amplifier following the comparator 146 was so adjusted that this time was about 2 minutes. The non-inverting input terminal of the comparator was then grounded, rendering the comparator output constant and equal to i t s peak value. The time required to integrate the comparator output to 8 v o l t s through the same amplifier was again noted. The local, gas f r a c t i o n was then obtained from the r a t i o of these two times. The bubble frequency was obtained, as mentioned e a r l i e r , by counting the pulses i n the comparator output either e l e c t r o n i c a l l y by a Darcy frequency counter or manually from the recording of the output. 3.3.4 Holdup studies i n three-phase f l u i d i z e d beds A t y p i c a l run was conducted by feeding a c a r e f u l l y weighed amount of well screened p a r t i c l e s into the t e s t section. Depending on the desired v e l o c i t y of l i q u i d through the column, the l i q u i d flow rate was measured by either of the flow meters. The l i q u i d was c i r c u l a t e d i n the column u n t i l a constant temperature was achieved and noted. A l l the manometer taps were c a r e f u l l y flushed to remove any a i r bubbles remaining i n the connecting l i n e s . Once the temperature and l i q u i d flow rates were s t a b i l i z e d , the expanded bed height was recorded. To determine the s t a t i c pressure p r o f i l e along the f l u i d i z e d bed, s t a t i c pressure drop readings were taken between tap 1 and other 147 taps above i t (see Figure 3.4). The open mercury manometer measured the absolute pressure near the top of the experimental section. The a i r was introduced by pressurizing the a i r l i n e and making l i q u i d flow rate adjustments to ensure that no s o l i d p a r t i c l e s were ejected out of the column during the introduction of the a i r stream. The back pressure i n the a i r l i n e was then so adjusted as to obtain the desired gas flow rate without any fluctuations i n the rotameter reading. The l i q u i d flow rate was then readjusted, so as to give a stable operation of the f l u i d i z e d bed. In order to determine the complete s t a t i c pressure p r o f i l e i n the t e s t section, s t a t i c pressure drop readings were taken between tap 1 and a l l other taps above i t . The measurements of s t a t i c pressure drop gradient below and above the t e s t section were recorded by separate U-tube manometers. A record was kept of the observed bed behaviour and the flow regime encountered i n the t e s t section. The open mercury manometer was used to measure the absolute pressure near the top of the experimental section. The two b a l l valves were then closed by actuating the l i n k rod manually. The gas flow rate was cut o f f by venting the a i r and the l i q u i d flow rate by switching o f f the motor. The s e t t l e d l i q u i d height i n various sections was measured (see Figure 8.2.3), and the absolute pressure near the top of the t e s t section was read from the open mercury manometer. A check of fixed bed height before and 148 afte r the run revealed i f any of the p a r t i c l e s were e l u t r i a t e d from the experimental column. If the proportion of p a r t i c l e s c a r r i e d out of the experimental section during a run was large, the run was discarded. From these measurements the expanded bed height, the sol i d s and gas holdup inside the three-phase f l u i d i z e d bed, and the gas holdup above and below the f l u i d i z e d bed were calculated as described under Data processing. The e l e c t r o - r e s i s t i v i t y probe was used to obtain the r a d i a l p r o f i l e of l o c a l gas f r a c t i o n inside the bed and was located 8 inches above tap 1 (29.0 cm above the bed support screen). The same procedure as used for measuring the gas holdup i n gas - l i q u i d flow was followed. 3.4 Data processing From the data obtained i n the 2 inch perspex column, the expanded bed height, L^, and the s o l i d s holdup i n the f l u i d i z e d bed, as well as the gas holdup i n the gas - l i q u i d and g a s - l i q u i d - s o l i d regions, were calculated as outlined i n the following sections. 3.4.1 Expanded bed height and s o l i d s holdup The longitudinal pressure drop p r o f i l e s for a l l the two-phase l i q u i d - s o l i d and the three-phase g a s - l i q u i d - s o l i d f l u i d i z a t i o n studies were measured up to a height of 46 inches 149 above tap 1 (Figure 3.4). The method of obtaining the expanded bed height, and thereby the so l i d s holdup, from these measurements i s discussed i n Appendix 8.2. Accordingly, the observed pressure drop, as represented by the U-tube manometer reading, i s plotted against the distance from tap 1. Then for a two-phase l i q u i d - s o l i d f l u i d i z e d bed, as shown i n Appendix 8.2, the following s t r a i g h t l i n e represents the pressure drop for Z < Z : — in 9.x H (pM - p±) = Z e3 (p 3 - px) (8.2.18) and the point of i n t e r s e c t i o n of t h i s s t r a i g h t l i n e with H = H s a t i s f i e s max H (pA. - p.) = Z e, (p, - p, ) (8.2.19) max KM 1 max 3 3 1 The measured lon g i t u d i n a l pressure drop p r o f i l e for a t y p i c a l l i q u i d - s o l i d f l u i d i z a t i o n experiment i s shown i n Figure 3.10. The s t r a i g h t l i n e for the f l u i d i z e d bed region i s obtained by f i t t i n g the best l i n e through the pressure drop data for Z « Z m a x by the method of l e a s t squares. The intercept of t h i s s t r a i g h t l i n e with the averaged value of H = H gives the value of Z , from which the expanded max ^ max' c bed height i s calculated as Z + 8.7 max (3.6) 150 o o o 60 55 -50 -45 -E 40 = 35 21 Q < cr cr LU h -LU o < 30 25 20 15 10 max WATER -Q5mm GLASS BEADS W = . l200gm. = 1.59 cm/sec. L-b.o = 34 .2 cm. Visually, L b = 5 4 . 5 cm. -<Zmax>cm +8-7 ".-54.8cm. . 5 10 15 20 25 30 35 40 45 DISTANCE FROM TAP 1, Z, in. FIGURE 3.10 TYPICAL PRESSURE DROP PROFILE IN LIQUID-SOLID FLUIDIZATION The expanded bed height so calculated was found gen-e r a l l y to be i n good agreement with the bed height measured by d i r e c t observation of the bed boundary. However, for the pa r t i c u l a r case of small glass beads at a large degree of bed expansion, the observed pressure drop data near Z = Z m a x deviated considerably from the i n i t i a l s t r a i g h t l i n e . This deviation i s believed to be caused by the non-uniformity of longitudinal s o l i d s d i s t r i b u t i o n a r i s i n g from s t r a t i f i c a t i o n by size of the imperfectly sized s o l i d s . In such cases the expanded bed height calculated by the s t r a i g h t l i n e i n t e r -section method was found to be smaller than the measured bed height by v i s u a l estimation of the bed boundary; nevertheless the former was used to calculate the sol i d s holdup i n the f l u i d i z e d bed. The s o l i d s holdup was then determined from the equation e 3 = W /P 3AL b (3.1) and also from the slope, S^, of the best s t r a i g h t l i n e through the pressure drop data. Then from equation 8.2.13, since £2 = ° ' E3 = S I ( P M ~ P 1 ) / ( P 3 " P 1 ) ( 3 , 7 ) The values of soli d s holdup obtained from equations 3.1 and 3.7 r e s p e c t i v e l y , were found to be i n good agreement, and an arithmetic mean of these two values i s reported as the so l i d s holdup. 152 The procedure for obtaining the expanded bed height of a three-phase f l u i d i z e d bed, i s e s s e n t i a l l y the same as described above. The observed pressure drop, as represented by the manometer reading, i s plotted against the distance from tap 1. As shown i n Appendix 8.2.1, the following s t r a i g h t l i n e represents the pressure drop data for Z « Z : c c max H I H (P M-P X) = Z [ e 3 ( p 3 ~ p l ) " £2 ( P i _ P 2 ) ] (8.2.12) while for Z > Z , — max' H ( pM" pl> = Z m a x t £ 3 ( p 3 - p l ) - ( £ 2 - £ 2 ) ( p l ~ p 2 ) ] - Z e 2 (P 1-P 2) (8.2.10) The i n t e r s e c t i o n of these two l i n e s s a t i s f i e d Hmax ( pM- pl> = W ^ a - P l * - e 2 ( p l - p 2 ) ] ( 8 ' 2 The measured longitudinal pressure drop p r o f i l e for a t y p i c a l g a s - l i q u i d - s o l i d f l u i d i z a t i o n experiment i s shown i n Figure 3.11. The l i n e for the f l u i d i z e d bed region i s obtained by f i t t i n g the best s t r a i g h t l i n e through the pressure drop data for Z « Z by the method of l e a s t squares. S i m i l a r l y max the l i n e for the region above the bed i s obtained by f i t t i n g the best s t r a i g h t l i n e through the pressure drop data for 153 551 —i 1 1 1 1 1 1 r~—-— 5 10 15. 20 25 30 35 40 45 50 DISTANCE FROM TAP I, Z, in. FIGURE 3.11 TYPICAL PRESSURE DROP PROFILE IN THREE-PHASE FLUIDIZATION (arrows indicate upper l i m i t of bed l e v e l as v i s u a l l y observed) 154 Z » .. Z max by l e a s t squares. The point of i n t e r s e c t i o n of these two st r a i g h t l i n e s determines Z 'max' from which the expanded bed height i s calculated: L, b Z max + 8.7 (3.6) The expanded bed height from equation 3.6 was found to be i n good agreement with the bed height measured by d i r e c t v i s u a l observation of the bed boundary at small gas flow rates, when thi s boundary could be c l e a r l y defined. However, for higher gas flow rates (> 4 cm/sec), the bed boundary was quite d i f f u s e and could no longer be located v i s u a l l y with any confidence and consistency. Under these circumstances the method outlined above provides a meaningful d e f i n i t i o n to the expanded bed height, and was used throughout t h i s study to obtain expanded bed heights with a high degree of repro-d u c i b i l i t y and confidence. The solids holdup i n the three-phase f l u i d i z e d bed was then obtained from Care was exercised i n a l l experiments to prevent e l u t r i a t i o n or e j ection of s o l i d p a r t i c l e s from the experimental column. Nevertheless some e l u t r i a t i o n , depending on the gas and the l i q u i d flow rates and the size of the p a r t i c l e s present i n H I = W/p 3AL b (3.1) 155 the column, did occur, as evidenced by the presence of p a r t i c l e s i n the screen c a t c h - a l l and by the reduction i n s t a t i c bed height. If the weight of p a r t i c l e s l o s t from the column during a p a r t i c u l a r run was disproportionately large (>5%), that run was discarded; otherwise the weight of p a r t i c l e s i n the system was taken as the mean of the weights of p a r t i c l e s i n the system before and af t e r the run. No such problem was encountered for the large and heavy p a r t i c l e s . I t i s also important to point out that no measurable f r i c t i o n a l pressure drop was observed when operating without solids i n the column with water as the test l i q u i d , at a l l the water flow rates investigated (2-39 cm/sec). However, with polyethylene glycol-water solution as the t e s t l i q u i d , f r i c t i o n a l pressure drop i n the column without s o l i d s was measurable for two of the flow rates studied (13.8 2 and 18.84 cm/sec). In such instances, the measured pressure drops for the longitudinal p r o f i l e s were corrected by subtracting the appropriate f r i c t i o n a l pressure drop from each of the measured values. The corrected pressure drop readings were than used to obtain the expanded bed height for determining the s o l i d s holdup. In the 20 mm glass-column the expanded bed height was measured.by locating the bed boundary v i s u a l l y . The small cross-section of the tube employed f a c i l i t a t e d observation of the bed boundary; nevertheless, the upper range of gas flow rate investigated was r e s t r i c t e d due to the d i f f i c u l t y 156 of defining the bed boundary at higher gas flow rates. The s o l i d s holdup was calculated from equation 3.1 as before. 3.4.2 Gas holdup The two main methods used to study the average gas holdup i n the two-and three-phase systems v i z . the measurement of s t a t i c pressure drop gradient and the measurement of l i q u i d l e v e l a f t e r i s o l a t i n g the sections by shutting o f f the valves, are described i n Appendix 8.2. I t i s shown there that due to changes i n k i n e t i c energy of the stream and f r i c t i o n a l pressure losses, which Neal and Bankoff [103] estimated to be only about 2.5% of the t o t a l s t a t i c pressure drop i n t h e i r own gas - l i q u i d system, the gas holdup obtained by s t a t i c pressure drop measurements i s subject to some error. However, i f these losses are properly accounted f o r , the pressure drop per unit length i n gas-liquid flow i s very nearly equal to the mean density of the two-phase flow stream. In such instances, the gas holdup i s given by e2 = " H ( p M _ p l ) / ( p l " P 2 ) Z (8.2.17) The gas holdups above and below the experimental section, are thus obtained by measuring the pressure drop with the U-tube manometer provided i n each of these sections. These measured gas holdups are then suitably corrected from the 157 midpoint pressure of the measuring section to a standard pressure of 760 mm of mercury. A l l gas holdups are reported at t h i s pressure. For the estimation of gas holdup i n a three-phase f l u i d i z e d bed, a complete longitudinal pressure drop p r o f i l e i s required. As shown i n Appendix 8.2, the slope of the strai g h t l i n e for the three-phase region i s given by S I = [ e3 ( p 3 " p l ) " e2 ( p l " p 2 ) ] ~ (8.2.13) ( PM~ P1 ) from which, on rearrangement, £2 = [ £ 3 ( p 3 - p l } - S I ( p M - p l ) ] , 1 , ( 3 ' 8 ) (Px-P2) where i s obtained from equation 3.1 as discussed i n the preceding section, and Sj i s obtained from the slope of the best st r a i g h t l i n e through the pressure drop data for Z « % m a K by the method of l e a s t squares. Alternately, i n f l u i d i z a t i o n one can u t i l i z e the fa c t that pressure drop across the bed i s equal to the weight of the bed per unit area [98]. In three-phase f l u i d i z a t i o n Hmax ^ s a m e a s u r e °f ^ e pressure drop across the bed of height Zmax a n c ^ s a t i s f i e s equation 8.2.11. Then by rearranging equation 8.2.11, the gas holdup i n the three-phase f l u i d i z e d bed i s given by 158 II e3(P3-Pi>. Z max - H max (3.9) e 2 ( P 1 - P 2 ) z max The method of measuring gas holdup by q u i c k l y s h u t t i n g o f f the v a l v e s and r e c o r d i n g the subsequent l i q u i d l e v e l i n each i s o l a t e d s e c t i o n i s d e s c r i b e d i n Appendix 8.2.2. As d i s c u s s e d t h e r e , the gas c o l l e c t e d i n each s e c t i o n i s a t a d i f f e r e n t p r e s s u r e depending on the h y d r o s t a t i c head above t h a t s e c t i o n . Therefore s u i t a b l e p r e s s u r e c o r r e c t i o n s g i v e n by equations 8.2.29 and 8.2.3 0 were a p p l i e d r e s p e c t i v e l y to the gas c o l l e c t e d i n and below the experimental s e c t i o n , so as to o b t a i n gas holdups i n these s e c t i o n s a t a standard p r e s s u r e o f 760 mm of mercury. For the gas holdup above the experimental s e c t i o n , no such c o r r e c t i o n f a c t o r i s n e c e s s a r y . The q u i c k v a l v e s h u t - o f f measurements of gas holdups i n t h i s s e c t i o n under s l u g flow c o n d i t i o n s were found to be u n r e l i a b l e due to the s h o r t l e n g t h of the s e c t i o n , and were t h e r e f o r e d i s c a r d e d . phase f l u i d i z e d bed r e g i o n by the q u i c k v a l v e s h u t - o f f method i s o b t a i n e d from As shown i n Appendix 8.2, the gas holdup i n the t h r e e -n i n n e0 = e_ + [e 2EC " £2] V L b (8.2.33) n where e~ i s the gas holdup i n the two-phase g a s - l i q u i d r e g i o n 159 above the bed and i s measured independently by a U-tube manometer located near the top of the experimental section. For low gas flow rates it.was not possible to measure the l i q u i d l e v e l i n the transparent experimental section. In such instances a mean of gas holdups obtained from equations 3.8 and 3.9 i s reported; otherwise, i n a l l other cases, a mean of gas holdups by the two pressure drop methods and by the quick valve shut-off method i s reported. The gas holdup i n the 20 mm glass column was measured mainly by observing the pressure drop across the bed on a U-tube manometer. Then the reduction i n manometer reading on introduction of the gas corresponds to the gas f r a c t i o n be-tween the taps. No attempt was made to s p e c i f i c a l l y calculate the gas f r a c t i o n inside the three-phase region; nevertheless the gas holdup measured i n this manner could reveal whether the presence of s o l i d p a r t i c l e s modified the two-phase gas-l i q u i d holdup s i g n i f i c a n t l y . 160 CHAPTER 4 RESULTS AND DISCUSSION In this chapter, the mathematical models derived i n Chapter 2 are f i r s t compared with e x i s t i n g models and data from the l i t e r a t u r e , and then the experimental data obtained i n t h i s study are used to evaluate the proposed models. 4.1 Comparison of proposed mathematical models with previous  work 4.1.1 Gas holdup i n gas-liquid flow Zuber and Findlay [39] derived equation 2.26, presented i n Section 2.1.2, for two-phase gas- l i q u i d flow; i n combin-ation with equation 2.24 i t can be written as v 2 = = C 0 ( < j 1 + j 2 » + _ 2 - 2 l _ ( 4 i l ) <a2> <a2> Equation 4.1 i s quite general and i s applicable to a l l the gas-liquid flow regimes i f the d i s t r i b u t i o n parameter and the weighted mean d r i f t v e l o c i t y can be obtained indepen-dently . The d i s t r i b u t i o n parameter, CQ, was shown t h e o r e t i c a l l y , with the help of equations 2.27a-c, to vary between 1.0 and 161 1.5 for most cases of gas-liquid flow. Using the data of Smissaert [113] for air-water flow i n a 2 inch v e r t i c a l pipe, Zuber and Findlay were also able to show empirically that for the churn-turbulent bubbly and slug flow regimes, CQ was equal to 1.2. N i c k l i n et a l . [114] used a d i f f e r e n t l i n e of argument and also found a value of 1.2 for CN to s a t i s f y a wide range of data for gas - l i q u i d flow, i f the Reynolds number based on the gas - l i q u i d f l u x through the conduit exceeded 8,000. For d r i f t v e l o c i t y , Zuber and Findlay proposed v 2 j = V a (1 - a 2 ) m (2.31) where the exponent m was found to vary between 0 and 3, depending on the bubble s i z e . I t was further noticed that the l o c a l d r i f t v e l o c i t y was constant for both slug flow and bubbly flow i n a turbulent stream ( i . e . , m = 0 ) ; then the weighted mean d r i f t v e l o c i t y i s simply <a„v„ . > 1 3 = 0.35 S~T (4.2) <a2> * for the slug flow regime and <a„v 0.> ag n — £ _ £ i _ = 1.53 [—V'^ (4.3) <a„> p l for the churn-turbulent bubbly flow regime. 1 6 2 The present author has proposed a model for d r i f t v e l o c i t y i n the bubble flow regime [76] which, as shown e a r l i e r , y i e l d s the equation V 2 j ~ V ~ "'T , r r t o c / - i , X 1/3 ( 2 . 3 7 ) J tanh [ 0 . 2 5 ( l / a 2 ) ' and which for the slug flow regime (r = R), applying equation 2.10 for large Eotvos number, s i m p l i f i e s to v 2 j = 0 . 3 5 ( 2 > 3 9 ) Thus for the slug flow regime the two models for d r i f t v e l -o c i t y (equations 2.39 and 4.2) are i d e n t i c a l , so that equation 4 . 1 , with CQ = 1.2, reduces to the slug flow re l a t i o n s h i p proposed by N i c k l i n [ 1 9 ] : — — = 1.2 (<j 1+j 2>) + 0 . 3 5 /gD~ ( 4 . 4 ) <a2> Various models for predicting the d r i f t v e l o c i t y i n the bubble flow regime are shown i n Figure 4 . 1 , along with Happel's [ 4 9 ] equation for sedimentation of s o l i d spheres. The following remarks are based on Figure 4 . 1 : (a) The discrepancy between the curves for bubble swarms and s o l i d p a r t i c l e s arises from the fac t that the tangential l i q u i d v e l o c i t y i s zero at the surface of 163 FIGURE 4.1 PROPOSED MODELS FOR DRIFT VELOCITY OF BUBBLE SWARMS 164 LEGEND FOR FIGURE 4.1 1. Sedimentation of s o l i d p a r t i c l e s by Happel's model. 2. Equation 2.31 with m = 3 for small bubbles (d^ < 0.5 mm) obeying Stokes law [74]. 3. Equation 2.31 with m = 2 recommended for bubble flow regime by Bhaga [1]. 4. Equation 2.31 with m = 1.5 for larger bubbles (1 < d b < 20 mm) [74]. 5. Equation 2.31 with m = 0 for churn turbulent-bubbly flow and slug flow regimes [39]. 6. Equation 2.34 proposed for bubble flow regime by Marrucci [75] . 7. Equation 2.37 proposed for bubble flow regime by Bhatia [76]. a s o l i d p a r t i c l e but i s not zero at the surface of a bubble. Consequently the energy d i s s i p a t i o n i s smaller and the d r i f t v e l o c i t y i s higher for the bubble swarm. (b) The model proposed by Marrucci [75] based on poten-t i a l flow shows a dependence of d r i f t v e l o c i t y on volumetric gas f r a c t i o n which i s very s i m i l a r to that of the Zuber-Findlay model with the exponent m equal to 1.5 . (c) The exponent m i n the Zuber-Findlay model (equation 2.31) was reported to vary between 0 and 3, depending on the bubble s i z e , the larger values corresponding to the smaller bubble s i z e s . The slope of the curve representing equation 2.37 shows a gradual reduction with increasing gas holdup. Since the stable bubble size and the gas holdup are i n t e r r e l a t e d , increases i n the l a t t e r accompanying increases i n the former, i t can be argued at l e a s t q u a l i t a t i v e l y that the present model has the v i r t u e of predicting the correct trend i n d r i f t v e l o c i t y of a bubble swarm over a wide range of operating variables. In order to tes t the quantitative a p p l i c a b i l i t y of the present model, a comparison of the predictions with some of the available l i t e r a t u r e data for bubble columns and for cocurrent gas-liquid flow i n v e r t i c a l pipes has been published by the present author [76] and i s presented 166 i n Appendix 8.8. I t was found that: (a) The model i s applicable to low v i s c o s i t y and com-paratively pure gas-liquid systems. (b) For bubble columns, the model s a t i s f i e d the data obtained i n small columns (D _< 2 inch) , systematic bulk c i r c u l a t i o n not being important i n such columns [59, 60]. The systematic c i r c u l a t i o n found i n columns with D ^ 4 inch [59, 60] would increase the bubble concentration i n the upward moving central core, thereby reducing the average gas holdup and increasing the value'of the d i s t r i b u t i o n parameter, CQ, by making the flow and gas holdup p r o f i l e s more pointed (as opposed to f l a t ) . (c) For cocurrent flow the present model i n conjunction with equation 2.20 for bubble diameter [69] agrees well with Hughmark's empirical c o r r e l a t i o n [52], which has been corroborated against experimental data by Dukler et a l . [72] . In order to check the general v a l i d i t y of the above model for determining the d r i f t v e l o c i t y of a bubble swarm, systematic data for average bubble diameter and gas-holdup as a function of gas and l i q u i d v e l o c i t i e s are needed. The model, which s t r i c t l y speaking applies only l o c a l l y , should be supplemented by equation 4.1 to account for any r a d i a l non-uniformities i n v e l o c i t i e s and holdups. 167 4.1.2 Gas holdup i n three-phase f l u i d i z e d beds Before a comparison of the general model for describ-ing the r i s e v e l o c i t y of bubble swarms i n three-phase f l u i d i z e d beds with the empirical correlations based on data of various investigators [14, 17, 79] can be attempted, i t i s necessary that a mathematical expression be obtained to predict the wake volume f r a c t i o n . As defined e a r l i e r , the wake f r a c t i o n i n a three-phase f l u i d i z e d bed i s given by H I £k = £2 ( > ( 4 * 5 ) where the r a t i o of wake volume to bubble volume i n the three-phase f l u i d i z e d bed may be represented by <nr> - K> f ( e ) (2.H7) B B f being a continuous function of bed voidage which approaches unity as the sol i d s f r a c t i o n , £ 3 / i n the three-phase f l u i d i z e d bed approaches zero. The simplest such function i s f(e) = ( l - e 3 ) P (4.6) 168 Combining equations 4.5, 2.117 and 4.6, the necessary mathematical expression for the wake f r a c t i o n i n a three-phase f l u i d i z e d bed i s given by ek = e2 ( } ( 1 _ e 3 ) P ( 4 * 7 ) ^k " where (^ —) may be estimated from the data of Letan and " B Kehat [61] for a l i q u i d - l i q u i d system, as shown previously. In order to evaluate the exponent p, simultaneous measurements of e^, and are needed. Such measurements are scarce. Nevertheless Efremov and Vakhrushev [16] presented an equation for e^, based on measurements of e!}1 and i n beds of glass beads (0.32 - 2.15 mm) f l u i d i z e d by a cocurrent stream of a i r and water: (JS) = 5.1 ( e ! ' ) 4 - U b [1-tanh {40 ( e l ' ) 1 0 ftn x ->2 x o - 3.32 ( e j ) 5 , 4 5 } ] (2.128) where i s the voidage i n the l i q u i d - s o l i d f l u i d i z e d bed before the introduction of gas and can be computed from equation 2.46. Tables 4.1 and 4.2 present the r a t i o of wake to bubble volume for two sizes of glass beads, f l u i d i z e d by a cocurrent stream of a i r and water, as predicted by equations 4.7 (for p=3), 2.128 and 1.13, respectively, along with some limited data of Rigby and 169 TABLE 4.1 RATIO OF WAKE TO BUBBLE VOLUME IN THREE-PHASE FLUIDIZED BED PARTICLES - Glass beads (dp=0.775 mm) p3=2.67 gm/cm3 D > 4 inches Liquid Flux, <j x> (cm/sec) Gas Flux, <^2> (cm/sec) (1) [xk=0.0] (2) [xk=0.0] (3) V ° B [xk=1.0j (4) [xk=0.0] 2.61 0.5 0 .689 0 .541 2.26 0.72 (1.80)* 1.0 0 .607 0.516 1.50 (1.08)* 2.0 0 .486 0 .466 0.99 (0.73)* 3.0 0 .401 0 .412 0.77 -4.0 0 .337 0 .367 0.65 -4.35 0.5 1.127 1.483 4 .19 -1.0 0.968 1.298 2.80 -2.0 0.745 0.899 1.85 -3.0 0 .594 0 .545 1.45 -4.0 0.483 0.298 1.22 -From generalized wake model, using equation 4.7 with p = 3 Equation 2.128 by Efremov and Vakhrushev [16] Equations 1.12 and 1.13 by 0stergaard [8] From Figure 2 of Rigby and Capes [8 0] Values i n brackets are for x, =1.0 (1) (2) (3) (4) (*) 17 0 TABLE 4.2 RATIO OF WAKE TO BUBBLE VOLUME IN THREE-PHASE FLUIDIZED BED PARTICLES - Glass beads (dp=2.0 mm) p^= 2.88 gm/cm3 D = 2.0 inches Liquid Flux, <J!> (cm/sec) Gas Flux, (cm/sec) (1) "k^B [xk=0.0] (2) [xk=0.0] (3) V ° B [xk=1.0] 3 .38 0.5 0 .364 0 .154 0.827 1.0 0.342 0.154 0 .551 2.0 0.301 0.152 0.364 3.0 0 .265 0 .151 0 .284 4.0 0.231 0 .150 0 .238 5.0 0 .199 0.149 0.207 6.0 0 .166 0 .148 0 .184 7.0 0 .131 0.147 0 .167 8.0 0.129 0 .146 0.153 9.0 0.128 0.145 0.142 8.51 0.5 0.99 1.02 6 .77 1.0 0 .91 0.99 4 .55 2.0 0.76 0.92 3.04 3.0 0.63 0.85 2.40 4.0 0.52 0.78 2^02 5.0 0.41 0 .71 1.77 6.0 0 .31 0 .64 1.59 7 .0 0.31 0.57 1.44 8.0 0.30 0.50 1.33 9.0 0 .29 0.44 1.24 (1) From generalized wake model, using equation 4.7 with p=3. (2) Equation 2.128 by Efremov and Vakhrushev [16]. (3) Equation 1.13 by 0stergaard [8]. Capes [80]. A comparison of these values reveal that: (a) The values predicted by equation 4.7 with p=3 show mod-erate agreement with the equation of Efremov and Vakhrushev [16] and the data of Rigby and Capes [80] . The wake was assumed to be free of s o l i d s i n the generalized wake model i n order to match the assumption inherent i n the equation of Efremov and Vakhrushev. (b) The values predicted by 0stergaard 1s [8] equation are generally higher than those by the other two equations; this discrepancy i s p a r t i a l l y due to 0stergaard's assumption that the concentration of s o l i d p a r t i c l e s i n the wake i s equal to the concentration i n the part i c u l a t e phase. The agreement of predictions by equations 4.7 with those by equation 2.128 was found to improve by assuming d i f f e r e n t values of the exponent p for d i f f e r e n t operating conditions. However, a value of 3 for the exponent gave a rough agreement with a l l available information, but i s not to be interpreted as either the best or the un i v e r s a l l y recommended value. Only a systematic inves t i g a t i o n of wakes behind bubbles i n three-phase f l u i d i z e d beds, with simultan-eous measurements of gas and solids holdups, could provide a j u s t i f i a b l e c o r r e l a t i o n . Nevertheless, i n the absence of suitable data for wake volume f r a c t i o n , equation 4.7, with p=3, w i l l be used i n the generalized wake model of Section 172 2.3 2 for the purpose of comparing i t s predictions of gas and s o l i d s holdups with experimental data. I t w i l l be seen that i n many instances the predictions are i n s e n s i t i v e to the wake f r a c t i o n and hence to the function f(e) i n equation 2.117. The more complex equation 2.128 w i l l be used simultaneously to provide guidelines within i t s range of a p p l i c a b i l i t y . Now, according to the model proposed i n Section 2.3.2, the r i s e v e l o c i t y of a bubble swarm i n a three-phase f l u i d i z e d bed i s given by C"» <j,+j 2> e" (l-e 2-e k> _ v 2 = ± — + - i i ± — — v™± (2.114) £ £ where v ^ i s calculated from equation 2.107 for D >_ 4 inches and from equation 2.108a for D < 4 inches. Since the value of the d i s t r i b u t i o n parameter i n three-phase f l u i d i z e d beds, CQ', cannot be estimated at present, therefore, i n a l l the calculations and discussions which follow, i t w i l l be t a c i t l y assumed that CQ = 1.0 and that the e f f e c t s of non-uniform r a d i a l p r o f i l e s can be lumped into the r e l a t i v e v e l o c i t y term, as has been done i n the past for gas - l i q u i d flow [19]• The gas holdup i s then obtained from <j 2>/v 2 (2.94) 173 Tables 4.3-A and 4.3-B present the values of /e2 a n < ^ v 2 ' respectively, predicted by the present gener-a l i z e d wake model [equations 2.114, 2.107 and 2.94] for 0.775 mm glass beads f l u i d i z e d by a cocurrent stream of a i r and water [80], along with the values calculated from the empirical correlations presented i n Section 2.3.2. An estimate of bubble length was obtained from the data of Rigby et a l . [79] (1 = 0.61 cm; then from equation 2.125, r e - 0.6 cm) and was used for calculations i n the present model as well as i n the c o r r e l a t i o n proposed by Rigby and Capes [80]. Equation 2.107 was used for c a l c u l a t i n g the r e l a t i v e v e l o c i t y of a bubble swarm i n the model because the empirical correlations presented i n Tables 4.3-A and -B were a l l based on data obtained i n columns of D > 4 inch. On comparing the predictions from the model with those from the correlations i n Table 4.3-A, i t can be seen that: (a) The predictions from the model are i n reasonable agreement with the re s u l t s of Efremov and Vakhrushev [16], who were forced to l i m i t t h e i r measurements to low gas v e l o c i t i e s because of the uncertainty of locating the three-phase boundary at high-gas flow rates, e s p e c i a l l y for large bed expansions. (b) Although the c o r r e l a t i o n reported by Michelsen and 0stergaard i s also based on data for low gas v e l o c i t i e s , i t too i s found to give r e s u l t s which are i n f a i r l y good agreement with the predictions of the model. 174 TABLE 4.3-A RATIO OF GAS HOLDUP IN THREE-PHASE FLUIDIZED BED TO GAS HOLDUP IN TWO-PHASE GAS-LIQUID FLOW (D > 4 INCHES) PARTICLES - Glass beads (d = 0 P .775 mm) P 3 = 2 .67 gm/cm3 Liquid Flux, ^1* (cm/sec) Gas Flux <j 2> (cm/sec) CD (2) (3) c in/,-, I I e 2 / e 2 (4) c in / c I I e 2 / e 2 2.61 0.5 0.602 0.542 0.556 0.340 1.0 0.596 0.542 0.576 0 .327 2.0 0.587 0 .542 0.567 0.311 3.0 0.581 0 .542 0 .533 0.302 4.0 0.579 0 .542 0.546 0.299 4.35 0.5 0 .716 0 .632 0 .734 0 .482 1.0 0.716 0.632 0 .734 0 .482 2.0 0 .696 0 .632 0.505 0.432 3.0 0 .689 0 .632 0.452 0.418 4.0 0.687 0.632 0.419 0.414 (1) From generalized wake model with x, = 0 (2) From equation of Efremov and Vakhrushev [16] (3) From equations 2.121 and 2.122, Michelsen and 0stergaard [14] (4) From equation 2.120, V a i l et a l . [17] 175 TABLE 4.3-B RISE VELOCITY OF BUBBLE SWARMS IN THREE-PHASE FLUIDIZED BED (D > 4 INCHES) PARTICLES - Glass beads (d = 0.775 mm) p 3 = 2.67 gm/cm3 Liquid Flux, (cm/sec) Gas Flux, <h> (cm/sec) (1) v 2 , cm/sec [r e=0.6 cm] (2) ^2 (3) ^2 (4) (5) ^2 [1=0.612 cm] 2.61 0.5 50 .24 27.50 50.00 52.05 36 .65 1.0 52.51 24 .31 52.63 62.85 34 .43 2.0 56 .90 21.12 58 .82 77 .05 32.04 3.0 61.04 19.26 61.22 86 .82 31.28 4.0 64.88 17 .93 66 .67 93.71 31.55 4 .35 0.5 44.72 29 .24 45.30 54 .65 92.67 1.0 46.74 26 .05 52.77 66.70 81.09 ^ 2.0 50.62 22.86 61.46 82.72 73.78 3.0 54.20 21.00 67 .19 93 .47 64 .61 4.0 57.40 19 .67 71.59 100 .54 64.11 (1) From generalized wake model with x^ = 0 (2) From equation 1.12, 0stergaard [8] (3) From equations 2.121 and 2.94, Michelsen-->and 0stergaard [14] (4) From equations 2.118 and 2.94, V a i l et a l . [17] (5) From equation 2.123, Capes et a l . [79] (c) The predictions from the model are consistently higher than the re s u l t s of V a i l et a l . [17]. This discrepancy could be p a r t i a l l y due to the approximate method used by V a i l et a l . to measure the so l i d s holdup and p a r t i a l l y due to th e i r method for measuring the gas holdup i t s e l f (see Appendix 8.2). Table 4.3-B presents the r i s e v e l o c i t i e s of bubble swarms calculated from various empirical c o r r e l a t i o n s . A comparison of these values with the predictions from the generalized wake model reveals that: (a) The o r i g i n a l 0stergaard c o r r e l a t i o n for estimating the r i s e v e l o c i t y of a bubble swarm, equation 1.12, gives values which are far out of l i n e from those given by the other correlations as well as from the model. The cor r e l a t i o n proposed l a t e r by Michelsen and 0stergaard [14], on the other hand, gives values which are i n good agreement with the predictions from the model. (b) The bubble r i s e v e l o c i t i e s calculated from the c o r r e l a -t i o n proposed by.Vail et a l . [17] are found to be larger than those predicted by the model, though the two exhibit similar trends with respect to increase i n gas v e l o c i t y . (c) The values calculated from the c o r r e l a t i o n proposed by Rigby and co-workers [79] show no s i m i l a r i t i e s with those of the model. At the smaller l i q u i d f l u x the bubble v e l o c i t i e s are smaller, whereas at the larger 177 l i q u i d flux the bubble v e l o c i t i e s are much larger, than predicted by the model. However, i t i s important to notice that the c o r r e l a t i o n of Rigby et a l . (equation 2.123) i s quite sensitive to the length of bubbles i n the swarm. Since only l i m i t e d data for bubble lengths are available [7 9], a f a i r comparison of equation 2.123 with the other correlations as well as with the model i s not possible at present. Since the model proposed here i s used l a t e r to test the 2 inch diameter column data obtained i n the present study, an attempt i s f i r s t made to compare the predictions of the model with data obtained from 2 inch diameter columns by e a r l i e r investigators. Although the bed voidage data for the 2 inch columns are available, no such data for gas hold-up are reported. Therefore the bed voidage data of 0stergaard and Theisen [18] , for 2 mm glass beads f l u i d i z e d by a cocurrent stream of a i r and water i n a 2 inch column, are used as a basis for obtaining the values of e'^/z^ a n c ^ v2 from empirical c o r r e l a t i o n s . For the model the r e l a t i v e v e l o c i t y i s obtained from equation 2.10 8a, where the d r i f t v e l o c i t y for the slug flow regime i s obtained by modifying equation 2.39 to account for non-uniformities i n r a d i a l p r o f i l e s , following the recommendation of N i c k l i n [19]: = 0.2 <j. + j 9> + 0.35 vfqD (4.8) 17 8 The calculated values are presented i n Tables 4.4-A and 4.4-B. A comparison of various values of e^/t 1^ ^ n T a b l e 4.4-A reveals that: (a) The predictions from the model are i n f a i r l y good agree-ment with the res u l t s of Efremov and Vakhrushev [16] for small l i q u i d v e l o c i t y , becoming poorer as the l i q u i d v e l o c i t y i s increased. (b) The predictions from the model agree with the re s u l t s of V a i l et a l . [17] only i n the i r respective trends, but not i n absolute values. (c) The predictions from the model do not agree with the values calculated from the Michelsen-0stergaard [14] co r r e l a t i o n , either absolutely or i n trends displayed. S i m i l a r l y a comparison of r i s e v e l o c i t i e s of bubble swarms i n three-phase f l u i d i z e d beds (Table 4.4-B) reveals that: (a) The c o r r e l a t i o n proposed by 0stergaard [8], equation 1.12, provides a poor estimate for the bubble r i s e v e l o c i t y , i f one gives any credence at a l l to the empirical correlations of references [14] and [17]. The co r r e l a -t i o n proposed l a t e r by Michelsen and 0stergaard [14] gives moderate agreement with the predictions from the model. (b) The bubble r i s e v e l o c i t i e s calculated from the c o r r e l a t i o n proposed by V a i l et a l . [17] do not agree with the pre-d i c t i o n s from the model. 179 TABLE 4.4-A RATIO OF GAS HOLDUP IN THREE-PHASE FLUIDIZED BED TO GAS HOLDUP IN TWO-PHASE GAS-LIQUID FLOW (D = 2 INCHES) PARTICLES - Glass beads (d = 2.0 mm) p, = 2.88 gm/cc Liquid Gas (1) (2) (3) (4) Flux, Flux, <h> £2 £2 e2 e2 e '"/e " 2 2 2 fc2 (cm/sec) (cm/sec) 3.38 0.5 0.471 0.436 0.216 0.645 1.0 0.485 0.436 0 .216 0 .532 2.0 0.484 0.436 0 .218 0 .442 3.0 0 .495 0.436 0.220 0.396 4.0 0.504 0.436 0 .225 0 .366 5.0 0.507 0 .436 0 .232 0 .345 6.0 0 .515 0.436 0 .241 0.328 7.0 0.532 0 .436 0 .253 0.315 8.0 0 .531 0 .436 0 .257 0 .304 9.0 0.535 0.436 0.260 0.294 8.51 0.5 0 .714 0 .560 0 .436 1.044 1.0 0.679 0 .560 0.433 0.868 2.0 0 .685 0 .560 0.430 0.720 3.0 0.705 0 .560 0.433 0.650 4.0 0.703 0 .560 0 .440 0 .597 5.0 0.713 0 .560 0 .455 0 .562 6.0 0 .720 0.560 0 .472 0.535 7.0 0.728 0 .560 0.475 0.514 8.0 0 .733 0 .560 0 .479 0.495 9.0 0.741 0 .560 0.482 0 .480 (1) From generalized wake model with x. = 0 (2) From empirical equation of Efremov and Vakhrushev [16] (3) From equation 2.120, V a i l et a l . [17] (4) From equations 2.121 and 2.122, Michelsen and 0stergaard [14] 180 TABLE 4.4-B RISE VELOCITY OF BUBBLE SWARMS IN THREE-PHASE FLUIDIZED BED (D = 2 INCHES) PARTICLES - Glass beads (d =2.0 mm) p, = 2.88 gm/cc Liquid Gas (1) (2) (3) (4) Flux, Flux, — _ _ _ V2 V2 V2 V2 (cm/sec (cm/sec) (cm/sec) 3.38 0.5 60.84 28 .27 49.74 100.2 1.0 61.80 25.08 57.93 116.7 2.0 63 .66 21.89 67 .47 134 .6 3.0 65.40 20 .03 73 .77 146.0 4.0 67 .00 18 .70 78 .59 152 .0 5.0 68.36 17 .68 82 .54 154 .8 6.0 69 .48 16.84 85.92 155.1 7.0 70.26 16.13 88.89 152.8 8.0 71.87 15.52 91.54 155.0 9.0 73 .54 14 .97 93 .94 157 .1 8.51 0.5 52.26 33.40 35.46 106.1 1.0 52.88 30.21 41.17 119 .6 2.0 54.11 27 .02 47 .95 140.3 3.0 55.28 25.16 52.42 152.3 4.0 56.36 23 .83 55.85 160 .0 5.0 57.32 22.81 58.66 162.2 6.0 58 .21 21.97 61.06 162.7 7.0 59 .38 21.26 63.16 167 .3 8.0 60.57 20.65 65.05 170 .8 9.0 61.79 20 .10 66 .75 174.2 (1) From generalized wake model with x, = 0 (2) From equation 1.12, 0stergaard [8] (3) From equations 2.121 and 2.94, Michelsen and 0stergaard [14] (4) From equations 2.118 and 2.94, V a i l et a l [17] 181 Thus i t can now be ten t a t i v e l y concluded that the present model, when used with a suitable c o r r e l a t i o n for the r e l a t i v e v e l o c i t y of bubble swarms i n three-phase f l u i d i z e d beds, provides an e f f e c t i v e method for estimating the gas holdups and bubble r i s e v e l o c i t i e s i n three-phase f l u i d i z -ation, since the predictions from the model generally agree with the available correlations of data from columns of D >_ 4 inches [14, 16] . Since the bubble dynamics for gas-l i q u i d flow i n small columns (D < 4 i n c h ) d i f f e r markedly from those i n larger ones [59, 60, 68], no single c o r r e l a t i o n can be successfully used for a l l column sizes unless these dynamics are properly accounted f o r . The present model does so by allowing a separate c o r r e l a t i o n for the r e l a t i v e v e l o c i t y of bubble swarms to be used for small diameter columns than for large columns. 182 4.1.3 Voidage i n three-phase f l u i d i z e d beds The measurement of voidage i n three-phase f l u i d i z e d beds has been carried out by various investigators, of whom.0stergaard and co-workers [8, 14, 18, 54] and Efremov and Vakhrushev [16] are most noteworthy because of the wide range of p a r t i c l e sizes which they investigated. Most of these measurements were obtained i n columns with diameter D >_ 4 inches; we note, however, that 0stergaard and Theisen [18] reported on limited data obtained i n a 2 inch diameter column. The number of th e o r e t i c a l analyses attempted for predicting the voidage i n a three-phase f l u i d i z e d bed has been li m i t e d . The wake model proposed by 0stergaard [18] and presented i n Chapter 1 was the f i r s t successful analysis to s a t i s f y the reported paradox [6, 9] of bed contraction i n three-phase f l u i d i z a t i o n . However this model f a i l e d to s a t i s f y the extensive data of 0stergaard and Theisen [18] qu a n t i t a t i v e l y . Efremov and Vakhrushev [16] then proposed and derived a model quite s i m i l a r to the wake model but with the assumption that the p a r t i c l e content of bubble wakes was zero. From simultaneous measurements of gas and S o l i d holdups, the wake f r a c t i o n i n a three-phase f l u i d i z e d bed was calculated from equations 2.126 - 2.127. The wake fractions so calculated were found to be adequately represented by equation 2.128. 183 Efremov and Vakhrushev then tested the a p p l i c a b i l i t y of equations 2.126 - 2.128 for predicting the data of 0stergaard and Theisen [18], and reported a s a t i s f a c t o r y agreement. Michelsen and 0stergaard [14] , i n a l a t e r study of three-phase f l u i d i z a t i o n of 1, 3 and 6 mm glass beads i n a 6 inch diameter column, reported data on bed voidage and gas holdup for a wider range of operating variables (<j-^ > up to 26.0 cm/sec and < J 2 > UP to 15.0 cm/sec). These data have not been tested hitherto by any of the th e o r e t i c a l analyses. Therefore the generalized wake model derived i n Section 2.3, which assumes various possible s o l i d s contents of the wake with consequent c i r c u l a t i o n of so l i d s i n the pa r t i c u -late phase, w i l l now be compared with the data of Michelsen and 0stergaard [14] and the model of Efremov and Vakhrushev [16] . I t i s important to note at the outset that the generalized wake model with = 0.0 i s , i n essence, i d e n t i c a l to the model formulated by Efremov and Vakhrushev [16], d i f f e r i n g only i n the exact expressions used to calculate the r i s e v e l o c i t y of a bubble swarm and the wake f r a c t i o n i n a three-phase f l u i d i z e d bed. From a comparison of the models with respect to these two quantities i n the pre-ceding section, i t was found that (i) the r i s e v e l o c i t y of a bubble swarm calculated from the generalized wake model i s i n general smaller than predicted by the Efremov-Vakhrushev equations, and 184 ( i i ) the r a t i o of wake f r a c t i o n to gas f r a c t i o n pre-dicted by the generalized wake model with x^ . = 0 shows a scattered agreement with equation 2.128 of Efremov and Vakhrushev. The f u l l comparison i s summarized i n Table 4.5. Michelsen and 0stergaard [14], who used two indepen-dent techniques (residence time d i s t r i b u t i o n by tracer studies and pressure drop measurement) to measure the gas holdup i n a three-phase f l u i d i z e d bed, found the techniques to give r e s u l t s that d i f f e r e d widely, e s p e c i a l l y for beds of large p a r t i c l e s . Hence, since no agreement i s found for the predicted values of r i s e v e l o c i t i e s of bubble swarms from the generalized wake model with those from the Efremov-Vakhrushev equations, and since the measurements of gas hold-up by Michelsen and 0stergaard are uncertain, a comparison between measured and predicted values i s made for l i q u i d f r a c t i o n , e^, rather than for bed voidage, e(=£^+£2)• Pre-dictions by the generalized wake model are compared here with those by the Efremov-Vakhrushev equations, as well as with the experimental data of Michelsen and 0stergaard. Another special case of the generalized wake model i s r e a l i z e d by assuming the wake volume f r a c t i o n , £^ ., to be i n s i g n i f i c a n t and n e g l i g i b l e . In that case equation 2.106 si m p l i f i e s to 185 TABLE 4.5 DEGREE OF AGREEMENT BETWEEN EQUATIONS 4.7 AND 2.128 FOR PREDICTING RATIO OF WAKE FRACTION TO GAS FRACTION IN A THREE-PHASE FLUIDIZED BED \ P a r t i c l e \ Size Bed >v Expansion Large (d >3mm) P Medium (l<dp<3mm) Small (dp<lmm) High (e > 0.8) adequate agreement at a l l gas flow rates poor agreement at a l l gas flow rates worst agreement at a l l gas flow rates Medium (0 .6<e<0.8) excellent agreement at a l l gas flow rates adequate agreement up to j 2 - 10 cm/ sec poor agreement at a l l gas flow rates Low (0.4<e<0.6) favorable agreement at a l l gas flow rates favorable agreement at a l l gas flow rates adequate agreement up to j 2 - 10 cm/ sec 186 <J!> 1/n £ i£ - I m- 1 which, when substituted into equation 2.91, gives the bed voidage as <j,> 1/n e = (1-e"') [ ^-ir, ] + e"« (4.9) Equation 4.9 i s i d e n t i c a l to equation 2.. 62 derived i n Section 2.3.1 for the gas-free model. Thus for a three-phase f l u i d i z e d bed, i f e^ . i s small, the generalized wake model approaches the gas-free model. Before any comparisons could be attempted i t was necessary to f i n d a c o r r e l a t i o n which could s a t i s f a c t o r i l y represent the bed voidage for l i q u i d - s o l i d f l u i d i z a t i o n . As shown i n Table 4.6, the Neuzil-Hrdina c o r r e l a t i o n (equation 2.51)predicts the data much more s a t i s f a c t o r i l y than does the Richardson-Zaki c o r r e l a t i o n (equation 2.46), even though the data of Michelsen and 0stergaard are out-side the recommended range of a p p l i c a b i l i t y of the former. I t i s of i n t e r e s t to point out too that the bed expansion data for 3 and 6 mm glass beads (ROp > 1000) also agree well with the predictions of Trupp [87], thereby supporting the hypothesis that turbulence may a f f e c t the bed expansion behaviour of a l i q u i d - s o l i d f l u i d i z e d bed. The empirical c o r r e l a t i o n (equation 2.51) recommended by Neuzil and Hrdina 187 TABLE 4.6 COMPARISON OF MEASURED AND PREDICTED BED VOIDAGES FOR LIQUID-SOLID FLUIDIZATION P a r t i c l e L iquid (1) (2) (3) (4) Diameter Flux (mm) <J1> £ £ £ £ (cm/sec) 1.25 3.0 0 .575 0.487 0 .552 _ (p -2.67 a» ) 4.2 0.645 0.555 0.625 -cm 5.4 0.705 0 .612 0.685 — 6.6 0 .750 0.662 0.739 -7.8 0.810 0 .706 0 .786 -9.0 0.850 0.747 0 .829 -2.95 6.6 0 .570 0 .494 0 .579 0 .563 (p -2.45 S* ) 8.4 0.630 0.546 0.633 0 .626 cm 11.0 0 .700 0.611 0.699 0.704 14 .0 0.770 0 .676 0.764 0 .783 16.0 0.805 0.715 0 .803 0.830 5.93 10.0 0 .580 0 .495 0 .577 0 .547 (p =2.63 SE» ) 14.0 0 .646 0.570 0 .653 0 .634 cm 20 .0 0.776 0 .662 0 .746 0 .741 26 .0 0.833 0.739 0.822 0 .832 (1) Measurements by Michelsen and 0stergaard [14] (2) Equation 2.46 by Richardson and Zaki [2] (3) Equation 2.51 by Neuzil and Hrdina [47] (4) Dimensional c o r r e l a t i o n proposed by Trupp [87] 188 [47] w i l l , nevertheless, be used to describe the bed expan-sion c h a r a c t e r i s t i c s of a l l p a r t i c l e sizes i n the p a r t i c u l a t e phase of a three-phase f l u i d i z e d bed. Thus for calculations of e£* from the generalized wake model i t i s assumed that (a) the r e l a t i v e v e l o c i t y of bubble swarms i s represented by equation 2.107 (since a l l the data reported i n references 14 and 16 were obtained i n columns of L\>4 in) , and (b) the voidage i n the pa r t i c u l a t e phase i s represented by the Neuzil-Hrdina c o r r e l a t i o n (equation 2.51). I t i s worth noting that since the voidage i n the par t i c u l a t e phase, for x^ = 0.0, i s given by 'If < ^ 1 > " v2 ek X / 2 [ — — - ] (2.106-a) V ^ l - e ^ ) and since the l i q u i d f r a c t i o n i n a three-phase f l u i d i z e d bed i s given by e l = £ l f ( 1 _ £ 2 " e k ) + ek (2.91-a) then for determining e£' accurately, should be calculable accurately (as normally > e^, and both e 2 and e^ . are very much smaller than u n i t y ) , and for determining 189 accurately, the product v 2 £^ need be known accurately. An overestimate of would r e s u l t from an underestimate of v 2 (as i n 0stergaard's equations), while an underestimate of would conversely r e s u l t from an overestimate of v 2 (as i n the Efremov-Vakhrushev equations). The predicted values of l i q u i d f r a c t i o n i n a three-phase f l u i d i z e d bed calculated from the generalized wake model for x^ . = 0.0 and from the gas-free model for the larger p a r t i c l e s , are shown i n Figures 4.2-4.4, along with the data of Michelsen and 0stergaard [14] and the predicted values from the equations of Efremov and Vakhrushev [16] and of 0stergaard [8]. Some re s u l t s are also presented i n Tables 4.7 and 4.8. Beds of 6_ mm p a r t i c l e s (Figures 4.2a, b and Table 4.7) The data of Michelsen and 0stergaard show that the l i q u i d holdup reduces gradually as the gas v e l o c i t y i s i n -creased. The predictions from the model for <j^> = 20 cm/sec show excellent agreement with the data up to < J2> ~ 6 cm/sec, and then only an agreement i n trend for j 2 > 8 cm/sec, whereas the predictions from the Efremov-Vakhrushev equations show only an agreement i n trend with the data up to < J2> = 7 cm/sec, and no agreement for j 2 > 8 cm/sec. The predictions from 0stergaard's model are at best q u a l i t a t i v e up to <J2> = 3 cm/sec. The discrepancy between the predictions of the 190 0.8 0.4 0 FIGURE 4.2a 2 4 6 8 10 .12 14 16 < j 2 > , cm/sec LIQUID FRACTION DATA OF MICHELSEN AND 0 S T E R G A A R D [14] FOR 6 MM GLASS BEADS ( O - j =2 0.0; B -j]_= 10.0; generalized wake model with x^ = 0; Efremov-Vakhrushev equations 2.12 6 -2.128, for J i = 20.0; 0stergaard 1s ' equations, for j i = 20.0; gas-free model) 191 LU < 9 o > O U J DO 0 2 6 8 10 12 14 r <j 2> , cm/sec FIGURE 4.2b BED VOIDAGE DATA OF MICHELSEN AND 0STERGAARD [14] FOR 6 MM GLASS BEADS ( H -j1=20.0; O -Jl= 10.0; generalized wake model with x^=0; gas-free model) TABLE 4.7 COMPARISON OF MEASURED AND PREDICTED GAS AND LIQUID FRACTIONS IN THREE-PHASE FLUIDIZED BEDS d P = 5 .93 mm, P3 = 2. 63 gm/cc - <j1> = 20.0 cm/sec, D = 6.0 i n c h Gas F l u x , < j?> (cm/sec) M e a s u r e d (1) P r e d i c t e d (2) P r e d i c t e d (3) P r e d i c t e d (4) P r e d i c t e d (5) 1 e2 e k / E2 e'" 1 k / 2 e "' 1 e 2 £ k /£2 e "' 1 E2 c IM 1 e el E2 0.0 0.730 _ 0 . 662 _ _ 0.744 _ _ 0.662 _ 0 .741 0.741 1.0 0.720 0 .018 1. 827 0 . 644 0.011 1.282 0 .730 0.015 .12.95 0.615 0 .024 0 .73 4 0 .752 2.0 0.710 0.040 1. 729 0 . 625 0.018 1.135 0.716 0.030 8.80 0 .601 0.052 0.725 0 .765 3.0 0.698 0 .060 1. 622 0 . 607 0.025 1.000 0.705 0.043 7.01 0.592 0 .082 0 .716 0 .776 4.0 0 .685 0.080 1. 507 0 . 590 0 .030 0.875 0.695 0.056 5.96 0 .587 0 .113 0 .707 0.787 5.0 0.673 0.100 1. 386 0 . 576 0.036 0.758 0.687 0.067 5.25 0.588 0 .146 0.699 0 .799 6.0 0.662 0.120 1. 262 0 . 564 0.041 0.650 0.680 0 .078 - - - 0.690 0 .810 7.0 0.650 0.138 1. 136 0 . 556 ,0.045 0.547 0.675 0.088 - - - 0.682 0 .820 8,0 0.639 0 .156 1. 012 0 . 551 0 .050 0.450 0.672 0.098 - - - 0.674 0.830 9.0 0.628 0 .17 2 0 . 893 0. 550 0.054 0.422 0 .665 0.107 - - - 0.667 0.839 10.0 0 .617 0.190 0 . 779 0 . 551 0.058 0.416 0 .65.8 0.115 - - - 0.659 0.849 11.0 0.604 0 .200 0 . 674 0 . 554 0.062 0.409 0 .651 0.123 - - - 0.654 0 .854 12.0 - 0 .214 0 . 578 0 . 558 0.066 0.403 0.644 0.131 - - 0.648 0.862 13.0 - - 0 . 492 0 . 563 0.070 0.397 0 .637 0.138 - - - - -14.0 - - 0 . 416 0 . 568 0.073 0.391 0.630 0.145 - - - - -15.0 - - 0 . 349 0 . 574 0.077 0 .386 0.624 0 .151 - - - -(1) Measurements by M i c h e l s e n and 0stergaard [14] (2) From e q u a t i o n s 2.126 and 2.128, Ef r e m o v and V a k h r u s h e v [16] (3) From g e n e r a l i z e d wake model (4) From e q u a t i o n s 1.10 - 1.13, 0stergaard [8] (5) From g a s - f r e e m o d e l , e q u a t i o n 4.11, u s i n g t h e v a l u e s o f e'l1 measured by M i c h e l s e n and 0stergaard H vo to 193 generalized wake model and the Efremov-Vakhrushev equations a r i s e s , i n part, from overestimation of v 2 (due to under-estimation of £ 2) ky t n e l a t t e r , the estimates of e^ . by these two methods being i n reasonable agreement with each other. However, i f i t i s assumed that the r o l e of bubble wakes i s i n s i g n i f i c a n t , the gas-free model can be used for c a l c u l a t i n g the l i q u i d f r a c t i o n by substituting Trupp 1s dimensional equation for into equation 2.60: <j,> 1/2.28 e'" = (l-e™) [ ^ r p ] (4.11) X Z 0.36 V„ (1-e*) and the bed voidage from e = ej' + e'2" (1.3) The l i q u i d fractions calculated from equation 4.11, using the gas holdup data as reported by Michelsen and 0stergaard [14], are presented i n Table 4.7 and also shown i n Figure 4.2a. These valuesoof l i q u i d f r a c t i o n appear to be i n as good agreement with the experimental values as those calcu-lated from the generalized wake model. The values of bed voidage calculated from the gas-free model (equations 4.11 and 2.91) are presented i n Figure 4.2b and also show only small percentage deviations from the experimental values [14]. 194 Thus for f l u i d i z e d beds of 6 mm p a r t i c l e s the ro l e of bubble wakes appears to be i n s i g n i f i c a n t and the bed voidage can be well represented by the simple gas-free model. Beds of 3 mm p a r t i c l e s (Figures 4.3a,b) The data of Michelsen and 0stergaard [14] for l i q u i d f r a c t i o n i n three-phase f l u i d i z e d beds show excellent agreement with the predicted values from the generalized wake model, p a r t i c u l a r l y i f the parameter x^ . i s suitably adjusted. On the other hand, the o v e r a l l void f r a c t i o n , e s p e c i a l l y at < j ^ > = 14.0 cm/sec, shows poor agreement with the predictions from the model. This discrepancy i s caused by the predicted gas holdup being considerably smaller at the higher l i q u i d flux and considerably larger at the lower l i q u i d flux than the measured values reported by Michelsen and 0stergaard. They observed the bubble behaviour for l i q u i d flux below 7 cm/sec to be markedly d i f f e r e n t than that for higher l i q u i d fluxes. Since i n c a l c u l a t i n g the l i q u i d f r a c t i o n from the generalized wake model i t was assumed that the r e l a t i v e v e l o c i t y for a l l gas fluxes i s well represented by equation 2.107 with r g =6 mm [79] r the discrepancy from the measured values i s not e n t i r e l y unexpected. To improve the predictions i t would be necessary to obtain a rel a t i o n s h i p for describing the average diameter of bubbles i n the swarm as a function of 195 0.8 0.41 . . _ i _ . « f . I 0 2 4 6 8 10 12 14 16 <j2> , cm /sec FIGURE 4.3a LIQUID FRACTION DATA OF MICHELSEN AND 0STERGAARD ' [14] FOR 3 MM GLASS BEADS ( p-j 1=14.0; 8-^=6.6 generalized wake model; —; gasrfree model) 196 uJ 0.7 < 9 o > Q UJ 00 0.5 X, = 0.6 0 - : ° ^ o o - o o h: 0 xk=o <L> > cm/sec 8 10 12 14 16 FIGURE 4.3b BED VOIDAGE DATA OF MICHELSEN AND 0STERGAARD [14] FOR 3 MM GLASS BEADS ( O-Ji=14.0; H-j-^6.6; ' generalized wake model; gas-free model) 197 gas and l i q u i d fluxes and p a r t i c l e properties. However, i f i t i s assumed again that the ro l e of bubble wakes i s i n s i g n i f i c a n t , the gas-free model can be used for describing the bed behaviour. Thus for the l i q u i d flow rate of 14.0 cm/sec, the l i q u i d fractions calculated from equation 4.11, using the measured values of gas holdup [14], are shown i n Figure 4.3a. These values are found to be i n good agreement with the experimental values up to a gas flow rate of 5.0 cm/sec, the agreement becoming poorer for larger gas flow rates. The values of bed voidage, calculated from equation 2.91 and presented i n Figure 4.3b, exhib i t a si m i l a r range of agreement with the measured values. However, for the l i q u i d flow rate of 6.6 cm/sec, the agreement of calculated values of l i q u i d f r a c t i o n and bed voidage with the corresponding experimental values was poor at almost a l l gas flow rates. Therefore for f l u i d i z e d beds of 3 mm glass beads, the ro l e of bubble wakes cannot be considered t o t a l l y i n s i g n i f i c a n t . Nevertheless, the bed voidage predicted by the gas-free model can be used as a f i r s t approximation. Beds of 1 mm p a r t i c l e s (Table 4.8 and Figure 4.4) The data of Michelsen and 0stergaard show a gradual reduction i n l i q u i d f r a c t i o n as the gas flux through the bed i s increased. For a volumetric l i q u i d f l u x of 7.8 cm/sec, T A B L E 4 . 8 COMPARISON OF MEASURED AND PREDICTED GAS AND L IQUID FRACTIONS FOR T H R E E - P H A S E F L U I D I Z E D BEDS d P = 1 .25 mm , P 3 = 2 . 67 g m / c c , L i q u i d F l u x , <J1> = 7 . 8 cm/sec, D = 6 .0 i n c h Gas F l u x , ( c m / s e c ) M e a s u r e d (1) P r e d i c t e d (2) P r e d i c t e d (3) P r e d i c t e d (4) e 1 " 1 P in 2 k / 2 1 P HI 2 e / e ' " k / e 2 k / E 2 e'" E l £"• 2 0 . 0 0 .780 - - 0 .728 - - 0 .786 _ _ 0 . 706 _ 1 .0 0 . 7 1 0 0 . 0 2 7 2 . 2 2 5 0 .667 0 .021 1 .298 0 .7 39 0 .021 4 .974 0 .610 0 . 0 3 4 2 . 0 0 .680 0 . 0 4 5 1 .539 0 . 6 3 5 0 .034 1 .026 0 .706 .040 3 .322 0 .570 0 .076 3 . 0 0 .663 0 .060 0 . 8 6 9 0 .644 0 .045 0 .826 0 .683 .057 2 . 6 1 5 0 .537 0 .123 4 . 0 0 . 6 5 0 0 .070 0 .416 0 .667 0 .054 0 .667 0 . 6 6 9 .072 2 .202 0 .506 0 . 1 7 3 5 . 0 0 . 6 4 0 0 . 0 7 9 0 . 1 8 1 0 .684 0 .063 0 . 5 3 2 0 .662 .087 1 .926 0 . 4 7 7 0 .226 6 . 0 0 .630 0 .088 0 . 0 7 5 . 0 ;693 0 .072 0 .411 0 .661 .101 - - -7 . 0 0 .625 0 .094 0 . 0 3 0 0 .696 0 .080 0 . 3 9 3 0 .645 .112 - - -8 . 0 0 . 6 2 0 0 .100 0 . 0 1 2 0 .700 0 .087 0 .377 0 .630 .122 - - -9 . 0 0 .615 0 .107 0 .005 0 .697 0 .094 0 .362 0 .616 .132 - - -1 0 . 0 0 . 6 1 2 0 . 1 1 2 0 .002 0 . 6 9 5 0 .101 0 .348 0 .602 .140 - -1 1 . 0 0 .609 0 .118 0 . 001 0 .693 0 .108 0 . 3 3 5 • 0 .590 .148 - -1 2 . 0 0 .608 0 .124 0 . 0 0 0 0 .690 0 .114 0 .324 0 .577 .154 - - -13 .0 0 .605 0 .128 0 .000 0 .688 0 .120 0 . 3 1 5 0 .564 .160 - - •-1 4 . 0 0 .600 - 0 .000 0 .686 0 .126 0 .307 0 .552 .166 - - -1 5 . 0 - - 0 .000 0 .684 0 .132 0 .298 0 .540 .171 - - -(1) M e a s u r e m e n t s b y M i c h e l s e n and 0stergaard [14] (2) P r e d i c t e d v a l u e s f r o m c o r r e l a t i o n s ( e q u a t i o n s 2 .126 - 2 .128) b y E f r e m o v a n d V a k h r u s h e v [16] (3) P r e d i c t e d v a l u e s f r o m g e n e r a l i z e d wake m o d e l w i t h x^ = 0 (4) P r e d i c t e d v a l u e s f r o m wake m o d e l b y 0stergaard [8] 12 14 16 < j > , cm/sec FIGURE 4.4 LIQUID FRACTION DATA OF MICHELSEN AND 0STERGAARD [14] FOR 1 MM GLASS BEADS ( a-j-^3.0; 0-J 1=7.8 generalized wake model with xj^O; Efremov-Vakhrushev equations, 2.126-2.128, for j-j=7.8; 0stergaard's equation, for j-^=7.8) 200 there i s considerable disagreement between the predicted values from both the generalized wake model and the Efremov-Vakhrushev equations, and the measured values, for j > 9 cm/sec. However the predictions from the model are in good quantitative agreement with the measured values up to < j 2 > = 6.0 cm/sec, whereas the predictions from the equations of Efremov and Vakhrushev show better q u a l i t a t i v e agreement with the measured values for J2 > 8 cm/sec. As can be noted from Table 4.5, for highly expanded beds of 1 mm p a r t i c l e s , the wake fractions estimated by the model show poor agreement with the values calculated from equation 2.128, the wake f r a c t i o n predicted by the model being smaller at lower gas fluxes and considerably larger at higher gas fluxes ( J2 > 6 cm/sec). This inequality of wake fractions overshadows the inequality of bubble r i s e v e l o c i t i e s i n t h i s case and i s probably the p r i n c i p a l source of discrepancy between the two predictions. I t therefore appears that the generalized wake model overestimates the wake f r a c t i o n whereas equation 2.128 underestimates i t for the larger gas v e l o c i t i e s . For the smaller l i q u i d flow rate of 3 cm/sec, the predictions from the model are i n excellent agreement with the measured values [14] up to <J2> = 9 cm/sec, where, due to the apparent overestimation of wake f r a c t i o n by the model, the predictions from i t s t a r t to deviate from the measured values. Since the gas holdups predicted by the model were 201 also found to be i n excellent agreement with the measured values up to < J 2 > = 9 cm/sec, i t can be stated that the wake fractions predicted by the model are at lea s t as r e a l i s t i c as those given by equation 2.128, with which they were found to be i n reasonable agreement (Table 4.5). Since the generalized wake model developed i n Section 2.3 w i l l subsequently be used to analyse the data obtained i n the 2 inch diameter column of t h i s study, i t s predictions are now compared with the limited data of 0stergaard and Theisen [18] for 2 mm glass beads i n a 2 inch diameter column and with the predictions of the Efremov-Vakhrushev equations, good agreement of the l a t t e r equations with the 0stergaard-Theisen data having been reported [16]. For c a l c u l a t i n g the bed voidage from the generalized wake model i t i s assumed that (a) the r e l a t i v e v e l o c i t y i s represented by equation 2.108a, the l o c a l d r i f t v e l o c i t y being obtained from equation 4.8 for the slug flow regime, and that (b) the voidage i n the pa r t i c u l a t e phase i s represented by the Neuzil-Hrdina c o r r e l a t i o n , equation 2.51. The bed voidages calculated from the generalized wake model for 2 mm glass beads f l u i d i z e d by a cocurrent stream of a i r and water are shown i n Figure 4.5a, along with the data of 0stergaard and Theisen [18] and the predicted values from the Efremov-Vakhrushev equations. For a volumetric l i q u i d flux of 11.0 cm/sec, the predicted values of bed 202 Q U J m 0.5 4 6 8 10 < i > , cm/sec FIGURE 4.5a BED VOIDAGE DATA OF 0STERGAARD AND THEISEN [18] FOR 2 MM GLASS BEADS ( A - J1 =3 . 3 8 ; 8-0 2 = 6.17; o-j-j_=11.0; — generalized wake model; Efremov-Vakhrushev equations, 2.126-2.128, for j-^11.0) 203 i l l 4 6 8 <j2X cm/sec 10 FIGURE 4.5b COMPARISON OF ^1 BY GENERALIZED WAKE MODEL WITH e"' BY EFREMOV-VAKHRUSHEV EQUATIONS FOR 2 MM GLASS BEADS AT <Ji>=11.0 cm/sec ( general ized wake model with xk=0; —• Ef remov-Vakhrushev equations, 2.126-2.128) T A B L E 4.9 COMPARISON OF MEASURED AND PREDICTED GAS AND L IQUID FRACTIONS FOR T H R E E - P H A S E F L U I D I Z E D BEDS d = 2 .0 mm, = 2 .88 g m / c c , <j-,> = 1 1 . 0 c m / s e c , D = 2 .0 i n c h G a s F l u x M e a s u r e d (1) P r e d i c t e d (2) P r e d i c t e d (3) P r e d i c t e d (4) < ^ 2 > ( c m / s e c ) V e 2 E l e" 2 V e 2 e" 1 e" 2 e" 1 e " 2 0 . 0 0 . 7 6 - 0 .708 - - 0 .744 - - 0 .708 -170 0 . 7 4 4 1 .306 0 .685 0 .014 1 .197 0 .717 0 .019 6 .73 0 . 6 3 3 0 .031 2 . 0 0 .740 1 .208 0 .664 0 .024 0 . 9 9 5 0 . 6 9 5 0 .038 4 .52 0 .600 0 .068 3 .0 0 .738 1 .104 0 . 6 4 4 . 0 . 0 3 2 0 .820 0 .679 0 .056 3 .57 0 . 5 7 2 0 . 1 0 9 4 .0 0 . 7 3 7 0 . 9 9 7 0 . 6 2 8 0 . 0 3 9 0 . 661 0 .667 0 . 0 7 3 3 .02 0 . 5 4 6 0 . 1 5 2 5 .0 - 0 .'888 0 . 6 1 6 0 .046 0 .509 0 .661 0 . 0 8 9 2 . 6 5 0 . 5 2 2 0 . 1 9 8 0 6 . 0 - 0.7.81 0 . 6 0 7 0 . 0 5 2 0 . 4 0 5 0 .656 0 .106 • - - ' -7 .0 0 .678 0 . 6 0 2 0 .058 0 . 3 9 5 0 .642 0 .121 - - -8 .0 0 . 5 8 2 0 .601 0 . 0 6 3 0 .385 0 .630 0 .136 - ' - -9 .0 - 0 . 4 9 4 0 . 6 0 2 0 .069 0 .375 0 .618 0 .150 - - -(1) M e a s u r e m e n t s b y 0stergaard and T h e i s e n [18] (2) P r e d i c t e d v a l u e s f r o m e q u a t i o n s 2 . 1 2 6 - 2 .128 o f E f r e m o v and V a k h r u s h e v [16] (3) P r e d i c t e d v a l u e s f r o m g e n e r a l i z e d wake m o d e l (4) P r e d i c t e d v a l u e s f r o m wake m o d e l o f ( J fs te rgaard [8] to o 205 voidage from the generalized wake model are found to be i n excellent agreement with the measured values, whereas the predicted values from the equations of Efremov and Vakhrushev ex h i b i t only a q u a l i t a t i v e agreement even though a quanti-t a t i v e agreement was claimed i n t h e i r publication [16]. Since the r a t i o s of wake f r a c t i o n to gas f r a c t i o n predicted by these two methods are i n r e l a t i v e l y good agreement with each other (Table 4.9), the discrepancy i n voidage between them shown i n Figure 4.5a can be ascribed to the differences i n the respective values of gas holdup predicted. This i s further i l l u s t r a t e d i n Figure 4.5b, where the values of l i q u i d f r a c t i o n i n a three-phase f l u i d i z e d bed of 2 mm glass beads predicted by the two methods are compared d i r e c t l y (no estimate of l i q u i d f r a c t i o n could be obtained from the data of 0stergaard and Theisen, since the gas holdup inside the bed was not measured). The respective values show adequate agreement with each other, although the values predicted by the Efremov-Vakhrushev equations are consis-ten t l y lower than those by the generalized wake model. The values for the bed voidage of 2 mm glass beads predicted by the generalized wake model at other l i q u i d flow rates are also found to be i n excellent agreement with the measured values, as shown i n Figure 4.5a. Thus the above comparison of predictions from the Efremov-Vakhrushev equations and with the measurements of Michelsen and 0stergaard shows that: (1) The agreement of the simple equation proposed here for the wake f r a c t i o n i n a three-phase f l u i d i z e d bed with equation 2.128 (cf. Table 4.5) suggests the former to be a r e a l i s t i c approach.to c o r r e l a t i n g wake fra c t i o n s which warrants further investigation. Equation 4.7 with p = 3 has been corroborated, at lea s t as a f i r s t approximation. (2) Since the reported values of gas holdup i n three-phase f l u i d i z e d beds exhi b i t a wide v a r i a t i o n for si m i l a r conditions, the authenticity of the model proposed here for the r i s e v e l o c i t y of a bubble swarm cannot be ascertained from the l i t e r a t u r e data. Nevertheless, since the proposed model i s based on the available i n -formation concerning the f l u i d dynamics of multi-phase flow, i t provides a better and simpler method for analysis of experimental data than hitherto a v a i l a b l e . (3) The reasonable agreement of the predicted values of l i q u i d f r a c t i o n from the generalized wake model with both the measured values [14] and the predicted values from the Efremov-Vakhrushev equations [16] ascertains the basic correctness of the present fluid-dynamic description of a three-phase f l u i d i z e d bed. In most instances, x^ . = 0.0 (equivalent to no p a r t i c l e s i n the wake, as o r i g i n a l l y postulated by Stewart and Davidson [7]) gives adequate agreement with the reported data. However, for beds of large p a r t i c l e s , for example 6 mm 207 glass beads, the voidage i s found to be as well represented by the simple gas-free model. What d i s -crepancies were observed between the predicted and the measured values of bed voidages could be due p a r t l y to the inaccuracies and uncertainties i n the measure-ment of gas holdup, thus necessitating the development of better techniques for the measurement of l o c a l and o v e r a l l gas holdup. This task was undertaken i n the present inve s t i g a t i o n . 208 4.2 Discussion of experimental re s u l t s and comparison  with t h e o r e t i c a l predictions 4.2.1 Evaluation of experimental techniques The t y p i c a l experimental errors involved i n various measured quantities are estimated i n Appendix 8.9. The two techniques used for measuring the gas holdup i n gas-liquid flow, v i z . the pressure drop gradient method and the valve shut-off technique, gave s a t i s f a c t o r y r e s u l t s . Thus, for measuring gas holdups greater than 0.1, either of the two techniques could be used with equal accuracy. However, for measurements concerning the l o c a l structure i n gas - l i q u i d flow other suitable methods have to be adopted [115]. One such method, the e l e c t r o - r e s i s t i v i t y probe, was developed and used successfully i n t h i s study and w i l l be discussed l a t e r i n d e t a i l . For measuring the gas holdups i n three-phase f l u i d i z e d beds, the f i r s t two methods above were found to be somewhat more erroneous than for gas - l i q u i d flow, es p e c i a l l y for measurements of small gas holdups (e^1 < 0.1), but were nevertheless considered adequate for present purposes. A complete and more accurate technique of measurement i s needed for better understanding of the l o c a l structure of gas flow i n three-phase f l u i d i z e d beds. The e l e c t r o - r e s i s t i v i t y probe developed and tested i n t h i s work was used for measuring l o c a l quantities, but the processing of the probe output with the available e l e c t r o n i c equipment, es p e c i a l l y for measuring the l o c a l gas f r a c t i o n , was cumber-209 some and subject to errors; other l o c a l q u a n t i t i e s ; e.g. bubble frequency and f i l m thickness at the w a l l , could be measured quite accurately. I t i s therefore not yet possible to c r i t i c a l l y appraise the s u i t a b i l i t y of the e l e c t r o - r e s i s t i v i t y probe for measurements concerning the l o c a l structure of gas flow i n three-phase f l u i d i z e d beds. The measurement of so l i d s holdup i n l i q u i d - s o l i d f l u i d i z e d beds was straightforward, and either of the two methods, v i z . by the expanded bed height and by the slope of the longitudinal pressure drop p r o f i l e , could be used with equal accuracy. However, for determining the s o l i d s holdup i n three-phase f l u i d i z e d beds, the knowledge of, as well the manner of defining, the expanded bed height was c r i t i c a l . The method for measuring the bed height developed and used i n t h i s work not only provided accurate and reproducible measurements, but also a meaningful d e f i n i t i o n . 4.2.2 Gas holdup re s u l t s 4.2.2.1 Gas holdup i n gas-liquid flow The purpose of obtaining gas holdup data i n cocurrent gas-liquid flow was two-fold: 1. To have r e l i a b l e data avai l a b l e for comparing l a t e r with gas holdup measurements i n three-phase f l u i d i z e d beds under si m i l a r flow conditions. Thus e f f o r t s were 210 made to obtain gas holdup data for a l l possible combin-ations of gas and l i q u i d flow rates that were used l a t e r i n the three-phase f l u i d i z a t i o n studies. 2. To check the a p p l i c a b i l i t y of the two-phase gas holdup model (equation 2.37 for bubbly flow and equation 2.39 for slug flow) to the 2 0 mm and 2 inch column data obtained i n t h i s study. The data from the 20 mm and 2 inch i . d . columns, respec-t i v e l y , are presented and discussed separately. The graphs and tables i n t h i s section are obtained from data presented in Appendix 8.7. (A) 20 mm glass column The v i s u a l observation of the column revealed that slug flow prevailed at a l l the gas and l i q u i d flow rates studied. The slugs appeared to t r a v e l independently of each other as no coalescence could be detected. The simplest scheme for analyzing the data i s to use equation 4.1 i n combination with equation 2.37 i n the bubbly flow regime and with equation 2.39 i n the slug flow regime. Thus, combining equations 4.1 and 2.39 for the slug flow regime, v 2 = C. <j> + 0.35 /gD (4.12) Then assuming C n to be 1.2 [19, 39], equation 4.12 for the 211 2 0 nun glass column becomes < j 2 > v„ = — — = 1.2 <j> + 15.50 (4.13) ^ £2 The predictions from equation 4.13 are compared with experimental data i n Figure 4.6, and e x h i b i t s u f f i c i e n t agreement with the data to j u s t i f y the assumed value for CQ. Thus the model for the slug flow regime, represented by equation 4.13, can be successfully used to predict the gas holdup for air-water flow i n 2 0 mm columns. (B) 2 inch perspex column The gas holdups i n the 2 inch i . d . perspex column were measured both by the pressure drop gradient method and by the valve shut-off technique, and supplemented i n part by the e l e c t r o - r e s i s t i v i t y probe. Data were obtained for a bubble column (<j^> = 0.0) as well as for cocurrent gas- l i q u i d flow. (i) Bubble column The flow regime i n bubble columns at low gas flow rates ( j 2 < 3 cm/sec) was mainly bubbly with l i t t l e or no evidence of coalescence. However, bubble clu s t e r s began to appear at a gas flow rate of about 5 cm/sec, a f t e r which coalescence increased progressively. F u l l y developed slugs were not observed u n t i l the gas flow rate was 7.8 cm/sec, but bubble conglomerates leading to slugs near the top of 212 FIGURE 4.6 COMPARISON OF AVERAGE BUBBLE RISE VELOCITIES PREDICTED BY EQUATION 4.13 WITH EXPERIMENTAL DATA IN 20 MM GLASS COLUMN i. 213 the test section were noticeable. As discussed i n Appendix 8.2, the hydrostatic pressure on a bubble decreases continuously during i t s ascent. This change i n hydrostatic head would cause the bubble size to change, thereby a f f e c t i n g the bubble r i s e v e l o c i t y , and both e f f e c t s together could change the gas holdup along the column. Yet for s i m p l i f i c a t i o n i t was assumed i n Appendix 8.2 that gas holdup along the column axis remained constant. The gas holdup at a given v e r t i c a l l e v e l of the experimental section could be calculated from the longitud-i n a l pressure drop p r o f i l e along the column w a l l . Figure 4.7 shows such longitudinal gas holdup p r o f i l e s , calculated from unsmoothed data. I t i s seen i n Figure 4.7 that the gas holdup for a l l gas flow rates greater than 2 cm/sec increases along the column axis, the increase, however, being almost i n s i g n i f i c a n t for a l l but the highest gas flow rates. Therefore i n each case an average of a l l the gas holdups was taken, and t h i s value was assumed to e x i s t at the mid-point of the column. The gas holdup was then corrected to a pressure of 760 mm of mercury. The gas hold-ups from the valve shut-off technique were s i m i l a r l y corrected to a pressure of 760 mm of mercury and are presented i n Appendix 8.7. As shown e a r l i e r , the simplest and most comprehensive scheme for analyzing the data i s provided by 214 T 1 r I i i 1 « -0 8 16 24 32 DISTANCE FROM T A P l , Z , i n . FIGURE 4.7 AXIAL VARIATION OF GAS HOLDUP IN BUBBLE COLUMN (<j1>=0.0) 215 < j < a 0 v 0 • > The second term on the right-hand side of equation 4.1 represents the "weighted mean d r i f t v e l o c i t y " of the two-phase mixture. Several competing models for the d r i f t v e l o c i t y have been proposed [1, 39, 75, 76], but i n general —Cjj- = f<1<V> (V~>B ( 4 ' 1 4 ) where f (<a2>) i s normally a monotonic function of <ot2> that approaches unity as <o&2> approaches zero. Then combining equations 4.1 and 4.14, we get ~ = C 0 « j 1 + j 2 » + f ( < a 2 » ( V j B (4.15) Now d i f f e r e n t i a t i n g equation 4.15 with respect to <a2>, the slope of the < J 2 > versus <o'2> curve at the o r i g i n , that i s as < J 2 > a n < ^ hence  <^2 > approaches zero, would be given by d<j 2> = Cn <j1> + ( V jn (4.16) <jj>- 0 and since for bubble columns <j^>=0, equation 4.16 s i m p l i f i e s to 216 d<j2> (4.17) d<a~> 2 <j 2> = 0 which was also derived by Bhaga [1] i n a d i f f e r e n t manner. The measured gas holdups for the bubble column are shown on the < J 2 > ~ e2 P^-ane Figure 4.8a, and from the slope of the curve at the o r i g i n , the r i s e v e l o c i t y of a bubble i s found to be ( V ^ B = 2 4.7, which corresponds from equation 2.10 to a bubble diameter of 9.4 mm. The observed bubble size i n the bubble column was between 5 and 10 mm. Zuber and Findlay [39] reported that for both churn-turbulent bubbly flow and slug flow, the weighted mean d r i f t v e l o c i t i e s were constant and given by equations 4.3 and 4.2 respectively. The above r i s e v e l o c i t y of 24.7 cm/sec compares favorably with the value of 24.91 cm/sec from equation 4.3 and 24.71 from equation 4.2. Since for the 2 inch i . d . column the d r i f t v e l o c i t i e s for the churn-turbulent bubbly flow and the slug flow regimes are p r a c t i c a l l y the same, equation 4.1 with C n = 1.2 [39] becomes > = 1.2 (<j 1+j 2 >) + 24.7 (4.18) v 2 Equation 4.18 i s equivalent to equation 4.4 proposed by N i c k l i n [19] from a t o t a l l y d i f f e r e n t approach. Now, since for bubble columns < j j _ > = 0, equation 4.18 further s i m p l i f i e s to 217 218 < j 2 > v 2 = = 1.2 <j 2> + 24.7 (4.19) The gas holdups predicted by equation 4.19 are compared i n Figure 4.8b with the experimental data and exhibit good agreement. Also shown i n Figure 4.8b i s the curve repre-senting the data obtained by E l l i s and Jones [60] i n a 2 inch i . d . column. The agreement of t h i s curve with the present data i s strong support for the data, whereas the curve representing the Efremov-Vakhrushev c o r r e l a t i o n [66] undoubtedly underestimates the gas holdup. The present model for the bubbly flow regime (represented by equation 4.20, which i s based on equation 2.37) and that recommended by Bhaga [1], although i n good agreement with each other, considerably overestimate the gas holdups at the gas v e l o c i t i e s of the present experiments. I t can then be concluded that the model for the slug flow regime, represented by equation 4.19, can be successfully used to predict the gas holdup i n 2 inch diameter bubble columns. ( i i ) Cocurrent flow The flow regimes encountered i n cocurrent gas^-liquid flow were bubbly and slug flow. The gas flow rates when the slugs were f i r s t observed i n the column for air-water flow are presented i n Table 4.10 for the various l i q u i d flow rates studied. I t can generally be stated that i n the present 2 inch diameter column, the slug flow regime occurred 219 CLUSTERS DEVELOP 0 4 6 8 10 < j2> , cm/sec 12 14 FIGURE 4.8b COMPARISON OF MEASURED AND PREDICTED GAS HOLDUPS IN 2 INCH BUBBLE COLUMNS ( — Bhaga [1] , E l l i s and Jones [60] , — Efremov and Vakhrushev [66]) 220 TABLE 4.10 TRANSITION FROM BUBBLY TO SLUG FLOW IN AIR-WATER FLOW Liquid Flux ^ 1 * (cm/sec) Gas Flux,  <J2 >' a t Transition From v i s u a l Observation From Equation 2.21 0.0 4.3 - 5.1 3.1 1.25 4.8 - 5.3 3.3 6.25 4.4 - 4.9 4.3 7.00 4.0 - 4.5 4.5 7.65 4.3 - 5.0 4.6 17.75 4.3 - 5.4 6.6 221 for a l l gas flow rates greater than 5 cm/sec i r r e s p e c t i v e of the l i q u i d flow rate, as was also noted by Reith et a l . [68]. Also presented i n Table 4.10 are the gas flow rates predicted for t r a n s i t i o n from bubbly to slug flow by equation 2.21 [60]. These correspond only roughly to the observed t r a n s i t i o n points. In air-polyethylene g l y c o l solution flow, the slug flow regime was encountered at a l l the gas and l i q u i d flow rates studied and therefore no data for the t r a n s i t i o n point were recorded. However i t was observed that at low l i q u i d flow rates coalescence was quite prominent, whereas at high l i q u i d flow rates the large spherical capped bubbles rose at regular i n t e r v a l s without much coalescence. The flow of air-water mixture at very low water flow rates d i f f e r e d v i s i b l y from that at a l l other water flow rates studied. Thus for a water rate of 1.25 cm/sec, the air-water flow was predominantly bubbly for a l l the gas flow rates studied, although for gas rates greater than 5 cm/sec coalescence i n the system became so much more active that bubble conglomerates were observed to r i s e through the column with a r o l l i n g action. No f u l l y developed slugs were observed even at the highest gas flow rate studied (<j2> = 8.44 cm/sec), but the presence of bubble conglomer-ates was taken to mean a change i n flow regime and the corresponding gas flow rate was recorded as a t r a n s i t i o n point i n Table 4.10. For the next higher l i q u i d flow rate 222 (<j^> = 1.87 cm/sec) sim i l a r bubble c l u s t e r s were observed at gas flow rates higher than 5 cm/sec, but these bubble clusters coalesced to form a well defined slug. At a l l other l i q u i d flow rates the upward r o l l i n g action of bubble clus t e r s was not observed and the slugs encountered for gas flow rates higher than 5 cm/sec were c l e a r l y defined. The a x i a l d i s t r i b u t i o n of gas holdups, calculated from the longitudinal pressure drop p r o f i l e s along the column wall, also revealed a d i f f e r e n t pattern for low l i q u i d flow rates. The gas holdup d i s t r i b u t i o n s for a water flow rate of 1.25 cm/sec are shown i n Figure 4.9(in which the l i n e s are drawn for making any possible trend i n the data perceptible, but have no other s i g n i f i c a n c e ) . I t i s seen.that the gas holdup for the gas flow rate of 3.81 cm/ sec remained p r a c t i c a l l y constant, whereas the gas holdup for the gas flow rate of 5.31 cm/sec decreased up the column, in d i c a t i n g that the bubble swarm was accelerating. For higher gas flow rates the reduction i n gas holdup with distance was even more pronounced. For the next higher l i q u i d flow rate of 1.87 cm/sec, Figure 4.10 shows the longitudinal gas holdup d i s t r i b u t i o n to have remained p r a c t i c a l l y constant throughout the experimental section except for the highest gas flow rate of 8.44 cm/sec, for which the gas holdup up the column once again showed a marked reduction. For the l i q u i d flow rate of 4.55 cm/sec, 223 CO CL Z > Q _ J O X 00 < 0.32 0.28 0.30 0.26 0.22 0.22 0.18 0.15 0.13 X T <j2> = 8.44 X L O J <j?> = 6.88 <L> =5.31 — 8 w <L> =3.81 12 20 28 36 DISTANCE FROM TAPl.Z.in FIGURE 4.9 AXIAL DISTRIBUTION OF GAS HOLDUP IN AIR-WATER FLOW AT <j 1> = 1.25 cm/sec 224 1 1 1 '<ja>=8,44 0.30 V o -0.26 - o o o ° ° c^ O.22 w CL o 8 Q — o o° o° • <j2> = 6.88_ O ° X c/) ^ , ^  < 0.18 o ) 0 o o ° o o o o — — 0.22 _o 8 o o 8 8 o o <j2>=e5.3l_ 0.18 ) o 1 o 1 o o 1 °o 1 1 5 10 15 20 25 30 35 40 DISTANCE FROM TAP 1,Z, in FIGURE 4.10 AXIAL DISTRIBUTION OF GAS HOLDUP IN AIR-WATER FLOW AT <j,> = 1.87 cm/sec 225 the longitudinal d i s t r i b u t i o n of gas holdup, shown i n Figure 4.11, remained constant for a l l gas flow rates used. Similar behaviour was found for a l l the other l i q u i d flow rates studied. For the air-polyethylene g l y c o l solution flow, the a x i a l d i s t r i b u t i o n of gas holdup i s shown i n Figure 4.12. At low l i q u i d flow rates i t d i f f e r e d s u b s t a n t i a l l y from that of air-water flow. In the case of the two lower l i q u i d flow rates, the gas holdup increased markedly up to 20 inches above Tap 1, but above that l e v e l , the change i n gas holdup was almost i n s i g n i f i c a n t . For two higher l i q u i d flow rates, however, the gas holdup remained p r a c t i c a l l y constant throughout the entire experimental section, as for air-water flow. The gas holdup was measured by two d i f f e r e n t techniques in each of the three d i f f e r e n t sections of the column. As discussed i n Appendix 8.2, the r e s u l t s could be misinter-preted i f the pressure at the location where the gas holdup was measured were not duly considered. This i s i l l u s t r a t e d i n Table 4.11, where the gas holdup data for a few t y p i c a l runs are presented both before and after the pressure corrections were applied. As shown e a r l i e r , no pressure correction was deemed necessary for gas holdup measurements above the t e s t section and these are therefore presented as measured. The gas holdups measured by the two techniques, i n both the t e s t section and the section below i t , were 0.26 CM * 0.22 Q_ 0.21 Q _ J O £ o.i 0.09 o o o & ° o o 0 n n o , 1 0 o 0 ' o L8 ° o ^ ° ° o * ° o ° ° u o <j2> =9.94 o o " o o° 0 o ° ° o O o U ^ o o n n u O - 8 ° 0 ° . O " ° o o <J2> =7.91 0 o 8 u u o " o u u y - o 8 u <j2> =3.31 • • 0 8 16 24 32 DISTANCE FROM TAP 1,Z ,in FIGURE 4.11 AXIAL DISTRIBUTION OF GAS HOLDUP IN AIR-WATER FLOW AT <j 1> = 4.55 cm/sec 227 4 .12 2 0 2 8 3 6 DISTANCE FROM TAP.1, Z , in FIGURE 4.12 AXIAL DISTRIBUTION OF GAS HOLDUP IN AIR-PEG SOLUTION FLOW T A B L E 4 . 1 1 COMPARISON OF GAS HOLDUPS IN VARIOUS SECTIONS OF THE COLUMN FOR AIR-WATER FLOW L i q u i d F l u x , G a s F l u x ( c m / s e c ) M e a s u r e d G a s H o l d u p i n D i f f e r e n t S e c t i o n s o f t h e C o l u m n C o r r e c t e d Gas H o l d u p i n D i f f e r e n t S e c t i o n s o f t h e Column Gas H o l d u p A b o v e The E x p e r i m e n t a l A v e r a g e Gas ( c m / s e c ) B e l o w t h e E x p e r i m e n t a l S e c t i o n b y I n t h e E x p e r i m e n t a l S e c t i o n b y Be low t h e E x p e r i m e n t a l S e c t i o n by In t h e E x p e r i m e n t a l S e c t i o n by S e c t i o n , M e a s u r e d b y H o l d u p (-) VSO PDM VSO PDM VSO PDM VSO PDM PDM 1 . 2 5 1 . 9 7 _ 0 . 0 4 9 _ 0 .068 _ 0 .061 _ 0 .076 0 . 0 7 7 0 . 0 7 6 3 . 8 1 0 . 0 9 3 0 . 0 9 1 0 . 1 4 0 0 . 1 4 3 0 . 1 2 3 0 .114 0 .157 0 .160 0 .146 0 .150 5 . 31 0 . 1 1 6 0 .12 3 0 . 201 0 . 2 1 1 0 .150 0 . 1 5 3 0 .224 0 .234 0 . 1 8 3 0 . 2 0 7 6 . 8 8 0 . 1 5 3 0 . 1 5 9 0 . 2 2 7 0 . 2 5 0 0 .198 0 .194 0 .252 0 .276 0 . 2 2 3 0 . 2 4 2 8 . 4 4 0 . 1 8 4 0 . 1 9 7 0 . 2 5 8 0 . 2 6 9 0 .242 0 .234 0 .284 0 .296 0 . 2 6 2 0 . 2 7 5 1 . 8 7 5 . 3 1 0 .114 0 . 1 2 7 0 . 2 0 5 0 .211 0 .148 0 .157 0 .227 0 .234 0 . 2 0 9 0 . 2 1 8 6 .88 0 .142 0 . 1 6 3 0 . 2 1 8 0 . 2 0 9 0 .184 0 .200 0 .241 0 .231 0 . 2 2 6 0 .228 8 . 4 4 0 . 1 7 0 0 . 1 9 2 . 0 . 2 7 3 0 .268 0 .217 0 .233 0 .301 0 . 2 9 5 0 . 2 6 0 0 .281 4 . 5 5 3 . 3 1 0 .072 0 . 0 7 6 0 .102 0 . 1 0 0 0 .096 0 .095 0 .114 0 .111 0 . 1 0 7 0 . 1 0 6 7 . 9 1 0 .140 0 . 1 5 8 0 . 1 8 6 0 . 2 0 0 0 . 1 8 3 0 . 1 9 5 0 .209 0 .221 0 . 1 9 9 0 . 2 0 4 9 . 9 3 0 . 1 8 8 0 . 1 9 8 0 . 2 2 5 0 . 2 3 4 0 . 2 4 3 0 .242 0 . 2 5 0 0 . 2 5 7 0 . 2 4 0 0 . 2 4 7 VSO - v a l v e s h u t - o f f PDM - p r e s s u r e d r o p m e a s u r e m e n t 229 compatible, even though the gas holdups by the valve shut-off technique were generally s l i g h t l y smaller than those by the pressure drop gradient method, e s p e c i a l l y below the test section. The uncorrected gas holdups i n d i f f e r e n t sections of the column, even by the same measurement technique, were s i g n i f i c a n t l y d i f f e r e n t . When proper pressure correc-tions (see Section 3.4) were employed, however, the agreement between gas holdups i n d i f f e r e n t sections of the column was improved, e s p e c i a l l y at the higher l i q u i d v e l o c i t i e s . The average gas holdup for a p a r t i c u l a r run was obtained by taking an arithmetic mean of the two corrected gas holdups in the experimental section, the two below the experimental section, and the measured gas holdup above the experimental section. I t i s important to note the near equality of the average gas holdup for a run to the measured gas holdup above the experimental section. This j u s t i f i e s the procedure of not applying any pressure correction to the l a t t e r . Similar agreement between the corrected values was achieved for a l l the other runs. The measured gas holdups obtained by varying the gas flow rate for various constant l i q u i d flow rates are shown in Figures 4.14 - 4.15, whereas the slopes at the o r i g i n for some of these runs are given i n Table 4.12. The r i s e v e l o c i t y of a single bubble i n cocurrent gas - l i q u i d flow was found to be unaffected by l i q u i d flow rate i n the sense that the r e l a t i v e v e l o c i t y of the bubble remained 230 TABLE 4.12 ESTIMATION OF DISTRIBUTION PARAMETER, C , FROM GAS HOLDUP MEASUREMENTS System d <j 2> C o (cm/sec) de„ Jo=0 ; From z Equation 4.16 (cm/sec) (-) Air-Water 0.0 24.7 — 6.25 32.2 1.20 7.00 32 .8 1.16 7.65 33.2 1.11 12.61 42.7 1.42 17.80 47.8 1.30 Air-PEG Solution 13.82 41.7 1.23 Average 1.24 , , the same as i n a quiescent stream [65]. For the bubble column, as shown e a r l i e r , the bubble r i s e v e l o c i t y was found from the slope at the o r i g i n to be equal to 24.7 cm/sec. Therefore using t h i s value for (V ) and the measured values of the slope at the o r i g i n , the values of the d i s t r i b u t i o n parameter, CQ, were calculated from equation 4.16 and are also presented i n Table 4.12. Even though t h i s method of estimating CQ may not be p e r f e c t l y accurate, i t nevertheless provides a good f i r s t approximation for CQ, and the f i n a l average value of 1.24 compares well with the recommended value of 1.2 [19, 39]. Equation 4.18 can then be j u s t i f i a b l y used for describing the data obtained in the 2 inch diameter column, including the data for high v i s c o s i t y polyethylene g l y c o l solution, since the v i s c o s i t y of the system was found from l i t e r a t u r e plots [27] to have l i t t l e or no e f f e c t on the dynamics of the slugs encountered. A comprehensive test of equation 4.18 i s ^ shown i n Figure 4.13. It can be seen that almost a l l the data f a l l within ±10% of the values predicted by equation 4.18, the p r i n c i p a l exception being the data for low l i q u i d flow rates (<j 1> = 1.25 cm/sec). Thus the model for the slug flow regime, represented by equation 4.18, s a t i s f i e s most of the data and can be used for estimating the gas holdup for most l i q u i d flow rates ( j 1 > 2 cm/sec). The gas holdup predicted e x p l i c i t l y by the slug flow model shows excellent agreement with the data, as i l l u s t r a t e d i n Figures 4.14 -4.15. < j > = < i i > + < j 2 > , c m / s e c FIGURE 4.13 COMPARISON OF EQUATION 4.18 WITH EXPERIMENTAL DATA IN 2 INCH PERSPEX COLUMN (arrows indicate observed change from bubbly flow to slug flow) 233 LEGEND FOR FIGURE 4.13 AIR-WATER AIR-PEG SOLUTION O < j x > -e- <jx> 0 <J1> ^ <jx> 0.0 cm/sec A <j±> £ <^i> •A" <Ji> ^ <jx> 0.26 cm/sec 1.25 cm/sec 1.01 cm/sec 1.87 cm/sec 13.82 cm/sec 4.55 cm/sec 18.8 4 cm/sec • <j^ > = 6.25 cm/sec X <J1> = 7.00 cm/sec jzf <J l> = 7.65 cm/sec + <j^ > =12.61 cm/sec O <j,> = 17.75 cm/sec <J2>, cm/sec < j2> , cm/sec <j2> , cm/sec 0.2 0. 0.0 T 1 1 —r <j.> = 12.61 cm/sec 0 FIGURE 4.14 J L Id) 2 4 6 .8 10 12 14 ' <j2> , cm/sec 0.2 0.0 2 4 6 8 10 12 <j2> , cm/sec > GAS HOLDUP FOR COCURRENT AIR-WATER FLOW IN 2 INCH.PERSPEX COLUMN ? (ERP measurements at: >f - 8 inch above Tap 1, ^ f - 38 inch above Tap 1; fo r other symbols, see•legend for Figure 4.13) to < j 2 > , cm/sec < j 2 > , c m / s e c FIGURE 4.15 GAS HOLDUP FOR COCURRENT AIR-PEG SOLUTION FLOW IN 2 INCH PERSPEX j COLUMN (ERP measurements at: i j L - 8 inch above Tap 1, ^  - 38 inch -above Tap 1; for other symbols, see legend for Figure 4.13) For the case of low l i q u i d flow rates, the flow regime encountered for most gas flow rates was predominantly bubbly. The d r i f t v e l o c i t y for the bubbly flow regime was shown to be given by equation 2.37 . Then assuming that CQ - 1.0 [1, 39], equation 4.1 in combination with equation 2.37 becomes V2 = I T = < j l + j 2 > + ( V J B / , r „ M / , 1 / 3 , 2 tanh 10.25 ( l / e 2 ) J " (4.20) where (V^) B = 2 4.7, corresponding to a bubble radius of 0.5 cm from equation 2.10. The values of gas holdup predicted by equation 4.20 for a l i q u i d flow rate of 1.25 cm/sec are compared i n Figure 4.14a with experimental data. The measured values are found to l i e between the predicted values from equation 4.20 for bubbly flow and equation 4.18 for slug flow. Since, as stated e a r l i e r from v i s u a l observations, the flow regime was neither t r u l y bubbly nor t r u l y slug flow, i t cannot be expected that either equation 4.2 0 for a single bubble si z e or equation 4.18 would predict the gas holdup accurately over a range of gas flow rate; and since the bubble size increases with increasing gas. flow rate [69], equation 4.20 for a bubble radius of 1.0 cm, also shown i n Figure 4.14a, gives a good compromise agreement with the experimental data. Hence the model for the slug flow regime, represented by equation 4.18, which provides excellent agreement with 237 most data, w i l l l a t e r be used for describing the r i s e v e l o c i t y of bubble swarms in three-phase f l u i d i z e d beds, for water flow rates greater than 2 cm/sec. E l e c t r o - r e s i s t i v i t y probe (ERP) measurements As mentioned e a r l i e r , the e l e c t r o - r e s i s t i v i t y probe was used to measure l o c a l gas f r a c t i o n i n air-water as well as air-polyethylene gly c o l solution flow. Even though the data obtained were limit e d , experimental findings are described below. (i) Air-water flow The measured r a d i a l p r o f i l e s of l o c a l gas f r a c t i o n for a water flow rate of 4.55 cm/sec are shown i n Figure 4.16. These r a d i a l p r o f i l e s show that the gas holdup was p r a c t i c a l l y uniform i n the central core of the column, decreasing gradually, maintaining a x i a l symmetry, to reach zero at the w a l l . For obtaining the cross-sectional average < a 2 > ' t n e s e p r o f i l e s were integrated according to equation 3.5, and the values of gas holdup so obtained are compared i n Table 4.13 with the average gas holdup, z^, determined d i r e c t l y from valve shut-off and pressure drop measurements. The agreement between the respective values of gas holdup i s excellent, i n contrast to other reported probes used for measuring l o c a l gas f r a c t i o n which gave average values that were consistently smaller than those o < o Q 0.5 ^ 0.4 o < cr LL_ oo < a 2 <J2>. M' o 0.123 3.31 9.46 A 0.242 9.92 5.52 •Eq. 2.27b with above values of M' O—O O .0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 DIMENSIONLESS RADIAL DISTANCE, R* to to co FIGURE 4.16 COMPARISON OF RADIAL GAS HOLDUP PROFILES COMPUTED BY EQUATION 2.27b WITH EXPERIMENTAL DATA (System: Air-water, <j,> = 4.55 cm/sec) TABLE 4.13 GAS HOLDUP BY ELECTRO-RESISTIVITY PROBE FOR AIR-WATER FLOW <j-> = 4.55 cm/sec Liquid Flux, Probe Location Gas Holdup Ratio M1 From Co From (cm/sec) Above Tap 1 (inches) a2C < a 2 > e2 ( e 2 / < a 2 > ) Equation 4.23 Equation 4.21 3.31 8 0.149 0.123 0.106 0.86 9.46 1.10 7.91 8 0.292 0.211 0.204 0.97 5.23 1.16 7.91 38 0.300 0.213 0.204 0.96 4.90 1.17 9.94 8 0.320 0.226 0.247 1.09 4.82 1.17 9 .92 38 0.327 0.242 0.246 1.02 5.52 1.15 0 .98 1.15 to co measured d i r e c t l y [80, 81, 104]. The a x i a l location of the probe i n the column was varied from 8 inches above Tap 1 to 38 inches above Tap 1 i n order to e s t a b l i s h the e f f e c t of hydrostatic head on the r a d i a l p r o f i l e s of gas f r a c t i o n , as well as on the average values. The r a d i a l p r o f i l e s as well as the averaged values remained unchanged, as shown i n Table 4.13, i n d i c a t i n g that for the given runs at l e a s t , the hydrostatic head had l i t t l e or no e f f e c t on either the r a d i a l or the a x i a l d i s t r i b u t i o n of gas holdup. I t was indicated previously that for determining the d i s t r i b u t i o n parameter, CQ, the r a d i a l p r o f i l e s of both l o c a l gas f r a c t i o n and l o c a l volumetric flux of the gas-l i q u i d mixture have to be measured simultaneously. However, an estimate for the value of CQ can be obtained i f we assume that the r a d i a l p r o f i l e of mixture flux i s s i m i l a r to that of gas f r a c t i o n , i . e . the exponents in equations 2.27a and 2.27b are equal. Zuber and Findlay [39] asserted that the assumption of equality of the two exponents i s not un-reasonable i f the volumetric f l u x of the mixture i s con-sidered to be greatly ( i f not mostly) affected by the volu-metric flux of the gas. With th i s assumption, equation 2.27c reduces to M1 + 2 Cn = (4.21) In order to evaluate M', l e t us substitute the value of a2 from equation 2.27b into equation 3.5, giving 1 M ' <a0> = 2 / a „p {l-(R*) } R* dR* (4.22) * 0 which can e a s i l y be integrated by parts to y i e l d <a0> M1 — — = (4.23) a2 c M' + 2 Impl i c i t i n equation 2.27b and hence equation 4.2 3 i s the assumption that gas f r a c t i o n at the wall i s zero. The exponent M' was calculated for each run from equation 4.23 using the values of <a2> and a2 c reported i n Table 4.13, and then CQ was evaluated from equation 4.21. I t i s seen from t h i s table that, as the gas flow rate was increased from 3.3 to 7.9 cm/sec CQ increased s l i g h t l y , but a further increase i n gas flow rate did not a f f e c t the value of CQ. The average value of CQ for these experiments was 1.15, thus again j u s t i f y i n g the assumption of CQ = 1.2, recommended elsewhere [19, 39]. The gas holdup p r o f i l e s computed by equation 2.27b, using the value of exponent M' calculated from equation 4.23, are compared with the experimental data i n Figure 4.16. The agreement between the computed and the experimental values confirms that the r a d i a l gas f r a c t i o n p r o f i l e s do indeed conform to equation 2.27b. ( i i ) A i r - PEG solution flow Some of the measured gas holdup p r o f i l e s are shown i n Figure 4.17. The p r o f i l e s were found to be a x i a l l y symmetric for a l l the gas and l i q u i d flow rates studied, becoming s l i g h t l y f l a t t e r with increasing gas flow rate and more pointed with increasing l i q u i d flow rate. Once again, for obtaining the cross-sectional average of gas holdup,  <®2 > ' these p r o f i l e s were integrated according to equation 3.5, and the values so calculated compared i n Table 4.14 with the average gas holdup, z^i measured by valve shut-off and pressure drop measurements. The agreement between the two values i s poor, becoming worse with increase i n flow rate of either phase, the average gas holdup from l o c a l measure-ments being consistently smaller. This discrepancy i s believed to be caused by the i n a b i l i t y of the e l e c t r o -r e s i s t i v i t y probe to penetrate the bubble front i n s t a n t l y i n the PEG solution, the gas bubble becoming markedly deformed i n the v i c i n i t y of the probe. Also since only slugs were found to e x i s t in t h i s viscous medium, the probe at r a d i a l locations receding from the center of the column tended to d e f l e c t the bubble and thus f a i l e d to penetrate the steep gas-liquid interface at these locations. Thus the probe generally suffered from the same drawbacks as encountered by Nassos and Bankoff [104] with t h e i r pointed needle probe. Therefore, while the change i n probe design improved the a p p l i c a b i l i t y of an e l e c t r o - r e s i s t i v i t y probe for the a i r -0 .30 CM ~ 0 .25 o cn ^ 0.15 CO ^ 0.101-< H 0 .05 0 <j|> <J2> 1.01 7.80 O 13.82 2.89 A 18.84 9.42 _.A—A-A-— 'O / / -o-o-o.- ^ / 1.0 0 .8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 DIMENSIONLESS RADIAL DISTANCE, R* FIGURE 4.17 RADIAL GAS HOLDUP PROFILES IN AIR-PEG SOLUTION FLOW to TABLE 4.14 GAS HOLDUP BY ELECTRO-RESISTIVITY PROBE FOR AIR-PEG SOLUTION FLOW Liquid Flux, (cm/sec) Gas Flux (cm/sec) Probe Location Above Tap 1 (inches) Gas Holdup Ratio (e2/<a2> M1 From Eq. 4.23 C o From Eq. 4.21 a2C <a 2> £2 0.26 2.03 8 0.112 0.061 0.077 1.26 2.37 1.30 7.78 8 0.253 0.141 0.238 1.69 2.52 1.28 7.83 22 0 .303 0.168 0.231 1.38 2.49 1.29 1.01 5.47 22 0.246 0.137 0.167 1.22 2.51 1.29 7.79 22 0.302 0.173 0.235 1.36 2.68 1.27 13.82 2.89 22 0.122 0.046 0.065 1.41 1.22 1.45 4.77 22 0.163 0.068 0.102 1.50 1.41 1.41 7.12 22 0.212 0.095 0.143 1.51 1.61 1.38 18.84 4.85 22 0.100 0.036 0.055 1.53 1.14 1.47 9.42 22 0.210 0.091 0.138 1.52 1.54 1.39 245 water system, i t s a p p l i c a b i l i t y for the highly viscous a i r -polyethylene g l y c o l solution system was s t i l l not s a t i s f a c t o r y . An estimate of CQ was obtained as before from the measured p r o f i l e s , and these calculated values are also presented i n Table 4.14. Based as they were on unsatisfactory e l e c t r o - r e s i s t i v i t y probe measurements, l i t t l e credence was given to these values, which a l l exceeded 1.2, e s p e c i a l l y since the value of 1.2 for CQ was found to s a t i s f a c t o r i l y describe the bubble r i s e v e l o c i t y i n Figure 4.13 and the gas holdup data i n Figure 4.15 (except the data for <j^> = 18.84 cm/sec). This value w i l l therefore be used for describing the bubble r i s e v e l o c i t i e s i n three-phase f l u i d i z e d bed, even with PEG solution as l i q u i d . Another important feature to be observed from the measured gas f r a c t i o n p r o f i l e s i s the existence of a l i q u i d f i l m near the column wall in which the gas f r a c t i o n i s p r a c t i c a l l y zero. The existence of such a f i l m i n the r i s e of a slug has been previously reported [27, 34], The values of f i l m thickness as obtained from these p r o f i l e s are reported i n Table 4.15. I t can be seen that the f i l m thickness changes l i t t l e with increase i n gas flow rate, but increases with increase i n l i q u i d flow rate. Due to the presence of t h i s l i q u i d f i l m , equation 2.27b cannot be used for describing the r a d i a l p r o f i l e s since the equation 246 TABLE 4.15 LIQUID FILM THICKNESS, <5*, FROM ELECTRO-RESISTIVITY PROBE MEASUREMENTS IN AIR-PEG SOLUTION FLOW <h> * <5 ! (cm/sec) (cm/sec) (-) 0.26 7 .79 0.09 1.01 5.47 0.11 7.80 0.12 13.82 2.89 0.19 4.77 0.19 7.12 0.19 18.84 4.85 0.22 9.42 0.17 247 i m p l i c i t l y assumes that the gas f r a c t i o n i s zero only at the wall ( i . e . = 0 only at R* = 1). An alternate scheme, which uses the minimum r a d i a l distance at which becomes zero, instead of the column radius, for making the r a d i a l distances dimensionless, i s recommended for l a t e r analysis of these p r o f i l e s . Radial p r o f i l e s of bubble frequencies were also measured by the e l e c t r o - r e s i s t i v i t y probe and were generally similar to the gas f r a c t i o n p r o f i l e s . An estimate of bubble size was obtained by using equation 8.3.6, and the calculated values are presented i n Table 4.16. However, i t i s not possible to confirm these values, as no independent measure-ments of bubble size were ca r r i e d out. Nevertheless an order of magnitude analysis can be made by converting the average bubble s i z e , r , to an average c y l i n d r i c a l slug length, X c , using (4.24) and then comparing the slug length so calculated with the values from v i s u a l observations. From the values recorded in Table 4.16, the slug lengths calculated from equation 4.24 appear to be at least of the r i g h t order of magnitude. I t i s therefore believed that measurements of bubble frequency p r o f i l e s can be successfully used for obtaining an estimate of average bubble si z e i n the swarm, and thereby for predicting the bubble r i s e v e l o c i t y . 248 TABLE 4.16 AVERAGE BUBBLE SIZE, J , PROM ELECTRO-RESISTIVITY PROBE MEASUREMENTS IN AIR-PEG SOLUTION FLOW ^1* (cm/sec) (cm/sec) r Q , From Eg. 8.3.6 (cm) Slug Length, A G From Eq. 4.24 (cm) From Vis u a l Observations (cm) 1.01 > 5.47 4.4 15.3 -7.79 4.2 17.7 -13.82 2.89 2.3 2.4 2.5 4.77 3.2 6.6 10.0 7.12 4.1 14.6 -18.84 4.85 3.8 11.1 4.0 9.42 5.4 33.2 -4.2.2.2 Gas holdup i n three-phase f l u i d i z e d beds The measurement of gas holdup i n three-phase f l u i d i z e d beds was conducted mainly i n the 2 inch i . d . perspex column. The l i m i t e d preliminary data obtained in the 2 0 mm glass column w i l l be discussed f i r s t . A l l the graphs and tables presented i n t h i s section are drawn from the data i n Appendix 8.7. (A) 20 mm glass column The v i s u a l observation of the transparent column showed that the slug flow regime existed i n t h i s column at almost a l l the gas and l i q u i d flow rates studied. However, at very small gas and l i q u i d flow rates some large and i r r e g u l a r bubbles were observed i n the column, but due to the small size of the column no clear d i s t i n c t i o n between the bubbly and slug flow regime could be made. The gas holdup data for one l i q u i d flow rate from each of the three systems studied are presented i n Figures 4.18-4.20. As can be seen, the agreement between the two techniques of gas holdup measurement i s generally quite s a t i s f a c t o r y . I t i s important to r e c a l l that the gas holdup i n the 20 mm glass column was measured i n a section of the column which included not only the three-phase f l u i d i z e d bed region, but also the preceding and the following two-phase gas-liquid regions (see Figure 3.1). A comparison of gas holdup so measured with the gas holdup in s o l i d s - f r e e gas-I 250 0.20 0.16 0.12 ZD Q _ J O i 0.08 in < <s> 0.04 0.00 1 -• • 1 i — i T -• ,Ec| 4.13 - • -T x V v • -/ v - X 7 -V v V / 1 1 » 0.0 1.0 2.0 3.0 4.0 5.0 <j > , cm/sec FIGURE 4.18 GAS HOLDUP IN 20 MM GLASS COLUMN FOR THREE-PHASE FLUIDIZATION BY AIR AND WATER OF 1 MM GLASS BEADS (j 3 = 6 . 02 cm/sec; W=50 gm; v - from pressure drop; V - from valve shut-off) 251 CM Q _ J O X CO < (3 0.20 0.16 0.12 0.08 0.04 0.00 <J,> w A -6.03 cm/sec 50 gm. A From pressure drop A From valve shut - off Eq 4.13 0.0 1.0 2.0 3.0 4.0 <j2> , cm/sec 5.0 FIGURE 4.19 GAS HOLDUP IN ^20 MM GLASS COLUMN FOR THREE-PHASE FLUIDIZATION BY AIR AND AQUEOUS GLYCEROL OF 1 MM GLASS BEADS / 252 <j2> , cm/sec FIGURE 4.20 GAS HOLDUP IN 20 MM GLASS COLUMN FOR THREE-PHASE FLUIDIZATION BY AIR AND WATER OF 1/2 MM SAND PARTICLES l i q u i d flow could reveal possible e f f e c t s of the presence of s o l i d s on the bubble dynamics. However, since no gas holdup for these combinations of gas and l i q u i d flow rates were obtained i n the absence o f . s o l i d s , equation 4.13 was used for predicting the gas holdup i n gas-liquid flow. In Figure 4.18 the gas holdups measured i n the presence of a bed of 1 mm glass beads f l u i d i z e d by a cocurrent stream of a i r and water aase compared with the values predicted from equation 4.13. The two sets of values were found to be i n good agreement, in d i c a t i n g that the presence of the 1 mm glass beads did not a f f e c t the gas holdup s i g n i f i c a n t l y . S i m i l a r l y , the measured gas holdups i n the presence of the same 1 mm glass beads f l u i d i z e d by a cocurrent stream of a i r and aqueous glycero l solution (u^ = 2.1 c.p.) showed good agreement with the predicted values from equation 4.13 (see Figure 4.19), i n d i c a t i n g that neither the presence of s o l i d p a r t i c l e s nor the small change i n l i q u i d v i s c o s i t y (y^ doubled) affected the gas holdup. However, the measured gas holdups i n the presence of a bed of 1/2 mm sand p a r t i c l e s f l u i d i z e d by a cocurrent stream of a i r and water at <j^> = 2.03 cm/sec, were found to be consistently smaller than the values predicted by equation 4.13, as shown i n Figure 4.2 0a. Thus the bed of 1/2 mm sand p a r t i c l e s , for r e l a t i v e l y low bed expansion, tended to reduce the gas holdup by promoting bubble coalescence within the bed. The gas holdups at higher l i q u i d flow rates showed good 254 agreement with the predicted values from equation 4.13 (see Figure 4.20b), in d i c a t i n g once again that the s o l i d p a r t i c l e s had l i t t l e or no e f f e c t on gas holdups. (B) 2 inch perspex column Experimental findings for each of the s o l i d species studied are presented separately below. (i) Air-water-1/4 mm glass beads The v i s u a l observation of the f l u i d i z e d bed and the region above the bed showed the existence of slugs i n the experimental section for gas flow rates greater than 3 cm/sec at a l l the l i q u i d flow rates studied. I t i s important to r e c a l l that, i n contrast, for two of the l i q u i d flow rates, <j^> .= 1.25 and 1.87 cm/sec, air-water runs performed i n the absence of s o l i d s showed only the bubbly flow regime at most gas flow rates. The gas holdups measured i n three-phase f l u i d i z e d beds are shown i n Figure 4.21 along with the curve for two-phase gas-liquid flow data reported e a r l i e r . I t can be seen in Figure 4.21 that, whereas the gas holdups in three-phase f l u i d i z e d beds for l i q u i d flow rates of 1.25 and 1.87 cm/sec are p r a c t i c a l l y i d e n t i c a l , those for a l i q u i d flow rate of 3.18 cm/sec are s l i g h t l y , but consistently, larger. A l l these gas holdups are, however, smaller than those for two-phase gas-liquid flow. 255 < j 2 > , c m / s e c FIGURE 4.21 GAS HOLDUP IN THREE-PHASE BEDS OF 1/4 MM GLASS BEADS FLUIDIZED BY AIR AND WATER ( O -D1=1.25 x - j 1 = 1 . 8 7 ; A-j-^3.18; two-phase air-water flow, J1=1.25; generalized wake model with xk=0.4: j =1.25, Jx=3.18) 256 The gas holdups i n three-phase f l u i d i z e d beds are compared i n Table 4.17 with the gas holdups in d i f f e r e n t g a s - l i q u i d regions i n the column, as well as with those i n gas-liquid flow alone, for the l i q u i d flow rate of 1.25 cm/sec: 1. Since the gas holdups i n the two-phase regions above and below the test section were comparable (the former being s l i g h t l y , but consistently, smaller), an average of a l l f i v e or s ix gas holdup measurements in the three gas-liquid regions was taken to represent the gas holdup in the two-phase regions i n the presence of a bed of s o l i d p a r t i c l e s . These gas holdups were s i g n i f i c a n t l y smaller than the gas holdups i n the corresponding g a s - l i q u i d flow alone. Thus the i n t r o -duction of a bed of s o l i d p a r t i c l e s seemed to a f f e c t the structure of gas-liquid flow not only downstream, but s u r p r i s i n g l y even upstream, from the bed, apparently causing reduction i n gas holdup by promoting bubble coalescence throughout the entire column. 2. The gas holdups i n the three-phase f l u i d i z e d beds as such were considerably smaller than those i n gas- l i q u i d flow alone, e s p e c i a l l y at gas flow rates greater than 4 cm/sec; however, the gas holdups i n the three-phase f l u i d i z e d beds represented on a s o l i d s - f r e e basis were s l i g h t l y larger than those i n gas-liquid flow for gas flow rates less than 3 cm/sec, and somewhat smaller T A B L E 4 . 1 7 COMPARISON OF GAS HOLDUP IN T H R E E - P H A S E F L U I D I Z E D BED TO THAT* IN TWO-PHASE REGIONS OF THE COLUMN S Y S T E M : A i r - W a t e r - 1 /4 mm G l a s s B e a d s , < j ,> = 1 . 2 5 c m / s e c , W = 1100 gm, e , | = 0 .237 3 l j 2 = 0 Gas F l u x , < ^ 2 > ( c m / s e c ) Gas H o l d u p A v e r a g e Gas H o l d u p i n Two-P h a s e R e g i o n s A v e r a g e Gas H o l d u p i n G a s -L i q u i d F l o w Gas H o l d u p i n T h r e e - P h a s e R e g i o n s B e l o w t h e T e s t S e c t i o n A b o v e t h e T e s t S e c t i o n E 2 (measured) On S o l i d s F r e e B a s i s , e £ / ( l - e 3 ) 2 . 0 3 0 . 0 6 1 0 .055 0 .054 0 . 0 7 7 0 .059 . 0 . 0 9 2 2 . 8 0 0 . 0 9 4 0 . 0 7 5 0 . 0 8 0 0 .104 0 .080 0 . 1 2 4 ' 4 . 1 7 0 . 1 2 8 0 .110 0 .118 0 . 1 5 5 0 .090 0 . 1 3 7 5 . 3 0 0 . 1 5 1 0 .135 0 . 1 4 0 0 .194 0 .113 0 . 1 7 3 6 .9 0 0 . 1 9 9 0 . 1 7 5 0 . 1 8 3 0 . 2 4 3 0 .122 0 . 1 8 9 8 . 7 1 0 . 2 3 4 0 . 2 1 3 0 . 2 1 5 0 .278 0 .150 0 . 2 3 3 1 0 . 0 0 0 . 2 6 6 0 .238 0 .244 0 .300 0 . 1 8 3 0 .278 M Cn —1 258 for gas flow rates greater than 4 cm/sec. A further comparison of gas holdups i n the three-phase f l u i d i z e d beds on a s o l i d s - f r e e basis with those i n the concurrent two-phase regions showed that, i n the slug flow regime ( j 2> 3 cm/sec), the gas-liquid r a t i o changed only mod-erately from the two-phase to the three-phase region, the gas f r a c t i o n being about 13% larger on the average in the three-phase region. Thus, for gas flow rates greater than 3 cm/sec, the presence of s o l i d p a r t i c l e s apparently caused the gas-liquid flow structure to be changed i n the column as a whole, but not exclusively (to any' significant;.extent) in the three-phase region. Similar trends were observed for the other l i q u i d flow rates as w e l l . As was shown e a r l i e r i n Section 2.3.2, V a i l et a l . [17] recommended that the r a t i o of gas holdup i n a three-phase f l u i d i z e d beds to that i n the corresponding two-phase gas-liquid flow, z^/z^, was only a function of the bed voidage and was well represented by equation 2.12 0. Efremov and Vakhrushev [16] used the following empirical c o r r e l a t i o n to represent the r a t i o z^/z^ from t h e i r measurements i n a 4 inch diameter column: 259 where ^ s the s o l i d s holdup i n the l i q u i d - s o l i d f l u i d i z e d bed before the gas i s introduced. The r a t i o s of experimen-t a l l y measured gas holdups i n three-phase f l u i d i z e d beds to those i n gas-liquid flow are compared i n Table 4.18 with the predictions from equations 2.120 and 4.25. Although the measured r a t i o s were found to vary somewhat with gas flow rates, the average values for gas flow rates between 2 and 10 cm/sec agreed c l o s e l y with the predicted values from equation 4.25. Predictions from equation 2.12 0 were s i g n i f i c a n t l y smaller, the c o r r e l a t i o n being based on data in three-phase f l u i d i z e d beds of 0.73 mm glass beads. Even though the measured absolute values of gas holdup did not agree with the reported data of Efremov and Vakhrushev either in g a s - l i q u i d flow or i n three-phase f l u i d i z e d beds, the equality of the gas holdup r a t i o s seems to indicate that the role of s o l i d p a r t i c l e s i n a f f e c t i n g the bubble dynamics i s s i m i l a r i n both small diameter (D < 4 inch) and large diameter (D >_ 4 inch) columns. As derived e a r l i e r i n Section 2.3*1, equation 2.114 provides a general expression for describing the r i s e v e l o c i t y of bubble swarms in three-phase f l u i d i z e d beds, i f a proper expression for c a l c u l a t i n g v ^ i s used. Equation 2.114, however, cannot be solved independently, but instead equation 2.112 was solved as an i n t e g r a l part of the general-ized wake model with the following assumptions: 260 TABLE 4.18 COMPARISON OF GAS HOLDUP IN THREE-PHASE FLUIDIZED BED TO THAT IN GAS-LIQUID FLOW SYSTEM: Air-Water- 1/4 mm Glass Beads <j-> = 2-10 cm/sec Liquid Flux, <V (cm/sec) E2/ E2 (Experimental) e2X 2 (Eq. 4.25) c i n / , . . n e2/ E2 (Eq. 2.120) p "i/p I I t2/ t2 (Model with x k = 0.4) 1.25 0.62 (0.5-0.77) 0.59 0.41 0.55 (0.52-0.60) 1.87 . 0.64 (0.53-0.86) 0.68 0.48 0.74 (0.72-0.78) 261 1. Equation 2.46 can be used for describing the voidage in the p a r t i c u l a t e phase of three-phase f l u i d i z e d beds, as w i l l be shown l a t e r i n Section 4.2.3.1. 2. Equation 2.112 can be used for c a l c u l a t i n g the r i s e v e l o c i t y of bubble swarms i n conjunction with equation 4.8 for the d r i f t v e l o c i t y i n the slug flow regime, slugs having been observed under most conditions. 3. Equation 4.7 can be used for estimating the wake volume f r a c t i o n , as discussed i n Section 4.1.2. Now for c a l c u l a t i n g the gas holdup and the bed voidage from the generalized wake model with these assumptions, the following eight equations have to be solved simultaneously for the eight unknowns involved (e^, e 2 , £3 /  e i f v2 ' v'V v 2 j ) : < ^ 2 > e0 = — — (2.94) ^2 < j 1 + j 2 > £ l f ( 1"£2 " £ k ) -v 2 = — - — — f f _ J ^ (2.112) '21 v i i (2.108a) v 2j = 0.2 <J1+ J2> + 0.35 /gb~ (4.8) £ ( l - e3) = in + £ in (1.3) 1 2 £ ill £k( l - Xk) + ^ ( l - C ^ + X ^ ) (2.91) 1 A t r i a l and error scheme was developed for solving these equations numerically. The convergence for small gas flow rates ( j 2 < 2 cm/sec), even for small l i q u i d flow rates (<j^> = 1.25 cm/sec), was quickly obtained; but for large gas flow rates and small values of x k ( x k ~ 0 ) / the t r i a l and error scheme adopted f a i l e d to converge for 1/4 mm glass beads, as w i l l be i l l u s t r a t e d l a t e r i n Section 4.2.3.2. I t was, however, found that neither the wake volume f r a c t i o n , £^, nor the r e l a t i v e p a r t i c l e content of the wake, x^, affected the value of gas holdup s i g n i f i c a n t l y . Thus the values of gas holdup predicted by the generalized wake model are shown i n Figure 4.21 for x^ = 0.4 because the t r i a l and error scheme f a i l e d to converge for smaller values of 262a x^ .. The agreement between the predicted and the measured values i s good enough to confirm the correctness of equation 2.112 for describing the r i s e v e l o c i t y of bubble swarms. The predicted values show the gas holdup to increase with increasing gas and l i q u i d flow rates. The r a t i o s of the predicted values of gas holdup i n three-phase f l u i d i z e d beds to the measured values in gas- l i q u i d flow, presented in Table 4.18, show that, although these r a t i o s varied s l i g h t l y with the gas flow rates, the average values at a given l i q u i d flux were i n reasonably good agreement with the predictions from empirical equation 4.25, as well as with the measurements, indi c a t i n g that equation 2.112 appropriately accounted for the r o l e of p a r t i c l e s i n a f f e c t i n g the bubble behaviour. ( i i ) Air-water- 1/2 mm glass beads The v i s u a l observation of the f l u i d i z e d bed and the region above the bed, at a l l l i q u i d flow rates studied, showed that for gas flow rates greater than 4 cm/sec, f u l l y developed slugs were present i n the column, while for gas rates less than 3 cm/sec, the flow was primarily bubbly with intermittent occurrence of large i r r e g u l a r bubbles due to coalescence. The bubble motion appeared to be quite chaotic and v i o l e n t even though the l i q u i d flow, except at the two highest rates, was not turbulent. The increase i n l i q u i d flow rate, or bed f l u i d i t y , had, i n general, a calming e f f e c t on the bubble motion. 263 The measured values of gas holdup i n three-phase f l u i d i z e d beds are shown i n Figure 4.22 along with the curve for two-phase gas-liquid flow data reported e a r l i e r . The gas holdups at the l i q u i d flow rate of 1.59 cm/sec were found to fluctuate for gas flow rates greater than 7 cm/sec, as p a r t i c l e bridging occurred frequently, causing the bed to be l i f t e d off the bed support screen. No bridging was observed for higher l i q u i d flow rates. The gas holdup was found to increase with increasing gas flow rate, as can be seen from Figure 4.22, but the similar e f f e c t of increasing l i q u i d flow rate was not as c l e a r l y d i s c e r n i b l e from the data. The values of gas holdup measured i n the three-phase f l u i d i z e d bed region are compared i n Table 4.19 with those i n the corresponding two-phase regions of the column, as well as with those i n two-phase ga s - l i q u i d flow, at the l i q u i d flow rate of 4.55 cm/sec, for W = 567 gm t W = 1200 gm. As can be seen from t h i s table, the weight of the p a r t i c l e s in the column (or the height of the three-phase f l u i d i z e d bed region) neither affected the gas holdup i n the two-phase regions nor that i n the three-phase f l u i d i z e d bed i t s e l f , even though the s o l i d s holdup was markedly affected, as w i l l be shown l a t e r i n Section 4.2.3.2. Other inferences from the table are: 264 0 . 3 1 — — r 1 — — 1 r ~ f < j 2 > > c m / s e c FIGURE 4.22 GAS HOLDUP IN THREE-PHASE BEDS OF 1/2 MM GLASS BEADS FLUIDIZED BY AIR AND WATER ( O -j,=1.59; x~j 1=3.52; •-J 1=4.55 (W=1200) ; X - J ^ 4 - 5 5 (W=567) ; AJ-,=5.71; two-phas"e air-water flow, j^=4.55; generalized wake model with x,=0i j,=1.59, j x=5.71) K X T A B L E 4 .19 COMPARISON OF GAS HOLDUP IN T H R E E - P H A S E F L U I D I Z E D BED TO THAT IN TWO-PHASE REGIONS OF THE COLUMN S Y S T E M : A i r - W a t e r - 1 /2 mm G l a s s B e a d s , < j 1 > = 4 . 5 5 c m / s e c G a s F l u x , <h> ( c m / s e c ) Gas H o l d u p B e l o w t h e A b o v e t h e T e s t S e c t i o n T e s t S e c t i o n A v e r a g e Gas H o l d u p i n T w o - P h a s e R e g i o n s A v e r a g e Gas H o l d u p i n G a s - L i q u i d F l o w Gas H o l d u p i n e" 2 (measured) T h r e e - P h a s e R e g i o n s On S o l i d s F r e e B a s i s , e 2 V ( l - e 3 ) W = 567 gm, e3|j2=o = ° - 1 3 4 2 , 0 0 0 . 0 5 4 0 . 0 5 2 0 . 0 5 4 0 .061 0 . 0 5 6 0 . 0 7 3 4 .54 0 . 1 1 6 0 . 1 1 2 0 .116 0 .128 0 . 0 9 3 0 .122 6 .17 0 . 1 5 2 0 .142 0 .147 0 . 1 6 3 0 .130 0 . 1 7 1 7 .92 0 . 1 8 6 0 . 1 6 4 0 .177 .0 .213 0 .154 0 .204 9 .94 0 . 2 2 7 0 . 2 1 3 0 . 2 2 0 0 .246 0 .172 0 . 2 2 5 1 1 . 2 3 0 . 2 6 5 0 .234 0 . 2 5 6 0 .258 0 . 1 7 6 0 .228 W = 1200 gm, £ 3 | J 2 = 0 - ° ' 1 4 8 2 .06 0 . 0 5 7 0 . 0 5 3 0 .057 0 .065 0 . 0 5 5 0 . 0 7 3 4 .51 0 . 1 3 2 0 . 1 0 2 0 .116 0 .128 0 . 1 0 1 0 . 1 3 9 6 .17 0 .158 0 . 1 3 8 0 . 1 5 0 0 . 1 6 3 0 . 1 2 5 0 . 1 7 3 7 .91 0 . 1 9 2 0 . 1 7 7 0 .182 0 .204 0 .146 0 . 2 0 0 9 . 9 4 0 .248 0 . 2 1 0 0 . 2 2 5 0 .246 0 .154 0 . 2 1 1 1 1 . 2 5 0 . 2 7 7 0 . 2 4 7 0 . 2 5 3 0 .258 0 .17 7 0 . 2 4 3 1. At a given gas and l i q u i d rate, gas holdups i n the two-phase regions above and below the test section are observed to be comparable. Therefore the average of a l l the gas holdup measurements i n the three two-phase regions was taken to represent the gas holdup in these two-phase regions for the given flow rates. These gas holdups are observed to be s l i g h t l y smaller than those i n the corresponding gas-liquid flow without sol i d s i n the column. Thus the presence of a bed of 1/2 mm glass beads affected the structure of gas-l i q u i d flow in the adjacent regions only s l i g h t l y . 2. The gas holdups in the three-phase f l u i d i z e d beds as such were smaller than those i n gas-liquid flow at gas fluxes exceeding 4 cm/sec; however, when represented on a s o l i d s - f r e e basis, they approximated both the gas holdups i n the adjacent two-phase regions and the gas holdups i n gas-liquid flow alone. Thus the gas-l i q u i d r a t i o remained p r a c t i c a l l y unchanged, from the two-phase region to the three-phase region, from which i t can be inferred that the structure of bubble flow through the bed of 1/2 mm glass beads i s l i t t l e d i f f e r -ent from that of the corresponding two-phase gas-l i q u i d flow. Similar trends were found for the other l i q u i d flow rates as wel l . 267 The r a t i o s of gas holdups measured i n three-phase f l u i d i z e d beds to those measured i n two-phase gas-liquid flow are compared with the predictions of equations 4.25 and 2.120 i n Table 4.20. Even though the r a t i o s varied somewhat with gas flow rates, the averages of these r a t i o s , for gas flow rates between 2 and 11 cm/sec, are seen to be in excellent agreement with the values predicted from equation 4.25. Predictions from equation 2.120 were s i g n i f i c a n t l y lower, as found previously for the 1/4 mm glass beads. The agreement with equation 4.25 once again indicates that the e f f e c t of p a r t i c l e s on the bubble dynamics i s the same i n both large (D >^  4 inches) and small (D = 2 inch) diameter columns. In order to check the a p p l i c a b i l i t y of the generalized wake model, the eight equations l i s t e d above (Equations 2.94, 2.112, 2.108a, 4.8, 2.106, 4.7, 1.3 and 2.91) were solved numerically. The convergence for a l l the gas and l i q u i d flow rates was in t h i s case quite s a t i s f a c t o r y , and the gas holdups calculated from the model with x^ = 0 are shown i n Figure 4.22 for the highest (<j^> = 5.71 cm/sec) and the lowest (^ j-|_> = 1.59 cm/sec) l i q u i d flow rates. I t was again observed that neither the wake volume f r a c t i o n , e^, nor the r e l a t i v e p a r t i c l e content of the wake, x^, affected the gas holdup s i g n i f i c a n t l y . However, an increase i n either the gas or the l i q u i d flow rate caused the gas holdup to increase. The r a t i o s of the predicted values of gas holdup 268 TABLE 4.20 COMPARISON OF GAS HOLDUP IN THREE-PHASE FLUIDIZED BED TO THAT IN GAS-LIQUID FLOW SYSTEM: Air-Water- 1/2 mm Glass Beads <j 9> = 2-11 cm/sec Liquid Flux, (cm/sec) _ I I I / - II 2 / 2 (Experimental) P in / c H fc2/fc-2 (Eq. 4.25) e"'/e" (Eq. 2.120) P i n P i i 2 2 (Model with x k = 0) 4 .55 0 .74 (0.67-0.84) 0.75 0.55 0.82 (0.80-0.83) 5.71 0.86 (0.76-0.90) 0.85 0.66 0.89 (0.86-0.91) 269 i n three phase f l u i d i z e d beds to the measured (for-"<j^> = 4.55 cm/sec) or the predicted(for < j ^ > = 5.71 cm/sec, at which no gas-l i q u i d flow measurements were made) values i n gas-li q u i d flow, presented i n Table 4.20, show that although t h i s r a t i o varied s l i g h t l y with gas flow rate, the averaged values were i n good agreement with the predictions from equation 4.25, as well as with the measurements, in d i c a t i n g again that equation 2.112 appropriately accounted for the role of p a r t i c l e s i n a f f e c t i n g the bubble behaviour i n three-phase f l u i d i z e d beds. The e l e c t r o - r e s i s t i v i t y probe developed for t h i s study and tested successfully i n two-phase air-water flow, was used for the purpose of measuring the gas holdup inside the three-phase f l u i d i z e d bed d i r e c t l y . The operation of the probe was more troublesome than i n air-water flow, as i t became quite d i f f i c u l t to i d e n t i f y the signal corresponding to the gas phase. The glass beads, being non-conductive, made the datum fluctuate considerably. Thus, i n order to assure that only the signal corresponding to the gas phase was integrated, the cut-off l e v e l i n the comparator had to be raised from 0.05 v o l t to 0.5 vo l t above the datum. Measured gas holdup p r o f i l e s are shown i n Figure 4.2 3 and integrated average gas holdups are presented in Table 4.21. It can be seen that the two gas holdup p r o f i l e s i n Figure 4.23 are both a x i a l l y symmetric and well represented by a 0 .3 1.0 0.8 0.6 0.4 0.2 0 0 .2 0.4 0.6 0.8 1.0 K DIMENSIONLESS RADIAL DISTANCE, R FIGURE 4.23 COMPARISON OF GAS HOLDUP PROFILES COMPUTED BY EQUATION 2.27b WITH EXPERIMENTAL DATA (System: Air-water-1/2 mm glass beads; j 1 = 4.55) 271 TABLE 4.21 GAS HOLDUP BY ELECTRO-RESISTIVITY PROBE IN THREE-PHASE FLUIDIZED BED SYSTEM: Air-Water- 1/2 mm Glass Beads, <j,> = 4.5 5 cm/sec Gas Flux, <h> (cm/sec) a2C <a2> £2 Ratio (•e2/«x2>) M1 From Equation 4.2 3 4.54 0.127 0 .058 0.093 1.60 1.70 7.92 0 .228 0.100 0.153 1.52 1.60 9.94 0.235 0.119 0.187 1.57 1.87 272 equation 2.27b. A comparison of these p r o f i l e s with those obtained i n gas-liquid flow for similar gas and l i q u i d flow rates (cf. Figure 4.16) shows the gas holdup p r o f i l e s i n three-phase f l u i d i z e d beds to be more pointed, thus i n d i c a t -ing that the presence of a bed of 1/2 mm glass beads renders the r a d i a l gas holdup d i s t r i b u t i o n more non-uniform. The average gas holdup was obtained, as before, by integrating these p r o f i l e s according to equation 3.5. The average gas holdups, so calculated, were found to be con-s i s t e n t l y smaller than those measured by the pressure drop gradient and valve shut-off techniques. These consistent discrepancies are believed to have been caused by being forced to employ a high cut-off l e v e l i n the comparator, thereby introducing a small error i n the residence time of each bubble as recorded by the integrator; since no indepen-dent investigation was undertaken to i d e n t i f y the various sources of errors, these values of gas holdups are not shown in Figure 4.22.,, I t i s nevertheless believed that with some further modifications i n the design of the probe, along with a better method of processing the probe output, the e l e c t r o -r e s i s t i v i t y probe can be successfully used for measuring the l o c a l properties of the gas bubble phase in three-phase f l u i d i z e d beds. 273 ( i i i ) Air-water- 1 mm glass beads Visual observation of the f l u i d i z e d bed and the region above the bed showed the existence of f u l l y developed slugs for gas flow rates greater than 4 cm/sec at a l l the l i q u i d flow rates studied. The measured values of gas holdup are shown in Figure 4.24 along with the data curve for gas-l i q u i d flow alone. It i s clear that the gas holdup i n the three-phase f l u i d i z e d bed increases with increase i n gas flow rate, but the e f f e c t of l i q u i d flow rate appears to be either small or n e g l i g i b l e . The gas holdups i n the three-phase f l u i d i z e d bed are compared i n Table 4.22 with those i n various two-phase regions of the column, as well as with those i n gas-liquid flow alone, for the l i q u i d flow rate of 12.8 0 cm/sec: 1. A comparison of the gas holdups i n the two-phase regions above and below the t e s t section show them to be almost i d e n t i c a l to each other, at a given gas and l i q u i d flow rate. Therefore the average of a l l the gas holdup measure-ments i n the two-phase regions was taken to represent the gas holdup in these regions for the given flow rates. These gas holdups i n turn are equal to those measured i n the corresponding g a s - l i q u i d flow without s o l i d s . Thus the presence of a bed of 1 mm glass beads does not a f f e c t the gas holdup, and therefore presumably not the flow structure, i n air-water flow. 274 <j 2> , c m / s e c FIGURE 4.24 GAS HOLDUP IN THREE-PHASE BEDS OF 1 MM GLASS BEADS FLUIDIZED BY AIR AND WATER ( X - average of 5 repeated runs; generalized wake model with xk=G; — : two-phase air-water flow) T A B L E 4 . 2 2 COMPARISON OF GAS HOLDUP IN T H R E E - P H A S E F L U I D I 2 E D BED TO THAT IN TWO-PHASE REGIONS OF THE COLUMN S Y S T E M : A i r - W a t e r - 1 mm G l a s s B e a d s , < j ,> = 1 2 . 8 0 c m / s e c , W = 4 8 7 . 0 gm, e , | . = 0 . 1 1 7 Gas F l u x , < j 2 > ( c m / s e c ) Gas H o l d u p A v e r a g e Gas H o l d u p i n T w o - P h a s e R e g i o n s A v e r a g e Gas H o l d u p i n G a s - L i q u i d F l o w Gas H o l d u p i n T h r e e - P h a s e R e g i o n s B e l o w t h e T e s t S e c t i o n A b o v e t h e T e s t S e c t i o n in b 2 (measured) On S o l i d s ' F r e e B a s i s , e £ / ( l - e 3 ) 4 . 8 5 0 .102 0 . 1 0 0 0 . 1 0 0 0 .105 0 . 0 7 9 0 . 0 9 3 6 . 3 3 0 .128 0 . 1 2 3 0 .126 0 . 1 3 3 0 . 1 0 0 0 . 1 1 7 8 . 4 2 0 .168 0 . 1 5 5 0 . 1 6 3 0 .169 0 .124 0 . 1 4 5 1 0 . 0 6 0 .194 0 . 2 0 4 0 .192 0 .194 0 . 1 4 9 0 . 1 7 3 1 1 . 6 6 0 .234 0 . 2 3 0 0 .232 0 . 2 1 6 0 . 1 6 2 0 .188 to 276 2. The gas holdups i n the three-phase f l u i d i z e d bed as such are somewhat smaller than those i n gas-liquid flow alone; however, when represented on a s o l i d s - f r e e basis, they become almost equal to those i n gas-liquid flow. This then indicates that the gas- l i q u i d r a t i o remains almost unchanged from the two-phase region to the three-phase region. Therefore the bubble flow behaviour within the bed of 1 mm glass beads would appear to be l i t t l e d i f f e r e n t than i n the corresponding two-phase gas-liquid flow, as also noted e a r l i e r for the 1/2 mm p a r t i c l e s . The r a t i o s of gas holdups in three-phase f l u i d i z e d beds to those i n gas-liquid flow, from experimental measure-ments, are compared i n Table 4.2 3 with the predicted values from equations 4.25 and 2.120. Also included i n the table are the gas holdup r a t i o s calculated from the experimental measurements of Michelsen and 0stergaard [14] for 1 mm glass beads at a water flow rate of 7.8 cm/sec. I t i s seen i n Table 4.23 that, while the r a t i o s of measured gas holdups for the l i q u i d flow rate of 7.65 cm/sec are i n reasonably good agreement with those of Michelsen and 0stergaard, both of these r a t i o s are considerably larger than those predicted by either equation 4.25 or equation 2.120«On the other hand, the measured gas holdup r a t i o s for the l i q u i d flow rate of 12.8 0 cm/sec are i n better agreement with the predicted values TABLE 4.23 COMPARISON OF GAS HOLDUP IN THREE-PHASE FLUIDIZED BED TO THAT IN GAS-LIQUID FLOW SYSTEM: Air-Water- 1 mm Glass Beads <j„> = 2-12 cm/sec Liquid Flux, (cm/sec) _ Ml /p I I E 2 7 2 (Experimental) e "Ve " 2' 2 (Eq. 4.25) 2/ 2 (Eq. 2.120) P in /p I I E 2 / e 2 (Model with x k = 0) 2 / £2 [14] 7 .65 0.80 (0.72-0.91) 0.59 0.53 0.77 (0.74-0.79) -7.80 - - - - 0.73 (0.64-0.80) 12.80 0.82 (0.73-0.94) 0.76 0 .72 0.92 (0.90-0.94) -278 from both equation 4.25 and equation 2.120. Therefore only at the larger l i q u i d flux can i t be surmised that the e f f e c t of 1 mm glass beads on the bubble dynamics i s about the same in both large and small diameter columns. In order to check the a p p l i c a b i l i t y of the generalized wake model, the eight equations mentioned e a r l i e r were again solved numerically. The convergence for a l l the gas and l i q u i d flow rates studied was good, and the gas holdups calculated from the model with x^ = 0 are shown i n Figure 4.24 for l i q u i d flow rates of 7.65 and 12.8 0 cm/sec. The agreement between the measured and predicted values was quite good, although the predicted gas holdups for the l i q u i d flow rate of 12.80 cm/sec were somewhat high. Once again i t was observed that neither the wake volume f r a c t i o n , e^, nor the r e l a t i v e p a r t i c l e content of the wake, x^, had any s i g n i f i c a n t e f f e c t on gas holdups. However, an increase i n either the gas or the l i q u i d flow rate caused the gas holdup to increase. The predicted r a t i o s of gas holdup i n the three-phase f l u i d -ized bed to gas holdup i n the corresponding gas - l i q u i d flow, presented i n Table 4.23, show that, although the r a t i o varied somewhat with gas flow rate, the averaged values at a given l i q u i d flux were i n reasonably good agreement with the measured values. This then indicates again that equation 2.112 appropriately considers the r o l e of p a r t i c l e s i n a f f e c t i n g the bubble behaviour. 279 (iv) Air-water- 2 mm lead shot The v i s u a l observation of the f l u i d i z e d bed as well as the region above the bed showed that both bubbly and slug flow regimes occurred, depending primarily on the gas flow rate. The t r a n s i t i o n from bubbly to slug flow was observed to occur between the gas flow rates of 6-8 cm/sec at a l l the l i q u i d flow rates studied. Below the gas flow rate of 6 cm/sec, i n the bubbly flow regime (d^ - 5-10 mm), very l i t t l e coalescence occurred, e s p e c i a l l y at the higher l i q u i d flow rates. The gas holdups measured i n the three-phase f l u i d i z e d beds are shown i n Figure 4.25a-d. The gas holdup increased with increase i n gas flow rate at a l l the l i q u i d flow rates studied. No abrupt change i n gas holdup due to change i n flow regime was c l e a r l y noticeable, nor could the e f f e c t of an increase i n l i q u i d flow rate on the gas holdup be c l e a r l y established from the measured data. The values of gas holdup measured i n the three-phase f l u i d i z e d beds are compared i n Table 4.24 with the measured values i n various two-phase regions of the column and with those in gas-liquid flow alone, for the l i q u i d flow rate of 26.4 cm/sec: 1. A comparison of gas holdups i n the two-phase regions above and below the bed showed that they were not only equal to each other but were also equal to the corresponding gas holdups i n gas-liquid flow as c a l -culated from equation 4.18. This then indicates that oo 0.08 < 0 2 4 6 8 10 12 . 14 16 18 20 <j 2> , cm/sec 0 .20 0.16 oT D 0.12 O I 1 o 0.08 X CO < 0.04 0.00 ( 8 10 12 14- 16 18 2 0 <j2>, cm/sec 0.18 0.14 = 0.10 — N oj- 0.06 Q _J O x 0.16 CO < ^ 0.12 0.08 0.04 - < j , > = 1 1 1 1 1 17.8 cm/sec , i ? - ° > • 06-' O / y . \SIugf low -^Bubb ly flow (b) ' < j , > - 8.26 cm/sec x x X X | ^ ' -I • / ^ - \ S l u g flow ' 1 \Bubbly flow ( a ) i, J. i i i 0 2 4 6 8 10 12 <j 2> , cm/sec to 00 o FIGURE 4.25 GAS HOLDUP IN THREE-PHASE BEDS OF 2 MM LEAD SHOT FLUIDIZED BY AIR AND WATER ( generalized wake model with x^ = 0; arrows indicate observed change from bubbly flow to slug flow) T A B L E 4 . 2 4 COMPARISON OF GAS HOLDUP IN T H R E E - P H A S E F L U I D I Z E D BED TO THAT IN TWO-PHASE REGIONS OF THE COLUMN S Y S T E M : A i r - W a t e r - L e a d S h o t , < j 1 > = 2 6 . 4 0 c m / s e c , W = 3 4 1 4 . 0 gm, . _ 0 = 0 . 3 2 0 Gas F l u x , <v ( c m / s e c ) Gas H o l d u p A v e r a g e G a s H o l d u p i n T w o - P h a s e R e g i o n s A v e r a g e Gas H o l d u p i n G a s - L i q u i d F l o w Gas H o l d u p i n T h r e e - P h a s e R e g i o n s B e l o w t h e T e s t S e c t i o n A b o v e t h e T e s t S e c t i o n E 2 (measured) • On S o l i d s F r e e B a s i s , E £ / ( 1 - E 3 ) 2 .56 0 . 0 4 2 0 . 0 4 9 0 . 0 4 3 0 .043 0 .026 0 . 0 3 7 3 . 7 7 0 . 0 6 0 0 . 0 6 0 0 .060 0 .062 0 .074 0 .104 4 . 7 3 0 . 0 7 7 0 .078 0 . 0 7 7 0 .076 0 .081 0 . 1 1 4 6 . 0 0 0 . 0 9 7 0 . 0 9 7 0 .100' 0 .094 0 .100 0 . 1 3 8 6 .91 0 . 1 1 3 0 . 1 1 4 0 .115 0 .107 0 .139 0 . 1 9 3 7 . 7 3 0 . 1 2 5 0 . 1 2 1 0 . 1 2 3 0 .118 0 .119 0 . 1 6 8 8 . 1 5 0 .134 0 .128 0 .132 0 .123 0 .090 0 .127 9 . 1 7 0 . 1 4 8 0 . 1 4 2 0 .148 0 .136 0 .118 0 .168 1 5 . 6 0 0 . 2 0 5 0 . 2 0 0 0 . 2 0 0 0 .208 0 .163 0 .227 2 0 . 5 0 0 . 2 6 6 0 .254 0 .262 0 .253 0 .209 0 . 2 8 6 CO 282 the gas-liquid r a t i o i n the two-phase regions remained unaffected by the presence of 2 mm lead shot. 2. A comparison of gas holdups i n the three-phase f l u i d i z e d beds to those i n the two-phase regions showed that the absolute values of the gas holdups i n the three-phase f l u i d i z e d beds as measured were inconsistently smaller than, equal to, or greater than the corresponding values i n gas-liquid flow up to a gas flow rate of 8 cm/sec, above which the three-phase values were consistently smaller. However, almost a l l the gas holdups i n the three-phase f l u i d i z e d beds, when represented on a s o l i d s -free basis, were found to be greater than those i n gas-l i q u i d flow alone, in d i c a t i n g that the gas-liquid r a t i o within the bed was considerably affected by the r e l a t i v e l y large and heavy s o l i d p a r t i c l e s . The r a t i o s of gas holdups measured i n the three-phase f l u i d i z e d beds to those measured (<j^> = 17.8 0 cm/sec) or predicted (<j^> = 26.40 cm/sec) i n gas - l i q u i d flow are com-pared i n Table 4.25 with the predicted values from equations 4.25 and 2.120. The former values were many times greater than those predicted by either equation 4.2 5 or equation 2.120. Since neither of these equations was based on experimental data for large and heavy p a r t i c l e s , and since no consideration was given to bubble dynamics i n t h e i r development, t h e i r f a i l u r e to predict for such systems was 283 TABLE 4.25 COMPARISON OF GAS HOLDUP IN THREE-PHASE FLUIDIZED BED TO THAT IN GAS-LIQUID FLOW SYSTEM: Air-Water- Lead Shot <j„> = 2.5-21.0 cm/sec Liquid Flux, ^ 1 * (cm/sec) e2 e2 (Experimental) b2 2^ (Eq. 4.25) ( . . i i _ II e2 e2 (Eq. 2.120) ( - H - i i e2 e2 (Model) 17 .8 0.92 0.13 0.37 0.73 (0.73-1.02) (0 .63-0.86) 26.4 0.93 0.17 0.50 0.84 (0.73-1.20) (0.75-0.99) 284 not unexpected. The measurement of gas holdup for a s i m i l a r system of large heavy p a r t i c l e s has not been reported by any of the e a r l i e r investigators, and therefore no further comparisons could be made. In order to check the a p p l i c a b i l i t y of the generalized wake model,.the following assumptions were made based on v i s u a l observations and experimental measurements: (i) The bed expansion c h a r a c t e r i s t i c s of lead shot f l u i d i z e d by water are well represented by the dimensional c o r r e l -ation of Trupp [87], as w i l l be shown i n Section 4.2.3.1. Therefore equation 2.106, used for describing the voidage i n the p a r t i c u l a t e phase of the three-phase f l u i d i z e d bed, i s modified for the air-water-lead shot system to 'If <j x> - v 2 d - x k ) £ k 0.36 v i - 1 8 ( l - e2- ek) 1/2.28 (4.26) ( i i ) Since both the bubbly and the slug flow regimes were observed to e x i s t , the d r i f t v e l o c i t y for the bubbly flow regime i s assumed to be given by V 2 j = ( V O O > B ^ tanh [0.25 (l/e») 1 / 3] ( 2 ' 3 7 ) where (V^)^ i s calculated from equation 2.10 for an assumed bubble radius, r& = 4 mm (d^ = 5-10 mm), while equation 4.8 i s used for the slug flow regime as before. 285 The other equations i n the model remained the same as before and a l l eight equations were solved numerically. The convergence for a l l the gas and l i q u i d flow rates was s a t i s f a c t o r y , and the gas holdups calculated from the model with = 0 are shown i n Figure 4.25a-d. The calculated values showed a break at the t r a n s i t i o n point between regimes because a constant rather than a steadily increasing bubble radius was assumed i n the bubbly flow regime. The measured values, on the other hand, showed a gradual s h i f t from the curve for the bubbly flow regime towards the curve for the slug flow regime, e s p e c i a l l y at high l i q u i d flow rates (Figure 4.25d). The agreement between the calculated and the predicted values was i n general good. The e f f e c t of l i q u i d flow rate on the predicted values of gas holdup was small. The r a t i o of the predicted gas holdup i n the three-phase f l u i d i z e d bed to that i n the corresponding gas-liquid flow, presented i n Table 4.25, was found to increase with increasing gas flow rate i n the bubble flow regime (approach-ing a value near unity at < j 2 > ~ 7 cm/sec), but remained p r a c t i c a l l y constant (but less than unity) i n the slug flow regime. The averaged values were also i n considerably better agreement with the measured values than those pre-dicted by either equation 4.25 or equation 2.120. This then indicates that equation 2.112 appropriately incorporates the e f f e c t of 2 mm lead shot on the air-water bubble behaviour in three-phase f l u i d i z e d beds. 286 (v) Air-PEG solution - 1 mm glass beads The gas holdup data obtained with t h i s system were limited because a stable three-phase f l u i d i z a t i o n could not be achieved due to consistent e l u t r i a t i o n of glass beads even at a gas flow rate as low as 1 cm/sec. A l l the possible gas holdup measurements were obtained from either the pressure drop measurements (gas holdups were too small to be measurable by the valve shut-off technique) or the e l e c t r o -r e s i s t i v i t y probe measurements, and are shown in Figure 4.26. Also shown i s the predicted curve from the generalized wake model, assuming slug flow with x^ = 0 (the same eight equations as l i s t e d under the section for 1/4 mm glass beads above were solved numerically). I t can be seen i n Figure 4.2 6 that, although the scatter i n the data i s quite large, the values predicted by the generalized wake model are i n reasonable agreement with the values obtained from the e l e c t r o - r e s i s t i v i t y probe measurements. The l a t t e r cannot, however, be validated i n view of the f a i l u r e of the probe to give r e l i a b l e r e s u l t s i n the air-water- 1/2 mm glass beads system. Some of the c h a r a c t e r i s t i c measurements obtained with the e l e c t r o - r e s i s t i v i t y probe are shown in Table 4.26 and Figure 4.27. The r e s u l t s support the optimistic view that, with further refinement, the e l e c t r o - r e s i s t i v i t y probe can be successfully used for measuring the l o c a l properties of the gas-bubble phase inside the three-phase 287 0.15 — CM MJ CL ZD Q - J O X 0.10 3 0.05 0.00 — : — , j — cm/sec x 0.41 o 0.88 A |.|4 1 measured by E R P • X -X o m O , ^y y o o $ y y ' yy S l — » 0 I <L> , cm/sec FIGURE 4.26 GAS HOLDUP IN THREE-PHASE BEDS OF 1 MM GLASS BEADS FLUIDIZED BY AIR AND PEG SOLUTION (generalized wake model with x k=0: j1=0.41, J!=1.14) 288 TABLE 4.2 6 CHARACTERISTIC MEASUREMENTS OF GAS-BUBBLE PHASE IN THREE-PHASE FLUIDIZED BED BY ELECTRO-RESISTIVITY PROBE SYSTEM: Air-PEG Solution - 1 mm Glass Beads, <j,> = 0.86 cm/sec Gas Flux, <j 2> (cm/sec) <a2> r e from Eq. 8.3.6 (cm) 6 (-) 0.58 0.017 1.56 0.16 0.71 0.021 1.21 0.17 1.63 0 .037 1.73 0.17 1.98 0.050 2.62 0.16 289 CJ cc 0.12 0.10 O 0.08 0.06 & 0.04 < Q 0.02 0.00 = 0.86 cm/sec. \ \ 1 A ^-A-A^ 6 \ \ 0 0.0 0.2 0.4 0.6 0.8 1.0 FIGURE 4.27 GAS HOLDUP PROFILES IN THREE-PHASE FLUIDIZED BED (System: air-PEG solution - 1 mm glass beads; o -J!=0.58; A - j ^ l . 9 8 ) 290 f l u i d i z e d bed. For example, the average bubble r a d i i shown in Table 4.26,.obtained from equation 8.3.6 using the bubble frequency data, are quite r e a l i s t i c and did compare well with the v i s u a l observations. (vi) Air-PEG s o l u t i o n - s t e e l shot The v i s u a l observation of the f l u i d i z e d bed and the region above the bed showed that slug flow occurred at a l l the gas and l i q u i d flow rates studied. The length of the slugs observed i n the t e s t section increased with increasing gas flow rate, reaching a length of about 15 cm for a gas flow rate of about 9 cm/sec, when excessive bed fluctuations were noted. No further measurements were therefore attempted. The gas holdups measured in the three-phase f l u i d i z e d bed are shown i n Figure 4.28 along with the data for two-phase air-PEG solution flow, at <jj> = 13.82 cm/sec. I t can be seen from this figure that the gas holdup increased quite markedly with increase i n gas flow rate but decreased, though only s l i g h t l y , with an increase i n the l i q u i d flow rate. The gas holdups measured i n the three-phase f l u i d i z e d beds for the l i q u i d flow rate of 13.82 cm/sec are compared in Table 4.27 with the gas holdups measured simultaneously in the two-phase regions of the column, as well as with those measured i n gas-liquid flow. I t i s important to point out that the measurement of gas holdup by the pressure drop gradient method was i n th i s case complicated by the presence of a measurable f r i c t i o n a l pressure drop during the flow of i 291 0 . 2 <j 2> , cm/sec FIGURE 4.28 GAS HOLDUP IN THREE-PHASE BEDS OF 3 MM STEEL SHOT FLUIDIZED BY AIR AND PEG SOLUTION (generalized wake model with xj^O: 13.82, •j1=18.84; two-phase a i PEG solut i o n flow) T A B L E 4.27 COMPARISON OF GAS HOLDUP IN T H R E E - P H A S E F L U I D I Z E D BED TO THAT IN TWO-PHASE REGIONS OF THE COLUMN S Y S T E M : A i r - P E G S o l u t i o n - S t e e l S h o t , <J1> = 13.82 c m / s e c ; W = 1078 gm, e,| . . = 0.139 3h2=0 Gas F l u x , ^2* ( c m / s e c ) G a s H o l d u p A v e r a g e Gas H o l d u p i n T w o - P h a s e R e g i o n s A v e r a g e Gas H o l d u p i n G a s - L i q u i d F l o w Gas H o l d u p i n T h r e e - P h a s e R e g i o n Be low t h e T e s t S e c t i o n A b o v e t h e T e s t S e c t i o n e" 2 (measured) On S o l i d s - F r e e B a s i s , e^/U-e-j) 4.89 0.063 0.053 0.062 0.104 0.033 0.039 7.37 0.097 0.106 0.105 0.146 0.094 0.113 9.33 0.134 0.130 0.138 0.178 0.152 0.183 J VO the l i q u i d phase alone at these v e l o c i t i e s . Although the gas holdups were calculated from these measurements by suitably subtracting the f r i c t i o n a l pressure drop for the l i q u i d phase alone [104], they were found to be consistently larger than the gas holdups by the valve shut-off technique. The averaged gas holdups i n the two-phase regions above and below the test section, though equal to each other, were s i g n i f i c a n t l y smaller than the corresponding gas holdups in g a s - l i q u i d flow without s o l i d s , as shown i n Table 4.27. This indicates that the introduction of 3 mm s t e e l shot reduced the gas holdup not only above, but even below, the bed. At the same time there was maintained an approximate equality of gas holdups i n the two-phase regions with those i n the three-phase f l u i d i z e d beds represented on a s o l i d s -free basis. The e l e c t r o - r e s i s t i v i t y probe was used for a few experiments to determine the average bubble size by means of equation 8.3.6 and measurements of bubble frequency. I t was found that the average bubble size i n the three-phase f l u i d -ized bed was smaller than that i n the corresponding gas-l i q u i d flow, as shown in Table 4.28. Apparently the decrease in gas holdup due to so l i d s i s accompanied i n th i s instance by a decrease i n slug length, an e f f e c t which i s known to increase the slug r i s e v e l o c i t y [27]. In order to check the a p p l i c a b i l i t y of the generalized wake model, the gas holdup was calculated by numerically 294 TABLE 4.28 COMPARISON OP AVERAGE BUBBLE SIZE IN THREE-PHASE FLUIDIZED BED TO THAT IN GAS-LIQUID FLOW SYSTEM: Air-PEG Solution - Steel Shot, <J 1 >= 13.82 cm/sec Gas Flux, Average Bubble Size , r , cm e < V (cm/sec) Three-Phase F l u i d i z e d Bed Gas-Liquid Flow ( c f . Table 4.16) 2.17 1.5 2.3 4.89 2.7 3.2 295 solving the eight equations l i s t e d under the section for 1/4 mm glass beads. The calculated values for x^=0 are shown in Figure 4 .28. The agreement with the measured values i s rather poor, but the calculated values nevertheless show the gas holdup to increase with increasing gas flow rate and to decrease with an increase i n the l i q u i d flow rate. This agreement with the trends of the measurements would seem to indicate the q u a l i t a t i v e correctness of the proposed model. 4.2.3 Voidage re suits In t h i s section the measurements of s o l i d s holdup (e^ = 1-e) i n l i q u i d - s o l i d and i n g a s - l i q u i d - s o l i d f l u i d i z e d beds are reported i n both graphical and tabular form. The experimental data from which the graphs and tables were pre-pared are presented i n Appendix 8.7. The physical properties of a l l the s o l i d s and l i q u i d s used i n t h i s study are recorded i n Appendix 8.6. 4.2.3.1 Voidage in l i q u i d - s o l i d f l u i d i z e d beds The measurements of s o l i d s holdup i n the 20 mm glass column as well as i n the 2 inch Perspex column were under-taken primarily to e s t a b l i s h the relationships for describing the expansion c h a r a c t e r i s t i c s of l i q u i d - s o l i d f l u i d i z e d beds, such that the same relationships could then be used for 296 describing the voidage i n the pa r t i c u l a t e phase of a three-phase f l u i d i z e d bed. (A) 20 mm glass column The measured bed voidages for the three systems studied, obtained from the measurements of s o l i d s holdup i n l i q u i d -s o l i d f l u i d i z e d beds, are presented i n Table 4.2 9 along with the predicted values from the Richardson-Zaki c o r r e l a t i o n , equation 2.46, as well as from the Neuzil-Hrdina c o r r e l a t i o n , equation 2.51. For beds of 1 mm glass beads f l u i d i z e d either by water or by aqueous g l y c e r o l , the measured values of bed voidage were on the average about 12% greater than the predicted values from equation 2.46, using equation 2.4 9 for evaluating the exponent n. The agreement was improved to within 5% i f the wall correction factor recommended by Richardson and Zaki [2] was employed. However, using the Neuzil-Hrdina c o r r e l a t i o n , equation 2.51, for predicting the bed voidage i n confined media, the agreement between the measured and the predicted values was found to be excellent, as shown i n Table 4.29. Therefore equation 2.51 was used subsequently for characterizing the expansion behaviour i n the p a r t i c u l a t e phase of a three-phase f l u i d i z e d bed of 1 mm glass beads. The measured values of bed voidage i n water f l u i d i z e d beds of 1/2 mm sand p a r t i c l e s were about 15% greater than the predicted values from equation 2.46, using equation 2.49 TABLE 4.29 EXPANSION RESULTS FOR LIQUID-SOLID FLUIDIZATION IN 20 MM COLUMN F l u i d i z a t i o n System d P (mm) Terminal Free S e t t l i n g Reynolds Number/ Re (-) P Liquid Flux (cm/sec) Measured Voidage,e (-) ' Predicted Voidage i e<2> Glass Beads Water 1 .08 191 .4 3 .04 0 .586 0 .512 0 .580 4 .01 0 .641 0 568 0 .642 4 .8.1 0 .700 0 .609 0 .687 5 .22 0 .707 0 .628 0 .708 6 .02 0 .754 0 .663 0 .747 6 .42 0 .769 0 680 0 .765 6 .82 0 .786 0 .696 0 .782 8 .02 0 .831 0 740 0 .830 Glass Beads Aqueous- 1 08 74 .58 4 .06 0 .724 0 657 0 .718 Glycerol 4 .86 0 777 0 699 0 .767 Solution 6 .03 0 .826 0 753 0 .831 Sand-Water 0 .458 34 .0 1 67 0 .751 0 630 0 .751* 2 03 0 .787 0 671 0 .751* 2 38 0 .823 0 706 0 .822* 2 71 0 .844 0 737 0 .850* 3 04 0 873 0. 764 0 .8 74* (1) Calculated from equation 2.46 with n = 4.45 Re P (2) Calculated from equation 2.51 * Calculated from modified Neuzil-Hrdina c o r r e l a t i o n , <J1>AM = 0.67 R e p ° ' 0 3 [1-1.27 (dp/D) 1* 1 5] e±3 , 9 5 (4.27) VO 298 for evaluating the exponent n. Equation 2.51 could not be used i n t h i s case since the free s e t t l i n g Reynolds number of the sand p a r t i c l e s was outside the recommended range of i t s applicability»• However the success of equation 2.51 i n predicting the voidage i n beds of 1 mm glass beads suggested the idealof modifying the exponent on i n equation 2.51 to f i t the experimental data. From a log-log p l o t of measured bed voidage against l i q u i d f l u x , Figure 4.29, the slope of the r e s u l t i n g straight l i n e was found to be 3.95 (as compared with the value of 3.21 predicted by equation 2.49). There-fore the Neuzil-Hrdina c o r r e l a t i o n was modified to = 0.67 R e 0 , 0 3 [1-1.27 (d / D ) 1 , 1 5 ] e3*95 (4.27) P P 1 The voidages predicted by equation 4.27 were found to be i n almost perfect agreement with the measured values, as shown in Table 4.29. Therefore equation 4.27, instead of equation 2.51, was used for characterizing the expansion behaviour in the p a r t i c u l a t e phase of a three-phase f l u i d i z e d bed of 1/2 mm sand p a r t i c l e s . (B) 2 inch perspex column The measurements of bed voidages, presented i n Table 4.30, were for a wide range of p a r t i c l e sizes (1/4 - 3 mm) 3 and p a r t i c l e densities (2.8 - 11.3 gm/cm ), and with the two 299 FIGURE 4.29 THE EXPANSION CHARACTERISTICS OF 1/2 MM SAND PARTICLES FLUIDIZED BY WATER TABLE 4.30 EXPANSION RESULTS FOR LIQUID-SOLID FLUIDIZATION IN 2 INCH COLUMN F l u i d i z a t i o n System d P Terminal Free S e t t l i n g Reynolds Number, Re (-) ' p Liquid Flux (cm/sec) Measured Voidage,e (-) Predicted Voidage e ( 2 ) Glass Beads Water 0.273 10.2 1.25 1.87 2.23 3.18 0.763 0.834 0.889 0.719 0.804 0.844 0.931 Glass Beads Water 0.456 28 .0 1.59 3.52 4.55 5.71 0.635 0.805 0.866 0.922 0.628 0.799 0.864 0.925 Glass Beads Water 1.08 2 02 .2 202 .2 191.4 202 .2 202 .2 6.25 6.97 7.02 7 .65 12 .8 0.706 0.733 0.738 0.760 0.879 0.680 0.706 0.724 0.730 0.875 Lead Shot-Water 2 .18 1762 .5 8.26 17 .80 26.40 38.77 0.417 0.583 0.683 0.808 0.385* 0.531* 0.626* 0.735* 0.407 0.570 0.677 0.801 Glass Beads PEG Solution 1.08 0.36 0.36 0.40 0.40 0.86 1.14 0 .732 0.846 0.886 0.7291 0.853t 0.893t Steel Shot PEG Solution 3.18 15.47 15.47 13.82 18 .84 0.865 0.938 0.837 0.901 (1) Calculated from equation 2.4 6 with n = (4.45 + 18 d /D) Re t with n = (4.35 + 17.5 pd /D) PRe"0.03 * with n = 2.39 P p (2) Calculated from equation 4.2 8 301 li q u i d s used, they represented a range of Reynolds number (0.36 < Re < 1770) from the Stokes 1 to the Newton's law p regimes. Since a l l the experiments were carr i e d out i n the one column, the r a t i o of p a r t i c l e to column diameter varied from 0.005 to 0.063. These ranges of variables investigated were well within the range of a p p l i c a b i l i t y of the Richardson-Zaki c o r r e l a t i o n (equation 2.46), but were outside the recommended range of a p p l i c a b i l i t y of the Neuzil-Hrdina c o r r e l a t i o n (equation 2.51). Therefore the l a t t e r could not be employed when wall e f f e c t s became important. Instead, the equations recommended by Richardson and Zaki [2] were used and these are: <JI>/V: = (2.46) where = [4.35 + 17.5 d /D] Re P P -0.03 0.2 < Re„ < 1 P n (2 .48a) = [4.45 + 18 d /D] Re P P -0.1 1 < Re < 200 P n (2.49a) n = 2.39 Re > 500 P (2.50) and - V M, the terminal v e l o c i t y of p a r t i c l e s i n an i n -f i n i t e medium estimated from standard relationships [27]. 302 The following discussion of experimental findings r e s u l t s from the measurements conducted for each of the f l u i d i z a t i o n systems studied. glass beads - water The values of bed voidage reported i n Table 4.30 were i n most cases averages of voidage obtained by two methods, v i z . the expanded bed height method and the pressure drop gradient method. The measurements i n f l u i d i z e d beds of 1 mm glass beads did not provide any d i f f i c u l t i e s and the voidage obtained by either method was s a t i s f a c t o r y . The s o l i d s holdup remained p r a c t i c a l l y constant from the bottom to the top of the bed, and the expanded bed height obtained from the longitudinal pressure drop p r o f i l e agreed with the measured bed height by v i s u a l observation (Figure 4.30). However, at large bed expansions i n f l u i d i z e d beds of 1/4 mm glass beads, the so l i d s holdup was found to vary along the bed, being higher than average near the bottom and lower near the top. The longitudinal pressure drop p r o f i l e and the solids holdup d i s t r i b u t i o n (calculated from the d i f f e r -e n t i a l pressure drop across a d i f f e r e n t i a l section of the bed from the unsmoothed data) for such a bed are shown i n Figure 4.31. As can be seen, the s o l i d s holdup along the bed decreases considerably with distance from the bottom and appears to be responsible for the marked difference i n 303 0.15 FIGURE 4.30 0 5 10 15 20 25 30 35 DISTANCE FROM TAP 1, Z, in. ESTIMATION OF EXPANDED BED HEIGHT AND SOLIDS HOLDUP DISTRIBUTION FROM LONGITUDINAL PRESSURE DROP PROFILE IN BED OF 1 MM GLASS BEADS FLUID-IZED BY WATER (W=536.4 gm; j1=12.8 0 cm/sec; L b , 0 = 1 5 , 4 c m ) 304 0.2 O.I -to 0.0 ~ 30 x 1 2 6 < UJ 18 UJ o o 14 10 x X -I r x x x o l 6 * 5 8 -Zmax Visually L b = 88 .3 cm . (Zmax)cm+ 8.7=71.4 cm FIGURE 4.31 5 10 .15 20 25 30 35 40 DISTANCE FROM TAP I, Z, in. ESTIMATION OF EXPANDED BED, HEIGHT AND SOLIDS HOLDUP DISTRIBUTION FROM LONGITUDINAL PRESSURE DROP PROFILE IN BED OF 1/4 MM GLASS BEADS FLUIDIZED BY WATER, (W=739.3 gm; j.^1.87 cm/sec; L^ Q=21.5 cm) 305 bed height obtained by the two methods. The dispersion i n s o l i d s holdup may have been caused by si z e segregation in the screen cut selected (0.25 - 0.175 mm). The expanded bed height chosen for determination of average bed voidage in this case, as well as i n other si m i l a r cases, was the one from the longitudinal pressure drop p r o f i l e . For f l u i d i z e d beds of 1/2 mm glass beads the differences were not so great (see Figure 3.10), but nevertheless the same procedure was adopted. The measured values of bed voidage for the glass beads-water system showed reasonable agreement with the predictions from the Richardson-Zaki c o r r e l a t i o n , with the exponent n calculated from equation 2.49a, and was therefore used for describing the voidage i n the p a r t i c u l a t e phase of three-phase beds of glass beads f l u i d i z e d by a i r and water. lead shot- water The two methods used for studying the bed voidage gave excellent agreement with each other, since the observed bed height was found to be i n almost perfect agreement with the bed height obtained from the longitudinal pressure prop p r o f i l e , as i l l u s t r a t e d i n Figure 4.32. The f l u i d i z e d bed was observed to expand smoothly, although the existence of agglomerates could not always be ruled out. The measured values of bed voidages were found to be about 10% higher than the predictions from equation 2.46 with n = 2.39. For f l u i d i z e d beds of such large heavy p a r t i c l e s , Trupp [87] 306 T 1 1 1 r~—•—I 5 10. 15 20 25 30 35 40 DISTANCE FROM TAP 1, Z, in. FIGURE 4.32 ESTIMATION OF EXPANDED BED HEIGHT FROM LONGI-TUDINAL PRESSURE DROP PROFILE IN BED OF 2 MM LEAD SHOT FLUIDIZED BY WATER (TBE=1,1,2,2-tetra-bromo-ethane; W=3245.0 gm; j1=38.77 cm/sec; L, A=25.6 cm) b, 0 307 recommended the c o r r e l a t i o n <j x> = 0.36 ( V j 1 , 1 8 e 2 , 2 8 (4.28) for Re > 500 and d /D < 0.06, based on the assumption P P that the turbulence generated by the p a r t i c l e s affected the bed expansion behaviour. The predictions from equation 4.28 were found to be i n excellent accord (Table 4.30) with the measured values, and therefore equation 4.2 8 was sub-sequently used for describing the voidage i n the p a r t i c u l a t e phase of three-phase f l u i d i z e d beds of lead shot. s o l i d p a r t i c l e s - PEG solution The f l u i d i z a t i o n of two s o l i d p a r t i c l e species (1 mm glass beads and 3 mm s t e e l shot) with PEG solution provided smooth and uniform bed expansion at a l l l i q u i d flow rates. As shown i n Table 4.30, the measured values of bed voidage were found to be i n good agreement with the values predicted from equation 2.46, using calculated values of the exponent n from equations 2.48a and 2.49a as appropriate. Therefore equation 2.4 6 was used subsequently for describing the voidage in the p a r t i c u l a t e phase of the corresponding three-phase f l u i d i z e d beds. 308 4.2.3.2 Voidage in three-phase f l u i d i z e d beds The voidage i n three-phase f l u i d i z e d beds of tes t p a r t i c l e s was measured i n both the 20 mm i . d . glass column and the 2 inch i . d . perspex column. For the studies i n the 2 0 mm glass column, the height of the three-phase f l u i d -ized bed was obtained by locating the bed boundary v i s u a l l y , while the bed height i n the 2 inch perspex column was evaluated from the measured pressure drop p r o f i l e . The values of bed height calculated by the l a t t e r method were found to be r e a l i s t i c and reproducible, even at the high gas flow rates when the bed boundary could no longer be located from v i s u a l observation with any degree of confidence. From the measurement of expanded bed height, L^, the s o l i d s holdup was calculated from equation 3.1 and then the bed voidage was obtained from equation 1.3. (A) 20 mm glass column The experimental findings for the three systems studied are described below. (i) Air-water - 1/2 mm sand The f l u i d i z a t i o n of sand p a r t i c l e s by water alone produced f a i r l y uniform beds. On introduction of a i r at small flow rates, keeping the l i q u i d flow rate constant, the bed was found to contract quite noticeably, the amount of con-t r a c t i o n being dependent on the i n i t i a l degree of bed 309 expansion of the l i q u i d - s o l i d f l u i d i z e d bed. Thus, for highly expanded beds (<j^> = 3.04 cm/sec), the observed bed contraction amounted to about 34% of the i n i t i a l bed height, whereas for moderately expanded beds (<j^> = 1.67 cm/sec) the observed bed contraction was only 14% (Figure 4.33) . The minimum value of bed voidage was observed to occur at a volumetric gas flux of about 1 cm/sec for a l l the l i q u i d flow rates studied. A further increase i n the gas flow rate, for a fixed l i q u i d flow rate, produced a slow but gradual bed expansion. Although no p a r t i c l e entrainment was observed, the bed boundary became quite d i f f u s e on increasing the gas flow rate above <J2> ~ 2.5 cm/sec. No data were therefore taken for much higher gas flow rates. In order to check the a p p l i c a b i l i t y of the generalized wake model, the eight equations l i s t e d e a r l i e r i n Section 4.2.2.2 (Equations 2.94, 2.112, 2.108a, 4.8, 2.106, 4.7, 1.3 and 2.91) were solved numerically with the same assumptions as c i t e d o r i g i n a l l y , except that the voidage i n the p a r t i c u l a t e phase was assumed to be described by equation 4.27 instead of equation 2.46, as discussed above i n Section 4.2.3.1. The curves of bed voidage calculated from the model for various assumed values of x^ are shown i n Figure 4.33.From a comparison of the predicted values of bed voidage with the measured values, i t i s seen that: 310 LU CD < Q O > U J 00. 0.88 0.84 0.80 0.76 0.72 0.84 0.80 0.76 0.72 0.68 /Xk=0.4 \ o ° — • \ \ ^s \<k=0.0 ^Xk=0.6-\ \ • D° V D /X,=0.4 \ cK • \ ^ -~ \ NXk=0.0 © \ ® ' • -~ \ ;s xxk=o.o \ s \ A A * *"A ""* \ \ \ \xk=o.o ^Xk=0.6 •/Xk=0.4 A •Xk=0,6 <] >, cm/sec A 1.67 ° 2.03 © 2.71 o 3.04 0 1 2 3 4 <j2> , cm/sec FIGURE 4.33 BED VOIDAGE IN THREE-PHASE BEDS OF 1/2 MM SAND PARTICLES FLUIDIZED BY AIR AND WATER IN 20 MM GLASS COLUMN ( generalized wake model) 1. The predictions also show minima in the gas voidage, but,these predicted minima occur at a volumetric gas flux of between 1 and 2 cm/sec, which i s up to twice the corresponding observed gas fluxes. 2. The measured bed voidages l i e between the predicted values for x^ = 0.0 and x^ = 0.6, the i n i t i a l bed contraction being closer to the predicted values for x^ = 0.0 and the l a t e r bed expansion f a l l i n g between the predicted values for x^ = 0.4 and x^ = 0.6. The implication of t h i s r e s u l t , according to the model, i s that p a r t i c l e entrainment i n the bubble wakes increases as gas flux increases. ( i i ) Air-water- 1 mm glass beads The behaviour of water f l u i d i z e d beds of 1 mm glass beads, on introduction of a i r at small flow rates, varied according to the l e v e l of i n i t i a l bed expansion. Thus for a s l i g h t l y expanded bed (<j^> = 4.01 cm/sec), no bed contraction was observed; the bed height remained nearly constant for < J 2 > UP t o 1 cm/sec, and thereafter the bed expanded smoothly as the a i r flow rate was further increased (Figure 4.34). For a l l other l i q u i d flow rates studied, the bed was found to contract when a i r was introduced at small flow rates; however, the amount of bed contraction was not as large as for the air-water-sand system. The 312 LLI < Q Q UJ CD 0.82 0.78 0.74 0.70 0.66 0.84 0.80 0.76 0.7.2 0.68 0.64 0.60 , X K = 0 .6 , - X K = 0 . 8 - X K = 0 . 6 X ,= 0 . 6 - - - X K = 0 . 4 - ~ X K = 0 . 2 A ' ' — — ""A '•A -~ X W = 0 . 6 — X K = 0 . 4 J© X K = Q 6 / 5-0°" / O Q-6 ° ^ = 1 . 0 X K = 0 . 8 ' O Cr P" 0 1 2 3 4 5 6 <j2> , cm/sec 7 8 FIGURE 4.34 BED VOIDAGE IN THREE-PHASE BEDS OF 1 MM GLASS BEADS FLUIDIZED BY AIR AND WATER • IN 20 MM GLASS COLUMN ( generalized wake model) 313 maximum bed contraction was only 15% of the i n i t i a l bed height for the highest l i q u i d flow rate (<j^> = 8.02 cm/sec) investigated. The minima i n the measured values of bed voidages occurred at gas flow rates of between 0.5 (for <j 1> = 4.81 cm/sec) and 1.5 cm/sec (for <jj> = 8.02 cm/sec). The upper bed boundary eventually became d i f f u s e on increas-ing the gas flow rate, e s p e c i a l l y for the i n i t i a l l y more expanded beds. For c a l c u l a t i n g the bed voidages from the generalized wake model, i t was assumed that the voidage i n the pa r t i c u -l a t e phase could be described by equation 2.51 instead of equation 2.46, as discussed e a r l i e r in section 4.2.3.1. Other assumptions were the same as those c i t e d previously in Section 4.2.2.2. The curves of bed voidage calculated from the model for various assumed values of x^ are shown i n Figure 4.34 . As can be seen, the agreement between the predicted and the measured values i s excellent i f a proper value of x, can be assumed. I t was found that, whereas a large value of x^ (- 1.0) was required to match the data for dense beds (<j-^> = 4.01 cm/sec), decreasingly smaller values were needed for progressively less dense beds, the highly expanded bed at <j-j_> = 8.02 being matched by a value of x^ - 0.2. The implication of t h i s r e s u l t i s that p a r t i c l e entrainment i n the bubble wakes increases with increasing p a r t i c l e concentration of the l i q u i d phase, an implication which i s q u a l i t a t i v e l y reasonable. However, 314 no d i r e c t measurements on the p a r t i c l e content of wakes were made i n t h i s study nor are such measurements available i n the l i t e r a t u r e . Nevertheless the generalized wake model does i l l u s t r a t e the e f f e c t of wake p a r t i c l e s on the general bed behaviour of three-phase f l u i d i z e d beds, i n contrast to the e a r l i e r models of 0stergaard [8], who assumed x k = 1.0 with no s o l i d s c i r c u l a t i o n , and Efremov and Vakhrushev [16], who assumed = 0. ( i i i ) Air-aqueous g l y c e r o l - 1 mm glass beads The f l u i d i z a t i o n of glass beads by aqueous gl y c e r o l solution gave a uniformly f l u i d i z e d bed which expanded smoothly as the volumetric l i q u i d flux was increased. On introduction of a i r into the l i q u i d - s o l i d f l u i d i z e d bed at small flow rates, the bed was found to contract at a l l l i q u i d flow rates. The extent of bed contraction was again found to depend on the i n i t i a l degree of bed expansion of the l i q u i d - s o l i d f l u i d i z e d beds; thus for highly expanded beds ( <j-j_ > = 6.03 cm/sec), the observed bed contraction was about 14% of the i n i t i a l bed height, whereas for moderately expanded beds (<j^> = 4.06 cm/sec), the observed bed contraction was only 7%. Although no entrainment by, or attach-ment to, the bubbles was observed, the bed boundary became quite d i f f u s e at comparatively smaller a i r flow rates than for the air-water-g-Iass beads system. The voidage was calculated from the generalized wake model i n the same manner as for the air-water-glass 315 beads system, and the predicted values for various assumed values of x^ . are shown i n Figure 4.35. With x^ again as an adjustable parameter, the agreement between the predicted and the measured values i s excellent, exhibiting the same trends as were found for the air-water-glass beads system. A d i r e c t comparison of the measured and the predicted values of bed voidages in the air-water-glass beads and the air-aqueous glycerol-glass beads systems, for si m i l a r values of <jn>/V or the i n i t i a l bed expansion, i s shown i n Figure 4.36. I t can be seen that both the measured and the predicted values for the two systems are i n excellent agree-ment with each other, for intermediate bed expansion. For lower bed expansion, the measured values i n the air-water-glass beads system are somewhat smaller than those i n the air-aqueous glycerol-glass beads system, as i s also the case for the predicted values. For higher bed expansion, the measured values i n the air-water-glass beads system are s l i g h t l y larger than those i n the air-aqueous glycerol-glass beads system, as i s again the case for the predicted values. Thus i t can be said that a r e l a t i v e l y small change (e.g. doubling) i n the v i s c o s i t y of the l i q u i d has l i t t l e or no e f f e c t on the voidage of three-phase f l u i d i z e d beds of 1 mm glass p a r t i c l e s , and that the generalized wake model can r e a l i s t i c a l l y predict the small e f f e c t observed. 316 0.86 0.82 0.78 0.74 0.70 0.66 vX XxX oo xk=0.2 x k=0.4 . ^ - - x k =0.8 x, =0.2 < j> , cm/sec o 4.06 x 4.86 A 6 .03 0 I 2 3 4 5 - 6 7 <j > , cm/sec FIGURE 4.35 BED VOIDAGE IN THREE-PHASE BEDS OF 1 MM GLASS BEADS FLUIDIZED BY AIR AND AQUEOUS GLYCEROL IN 20 MM GLASS COLUMN ( generalized wake model) 317 0.88 0.84 0.80 * 0.76 ca 0.72 < Q O > Q UJ 00 0.82 0.78 0.74 0.70 0.66 (a ) IB • • D ? b o o ° CO o o © © <j.>, cm/sec < J , > / V o o ® 5.22 0.294 o 4.06 0.297 6.42 0.362 * 4.86 0.355 H 8.02 0.453 • 6.03 0.441 J i i o I 2 3 4 5 <j2> , cm/sec 6. FIGURE 4.36 COMPARISON OF MEASURED (a) AND PREDICTED (b) VALUES OF BED VOIDAGES IN THREE-PHASE BEDS OF 1 MM GLASS BEADS FLUIDIZED BY AIR AND WATER WITH THOSE FLUIDIZED BY AIR AND AQUEOUS GLYCEROL (generalized wake model: air-water - 1 mm glass beads, air-aqueous g l y c e r o l - 1 mm glass beads; s o l i d symbols are for water and open symbols are for aqueous glycerol) 318 (B) 2 inch perspex column The primary objective of t h i s study was to e s t a b l i s h the e f f e c t of size and density of the p a r t i c l e s on the bed voidage i n three-phase f l u i d i z a t i o n . In order to achieve t h i s objective the same wide range of p a r t i c l e sizes (1/4 -3 mm) and p a r t i c l e densities (2.8 - 11.3 gm/cm ) were investigated as i n l i q u i d - s o l i d f l u i d i z a t i o n , again covering a free s e t t l i n g p a r t i c l e Reynolds number range of 0.36 to 177 0. Attempts to investigate the e f f e c t of l i q u i d phase v i s c o s i t y were not successful, as a stable bed operation with small p a r t i c l e s could not be achieved i n the present setup using high v i s c o s i t y l i q u i d s . The experimental f i n d -ings are discussed i n the following sections. (i) Air-water-1/4 mm glass beads A great deal of attention and care had to be exercised in order to obtain a reproducible set of data with such small p a r t i c l e s because the entrainment of p a r t i c l e s was found to be an unavoidable phenomenon at large gas flow rates, even for the lowest bed expansion studied. A few runs were carr i e d out i n three-phase beds of 1/4 mm glass beads (d = 0.323 mm for these runs) to check on whether the P entrainment of p a r t i c l e s from the system occurred system-a t i c a l l y or a c c i d e n t a l l y . Figure 4.37 shows the v a r i a t i o n in observed pressure drop at a p a r t i c u l a r location i n the 319 O e o X Q < LJJ cr LU LU O IZ < + 10 0 - 10 - 2 0 - 3 0 - 4 0 - 5 0 o - - -o - Z = 7 ' - o - - Z =16 ^ - o - - Z = 2 2 0k. o - - Z = 3 0 - o . - O k -<y~ Z = 3 8 - - o - Z = 4 6 0 4 0 8 0 120 160 2 0 0 2 4 0 2 8 0 TIME IN- MINUTES FIGURE 4.37 VARIATION IN OBSERVED PRESSURE DROP FOR ELUTRIATING THREE-PHASE BED OF 1/4 MM GLASS BEADS (d =0.323 mm) FLUIDIZED BY AIR (<J2 > = 18.0 cm/sec) AND WATER (<Ji> = 3.18 cm/sec) (Z = distance i n inches from tap 1) 320 test section as a function of elapsed time, for one such run (<j^> = 3.18 cm/sec and < J 2 > ~ 18.0 cm/sec). From these measurements i t was confirmed that the p a r t i c l e entrainment did indeed occur systematically (1 gm/min for the quoted experiment). However, these and other similar runs where p a r t i c l e entrainment was disproportionately large were not included i n the r e s u l t s , as discussed i n Section 3.3, and w i l l be excluded from further discussions. The introduction of an a i r stream into a nearly uniform f l u i d i z e d bed of 1/4 mm glass beads caused a large reduction in bed height, the ex