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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 U n i v e r s i t y , I n d i a , 1963 M. Tech.  (Chem. Eng.), I.I.T., Kharagpur, I n d i a , 1965  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n the Department of CHEMICAL  ENGINEERING  We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA December, 1972  In  presenting this  requirements  thesis  in partial  fulfillment  of  the  f o r an a d v a n c e d d e g r e e a t t h e 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  freely  I further  available  that permission  f o r r e f e r e n c e and  f o r extensive copying  s c h o l a r l y p u r p o s e s may D e p a r t m e n t o r by  be  granted  thesis  o u t my  by  of t h i s  for financial  It is  understood  i n whole, or the copying  g a i n s h a l l n o t be  VINAY K.  The  University  Vancouver  8,  Engineering  of B r i t i s h  Canada  for  my  allowed  written permission.  Department of Chemical  agree  thesis  t h e Head o f  his representatives.  that p u b l i c a t i o n , i n part or this  study.  Columbia  BHATIA  of  with-  ABSTRACT  P r e v i o u s r e s e a r c h i n three-phase  f l u i d i z a t i o n has  p r o v i d e d some experimental data on i n d i v i d u a l ups, but no u n i f i e d bed model f o r p r e d i c t i n g holdups under a v a r i e t y direction  of circumstances.  0 s t e r g a a r d ,  Capes. 1.  these  A step i n t h i s  i s made here w i t h the development of a " g e n e r a l -  i z e d wake model", which b u i l d s of  phase h o l d -  upon the e a r l i e r a n a l y s e s  of Efremov and Vakhrushev, and o f Rigby and  The p r e s e n t a n a l y s i s  takes i n t o  account  the e f f e c t o f s i z e and p a r t i c l e content o f the bubble wakes,  2.  the c i r c u l a t i o n o f s o l i d s entrainment  3.  i n i t i a t e d by p a r t i c l e  i n the bubble wakes, and  the r e l a t i v e motion  between the continuous phase  and the d i s p e r s e d phases. I t does not take i n t o account any s u r f a c e e f f e c t s , the s o l i d s w e t t a b i l i t y .  The e x p r e s s i o n used  the wake volume f r a c t i o n was n e c e s s a r i l y  especially  for estimating  arbitrary,  due t o  the p a u c i t y of r e l e v a n t i n f o r m a t i o n on bubble wakes, especially  i n the presence o f s o l i d s .  Comparison o f the g e n e r a l i z e d wake model w i t h p r e v i o u s analyses indicates  t h a t the e a r l i e r models a r e s p e c i a l  cases o f the g e n e r a l i z e d Wake model.  Where the wake volume  f r a c t i o n can be n e g l e c t e d , the g e n e r a l i z e d model reduces  iii to the  " g a s - f r e e model," which  p r o p o s e d by V o l k . prediction oh  of  This  solids  s i m p l i f i e d model g i v e s  holdup  three-phase f l u i d i z a t i o n  for previous  i s no  longer  out  good  experimental  c o n t r a c t i o n on  data  particles,  i n t r o d u c t i o n of  observed.  Experiments to t e s t the carried  mechanism  of heavy and/or l a r g e  i n w h i c h t h e p a r a d o x i c a l bed gas  f o l l o w s from the  over a p a r t i c l e  g e n e r a l i z e d wake model were - 3  d i a m e t e r r a n g e o f 1/4  mm  3  and  a p a r t i c l e d e n s i t y range of  water  (1 c . p . ) , a q u e o u s g l y c e r o l  polyethylene  glycol  The  liquid  cm/sec and 21.0  i n 20 mm  and  the  gas  to  by  i n the  the valve  gm/cm , u s i n g  c.p.)  and  - 1800.  The  aqueous  particle e x p e r i m e n t s were  2 i n c h diameter transparent  Holdups i n the gas-liquid  velocity  regions  above and  2 0 mm  trary visual  two  straight  and  one  obtained  l i n e s , one  of n e g a t i v e  obtained  of p o s i t i v e  the  A t t e m p t s were made  e x p a n d e d bed  by  height  the  height  arbi-  in  the  intersection  (three-phase  (two-phase r e g i o n )  for  f r o m somewhat  o b s e r v a t i o n s , t h e e x p a n d e d bed  2 i n c h p e r s p e x c o l u m n was  to  d r o p g r a d i e n t method  T h u s , whereas the  g l a s s c o l u m n was  39  bed,  below  a n a l y z e , a s w e l l as t o m o d i f y , t h e methods u s e d  the  to  f r o m 0.2  three-phase f l u i d i z e d  the pressure  columns.  v a r i e d f r o m 0.4  shut-off technique.  measuring holdups. in  0.36  (air) s u p e r f i c i a l  b e d , were m e a s u r e d by and  (2.1  s u p e r f i c i a l v e l o c i t y was  cm/sec.  as w e l l a s  - 11.1  (63 c . p . ) , c o v e r i n g t h e  R e y n o l d s number r a n g e o f performed  2.5  slope,  of  region)  resulting  iv 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 direction.  S i m i l a r l y , attempts t o improve  h o l d u p measurement t e c h n i q u e s p r o d u c e d probe w i t h a s l i g h t v a r i a t i o n by N a s s o s water  i n the  and B a n k o f f , w e l l  flow.  gas h o l d u p s  The  upon t h e  gas  electro-resistivity  i n d e s i g n from t h a t  suited  subsequent  an  axial  employed  f o r measurements i n a i r -  use o f the probe  i n three-phase f l u i d i z e d  f o r measuring  b e d s was  not  as  successful. F o r t h e beds o f s m a l l  light particles/  o f wake c h a r a c t e r i s t i c s — t h e  size  the  knowledge  as w e l l a s t h e  particle  c o n t e n t o f t h e b u b b l e w a k e s — w a s shown t o be o f  critical  importance  little  for solids  consequence for  f o r gas h o l d u p p r e d i c t i o n s .  the large  sufficed  h o l d u p p r e d i c t i o n s and  for predictions  of solids  holdup, thereby suggest-  o f b u b b l e wakes i n t h e s e  operation of the three-phase f l u i d i z e d  S t o k e s r e g i m e , though similarly  subject to p a r t i c l e  showed no a p p a r e n t e f f e c t  g e n e r a l i z e d wake m o d e l was a s was  elutriation,  of bubble wakes.  f o r bubble r i s e  o r p r e s e n c e o f s o l i d s , w h i c h was  i n t o the model.  systems.  beds i n the  t h u s v i n d i c a t e d by  a proposed c o r r e l a t i o n  the absence  the other hand,  o r h e a v y p a r t i c l e s , t h e g a s - f r e e model a g a i n  ing the i n s i g n i f i c a n t r o l e The  On  of  the  The  experiments,  velocity in incorporated  V  ACKNOWLEDGEMENTS  I g r a t e f u l l y acknowledge the m e a n i n g f u l provided  by D r . Norman E p s t e i n , u n d e r whose q u i d a n c e  i n v e s t i g a t i o n was c o n d u c t e d . viewing  assistance  the progress  I am i n d e b t e d  this  t o him f o r r e -  o f t h i s work b o t h p a t i e n t l y and  critically. T h a n k s a r e due t o D r . S.D. C a v e r s f o r h e l p i n g t h e wake s i z e d a t a  in liquid-liquid  s y s t e m s , t o D r s . R.M.R.  B r a n i o n and G.H. N e a l e f o r h e l p i n g w i t h analyses, at various  stages  of preparation  t h e Workshop S t a f f  parts  some  theoretical  and t o D r . K.B. M a t h u r f o r h e l p f u l d i s c u s s i o n s  I acknowledge w i t h  facilities  of t h i s  thesis.  g r a t i t u d e the help  r e n d e r e d by  i n construction of experimental  at various  stages  o f t h e work and i n b u i l d i n g  of the apparatus. The  financial  assistance provided  by N a t i o n a l  C o u n c i l o f Canada and t h e U n i v e r s i t y o f B r i t i s h are  with  gratefully My v e r y  and most c o r d i a l l y special  timely assistance  Columbia  acknowledged.  t h a n k s a r e due t o M r . P.M.G. Rao f o r  i n preparation  this dissertation.  o f a l l the diagrams i n  L a s t b u t n o t t h e l e a s t , t h a n k s a r e due  t o my c o l l e a g u e s  a n d f r i e n d s f o r c o n t r i b u t i n g t o my  "knowledge" o v e r  the y e a r s .  :  And  t o my  Research  family, I  owe  i t a l l .  vi TABLE OF CONTENTS PAGE ABSTRACT  i i  ACKNOWLEDGEMENTS  V  LIST OF FIGURES  xii  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 f o r three-phase f l u i d i z e d beds . .  12  1.4  Importance o f t u r b u l e n c e phenomena i n three-phase f l u i d i z e d beds  23  1.5  Scope o f r e s e a r c h  26  2. THEORY  30  2.1  Holdup i n g a s - l i q u i d flow  30  2.1.1  31  Holdup s t u d i e s  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 c o c u r r e n t flow  46  2.1.2  Models f o r gas holdup p r e d i c t i o n s  2.1.3  C i r c u l a t i o n and t u r b u l e n c e i n twophase g a s - l i q u i d flow  . .  49 58  vii CHAPTER  2.2  PAGE  Voidage 2.2.1  in liquid-solid  beds  . . . .  fluidized  Holdup i n g a s - l i q u i d - s o l i d 2.3.1  Gas h o l d u p beds  2.3.3  Voidage  beds  fluidized  Models f o r three-phase beds (A) The g a s - f r e e m o d e l (B) T h e wake m o d e l (C) T h e c e l l m o d e l  2.3.2  66 beds  . .  .  i n three-phase  102  i n three-phase  fluidized 107 110  Apparatus  I l l  3.1.1  T h e 2 0 mm b e n c h t o p g l a s s c o l u m n  3.1.2  The 2 i n c h p e r s p e x column  3.1.2.1  Liquid  3.1.2.2  Gas c y c l e  3.1.4  cycle  and t e s t  . . . . . . . . section  and b u b b l e n o z z l e  Electro-resistivity  . . .  . . .  local  probe  126  129 12 9  gas f r a c t i o n  12 9  frequency  131  (b) b u b b l e Analysis  117  . . . . 122  Description of a u x i l i a r y c i r c u i t s f o r measurement o f l o c a l g a s h o l d u p and . bubble frequency  (a)  112 117  Q u a n t i t i e s measured  3.2  70 70 76 100  fluidized  3 . EXPERIMENTAL  3.1.3  69  fluidized  beds  3.1  59  E f f e c t o f t u r b u l e n c e on v o i d a g e i n liquid-solid  2.3  fluidized  o f t h e probe  Range o f v a r i a b l e s  signal  132  studied  (A) Gas h o l d u p i n two-phase g a s - l i q u i d f l o w . (B) S o l i d s and g a s h o l d u p i n a t h r e e - p h a s e f l u i d i z e d bed  134 .  135 136  viii CHAPTER  3.3  PAGE  Experimental procedure 3.3.1  3.3.2  3.3.3  3.3.4  138  P h y s i c a l p r o p e r t i e s of the used  liquids  P h y s i c a l p r o p e r t i e s o f the used . .  solids  Measurement o f gas h o l d u p l i q u i d flow Holdup s t u d i e s  138  140 i n gas143  i n three-phase  f l u i d i z e d beds 3.4  Data p r o c e s s i n g 3.4.1  3.4.2 4 . RESULTS AND 4.1  146 148  E x p a n d e d b e d h e i g h t and  solids  holdup  148  Gas  holdup  156  DISCUSSION  160  Comparison  of proposed mathematical  models  w i t h p r e v i o u s work 4.1.1  Gas  4.1.2  Gas h o l d u p i n t h r e e - p h a s e f l u i d i z e d beds Voidage i n three-phase f l u i d i z e d beds Beds o f 6 mm p a r t i c l e s ( F i g u r e s 4 . 2 a , b and T a b l e 4.7)  4.1.3  holdup  160 in gas-liquid  Beds o f 3 mm 4 . 3 a , b)  particles  Beds o f 1 mm p a r t i c l e s and F i g u r e 4.4) 4.2  flow  . . . .  167 18 2 189  (Figures 194 (Table  4.8  D i s c u s s i o n o f e x p e r i m e n t a l r e s u l t s and comparison with t h e o r i t i c a l p r e d i c t i o n s . 4.2.1  160  Evaluation of experimental techniques  197  . .  2 08  2 08  ix CHAPTER  PAGE 4.2.2  Gas holdup r e s u l t s  4.2.2.1  2 09  Gas holdup i n g a s - l i q u i d flow . . . (A) 20 mm g l a s s column (B) 2 i n c h perspex column (i) Bubble column ( i i ) C o c u r r e n t flow E l e c t r o - r e s i s t i v i t y probe measurements (i) A i r - w a t e r flow ( i i ) Air-PEG s o l u t i o n flow . .  4.2.2.2  Gas holdup i n three-phase beds  shot  4.2.3.2  2 36 2 36 241 249 249 254 254 262 27 3 279 286 290  Voidage r e s u l t s  4.2.3.1  210 211 211 218  fluidized  (A) 20 mm g l a s s column (B) 2 i n c h perspex column (i) Air-water-1/4 mm g l a s s beads ( i i ) Air-water-1/2 mm g l a s s beads . ( i i i ) A i r - w a t e r - 1 mm g l a s s beads (iv) A i r - w a t e r - l e a d shot (v) Air-PEG s o l u t i o n - 1 mm g l a s s beads (vi) Air-PEG s o l u t i o n - s t e e l  4.2.3  209  295  Voidage i n l i q u i d - s o l i d f l u i d i z e d beds (A) 20 mm g l a s s column (B) 2 i n c h perspex column g l a s s beads-water l e a d shot-water s o l i d particles-PEG solution. .  295 296 298 302 305 307  Voidage i n three-phase beds  308  fluidized  (A) 20 mm g l a s s column (i) Air-water-1/2 mm sand . . . . ( i i ) A i r - w a t e r - 1 mm g l a s s beads. . ( i i i ) Air-aqueous g l y c e r o l - 1 mm g l a s s beads (B) 2 i n c h perspex column  308 308 311 314 318  X  CHAPTER  PAGE (i) Air-water-1/4 mm g l a s s beads ( i i ) Air-water-1/2 mm g l a s s beads ( i i i ) A i r - w a t e r - 1 mm g l a s s beads (iv) A i r - w a t e r - l e a d shot (v) Air-PEG s o l u t i o n - 1 mm g l a s s beads (vi) Air-PEG s o l u t i o n - s t e e l shot  318 327 338 343 351 354  5.  CONCLUSIONS  358  6.  NOMENCLATURE  365  7.  LITERATURE CITED  370  8.  APPENDICES 8.1  Al  A c e l l model f o r three-phase f l u i d i z a t i o n .  .  Al  1. The case of a s t a t i o n a r y 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 . . .  8.2  8.3  A4  2. The case o f a r i s i n g bubble i n a c o c u r r e n t l i q u i d flow  A10  Results  A17  Measurement techniques f o r gas holdup (and bed height) i n three-phase f l u i d i z e d beds and r e l a t e d systems  A2 3  1. M o d i f i e d p r e s s u r e drop g r a d i e n t method. .  A24  2. D i r e c t v o l u m e t r i c measurements u s i n g quick c l o s i n g valves  A33  E s t i m a t i o n of bubble s i z e from  local  bubble frequency measurements 8.4  C a l i b r a t i o n of l i q u i d  8.5  C a l i b r a t i o n o f rotameters  8.6  P h y s i c a l p r o p e r t i e s of m a t e r i a l s used  A  flow meters  . . .  ^3  A  4  6  A  4  9  A  ^2  xi CHAPTER  PAGE (A) S o l i d d e n s i t i e s and p a r t i c l e sizes (B) D e n s i t i e s and v i s c o s i t i e s of l i q u i d s .  A  . .  5  2  A53  8.7  Processed data and r e s u l t s  A58  8.8  Publications  A70  8.9  Error Estimation  A71  Calculations  A74  8.10  xii  L I S T OF FIGURES FIGURE  2.1  2.2  2.3  2.4  3.1  PAGE  Gas h o l d u p d a t a f r o m l i t e r a t u r e f o r a i r w a t e r s y s t e m i n b u b b l e c o l u m n s o f 1-42 i n c h diameter  44  Schematic r e p r e s e n t a t i o n model  72  of the gas-free  S c h e m a t i c r e p r e s e n t a t i o n o f t h e wake model  78  S c h e m a t i c o f wake s t r u c t u r e de N e v e r s and Wu [61]  96  S c h e m a t i c d i a g r a m o f 2 0 mm apparatus  s u g g e s t e d by  glass  column 113  3.2  The 20 mm  3.3  Schematic diagram o f 2 i n c h perspex apparatus .  3.4  3.5  g l a s s column  116 column 118  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 section Location for  of pressure  t a p s and b a l l  valves  g a s h o l d u p measurements  12 3  3.6  D e s i g n o f gas i n l e t and d i s t r i b u t o r  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 E l e c t r o - r e s i s t i v i t y p r o b e and mount f o r t r a v e r s i n g mechanism  3.8  3.9  3.10  121  125  127 130  Schematic diagram o f t h e 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 bubblef r e q u e n c y and g a s f r a c t i o n  133  T y p i c a l pressure drop p r o f i l e solid fluidization  150  in liquid-  xiii FIGURE  3.11  4.1  4.2a  4.2b  4.3a  4.3b  4.4  PAGE  T y p i c a l pressure drop p r o f i l e phase f l u i d i z a t i o n Proposed models f o r d r i f t b u b b l e swarms  i n three153  velocity of 163  L i q u i d f r a c t i o n d a t a 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 6 mm g l a s s b e a d s  190  Bed v o i d a g e d a t a 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 6 mm g l a s s b e a d s  191  L i q u i d f r a c t i o n d a t a 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 3 mm g l a s s b e a d s  195  Bed v o i d a g e d a t a 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 3 mm g l a s s b e a d s  196  L i q u i d f r a c t i o n d a t a 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 b e a d s  199  4.5a  Bed v o i d a g e d a t a o f 0 s t e r g a a r d and T h e i s e n [18] f o r 2 mm g l a s s b e a d s  4.5b  C o m p a r i s o n o f eV' by g e n e r a l i z e d wake m o d e l w i t h e £ by E f r e m o v - V a k h r u s h e v e q u a t i o n s f o r 2 mm g l a s s b e a d s a t <j1 > = 11.0 cm/sec  4.6  4.7  Comparison o f 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 b y e q u a t i o n 4.13 w i t h e x p e r i m e n t a l d a t a i n 20 mm g l a s s c o l u m n Axial variation column  2  0  2  2  03  1  2  ( < j 1 > = 0.0)  2  14  Gas h o l d u p i n b u b b l e c o l u m n  4.8b  C o m p a r i s o n o f m e a s u r e d and p r e d i c t e d g a s holdups i n 2 i n c h b u b b l e columns  2  A x i a l d i s t r i b u<t i o n> o f g a s h o l d u p w a t e r f l o w a t J 1 = 1.25 cm/sec  2 2  4.10  2  o f gas h o l d u p i n b u b b l e  4.8a  4.9  2  i na i r -  A x i a l d i s t r i b u t i o n o f gas h o l d u p i n a i r w a t e r f l o w a t <j-,> = 1.87 cm/sec  IV  19 3  2 2  4  xiv FIGURE  4.11  4.12  4.13  4.14  4.15  4.16  4.17  4.18  4.19  4.20  4.21  4.22  4.23  4.24  4.2 5  PAGE  Axial water  d i s t r i b u <t i o n> flow at j,2  Axial distribution PEG s o l u t i o n f l o w  =  o f gas h o l d u p 4.55 cm/sec  inair-  o f gas h o l d u p  ina i r -  22 6  227  C o m p a r i s o n o f e q u a t i o n 4.18 w i t h e x p e r i mental d a t a i n 2 i n c h p e r s p e x column  2 33  Gas h o l d u p f o r c o c u r r e n t a i r - w a t e r i n 2 i n c h p e r s p e x column  23 4  Gas h o l d u p f o r c o c u r r e n t a i r - P E G f l o w i n 2 i n c h p e r s p e x column C o m p a r i s o n o f r a d i a l gas h o l d u p computed by e q u a t i o n 2.27b w i t h mental data R a d i a l gas h o l d u p p r o f i l e s s o l u t i o n flow  flow  solution 235 profiles experi238  i n air-PEG 24 3  Gas h o l d u p i n 2 0 mm g l a s s c o l u m n f o r t h r e e - p h a s e f l u i d i z a t i o n by a i r and w a t e r o f 1 mm g l a s s b e a d s  250  Gas h o l d u p i n 20 mm g l a s s c o l u m n f o r t h r e e - p h a s e f l u i d i z a t i o n by a i r and a q u e o u s g l y c e r o l o f 1 mm g l a s s b e a d s  251  Gas h o l d u p i n 20 mm g l a s s c o l u m n f o r t h r e e - p h a s e f l u i d i z a t i o n by a i r and w a t e r o f 1/2 mm s a n d p a r t i c l e s  2 5 2  Gas h o l d u p i n t h r e e - p h a s e b e d s o f 1/4 mm g l a s s b e a d s f l u i d i z e d . b y a i r and w a t e r . . . .  255  Gas h o l d u p i n t h r e e - p h a s e b e d s o f 1/2 mm g l a s s b e a d s f l u i d i z e d by a i r and w a t e r . . . .  264  C o m p a r i s o n o f gas h o l d u p p r o f i l e s computed b y ^ e q u a t i o n 2.27b w i t h e x p e r i m e n t a l d a t a . . .  270  Gas h o l d u p i n t h r e e - p h a s e b e d s o f 1 mm g l a s s b e a d s f l u i d i z e d by a i r and w a t e r . . . .  274  Gas h o l d u p i n t h r e e - p h a s e b e d s o f 2 mm l e a d s h o t f l u i d i z e d by a i r and w a t e r  28 0  XV  FIGURE  4.2 6  PAGE  Gas h o l d u p i n t h r e e - p h a s e b e d s o f 1 mm g l a s s b e a d s f l u i d i z e d by a i r and PEG solution  4.27  Gas h o l d u p p r o f i l e s f l u i d i z e d bed  4.28  Gas h o l d u p i n t h r e e - p h a s e b e d s o f 3 s t e e l s h o t f l u i d i z e d by a i r and PEG solution  4.2 9  4.30  4.31  4.32  4.3 3  4.34  4.35  4.36  in  three-phase  The e x p a n s i o n c h a r a c t e r i s t i c s o f 1/2 s a n d p a r t i c l e s f l u i d i z e d by w a t e r  287 2 8 9  mm  291 mm  299  E s t i m a t i o n o f e x p a n d e d 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 d r o p p r o f i l e i n bed o f 1 mm g l a s s b e a d s f l u i d i z e d by w a t e r  303  E s t i m a t i o n o f e x p a n d e d 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 d r o p p r o f i l e i n bed o f 1/4 mm g l a s s b e a d s f l u i d i z e d by w a t e r  304  E s t i m a t i o n o f e x p a n d e d bed h e i g h t f r o m 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 o f 2 mm l e a d s h o t f l u i d i z e d by w a t e r . . . . .  306  Bed v o i d a g e i n t h r e e - p h a s e b e d 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 a i r and w a t e r i n 20 mm g l a s s column  310  Bed v o i d a g e i n t h r e e - p h a s e b e d s o f 1 mm g l a s s b e a d s f l u i d i z e d by a i r and w a t e r i n 20 mm g l a s s c o l u m n  312  Bed v o i d a g e i n t h r e e - p h a s e b e d s o f 1 mm g l a s s b e a d s f l u i d i z e d by a i r and a q u e o u s g l y c e r o l i n 20 mm g l a s s c o l u m n s  316  C o m p a r i s o n o f m e a s u r e d (a) and p r e d i c t e d (b) v a l u e s o f b e d v o i d a g e s i n t h r e e - p h a s e b e d s o f 1 mm g l a s s b e a d s f l u i d i z e d by a i r and w a t e r w i t h t h o s e 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  XVI  FIGURE 4.37  PAGE V a r i a t i o n i n observed p r e s s u r e drop f o r e l u t r i a t i n g three-phase bed of 1/4 mm g l a s s beads (dp = 0.323 mm) f l u i d i z e d by a i r (<j > = 18.0 cm/sec) and water j = 3.18 cm7sec . . . . . .  319  Bed voidage 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 . . .  321  <  5  4.38 4.39  4.40 4.41  4.42  4.43 4.44  4.45  4.46  >  1  Comparison o f e , c a l c u l a t e d from equations 1.3, 2.91, 2.94 and 2.106 w i t h x^ = 0.2, u s i n g the measured v a l u e s 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 g l a s s beads, w i t h p r e d i c t e d v a l u e s ( ) from e q u a t i o n 4.7 w i t h p = 3 . . . .  327  Bed voidage 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 . . .  329  Comparison o f e , c a l c u l a t e d from e q u a t i o n s 1.3, 2.91, 2.94 and 2.106 w i t h x ^ O , u s i n g the measured v a l u e s of gas and s o l i d s h o l d up i n three-phase f l u i d i z e d beds of 1/2 mm g l a s s beads, w i t h p r e d i c t e d v a l u e s ( ) from equation 4.7 with. p=3  335  Comparison o f bed voidage data f o r 1/2 mm sand p a r t i c l e s i n 20 mm g l a s s column w i t h those f o r 1/2 mm g l a s s beads i n 2 i n c h perspex column  336  Bed voidage i n three-phase beds o f 1 mm g l a s s beads f l u i d i z e d by a i r and water . . .  338  Comparison o f c a l c u l a t e d from equations 1.3, 2.91, 2.94 and 2.106, u s i n g the measured v a l u e s 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 mm g l a s s beads, w i t h p r e d i c t e d v a l u e s ( ) from equation 4.7 w i t h p=3  342  Comparison o f bed voidage data f o r 1 mm g l a s s beads i n 20 mm g l a s s column w i t h those i n 2 i n c h perspex column  344  Bed voidage i n three-phase beds o f 2 mm l e a d shot f l u i d i z e d by a i r and water  346  xvii PAGE  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  Bed v o i d a g e i n t h r e e - p h a s e b e d s o f 1 g l a s s b e a d s f l u i d i z e d by a i r and PEG solution  mm  Bed v o i d a g e i n t h r e e - p h a s e b e d s o f 3 s t e e l s h o t f l u i d i z e d by a i r and PEG solution  mm  Schematic of c e l l fluidization  353  357  model f o r t h r e e - p h a s e A3  S o l u t i o n o f c e l l model f o r t h r e e - p h a s e 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 conditions  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 t h e c e l l model  A20  Schematic of s t a t i c p r e s s u r e drop along the t e s t s e c t i o n  A29  gradient  P l o t o f manometric e q u a t i o n s : threephase f l u i d i z e d bed; — - — liquid-solid f l u i d i z e d bed; — -two-phase g a s 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 m e n t a l column (a) u n d e r r u n n i n g 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  Longitudinal d i s t r i b u t i o n of absolute p r e s s u r e i n the e x p e r i m e n t a l column w i t h v a l v e s open .  A38  A p l a n view probe  A44  of bubble  traverse over  the  C a l i b r a t i o n curves f o r water flow meters i n 2 inch 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 r v e s f o r PEG s o l u t i o n meters i n 2 i n c h c i r c u l a t i o n loop  A48  flow  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 c o l u m n s e t u p  A50  xviii FIGURE 8.5.2 8.5.3 8.6.1  PAGE C a l i b r a t i o n curve f o r water rotameter i n 20 mm g l a s s column setup C a l i b r a t i o n curves f o r a i r rotameter i n 2 i n c h c i r c u l a t i o n loop Dynamic 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 solutions  A  ^0  A  51  A  ^7  xix  L I S T OF TABLES TABLE  2.1  3.1  3.2  3.3  3.4  4.1  4.2  4.3-A  4.3- B  4.4- A  4.4-B  4.5  4.6  PAGE  R a t i o o f wake t o b u b b l e volume f o r v a r i o u s v a l u e s o f d i s p e r s e d p h a s e h o l d u p i n twophase f l u i d systems  98  E x p e r i m e n t a l c o n d i t i o n s f o r two-phase g a s l i q u i d f l o w i n t h e 20 mm g l a s s c o l u m n  137  E x p e r i m e n t a l c o n d i t i o n s f o r two-phase g a s l i q u i d flow i n the 2 inch perspex column  137  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 c o l u m n  139  Experimental conditions f o r three-phase f l u i d i z a t i o n i n 2 i n c h p e r s p e x comumn  139  R a t i o o f wake t o b u b b l e v o l u m e i n t h r e e p h a s e f l u i d i z e d b e d (D_>4 i n c h e s )  169  R a t i o o f wake t o b u b b l e volume i n t h r e e p h a s e f l u i d i z e d b e d (D=2 i n c h ) . .  17 0  R a t i o o f gas h o l d u p i n t h r e e - p h a s e f l u i d i z e d b e d t o g a s h o l d u p i n twop h a s e g a s - l i q u i d f l o w (D>_4 i n c h e s )  174  R i s e v e l o c i t y o f b u b b l e swarms i n t h r e e p h a s e f l u i d i z e d b e d (D>4 i n c h e s )  175  R a t i o o f gas holdup i n t h r e e - p h a s e f l u i d i z e d b e d t o gas h o l d u p i n two-phase g a s l i q u i d f l o w (D=2 i n c h e s )  17 9  R i s e v e l o c i t y o f b u b b l e swarms i n t h r e e p h a s e f l u i d i z e d b e d (D=2 i n c h e s )  180  D e g r e e o f a g r e e m e n t b e t w e e n e q u a t i o n s 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 t o gas f r a c t i o n i n a t h r e e - p h a s e f l u i d i z e d bed  185  Comparison o f 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  4.7  4.8  4.9  4.10  4.11  4.12  4.13  4.14  4.15  4.16  4.17  4.18  4.19  PAGE  C o m p a r i s o n o f m e a s u r e d 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 t h r e e - p h a s e f l u i d i z e d beds  192  C o m p a r i s o n o f m e a s u r e d and p r e d i c t e d and l i q u i d f r a c t i o n s i n t h r e e - p h a s e f l u i d i z e d beds  198  Comparison and l i q u i d i z e d beds  gas  o f m e a s u r e d and p r e d i c t e d gas f r a c t i o n i n three-phase f l u i d -  T r a n s i t i o n from bubbly t o s l u g water flow  204 flow  ina i r 220  C o m p a r i s o n o f gas h o l d u p s i n v a r i o u s s e c t i o n s o f t h e column f o r a i r - w a t e r flow Estimation of d i s t r i b u t i o n parameter, f r o m gas h o l d u p measurements  229 CQ, 231  Gas for  h o l d u p by e l e c t r o - r e s i s t i v i t y air-water flow  probe  Gas air  h o l d u p by e l e c t r o - r e s i s t i v i t y - PEG s o l u t i o n f l o w  probe f o r  239  244  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 p r o b e measurements i n a i r PEG s o l u t i o n f l o w  246  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 p r o b e measurements i n a i r PEG s o l u t i o n f l o w  248  C o m p a r i s o n o f gas h o l d u p i n t h r e e - p h a s e f l u i d i z e d b e d t o t h a t i n two-phase r e g i o n s of t h e column  257  C o m p a r i s o n o f gas h o l d u p i n t h r e e - p h a s e 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 f l o w C o m p a r i s o n o f gas h o l d u p i n t h r e e - p h a s e 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  . . .  260  2 65  xx i TABLE  4.2 0  4.21  4.22  4.23  4.24  4.25  4.26  4.27  4.28  4.29  4.30  4.31  4.32  PAGE  C o m p a r i s o n o f gas h o l d u p i n t h r e e - p h a s e 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 f l o w Gas h o l d u p by e l e c t r o - r e s i s t i v i t y i n t h r e e - p h a s e f l u i d i z e d bed  . . .  268  probe 271  C o m p a r i s o n o f gas h o l d u p i n t h r e e - p h a s e f l u i d i z e d b e d t o t h a t i n two-phase r e g i o n s of t h e column  275  C o m p a r i s o n o f gas h o l d u p i n t h r e e - p h a s e 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 f l o w  277  . . .  C o m p a r i s o n o f gas h o l d u p i n t h r e e - p h a s e f l u i d i z e d b e d t o t h a t i n two-phase r e g i o n s o f t h e column  281  Comparison o f gas holdup i n t h r e e - p h a s e 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  283  . . .  C h a r a c t e r i s t i c measurements o f g a s b u b b l e p h a s e i n t h r e e - p h a s e 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  C o m p a r i s o n o f gas h o l d u p i n t h r e e - p h a s e f l u i d i z e d b e d t o t h a t i n two-phase r e g i o n s of t h e column  292  Comparison o f average bubble s i z e i n t h r e e 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  294  Expansion r e s u l t s f o r l i q u i d - s o l i d i z a t i o n i n 20 mm c o l u m n  fluid297  Expansion r e s u l t s f o r l i q u i d - s o l i d i z a t i o n i n 2 i n c h column  fluid-  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 f r o m g e n e r a l i z e d wake model w i t h xk=0.2, t o wake volume f r a c t i o n , e^, i n t h r e<e - > p h a s e b e d s o f 1/4 mm g l a s s b e a d s a t j - j _ = 1.87 cm/sec E s t i m a t e d v a l u e s o f wake volume f r a c t i o n , ek, from e x p e r i m e n t a l d a t a i n three-phase b e d s o f 1/4 mm g l a s s b e a d s f l u i d i z e d b y a i r and w a t e r  300  323  xxii TABLE  4.33  4.34  4.35  4.3 6  4.37  PAGE  S e n s i t i v i t y o f b e d v o i d a g e , e, p r e d i c t e d f r o m g e n e r a l i z e d wake model w i t h x^=0.0 t o wake volume f r a c t i o n , ej^, i n t h r e e p h a s e b e d s o f 1/2 mm g l a s s b e a d s  332  E s t i m a t e d v a l u e s o f wake volume f r a c t i o n , e^, f r o m e x p e r i m e n t a l d a t a i n t h r e e - p h a s e b e d s o f 1/2 mm g l a s s b e a d s f l u i d i z e d b y a i r and w a t e r .  333  E s t i m a t e d v a l u e s o f wake volume f r a c t i o n , ek, from e x p e r i m e n t a l d a t a i n t h r e e - p h a s e b e d s o f 1 mm g l a s s b e a d s f l u i d i z e d b y a i r and w a t e r  341  Comparison o f measured bed v o i d a g e s i n t h r e e - p h a s e b e d o f 2 mm l e a d s h o t f l u i d i z e d by a i r and w a t e r w i t h t h e v a l u e s p r e d i c t e d by t h e g a s - f r e e model  350  Comparison o f measured bed v o i d a g e s i n t h r e e - p h a s e b e d o f 1 mm g l a s s b e a d s f l u i d i z e d b y a i r and PEG s o l u t i o n w i t h t h e v a l u e s p r e d i c t e d b y t h e g a s - f r e e model  355  8.6.1  Solid densities  8.6.2  D e n s i t y and v i s c o s i t y solution  8.6.3  8.6.4  8.6.5  8.7.1  8.7.2  8.7.3  and p a r t i c l e  C a l i b r a t i o n o f Canon (300-H-304)  sizes  of water-glycerol  viscometer  A 5  A  A  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  D e n s i t y and v i s c o s i t y glycol-water solution  of polyethylene (II)  A  Gas h o l d u p d a t a f o r a i r - w a t e r f l o w i n 20 mm g l a s s column  A  Gas h o l d u p d a t a i n 2 i n c h p e r s p e x c o l u m n f o r (A) a i r - w a t e r f l o w (B) a i r - P E G s o l u t i o n f l o w 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 c o l u m n  2  53  54  A A  A  ^5  56  59  66 00  61  xxiii TABLE  PAGE  8.7.4  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 2 i n c h perspex column  8.7.5  Gas and s o l i d s holdup data i n 20 mm g l a s s column f o r three-phase beds of (A) air-water-1/2 mm sand (B) a i r - w a t e r - 1 mm g l a s s beads (C) air-aqueous g l y c e r o l - 1 mm g l a s s beads . .  A6 3 A64 A63  Gas and s o l i d s holdup data i n 2 i n c h perspex column f o r three-phase beds o f (A) a i r - w a t e r - l / 4 mm g l a s s beads (B) air-water-1/2 mm g l a s s beads (C) a i r - w a t e r - 1 mm g l a s s beads (D) a i r - w a t e r - 2 mm l e a d shot (E) air-PEG s o l u t i o n - 1 mm g l a s s beads . . . . (F) air-PEG s o l u t i o n - s t e e l shot  A65 A66 A67 A68 A69 A69  Estimated e r r o r s i n experimental r e s u l t s  A7 3  8.7.6  8.9.1  . . . .  . .  A62  NOTATION  The phases in  analysis o f multiphase  flow  s i m u l t a n e o u s l y , h a s drawn c o n s i d e r a b l e  recent years.  multiphase  f l o w , i n which s e v e r a l  F o r two-phase f l o w , t h e s i m p l e s t c a s e o f  flow, the information a v a i l a b l 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 . standard  definitions  multiphase  attention  o f t h e terms u s e d  i n the l i t e r a t u r e , S i n c e no  f o r d e s c r i b i n g the  f l o w phenomenon e x i s t , t h e t e r m i n o l o g y  a d o p t e d 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 v a g u e and sometimes v e r y ing.  Z u b e r and F i n d l a y  terminology terms.  [27].  closely  to develop  a rigorous  w h i c h g i v e s a p h y s i c a l meaning t o e a c h o f t h e  L a t e r t h e same t e r m i n o l o g y  Wallis  Bhaga  [39] t r i e d  confus-  The t e r m i n o l o g y  was u s e d e x t e n s i v e l y b y  employed i n t h i s  thesis  follows  t h a t a d o p t e d by W a l l i s , and r e c e n t l y e x t e n d e d by -  [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 t h e more common t e r m s i s g i v e n h e r e i n order  to provide  ships  among t h e s e The  the simple  involved i n three-phase  fluidization  by s u b s c r i p t s 1, 2 and 3 i n g e n e r a l .  the continuous  liquid  Subscript  phase, while s u b s c r i p t s  2 and 3 r e p r e s e n t t h e d i s p e r s e d g a s and s o l i d respectively.  relation-  terms.  t h r e e phases  are, i d e n t i f i e d describes  some f a m i l i a r i t y w i t h  For the three-phase  phases  system c o n s i d e r e d  i n this  XXV  study the gas and l i q u i d phases flow c o c u r r e n t l y 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 cylindrical  coordinate  system i s used i n which Z r e p r e s e n t s the d i s t a n c e measured v e r t i c a l l y upwards and r denotes the r a d i a l d i s t a n c e from the pipe a x i s . The v o l u m e t r i c flow r a t e i s r e p r e s e n t e d by symbol Q.  The  t o t a l v o l u m e t r i c flow r a t e i s then the sum  the i n d i v i d u a l  Q  =  the  component f l o w s :  Q-L + Q  2  + Q  (i)  3  S i n c e i n t h i s i n v e s t i g a t i o n flow of s o l i d p a r t i c l e s considered,  Q  i s zero and equation  =  Q  1  a t any  (i) reduces  (ii)  f l u i d i z e d bed,  every p a r t of the  r e g i o n w i l l be occupied by one  i n s t a n t of time.  i s not  to  + Q2  In a three-phase three-phase  of  I f we  phase or  another  c o n s i d e r an element of volume  which i s v e r y much s m a l l e r than the volume of e i t h e r a gas bubble or a s o l i d p a r t i c l e , fraction  then a ( t ) , r e p r e s e n t i n g the  of elemental volume occupied by one phase, can  practically t h i s event  o n l y be e i t h e r 0 o r 1. (occupation by one  The  temporal  average o f  phase) o c c u r r i n g over a long  p e r i o d w i l l r e p r e s e n t the s t a t i s t i c a l average volume of  the g i v e n phase f o r t h a t s p a t i a l l y f i x e d  fraction  elemental volume:  xxv i  a  =  ^ / a (t) d t 0  ( i i i )  T  1  F o r homogeneous d i s t r i b u t i o n o f t h e g i v e n p h a s e , <a> = a. However, i f t h e t h r e e - p h a s e r e g i o n i s n o t o f 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 , obtain a truly  s i g n i f i c a n t value  f r a c t i o n , that the averaging s p a c e and t i m e .  ~  < a >  Thus i n o r d e r  of the average  process  be c a r r i e d  i n order to volumetric o u t both i n  Then  //q(rft)rdr /rdr /dt  dt  U  v  .  )  t o a v o i d a n y a m b i g u i t i e s a b o u t t h e measurement  of  average v o l u m e t r i c  to  define e x p l i c i t l y  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 how t h e a v e r a g i n g  p r o c e s s was  carried  out. F o r many p u r p o s e s t h e d i s t r i b u t i o n o f p h a s e s i n t h e entire be  field  done o v e r  i s not r e q u i r e d ; t h e r e f o r e the averaging can a much l a r g e r  and  a then  represents  up)  o f one p h a s e .  volume o f t h e t h r e e p h a s e r e g i o n ,  the average v o l u m e t r i c  In this  case  a i s measured o v e r  c r o s s - s e c t i o n o f t h e c o n d u i t and o v e r e l i m i n a t e any s p u r i o u s a batch  of solid  longitudinal  particles  fraction  (hold-  the e n t i r e  s u f f i c i e n t length to  variations.  Thus, i f  o f m a s s , W, i s e x p a n d e d t o 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 volumetric fraction of solids region w i l l  be  ( s o l i d s holdup)  i n the flow  xx v i i  <a_,>  =  W/p  Similarly by c u t t i n g  A  3  L  (v)  b  i f we  i s o l a t e the three-phase flow r e g i o n  o f f the gas and  the l i q u i d flow r a t e s  then the average v o l u m e t r i c f r a c t i o n o f gas  simultaneously,  (gas holdup)  be obtained by measuring the u l t i m a t e volume of gas  0, - ,  can  collected,  a t the top:  =  The  n / 2  A  L  (vi)  b  symbol j i s used to r e p r e s e n t the v o l u m e t r i c  f l u x or v o l u m e t r i c flow r a t e per u n i t area of c o n d u i t , and i s e q u i v a l e n t to what has been commonly c a l l e d velocity  i n the l i t e r a t u r e .  "superficial"  F l u x i s indeed a v e c t o r q u a n t i t y ,  but i n t h i s study j w i l l be used to r e p r e s e n t the  scalar  component i n the d i r e c t i o n  by  along the p i p e .  Then  definition,  Q /A  (vii)  Q /A  (viii)  L  2  Q /A 3  = 0  and the t o t a l average v o l u m e t r i c f l u x i s  (ix)  XXVI11  The l o c a l v o l u m e t r i c f l u x i s r e l a t e d f r a c t i o n and v e l o c i t y  h  and  =  ±  a  J  2  =  j  3  =  a  to l o c a l phase v o l u m e t r i c  as f o l l o w s :  i  v  < > xi  2 2  (xii)  v  a  v  3  (xiii)  3  the t o t a l l o c a l v o l u m e t r i c f l u x i s j  a s  -3i+J2  3  +  (xiv)  3  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 d e f i n e d as  V  21 = 2  "  v  V  l  =  _ V  12  ( X V )  D r i f t v e l o c i t i e s are d e f i n e d as the d i f f e r e n c e between the phase v e l o c i t i e s and the t o t a l v o l u m e t r i c f l u x . v  V  =  x j  2j  =  v j  v  v  2  =  3  x  Thus  - j  (xvi)  " ^  v  3  (xvii)  - j  (xviii)  The d r i f t f l u x o f phase i r e p r e s e n t s the v o l u m e t r i c of t h a t phase based on d r i f t  j  ij  =  a  i  V  i i  =  a  i< i-3) v  flux  velocity:  <  x i x )  xx i x The in a pipe.  symbol Ap i s used to d e s c r i b e the p r e s s u r e drop I f dp/dZ r e p r e s e n t s the r a t e a t which pressure  i n c r e a s e s w i t h d i s t a n c e i n the Z d i r e c t i o n , then drop over a l e n g t h o f pipe L w i l l  - Ap  =  - / 0  L  (dp/dZ) dZ  pressure  be  (xx)  XXX  The  following  presented  •  • •  brief  to  One  citation  illuminate  on  Indian  was  knowledge."  "thought":  the  question  o f pramana —  F o u r means o f r e l i a b l e  perception  {pratyaksa)  , inference  analogy or  comparison  (upmana) , and  pronouncement of materialists own  is  o f t h e most i m p o r t a n t t o p i c s o f o r t h o d o x  philosophy  her  Philosophy  reliable  allowed only  system of  logic  "means o f  k n o w l e d g e were  reliable recognized:  (anumana) , i n f e r e n c e  p e r c e p t i o n , but study of  by  (sabda) , t h e  "word"  a u t h o r i t y , s u c h as  i n the  Indian  the Vedas.  India  the  The  developed  process  of  inference. A  c o r r e c t i n f e r e n c e was  which the  Indian  e s t a b l i s h e d by  (pancavayava)  form  proposition  {pratijna)  , reason  application  (upnaya),  and  example of 1.  Indian  There i s f i r e  on  2. b e c a u s e t h e r e 3. and  instance, in a  5. The  and  Indian  logic  conclusion  the  there  the  is fire the  ( 1 ) ) , the  and  s e c o n d c l a u s e s , e s t a b l i s h e d by  of the  essential part  finally  two  kitchen)  of the  follows:  as,  for  on i t .  the  that of  stated general  c l i n c h e d by  clauses. was  of  classical  ( 3 ) , minor premiss  argument b e i n g  and  first  above s y l l o g i s m the  is fire,  order  major premiss  third,  as  classical  mountain,  conclusion  repetition  paraphrased  The  kitchen;  and  example i n t h e  (nigamana).  i s smoke t h e r e  syllogism reversed  (Aristotelian:  (udaharana) ,  example  mountain,  case with  therefore  be  members:  i s smoke above i t ,  where t h e r e  4. s u c h i s t h e  comprises f i v e  (hetu),  s y l l o g i s m may  s y l l o g i s m , of  The  i n the rule  first  and  virtual  "example"  g e n e r a l l y looked  a r g u m e n t , and  (2),  on  ( i n the as  helped to strengthen  an its  xxxi  rhetorical  force.  "where there  But  b a s i s of g e n e r a l i z a t i o n  i s smoke there  ence r e s t s was concomitance  The  (for example  i s f i r e " ) on which every  infer-  b e l i e v e d t o be the q u a l i t y of u n i v e r s a l  (vyapti)  . . . .  the world i s more complex and  it,  and  may  a t the same time be  s u b t l e than we  t h a t what i s t r u e of a t h i n g i n one  of i t s aspects  f a l s e i n another.  Basham, A.L., "The wonder t h a t was Inc., New York (1959).  think  I n d i a , " Grove  Press,  1 CHAPTER 1 INTRODUCTION  1.1  Three-phase  fluidized  beds  A l a r g e number o f c h e m i c a l and p r o c e s s 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 and  solid)  into  (gas, l i q u i d  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 .  chemical c h a r a c t e r i s t i c s by  two o r more p h a s e s  o f the phases  involved  The  are determined  the r e q u i r e m e n t s o f the p r o c e s s i t s e l f , b u t the dynamic  behaviour o f these phases relative  m o t i o n between t h e l i g h t e r  Fluidization between a f l u i d  phase  and t h e h e a v i e r  and a s o l i d p h a s e  phases.  relative  i n s u c h a manner  fluid.  Under t h e s e  f o r c e s on t h e p a r t i c l e s , m o d i f i e d by buoy-  a n c y , a r e b a l a n c e d by d r a g f o r c e s m o t i o n between t h e f l u i d The g e o m e t r i c a l  arising  and t h e s o l i d  from  the r e l a t i v e  particles.  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 d e p e n d e n t o n t h e n a t u r e o f t h e f l u i d i z i n g medium. the  fluid  used i s a l i q u i d , a f l u i d i z e d  spherical particles  bed o f s i n g l e  w i l l g e n e r a l l y appear  If sized  t o be homogeneously  distributed.  T h e r e a r e t h e n no i n t e r - p a r t i c l e  each p a r t i c l e  a p p e a r s t o have i t s i n d i v i d u a l  therefore  motion  a r e s u p p o r t e d and m a i n t a i n e d i n a  s t a t e by upward f l o w i n g  conditions gravity  g o v e r n e d by t h e  i s a phenomenon i n v o l v i n g  that s o l i d p a r t i c l e s suspended  i s principally  collisions,  identity  and  s u c h a l i q u i d - s o l i d f l u i d i z e d b e d i s d e s c r i b e d as  2 "particulately fluidized".  The gross expansion c h a r a c t e r i s -  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 c o n d i t i o n s can be p r e d i c t e d t h e o r e t i c a l l y by v a r i o u s models  (Happel/ Kuwabara),  empirical relation  cell  b u t i s b e s t r e p r e s e n t e d by an  developed by R i c h a r d s o n and  Zaki [ 2 ] .  However, i f the f l u i d used i s a gas, l a r g e o r s m a l l  solid-  f r e e aggregates u s u a l l y r i s e up through the bed, g i v i n g i t the appearance o f a b o i l i n g l i q u i d , and t h e r e f o r e such a g a s - 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 " a g g r e g a t i v e l y ized".  The s o l i d - f r e e aggregates o r gas bubbles g i v 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 g a s - s o l i d  fluidrise  fluidized  beds, and t h e r e f o r e the d e s c r i p t i o n 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 o f 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 m a i n t a i n 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 s o l i d ) are thereby brought i n t o c o n t a c t s i m u l t a n e o u s l y . The term  'three-phase f l u i d i z a t i o n  1  has been used r a t h e r  l o o s e l y to d e s c r i b e the process of b r i n g i n g gas, l i q u i d suspended  and  s o l i d p a r t i c l e s i n t o c o n t a c t , where the gas and  the l i q u i d may currently.  be f l o w i n g e i t h e r  cocurrently or counter-  As d e f i n e d i n t h i s t h e s i s , 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 f l o w i n g  c o c u r r e n t l y and v e r t i c a l l y upwards  i  The term g a s - l i q u i d * f l u i d i z a t i o n has a l s o been used to d e s c r i b e t h i s mode of c o n t a c t i n g .  The p r e d i c t i o n of holdup  3 f o r each phase of a three-phase f l u i d i z e d bed  requires  knowledge of e q u i l i b r i u m c o n d i t i o n s between the three which i n t u r n i s governed by between the phases.  The  the l o c a l r e l a t i v e  geometrical  appearance o f the bed w i l l depend on each phase i n the bed. f l u i d i z e d a t low  a phases,  velocities  s t r u c t u r e and  physical  the l o c a l holdup o f  The bed w i l l appear to be p a r t i c u l a t e l y  g a s - t o - l i q u i d r a t i o s [7,8]  and  aggregatively  f l u i d i z e d a t l a r g e g a s - t o - l i q u i d r a t i o s , as has  been v i s u a l l y  observed by v a r i o u s i n v e s t i g a t o r s [13]. Jackson Bhatia  [4] observed q u a l i t a t i v e l y  granular  and  that f o r suspension o f  p a r t i c l e s i n a p o o l o f l i q u i d by gas  r e s u l t i n g bed  [3]  injection,  c o n s i s t s of s o l i d p a r t i c l e s dispersed  the  in a  continuous l i q u i d phase, w i t h the gas r i s i n g through  the  medium as d i s c r e t e bubbles, t r a n s f e r r i n g i t s momentum to the l i q u i d phase d u r i n g  the ascent.  t h a t i n a three phase f l u i d i z e d bed supported e n t i r e l y by  Thus i t i s envisaged the s o l i d p a r t i c l e s are  the l i q u i d phase alone, which suggests  t h a t the c h a r a c t e r i s t i c p r o p e r t i e s o f a three-phase f l u i d i z e d bed  can be  simpler  synthesized  on the b a s i s o f the p r o p e r t i e s o f  two-phase systems, namely,  flow and  liquid-solid  The  cocurrent g a s - l i q u i d  fluidization.  a n a l y s i s o f 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 above o b s e r v a t i o n s ,  s i n c e the holdup of l i q u i d - s o l i d  beds i s very w e l l d e s c r i b e d Richardson and  two  by  the fluidized  the e m p i r i c a l r e l a t i o n o f  Z a k i , as w e l l as by other  similar correlations.  4 However, knowledge o f l o c a l holdup d i s t r i b u t i o n s of the and  the l i q u i d phases, and  important The  gas  t h e i r i n t e r a c t i o n , i s a l s o very  i n order to a n a l y s e the phenomenon completely.  study of the d i s t r i b u t i o n o f gas and l i q u i d phases i n  three-phase  f l u i d i z e d beds r e q u i r e s knowledge of two-phase  g a s - l i q u i d flow and  the e f f e c t o f s o l i d p a r t i c l e s i n  p e r t u r b i n g the d i s t r i b u t i o n of phases i n the g a s - l i q u i d  flow.  The f i e l d of g a s - l i q u i d flow has been i n v e s t i g a t e d by a l a r g e number o f r e s e a r c h e r s , 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 r e s p e c t i v e flow and holdup p r o f i l e s i s f a r from complete incomplete  understanding  [39].  of the two phase g a s - l i q u i d  phenomenon puts attempts to analyse the behaviour phase f l u i d i z e d beds a t some  fluidization  i n t o c e r t a i n process  i n v o l v i n g g a s - l i q u i d r e a c t i o n i n the presence solid catalyst.  flow  of t h r e e -  disadvantage.  N e v e r t h e l e s s , the three-phase has a l r e a d y found i t s way  This  technique  industries of a suspended  In such r e a c t i o n systems, where interphase_  mass t r a n s f e r i s not the c o n t r o l l i n g " r e s i s t a n c e , c o c u r r e n t g a s - l i q u i d flow i s employed s i n c e much h i g h e r can be achieved without  throughputs  f l o o d i n g than f o r c o u n t e r c u r r e n t .  flow, w h i l e the higher mass t r a n s f e r d r i v i n g f o r c e o b t a i n a b l e from the l a t t e r i s not r e q u i r e d . The a p p l i c a b i l i t y o f the three-phase  fluidization  technique o f b r i n g i n g t h r e e phases i n t o c o n t a c t has not been completely  i n v e s t i g a t e d [54] but  simultaneously  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 o f being employed i n any o f the f o l l o w i n g combinations: 1.  gas r e a c t i n g w i t h l i q u i d i n the presence of  solids  as the c a t a l y s t , v i z . hydrogenation o f v e g e t a b l e oil  i n the presence of n i c k e l or o t h e r powdered  catalysts. 2.  gas and l i q u i d r e a c t i n g w i t h s o l i d p a r t i c l e s ,  e.g.  i n the p r o d u c t i o n o f c a l c i u m b i - s u l f i t e cooking l i q u o r from s o l i d limestone p a r t i c l e s i n the presence o f water and s u l f u r d i o x i d e [55]. 3.  gas r e a c t i n g w i t h s o l i d p a r t i c l e s i n the presence o f l i q u i d catalyst; certain e s t e r i f i c a t i o n reactions c o u l d be c l a s s i f i e d as s p e c i f i c  4.  examples.  gas r e a c t i n g w i t h 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 a p p l i c a t i o n s i n chemical r e a c t i o n  systems,  three-phase f l u i d i z a t i o n can a l s o be employed f o r p h y s i c a l operations.  Thus a three-phase c r y s t a l l i z e r has been designed  to r e g u l a t e 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 o f three-phase f l u i d i z e d beds The a c t u a l and p o t e n t i a l uses o f three-phase  fluidiz-  a t i o n i n v a r i o u s i n d u s t r i a l processes have l e d v a r i o u s i n v e s t i g a t o r s to study i t e x p e r i m e n t a l l y .  Thus, Turner  [6]  6 suggested the p o s s i b i l i t y o f u s i n g the three-phase  fluidiz-  a t i o n technique i n d e s u l f u r i z i n g h i g h e r m o l e c u l a r weight r e s i d u a l petroleum f e e d s t o c k s .  He t h e r e f o r e c a r r i e d  out  p r e l i m i n a r y hydrodynamic experiments on the a i r - w a t e r - s a n d system and thus demonstrated  f o r the f i r s t  behaviour of three-phase f l u i d i z e d  time the p e c u l i a r  beds.  Turner observed t h a t i n t r o d u c i n g a s m a l l flow r a t e of gas to a l i q u i d - s o l i d  fluidized  bed, keeping the l i q u i d  v e l o c i t y c o n s t a n t , r e s u l t e d i n c o n t r a c t i o n o f the bed. [7], 0stergaard  Stewart and Davidson Thompson [9]  [8] , and A d l i n g t o n and  then i n v e s t i g a t e d t h i s unexpected  three-phase f l u i d i z e d  behaviour of  beds, u s i n g a i r and water as gas  l i q u i d r e s p e c t i v e l y , with s p h e r i c a l , s i n g l e - s i z e d p a r t i c l e s of v a r i o u s d e n s i t i e s . contraction of a l i q u i d - s o l i d  solid  They too observed  fluidized  bed on  and  the  introduction  of gas a t c o n s t a n t l i q u i d flow r a t e i n a l l the systems which they i n v e s t i g a t e d .  The degree of c o n t r a c t i o n 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  fluidized  bed.  A s i m i l a r i n v e s t i g a t i o n had been c a r r i e d out by V o l k  earlier  [10], u s i n g n i t r o g e n as gas, heptane as l i q u i d  extruded c y l i n d r i c a l  and  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 v a r y i n g from 0.625 i n c h to 6 inches i n diameter. V o l k , however, observed t h a t under a l l c o n d i t i o n s s t u d i e d the l i q u i d - s o l i d  fluidized  bed expanded f u r t h e r on  of gas a t c o n s t a n t l i q u i d flow r a t e .  introduction  This contradictory  behaviour o f three-phase f l u i d i z e d  beds can perhaps be  a s c r i b e d to the d i f f e r e n c e i n p h y s i c a l p r o p e r t i e s and e s p e c i a l l y s u r f a c e t e n s i o n o f the l i q u i d phase used. The  e f f e c t of l i q u i d p r o p e r t i e s on the behaviour of  three-phase f l u i d i z e d has not been i n v e s t i g a t e d  systematic-  a l l y , b u t r e c e n t l y Dakshinamurty e t a l . [11], u s i n g  kerosene  as the l i q u i d phase, a i r as the gas phase and s i n g l e - s i z e d g l a s s beads as the s o l i d phase, observed t h a t 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 f u r t h e r on i n t r o d u c t i o n o f gas a t c o n s t a n t  liquid velocity.  Contrary  behaviour was observed w i t h water as l i q u i d . I t should be noted t h a t the o r g a n i c  l i q u i d s used by  Volk and by Dakshinamurty e t a l . have s u r f a c e t e n s i o n s one-third  t h a t o f water, and t h a t these i n v e s t i g a t o r s p a i d  no a t t e n t i o n to the p o s s i b l e presence o f t r a c e i n t h e i r t e c h n i c a l grade l i q u i d s . organic  about  impurities  I t i s known t h a t minute  i m p u r i t i e s could change the s u r f a c e 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 r e n d e r i n g by the l i q u i d phase. was p o i n t e d  them non-wettable  The importance o f s o l i d s w e t t a b i l i t y  out e a r l i e r by Guha e t a l . [13] i n t h e i r  study  o f the suspension o f 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 o f w e t t a b i l i t y o f 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 t h a t a' bed o f l i q u i d and  w e t t a b l e p a r t i c l e s c o n t r a c t e d , w h i l e the same bed w i t h the  8 p a r t i c l e s rendered non-wettable expanded f u r t h e r , on d u c t i o n o f gas  at a fixed l i q u i d rate.  intro-  I t i s , then, b e l i e v e d  t h a t the 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 , e s p e c i a l l y s u r f a c e t e n s i o n , and  the nature o f the p a r t i c l e s u r f a c e are  probably  i n t e r r e l a t e d and p l a y an important r o l e i n determining behaviour of three-phase f l u i d i z e d beds s i n c e no systematic  i n v e s t i g a t i o n has  to e l u c i d a t e the e f f e c t o f l i q u i d and  [83].  the  However,  y e t been c a r r i e d out. s o l i d surface  properties,  no g e n e r a l i z a t i o n s can be made w i t h any  confidence.  all  have been c a r r i e d out  s t u d i e s , i n c l u d i n g the present  using  one,  Almost  s o l i d p a r t i c l e s which were f u l l y w e t t a b l e by the  liquid  phase. M i c h e l s e n and experiments on 3 mm  0 s t e r g a a r d  and  6 mm  [14], i n conducting d e t a i l e d  g l a s s beads, found t h a t water  f l u i d i z e d beds c o n s i s t i n g of such r e l a t i v e l y d i d not show any  c o n t r a c t i o n when a i r was  observed f o r s m a l l e r p a r t i c l e s .  a t h i g h gas  introduced  Almost a l l o f the  works were c a r r i e d out u s i n g o n l y very s i n c e i t was  large p a r t i c l e s  small gas  M i c h e l s e n and  0 s t e r g a a r d  earlier  flow  d i f f i c u l t to d i s c e r n the expanded bed rates.  as  rates,  height  extended  the  range of gas v e l o c i t i e s s t u d i e d , though they d i d not i n d i c a t e how  the h e i g h t o f expanded bed was  measured.  t h a t a l i q u i d - s o l i d f l u i d i z e d bed o f 1 mm t r a c t e d on i n t r o d u c i n g the gas a t small  They observed  p a r t i c l e s con-  flow r a t e s .  On  f u r t h e r i n c r e a s i n g the gas v e l o c i t y a t a f i x e d l i q u i d v e l o c i t y , the bed  height  tended to r e a c h a d e f i n i t e minimum  9 and then i n c r e a s e w i t h i n c r e a s e of gas v e l o c i t y . Another important a s p e c t o f three-phase f l u i d i z e d bed o p e r a t i o n i s the behaviour o f the gas phase i n s i d e the b e d . Numerous attempts have been made to study t h i s a s p e c t , b u t i t i s not y e t f u l l y u n d e r s t o o d . s t u d i e d the r i s e v e l o c i t y  M a s s i m i l l a e t a l . [15]  o f a s i n g l e 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, u s i n g water as the l i q u i d . t h a t the r i s e v e l o c i t y  They observed  was a f u n c t i o n o f the bubble d i a m e t e r ,  but t h a t the f u n c t i o n a l dependence f o r s m a l l bubbles diameter up to 8mm), was r a d i c a l l y d i f f e r e n t water.  than i n pure  However, f o r l a r g e r bubbles the r i s e v e l o c i t y  l i q u i d - s o l i d f l u i d i z e d beds approached pure w a t e r .  (bubble  i n the  the r i s e v e l o c i t y i n  Another important o b s e r v a t i o n to be made from  t h e i r measurements i s t h a t , although the r i s e v e l o c i t y o f bubbles i n pure water i s p r a c t i c a l l y  c o n s t a n t 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 o f 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 f o r the same range o f  bubble diameters i n c r e a s e s m o n o t o n i c a l l y .  This observation  then l e a d s 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 c o n t a i n s bubbles of v a r i o u s s i z e s . istic the  Thus l a r g e b u b b l e s , w i t h t h e i r  h i g h v e l o c i t y , would be p r e v a l e n t i f c o a l e s c e n c e were  predominant  with their  phenomenon i n s i d e the bed, w h i l e s m a l l b u b b l e s ,  characteristic  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 of  character-  the s o l i d p a r t i c l e s  phenomenon.  The r o l e  can then be c h a r a c t e r i z e d by n o t i c i n g  10 how the gas holdup, d e f i n e d as the v o l u m e t r i c f r a c t i o n o f the bed occupied by the gas bubbles, i s a f f e c t e d . To compare the gas holdup i n a three-phase f l u i d i z e d bed w i t h t h a t i n two-phase g a s - l i q u i d flow i t i s necessary t h a t they both be r e f e r r e d ^ to a common s o l i d s f r e e b a s i s . 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 f r e e b a s i s i s g i v e n by  Ml  e  =  2  e /(l-e ) 2  (1.1)  3  The gas holdup from e q u a t i o n 1.1 can then be compared t o the II  gas holdup i n two-phase g a s - l i q u i d flow, e , 2  e f f e c t o f the s o l i d  t o  e l u c i d a t e the  particles.  Most i n v e s t i g a t o r s  [14,16,17] have, however, compared  the d i r e c t l y measured a b s o l u t e gas holdup i n a three-phase f l u i d i z e d bed, e , w i t h t h a t i n two-phase g a s - l i q u i d flow, 2  a t the same flow r a t e s of both the gas and the l i q u i d  phases.  Thus from such a comparison M i c h e l s e n and 0stergaard [14], who o b t a i n e d the gas holdup i n s i d e the bed from the s t a t i c p r e s s u r e drop measurements i n a 6 i n c h diameter column, concluded t h a t i n beds o f 3 mm and 6 mm g l a s s beads break-up of bubbles takes p l a c e i n the lower p o r t i o n o f the bed, whereas i n beds o f 1 mm g l a s s beads c o a l e s c e n c e takes p l a c e i n the same r e g i o n .  No m e c h a n i s t i c c o r r e l a t i o n was, however, o b t a i n e d  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 g a s - l i q u i d f l o w .  11 Efremov and Vakhrushev  [16] used the same p r i n c i p l e  o f s t a t i c p r e s s u r e drop measurement to o b t a i n the gas  holdup  i n s i d e the bed i n a 10 cm diameter column, although the accuracy o f t h e i r s t a t i c p r e s s u r e drop measurement technique, u s i n g 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 f r a c t i o n s o f g l a s s  beads w i t h mean diameters r a n g i n g between 0.32 mm  mm  and  and they observed t h a t f o r a l l s i z e s o f p a r t i c l e s  2.15 studied,  the gas holdup i n a three-phase f l u i d i s e d bed was c o n s i d e r a b l y s m a l l e r than i n two-phase g a s - l i q u i d flow under gas and l i q u i d flow r a t e s , r e s p e c t i v e l y .  identical  They were f u r t h e r -  more a b l e to. e m p i r i c a l l y c o r r e l a t e the gas holdup i n s i d e the three-phase f l u i d i z e d bed w i t h the gas holdup  i n two phase  g a s - l i q u i d flow. V a i l e t a l . [17] used the method o f i s o l a t i n g the s e c t i o n by s i m u l t a n e o u s l y c u t t i n g o f f the gas and the  test  liquid  flows and then r e c o r d i n g the amount of gas c o l l e c t e d a t the top of the bed. Technique,"  As w i l l be d i s c u s s e d under "Experimental  t h i s method has i n h e r e n t e r r o r s , which were not  f u l l y c o r r e c t e d f o r by these i n v e s t i g a t o r s . 0.73  mm  They s t u d i e d  g l a s s 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 o u t i n a 14.6 diameter column.  They found t h a t f o r a l l the t h r e e s o l i d  p a r t i c l e s s t u d i e d , the gas holdup i n three-phase was  cm  fluidization  always s m a l l e r 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 c o n d i t i o n s .  They a t t r i b u t e d  12 t h i s r e s u l t to the f a c t t h a t the s o l i d phase d i s p l a c e s p a r t of  the l i q u i d , w h i l e the gas bubbles can r i s e and e x i s t o n l y  w i t h i n the l i q u i d phase. Based on t h i s reasoning,an e m p i r i c a l correlation  between the gas holdup i n a three-phase f l u i d i z e d  bed and i n two-phase g a s - l i q u i d  flow was  presented.  In view o f the r e p o r t e d works [14,15,79] i t would seem to be c o r r e c t t h a t i f the p a r t i c l e s i z e i s s m a l l as compared to the bubble s i z e , bubble c o a l e s c e n c e w i l l whereas i f the p a r t i c l e s i z e i s comparable  result,  o r b i g g e r than  the bubble s i z e , bubble break-up w i l l o c c u r , i n the bed. 0 s t e r g a a r d  and Theisen [18] observed t h a t t h i s does not seem  to a f f e c t the c o n t r a c t i o n or expansion behaviour of the bed, s i n c e p a r t i c l e s i z e does not seem to i n f l u e n c e 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 phase and the g a s - l i q u i d detailed  (bubble) phase.  study has been done  (particulate)  However, s i n c e no  of flow d i s t r i b u t i o n o r  solids  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 r a t h e r important to understand the c h a r a c t e r i s t i c each phase and the mutual  interaction  behaviour o f  between phases, a t l e a s t  qua1i ta t i v e l y .  1.3  Wake model f o r 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  dividual  of the behaviour of i n -  phases can p r o v i d e the b a s i s f o r a mathematical model  d e s c r i b i n g the g r o s s 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 f l u i d i z e d beds which, although i t plays an important r o l e i n determining the behaviour of such beds, has not been s u f f i c i e n t l y investigated, i s the phenomenon of wake formation behind the dispersed gas phase.  I t i s very well recognized now that a dispersed phase,  when moving through a continuous medium, c a r r i e s along with i t some continuous phase as i t s wake. The average amount of continuous phase c a r r i e d 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,  i n a three phase f l u i d i z e d bed, gas bubbles r i s i n g through the continuous l i q u i d medium w i l l carry part of the l i q u i d phase i n their wake, making that part of the l i q u i d phase unavailable for support of the s o l i d p a r t i c l e s .  On the  basis of this wake phenomenon, Stewart and Davidson  [7] were  able to explain the observed contraction i n three-phase fluidized  beds.  Using a two-dimensional l i q u i d - f l u i d i z e d bed contained between perspex plates spaced 0.25 inch apart, Stewart and Davidson observed photographically that when an a i r bubble r i s e s through the bed, some l i q u i d follows the bubble as i t s wake.  They also observed that the l i q u i d wake was p r a c t i c a l l y  free of s o l i d p a r t i c l e s .  The r e s u l t i n g combined gas-liquid bubble  r i s e s through the bed at a much greater v e l o c i t y than the  14 average v e l o c i t y o f l i q u i d through the i n t e r s t i c e s o f the bed, thus removing  some l i q u i d from the continuous phase and,  a c c o r d i n g to the r e a s o n i n g o f these i n v e s t i g a t o r s , thereby r e d u c i n g the o v e r a l l f l u i d i z i n g  force.  Because o f t h i s r e -  d u c t i o n , 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 r e p r e s e n t the observed bed behaviour. The model subsequently proposed by 0stergaard  [8] was  i n i t s main f e a t u r e s the same as t h a t suggested by Stewart and Davidson, b u t i t was based on r e p r e s e n t i n g a three-phase f l u i d i z e d bed as c o n s i s t i n g o f a l i q u i d - f l u i d i z e d phase, a gas bubble phase and a wake phase.  particulate  The wake phase  was assumed to f o l l o w the bubble phase a t the bubble v e l o c i t y and have a p o r o s i t y i d e n t i c a l to t h a t o f the p a r t i c u l a t e  phase.  T h i s l a s t assumption, based on photographic o b s e r v a t i o n o f 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 b e i n g  f o l l o w e d by a long t r a i l  c o n t a i n i n g s o l i d s , c o n t r a d i c t s the  o b s e r v a t i o n o f Stewart and Davidson t h a t a bubble i s f o l l o w e d by a wake o f l i q u i d devoid o f p a r t i c l e s .  There i s , u n f o r t u n a t e l y ,  no c o n c l u s i v e evidence a v a i l a b l e y e t to support or r e p u d i a t e either claim.  The model proposed by 0stergaard i s presented  here i n i t s e n t i r e t y , s i n c e i t i s f e l t t h a t t h i s model attempts t o c o n s i d e r the fundamental d i s t r i b u t i o n o f the l i q u i d between the p a r t i c u l a t e phase and the wake phase. F u r t h e r , i t i s hoped t h a t d e f i c i e n c i e s o f the model can be i d e n t i f i e d so t h a t m o d i f i c a t i o n s can be i n c o r p o r a t e d to  15 explain  three-phase bed behaviour more r e a l i s t i c a l l y . L e t us c o n s i d e r  expanded to a h e i g h t ,  a bed o f p a r t i c l e s o f weight, W, L^, i n a column o f c r o s s - s e c t i o n a l  area, A, under the i n f l u e n c e of gas and l i q u i d fluxes  <  j-j_  volumetric  >  and J 2 <  respectively.  >  solids  fraction  o r s o l i d s holdup w i l l be  (1.2)  Ps^b  3 The  Then the average  —  =  e,  e , defined  bed p o r o s i t y ,  volumetric  as the f r a c t i o n o f 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)  I f i t i s assumed t h a t the p o r o s i t y of the wake phase i s i d e n t i c a l to the p o r o s i t y of the p a r t i c u l a t e solid  f l u i d i z e d ) phase, i t f o l l o w s  £ = £ (l-£ ) + £ 1  where  2  ( i . e . the l i q u i d -  that  (1.4)  2  i s the volume f r a c t i o n o f l i q u i d  phase and i s r e l a t e d to the v o l u m e t r i c  i n the p a r t i c u l a t e  f l u x through t h i s  n  region,  j ^ , by the w e l l known Richardson - Z a k i  [2] equation,  i.e., *1  =  " ... \ 1/n <3i/Vj  ( 1  .  5 )  16 where =  n  f (Re , d /D) p' p  (1.6)  L e t us assume t h a t the volume f r a c t i o n of the t h r e e phase bed o c c u p i e d by the wake phase i s e^. the three-phase  I f we c o n s i d e r  f l u i d i z e d bed to be m a c r o s c o p i c a l l y homogeneous,  then the f r a c t i o n of any c r o s s - s e c t i o n a l area occupied by the gas bubble phase and the wake phase w i l l be e respectively.  e  a n < 2  ^ yi  Therefore on the b a s i s o f the p h y s i c a l p i c t u r e  assumed by 0stergaard f o r a three-phase  f l u i d i z e d bed, the  area occupied by the p a r t i c u l a t e o r l i q u i d - s o l i d  fluidized  phase i n the plane p e r p e n d i c u l a r to the p r i n c i p l e flow a x i s w i l l be  A  LS  =  ( 1  - 2- k £  £  (1.7)  ) A  The d i s t r i b u t i o n o f 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 o b t a i n e d by c a r r y i n g o u t a m a t e r i a l balance a c r o s s any c r o s s - s e c t i o n i n s i d e the three-phase bed:  T o t a l v o l u m e t r i c flow r a t e o f l i q u i d through the column  V o l u m e t r i c flow r a t e o f l i q u i d through the p a r t i c u l a t e phase )*  (  V o l u m e t r i c flow r a t e o f l i q u i d through the wake phase  17 or Q  l  =  Q  l f  +  Q  l k  ( 1  '  8 )  I f i t i s assumed t h a t the wake phase t r a v e l s w i t h the gas bubble phase a t the bubble v e l o c i t y , a good assumption i f one d i s r e g a r d s the wake shedding phenomenon, then e q u a t i o n 1.8, w i t h the a i d o f e q u a t i o n 1.7, may be r e w r i t t e n as  _  II  <j > 1  A  =  J (l-e -e )A + v 1  2  k  II 2  e  k  A e  1  (1.9)  II  S o l v i n g e q u a t i o n 1.9 f o r  g i v e s the v o l u m e t r i c f l u x through  the p a r t i c u l a t e phase as _  3  ±  =  II  —  (1.10) 1 _ e  2- k e  In o r d e r to p r e d i c t the behaviour o f a three-phase f l u i d i z e d bed from the above s e t o f e q u a t i o n s , one r e q u i r e s independent knowledge o f e  2  and e^.  The f r a c t i o n o f the  bed volume occupied by the gas phase i s r e l a t e d to the average r i s e v e l o c i t y o f the bubbles by  =  <  Jo /v  (1.11)  >  9  As d i s c u s s e d i n the p r e c e d i n g s e c t i o n , the r i s e of a s i n g l e bubble  in a l i q u i d - s o l i d  velocity  f l u i d i z e d bed was  18 s t u d i e d by M a s s i m i l l a  [15].  However, no d a t a 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 bed are a v a i l a b l e .  fluidized  t h e r e f o r e o b t a i n e d an  0 s t e r g a a r d  e s t i m a t e o f bubble r i s e v e l o c i t y i n a three-phase bed from the data of N i c k l i n  fluidized  [19] f o r two-phase g a s - l i q u i d  flow, which 0 s t e r g a a r d c o r r e l a t e d by the e m p i r i c a l equation,  v  2  =  21.7  - 4.6  However, s i n c e e q u a t i o n 1.12  In <j > + <j > 2  (1.12)  x  i s based on two-phase d a t a , i t  c o u l d h a r d l y be expected t o a c c u r a t e l y r e p r e s e n t the gas bubble phase i n a three-phase  bed.  The phenomenon of wake f o r m a t i o n behind a gas has not been s t u d i e d e x t e n s i v e l y .  bubble  The importance o f the wake  phenomenon i n c o n t r o l l i n g t r a n s p o r t p r o c e s s e s was r e a l i z e d i n l i q u i d - l i q u i d o p e r a t i o n s [36].  first  Subsequently,  t h e r e f o r e , attempts were made to o b t a i n i n f o r m a t i o n r e g a r d i n g the shape and s i z e o f wakes behind l i q u i d drops  [84], w h i l e  study of the mode, frequency and r a t e of wake shedding i n liquid-liquid  systems  i s c u r r e n t l y i n p r o g r e s s [85].  G e n e r a l i z a b l e q u a n t i t a t i v e i n f o r m a t i o n r e g a r d i n g the wake characteristics i n liquid-liquid  systems, however, i s  r a t h e r l i m i t e d , whereas even s p e c i f i c i n f o r m a t i o n 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 n three-phase systems  i s minimal  almost n o n - e x i s t e n t .  and  Therefore i n  19 order to o b t a i n an estimate f o r the volume f r a c t i o n o c c u p i e d by the wakes, e^.,  [8] postulated, q u a l i t a t i v e l y  0 s t e r g a a r d  that i n c r e a s e s w i t h i n c r e a s i n g z^i  1.  the r a t e o f  i n c r e a s e i n e^. being slower a t l a r g e v a l u e s of  2.  i n c r e a s e s w i t h i n c r e a s i n g l i q u i d flow r a t e , t h a t i s , w i t h i n c r e a s i n g bed  fluidity.  These two assumptions were then i n c o r p o r a t e d by t r i a l e r r o r i n t o the f o l l o w i n g equation i n such a way  and  as to s a t i s f y  the data f o r bed c o n t r a c t i o n :  e  where j ] _ <  =  k  > m  f  0.14  1  S  e  0 2  -  5  (<j _> - < j > ]  1  m f  )  (1.13)  the minimum v o l u m e t r i c l i q u i d f l u x r e q u i r e d  to i n i t i a t e f l u i d i z a t i o n o f the s o l i d 0 s t e r g a a r d  particles.  thus presented the s e t of equations 1.2  1.10,  which when used  1.13,  s a t i s f i e d a l i m i t e d q u a n t i t y of experimental d a t a on  bed  contraction.  0 s t e r g a a r d  by  and  0 s t e r g a a r d  i n c o n j u n c t i o n w i t h equations 1.11  -  But l a t e r , i n a more d e t a i l e d  investigation,  Theisen [18] r e p o r t e d t h a t the model c o u l d not s a t i s f a c t o r i l y p r e d i c t the  c o n t r a c t i o n of three-phase of o p e r a t i n g v a r i a b l e s .  proposed observed  f l u i d i z e d beds over a wider  Furthermore,  the model f a i l s  d e s c r i b e f u l l y the observed bed behaviour of  -  range to  three-phase  20 f l u i d i z e d beds as o u t l i n e d i n the p r e c e d i n g s e c t i o n , v i z . t h a t the bed h e i g h t drops on i n t r o d u c t i o n of gas a t a cons t a n t l i q u i d flow r a t e , then reaches a d e f i n i t e minimum on i n c r e a s i n g the gas flow r a t e , and f i n a l l y s l o w l y expands again as the gas flow r a t e i s f u r t h e r i n c r e a s e d . i n i t i a l bed c o n t r a c t i o n i s p r e d i c t e d by the model. less  0 s t e r g a a r d ' s  Only the Neverthe-  model f o r a three-phase f l u i d i z e d bed does  have the v i r t u e of d e s c r i b i n g , a l b e i t approximately, the d i s t r i b u t i o n o f 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 f o r e s t i m a t i n g the  v o l u m e t r i c gas f r a c t i o n and the v o l u m e t r i c wake f r a c t i o n cannot be expected to a c c u r a t e l y p r e d i c t these q u a n t i t i e s i n a three-phase f l u i d i z e d bed, s i n c e they were o b t a i n e d from a v e r y l i m i t e d range o f d a t a .  Even i n two-phase gas-  l i q u i d flow the i n f o r m a t i o n on gas holdup i s n o n - c o n c l u s i v e and ambiguous, w h i l e the i n f o r m a t i o n on v o l u m e t r i c wake f r a c t i o n and i t s r o l e i n d e t e r m i n i n g v e l o c i t y p r o f i l e s i n gasl i q u i d flow i s almost n o n - e x i s t e n t . I t i s u s e f u l a t t h i s p o i n t to d i g r e s s from  0 s t e r g a a r d ' s  model and c o n s i d e r the simple model proposed by Davidson to e x p l a i n the bubble phenomenon i n gas s o l i d  [20]  fluidization,  and the m o d i f i c a t i o n i n t r o d u c e d to t h i s simple model by K u n i i and L e v e n s p i e l [21] to e x p l a i n bubble behaviour i n g a s - s o l i d 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  p o s t u l a t e d t h a t a b u b b l i n g g a s - s o l i d f l u i d i z e d bed can be c o n s i d e r e d as b e i n g c o n s t i t u t e d of a gas bubble phase and a  21 g a s - s o l i d emulsion phase. i n the bubble-phase  Davidson a l s o assumed the bubbles  to be s p h e r i c a l and the flow around the  s p h e r i c a l c a v i t i e s to be i r r o t a t i o n a l and i n c o m p r e s s i b l e . The flow p a t t e r n o f gas and s o l i d and the p r e s s u r e d i s t r i b u t i o n i n the v i c i n i t y o f the r i s i n g bubble p r e d i c t e d by t h i s model have been shown to be e s s e n t i a l l y c o r r e c t [21]. However, Rowe and P a r t r i d g e  [22] observed t h a t the r i s i n g  bubbles each c a r r y a wake behind them c o n t a i n i n g particles.  solid  The reason g i v e n f o r the presence o f t h i s wake  was t h a t the p r e s s u r e i n the lower p a r t o f the bubble i s l e s s than i n the nearby emulsion phase, a reason p r e d i c t a b l e by Davidson's model.  Gas i s thereby drawn i n t o 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 c o l l a p s e o f the bubble, and t u r b u l e n t mixing behind i t .  This turbulence r e s u l t s  i n s o l i d s b e i n g drawn up behind the bubble and forming a wake.  The wake o f the bubble exchanges  solid  material  c o n t i n u a l l y d u r i n g i t s r i s e , depending on the mode, frequency and r a t e o f wake shedding, b u t u l t i m a t e l y the s o l i d  particles  c a r r i e d i n the wake o f the bubble a r e d e p o s i t e d a t the bed s u r f a c e when the bubble emerges from the bed, thus g i v i n g r i s e to downward s o l i d movement i n the emulsion phase. K u n i i and L e v e n s p i e l  [21] m o d i f i e d Davidson's simple  model by i n c o r p o r a t i n g the wake phenomenon.  They c o n s i d e r e d  the b u b b l i n g g a s - s o l i d f l u i d i z e d bed as c o n s i s t i n g o f a gas bubble phase, a g a s - s o l i d emulsion phase, and a wake phase. T h i s model i s analogous t o t h a t proposed by 0stergaard f o r a  22 three-phase f l u i d i z e d bed. out  K u n i i and L e v e n s p i e l p o i n t e d  t h a t 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 a l s o  showed t h a t  s o l i d s movement d i d not a f f e c t the behaviour of the bubble phase markedly, but t h a t the movement o f the e n t i r e  emulsion  phase c o u l d be r e v e r s e d 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 p i c t u r e d  in  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 t h a t would  the  expansion o r c o n t r a c t i o n o f the bed.  the  d i s t r i b u t i o n o f l i q u i d between the p a r t i c u l a t e  control  0 s t e r g a a r d  phase  and the wake phase, but the c i r c u l a t o r y motion o f the i n the p a r t i c u l a t e not  considered.  considered  solids  phase as induced by the gas bubbles,  By analogy w i t h g a s - s o l i d  was  fludization, 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 a l s o 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 t h a t the wake  accompanying a gas bubble c o n t a i n s s o l i d p a r t i c l e s . Thus the main drawbacks i n the simple but e l e g a n t model proposed by  0 s t e r g a a r d  (i) The assumption is.equal  seem to be:  t h a t the p o r o s i t y o f the wake phase  to t h a t of the p a r t i c u l a t e  phase,  ( i i ) The n e g l e c t of s o l i d s c i r c u l a t i o n induced by the motion o f gas bubbles c a r r y i n g solid  particles.  wakes c o n t a i n i n g  (iii)  The q u a n t i t a t i v e r e p r e s e n t a t i o n o f wake volume f r a c t i o n by equation 1.13 v e l o c i t y by equation  1.4  and o f bubble r i s e  1.12.  Importance of t u r b u l e n c e phenomena i n  three-phase  f l u i d i z e d beds Turbulence momentum t r a n s f e r  i s known to e x e r t s i g n i f i c a n t i n f l u e n c e on (and o t h e r t r a n s f e r processes)  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 p a r t i c l e .  from a  the flow f i e l d  around  However, very l i t t l e e f f o r t has gone i n to  quantitatively correlating  these e f f e c t s w i t h the measureable  fundamental p r o p e r t i e s of a t u r b u l e n t f i e l d , v i z . i n t e n s i t y and  scale of turbulence.  f l u i d i z e d bed  The importance o f t u r b u l e n c e i n a  can be a p p r e c i a t e d i f we  c o n s i s t s of s o l i d p a r t i c l e s f i e l d i s developed  of f i n i t e  c o n s i d e r t h a t the bed size.  around each p a r t i c l e due  motion between the f l u i d  and  Then a flow to the  the p a r t i c l e and  relative  the no  c o n d i t i o n to be s a t i s f i e d a t the p a r t i c l e s u r f a c e . l a t t e r causes g e n e r a t i o n o f v o r t i c i t y a t the  whether the  flow f i e l d near the p a r t i c l e i s e i t h e r t u r b u l e n t or  convected  At high r e l a t i v e v e l o c i t y , downstream w i t h the flow and  The  particle  s u r f a c e , the growth and decay of which determines  turbulent.  slip  non-  the v o r t i c i t y i s i s concentrated  at  the r e a r o f the p a r t i c l e , c a u s i n g a backward flow to be induced near the s u r f a c e .  T h i s 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 r e a r ,  24 s t r e n g t h e n i n g the r o t a t i o n a l The  motion i n the s t a n d i n g eddy.  term wake i s commonly a p p l i e d to t h i s whole r e g i o n of  non-zero v o r t i c i t y on the downstream s i d e of the p a r t i c l e . A t s t i l l h i g h e r 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 o f  fluid.  Thus f o r m a t i o n of a s u f f i c i e n t l y h i g h v o r t i c i t y wake behind a p a r t i c l e can be c o n s i d e r e d as the onset o f a t u r b u l e n c e field  i n the f l u i d medium around the p a r t i c l e .  The  turbulence  field  i n a f l u i d i z e d bed, which i s c o n s t i t u t e d of an assemblage  of p a r t i c l e s , can then be c o n s i d e r e d as the composite of  the wakes o f i n d i v i d u a l  particles  However, i n three-phase phase g a s - l i q u i d  effect  [87].  f l u i d i z e d beds as i n  two-  systems, i t i s the gas bubbles which p l a y  the dominant r o l e i n c r e a t i n g t u r b u l e n c e i n the f l u i d  phase.  The mechanism f o r g e n e r a t i o n o f t u r b u l e n c e i s p r i n c i p a l l y the same as d e s c r i b e d above, v i z . the f o r m a t i o n of wakes behind  the bubbles  and consequent wake shedding  relative velocities.  Thus a c o c u r r e n t g a s - l i q u i d  non-laminar flow c o n d i t i o n s may which generates  a t higher  be c o n s i d e r e d as a system  f l u i d phase t u r b u l e n c e through  of randomly moving bubbles.  flow under  The presence  of s o l i d  i n such a t u r b u l e n c e g e n e r a t i n g system may  suppress  l i q u i d phase t u r b u l e n c e i n v a r y i n g degrees, 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  presence particles the  depending on  the i n t e n s i t y  of  turbulence. and  A l a r g e d e n s i t y d i f f e r e n c e between t h e p a r t i c l e  the l i q u i d w i l l  t e n d t o damp o u t t u r b u l e n c e  [88] , so  that the liquid-phase i n t e n s i t y of turbulence i n a  three-phase  f l u i d i z e d b e d may b e much s m a l l e r t h a n i n t h e c o r r e s p o n d i n g cocurrent gas-liquid  flow, f o r equal  f l u i d phase  A systematic i n v e s t i g a t i o n o f the expansion of  a fluidized  generation  bed s h o u l d then i n v o l v e  velocities.  characteristics  a study of turbulence  i n t h e b e d and t h e 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 t h e d r a g c o e f f i c i e n t o f an i n d i v i d u a l particle. The and  k n o w l e d g e o f wake f o r m a t i o n b e h i n d  t h e mode, f r e q u e n c y and r a t e o f wake s h e d d i n g  b e e n examined t o a n y e x t e n t . their  Stewart  i n v e s t i g a t i o n o f a three-phase  dimensional column, observed photographically.  Rigby  and C a p e s  bed,  has n o t  and D a v i d s o n fluidized  [7],i n  b e d i n a two  t h e wake s h e d d i n g  i n v e s t i g a t i o n of a two-dimensional  on  the bubble  phenomenon  [8 0],. f o l l o w i n g up three-phase  s u c c e s s f u l l y e x p l a i n e d the observed  fluidized  contraction  t h e b a s i s o f a few m e a s u r e m e n t s o f wake s h e d d i n g  consequent  r i s e o f shed v o r t i c e s  work t h u s i n d i r e c t l y d e m o n s t r a t e s  through  the bed.  bubbles, i n the study o f three-phase  have a l s o demonstrated  and t h e Their  the r i s i n g  f l u i d i z e d beds.  i n three-phase  indirectly  phenomenon  the p o s s i b l e relevance o f  t u r b u l e n c e , g e n e r a t e d b y s h e d d i n g o f wakes f r o m  [14,17] o n t h e s t a t e o f m i x i n g  their  Studies  fluidized  t h e dominant r o l e o f  beds  t u r b u l e n c e i n multiphase  flow.  However, attempts to g a i n  b e t t e r i n s i g h t i n t o the mechanisms c o n t r o l l i n g o t h e r t r a n s p o r t 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 flow f i e l d s around both the d i s p e r s e d phases are  the  fully  understood. Thus we  see t h a t t u r b u l e n c e probably p l a y s a r o l e i n  d e f i n i n g the behaviour  of three-phase  f l u i d i z e d beds.  the p r e s e n t s t a t e o f knowledge of three-phase  Given  systems t  however, i t appears s u f f i c i e n t to know the s i z e and shape o f a wake behind bubbles  an i s o l a t e d bubble,  the i n f l u e n c e of o t h e r  and p a r t i c l e s on the s i z e and  shape o f the wake, and  the mode, frequency and r a t e of wake s h e d d i n g — i n q u a n t i f y the c o n t r a c t i o n phenomenon.  order to  N e v e r t h e l e s s , i n order  to g a i n complete knowledge o f the f l u i d dynamics of t h r e e phase f l u i d i z e d beds, i t i s important  t h a t the  phenomenon be s y s t e m a t i c a l l y examined.  turbulence  The p r e s e n t s t a t e of  knowledge of t u r b u l e n c e can be used to d e s c r i b e s i n g l e phase flow, but i t has not advanced enough to p r e d i c t fundamental q u a n t i t i e s f o r multiphase  1.5  flow.  Scope of r e s e a r c h The i n f o r m a t i o n a v a i l a b l e on three-phase  beds i s r a t h e r scanty. r e s e a r c h was  fluidized  T h e r e f o r e the primary aim of t h i s  to d e v i s e and  c a r r y out an  experimental  programme f o r c o l l e c t i n g r e l i a b l e and a c c u r a t e data on holdup of s o l i d s and gas i n a three-phase  fluidized  bed  the  under a wide range o f flow c o n d i t i o n s . earlier  As has been suggested  [3,4], a three-phase f l u i d i z e d bed can be c o n s i d e r e d  as a complex  system, the p r o p e r t i e s of which are a composite  of two simpler systems: liquid-solid  g a s - l i q u i d c o c u r r e n t flow and a  f l u i d i z e d bed.  The parameters to be s t u d i e d  were s e l e c t e d on the b a s i s of i n f o r m a t i o n about the systems.  two-phase  Thus, s i n c e i t had been e s t a b l i s h e d i n v a r i o u s  studies  on two-phase g a s - l i q u i d flow t h a t v a r i a t i o n i n the p r o p e r t i e s of the gas phase does not p l a y an important r o l e under normal atmospheric c o n d i t i o n s , i t was decided to use atmospheric a i r as the gas phase f o r the e n t i r e programme. Water was chosen as the l i q u i d phase f o r most of the study f o r s i m p l i c i t y and f o r purposes o f comparing the d a t a c o l l e c t e d i n t h i s study w i t h the data a v a i l a b l e i n the l i t e r a t u r e from the work o f v a r i o u s i n v e s t i g a t o r s , most o f 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 c o n d i t i o n s where t u r b u l e n c e i n the continuous phase i s i n s i g n i f i c a n t , water was r e p l a c e d by a h i g h v i s c o s i t y l i q u i d .  A 30%  s o l u t i o n of p o l y e t h y l e n e g l y c o l i n water was i t gives high v i s c o s i t y  (-60  (by weight)  s e l e c t e d because  c.p.) w i t h o u t a f f e c t i n g  and s u r f a c e t e n s i o n of the s o l u t i o n markedly.  density  Surface  t e n s i o n o f 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 f o r three-phase systems  [11], was not d e l i b e r a t e l y v a r i e d  because o f the experimental d i f f i c u l t y o f keeping t r a c e s of  28  i m p u r i t i e s from e n t e r i n g the system and thus r a d i c a l l y i n g the s t a t i c e q u i l i b r i u m  chang-  (contact angle) between the phases.  S o l i d p a r t i c l e s s e l e c t e d f o r the study had t o be nonr e a c t i n g w i t h the l i q u i d s chosen and of w e l l d e f i n e d  shape.  The behaviour o f 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 t h e r e f o r e closely sized spherical particles of various densities and s i z e s were chosen.  A l s o the behaviour o f p a r t i c u l a t e l y  f l u i d i z e d beds w i t h s p h e r i c a l p a r t i c l e s l e n d s i t s e l f t o s a t i s f a c t o r y e x p l a n a t i o n by simple mathematical models. The secondary aim of t h i s r e s e a r c h was to d e r i v e a mathematical model which, when coupled w i t h some e m p i r i c a l i n f o r m a t i o n , would y i e l d b e t t e r understanding o f the behaviour o f a three-phase f l u i d i z e d bed. the wake model proposed by  0 s t e r g a a r d  For t h i s  purpose  was chosen as a s t a r t i n g  p o i n t , s i n c e i t d e s c r i b e s 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 p o i n t e d  out e a r l i e r , i t was expedient to i n c o r p o r a t e i n t o the model the r e c i r c u l a t i o n of s o l i d s induced by movement o f gas bubbles i n o r d e r to e x p l a i n not o n l y the observed c o n t r a c t i o n , b u t a l s o the subsequent expansion, o f three-phase f l u i d i z e d To develop a model f o r three-phase f l u i d i z e d  beds.  beds,  knowledge of gas holdup i n c o c u r r e n t g a s - l i q u i d flow i s essential.  S i n c e the models a v a i l a b l e i n the l i t e r a t u r e  f o r p r e d i c t i n g gas holdup i n g a s - l i q u i d flow a r e mostly e m p i r i c a l i n nature, t h e r e i s q u i t e a v a r i a b i l i t y and ambiguity i n t h e i r range o f a p p l i c a b i l i t y .  I t t h e r e f o r e became  necessary  to study two-phase g a s - l i q u i d c o c u r r e n t flow from  the p o i n t of view of d e v e l o p i n g a l o g i c a l and  reasonable  p h y s i c a l model f o r gas holdup i n two-phase g a s - l i q u i d flow. Attempts were a l s o made to develop a model f o r t h r e e phase f l u i d i z e d beds under c o n d i t i o n s i n which the  turbulence  phenomenon can be n e g l e c t e d . Experimental measurements c a r r i e d out w i t h small p a r t i c l e s i n a h i g h l y v i s c o u s l i q u i d  to  support such a model were not e n t i r e l y s u c c e s s f u l , but  the  model .-is. presented h e r e i n f o r p o s s i b l e f u t u r e i n v e s t i g a t i o n s .  30  CHAPTER 2  THEORY  This chapter i s divided into with  holdup i n g a s - l i q u i d flow,  phase o p e r a t i o n s  past to v a r y i n g  extents.  sections—dealing  in liquid-solid  b e d s and i n t h r e e - p h a s e f l u i d i z e d dispersed  three  fluidized  beds, r e s p e c t i v e l y .  have been i n v e s t i g a t e d i n the Mathematical models  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  purporting  have been p r e s e n t e d ,  w i t h o u t any u n d e r s t a n d i n g o f t h e phenomena i n v o l v e d . f o r e , an a t t e m p t i s made t o o u t l i n e t h e mechanisms phenomena f r o m t h e k n o w l e d g e  a v a i l a b l e i n the  and t h e n , on t h e b a s i s o f t h e s e m e c h a n i s m s , e i t h e r new order  models o r m o d i f i c a t i o n s  to d e s c r i b e  phase o p e r a t i o n s  2.1  Thereof these  literature  to propose  of the dispersed  satisfactorily.  Holdup i n g a s - l i q u i d f l o w  Two-phase g a s - l i q u i d f l o w years, in  often  to e x i s t i n g models, i n  the c h a r a c t e r i s t i c s more  These  so t h a t t h e amount o f i n f o r m a t i o n  the l i t e r a t u r e ,  Excellent treatises subject  has been s t u d i e d  f o r many  currently available  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 . [23,24,25] h a v e b e e n w r i t t e n o n t h e  and a n i n d e x o f o v e r 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 f l o w h a s b e e n p r e p a r e d by Gouse [ 2 6 ] .  31 Recently, the book by W a l l i s  [27] has put the s u b j e c t i n t o  some p e r s p e c t i v e . Due to the complex nature o f two-phase flow phenomena, i t has become necessary to r e l y h e a v i l y on experimental d a t a , s i n c e a r e a l i s t i c a n a l y s i s has been l a c k i n g . For example, the presence o f wakes behind a r i s i n g  bubble  swarm has been r e c o g n i z e d but as y e t no a n a l y s i s has adequately taken the wake phenomenon i n t o c o n s i d e r a t i o n .  I t i s , there-  f o r e , necessary t h a t a l l the b a s i c i n f o r m a t i o n a v a i l a b l e be c a r e f u l l y s t u d i e d , and t h a t the s a l i e n t f e a t u r e s found to be c o n t r o l l i n g the behaviour o f g a s - l i q u i d flow i n a g i v e n 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 observations.  found t o be r e l e v a n t i n the p h y s i c a l  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 o f  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 e x p e r i m e n t a l d a t a w i l l be r e q u i r e d .  Only an approach  which i s a p p r o p r i a t e l y balanced between t h e o r e t i c a l and experimental e f f o r t s w i l l l e a d t o b e t t e r understanding o f two-phase g a s - l i q u i d f l o w .  2.1.1  Holdup s t u d i e s When we a r e c o n s i d e r i n g a d i s p e r s e d two-phase system  i n which the gas i s u n i f o r m l y d i s t r i b u t e d i n a l i q u i d medium as d i s c r e t e bubbles, the r i s e v e l o c i t y o f the swarm o f bubbles i s s u b j e c t t o two i n f l u e n c e s , one a r i s i n g from the motion o f the bubbles and the o t h e r from t h e i r presence.  The  32 r e l a t i o n s h i p between the average r i s e v e l o c i t y o f the bubble swarm, v , and the average volume f r a c t i o n of gas i n the 2  swarm, <a > / i s simply 2  v  9  =  <j > / <a„>  (2.1)  9  where <J2> i s the average v o l u m e t r i c f l u x o f the gas through the  system.  Thus i n o r d e r t o p r e d i c t the r i s e v e l o c i t y o f  the  bubble swarm, and consequently the gas holdup, i t i s  necessary to understand the motion o f a bubble and how i t i s a f f e c t e d by the p r o x i m i t y o f o t h e r bubbles. 2.1.1.1  Bubble  dynamics  The r i s e v e l o c i t y o f a s i n g l e bubble i n an i n f i n i t e medium has been s t u d i e d e x t e n s i v e l y . [29] presented a comprehensive up to 1956.  review o f bubble motion  studies  Although the work o f Haberman and Morton [28]  and of Peebles and Garber the  Haberman and Morton  [30] e l u c i d a t e s the importance o f  p h y s i c a l p r o p e r t i e s o f the l i q u i d  (the p r o p e r t i e s o f 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 o f a bubble, most o f the experimental d a t a a v a i l a b l e f o r the r i s e v e l o c i t y of s i n g l e bubbles i s f o r bubble motion i n water. the  In the f o l l o w i n g  data o b t a i n e d f o r bubble motion i n water  description  [31] are used  to i l l u s t r a t e the important a s p e c t s o f the r i s e o f a gas bubble through a p o o l o f stagnant l i q u i d .  I t has been observed t h a t s m a l l gas bubbles  (r < 0 . 4  mm)  which are almost p e r f e c t spheres because o f the dominant s u r f a c e t e n s i o n f o r c e s , behave v e r y much l i k e s m a l l particles.  solid  However, the Stokes s o l u t i o n f o r the t e r m i n a l  r i s e v e l o c i t y of a bubble, V s m a l l e r bubbles  , can be used o n l y f o r s t i l l  < 0 . 2 mm,  (r  V„ =  g  Re^ < 2 ) :  (Pl-p2)/18  u  (2.2)  1  E q u a t i o n 2 . 2 assumes t h a t the l i q u i d v e l o c i t y a t the bubble s u r f a c e r e l a t i v e to the bubble i s z e r o , an assumption which, however, breaks down f o r bubbles w i t h i n t e r n a l Hadamard [32]  and Rybczynski [33]  for perfect f l u i d  m o d i f i e d the above e q u a t i o n  spheres w i t h complete t r a n s f e r e n c e o f shear  s t r e s s a t the b u b b l e - l i q u i d i n t e r f a c e , and  d V  oo  b  which f o r y^>>  Vo  g^i-p  =  = d*  ) 2  18y J L u 2  circulation.  3  *  ^ i  2y1  + 3 l i  +  obtained  2  „ (2-3)  2u 2  reduces to  g(p1-p2)/12u1  (2.4)  E q u a t i o n 2.4 a p p l i e s to s m a l l bubble s i z e s i n the complete absence o f s u r f a c e  (Re^<2), but o n l y  impurities.  A t the other extreme, when the bubbles are v e r y l a r g e (r  > 9 . 0 mm,  Re,  > 5000)  and show a s p h e r i c a l cap  shape,  the  Reynolds  and Weber numbers a r e known t o c h a r a c t e r i z e  the motion  o f such b u b b l e s .  the  i n t h i s regime, the surface  bubble  F o r p r e d i c t i n g t h e shape o f  forces are normally considered t o t h e g r a v i t y and i n e r t i a  around and  the f r o n t stagnation  Based  on these  [34] c o n s i d e r e d  point  t o be  a s compared  the motion  irrotational  obtained  where R  s  i s the radius of curvature  front stagnation although behind  point.  o f the bubble a t the  I t i s important t o note here  D a v i e s and T a y l o r o b s e r v e d a s i z a b l e wake  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  was a p p l i e d only  t o be n e g l i g i b l e  forces.  a s s u m p t i o n s , D a v i e s and T a y l o r  t e n s i o n and v i s c o u s  to the region  around  to p r e d i c t the terminal (r  > 9.0 mm)  the  bubble v e l o c i t y  rise  region model  the f r o n t stagnation  and no a t t e m p t was made t o c o n s i d e r  s t r u c t u r e o f t h e wake i t s e l f .  flow  that  point  the d e t a i l e d  The a b i l i t y  o f equation  velocity of large  2.5  bubbles  stems f r o m t h e f a c t t h a t t h e b u b b l e  shape,  and t h e r a t e o f energy d i s s i p a t i o n a r e  inter-related. For  the intermediate  where t h e s u r f a c e  t e n s i o n and v i s c o u s  t o t h e g r a v i t y and i n e r t i a velocity  bubble sizes(0.6  are d i f f i c u l t  g  < 9.0  mm),  f o r c e s a r e comparable  f o r c e s , the bubble  to model.  < r  s h a p e and b u b b l e  In t h i s range  t h e gas  35 bubbles are n e i t h e r s p h e r i c a l nor do they r i s e  rectilinearly.  Although no study has c o n s i d e r e d the wake s t r u c t u r e behind a r i s i n g bubble i n t h i s range, the s t u d i e s on r i s e o r f a l l of a l i q u i d drop through a l i q u i d medium w i t h which i t i s i m m i s c i b l e by Edge and Grant and Magarvey and Bishop  [35], Letan and Kehat [36],  [37] a l l suggest t h a t wake a c t i v i t y  i n t h i s r e g i o n o f Reynolds number i s q u i t e predominant. bubbles w i t h r  g  The  > 0.8 mm are q u i t e n o t i c e a b l y deformed and  t h e i r path o f ascent i s h e l i c a l .  Because o f the h e l i c a l  path, the r i s e v e l o c i t y o f these bubbles i n the v e r t i c a l d i r e c t i o n decreases s l o w l y as the bubble s i z e i n c r e a s e s u n t i l r  g  - 2.4 mm,  which corresponds t o the minimum i n the r i s e  v e l o c i t y v s . bubble s i z e r e l a t i o n s h i p . [30],  from t h e i r  Peebles and Garber  e x t e n s i v e experimental d a t a , observed  the average bubble r i s e v e l o c i t y f o r 1.0 < r  g  that  < 2.4 mm i s  b e s t r e p r e s e n t e d by the e q u a t i o n  =  For  bubbles w i t h r  g  1.35(o-/ r ) * 0  P l  > 2.4 mm,  (2.6)  5  e  the bubble  shape i s not  r e g u l a r b u t p u l s a t e s around an o b l a t e s p h e r o i d .  The path  of r i s e o f such bubbles becomes l e s s h e l i c a l and t h e r e f o r e the v e r t i c a l r i s e v e l o c i t y o f the bubble once a g a i n begins to s l o w l y i n c r e a s e w i t h i n c r e a s i n g bubble s i z e .  Nevertheless  Peebles and Garber  [30] , Haberman and Morton  Levich  the bubble movement t o be t u r b u l e n t i n  [38] termed  [28] , and  this is  r a n g e and  found t h a t the average v e r t i c a l  g o v e r n e d m a i n l y by  liquid for  system  a given  and  rise  the p h y s i c a l p r o p e r t i e s of the gas-  i s therefore approximately a constant  system.  The  empirical equation  s u g g e s t e d by  Haberman and M o r t o n , w h i c h has b e e n v i n d i c a t e d [ 3 9 ] liquid  systems w i t h  2.4  =  For  < rQ  1.53  < 4.0  ag (—) H l  mm  f o r gas-  is  0.25 (2.7)  bubbles with r  spheroid  > 4 mm, t h e t r a n s i t i o n f r o m an o b l a t e e t o a s p h e r i c a l b u b b l e c a p becomes n o t i c e a b l e and i s  m a n i f e s t e d by an a l m o s t r e c t i l i n e a r  path of r i s e .  with r  and h a v e a w e l l  > 9.0  e  mm  rise  r e c t i l i n e a r l Jy  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 velocity by  velocity  i s controlled  n o t by  tail.  Bubbles  Their  defined  rise  the volume o f the b u b b l e  t h e c u r v a t u r e o f t h e l e a d i n g edge o f t h e s p h e r i c a l  but cap.  Rigorous t h e o r e t i c a l analyses o f bubble motion i n this met  i n t e r m e d i a t e bubble s i z e range with l i t t l e  t h e shape o f t h e b u b b l e and  the  bubble.  Levich  the energy i s d i s s i p a t e d  Moore  layer  < 9.0  t h a t i n low  mm)  around  viscosity  i n t o a t h i n boundary  the flow f i e l d  outside  the  layer  thin  i s e s s e n t i a l l y u n a f f e c t e d by t h e b u b b l e .  [40] p r e s e n t e d a m o d e l b a s e d on t h i s p o s t u l a t e  found i t to p r e d i c t the r i s e v e l o c i t y suitably  have  i n defining  (b) t h e f l o w f i e l d  [38] p o s t u l a t e d  a r o u n d t h e b u b b l e and boundary  g  success because o f the d i f f i c u l t y  (a)  systems  (0.4 < r  and  satisfactorily for  s i z e d b u b b l e s w i t h f o r e and a f t symmetry i n a h i g h  37 s u r f a c e t e n s i o n 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 o f d e f i n i n g the bubble boundary,  the a p p l i c a b i l i t y  o f t h i s model i s l i m i t e d t o bubble s i z e s o f r  < 2.0 e —  Most o f the bubble s i z e s encountered i n two-phase flow are g e n e r a l l y g r e a t e r than 2.0 mm.  mm.  gas-liquid  T h e r e f o r e we cannot  expect Moore's t h e o r e t i c a l model t o be o f much p r a c t i c a l use. Mendelson l a r g e bubbles  [41], on the o t h e r hand, c o n s i d e r e d the  (r  > 1.5 mm)  as b e i n g analogous to i n t e r f a c i a l  d i s t u r b a n c e s whose dynamic motion i s assumed to be s i m i l a r to those of waves on an i d e a l l i q u i d , because o f the i n v i s c i d nature o f the motion o f l a r g e bubbles  [38].  F o r waves o f  small wave l e n g t h , X, compared t o the depth o f the l i q u i d , the wave v e l o c i t y , C^, i s g i v e n by Lamb  CCO  2ira Xp^  [42] as  (2.8)  g_X 2TT  Mendelson r e p l a c e d the wave l e n g t h i n e q u a t i o n 2.8 by the circumference o f the e q u i v a l e n t bubble d e f i n e d by the relation  X  2  (2.9)  TO:  e  and o b t a i n e d the bubble r i s e v e l o c i t y , •*  V 00  a  '  V  (-C ) as CO  CO  (2.10) + gr  e  38 On  comparing the r i s e v e l o c i t y p r e d i c t e d by e q u a t i o n  w i t h experimental data  2.10  [28,30], Mendelson observed t h a t  this  s i m p l i s t i c model p r e d i c t s 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 q u i t e s a t i s f a c t o r i l y f o r r Therefore  g  > 1.5  i t i s recommended t h a t , i n order  mm.  to p r e d i c t  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  f o l l o w i n g r e l a t i o n s be used :  T h e o r e t i c a l s o l u t i o n s of Moore [40], r < 1.5 e V  = co  (gr + a/p,r .»= Q < The  ) * 0  r  5  e  > 1.5  mm  and  mm  and  Re,  e  > 800  (2.10)  r i s e v e l o c i t y o f l a r g e gas bubbles i n narrow ducts of two-phase g a s - l i q u i d  flow, as i t forms a flow regime q u i t e d i s t i n c t from d i s c r e t e bubble flow regime, and [43]  800.  b  i s y e t another i n t e r e s t i n g aspect  trescu  Re,< fc>  and  by Davies and  the  has been s t u d i e d by Dumi-  Taylor  [34].  These bubbles  occupy almost the e n t i r e c r o s s - s e c t i o n of the d u c t and called 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 s l u g i s  g i v e n by Dumitrescu's  V  = oo  are  equation,  k, 1  where k-^ i s i n g e n e r a l  (2.11) gD a complex f u n c t i o n of v i s c o s i t y and  s u r f a c e t e n s i o n , but f o r a low v i s c o s i t y l i q u i d i s w e l l approximated by a constant  v a l u e o f 0.35  [44].  The r i s e v e l o c i t y o f a p a r t i c u l a r bubble i n a bubble swarm, w i t h r e s p e c t to the column boundaries, i s i n f l u e n c e d by the w a l l s of the c o n t a i n i n g v e s s e l as w e l l as by the bubble around i t .  The r i s e o f a gas bubble i n a c o n f i n e d  liquid  medium i s somewhat analogous to the c o r r e s p o n d i n g sediment a t i o n o f a s o l i d ' p a r t i c l e ; but t h i s analogy has been misi n t e r p r e t e d i n the p a s t by v a r i o u s authors who  were t h e r e f o r e  o b l i g e d e i t h e r to s e t narrow o p e r a t i n g 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] o r to c o r r e c t them by means o f e m p i r i c factors  [46].  These authors used the b a s i c e q u a t i o n d e s i r e d  f o r the s e d i m e n t a t i o n of a s o l i d p a r t i c l e i n a c o n f i n e d  liquid  medium on the assumption t h a t , due to the r e t u r n flow caused by the displacement of the l i q u i d by the f a l l i n g  particle,  a d d i t i o n a l r e s i s t a n c e would be encountered by the p a r t i c l e and thus i t would  V/V^  For dp/D  s e t t l e a t a lower v e l o c i t y .  OC  (l-d /D)  Therefore  (2.12)  k  p  1, i t can be seen from e q u a t i o n 2.12  t h a t V -> zero.  However, f o r l a r g e bubbles r i s i n g i n a narrow c o n d u i t it  (d^D)  i s w e l l known t h a t the bubble r i s e v e l o c i t y i s not zero  but i s g i v e n by e q u a t i o n 2.11.  Thus i t i s not s u r p r i s i n g  t h a t an e m p i r i c a l c o r r e l a t i o n o f the b a s i c form o f e q u a t i o n 2.12  does not s a t i s f a c t o r i l y p r e d i c t the r i s e v e l o c i t y o f a  gas bubble i n a r e s t r i c t e d l i q u i d medium, d e s p i t e the f a c t t h a t the above form has been s u c c e s s f u l l y used to p r e d i c t  the w a l l e f f e c t and M a n e r i for  in liquid-solid  fluidization  [48] , t h e r e f o r e , questioned  such f o r m u l a t i o n s , and suggested  [47].  Mendelson  the t h e o r e t i c a l  basis  a g a i n t h a t a bubble  can  be c o n s i d e r e d as an i n t e r f a c i a l d i s t u r b a n c e whose dynamic behaviour ideal  i s analogous  liquid.  The e x t e n s i o n of t h i s analogy  wall proximity effects similarity  to the motion o f s u r f a c e waves on  was  to account f o r  o b t a i n e d by a r g u i n g t h a t a dynamic  should e x i s t between the p r o p a g a t i o n o f waves over  shallow water and the r i s e v e l o c i t y medium.  an  In shallow l i q u i d s ,  of a bubble  the wave v e l o c i t y  in a  restricted  i s g i v e n by  [42]  (2.13)  where h i s the depth of the u n d i s t u r b e d l i q u i d . for  the wave l e n g t h i n e q u a t i o n 2.13,  Substituting  as b e f o r e , the  f e r e n c e of the e q u i v a l e n t bubble d e f i n e d by equation Mendelson and Maneri[4 8] o b t a i n e d the bubble r i s e V(=C)  i n c o n j u n c t i o n w i t h equation 2.10  circum2.9,  velocity  as  (2.14)  The parameter h, which by analogy effective  l i q u i d depth, was  to be d i r e c t l y  should be r e l a t e d to some  assumed by Mendelson and  p r o p o r t i o n a l , t o the tube r a d i u s .  Maneri  Then  41  C (R/r )  tanh  The  1  (2.15)  e  v a l u e o f the c o n s t a n t  was  then o b t a i n e d  from the known  r i s e v e l o c i t y o f s l u g s , assuming t h a t l a r g e bubbles w i t h r  e  = R behave l i k e s l u g s , an assumption a l r e a d y  v a l i d by Dumitreseu  [43]  and many o t h e r  investigators.  c o m p a r a t i v e l y l a r g e tubes, a t l a r g e N ,  containing  Eq  v i s c o s i t y and was  high  found to be  surface  0.25.  bubble i n a c o n f i n e d  shown to  tension l i q u i d ,  the  be For  low  constant  Thus the r i s e v e l o c i t y of a s i n g l e medium i s g i v e n  by  (2.16)  The  e f f e c t of the presence o f o t h e r  of a bubble has  not been p o s s i b l e s i n c e the  around the conglomeration of bubbles can  except f o r v e r y s m a l l  s p h e r i c a l bubbles.  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 flow w i t h i n a c o n f i n e d  space has  by Happel, Brenner and  co-workers  techniques.  fluid  A flow  not be  defined  However, the motion  i n groups undergoing laminar been e x t e n s i v e l y [49]  utilizing  studied c e l l model  T h i s i n v o l v e s the concept t h a t an assemblage of  s o l i d p a r t i c l e s can be d i v i d e d i n t o a number of cells,  the motion  not been i n v e s t i g a t e d s y s t e m a t i c a l l y .  t h e o r e t i c a l a n a l y s i s has field  bubbles on  each o f which c o n t a i n s envelope c o n t a i n i n g  identical  a p a r t i c l e surrounded by  a volume of f l u i d s u f f i c i e n t  a to  make the f r a c t i o n a l v o i d volume i n the c e l l t h a t i n the e n t i r e assemblage.  i d e n t i c a l to  The c e l l model technique  was  found to apply w i t h g r e a t e s t success f o r concentrated assemblages where the p a r t i c l e s i n the assemblage are  distri-  buted more or l e s s randomly and the e f f e c t of the c o n t a i n e r w a l l s i s not important.  Happel  envelope to be s p h e r i c a l .  E  Happel and A s t envelope cell  3  "  (  [4 9]  assumed a t y p i c a l  Then f o r a s p h e r i c a l  V cell> " r  particle,  <^»  3  <- >  3  2  [50] , however, c o n s i d e r e d the t y p i c a l  to be c y l i n d r i c a l .  e q u a t i o n 2.17,  17  cell  To c h a r a c t e r i z e the i n d i v i d u a l  completely, both the c e l l r a d i u s and the c e l l  then r e q u i r e d .  cell  l e n g t h are  They s t i l l assumed the a p p l i c a b i l i t y of so t h a t the l e n g t h of t h e i r c y l i n d r i c a l  would have t o be 4/3 (  r c e  -^).  cell  They found t h a t the p r e d i c t e d  values of s e t t l i n g v e l o c i t i e s f o r t h e i r c y l i n d r i c a l  cell  model agreed r e a s o n a b l y w e l l w i t h the v a l u e s p r e d i c t e d by the c o n c e n t r i c sphere c e l l model up to a s o l i d s holdup of e3  = 0.216  (corresponding to r /  r  g  c  e  l  l  - 0.6).  concluded t h a t the shape o f the f l u i d envelope  Thus Happel to be  used  i n such c e l l u l a r r e p r e s e n t a t i o n s of assemblages i s o f no s i g n i f i c a n t importance up to s u b s t a n t i a l s o l i d s  holdups.  A s i m i l a r c e l l model r e p r e s e n t a t i o n i s subsequently In  the present, work, t o swarms of bubbles.  applied  43 2.1.1.2  Bubble  column  I f a two-phase g a s - l i q u i d o p e r a t i o n i s c a r r i e d out i n a v e r t i c a l column under c o n d i t i o n s o f zero net l i q u i d  flow  r a t e through the column, then the c o n t a c t i n g d e v i c e i s c a l l e d a "Bubble Column".  The c o n t r o l l i n g parameters  f o r the  o p e r a t i n g c h a r a c t e r i s t i c s of a bubble column are the r e s i d e n c e time o f 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 s i z e 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 s t r u c t u r e of the c o n t a i n i n g v e s s e l ) . Freedman and Davidson  [51] presented a summary of data  o b t a i n e d i n bubble columns of diameters r a n g i n g from 1 i n c h to 42 i n c h which i s shown i n F i g u r e 2.1.  As can be seen, a  wide spread i n gas holdup e x i s t s but the data can mainly be d i v i d e d i n t o two d i s t i n c t r e g i o n s , t h a t i s , a r e g i o n i n which l i q u i d c i r c u l a t i o n i s p r e v a l e n t (column diameter >^ 4 inch) and a r e g i o n 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) . of Shulman and Molstad  The s t u d i e s of Hughmark [52] and  [53] f o r s m a l l diameter columns  (1 i n c h  to 4 inch) have shown t h a t the gas holdup i s p r i m a r i l y a f u n c t i o n of the v o l u m e t r i c f l u x of gas, J 2 / <  >  through the  column and can be p r e d i c t e d from a knowledge of bubble i n the swarm.  size  However, the i n f o r m a t i o n c o n c e r n i n g bubble  s i z e s i s not r e a d i l y a v a i l a b l e except a t s m a l l gas flow r a t e s . At h i g h e r gas flow r a t e s , d i s p e r s i o n of the gaseous phase  0.30 0.26 » 0.22 Q.  3  Q  o  0.18 0.14 V  § 0.10 0.06 0.02 V 3  4  5  6  < j > , cm/sec 2  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 w i t h references)  i n t o the l i q u i d phase i s brought about by induced t u r b u l e n c e , and the breakup  and c o a l e s c e n c e of the d i s p e r s e d phase  occurs c o n t i n u o u s l y . The s i z e of bubbles i n the swarm under these t u r b u l e n t c o n d i t i o n s had not been s t u d i e d  conclusively  but i t i s b e l i e v e d t h a t i n columns of diameter up t o 4 i n c h , the bubble s i z e i n c r e a s e s w i t h gas flow r a t e bubbles are l a r g e enough (r capped  [58] .  >_ 9.0 mm)  [53] u n t i l  the  to become s p h e r i c a l l y  Then, i f the l i q u i d p o o l i s deep enough, these  s p h e r i c a l l y capped bubbles c o a l e s c e t o form s l u g s which  occupy  almost the e n t i r e c r o s s - s e c t i o n of the column. T o w e l l e t a l . [59] s t u d i e d the bubble s i z e and holdup i n a 16 i n c h diameter column. speed c i n e photography,  gas  With the h e l p of h i g h  they observed t h a t l i q u i d  circulation  i n such l a r g e diameter columns i s very s i g n i f i c a n t and i s r e s p o n s i b l e f o r a h i g h degree of mixing and a l o w e r i n g o f gas holdup r e l a t i v e t o s m a l l e r columns. columns  Small diameter  (up to 4 i n c h i n d i a m e t e r ) , on the c o n t r a r y , were  shown [60] not t o 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 were, t h e r e f o r e , found t o e x h i b i t v e r y i n s i g n i f i c a n t mixing and much l a r g e r gas holdups.  axial  Various explanations  have been p r o v i d e d f o r the presence o f l i q u i d i n l a r g e diameter columns.  and  circulation  Freedman and Davidson  believe  t h a t i t i s caused by the m a l d i s t r i b u t i o n o f the gas a t the bottom o f the column.  De Nevers  [61] suggests t h a t d e n s i t y  d i f f e r e n c e s between those p a r t s o f the column which are r i c h and those which are poor i n d i s p e r s e d phase cause the  l i q u i d c i r c u l a t i o n to s e t i n .  [62]  Yoshitome and S h i r a i  measured the i n t e n s i t y o f c i r c u l a t i o n i n a 15  cm  diameter  column and found a s t r o n g upward flow of continuous phase i n the c e n t r a l core r e g i o n and a downward flow near the w a l l s of the column. [59-62], the t u r b u l e n t  As observed by these authors activity  i n l a r g e columns i s q u i t e s i g n i f i c a n t and under  such c o n d i t i o n s the s i z e of the bubbles i n the swarm i s c o n t r o l l e d 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 s t u d y i n g the d i s p e r s i o n o f gas  i n a m e c h a n i c a l l y s t i r r e d tank, u t i l i z e d  t h i s concept to  o b t a i n the average bubble s i z e i n the tank by b a l a n c i n g the s u r f a c e t e n s i o n f o r c e s w i t h the t u r b u l e n t energy  dissipation.  Towell e t a l . recommended t h a t C a l d e r b l a n k s c o r r e l a t i o n 1  be extended  to p r e d i c t average bubble s i z e f o r bubble columns  by assuming t h a t the power d i s s i p a t e d per u n i t volume i n a bubble column can be taken as  <  Power i n p u t per u n i t volume =  They then s u b s t i t u t e d e q u a t i o n 2.18  J2  >  (2.18)  into Calderblank s 1  c o r r e l a t i o n f o r s t i r r e d tanks and o b t a i n e d the f o l l o w i n g r e l a t i o n s h i p t o p r e d i c t the average bubble s i z e i n a swarm:  d  b  = 0.25  [ (<j >)" 2  0 , 4  (a/p) * ] 0  6  e  0 2  *  5  + 0 .09  (2.19)  Good agreement was limited  r e p o r t e d between e q u a t i o n 2.19  and  the  amount of data o b t a i n e d by Towell e t a l . p h o t o g r a p h i c -  a l l y for large 2.1.1.3  (> 4 inch) diameter columns.  Vertical  c o c u r r e n t flow  The flow of the gas and the l i q u i d phases in a vertical  cocurrently  c o n d u i t has been s t u d i e d and a number of methods  have been suggested to p r e d i c t the gas holdup.  I t i s important  to note t h a t 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 o f the l i q u i d phase.  Baker and Chao [65] observed t h a t 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 a f f e c t e d by the v e l o c i t y o f the l i q u i d  stream inasmuch as the  relative  v e l o c i t y o f the bubble i s found to be the same as the bubble v e l o c i t y i n the q u i e s c e n t l i q u i d stream. observed  I t has a l s o been  [66] t h a t bubble f o r m a t i o n 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 u n a f f e c t e d by the v e l o c i t y of  the l i q u i d stream.  Thus the phenomenon of  relative  v e l o c i t y c o u l d be used to g e n e r a l i z e the behaviour of phase flow, as suggested by Lapidus and E l g i n The e f f e c t  [67].  of column diameter on the mixing c h a r a c t e r -  i s t i c s and gas holdup i n c o c u r r e n t g a s - l i q u i d flow i n v e s t i g a t e d by R e i t h e t a l [68] i n 5, 14 and 29 cm They found t h a t a 5 cm column d i s p l a y s very l i t t l e mixing and h i g h gas holdup, as was columns.  two-  observed e a r l i e r  was columns. axial f o r bubble  But f o r l a r g e r v e s s e l s , although no s y s t e m a t i c  48 c i r c u l a t i o n o f the l i q u i d stream was observed, the r a t e s o f a x i a l mixing were found t o be much h i g h e r , and the gas holdup much lower, than f o r the 5 cm column.  T h e r e f o r e they  suggested t h a t a x i a l mixing i n l a r g e columns i s caused by the g e n e r a t i o n o f l a r g e s c a l e eddies i n the l i q u i d phase due  to passage o f the bubbles. The average bubble s i z e i n c o c u r r e n t g a s - l i q u i d flow  has n o t been i n v e s t i g a t e d s y s t e m a t i c a l l y . [69],  However, P a t r i c k  based on a l i m i t e d amount o f data o b t a i n e d i n a 5 cm  column, r e p o r t e d t h a t the average bubble s i z e i n two-phase g a s - l i q u i d flow i s a f u n c t i o n o f the average l i n e a r  liquid  v e l o c i t y , the f o l l o w i n g r e l a t i o n s h i p r e p r e s e n t i n g the data for  the c o c u r r e n t bubble flow regime:  d  b  = 5.074/v  0.666  15 < v  1  (2.20)  < 150  The t r a n s i t i o n p o i n t from the bubble flow regime flow regime has been s t u d i e d o n l y q u a l i t a t i v e l y . al.  to the s l u g Reith et  suggested t h a t f o r a 5 cm column, s l u g flow o c c u r r e d a t  gas v e l o c i t i e s above  <  J  > 2  =  5  cm/sec.  Ellis  and Jones  [60]  observed the t r a n s i t i o n p o i n t v i s u a l l y and found the f o l l o w i n g r e l a t i o n s h i p to d e s c r i b e the t r a n s i t i o n approximately:  > = 0.2 <j.> + 3 .05  (2.21)  Although no s l u g s were a c t u a l l y observed i n p i p e s g r e a t e r than 10 cm i n diameter up to <j > = 45 cm/sec [68], E l l i s 9  49 and Jones c l a s s i f i e d if  the average  the flow to be i n the s l u g flow regime  r e l a t i v e v e l o c i t y of the gas phase was  to be g r e a t e r than the v e l o c i t y of r i s e of a s i n g l e as g i v e n by Dumitrescu's g i v e n column  r e l a t i o n , equation 2.11,  slug  f o r the  diameter.  S e v e r a l methods have been suggested gas holdup  found  i n cocurrent g a s - l i q u i d flow.  to p r e d i c t  the  Of a l l the e m p i r i c a l  c o r r e l a t i o n s a v a i l a b l e , t h a t of L o c k h a r t and M a r t i n e l l i o r i g i n a l l y developed  f o r h o r i z o n t a l flow but  a p p l i e d a l s o to v e r t i c a l flow, i s s t i l l  [70],  subsequently  the most convenient  to use when i n f o r m a t i o n concerning the d e t a i l e d flow s t r u c t u r e i s e i t h e r not a v a i l a b l e or not d e s i r e d [71], [72] compiled  Duckler e t a l .  a l l the a v a i l a b l e data and checked  the v a l i d i t y  of v a r i o u s e m p i r i c a l c o r r e l a t i o n s t h a t have been recommended i n the l i t e r a t u r e .  They found t h a t Hughmark's  [73]  c o r r e l a t i o n r e p r e s e n t s most o f the data r e p o r t e d over a wide range of o p e r a t i n g l i m i t s v e r y s a t i s f a c t o r i l y .  Therefore,  Hughmark's c o 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 o f the model proposed 2.20  2.1.2  and  herein, i n conjunction with  equations  2.21.  Models f o r gas holdup p r e d i c t i o n s The r e l a t i v e v e l o c i t y between the d i s p e r s e d and  continuous phases has been suggested by Lapidus and  the  Elgin  [67] as the s i n g l e parameter r e q u i r e d to completely d e s c r i b e the behaviour of an i d e a l d i s p e r s e d phase system.  They d i d  50 not  p r o v i d e a d e f i n i t i o n o f an i d e a l system, but i t can be  i n f e r r e d from t h e i r work t h a t i f the d i s p e r s e d phase i s uniformly d i s t r i b u t e d  throughout the continuous phase i n  such a manner t h a t each p a r t i c l e has an i n d i v i d u a l  identity,  y e t i t s behaviour i s i d e n t i c a l to t h a t of a l l the other p a r t i c l e s , then the d i s p e r s e d phase system i s i d e a l . local relative velocity  The  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  n o n - i d e a l system the concept o f r e l a t i v e v e l o c i t y can only be employed are  locally.  S i n c e the data on l o c a l r e l a t i v e v e l o c i t y  not e a s i l y a v a i l a b l e ,  Zuber and F i n d l a y  [39] recommended  t h a t the concept o f l o c a l d r i f t v e l o c i t y be used i n s t e a d . p r o p e r t y of constancy o f d r i f t v e l o c i t y f o r c e r t a i n regimes can then be u t i l i z e d , velocity  specific  the i n f o r m a t i o n on r e l a t i v e  i n v a r i o u s flow regimes b e i n g o f t e n vague and  incomplete. The l o c a l d r i f t v e l o c i t y , as d e f i n e d e a r l i e r , r e p r e sents the l o c a l v e l o c i t y of the bubble w i t h r e s p e c t to the l o c a l v o l u m e t r i c f l u x of the m i x t u r e :  V  2j  =  v  2  "  j  (2.22)  In a two-phase system, data on the average v a l u e s are more readily available;  thus the average v o l u m e t r i c f l u x of the  gas phase, <j >, which i s r e a d i l y measurable, i s 9  The  51  <j > = <a v > = Q /A 2  2  2  (2.23)  2  The weighted mean v e l o c i t y of the gas phase, v , which i s 2  o b t a i n e d by i n t e g r a t i o n across the c r o s s - s e c t i o n , i s g i v e n by <a v >  <j > v  2  —  =  —  9  =  2  —=-=< A  <c*2>  2  (2.24)  >  In view o f e q u a t i o n 2 . 2 2 , the weighted mean v e l o c i t y o f the gas phase can a l s o be expressed as  v  2  <a93> <a0 v0 -> = — - — + —£_j£2_ < A  <a2>  2  (2.25)  >  E q u a t i o n 2 . 2 5 can be put i n s e v e r a l a l t e r n a t e forms which are most u s e f u l f o r a n a l y z i n g experimental data and f o r d e t e r m i n i n g the average v o l u m e t r i c gas f r a c t i o n , <a >. 2  Thus, m u l t i p l y i n g and d i v i d i n g the f i r s t  term on the r i g h t  hand s i d e by <j>, we o b t a i n <a v ,> 2  v  2  = C  0  <  j  >  2  +  ( 2  '  2 6 )  <a2>  where C Q i s the d i s t r i b u t i o n parameter  C0  =  —  <a9><j>  [ 3 9 ] and i s g i v e n by  (2.27)  The d i s t r i b u t i o n parameter takes i n t o account  t h a t both  v o l u m e t r i c f l u x o f the mixture and the gas holdup  are not  uniform over the c r o s s - s e c t i o n . However, i f e i t h e r v o l u m e t r i c f l u x o f the mixture or the gas holdup  t h a t CQ w i l l be u n i t y .  e  Equation 2.26  v  2  Combination  < a  2  >  a  2  2  x  (2.28)  to  2  (2.29)  2  of equation 2 . 2 9 w i t h e q u a t i o n 2 . 2 4  yields  e (<j + v ^>) — ±3 1 - e >  2  =  + <v j> = <j > + <j > + <v j>  9  <j >  equation  Thus where  can be s i m p l i f i e d  = <j>  2  =  the  i s uniform  over the c r o s s - s e c t i o n , i t can be e a s i l y seen from 2.27  =  the  <  1  9  (2.30)  2  For i d e a l bubbly  flow, a l l the averaging b r a c k e t s on  v e l o c i t i e s can a l s o be dropped.  the  I f , however, n e i t h e r the  v o l u m e t r i c f l u x of the mixture nor the v o l u m e t r i c gas  fraction  i s uniform over the c r o s s - s e c t i o n , the d i s t r i b u t i o n parameter f o r 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 j — =>c  =  1 -  * M (R )  to be  (2.27a)  53 and —^— 2C  =  1 - (R )  (2.27b)  a  r e s p e c t i v e l y , i s g i v e n by  „  C  Q  M+M  =  M  For p o s i t i v e v a l u e s of  [39]  1  +4 + 2  M*  +  _ (2.27c)  / 0  (M + M ' ) ,  CQ  0  x  i s obviously greater  than u n i t y . Zuber and F i n d l a y have p o i n t e d out t h a t the main problem i n two-phase flow i s determining the c o r r e c t equation for  the d r i f t v e l o c i t y , v  flow regime.  9  ., w i t h p a r t i c u l a r regard to the  In g e n e r a l , the l o c a l d r i f t v e l o c i t y i s found  to be a f f e c t e d by the bubble for  s p a c i n g or the gas holdup.  Thus  the bubble flow regime, Zuber and Hench [74] r e p o r t e d  that  v  where m bubble  was  2 j  =  V  w  (1 - a )  (2.31)  m  2  found to v a r y between 0 and  3 depending on the  size. As a b a s i s f o r c o n s i d e r i n g the more complicated  of  three-phase  flow, Bhaga [1] developed  a model f o r  phase g a s - l i q u i d flow by c o n s i d e r i n g the r e l a t i v e between the phases. to  case two-  velocity  The l o c a l r e l a t i v e v e l o c i t y i s r e l a t e d  the l o c a l d r i f t v e l o c i t y by the e q u a t i o n  54  (1 - a ) v 2  Thus, combining equations  v  =  2 1  (2.32)  21  2.31 and 2.32, we g e t  m-1 V „ (1 - a )  (2.33)  2  which i s the r e l a t i o n s h i p used by Bhaga [1] who the exponent  (m-1) to vary between -1 and 2.  data t e s t e d , however, Bhaga found  expected  F o r the l i m i t e d  that a value of m = 2 best  r e p r e s e n t e d h i s r e s u l t s f o r both c o c u r r e n t and c o u n t e r c u r r e n t a i r - w a t e r flow i n a v e r t i c a l c o n d u i t .  N e v e r t h e l e s s , the  model proposed by Bhaga p r o v i d e s no advantage over  the o r i g i n a l  model proposed by Zuber and F i n d l a y , a t l e a s t f o r two-phase flow. Marrucci  [7 5] u t i l i z e d  p a t i o n i n i r r o t a t i o n a l flow  the concept o f energy  [38,40] i n c o n j u n c t i o n w i t h the  c e l l u l a r r e p r e s e n t a t i o n o f a bubble swarm suggested [49]  dissi-  by Happel  to p r e d i c t the r i s e v e l o c i t y of a bubble swarm.  His  method, a p p l i c a b l e i n the bubble flow regime, assumes t h a t the v o r t i c i t y generated  i n the wake of a bubble i s not  t r a n s f e r r e d f a r enough downstream to a f f e c t the motion o f any downstream bubbles.  Based on these assumptions he found  t h a t the energy-destroying v e l o c i t y or the d r i f t v e l o c i t y , i s g i v e n by  (due to buoyancy a l o n e ) ,  55  V„  He n o t e d q u a l i t a t i v e l y  (1 - a  2.34.  2  /d  - a  a relatively  velocity with increasing  equation  )  5 / 3 2  (2.34)  )  t h a t t h e s p a r s e d a t a o f N i c k l i n [19]  and h i s own d a t a i n d i c a t e d rise  2  small decrease i n  a 2 , i n conformity with  A l t h o u g h no e x p l i c i t m e n t i o n  i s made o f t h e  r a d i u s o f b u b b l e s w h i c h make up t h e swarm, M a r r u c c i ' s m o d e l does i m p l i c i t l y rise  velocity  indicate  t h e e f f e c t o f b u b b l e r a d i u s on t h e  o f t h e swarm t h r o u g h t h e t e r m V  A n o t h e r model h a s b e e n p r o p o s e d [7 6]  f o rpredicting  regime, based predicting  the d r i f t v e l o c i t y  .  by t h e p r e s e n t a u t h o r i n the bubble  on t h e method o f M a n e r i and M e n d e l s o n  the r i s e v e l o c i t y o f l a r g e bubbles  flow [48]  for  i n confined 2  media.  The l a t t e r  that i s , f o r large rise  velocity  a u t h o r s showed t h a t f o r l a r g e N £ o  t u b e d i a m e t e r s s u c h t h a t 1/N^,Q << 1, t h e  o f a s i n g l e bubble  V/V '  00  (=gp^R / a ) ,  = / tanh  i s g i v e n by  (2.35)  [0.25 ( V r )]  E q u a t i o n 2.35 was f o u n d b y M a n e r i  and M e n d e l s o n t o c o r r e l a t e  e x p e r i m e n t a l d a t a w e l l f o r 1/y = R / r e b e t w e e n 1 and 1 0 , t h e lower l i m i t o f which gives  corresponds to a s l u g .  the d r i f t v e l o c i t y  cylindrical  tube.  Happel  of a single  This equation  s p h e r i c a l bubble  ina  a n d A s t [50] s u g g e s t e d a s p h e r e -  i n - c y l i n d e r - t y p e c e l l model t o r e p r e s e n t a n a s s e m b l a g e o f s o l i d  particles.  Thus, i f we use the approach of Happel and A s t ,  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 f o l l o w s :  a2  Now,  combining  Y  3  (2.36)  equations 2.35  and  2.36,  we  get  (2.37)  Thus equation 2.37  would p r e d i c t the d r i f t v e l o c i t y  [or the  e n e r g y - d e s t r o y i n g v e l o c i t y as d e f i n e d by N i c k l i n  [19]] of a  bubble  viscosity  swarm i n tubes of l a r g e diameters  systems e.g. water.  f o r low  I t should, however, be p o i n t e d o u t t h a t  e q u a t i o n 2.37  will  not g i v e the d r i f t v e l o c i t y f o r s l u g s , but  e q u a t i o n 2.35  with r  Q  = R does r e p r e s e n t the r i s e v e l o c i t y of  s l u g s i n a q u i e s c e n t medium, f o r which i t then [48] i n combination w i t h equation 2.10  v  gr  2j  e  (tanh  simplifies  f o r l a r g e N„  0.25)  to  (2.38)  or  v  2j  =  0.35  which i s the Dumitrescu  /~gD  (2.39)  e q u a t i o n f o r d r i f t v e l o c i t y i n the  s l u g flow regime recommended by s e v e r a l i n v e s t i g a t o r s  [39,27]  E q u a t i o n 2.26, d e r i v e d o r i g i n a l l y by Zuber and F i n d l a y , i s q u i t e g e n e r a l and a p p l i c a b l e to a l l g a s - l i q u i d flow regimes if  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 o b t a i n e d i n d e p e n d e n t l y .  This r e q u i r e s  simultaneous measurements o f v e l o c i t y and gas holdup p r o f i l e s , which have n o t g e n e r a l l y been made, and thus l o c a l v e l o c i t y p r o f i l e s are not a v a i l a b l e .  drift  The l a c k o f experimental  measurements o f l o c a l p r o p e r t i e s thus n e c e s s i t a t e s  suitable  assumptions f o r 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 o r d e r t o advance a meaningful e m p i r i c a l or  s e m i - e m p i r i c a l model f o r two-phase g a s - l i q u i d f l o w .  Zuber  and F i n d l a y assumed smooth and symmetric p r o f i l e s f o r 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 e q u a t i o n 2.27c).  They d i d n o t  a p p r e c i a t e the p o s s i b i l i t y o f s y s t e m a t i c 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 n e i t h e r smooth nor symmetric.  Thus, the models  above cannot be used s u c c e s s f u l l y to p r e d i c t 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 .  A t the p r e s e n t  time n o t enough i n f o r m a t i o n i s a v a i l a b l e t o p r e d i c t the r a t e of the  c i r c u l a t i o n , b u t a p o s s i b l e mechanism i s suggested i n next s e c t i o n .  58 2.1.3  C i r c u l a t i o n and t u r b u l e n c e i n two-phase g a s - l i q u i d flow I t has been shown above t h a t f o r a bubble column g r e a t e r  than 10 cm i n diameter,  a s y s t e m a t i c c i r c u l a t i o n develops i n  the l i q u i d phase w i t h an upward l i q u i d  flow i n the c e n t r a l  core o f the column and a downward l i q u i d From a l l the evidence presented,  flow near the w a l l s .  the f o l l o w i n g d e s c r i p t i o n o f  c i r c u l a t i o n can be g i v e n : A t v e r y s m a l l gas flow r a t e s and low Reynolds number (Re^ < 10) the bubbles  are u n i f o r m l y d i s t r i b u t e d and r i s e i n  d i s t i n g u i s h a b l e bubble chains  [77]. C r a b t r e e and Bridgwater  [78] showed t h a t the r i s i n g bubble c h a i n s drag l i q u i d by v i s c o u s 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.  T h i s c i r c u l a t i o n i s f u r t h e r enhanced by the presence o f both low d e n s i t y (high gas f r a c t i o n ) and h i g h d e n s i t y gas f r a c t i o n ) r e g i o n s .  A t higher Reynolds number  a d i s t i n g u i s h a b l e wake behind  the bubbles  (low o r zero (Re^ - 400),  appears and the  l i q u i d i n the attached wake t r a v e l s a t the bubble v e l o c i t y . The l i q u i d i n the wake i s d e p o s i t e d near the s u r f a c e of the main l i q u i d when the bubble reaches  t h i s s u r f a c e and  breaks up.  Because o f the r i s e o f l i q u i d i n the wake o f  the bubble,  the c i r c u l a t i o n becomes q u i t e i n t e n s e , 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 g i v i n g r i s e t o more r e g i o n s of h i g h and low d e n s i t y , which h e l p to s u s t a i n the circulation.  At s t i l l  h i g h e r Reynolds number, i t i s b e l i e v e d  t h a t the wake no longer remains attached to the bubble but i s shed a t r e g u l a r i n t e r v a l s  [79] , as observed  Kehat ^[36] f o r drops i n l i q u i d - l i q u i d systems.  by Letan  and  Under these  c o n d i t i o n s the l i q u i d c i r c u l a t i o n appears to be c h a o t i c as i t i s superimposed on random eddying  [59] , but i n f a c t a  s y s t e m a t i c 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 c e n t r a l  r e g i o n and downwards near the w a l l .  The bubbles mainly r i s e i n the h i g h 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 i n t e n s e t h a t a bubble c o u l d be trapped i n the downward flow r e g i o n or even swept back w i t h the downward flow, as observed A similar gas-liquid  flow.  [51].  p h y s i c a l model can be advanced f o r c o c u r r e n t Baker and Chao [65] have shown t h a t the  dynamics of an i n d i v i d u a l wards l i q u i d v e l o c i t y . a similar  by Freedman and Davidson  bubble i s not a f f e c t e d by the  I t i s t h e r e f o r e to be expected  upthat  g e n e r a l model should be a p p l i c a b l e to c o c u r r e n t  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 However, R e i t h e t a l . [68] observed  no s y s t e m a t i c  direction.  circulation  of l i q u i d i n c o c u r r e n t flow; n e v e r t h e l e s s , a x i a l mixing found to occur and was ^apparently .caused shedding  phenomenon.  by Letan and  The  the wake  same e f f e c t has a l s o been  observed  Kehat [36] f o r the study of a x i a l mixing i n  l i q u i d - l i q u i d e x t r a c t i o n (spray) columns. probable  by  was  I t i s therefore  t h a t i n c o c u r r e n t flow the c i r c u l a t i o n i s contained  in c e l l u l a r regions.  60 I t has been observed f o r the r i s e o r f a l l o f a drop through a l i q u i d medium t h a t the wake behind the drop i s shed a t r e g u l a r i n t e r v a l s  [36].  Though v o r t e x shedding  from  behind a bubble has n o t been s t u d i e d , i t i s b e l i e v e d to occur on v e r y s i m i l a r p a t t e r n s as behind a drop.  The shed v o 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 a t the bubble but w i l l  velocity,  subsequently be d i s s i p a t e d by v i s c o u s s t r e s s e s [79].  The i n t e r a c t i o n o f the mean flow w i t h these e j e c t e d v o r t i c e s would c r e a t e i n t e n s e and c h a o t i c v e l o c i t y f l u c t u a t i o n s i n the mean f l o w .  These f l u c t u a t i o n s are b e l i e v e d to be respon-  s i b l e f o r the g e n e r a t i o n and maintenance o f t u r b u l e n c e i n the l i q u i d phase a t the expense o f the energy o f the mean flow.  Delhaye  [81] had r e p o r t e d some measurements o f i n t e n s i t y  o f t u r b u l e n c e induced i n a l i q u i d due to passage bubbles.  of gas  He observed t h a t even a t s m a l l gas flow r a t e s , the  t u r b u l e n c e generated i n the l i q u i d phase by the gas bubbles i s quite  2.2  significant.  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 o f f l u i d flow through a bed o f g r a n u l a r m a t e r i a l has been r e s e a r c h e d q u i t e i n the p a s t .  successfully  B a s i c a l l y , t h r e e d i s t i n c t approaches  have been  developed and used t o study the hydrodynamics o f f l u i d i z e d beds:  ( i ) U s i n g t h e r e l a t i o n s h i p b e t w e e n p r e s s u r e d r o p and fluid  (ii)  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  transition  p o i n t to study i n c i p i e n t  subsequent  bed e x p a n s i o n [ 5 6 ] .  f l u i d i z a t i o n and  U s i n g t h e c o n c e p t s u g g e s t e d by L a p i d u s and E l g i n [67] that the r e l a t i v e solid  particles  liquid-solid  between t h e l i q u i d  and t h e  alone can p r e d i c t the voidage i n  fluidized  relationship bed  velocity  beds.  The a c t u a l  form o f t h e  b e t w e e n t h e r e l a t i v e v e l o c i t y and t h e  v o i d a g e h a s , however, t o be d e t e r m i n e d  experimen-  tal ly. (iii)  U s i n g a c e l l model t o r e p r e s e n t a n a s s e m b l a g e o f p a r ticles  as suggested by Happel  solving cell  the f u l l  analytically  and B r e n n e r  [49]; then  Navier-Stokes equations w i t h i n the f o r low R e y n o l d s  numbers  (Re  < 1),  P when t h e i n e r t i a l cally All  terms a r e n o t i m p o r t a n t , o r numeri-  f o r moderately h i g h Reynolds  numbers  t h r e e t e c h n i q u e s have b e e n f o u n d  [65,66] .  to p r e d i c t the  r e l a t i o n s h i p b e t w e e n v o i d a g e and v o l u m e t r i c l i q u i d  flux  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 . all  t h e s e t e c h n i q u e s have b e e n t r e a t e d q u i t e  various  investigators  [ 5 6 , 2 , 4 9 ] , no a t t e m p t  p r e s e n t them i n d e t a i l .  quite Since  e l o q u e n t l y by i s made h e r e t o  However, a summary o f t h e s e  techniques i s given below. Andersson sure drop  [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 pres-  e q u a t i o n f o r f i x e d beds t o o b t a i n a  relationship  between the l i q u i d v e l o c i t y and the bed voidage s o l i d f l u i d i z e d beds.  for liquid-  The a p p l i c a b i l i t y of t h i s  mainly depends on the e x p e r i m e n t a l l y observed  technique  fact  that  the p r e s s u r e 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 u n i t area, i . e . ,  -Ap  Ergun  =  e  3  ( p  3  - p  1  )  L  b  g  (2.40)  [57] s u c c e s s f u l l y r e p r e s e n t e d the p r e s s u r e drop  through  a f i x e d bed of s p h e r i c a l 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 p o i n t Ap  to f l u i d i z a t i o n , i s  d  (  2  —j) 9  P l ^ ^  L  (  e? — ) 3  £  u =  1 5 0  (hLf  ^l^lS  mf  +  1  '  7 5  (2.41) Since the p r e s s u r e drop through  a f i x e d bed o f 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 o r d e r to i n i t i a t e f l u i d i z a t i o n , combination 2.41  p r o v i d e s the d e s i r e d  of equations  2.40  and  r e l a t i o n s h i p between the v o i d  f r a c t i o n and the v o l u m e t r i c l i q u i d f l u x .  However, the knowl-  edge of bed voidage a t i n c i p i e n t f l u i d i z a t i o n , ( ] _ ) f / e  1 S  m  still  required.  particles  A v a l u e o f 0.4  [21,49] .  the voidage  Neuzil  i s recommended f o r s p h e r i c a l  and Hrdina  [47] observed  a t minimum f l u i d i z a t i o n , ( i ) f £  m  the d. ./D r a t i o and recommended the f o l l o w i n g J? spherical p a r t i c l e s :  ^  s  that  affected  by  relationship for  63  ^l^mf  The  °'  =  4 0 4  °-  +  4 2 9  ( /°)  (2.42)  d  p  second approach u t i l i z e s the concept o f 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 metric  the r e l a t i o n s h i p between the voidage and the v o l u -  liquid flux.  Richardson and Zaki  [2], based on a  d i m e n s i o n a l a n a l y s i s of the r e l e v a n t v a r i a b l e s i n v o l v e d i n f l u i d i z a t i o n , demonstrated  v  i3  V  v  r 3 v  V  that  = f, (Re , e,, d /D) 1 p 1 p  (2.43)  00  OO  Since i n batch f l u i d i z a t i o n the n e t v e l o c i t y o f the s o l i d particles and  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  the s o l i d p a r t i c l e s i s simply  v  1 3  = v  x  = <j >/e 1  1  (2.44)  Now,combining equations 2.43 and 2.44, the f u n c t i o n a l r e l a t i o n s h i p can be w r i t t e n as  = f  On  2  (Re , e p  ±f  d /D) p  the b a s i s o f a l a r g e amount o f d a t a o b t a i n e d  (2.45)  f o r sedimen-  t a t i o n and f l u i d i z a t i o n o f p a r t i c l e s f o r a wide range o f  system v a r i a b l e s , Richardson and  Zaki  [2] found t h a t  f o l l o w i n g simple r e l a t i o n s h i p represented  the  the r e s u l t s i n the  b e s t manner:  =  <J!>/v;  where V*  = V  00  and  ej  (2.46)  exponent n was  CO  p a r t i c l e Reynolds number  (R©p)  was  The  s u f f i c i e n t l y small.  reported  found to be a f u n c t i o n o f  ^  by Richardson and  alone,  o n l y i f the r a t i o  f o l l o w i n g values  dp/D  o f n were  Zaki over a wide range of Re  , in  the absence of w a l l e f f e c t :  n=4.65  Re<0.2 P  n = 4.35  Re"* P  n = 4.45  Re" P  03  0.2<Re KRe  0 , 1  n = 2.39  500  Richardson and dp/D  Zaki  P P  (2.47) '  < 1  (2.48)  < 500  (2.49)  < Re  (2.50)  P  [2] a l s o observed t h a t the  a f f e c t e d the f l u i d i z a t i o n q u i t e markedly.  N e u z i l and  Hrdina  [47]  ratio  However,  found t h a t the c o r r e c t i o n f a c t o r s  r e p o r t e d by Richardson and account were not adequate.  Zaki to take w a l l e f f e c t s i n t o They, t h e r e f o r e ,  b a s i s on which Richardson and effect correction factors.  Zaki had  N e u z i l and  questioned  the  formulated t h e i r w a l l Hrdina, f o l l o w i n g  an  approach s i m i l a r to t h a t suggested  by S t e i n o u r  [93] , o b t a i n e d  the f o l l o w i n g s e m i - e m p i r i c a l r e l a t i o n s h i p to d e s c r i b e the  bed  expansion:  <  J l  >  /  v  = °-  = o  6 7  R e  p'°  [1-1.27 ( d / D ) *  3  1  1 5  ]  el'  1  (2.51)  which i s based on a l a r g e q u a n t i t y of experimental the range 0.0454 < d /D  < 0.3  p  and 75.5  < Re  p  < 1795.  c o r r e l a t i o n f o r minimum f l u i d i z a t i o n of s p h e r i c a l was  obtained by combining equations  2.42  and  data i n The  particles  2.51,  the r e s u l t  being  J ±  m  V  f  =  M  [1-1.27 (d / D ) p'  1 , 1 5  ] [0.348+0.370 (d /D) ] * Re° * P - P 2  7  0 3  (2.52) However when the w a l l e f f e c t s are not important, e.g. f o r d /D p  = 0.1  the r e l a t i v e e r r o r i n the l i q u i d phase volume  f r a c t i o n i s o n l y 3.5% Richardson  and  Zaki  [47]. E q u a t i o n  2.46  proposed  by  [2], w i t h the v a l u e of the exponent n  o b t a i n e d from equations  2.4 7 -2.50, should be used because  i t has been checked f o r a wider v a r i e t y o f d a t a and i s g e n e r a l l y more a c c e p t a b l e . Thus the equation proposed by N e u z i l and Hrdina, when the w a l l e f f e c t s are important, by Richardson  and  and the equations  Z a k i , when the w a l l e f f e c t s are  proposed negligible,  w i l l be used i n t h i s work to p r e d i c t the r e l a t i o n s h i p between  66 the voidage and the v o l u m e t r i c l i q u i d f l u x f o r the l i q u i d s o l i d systems i n v e s t i g a t e d . The t h i r d method 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 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 r e p r e s e n t i n g an assemblage o f p a r t i c l e s by a u n i t cell  c o n s i s t i n g o f one s p h e r i c a l p a r t i c l e surrounded by a  c o n c e n t r i c s p h e r i c a l envelope, which c o n t a i n s a volume o f l i q u i d such t h a t 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 t h a t f o r the e n t i r e assemblage approach  [49].  This  then r e q u i r e s s o l v i n g the f u l l Navier-Stokes equa-  t i o n s o f motion w i t h i n the c e l l conditions.  f o r a p p r o p r i a t e boundary  The Navier Stokes e q u a t i o n s , w i t h the r e q u i r e d  boundary c o n d i t i o n s , can be s o l v e d 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 n e g l e c t e d (Re  < 0.2).  P  However, when the i n e r t i a l terms are important, s o l u t i o n o f the complete  Navier-Stokes equations can o n l y be o b t a i n e d by  numerical t e c h n i q u e s .  Such techniques have been developed  and used by M a s l i y a h and E p s t e i n Hamielec  [82,] and by L e c l a i r and  [90] t o o b t a i n s o l u t i o n s f o r an assemblage o f spher-  i c a l o r s p h e r o i d a l p a r t i c l e s up to q u i t e l a r g e numbers (Re - 100) . P. 2.2.1  Reynolds  E f f e c t o f t u 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 Free-stream t u r b u l e n c e 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 o f a s i n g l e  particle  [94-96] . found  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  to be i n f l u e n c e d by the f r e e - s t r e a m Reynolds number,  based on the r e l a t i v e velocity.,of  free  stream w i t h r e s p e c t  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 t u r b u l e n c e , v i z . , the i n t e n s i t y  o f the  of turbulence  free-stream  [94,95] and  the s c a l e o f t u r b u l e n c e r e l a t i v e to the p a r t i c l e s i z e  [96].  Torobin and Gauvin  aero-  [95] measured drag c o e f f i c i e n t s  d y n a m i c a l l y smooth spheres by s t u d y i n g t h e i r history  velocity  i n a wind t u n n e l , i n which t u r b u l e n c e was  by s c r e e n g r i d s .  They observed  generated  t h a t , a t c o n s t a n t Reynolds  number, an i n c r e a s e i n the i n t e n s i t y produced a moderate i n c r e a s e and  of turbulence at  first  then a sharp decrease  the drag c o e f f i c i e n t o f the p a r t i c l e . assumed to be caused  of  The i n c r e a s e  in  was  by the d i s r u p t i o n of the p a r t i c l e wake,  whereas the sharp decrease was  b e l i e v e d to be caused by  premature t r a n s i t i o n o f the laminar boundary l a y e r t u r b u l e n t boundary l a y e r  and  into  i t s consequent reattachment  the a to  the p a r t i c l e , thereby reducing the form drag, which constitutes  the main p o r t i o n o f the t o t a l drag f o r c e under these  conditions.  The v a l u e o f 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  at which t h i s t r a n s i t i o n of  Torobin and Gauvin  t h a t the c r i t e r i o n f o r t r a n s i t i o n was relationship  the p a r t i c l e ,  took p l a c e v a r i e d w i t h the  f r e e - s t r e a m t u r b u l e n c e , 1^..  by the  and  adequately  intensity found  described  68 =  Re  45  (2.53)  where Re  c  If the f r e e - s t r e a m Reynolds number was now i n c r e a s e d beyond R e , f o r a f i x e d i n t e n s i t y o f f r e e - s t r e a m t u r b u l e n c e , the c  drag c o e f f i c i e n t o f 'the p a r t i c l e decreased  to a d e f i n i t e  minimum a f t e r which i t i n c r e a s e d again q u i t e n o t i c e a b l y [ 9 4 ] . 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  in a  suspended s t a t e because t h e i r weight, m o d i f i e d by buoyancy, i s balanced  e x a c t l y by the drag f o r c e s due t o the r e l a t i v e  motion o f the l i q u i d w i t h r e s p e c t 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 t u r b u l e n t and the c r i t e r i o n e s t a b l i s h e d by equation 2.53 be s a t i s f i e d on i n c r e a s i n g the flow r a t e , the drag c o e f f i c i e n t as w e l l as the t o t a l drag f o r c e 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 c o n t r a c t i o n o f the bed, t h a t i s , r e d u c t i o n o f bed voidage.  For a l l o t h e r ranges o f Reynolds number, any  i n c r e a s e o f t u r b u l e n c e i n the f l u i d i z i n g l i q u i d stream on i n c r e a s i n g the flow r a t e would g i v e r i s e to an i n c r e a s e i n the drag c o e f f i c i e n t over and above t h a t caused i n c r e a s e d flow i t s e l f ,  by the  thereby c a u s i n g f u r t h e r expansion o f  the l i q u i d - s o l i d f l u i d i z e d bed. No d e t a i l e d  study has been conducted  the e f f e c t s of t u r b u l e n c e on the expansion  to e l u c i d a t e characteristics  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  i n v o l v i n g water f l u i d i z a t i o n of very l a r g e and v e r y heavy spheres,  carried  out by Trupp  [87], r e v e a l e d t h a t  f l u i d i z e d beds expanded more than p r e d i c t a b l e by Richardson - Z a k i c o r r e l a t i o n s .  generates f l u i d - p h a s e t u r b u l e n c e [97]. characteristics  The  particles  of t h i s induced t u r b u l e n c e , v i z . , the  intensity  Therefore  stage i t can o n l y be surmised t h a t the t u r b u l e n c e o f  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 behaviour o f a l i q u i d - s o l i d  in fluidization, characteristics  i n f l u e n c e s the  f l u i d i z e d bed.  understanding of the importance  But f o r a b e t t e r  of the t u r b u l e n c e phenomenon  i t i s necessary to q u a n t i t a t i v e l y study the of the t u r b u l e n c e and t h e i r  voidage of l i q u i d - s o l i d  2.3  fluidized  quantitative  and the s c a l e , have not, however, been measured. at t h i s  the  I t i s known t h a t a  bed o f randomly spaced and randomly moving s o l i d  such  fluidized  Holdup i n g a s - l i q u i d - s o l i d  i n f l u e n c e on  bed  beds.  fluidized  The a c t i v e academic i n t e r e s t  beds  i n the study of t h r e e -  phase f l u i d i z e d beds o r i g i n a t e d from the p r e l i m i n a r y i n v e s t i g a t i o n s undertaken by Turner [9], who  [6] and A d l i n g t o n and Thompson  r e p o r t e d t h a t on slow a d d i t i o n o f gas to a l i q u i d -  solid fluidized  bed, the bed c o n t r a c t e d .  i n an e a r l i e r and d e t a i l e d had observed  Volk  study o f three-phase  t h a t the l i q u i d - s o l i d f l u i d i z e d  [10], however, fluidization,  bed expanded  smoothly on i n t r o d u c t i o n o f the gas phase. contrary observations,  i t was  three-phase f l u i d i z e d bed  postulated  these  [7,8,10] t h a t i n a  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 the bed  Based on  travelled  as d i s c r e t e p a r t i c l e - f r e e b u b b l e s .  This  through postulate  forms the b a s i s o f v a r i o u s mathematical models formulated d e s c r i b e the expansion behaviour o f three-phase beds, three of which are c o n s i d e r e d  to  fluidized  i n the f o l l o w i n g  section.  2.3.1  Models f o r 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 p r e d i c t the v o l u m e t r i c  f r a c t i o n of the  u a l phases i n a three-phase f l u i d i z e d bed f r e e model, (B) the wake model and  are:  individ-  (A) the  i t s modifications,  gas-  and  (C) the c e l l model. A f o u r t h 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 w i t h a p p r o p r i a t e l y averaged f l u i d p r o p e r t i e s and v e l o c i t y , was  considered  by V o l k  [10].  T h i s homogeneous model i s r e j e c t e d here on the grounds t h a t it  i s u n s u i t a b l e even f o r two-phase g a s - l i q u i d flow i n most  i n s t a n c e s , e s p e c i a l l y f o r the s l u g flow regime. (A) The g^as-f'ree model Volk  [10] , who  made an e x t e n s i v e  study of the behaviour  o f three-phase f l u i d i z e d beds u s i n g n i t r o g e n 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 p a r t i c l e s , r e p o r t e d t h a t the bed expanded smoothly  solid  on  i n c r e a s i n g the gas flow r a t e a t a f i x e d l i q u i d flow r a t e .  He  considered the three-phase f l u i d i z e d bed to c o n s i s t o f a b u b b l i n g 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 r a t e through which i s m o d i f i e d due to was  the presence o f the bubbles.  The observed bed  expansion  c o n s i d e r e d due to (i) the i n c r e a s e i n bed volume caused by the presence of  the gas bubbles, and  ( i i ) the i n c r e a s e i n i n t e r s t i t i a l  liquid  velocity  caused by the r e d u c t i o n i n area a v a i l a b l e f o r liquid  flow.  These two f a c t o r s are i n t e r - d e p e n d e n t . scheme used by Volk was in error.  The data p r o c e s s i n g  e m p i r i c a l i n i t s nature and  T h e r e f o r e , by t a k i n g the two f a c t o r s  grossly  mentioned  above i n t o account, a simple mathematical model, along the l i n e s suggested by V o l k , i s developed here. demonstrates Let  F i g u r e 2.2  the model s c h e m a t i c a l l y .  A be the c r o s s - s e c t i o n a l area o f the column and  be the average v o l u m e t r i c f r a c t i o n o c c u p i e d by the i phase.  Then 3 Z i=l  e  i  =  1  (2.54)  72  <i,>  FIGURE 2.2  <J > 2  SCHEMATIC REPRESENTATION OF THE GAS-FREE MODEL  73 Let  us assume t h a t each phase i s homogeneously  distributed  so t h a t the area occupied by each phase i s e.A.  Area occupied by the gas phase  =  Then the  £ A  (2.55)  2  and the Area occupied by the l i q u i d - s o l i d  fluidized  phase = A ( l - e ) 2  (2.56) Since a l l the l i q u i d i s assumed to flow through solid fluidized  the l i q u i d -  bed r e g i o n , a m a t e r i a l balance f o r 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 A <j >  <j >  1  A  gas-free region =  x  (l-e )  ( " 2 1  2  e  (2.57) )  S i m i l a r l y a m a t e r i a l balance o f gas through  <j > 2  where v  2  =  v  2  e  the bed g i v e s  (2.58)  2  i s the average r i s e v e l o c i t y of bubbles  through the  bed. L t has been assumed. [8,16,79] t h a t bed expansion o f the l i q u i d - s o l i d the Richardson  fluidized  phase can be w e l l r e p r e s e n t e d by  - Zaki c o r r e l a t i o n , u s i n g the m o d i f i e d  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 r e g i o n . 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  Then the volume  phase, e|'^ i s g i v e n by  1/n 'If  (2.59)  e)  J  2  where the exponent n i s a f u n c t i o n o f p a r t i c l e Reynolds number, Re , w h i l e E " i s r e l a t e d p If  to the o v e r a l l v o l u m e t r i c  f r a c t i o n o f l i q u i d i n the three-phase  z^/  'If  Combining equations  (2.60)  (l-e ) 2  2.59 and 2.60, E ^ i s g i v e n by  <3i> £  1  V  2  (1 - £„)  e, i s g i v e n by <  (l-e ) 3  £  1  +  e  (2.61)  (l-e )  =  and the t o t a l bed voidage,  e =  f l u i d i z e d bed, e^, by  2 =  3^  1/n  >  (l-e ) + e 2  V  0  3  = (l-e ) 2  i n the t h r e e -  can be w r i t t e n as <  e  (2.62)  2  co  By r e a r r a n g i n g equation 2.62, the s o l i d s holdup phase f l u i d i z e d bed, e^,  2  (l-£ )  1 L  V  L.  3^  1/n ,  >  (1-£ )J 2  (2.63)  0  00  v  Thus e q u a t i o n 2.62, i n combination w i t h e q u a t i o n 2.58, p r o v i d e s a simple scheme f o r d e s c r i b i n g the voidage and bubble  velocity  i n a three-phase f l u i d i z e d bed.  With the h e l p o f t h i s model  and a measurement or e s t i m a t e o f the v o l u m e t r i c f r a c t i o n of one o f the three phases a t d i f f e r e n t gas and rates,  necessary  fractions.of  i n f o r m a t i o n about the average v o l u m e t r i c  the other i n d i v i d u a l  f l u i d i z e d bed  l i q u i d flow  can be o b t a i n e d .  phases i n a three-phase  The d e t e r m i n a t i o n of  holdup, by measuring the bed h e i g h t , L^, i s probably s i m p l e s t procedure,  as the s o l i d s holdup, e^,  the bed h e i g h t , L, , by the f o l l o w i n g  e  The  3  solids the  i s related  to  relationship:  (2.64)  =  average v o l u m e t r i c f r a c t i o n of gas,  o b t a i n e d , by t r i a l and e r r o r ,  e/ 2  from e q u a t i o n  can then 2.63,  average v o l u m e t r i c f r a c t i o n of l i q u i d , e^, from  and  be the  equation  2.54. Thus a simple mathematical model can be obtained the b a s i s of the d e s c r i p t i o n g i v e n by V o l k . model has  o f a three-phase f l u i d i z e d  However, i t i s important  on bed  to note t h a t t h i s  the i n h e r e n t assumption t h a t no l i q u i d i s a s s o c i a t e d  w i t h the gas phase.  Such l i q u i d , by d e f i n i t i o n , would not  p r o v i d e any e f f e c t i v e f o r c e f o r 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 c o n t r i b u t e to the expansion the bed.  I t i s t h e r e f o r e 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 by p s t e r g a a r d  phase.  The wake model d e r i v e d o r i g i n a l l y  [8], as d e s c r i b e d i n s e c t i o n  1.3,  considers  of  76 t h i s d i s t r i b u t i o n by assuming  t h a t the l i q u i d  associated  w i t h the gas phase i s r e p r e s e n t e d by the wake o f the bubbles. (B)  The wake model Stewart and Davidson  [7] and  [8] each p r o -  0 s t e r g a a r d  posed a mechanism f o r the observed bed c o n t r a c t i o n i n t h r e e phase, f l u i d i z a t i o n  [6,9] by d e s c r i b i n g a three-phase  fluidized  bed as c o n s i s t i n g o f  ( i ) a gas phase,  ( 2 ) , ( i i ) a wake  phase,  (k), and ( i i i )  a liquid-solid fluidized  phase,  ( f ) . The mechanisms proposed were s i m i l a r i n most  (or p a r t i c u l a t e )  r e s p e c t s ; however, one d i f f e r e n c e was r e p o r t e d i n the r e s p e c t i v e o b s e r v a t i o n s o f these e a r l y i n v e s t i g a t o r s .  Stewart  and Davidson observed p h o t o g r a p h i c a l l y t h a t the bubble wake i n a two-dimensional bed was e s s e n t i a l l y f r e e o f s o l i d p a r t i c l e s , whereas bubbles emerging  0 s t e r g a a r d ' s  photographic o b s e r v a t i o n s o f  from a three-phase f l u i d i z e d bed showed them  to be f o l l o w e d by a l o n g t r a i l c o n t a i n i n g s o l i d 0 s t e r g a a r d ,  particles.  i n p r o p o s i n g the mathematical model p r e s e n t e d i n  s e c t i o n 1.3, t h e r e f o r e assumed t h a t the volume f r a c t i o n o f s o 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 The main weaknesses i n the model proposed by  phase.  0 s t e r g a a r d ,  as  s t a t e d i n s e c t i o n 1.3, a r e (i) The assumption  t h a t the p o r o s i t y o f the wake phase  i s equal to t h a t o f the p a r t i c u l a t e  phase,  ( i i ) The n e g l e c t o f s o l i d s c i r c u l a t i o n induced by the motion o f gas bubbles c a r r y i n g wakes c o n t a i n i n g solid  particles.  (iii)  The q u a n t i t a t i v e  r e p r e s e n t a t i o n o f bubble  rise  v e l o c i t y by e q u a t i o n 1.12 and o f wake volume f r a c t i o n by equation 1.13. M o d i f i c a t i o n s to remedy these shortcomings  a r e proposed  below. In view o f the c o n t r o v e r s y about the s o l i d s  content  of the wake, the wake model i s r e d e r i v e d here f o r a wake s o l i d s content ranging from the p a r t i c u l a t e  zero to the v a l u e p r e v a i l i n g i n  phase, i n order to e s t a b l i s h  the bed c h a r a c t e r i s t i c s .  i t s e f f e c t on  F i g u r e 2 .3.. demonstrates the model  schematically. Consider the d i s t r i b u t i o n o f l i q u i d between the wake phase, k, and the p a r t i c u l a t e  phase, f .  L e t us assume t h a t  Volume o f l i q u i d i n wake phase =  (2.65)  Volume o f l i q u i d i n p a r t i c u l a t e  (2.66)  phase =  so t h a t the t o t a l volume o f l i q u i d i n the three-phase bed  fluidized  i s g i v e n by  ^1  and the average  =  lk  fi  +  R  l f  (2.67)  v o l u m e t r i c f r a c t i o n o f l i q u i d i n the t h r e e -  phase f l u i d i z e d bed i s  78  — Ill  VIf /  *  s  1  © 1  SOLID  €  •—  3f  1 1 1 1 .LIQUID 1  | I I I I I I i  (D  ®  WAKE G A S  <*  *1f  €  3k  l  _  V1f  FIGURE 2.3  III  Vo  SCHEMATIC REPRESENTATION OF THE WAKE MODEL  79  £  1  ^ l / ^ b  =  (2.68)  L e t us now d e f i n e the average v o l u m e t r i c f r a c t i o n o f wake l i q u i d i n the bed as  e,,  and  =  -±± AL. b  (2.69)  the average v o l u m e t r i c f r a c t i o n o f p a r t i c u l a t e  phase  l i q u i d as  "if e  — AL, b  =  1 f  (2.70)  Then i t can e a s i l y be seen t h a t  £  1  Similarly  -  £  lk  +  £  l f  ( 2  '  7 1 )  l e t us c o n s i d e r the d i s t r i b u t i o n o f s o l i d s  between the wake phase, k, and the p a r t i c u l a t e  phase, f . L e t  us assume.that W^. i s the weight o f s o l i d p a r t i c l e s i n the wake phase and particulate  the weight o f s o l i d p a r t i c l e s  phase.  i n the  Then the t o t a l weight o f s o l i d  particles  i s g i v e n by  W  =  W  f  +  w  k  (2.72)  80 and  the average'volumetric  f r a c t i o n o f s o l i d s i n the  three-  phase f l u i d i z e d bed i s  e  Let  =  3  W/p  AL  3  us d e f i n e the average v o l u m e t r i c f r a c t i o n of wake s o l i d s  i n the bed  as  W  £  3k  =  k  —p I T A"L 3  and  (2.64)  b  ( 2  7 3 )  b  the average v o l u m e t r i c f r a c t i o n of p a r t i c u l a t e  solids  '  phase  as  W  e  f  =  3 f  (2.74) p^ AL, H3  rtJ-<b  I t can e a s i l y be seen t h a t  £  =  3  £  3f  +  £  (2  3k  '  75)  Since a l l the gas passes through the bed as d i s c r e t e bubbles,  the average v o l u m e t r i c f r a c t i o n of gas  phase f l u i d i z e d bed,  z^,  is indivisible.  i n the t h r e e -  Then by m a t e r i a l  balance,  £  1  +  £  2  +  £  3  ~  1  (2.76)  81 Combining equations  £  lk  +  e  l f  +  £  2  2.71, 2.75 and 2.76 we g e t  +  £  3k  +  3f  £  =  Now the t o t a l volume occupied  1  ( 2  '  7 7 )  by the wake phase  = volume o f s o l i d s i n wake phase + volume o f l i g u i d i n wake phase W  k  =  and  +  °lk  ^b  =  3k  ( e  the average v o l u m e t r i c  t h r e e - phase f l u i d i z e d  +£  lk  )  f r a c t i o n o f the wake phase i n the  bed,  (= volume o f wakes/bed volume),  i s t h e r e f o r e g i v e n by  e  k  =  Combining equations  £  lf  +  £  lk  +  £  3k  ( 2  '  7 8 )  2.77 and 2.78, and r e a r r a n g i n g , we g e t  £  3f  =  ' 2 " k  1  £  £  ( 2  '  7 9 )  or  -Al— 1 _ £  +  2- k £  Z  1  3  =  f  i  (2.80)  - 2- k £  £  In o r d e r to e s t a b l i s h a r e l a t i o n s h i p between the particulate liquid-solid  (or l i q u i d - s o l i d  fluidized  bed) r e g i o n and the  wake r e g i o n , l e t us f i r s t c o n s i d e r the p a r t i c u -  82 late region.  L e t us d e f i n e the average v o l u m e t r i c  fraction  of s o l i d s i n t h i s r e g i o n , e^'f* as volume o f s o l i d s i n p a 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  "  "  Dividing  =  V 3 p  W /p  3 f  f  3  +  0  l f  both numerator and denominator by the volume of the  three-phase f l u i d i z e d bed we g e t (W-/p-AL. )  —  e 3  1  J  a  J  (N /p AL ) + (fi /AL )  f  f  substituting  I  e  (2.81)  = 3  b  l f  b  e  3 f  +  e q u a t i o n 2.79.into, equation 2.81  "  3f — (l- -  e  l f  yields  £  =  £ 2  (2.82) E k  )  S i m i l a r l y i t can be shown t h a t  the average v o l u m e t r i c 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 — - 2- k e  =  l f  ( 1  £  £  (2.83) )  Then from equations 2.80, 2.8 2 and 2.83, we can o b t a i n  n  II £  lf  +  £  3f  1  that  (2.84)  83 which a l s o f o l l o w s from the d e f i n i t i o n s o f  and  z'^f  For the l i q u i d - s o l i d wake r e g i o n l e t us d e f i n e 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 i n the wake as volume of s o l i d s i n w a k e / t o t a l volume o f wake o r  V 3 p  £  wk/p3 + n l k  3k  D i v i d i n g both numerator and denominator by the volume o f the three-phase f l u i d i z e d bed we g e t  e' 3  k  y^p—s^^—b( V ^ ^ V + ( "lk/ A L b )  =  =  =  3 k— £ lk + 3k £  (2.85)  £  S u b s t i t u t i n g e q u a t i o n 2.78 i n t o equation 2.8 5 y i e l d s  £  3k  "  i  f  <- > 2  86  S i m i l a r l y i t can be shown t h a t the average v o l u m e t r i c f r a c t i o n o f l i q u i d i n the wake i s e.  II  k Then from equations 2.78, 2.86 and 2.87, we can o b t a i n t h a t  II £  lk  II +  £  3k  =  1  (2.88)  84 II  which a l s o Now  f o l l o w s from the d e f i n i t i o n s o f £ ^  3 k  and  k  £  .  3 k  s i n c e the amount o f s o l i d s i n the wake o f a  bubble i s not known a p r i o r i , that e  II  i s related  e  3k  to  i t has been expedient to assume  via  = k 3f X  £  ( 2  *  8 9 )  where x ^ i s a c o e f f i c i e n t of p r o p o r t i o n a l i t y  which can take  any p o s i t i v e v a l u e between zero and u n i t y .  Then the case o f  Xj^. = 0 s i g n i f i e s t h a t t h e r e 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 t h a t the s o l i d s f r a c t i o n i n the wake i s the  same as t h a t i n the p a r t i c u l a t e ^stergaard).  phase  (as p o s t u l a t e d by  I t can e a s i l y be seen t h a t  e  lk  -  1  - k 3f x  e  =  ^ W l f  ( -90) 2  and t h e r e f o r e 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  £  == ( l - e ) 3  =  e  2  +  £  k^  1 - x  "2% 8'7 "and 2r. 90, i s  r  k^  +  e  l f( k k x  e  +1  ~ 2 e  ~ k^ e  (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 particulate  phase, l e t us assume t h a t the t o t a l v o l u m e t r i c  flow r a t e o f l i q u i d through the column i s Q, and the t o t a l  85 v o l u m e t r i c flow r a t e o f gas through the column i s Q . 2  the average l i q u i d  <Ji>  Then  f l u x through the column i s  =  (2.92)  Q-./A  and the average gas f l u x through the column i s  <Jo>  =  (2.93)  Qo/A  Now, s i n c e a l l the gas through the column t r a v e l s as d i s c r e t e bubbles, a simple m a t e r i a l balance over any c r o s s - s e c t i o n p e r p e n d i c u l a r t o the flow path w i l l g i v e  <j > 2  Let  =  e  2  v  (2.94)  2  us now c o n s i d e r a s i m i l a r m a t e r i a l balance f o r the l i q u i d  over a c r o s s - s e c t i o n p e r p e n d i c u l a r to the flow path.  v o l u m e t r i c flow rate of l i q u i d through the column  Then  (  / l i q u i d flux\ | through the J \ particulate J Vphase /  cross^sectional area o c c u p i e d by the p a r t i c u l a t e phase  x  (  liquid flux\ through the j x wake phase /  cross-sectional I area o c c u p i e d by \ the wake phase  or <j > A 1  =  A (l-e -e )] +[(v 2  k  2  e ) . (A e )] l k  fc  (2.95)  86 Substituting  equation  2.90 i n t o equation  2.95 and r e a r r a n g i n g  gives <j > - v x  j-^  2  (1-x +x ^ ) e k  k  f  k  -  (2.96) (l-e -e ) 2  k  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 can be r e l a t e d  to the average v o l u m e t r i c f r a c t i o n o f l i q u i d i n the particulate  phase,  by the Richardson  II  e  lf  =  (  V J  - Zaki  correlation,  1/n  V  (2  -  97)  where n = f (Re )' and the v a l u e s o f n f o r v a r i o u s Re a r e P P g i v e n by equations  2.47 to 2.50.  Thus f o r a d i s t r i b u t i o n o f  s o l i d p a r t i c l e s between the wake phase and the p a r t i c u l a t e phase as g i v e n by equation  2.89, equations  1.4 and 1.10 d e r i v e d  e a r l i e r f o r J^stergaard s model have been m o d i f i e d to 1  equations  2.91 and 2.96,  Another d e f i c i e n c y by 0stergaard Vakhrushev  respectively. o f the wake model, as p o s t u l a t e d  [8] and used subsequently  by Efremov and  [16] as w e l l as Rigby and Capes  [80] , has been the  n e g l e c t o f s o l i d s c i r c u l a t i o n i n the p a r t i c u l a t e  phase. A l -  though the e x i s t e n c e o f s o l i d s c i r c u l a t i o n was not r e p o r t e d by these authors, bed  the bubbles r i s i n g i n a three-phase f l u i d i z e d  have been observed  some p a r t i c l e s . then suggest  to c a r r y .sizeable wakes- c o n t a i n i n g  A simple c o n s i d e r a t i o n o f c o n t i n u i t y  would  a downward motion o f s o l i d p a r t i c l e s i n the  87 particulate  phase as a r e s u l t o f t h e i r upward motion i n the  wakes o f the bubbles.  Based on t h i s simple mechanism f o r  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, a model w i l l now be developed f o r i t . L e t us assume t h a t each bubble i n a three-phase f l u i d i z e d bed c a r r i e s w i t h i t an a t t a c h e d wake c o n t a i n i n g solid particles.  Then the r a t e a t which s o l i d s r e a c h the  s u r f a c e o f the three-phase f l u i d i z e d bed  Solids flux through .wake  x  c r o s s - s e c t i o n a l area occupied by wake phase  [  or  'volumetric flow r a t e of s o l i d s to bed ] = surface  After  [*2 3 k ] £  x  [  A £  k]  <' > 2  a bubble emerges from the bed, s o l i d s  98  carried  i n i t s wake w i l l be washed o u t o f i t by continuous exchange w i t h the surrounding l i q u i d the  s o l i d s w i l l be c a r r i e d  [80] .  The d i s t a n c e f o r which  above the bed would then depend  on the exchange r a t e 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  l e a v i n g the  wake, would move downwards to the bed and downwards i n the bed to compensate f o r the upward movement i n the wake. fore  There-  the downward flow r a t e o f s o l i d p a r t i c l e s i n the p a r -  88 t i c u l a t e phase  Solids  f l u x through  particulate  = [- v.  phase  [A(l-e -e )]  x  '3f "  c r o s s - s e c t i o n a l area occupied by p a r t i c u l a t e phase  x  2  (2.99)  k  Since there i s no n e t flow o f s o l i d p a r t i c l e s through the bed, the upward  flow o f s o l i d s i n the wake must e x a c t l y  equal the downward flow o f s o l i d s i n the p a r t i c u l a t e Hence, equating equations  phase.  2.98 and 2.99 and r e a r r a n g i n g , the  l i 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 p a r t i c u l a t e  phase  i s g i v e n by  v  ..  v „ e,_ e. ^—-———  =  3  e  3f  and i n combination _„ V  3  _ 2 k V  =  "  £  ( 1  (2.100)  " 2- k e  £  )  w i t h e q u a t i o n 2.89, X  k  (l-e -e ) 2  ( 2  k  *  1 0 1 )  From e q u a t i o n 2.101 i t can be seen t h a t i f t h e r e a r e 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 o f 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 z e r o . Thus no c i r c u l a t i o n would e x i s t by the mechanism here.  postulated  89 The presence  of o t h e r more complex modes of c i r c u l a t i o n i s  recommended f o r f u r t h e r  investigations.  F o r the p r e s e n t  i t w i l l be assumed t h a t e q u a t i o n 2.101 d e s c r i b e s the s o l i d s c i r c u l a t i o n phenomenon adequately As has been s t a t e d  f o r x^ > 0.  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 controls  the bed expansion  [67]. The Richardson  particles  - Zaki  c o r r e l a t i o n , equation 2.97, can be m o d i f i e d and r e p r e s e n t e d i n terms o f the r e l a t i v e v e l o c i t y : _  e  „  =  l f  II  v, , l / ( r i - l ) ( )  (2.102)  CO  The l i q u i d f l u x through equation 2.96. through  Therefore the l i n e a r v e l o c i t y o f l i q u i d  the p a r t i c u l a t e phase w i l l be  v  =  ±  *1 £  and  the p a r t i c u l a t e phase i s g i v e n by  <  = —  ^l - 2 >  V  { 1  - k x  + £  n  lf  £  l f  ( 1  lf k k '• x  ~ 2- k £  £  ) £  (2.103)  )  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  V  _.. ... 13 = l ~ 3 v  v  =  <  i >-v (l-V lf k • lf - 2- k x  £  1  2  £  ( 1  £  £  }  ) e  k  ^ 2 k. k V  £  x  + ( 1  - 2- k £  £  }  (2.104)  90  By s i m p l i f y i n g  equation 2.10 4 we  :  v 13  j  l £  >  "  lf  v  2  ^  ( 1 _ x  1 - £  k  ) e  get  }  (2.105)  2~ k^ e  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  lf'  w  ^  ation, 2.105  k  e  g i v e n by the m o d i f i e d Richardson - Zaki c o r r e l -  equation 2.102, which when combined w i t h equation reduces to  "  =  r!Mr  <j >  n _ 1 )  - v (l-x )  1  2  k  (l-e -e, ) V 2 k 0  0  1/n £ ]  0  (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 f l u i d i z e d bed, equations.2.97 r e p l a c e d by equation 2.106  and  2.96  three-phase  are m o d i f i e d and  to p r e d i c t the average v o l u m e t r i c II  phase, ^-^f  f r a c t i o n o f l i q u i d i n the p a r t i c u l a t e  ^ ne  average v o l u m e t r i c f r a c t i o n of gas, £ , and the t o t a l voidage, 2  e, i n the bed are s t i l l  g i v e n by equations 2.94  and  2.91  respectively. In order to complete  the model, independent  relation-  s h i p s f o r the r i s e v e l o c i t y o f a bubble swarm and the volume f r a c t i o n occupied by the wakes i n a three-phase  fluidized  91 bed have to be developed. Lapidus and E l g i n  For d i s p e r s e d phase o p e r a t i o n s ,  [67] have demonstrated  t h a t the r e l a t i v e  v e l o c i t y between the phases determines the holdup o f e i t h e r phase.  Thus i f we p o s t u l a t e t h a t the r e l a t i v e  between the gas and the l i q u i d i n a three-phase  velocity fluidized  bed can be p r e d i c t e d by the same c o r r e l a t i o n s as f o r twophase g a s - l i q u i d flow, the problem then reduces to c o r r e c t l y f o r m u l a t i n g the r e l a t i v e v e l o c i t y i n two-phase g a s - l i q u i d flow, s e v e r a l competing models having been proposed ( s e c t i o n 2.1.2).  Towell e t a l . [59] have suggested r a t h e r  a r b i t r a r i l y t h a t 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 l a r g e diameter  (> 4 inch) columns-, where  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,  II  v 21  =  v  (2.107)  t e s t e d and confirmed by R e i t h  e t a l . [68] f o r l a r g e diameter columns. diameter columns  i s g i v e n by  + 2 <j~>  T h i s e m p i r i c a l c o r r e l a t i o n was  system-  However, f o r s m a l l  (< 4 i n ) , i n the absence o f s y s t e m a t i c 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 f o r the bubble flow regime can be o b t a i n e d by combining equations 2.32  and 2.3 7  to g i v e  n  v 21  (2.108) II  92 I t has been shown [76] t h a t the model presented i n s e c t i o n 2.1.2  s a t i s f a c t o r i l y r e p r e s e n t s a wide v a r i e t y of data f o r  bubble.columns and f o r c o c u r r e n t 2.108, which i s based  gas-liquid flow.  on t h i s model, i s t h e r e f o r e  Equation recommended  f o r p r e d i c t i n g the r e l a t i v e v e l o c i t y between gas and i n s m a l l columns (< 4 inch) .  liquid  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 l a r g e r columns, the d e s c r i p t i o n of bubble motion i n such columns i s incomplete and successful.  t h e r e f o r e m o d e l l i n g attempts  N e v e r t h e l e s s , the e m p i r i c a l  have not been  c o r r e l a t i o n developed  by Towell e t a l . [59], e q u a t i o n 2.107, has been confirmed f o r l a r g e columns [68] and i s t h e r e f o r e  recommended.  The r e l a t i v e v e l o c i t y i n three-phase defined  fluidization is  as the d i f f e r e n c e 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  -"» 21  V  _ " 2  "  v  V  l  i s g i v e n by  < j  2 £  >  l "~ l f  j  2  £  (2.109)  II  Substituting 2.109  we  the v a l u e of  from equation 2.96  get <  v  into equation  2 1  =  = £  2  h  >  n—— £  l f  ( 1  (2.110) " 2- k e  £  )  which a f t e r s u b s t i t u t i o n s and rearrangements s i m p l i f i e s to  93  v  n  =  2 1  £  2  £  (2.111)  lf  - 2- k  ( 1  £  £  S u b s t i t u t i n g the v a l u e o f  )  from e q u a t i o n 2.76 i n t o  e q u a t i o n 2.111 and r e a r r a n g i n g  <j  >  Jl  <  2  >  <  +  Jo  = _± £  II  Tf  ( 1  " 2" k £  £  £_ + _ i i  (l-e )  2  -•'«  )  f—JL_  v  (l-e )  3  Bhaga by  E  >  gives  [ 1 ] , by a method v e r y s i m i l a r  Zuber and F i n d l a y  one-dimensional  (2.112) 1  3  [39] f o r g a s - l i q u i d  simultaneous  to the one employed  flow, o b t a i n e d f o r  flow o f 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 > "— <a > 9  =  C  „, o 3 +J 1  <a,a v ,> + — — — — 2 9  <  > 2  < a  2  9  (2.113)  >  in  where C  i s the d i s t r i b u t i o n parameter f o r c o c u r r e n t gas-  Q  l i q u i d - s o l i d flow and i s g i v e n by <a (j,+j )> 2  C  Q  =  2  1  (2.27d)  Z  <a ><j j > 2  1 +  2  Comparing e q u a t i o n 2.113 w i t h e q u a t i o n 2.26 i t can be seen in  t h a t the d i s t r i b u t i o n parameter, C , r e p r e s e n t s the e f f e c t Q  of r a d i a l d i s t r i b u t i o n of both v o l u m e t r i c f l u x o f the mixture  and i n s i t u volume f r a c t i o n o f the gas ( i . e . gas holdup) on the r i s e v e l o c i t y o f a bubble swarm. f o r g a s - l i q u i d flow, a v a l u e o f C  Q  As was observed  g r e a t e r than u n i t y  i n d i c a t e s t h a t n e i t h e r the gas holdup nor the v o l u m e t r i c f l u x o f the g a s - l i q u i d stream a r e c o n s t a n t over the c r o s s - s e c t i o n . T h e r e f o r e , f o r a three-phase f l u i d i z e d bed, i f both the gas holdup and the v o l u m e t r i c f l u x of the g a s - l i q u i d stream v a r y r a d i a l l y a t a g i v e n v e r t i c a l l e v e l i n the bed, then e q u a t i o n 2.112 should be m o d i f i e d to i n c l u d e the d i s t r i b u t i o n MI  parameter, C . Q  <J2>  e2  =  V ^ l *  Thus  +  ^2  (1-63)  "  >  }  ,  £  l f - 2" k ( 1  £  £  }  (l-e ) 3  -»'  (2.114) ^  Since i t has been assumed t h a t the r e l a t i v e v e l o c i t y i n t h r e e _ 11 j  phase f l u i d i z a t i o n , v ^ , can be r e p r e s e n t e d by the c o r r e l 2  M •  a t i o n s developed f o r two-phase g a s - l i q u i d flow, v ^ can be 2  o b t a i n e d from e q u a t i o n 2.107 f o r l a r g e diameter columns (_> 4 inch) and from e q u a t i o n 2.108 f o r s m a l l diameter columns (< 4 i n c h ) .  Both these c o r r e l a t i o n s r e q u i r e a knowledge o f V^,  which can be estimated from the average diameter o f bubbles i n the swarm. Since the s t r u c t u r e and s i z e o f wakes behind gas bubbles has not.been i n v e s t i g a t e d s y s t e m a t i c a l l y i n e i t h e r  two-phase  g a s - l i q u i d flow o r 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 f o r the volume f r a c t i o n o f wakes i n a three-phase f l u i d i z e d bed.  L e t a n and Kehat  [36] have  presented a l i m i t e d s e t o f data f o r the volume f r a c t i o n o f wake a t v a r i o u s v a l u e s o f the d i s p e r s e d phase holdup i n a liquid-liquid  system.  In the absence o f g a s - l i q u i d d a t a ,  the L e t a n - Kehat r e s u l t s w i l l be used i n t h i s t h e s i s  on  the assumption t h a t the wake behind a gas bubble has s i m 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 t e n t a t i v e model i s suggested based on the wake s t r u c t u r e p o s t u l a t e d by de Nevers and Wu (d  [61]. g  From the v i s u a l o b s e r v a t i o n of l a r g e a i r bubbles  =• 1-2 cm), which were e s s e n t i a l l y h e m i s p h e r i c a l i n shape,  r i s i n g i n e i t h e r g l y c e r i n e o r water, these i n v e s t i g a t o r s i n f e r r e d t h a t the bubbles were each f o l l o w e d by a c o n i c a l wake whose i n f l u e n c e extended to a d i s t a n c e 1, behind the bubble. k Thus, i f the t r a i l i n g bubble f o l l o w e d a t a d i s t a n c e g r e a t e r than 1^. i t . w o u l d n o t be a f f e c t e d by the wake o f the l e a d i n g bubble.  On the b a s i s o f such a s t r u c t u r e f o r the bubble  and the wake, de Nevers and Wu found t h a t a d i m e n s i o n l e s s d i s t a n c e , l^/Rg/ o f seven s a t i s f i e d t h e i r d a t a f o r bubbles r i s i n g and c o a l e s c e n c i n g i n both the a i r - w a t e r and a i r glycerine  systems.  I f we v i s u a l i z e a s i m i l a r s t r u c t u r e f o r the r i s e o f a n o n - c o a l e s c i n g swarm o f bubbles, we can t h e r e f o r e e s t i m a t e t h a t the v e r t i c a l d i s t a n c e between any two bubbles i n the swarm should be a t l e a s t  (1, + R ) .  Then the volume o f the  FIGURE 2.4  SCHEMATIC OF WAKE STRUCTURE SUGGESTED dE NEVERS AND WU [61]  97 wake as compared to the volume o f the bubble i s g i v e n by  k  V ^ k  B  2/3TTRJ  2R  (2.115) s  For a l i q u i d - l i q u i d system, both Letan and Kehat [36] and Hendrix e t a l . [99] have shown t h a t the wake s i z e i s markedly a f f e c t e d by the presence o f o t h e r drops.  The  e f f e c t o f gas volume f r a c t i o n on the wake s i z e can be estimated if  i t i s assumed f o r computational purposes  that:  (a) the bubbles have an o r d e r l y d i s t r i b u t i o n i n the swarm, and (b) the wake l e n g t h i s equal to the v e r t i c a l d i s t a n c e between the bubbles, so t h a t the wake o f the l e a d i n g 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 c u b i c d i s t r i b u t i o n f o r the bubbles i n -the swarm, w i t h the c e l l ,  (1, + R ) as the c h a r a c t e r i s t i c s  length of  i t can e a s i l y be shown t h a t the r a t i o o f wake s i z e  to bubble s i z e i s g i v e n by 1/3 - 1]  (2.116)  I n Table 2.1 a r e presented v a l u e s o f wake to bubble volume r a t i o f o r a c u b i c and two other .sys'tematic bubble  distri-  b u t i o n s , as w e l l as the v a l u e s 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 D i s p e r s e d phase holdup, e 2  V"B  0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.6  V"B  (1)  0.878 0.704 0.594 0.516 0.456 0.408 0.368 0.335 0.306 0.259  27T  K ( 2 )  ( 3 )  Orthorhombic  = 1/2  /  1.047 0.852 0 .728 0.640 0 .573 0.519 0.475 0.437 0.405 0.351  3  - II  [(=-?—) 2  a  Cell -  1  fl./fi  K  B  B  = 1/2  Rhombohedral C e l l -fi./n_ = 1/2 K  [ (—  Continuous phase: D i s p e r s e d phase:  1/3  3/3~e2  )  [(^IL!^) ^ 1  _ 3£  B  From F i g u r e 6 o f Letan and Kehat  ( 3 >  V°B  0.946 0.763 0.648 0.565 0.503 0.452 [ 0.411 0.376 0 .346 0.296  (1) * 'Cubic C e l l - n./nn  (2)  2  [36]  distilled kerosene  water  - 1]  3  -i]  vv  (*)  0.93 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.77 0.61  Kehat  [36] f o r wakes behind l i q u i d drops.  As can be seen  from Table 2.1, the model improvised here f o r e s t i m a t i o n o f the volume f r a c t i o n of wakes i n g a s - l i q u i d systems,  though  undoubtedly o v e r s i m p l i f i e d , are matched by the trend :.of the l i q u i d - l i q u i d d a t a " c l o s e l y enough  to^ justify  the use o f these data f o r 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  e x p o n e n t i a l wake s t r u c t u r e recommended by de Nevers and [61] and by C r a b t r e e  Wu  and Bridgwater [78] can be developed  s i m i l a r l y , b u t l a c k of d e t a i l e d data f o r wake s i z e s i n gasliquid  systems does not warrant such complicated models a t  t h i s stage. The presence o f s o l i d p a r t i c l e s i n three-phase  fluidiz-  a t i o n would a f f e c t the wake s i z e , as i n f e r r e d by 0stergaard [8] and c l e a r l y demonstrated by Rigby and Capes [80] and by Efremov and Vakhrushev recommendations,  [16].  Thus, f o l l o w i n g the l a t t e r ' s  i t i s p o s t u l a t e d t h a t the volume f r a c t i o n  o f wakes i n three-phase f l u i d i z a t i o n w i l l be r e p r e s e n t e d by  (P /P )"' k  B  =  (n /n ) k  B  n  f(e)  (2.117)  The e x a c t form of the f u n c t i o n f can o n l y be developed from e x p e r i m e n t a l l y observed v a l u e s of wake s i z e s i n two-phase g a s - l i q u i d 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 s i 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 d i f f e r e n c e  100 being the i n c l u s i o n o f s o l 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 p o s s i b l e mechanism f o r  s o l i d s c i r c u l a t i o n , based on the r i s e o f s o l i d  particles  i n the wake o f r i s i n g bubbles, i s suggested, a l o n g w i t h r e l a t i o n s h i p s f o r 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  fluidized  bed. (C) The c e l l model As s t a t e d e a r l i e r , the c e l l model technique has been s u c c e s s f u l l y used to d e s c r i b e v a r i o u s d i s p e r s e d phase operations  [49,90,86],  an assemblage  This technique c o n s i s t s o f r e p r e s e n t i n g  o f p a r t i c l e s by a s p h e r i c a l  (or sometimes a  c y l i n d r i c a l ) c e l l c o n t a i n i n g a s i n g l e p a r t i c l e and l i q u i d i n such a p r o p o r t i o n t h a t the voidage i n the c e l l the  voidage o f the e n t i r e assemblage.  i s equal to  The Navier-Stokes  equations o f motion are then s o l v e d f o r the c l o s e d c e l l , w i t h adequate  and c o n s i s t e n t boundary  c o n d i t i o n s , to o b t a i n the  r e l a t i v e v e l o c i t y o f the p a r t i c l e w i t h r e s p e c t t o the l i q u i d i n the c e l l . The proposed c e l l model f o r three-phase c o n s i s t s o f r e p r e s e n t i n g a bubble and a s o l i d i n d i v i d u a l l y by two separate s p h e r i c a l c e l l s .  fluidization particle Smith [91]  used such a model to a n a l y s e the r a t e o f s e d i m e n t a t i o n of p a r t i c l e s o f two d i f f e r e n t s p e c i e s .  In o r d e r to form a  c o n s i s t e n t s e t , Smith matched the two s p h e r i c a l c e l l s  each  101 r e p r e s e n t i n g an i n d i v i d u a l s p e c i e s , by u s i n g the boundary c o n d i t i o n s o f (a) equal t a n g e n t i a l v e l o c i t i e s a t the equator . o f both envelopes and (b) equal p r e s s u r e g r a d i e n t s at  a l l . s u r f a c e s o f both envelopes.  The proposed model, which  i s p r e s e n t e d i n i t s e n t i r e t y i n Appendix s l i g h t l y m o d i f i e d boundary  8.1, uses s i m i l a r but  c o n d i t i o n s to match the two  s p h e r i c a l c e l l s v i z . (a) e q u a l i t y o f t a n g e n t i a l  velocities  at  the e q u a t o r i a l s u r f a c e of both envelopes and (b) e q u a l i t y  of  drag p e r u n i t volume i n each c e l l  pressure gradient  such t h a t the o v e r a l l  (-Ap/L) i s the same f o r both the c e l l s  and throughout the bed. The s o l u t i o n o b t a i n e d f o r the c e l l model, u s i n g the c r e e p i n g flow equations w i t h the above boundary match the two s p h e r i c a l c e l l s , and H a p p e l s 1  c o n d i t i o n s f o r each i n d i v i d u a l c e l l  c o n d i t i o n s to  [49] boundary  ( i t i s realized  these may n o t be the unique s e t of boundary  that  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 i n t r o d u c t i o n o f gas.  S i n c e the wake behind a gas bubble i s  n o n - e x i s t e n t i n c r e e p i n g flow, t h i s r e s u l t i s not s u r p r i s i n g because, as e x p l a i n e d by 0stergaard Davidson  [8] and by Stewart and  [ 7 ] , i t i s the wake behind the gas bubble t h a t i s  r e s p o n s i b l e f o r the observed c o n t r a c t i o n i n three-phase fluidization.  I t i s t h e r e f o r e recommended t h a t the f u l l  Navier-Stokes  equations be s o l v e d by e x i s t i n g n u m e r i c a l  techniques c e l l model.  [ 8 2,, .90] , f or  a c r i t i c a l e v a l u a t i o n o f the proposed  102 2.3.2  Gas holdup i n three-phase f l u i d i z e d beds The behaviour o f bubbles i n a three-phase f l u i d i z e d  bed has been the s u b j e c t o f l i m i t e d study [15,79], although many i n v e s t i g a t o r s  [14,18,100] have observed t h a t i n w e l l  expanded beds o f small s o l i d p a r t i c l e s bubble c o a l e s c e n c e predominates, whereas i n s l i g h t l y expanded  ( c l o s e to packed  bed voidage) beds of l a r g e p a r t i c l e s bubble breakup occurs.  A number of attempts  generally  [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 y e t no r e a l i s t i c model has been formulated to i n c o r p o r a t e the q u a l i t a t i v e o b s e r v a t i o n s and to p o i n t out the areas r e q u i r e d f o r f u r t h e r study, i n order to complete the unders t a n d i n g of gas bubble b e h a v i o u r .  Nevertheless, empirical  c o r r e l a t i o n s have been suggested by v a r i o u s Thus V a i l e t a l . [17], who 146 mm  investigators.  measured the gas holdup i n a  diameter column by q u i c k l y s h u t t i n g o f f the gas and  l i q u i d flow r a t e s s i m u l t a n e o u s l y , thus i s o l a t i n g the e x p e r i mental s e c t i o n , recommended the f o l l o w i n g c o r r e l a t i o n f o r a bed o f 0.77  mm  g l a s s beads f l u i d i z e d by a i r and water:  e'" = 0.1026 ( l - e . )  for  .1.5  <j < 1  9.0,  4.0  2 , 0 9  (<j >/<j >) o  <j < 2  1  0 , 7 8  (2.118)*  20  *  changed  The exponents 2.09 and 0.78 were e r r o n e o u s l y i n t e r i n the o r i g i n a l paper [17].  103 They f u r t h e r suggested t h a t 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 p r e d i c t e d the gas holdup i n two-phase gasl i q u i d flow.  e  ,, 2  Therefore  =  0.1026 (<j >/<j >) 2  0.78  and combining the above two equations we .., e  2  .. =  e  Equation 2.120  2  (2.119)  1  see t h a t  2.09 " 3  ( 1  e  (2.120)  )  thus shows the importance of bed expansion,  which had been i m p l i e d by o t h e r i n v e s t i g a t o r s but not formulated l o g i c a l l y .  However, e q u a t i o n 2.120  i s o n l y an  e m p i r i c a l c o r r e l a t i o n and cannot be e x t r a p o l a t e d beyond  the  range of i t s s u p p o r t i n g d a t a w i t h o u t the r i s k o f s e r i o u s error. M i c h e l s e n and by measuring  [14] measured gas holdups  0 s t e r g a a r d  the s t a t i c p r e s s u r e drop a c r o s s the l e n g t h of  the bed and a l s o by employing a t r a c e r i n j e c t i o n t e c h n i q u e . They proposed the f o l l o w i n g c o r r e l a t i o n f o r a bed of 1 g l a s s beads f l u i d i z e d by a i r and water i n a 216 mm  mm  diameter  column:  ,v n n  e2 = 0.011 for  2 <J -< 1  7.5  x ° - 3 7 ^ . .0.78, <D-j. ^2 and 0.35  <J  2  <  (2.121)  2.2  104 and f o r two-phase g a s - l i q u i d flow o f the a i r - w a t e r system,  e ' = 0 .0394 < j > " ' 0  2  for  <j >  1 6  1  0.7  < 17.0  (2.122).  1, 0 5  2  and 0.35  The format of e q u a t i o n 2.121  <J2  <  2.2  contributes l i t t l e  to the  s t a n d i n g o f bubble behaviour i n three-phase f l u i d i z e d but the e q u a t i o n i t s e l f  i s found to be i n q u a n t i t a t i v e  ment w i t h e q u a t i o n 2.118 of  w i t h i n the l i m i t s o f  underbeds, agree-  applicability  e q u a t i o n 2.121. Capes e t a l . [7 9]  bubbles i n three-phase  s t u d i e d the l o c a l p r o p e r t i e s o f gas  ( a i r - w a t e r - g l a s s beads)  beds by l o c a t i n g 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 s h o r t d i s t a n c e , i n s i d e the bed. bubble r i s e v e l o c i t i e s  fluidized  The measured  were c o r r e l a t e d w i t h the measured  voidage i n the bed and the average l e n g t h , ! , of bubbles i n the swarm, by means o f the f o l l o w i n g  v 2  for  0.03  C<j +j >) = 1  <j  x  2  < 2.61  32.5  l  1  and 0.5  '  5  3  <J2  relationship:  ( ^ )  <  2  (2.123)  2.0  They a l s o suggested t h a t i f a s p h e r i c a l cap bubble w i t h a f l a t base i s assumed to have an i n c l u d e d wake angle of 135°, then the average l e n g t h and the e q u i v a l e n t r a d i u s o f the  105 bubble can be c o r r e l a t e d by  1  =  1.14 r  (2.124)  e  I t has been found, however, by the p r e s e n t author t h a t f o r an i n c l u d e d wake angle o f 135° the above r e l a t i o n s h i p i s i n c o r r e c t and should read  1  =  1.02 r  (2.125)  e  w h i l e o n l y f o r an i n c l u d e d wake angle o f 158° does e q u a t i o n 2.124 g i v e the c o r r e c t r e l a t i o n s h i p between 1 and r . E q u a t i o n 2.123 i n combination w i t h e q u a t i o n 2.125 shows the  i n f l u e n c e o f bubble diameter and the important e f f e c t of  bed voidage on bubble v e l o c i t y .  E q u a t i o n 2.123 i s found  to be i n q u a n t i t a t i v e agreement w i t h e q u a t i o n 2.118 on assuming  a r e a l i s t i c bubble diameter, no measurements o f bubble  diameter having been r e p o r t e d by V a i l e t a l .  Thus a l l these  e m p i r i c a l c o r r e l a t i o n s a r e q u a n t i t a t i v e l y compatible and demonstrate the importance o f bed voidage and average  bubble  diameter i n d e t e r m i n i n g the r i s e v e l o c i t y o f a bubble swarm, and thereby a l s o the gas holdup, i n three-phase  fluidization.  From the g e n e r a l i z e d wake model f o r a three-phase f l u i d i z e d bed, presented i n s e c t i o n 2.3.1, the r i s e of bubbles i s g i v e n by  velocity  106 IM  v  2  II  = -°-  ±—f_ + - J i _  £_JL_ v  (2.114)  2 1  where _ lit  v  2 1  = (VJ  B  + 2 <j >  (2.107)  2  f o r D > 4 i n c h and j  v  2 1  (VJ = —  B  "» 1/3 [0.25 ( l / e ) " ] _ v - ~Tfr~  "tanh •  L / J  2  2 j  (2.108)  m  f o r D < 4 i n c h , where (V ) o  B  i s the r i s e ' v e l o c i t y of a s i n g l e  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 o b t a i n e d  from  e  2  < =  >  ^ 2 ^ 2  (2.94)  Thus e q u a t i o n 2.114 i n combination w i t h e q u a t i o n 2.108  (or e q u a t i o n 2.107) and e q u a t i o n 2.94 p r o v i d e s a g e n e r a l  model f o r d e s c r i b i n g  the gas bubble behaviour i n three-phase  f l u i d i z e d beds.  Qualitatively  t h i s model i s b e t t e r  than any  of the e m p i r i c a l  correlations,  as i t not o n l y takes i n t o  account the e f f e c t o f bed voidage and bubble diameter on the r i s e v e l o c i t y o f bubbles, but a l s o of i n v e s t i g a t i n g  i l l u s t r a t e s the n e c e s s i t y  the r a d i a l p r o f i l e s o f 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 ) and the phenomenon o f wake f o r m a t i o n (to determine Q  £jj , i n order to b e t t e r the understanding o f 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 o f independent models p o s t u l a t e d f o r o v e r -  a l l voidage 0stergaard  ( l - e ^ ) i n three-phase f l u i d i z e d beds  is-limited.  [8] d e r i v e d a wake model, which has subsequently  been used and m o d i f i e d by v a r i o u s i n v e s t i g a t o r s w i t h o u t s i g n i f icantly  enhancing the understanding o f bed expansion behaviour  i n three-phase f l u i d i z a t i o n . [16],  Thus Efremov  and Vakhrushev  who used the wake model on the assumption t h a t no  particles  a r e p r e s e n t i n the wake [7], p o s t u l a t e d t h a t the  bed voidage i s g i v e n by  <i > - ( y n ) B  x  £ =  <j > 2  :  < - 2~ k 1  e  e  )  +  £  2  +  e  k  00  (2.126) where  £  and n  1  k  =  e  2.fe  < ' 2  B  (=Re^* /4.4 5) 1  1 2 7 )  i s the improvised Richardson - Zaki  exponent e v a l u a t e d by u s i n g the Reynolds number, Re, based on 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 n s t e a d o f the  f r e e s e t t l i n g Reynolds number, Re / as suggested by  Richardson and Zaki  p  [2].  The r a t i o o f wake volume to bubble  108 volume,  fij,/fl , B  as determined from the measured v a l u e s o f  e and e 2 u s i n g equations correlated.to f i t following  was then e m p i r i c a l l y  the observed bed voidage data by the  relation:  = 5.Me, r K  2.126 and 2.127,  , 0 D  [l  - tanh  {40 -3-  (e.)  <J  J-  X  B  2^  - '3 . 2 ( e , ) • ^O] 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  b e f o r e the i n t r o d u c t i o n o f the gas. The o t h e r  noteworthy  attempt t o improve  i s due to Rigby and Capes [80].  the wake model  They t e s t e d t h i s model to  f i n d o u t the e f f e c t of assumed p a r t i c l e c o n t e n t o f the wake by c o n s i d e r i n g the two extremes Davidson  [7] and 0stergaard  suggested by Stewart and  [8].  They concluded t h a t the  presence o f p a r t i c l e s i n the wake had a marked i n f l u e n c e on the wake volume, which was a l s o found to be a f f e c t e d by the bed voidage.and  to a l e s s e r degree by the p a r t i c l e  No phenomenological  equation  size.  was presented to c o r r e l a t e  the wake volume w i t h v a r i o u s bed parameters. Other e m p i r i c a l c o r r e l a t i o n s f o r the voidage i n t h r e e phase f l u i d i z e d beds are e i t h e r u n n e c e s s a r i l y complicated o r o u t r i g h t l y m i s l e a d i n g [11], included i n further  [10]  and w i l l t h e r e f o r e not be  discussions.  From the g e n e r a l i z e d wake model f o r three-phase f l u i d i z e d - b e d s presented i n s e c t i o n i s g i v e n by  2.3.1,  the bed voidage  109  e = e  2  +  £ k  ( l - x ) + e[f ( x e + k  k  k  1 - e  2  - e>  (2.91)  k  II  where e ^ . i s g i v e n by e q u a t i o n 2.106.  The terms *  k  and e  k  are the two q u a n t i t i e s f o r which estimates have to be obtained  e x p e r i m e n t a l l y , s i n c e no r e l i a b l e i n f o r m a t i o n con-  c e r n i n g them e x i s t  i n the three-phase f l u i d i z a t i o n  literature.  As has a l r e a d y been p o i n t e d o u t i n s e c t i o n 2.3.1, the improvised wake model takes i n t o wake f o r m a t i o n behind of  c o n s i d e r a t i o n not only  the bubbles and the p a r t i c l e  the wake, b u t a l s o p a r t i c l e  circulation  on the c o n t r a c t i o n - e x p a n s i o n c h a r a c t e r i s t i c s phase f l u i d i z e d bed.  content  and i t s e f f e c t of a three-  110  CHAPTER 3 EXPERIMENTAL  The main aim o f t h i s work was t o e s t 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 p r o p e r t i e s on holdups o f 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. holdup, o r the volume f r a c t i o n of s o l i d s i n s i d e can  be d i r e c t l y c a l c u l a t e d  The s o l i d s  the bed, e , 3  i f the weight o f s o l i d p a r t i c l e s  p r e s e n t i n the bed, W, i s known and i f the expanded bed h e i g h t , L^, can be measured.  e  3  =  W/p AL 3  Thus  (3.1)  b  However, the expanded bed h e i g h t , L^, cannot always be measured d i r e c t l y . - A t low gas flow r a t e s ,  especially  in a  bed  o f l a r g e or heavy p a r t i c l e s , the upper boundary o f the  bed  i s v e r y w e l l d e f i n e d and can be measured e a s i l y by v i s u a l  o b s e r v a t i o n through t r a n s p a r e n t column w a l l s . gas flow r a t e s ,  But a t l a r g e  the upper bed l e v e l i s not so c l e a r l y  d e l i n e a t e d and i t becomes d i f f i c u l t t o e s t a b l i s h bed  height v i s u a l l y .  the expanded  I t i s t h e r e f o r e necessary t o develop  a c r i t e r i o n to d e f i n e the expanded bed h e i g h t  consistently  under a l l c o n d i t i o n s o f gas and l i q u i d flow r a t e s .  Such  a c r i t e r i o n was developed and i s d i s c u s s e d i n d e t a i l i n  Ill Appendix  Thus w i t h the knowledge of W and L^,  8.2.  be c a l c u l a t e d from.equation  can  3.1.  Since  £  1  +  £  2  +  £  3  =  1  *°  (3.2)  becomes known i f the gas holdup i n s i d e the.bed, e ' c o u l d 2  be measured.  Of the v a r i o u s techniques a v a i l a b l e f o r measure-  ment of gas holdup, the f o l l o w i n g two were chosen f o r t h i s work: 1.  d i r e c t v o l u m e t r i c measurements u s i n g q u i c k c l o s i n g valves,  2.  the measurement o f s t a t i c p r e s s u r e drop g r a d i e n t .  A t h i r d technique, t h a t of measuring the l o c a l gas by an e l e c t r o - r e s i s t i v i t y probe, was  fraction  l a t e r developed and  used i n p a r t o f the work. The experimental equipment used was  designed to i n -  c o r p o r a t e these techniques f o r measuring the expanded bed h e i g h t and gas holdup i n t o t h i s study.  3.1  Apparatus The three-phase f l u i d i z a t i o n s t u d i e s were c a r r i e d out  i n two columns:  ( i ) a 20 mm  i n c h i . d . perspex column.  i . d . g l a s s column and  (ii) a 2  The two e x p e r i m e n t a l s e t ups are  d i s c u s s e d s e p a r a t e l y i n the f o l l o w i n g  sections.  112 3.1.1  The  20 mm  The  design of the  the apparatus Andersson  bench top g l a s s column  20 mm  g l a s s c o l u m n was  used f o r l i q u i d - s o l i d  [56] , w i t h  f l u i d i z e d bed  fluidization  suitable modifications for  operation.  The  overall  column c o n s i s t e d o f a 660  g l a s s column w i t h A  a s t r a i g h t 168  60 mesh c o p p e r s c r e e n , S , was  calming to  section D  a c t as a bed  The  support. 6 mm  Two  a centrifugal  liquid  The  i-.d.  section.  cm  the  also apart,  , t o measure  bed.  f e e d t a n k was  a 1/15  main  s e c t i o n E and  U - t u b e manometer,  pump d r i v e n by  equip-  to separate  t a p s , 4 9.3  by  three-phase  long entrance  pressure  from the  studies  l o n g , 2 0 mm  push-fitted  p r e s s u r e drop a c r o s s the test  mm  from the e x p e r i m e n t a l  were p r o v i d e d w i t h a the s t a t i c  mm  on  l a y o u t of the  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. experimental  based  circulated  h o r s e p o w e r .motor.  by  A  b y p a s s was  p r o v i d e d t o r e g u l a t e t h e f l o w and  a needle  valve  to  the  column.  The  control  liquid R-^.  f l o w r a t e was  The  overflow to  the  flow r a t e  liquid  m e a s u r e d by  i n t o an  exit  s e c t i o n , X,  feed t a n k , thus  kept c l o s e  was  completing  l i q u i d was  u s e d t o keep t h e  liquid  allowed  f r o m w h i c h i t was the  liquid  measured by  l i q u i d , an  to  returned  cycle.  The  a thermometer, adding  H o w e v e r , when t h e  u s e d as t h e t e s t  rotameter,  c o l u m n was  t o t h e room t e m p e r a t u r e by  tap water o c c a s i o n a l l y . s o l u t i o n was  a calibrated  from the experimental  temperature of the and  to the experimental  water  some  T,  fresh  glycerol  immersion c o o l e r  t e m p e r a t u r e w i t h i n ± 1°F o f  the  FIGURE 3.1  SCHEMATIC DIAGRAM OF 20 MM GLASS COLUMN APPARATUS  LEGEND FOR FIGURE 3.1  a i r source  (85 p s i g )  buffer bottle 1 mm g l a s s 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 flask  three-way g l a s s ground perspex  stop-cocks  ball  carbon t e t r a c h l o r i d e U-tube manometer open mercury manometer pressure  regulator  rotameters 60 mesh copper  s c r e e n bed support  thermometer liquid exit  reservoir  section  115 room temperature. t u r n o f f the l i q u i d  The three-way stopcock  was used to  flow.  The a i r supply was obtained from the l a b o r a t o r y o u t l e t through a f i l t e r - r e d u c e r v a l v e assembly, P.  The a i r  flow r a t e to the column was c a r e f u l l y c o n t r o l l e d by a needle v a l u e and measured by the c a l i b r a t e d rotameter, R . 2  The  a i r entered a t the base of the g l a s s column through a 5 cm l o n g , 1 mm g l a s s c a p i l l a r y , h e l d i n a v e r t i c a l p o s i t i o n c l o s e to the column a x i s by a spacer f i x e d to the column w a l l . A 5 cm l o n g , 1/2 mm g l a s s c a p i l l a r y was a l s o used f o r the low a i r flow r a t e s s t u d i e d .  especially  To damp o u t the  f l u c t u a t i o n s i n the a i r l i n e a damper b o t t l e , B, was  used.  The p r e s s u r e a t which the a i r was s u p p l i e d to the column was measured by the open mercury manometer, M . 2  stop-cock, G , was used 2  The three-way  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 q u i t e snugly i n t o a ground g l a s s j o i n t , was used as a s t o p - v a l v e to i s o l a t e the experimental s e c t i o n , once the gas and the l i q u i d flows had been c u t o f f .  The gas i n s e c t i o n D  c o l l e c t e d below t h e . s c r e e n S and the gas i n the experimental s e c t i o n c o l l e c t e d near the top o f the g l a s s column. the l a t t e r r e a d i n g was r e c o r d e d .  Only  The d e t a i l s o f the 20 mm  g l a s s column a r e g i v e n i n F i g u r e 3.2.  ~7V 116 CO  7F  06 2-0  co CO  T  ,60 MESH  COPPER  SCREEN  LO CO  CD  06  2-5cm GROUND P E R S P E X  C\J  FIGURE 3.2-  THE 20 MM GLASS COLUMN  BALL  117 3.1.2  The 2 i n c h perspex column The main bulk o f the three-phase f l u i d i z a t i o n  study  was c a r r i e d o u t i n a 2 i n c h perspex column, the schematic drawing o f which i s g i v e n i n F i g u r e 3.3.  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 The d e t a i l s o f the bulk o f the equipment have been  g i v e n by L e G l a i r [101], who designed same apparatus f o r an e a r l i e r study.  and used most o f the T h e r e f o r e o n l y the main  f e a t u r e s o f the equipment a r e d i s c u s s e d here. c i r c u l a t i o n loop i s made from 152  seamless copper t u b i n g .  i n c h long s t r a i g h t r u n of the 2 i n c h copper t u b i n g  ceding the experimental The  The l i q u i d A pre-  s e c t i o n a c t s as the calming s e c t i o n .  t e s t l i q u i d i s 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 3 horse-power motor.  A bypass i s p r o v i d e d t o r e g u l a t e  the p r e s s u r e a t which the l i q u i d  i s pumped.  With water  as the t e s t l i q u i d , the s e t t i n g of the bypass v a l v e i s n o t important,  b u t when p o l y e t h y l e n e - g l y c o l s o l u t i o n i s used,  the s e t t i n g i s found  t o be c r i t i c a l s i n c e an e x c e s s i v e  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 q u i t e frothy.  T h e r e f o r e the opening o f the bypass v a l v e was so  r e g u l a t e d as to m a i n t a i n a pump d e l i v e r y p r e s s u r e o f over 4 0 p s i g when u s i n g p o l y e t h y l e n e - g l y c o l s o l u t i o n as the t e s t liquid.  The l i q u i d from the pump flows through a heat  exchanger where i t i s cooled t o m a i n t a i n a steady  temperature  118  X  B, T  0  0, 0,  L,  feed tank  heat exchanger pump  N IV  s  t^Xj— FIGURE 3.3  -SCHEMATIC DIAGRAM OF 2 INCH PERSPEX COLUMN APPARATUS  119 LEGEND FOR FIGURE 3.3  A  -  A i r source  (35 p s i g )  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  G  -  P r e s s u r e gauge  I  -  G l a s s tube l e v e l i n d i c a t o r  N  -  A i r i n l e t cone  O^/OpyO^ P R  -  -  O r i f i c e meters  A i r f i l t e r and p r e s s u r e  1' 2  ~  R  -  Entry section  T  -  Thermometer  X  -  Exit section  -  regulator  Rotameters  S  Lj  test section  Lever arm p o s i t i o n when b a l l v a l v e s a r e f u l l y open  L  c  -  Lever arm p o s i t i o n when b a l l v a l v e s a r e f u l l y 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 a t a l o c a t i o n downstream from the measuring by a thermometer, T.  station  The l i q u i d then e n t e r s the main  experimental column through an annular e n t r y s e c t i o n ,  S.  The l i q u i d from the experimental column overflows i n t o e x i t s e c t i o n , X, whence i t i s r e t u r n e d to the feed thus completing  the l i q u i d  the  tank,  cycle.  The l i q u i d flow r a t e to the column i s measured a t the measuring s t a t i o n , e i t h e r by one of the three o r i f i c e meters O^,  C>2 o r 0^ / o r by the c a p i l l a r y - t u b e meter, C.  The c a l i b r a t i o n curves f o r these flow meters are g i v e n i n Appendix  8.4.  The diameter  t e s t section c o n s i s t s of a 5 f t long 2 i n . i n s i d e  perspex  tube.  A l l along the t e s t s e c t i o n  carefully  d r i l l e d p r e s s u r e taps are p r o v i d e d , each of which houses a c a r e f u l l y shaped 1/4 opening  i n c h copper  tube w i t h a 1/16  inch  i n t o the column, to f i t f l u s h w i t h the i n s i d e of  the t e s t s e c t i o n . 100 cm l o n g , 8 mm m a n i f o l d system  The p r e s s u r e taps are connected  i . d . U-tube manometer through a p r e s s u r e (Figure 3.4)  which permits the p r e s s u r e  to be measured between any two  taps.  dyed w i t h a c r y s t a l o f potassium  permanganate  manometer was  used f o r the r e s t .  drop  Carbon t e t r a c h l o r i d e was  the manometric f l u i d f o r most of the study, w h i l e bromo-ethane was  to a  used  as  tetra-  An open mercury  a l s o l o c a t e d i n the t e s t s e c t i o n to measure  the a b s o l u t e p r e s s u r e i n the column,  s  A screen t r a p i s clamped on to the opening of the l i q u i d r e t u r n l i n e i n t o the feed tank, t o c a t c h 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  OPEN MERCURY MANOMETER  --DATUM  0  DISTANCE IN INCHES FROM DATUM  FIGURE 3.4  PRESSURE DROP APPARATUS FOR MEASURING LONGITUDINAL PRESSURE DROP PROFILE IN THE EXPERIMENTAL SECTION  122 The  t e s t s e c t i o n i s separated from the calming  by a 60 mesh copper  section  screen h e l d i n the r e c e s s of a rubber  gasket, which i n t u r n i s h e l d between two  flanges.  Since  t h i s screen a l s o acted as the bed support, f o r f l u i d i z i n g 0.25  mm  g l a s s beads another f i n e r screen  (100 mesh) was  used  on top o f the 60 mesh screen to prevent the s m a l l p a r t i c l e s from - f a l l i n g through. were used  Two  2 i n c h f u l l bore b a l l v a l v e s  to t r a p the f l o w i n g mixture i n the t e s t s e c t i o n by  c l o s i n g them s i m u l t a n e o u s l y .  One v a l v e was  located 5 feet,  below the t e s t s e c t i o n and the o t h e r a t the top o f the t e s t section.  The  two v a l v e s were connected  through  by a l i n k rod and c o u l d be shut completely and by q u i c k l y r o t a t i n g the b a l l s through  l e v e r arms simultaneously  90° v i a the l i n k r o d .  C a r e f u l l y d r i l l e d taps were p r o v i d e d i n the s e c t i o n below the t e s t s e c t i o n f o r 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 s e c t i o n f o r s t a t i c p r e s s u r e drop measurements. taps.  F i g u r e 3.5  Again carbon t e t r a - c h l o r i d e dyed w i t h potassium  manganate was 3.1.2.2  used as the manometric  Gas c y c l e and bubble  Air  was  through a 1/2 12-16  shows the l o c a t i o n of these p r e s s u r e  fluid.  nozzle  taken from the l a b o r a t o r y supply a t 35 p s i g i n c h copper  l i n e and reduced  p s i g by a p r e s s u r e r e g u l a t o r and  which maintained desired l e v e l .  per-  the reduced  to a p r e s s u r e of  filter  assembly,  P,  supply p r e s s u r e c o n s t a n t a t any  T h i s p r e s s u r e was  read downstream from  the  123  IO  CO  -ixi-  ro in  4*.  Hr ALL  ±1  DIMENSIONS IN CM.  FIGURE 3.5  -LOCATION OF PRESSURE TAPS AND BALL VALVES FOR GAS HOLDUP MEASUREMENTS  12 4 measuring s t a t i o n on a Bourdon tube type p r e s s u r e gauge, G. A i r was  then brought  column through a 1/2  to the bottom of the main experimental i n c h copper  the gas d i s t r i b u t o r , N. column a t the top was  l i n e and admitted  The a i r l e a v i n g the  through  experimental  vented to the atmosphere.  The a i r flow r a t e to the experimental column  was  measured by e i t h e r of the two c a l i b r a t e d rotameters, R. 2  Yet another rotameter was  used f o r p a r t o f t h i s  to measure v e r y s m a l l gas flow r a t e s .  study  The c a l i b r a t i o n  f o r these rotameters are g i v e n i n Appendix  curves  8.5.  The gas entered the column through a gas the d e t a i l s of which are g i v e n i n F i g u r e 3.6. bubble n o z z l e , N, was  and  distributor, The main  turned from a b r a s s b l o c k to house  v a r i o u s gas d i s t r i b u t o r s t h a t can be screwed on to i t .  In  o r d e r to d i s t r i b u t e the gas u n i f o r m l y , a p e r f o r a t e d  1/4  i n c h t h i c k perspex  square  cm  [4]  was  p l a t e d i s t r i b u t o r w i t h 1 h o l e per  designed.  Similar perforated plate  distri-  b u t o r s w i t h fewer h o l e s were a l s o designed i n order to check any e f f e c t o f the gas d i s t r i b u t o r d e s i g n on the i n two-phase g a s - l i q u i d flow.  gas;,:holdup  Preliminary investigations  r e v e a l e d 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 t h e r e f o r e a p e r f o r a t e d 1/4 d i s t r i b u t o r w i t h f o u r 1/16  i n c h t h i c k perspex  i n c h h o l e s was  plate  used i n most of  the s t u d i e s . The gas d i s t r i b u t o r was  l o c a t e d 12 f e e t below the bed  support'screen with-the. hopeirithat the flow "and gas  distribution  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 t e s t  section.  V i s u a l o b s e r v a t i o n of the t e s t s e c t i o n , however, showed bubble c o a l e s c e n c e o c c u r r i n g a t d i f f e r e n t gas flow r a t e s . I t was  t h e r e f o r e c o n s i d e r e d d o u b t f u l t h a t the gas  b u t i o n p r o f i l e s were f u l l y developed. gas d i s t r i b u t o r was  distri-  N e v e r t h e l e s s the  l e f t a t the f o o t o f the column through-  out the e n t i r e study s i n c e i t p r o v i d e d a two-phase gasl i q u i d r e g i o n p r e c e d i n g the three-phase f l u i d i z e d region; since a g a s - l i q u i d  bed  zone a l s o f o l l o w e d the f l u i d i z e d  bed, the e f f e c t o f the presence o f s o l i d p a r t i c l e s i n the t e s t s e c t i o n on the gas holdup i n the two-phase r e g i o n above i t c o u l d t h e r e f o r e be  3.1.3  Electro-resistivity  determined.  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] f o r measuring  gas f r a c t i o n i n mercury-nitrogen f l o w .  the l o c a l v o l u m e t r i c The s e n s i n g element  of t h e i r probe c o n s i s t e d o f an i n s u l a t e d sewing needle w i t h i t s exposed  t i p p o i n t i n g i n t o the f l o w .  s u p p l i e d w i t h a D.C.  The probe  p o t e n t i a l and grounded  continuous phase to complete s c h e m a t i c a l l y i n F i g u r e 3.7.  was  through the  the c i r c u i t , which i s shown 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 , r e s u l t e d i n a n e a r l y square wave o u t p u t .  which  Nassos and Bankoff  [104] t e s t e d 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-  200 K  4 00 K  FIGURE 3.7  Probe  O/P  CIRCUIT DIAGRAM FOR ELECTRO-RESISTIVITY PROBE  128 water flow and found t h a t , due t o d e f l e c t i o n o f bubbles away from the probe, the average gas f r a c t i o n o b t a i n e d by i n t e g r a t i n g the l o c a l gas f r a c t i o n p r o f i l e was s m a l l e r than the v a l u e o b t a i n e d by s t a t i c p r e s s u r e drop measurements. Some m o d i f i c a t i o n s were suggested to improve  the agreement  [104], b u t s i n c e the d e f l e c t i o n of a bubble from a p o i n t e d sensing element remained as a b a s i c problem, i t was d e c i d e d to change the d e s i g n o f the probe s l i g h t l y f o r the p r e s e n t study. The e l e c t r o - r e s i s t i v i t y probe used i n t h i s study c o n s i s t e d o f two e l e c t r o d e s h e l d a t a s m a l l b u t f i x e d distance apart.  The a r r i v a l o f an i n d i v i d u a l bubble i s sensed  by the passage o f the bubble through the gap.  Although the  probe supports c o u l d d e f l e c t a bubble i n t o o r away from the gap, i t was n e v e r t h e l e s s b e l i e v e d t h a t the p r o b a b i l i t y o f r e g i s t e r i n g an impinging bubble would be i n c r e a s e d over t h a t of the o r i g i n a l needle probe.  Since the diameter o f the  bubbles encountered i n the g a s - l i q u i d flow study was always l a r g e r 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 l u t i o n o f the probe c o u l d be c o n s i d e r e d good. However, it  i s b e l i e v e d t h a t a q u i c k p e n e t r a t i o n o f the bubble on  impingement remains a problem and would become a major source of e r r o r when the probe i s used i n more v i s c o u s  liquids.  The probe used was o r i g i n a l l y a m i n i a t u r e h o t - f i l m probe  (1270-20W-6) s u p p l i e d by Thermo Systems Inc., from  which the h o t - f i l m f i l a m e n t was c a r e f u l l y c u t o f f so as to  expose the two  e l e c t r o d e s , l e a v i n g a gap  t a i l s of the probe are g i v e n  i n Figure  of 1 mm.  3.8.  The  The  support  needles are epoxy coated to i n s u l a t e them from the phase.  The  probe was  de-  continuous  mounted i n the experimental s e c t i o n E  through a t r a v e r s i n g mechanism to a l l o w f o r a r a d i a l of the column to p o s i t i o n s very  traverse  c l o s e to the column w a l l s .  The d e t a i l s of the mounting mechanism are shown i n F i g u r e 3.8. D.C. was  One  o f the e l e c t r o d e s was  maintained a t a  p o t e n t i a l w i t h r e s p e c t to the other  e l e c t r o d e , which  grounded through a 5 meter c o a x i a l c a b l e .  applied  (2-3 v o l t s ) was  so a d j u s t e d  an amplitude o f about 0.22  The  a 100,000  of  ohm  e l e c t r o n i c c i r c u i t used  to analyse the probe s i g n a l i s d e s c r i b e d  3.1.4  The p o t e n t i a l  as to produce p u l s e s  v o l t s across  r e s i s t o r connected i n s e r i e s .  constant  i n the next s e c t i o n .  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 o f l o c a l gas holdup and  bubble frequency  Before d i s c u s s i n g the c i r c u i t s used, i t i s important to c l e a r l y d e f i n e the v a r i a b l e s being  measured.  Q u a n t i t i e s measured (a) l o c a l gas f r a c t i o n The  l o c a l volumetric r - -  gas  fraction  is  defined  '  as the p r o b a b i l i t y t h a t gas w i l l e x i s t a t a p o i n t under consideration. (quasi-steady the gas  For flow w i t h s t a t i o n a r y time-averaged  properties  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 o f time  e x i s t s at that point  [103] .  Thus  DETAIL OF A (SUPPORT  NEEDLES)  152 ~7  7.25  Mi  l  TVVV  1< 6 - 3 >|  <  19  >  ALL FIGURE 3.8  DIMENSIONS IN  mm  ELECTRO-RESISTIVITY PROBE AND MOUNT FOR TRAVERSING MECHANISM  131  a  where t  2  2r  =  fc  2  /T  i s the time the probe i s exposed  and T i s the t o t a l sample i n t e r v a l .  ( 3 , 3 )  to the gas phase  In o r d e r to o b t a i n a  t r u e s t a t i s t i c a l average, the sample i n t e r v a l must be compared to the time s c a l e of flow o s c i l l a t i o n s , where n  1  i s the l o c a l bubble frequency.  steady flow, the l o c a l gas f r a c t i o n  a2r  1 |  =  large  1/n , 1  Thus,for a q u a s i -  can be expressed as  N  I t i=l  (3.4)  ±  In o r d e r t h a t the gas f r a c t i o n measured l o c a l l y by t h i s technique c o u l d be compared w i t h 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 p r e s s u r e drop g r a d i e n t , a t r a v e r s e of  the probe was made to o b t a i n 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 i n t e g r a t e d over the  c r o s s - s e c t i o n to p r o v i d e the o v e r a l l  average gas  fraction,  as g i v e n by  <a2>  where R  =  e2  =  2  y  1  a2r  R  *  dR  *  (3.5)  i s the d i m e n s i o n l e s s d i s t a n c e from the c e n t e r o f  the p i p e . (b) bubble  frequency  The bubble frequency a t a p o i n t , n^,, i s d e f i n e d as the number o f bubbles p a s s i n g through t h a t p o i n t per u n i t  time:  132  n  N/T  r  (3.6)  where N i s the t o t a l number o f bubbles t h a t pass through the p o i n t i n time T.  The time T must be l o n g enough t o o b t a i n  a r e p r e s e n t a t i v e sample, which i m p l i e s t h a t N>>1.  Normally  100 - 1000 bubbles, depending on the r a d i a l l o c a t i o n o f the probe, were counted i n o r d e r to o b t a i n the bubble frequency. A n a l y s i s o f the probe  signal  A simple e l e c t r o n i c analogue l o g i c c i r c u i t was designed to o b t a i n these q u a n t i t i e s from the probe s i g n a l and i s shown s c h e m a t i c a l l y i n F i g u r e 3.9.  The p r i n c i p a l component o f 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 t o t r i g g e r p u l s e s o f width equal to the r e s i d e n c e time of an i n d i v i d u a l bubble, u t i l i z i n g  the f o l l o w i n g  character-  i s t i c o f the comparator:  X  Y  •  •  NON-INVERTING I/P INVERTING  I/P IF X + Y > 0 , Z = I IF X + Y < 0 , Z = 0  • Z  IOK P85AU  20K  •MWV  Darcy Frequency f—| Meier  1  100 K 20K —wvw-  HOOK  P45ALU  P35A  |sP6o6  :IOOK  P35AU  P35A P35AU P85AU P45A P45ALU SP656  FIGURE 3.9  — -• —  10 K  iVWSV  1  -I5V  Philbrick solid state amplifier Philbrick solid state amplifier Philbrick solid state amplifier Philbrick solid state amplifier Philbrick solid state amplifier Philbrick photochopper stabilized solid state amplifier  ^XA7I0 ^ -=•  — — —  Fairchild logic differential comparator To high quality ground To power common  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 u n i f o r m a m p l i t u d e t h u s t r i g g e r e d by t h e comp a r a t o r were t h e n i n t e g r a t e d probe  i s exposed  f r a c t i o n was  t o t h e gas phase/  calculated  c i r c u i t preceding probe  to obtain  from which  time the  the l o c a l  by means o f e q u a t i o n 3.4.  gas  The  t h e c o m p a r a t o r was d e s i g n e d t o a m p l i f y t h e  s i g n a l , b u t most i m p o r t a n t l y  circuit  the t o t a l  to isolate  the measuring  f r o m t h e p r o b e , s o as n o t t o c r e a t e any f e e d b a c k s  [105] . The b u b b l e f r e q u e n c y was o b t a i n e d b y c o u n t i n g t h e number o f p u l s e s  t r i g g e r e d by t h e c o m p a r a t o r  frequency counter f o r a f i x e d  on a Darcy  t i m e o f 10 s e c o n d s .  The  total  number o f p u l s e s c o u n t e d were t h e n r e a d f r o m t h e e l e c t r o n i c d i s p l a y of the counter.  Alternatively, a strip  r e c o r d e r was u s e d t o r e c o r d  the comparator  chart  output.  The  number o f p u l s e s were t h e n c o u n t e d f r o m t h e r e c o r d i n g o f over a minute. f r e q u e n c y was  E i t h e r method o f o b t a i n i n g f o u n d t o be s a t i s f a c t o r y  the bubble  and u s e d  c h a n g e a b l y , depending on t h e a v a i l a b i l i t y  o f t h e equipment,.  The b u b b l e f r e q u e n c y m e a s u r e m e n t s were u s e d estimate o f average bubble s i z e method p r e s e n t e d i n A p p e n d i x  3.2  Range o f v a r i a b l e s  The was d i v i d e d  two  t o o b t a i n an  i n t h e t e s t s e c t i o n by t h e  8.3.  studied  e x p e r i m e n t a l programme into  inter-  parts:  for collecting  the data  135 (A) The study o f gas holdup i n two-phase g a s - l i q u i d f l o w , and (B) The study of s o l i d s and gas holdup i n a three-phase fluidized  bed.  The main experimental programme was  c a r r i e d o u t i n the 2 i n c h  i . d . perspex column l o c a t e d i n a 2 i n c h diameter f o r c e d c i r c u l a t i o n loop.  However, the 20 mm  i . d . g l a s s column  was  used to c a r r y out.a p r e l i m i n a r y study to e s t a b l i s h the r e l e v a n c e o f v a r i o u s parameters i n v o l v e d . l i m i t e d amount o f d a t a was a p p r e c i a b l e range was  Although o n l y a  o b t a i n e d i n the l a t t e r ,  an  i n v e s t i g a t e d and t h e r e f o r e the r e s u l t s  o b t a i n e d are i n c l u d e d . (A) Gas holdup i n two-phase g a s - l i q u i d flow The need to study two-phase g a s - l i q u i d flow arose from the l a c k o f e s t a b l i s h e d and r e l i a b l e methods to p r e d i c t the gas holdup f o r such f l o w .  The purpose o f 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 o f the mathematical model proposed i n s e c t i o n 2.1.2, and ( i i ) to o b t a i n data t h a t c o u l d be used l a t e r f o r comparing w i t h the d a t a on gas holdup i n t h r e e 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 e i t h e r the coalescence o r breakup of bubbles i n three-phase fluidized  beds.  136 Therefore and  3.2  the scope of t h i s study was  l i m i t e d , Tables  summarizing the range of v a r i a b l e s  (B) S o l i d s and  3.1  studied.  gas holdup i n a three-phase f l u i d i z e d  bed  As has been o u t l i n e d 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  p r o p e r t i e s on the i n d i v i d u a l gas, in The  a three-phase f l u i d i z e d bed,  l i q u i d and  solid-phase solid  holdups  f o r a wide range of  c h o i c e o f f l u i d s s e l e c t e d f o r t h i s study was  conditions.  guided  by  the f i n d i n g s i n c o r r e s p o n d i n g two-phase g a s - l i q u i d s t u d i e s . Thus, a i r was  conveniently  chosen as the gas  phase and  throughout the study, s i n c e i t has been shown [108] the p r o p e r t i e s of the gas  phase had  under normal atmospheric c o n d i t i o n s .  used  that  l i t t l e or no e f f e c t Ordinary  tap water  was  used as the l i q u i d phase f o r the most p a r t , so t h a t the data c o l l e c t e d i n t h i s study c o u l d be compared w i t h investigations.  In the l a t e r p a r t o f the work, an aqueous  polyethylene-glycol  s o l u t i o n was  e f f e c t of l i q u i d v i s c o s i t y . was  earlier  The  used to i n v e s t i g a t e  the  polyethylene-glycol  solution  chosen because i t i s a very v i s c o u s l i q u i d , the Newtonian  behaviour of which has been v e r i f i e d d e n s i t y and  [101] , and  because i t s  s u r f a c e t e n s i o n are o n l y a l i t t l e d i f f e r e n t from  t h a t o f water.  For  the s o l i d phase, e q u i - s i z e d s p h e r i c a l  g l a s s beads, l e a d shot and  s t e e l b a l l b e a r i n g s were chosen  to g i v e 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 Velocity, (cm/sec)  Gas Velocity, j  2  (cm/sec) 0.0 - 18.0  Liquid V i s c o s i t y , y, (cp) 1  5.0 - 18 .0  1.0  Gas Holdup, £~ Z (-)  0.20 - 0.39  TABLE 3.2 EXPERIMENTAL CONDITIONS FOR TWO-PHASE GAS-LIQUID FLOW IN 2 INCH PERSPEX COLUMN  Liquid Gas Velocity, j . Velocity, J (cm/sec) (cm/sec)  0.0 - 19.0  1.5 - 13.0  2  Liquid Gas V i s c o s i t y , y, Holdup,£1 (c ) (-) P  Flow Regime  1.0 & 69.0  bubbleslug  1  0.05 - 0.28  138 Tables 3.3 and 3.4 i n both the 20 mm  l i s t the range of v a r i a b l e s s t u d i e d  g l a s s column and the 2 i n c h perspex  column.  3.3  E x p e r i m e n t a l procedure The experimental procedure used to o b t a i n data i n  the 20 mm  g l a s s column and the 2 i n c h perspex column were  essentially  similar.  adopted are d e s c r i b e d  3.3.1  The s a l i e n t f e a t u r e s o f the procedure i n the f o l l o w i n g s e c t i o n s .  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 For the major p a r t o f t h i s work water was  test liquid.  used as the  O r d i n a r y tap water c o n t a i n i n g 0.2% by weight  sodium dichromate and 0.05% corrosion inhibitors  by weight sodium hydroxide as  [101] was  i n the 2 i n c h perspex column.  t r i e d f o r the e a r l y runs S i n c e the a d d i t i v e s d i d not  i n h i b i t c o r r o s i o n as e f f e c t i v e l y as had been hoped f o r , o r d i n a r y tap water w i t h o u t a d d i t i v e s was used t h e r e a f t e r . T h i s r e q u i r e d f r e q u e n t c l e a n i n g o f the mercury manometer t r a p s and the copper bed support s c r e e n . the 20 mm  For the s t u d i e s i n  g l a s s column o r d i n a r y tap water was used w i t h o u t  any problems.  The d e n s i t y of the water was  checked  a l l y , but i n the f i n a l p r o c e s s i n g o f the d a t a  occasion-  collected,  both the d e n s i t y and v i s c o s i t y of water were o b t a i n e d from Perry  [106] .  S u r f a c e t e n s i o n too was measured f o r the e a r l y  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 Velocity, (cm/sec)  1.7 - 8.1  Gas Velocity,J (cm/sec)  0.2 - 8.2  2  Liquid Viscosity,y (cp)  1  1  1.0 & 2.1  Particle Solids Diameter,d D e n s i t y , (mm) P (gm/cc)  2.5 - 3.0  0.5 - 1.0  Solids Holdup, £_ (-)  Gas Holdup, e •(-)  0.5 - 0.2  0.05 - 0.15  3  0  TABLE 3.4 EXPERIMENTAL CONDITIONS FOR THREE-PHASE FLUIDIZATION IN 2 INCH PERSPEX COLUMN Liquid Velocity,j. (cm/sec)  Liquid Gas Velocity,j2 Viscosity,u (cm/sec) (cp)  0.4 - 39.0  0.4 - 21.0  1.0 & 63.3  1  Particle Diameter,d (mm)  p  0.25 - 3.2  Solids Solids D e n s i t y , p _ Holdup, (gm/cc) (-)  3  2.9 - 11.1 0.5 - 0.1  Gas Holdup, £„ (-)  Flow Regime  2  0.05 - 0.25  bubbleslug  H1 CO VD  140 The v i s c o s i t y o f the p o l y e t h y l e n e g l y c o l  solution  was measured by a Cannon Viscometer (H-304) which was c a l i b r a t e d w i t h ASTM Standard O i l No. S-20 and No. S-60, a c c o r d i n g to the procedure recommended i n the ASTM manual (D445-53T).  The v i s c o s i t y o f the s o l u t i o n was then measured  by the c a l i b r a t e d v i s c o m e t e r , f o l l o w i n g the procedure recommended, and these measurements too a r e r e p o r t e d i n Appendix  8.6.  A p l o t o f dynamic v i s c o s i t y a g a i n s t the  i n v e r s e o f the a b s o l u t e temperature i s presented as F i g u r e 8.6.1 o f Appendix 8.6.  T h i s p l o t was used to o b t a i n the  v i s c o s i t y of the s o l u t i o n a t the measured temperature i n the  f i n a l a n a l y s i s o f the d a t a . The s u r f a c e t e n s i o n of p o l y e t h y l e n e g l y c o l  was a l s o checked and was found t o be 63 dynes/em. i t i s n o t v e r y d i f f e r e n t from t h a t o f pure water  solution Since (70 dynes/cm)*  no f u r t h e r measurements o f the s u r f a c e t e n s i o n were made or  reported.  3.3.2  P h y s i c a l p r o p e r t i e s o f the s o l i d s used G l a s s beads o f three d i f f e r e n t s i z e s , 0.25, 0.5 and  1.0 mm,  l e a d shot, and s t e e l b a l l b e a r i n g s were used f o r  s t u d i e s i n the.2 i n c h perspex column; washed g r a n u l a r sand and 1.0 mm g l a s s beads were used f o r s t u d i e s i n the 20 mm g l a s s column.  F o r a l l the g l a s s beads, l e a d shot and  washed sand, a c a r e f u l l y screened c u t was s e l e c t e d from the s c r e e n a n a l y s i s and the average p a r t i c l e s i z e was taken  as the a r i t h m e t i c mean of the two c o n s e c u t i v e s i e v e The diameter o f l e a d shot was diameter o f some 50 randomly meter.  The chrome-plated  sizes.  a l s o checked by measuring  the  chosen p a r t i c l e s by a m i c r o -  s t e e l b a l l b e a r i n g s were of  p r e c i s e l y ground grade; t h e r e f o r e the quoted diameter taken as the s i z e o f the p a r t i c l e s .  A random check on the  diameter o f a few s t e e l b a l l s w i t h a micrometer d i f f e r e n c e i n s i z e from the quoted  was  showed no  diameter.  The d e n s i t y o f g l a s s beads and sand was measured by the s p e c i f i c g r a v i t y 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 p l a c e d i n a 10 ml s p e c i f i c g r a v i t y and weighed c a r e f u l l y on a b a l a n c e .  The b o t t l e was  bottle then  c a r e f u l l y f i l l e d w i t h d i s t i l l e d water to the mark and weighed a g a i n .  The d e n s i t y of the d i s t i l l e d water  measured s e p a r a t e l y i n another 10 ml, bottle.  specific  The d e n s i t y o f the p a r t i c l e s was  was  gravity  then c a l c u l a t e d  from these measurements and i s r e p o r t e d i n Appendix  8.6.  The d e n s i t y of the l e a d shot and the s t e e l b a l l s measured by weighing some 50 randomly  selected  was  particles  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 a d j u s t e d balance, and i n the case of the l e a d shot by measuring  the  p a r t i c l e s i z e i n two p e r p e n d i c u l a r d i r e c t i o n s w i t h a m i c r o meter.  The d e n s i t i e s c a l c u l a t e d from these measurements  are r e p o r t e d i n Appendix  8.6.  142 The d e n s i t y o f the 25-75 g l y c e r o l - w a t e r s o l u t i o n  used  i n the 20 mm g l a s s column was measured both b e f o r e and a f t e r each r u n .  Since the measured d e n s i t i e s agreed w i t h the  p u b l i s h e d v a l u e s , the d e n s i t y o f the s o l u t i o n used was subsequently obtained from P e r r y  [106] .  The v i s c o s i t y o f the  25-75 g l y c e r o l water s o l u t i o n was taken from Mathur  [107] and  i s r e p o r t e d i n Appendix 8.6. A 33% by weight s o l u t i o n o f p o l y e t h y l e n e g l y c o l i n water was used f o r the measurements i n the 2 i n c h perspex column.  The s o l u t i o n was found t o be q u i t e a c i d i c and i t  corroded the mechanical was  s e a l s o f the c e n t r i f u g a l pump. I t  then decided t o n e u t r a l i z e the s o l u t i o n w i t h a d i l u t e  s o l u t i o n o f sodium hydroxide; 0.2% by weight o f sodium dichromate  was a l s o added to i n h i b i t c o r r o s i o n .  I t was a l s o  found t h a t f o r r e s t a r t i n g the pump a f t e r a l o n g shut-down, the mechanical  s e a l s should be thoroughly washed w i t h f r e s h water  so as to remove from them any d e p o s i t s o f s o l i d i f i e d p o l y ethylene g l y c o l .  The c l e a r orange-yellow  s o l u t i o n turned  dark brown w i t h usage and was r e p l a c e d w i t h f r e s h  solution.  I t was found t h a t the d e t e r i o r a t i o n i n c o l o u r o f the c l e a r s o l u t i o n was due t o the suspended c o r r o s i o n p r o d u c t s .  I f the  s o l u t i o n was allowed to stand u n d i s t u r b e d , i t became c l e a r once a g a i n as the c o r r o s i o n products s e t t l e d o u t . The d e n s i t y o f t h e p o l y e t h y l e n e g l y c o l s o l u t i o n was measured by a s p e c i f i c g r a v i t y b o t t l e , and t h e measurements are r e p o r t e d i n Appendix 8.6.  143 3.3.3  Measurement o f gas holdup  i n gas-liquid  For s t a r t i n g a run, the l i q u i d was column u n t i l a c o n s t a n t temperature  was  flow  c i r c u l a t e d i n the  achieved and  All  the runs were conducted  a t about the room  All  the manometer taps i n the experimental  noted.  temperature.  section,  and  above and below the experimental s e c t i o n , were c a r e f u l l y f l u s h e d to remove any a i r bubbles The l i q u i d  flow r a t e was  d e s i r e d v e l o c i t y through s o l u t i o n was  i n the c o n n e c t i n g  lines.  then a d j u s t e d t o o b t a i n the the column.  When p o l y e t h y l e n e g l y c o l  used as the t e s t l i q u i d ,  static  pressure  drop  r e a d i n g s were taken on a l l the manometers i n order to determine  the f r i c t i o n a l p r e s s u r e drop i n s i n g l e phase flow.  The a i r was  then i n t r o d u c e d by p r e s s u r i z i n g the a i r  l i n e , and the back-pressure was  so a d j u s t e d t h a t no  t i o n s i n the rotameter r e a d i n g were o b s e r v a b l e . p r e s s u r e i n the a i r l i n e was r a t e was  recorded.  fluctua-  T h i s back-,  The l i q u i d  flow  once a g a i n a d j u s t e d to the d e s i r e d flow r a t e  the s t a t i c  p r e s s u r e drop measurements along the  and  experimental  s e c t i o n , as w e l l as above and below i t , were r e c o r d e d . a b s o l u t e p r e s s u r e near the top of the experimental was  The  section  recorded w i t h the help of the open mercury manometer.  The v i s u a l o b s e r v a t i o n s o f bubble regime encountered  s i z e d i s t r i b u t i o n and  flow  were a l s o r e c o r d e d .  The two b a l l v a l v e s were then shut o f f by a c t u a t i n g the l i n k rod connecting them.  manually  The gas flow  was  144 c u t o f f by v e n t i n g the a i r to the atmosphere and flow by s w i t c h i n g o f f the motor. i n the experimental  s e c t i o n was  The  the  liquid  settled l i q u i d height  measured d i r e c t l y , t h a t i n  the s e c t i o n below i t by n o t i n g the l i q u i d l e v e l i n the g l a s s tube i n d i c a t o r experimental  (see F i g u r e 3.3)  and  t h a t above the  s e c t i o n by a d i r e c t d i p - r o d measurement.  a b s o l u t e p r e s s u r e near the top of the experimental was  section  once a g a i n recorded w i t h the h e l p o f the open mercury  manometer. was  The  The b a l l v a l v e a t the top of experimental  section  then opened and the s e t t l e d l i q u i d h e i g h t below the  experimental copper  s e c t i o n was  checked  again to ensure  screen allowed no l i q u i d to l e a k  t h a t the  through.  From these measurements the gas holdup was  calculated  as d e s c r i b e d subsequently under Data P r o c e s s i n g . During the l a t e r p a r t o f the work an i t y probe was  developed mainly to study the gas holdup i n  the three-phase were conducted  electro-resistiv-  f l u i d i z e d bed r e g i o n .  However, a few  runs  to check the a p p l i c a b i l i t y o f the probe f o r  measurements of l o c a l gas f r a c t i o n s i n a i r - w a t e r and a i r polyethylene g l y c o l solution  flow.  In o r d e r to use the e l e c t r o n i c c i r c u i t d e s c r i b e d above, the a m p l i f i e r s were warmed f o r 20 minutes under zero l o a d c o n d i t i o n s and then checked c i r c u i t d e s c r i b e d i n the manual  f o r any o f f s e t by [105].  the  The probe was l o c a t e d  i n the experimental s e c t i o n so t h a t the gap between the e l e c t r o d e s was  n e a r l y h o r i z o n t a l and p e r p e n d i c u l a r to the  145 flow d i r e c t i o n .  T h i s adjustment was  i n these measurements.  The probe was  p o t e n t i a l from a c o n s t a n t D.C. divider.  critical  then s u p p l i e d a  D.C.  source, through a p o t e n t i a l  The a p p l i e d v o l t a g e was  p u l s e s o f approximately 0.22 ohm  not found to be  so r e g u l a t e d as to produce  v o l t s amplitude a c r o s s a 100,000  resistor i n series. As has been s t a t e d e a r l i e r , the l o g i c  comparator  was  differential  the c e n t r a l component of the measuring  The output o f the probe was  c a r e f u l l y amplified  to  circuit.  produce  p u l s e s o f approximately 3.2 v o l t s amplitude, which were then fed as the n o n - i n v e r t i n g i n p u t to the comparator. i n v e r t i n g i n p u t to the comparator p o t e n t i a l o f approximately -3.0 a c o n s t a n t D.C.  was  D.C.  v o l t s amplitude, taken from  source through a p o t e n t i a l d i v i d e r .  r e f e r e n c e v o l t a g e was  v o l t s above the datum.  The  a d j u s t e d i n such a manner t h a t the  c u t t i n g o f f l e v e l o f the p u l s e s was  the comparator  a reference  The  a t approximately  0.2  Both the i n p u t and the output o f  were monitored c o n t i n u o u s l y on a d u a l beam  o s c i l l o s c o p e to ensure t h a t the c u t - o f f l e v e l i n the comparator  was  such t h a t no p u l s e s were t r i g g e r e d from the  i n p u t s i g n a l c o r r e s p o n d i n g to the l i q u i d The comparator  output was  phase.  i n t e g r a t e d by the i n t e g r a t -  i n g c i r c u i t w i t h a time c o n s t a n t of n e a r l y one second. time r e q u i r e d to i n t e g r a t e the comparator was  noted.  The  o u t p u t to 8 v o l t s  The g a i n of the a m p l i f i e r f o l l o w i n g the  comparator  146 was  so a d j u s t e d t h a t t h i s time was  about 2 minutes.  n o n - i n v e r t i n g i n p u t t e r m i n a l of the comparator grounded, r e n d e r i n g the comparator to i t s peak v a l u e . comparator  was  The then  output c o n s t a n t and equal  The time r e q u i r e d to i n t e g r a t e the  output to 8 v o l t s through the same a m p l i f i e r  again noted.  The l o c a l , gas f r a c t i o n was  the r a t i o o f these two  was  then o b t a i n e d from  times.  The bubble frequency was  o b t a i n e d , as mentioned  by c o u n t i n g the p u l s e s i n the comparator  earlier,  output e i t h e r  e l e c t r o n i c a l l y by a Darcy frequency counter or manually from the r e c o r d i n g of the o u t p u t .  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 A t y p i c a l run was  conducted by f e e d i n g a c a r e f u l l y  weighed amount o f w e l l screened p a r t i c l e s i n t o the t e s t section.  Depending on the d e s i r e d v e l o c i t y of l i q u i d  the column, the l i q u i d the flow meters.  flow r a t e was  The l i q u i d was  u n t i l a c o n s t a n t temperature was  through  measured by e i t h e r o f  c i r c u l a t e d i n the column achieved and noted.  A l l the  manometer taps were c a r e f u l l y f l u s h e d to remove any a i r bubbles remaining i n the c o n n e c t i n g l i n e s . Once the temperature stabilized,  and l i q u i d flow r a t e s were  the expanded bed h e i g h t was  recorded.  To  the s t a t i c p r e s s u r e p r o f i l e along the f l u i d i z e d bed, p r e s s u r e drop r e a d i n g s were taken between tap 1  determine static  and other  147 taps above i t (see F i g u r e 3.4).  The open mercury manometer  measured the a b s o l u t e p r e s s u r e near experimental  the top o f the  section.  The a i r was and making l i q u i d  i n t r o d u c e d by p r e s s u r i z i n g the a i r l i n e flow r a t e adjustments  to ensure  t h a t no  s o l i d p a r t i c l e s were e j e c t e d o u t of the column d u r i n g the i n t r o d u c t i o n of the a i r stream. l i n e was  then so a d j u s t e d as to o b t a i n the d e s i r e d gas  r a t e w i t h o u t any liquid  The back p r e s s u r e i n the a i r  f l u c t u a t i o n s i n the rotameter  flow r a t e was  flow  reading.  The  then r e a d j u s t e d , so as to g i v e a s t a b l e  o p e r a t i o n of the f l u i d i z e d bed.  In order to determine  complete s t a t i c p r e s s u r e p r o f i l e i n the t e s t s e c t i o n ,  the static  p r e s s u r e drop readings were taken between tap 1 and a l l o t h e r taps above i t .  The measurements of s t a t i c p r e s s u r e  drop  g r a d i e n t below and above the t e s t s e c t i o n were recorded separate U-tube manometers. bed behaviour and section.  A r e c o r d was  kept of the  the flow regime encountered  The open mercury manometer was  a b s o l u t e p r e s s u r e near  used  i n the  by  observed test  to measure the  the top o f the experimental  section.  The two b a l l v a l v e s were then c l o s e d by a c t u a t i n g the l i n k rod manually. v e n t i n g the a i r and the motor. was  measured  The gas flow r a t e was  c u t o f f by  the l i q u i d flow r a t e by s w i t c h i n g o f f  The s e t t l e d l i q u i d h e i g h t i n v a r i o u s s e c t i o n s (see F i g u r e 8.2.3), and the a b s o l u t e p r e s s u r e  near the top of the t e s t s e c t i o n was mercury manometer.  read from the open  A check of f i x e d bed h e i g h t b e f o r e  and  148 a f t e r the run r e v e a l e d i f any o f 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.  I f the p r o p o r t i o n of p a r t i c l e s  c a r r i e d out o f the experimental s e c t i o n d u r i n g a run was l a r g e , the run was  discarded.  From these measurements the expanded bed h e i g h t , the s o l i d s and gas holdup i n s i d e 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 c a l c u l a t e d as d e s c r i b e d under Data p r o c e s s i n g . The e l e c t r o - r e s i s t i v i t y probe was  used to o b t a i n the  r a d i a l p r o f i l e o f l o c a l gas f r a c t i o n i n s i d e the bed and  was  l o c a t e d 8 i n c h e s above tap 1 (29.0 cm above the bed support screen).  The same procedure as used f o r measuring the gas  holdup i n g a s - l i q u i d flow was  3.4  followed.  Data p r o c e s s i n g From the data o b t a i n e d i n the 2 i n c h perspex column,  the expanded bed h e i g h t , L^, and the s o l i d s holdup i n the f l u i d i z e d bed, as w e l l as the gas holdup i n the g a s - l i q u i d and g a s - l i q u i d - s o l i d r e g i o n s , were c a l c u l a t e d as o u t l i n e d i n the f o l l o w i n g  3.4.1  sections.  Expanded bed h e i g h t and s o l i d s holdup The 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 s f o r 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 s t u d i e s were measured up to a h e i g h t o f 46 inches  149 above tap 1  (Figure 3.4).  expanded bed h e i g h t , and  The method o f o b t a i n i n g the thereby the s o l i d s  measurements i s d i s c u s s e d i n Appendix 8.2. observed  p r e s s u r e drop, as r e p r e s e n t e d by  holdup,  from  A c c o r d i n g l y , the the U-tube manometer  r e a d i n g , i s p l o t t e d a g a i n s t the d i s t a n c e from tap 1. f o r a two-phase l i q u i d - s o l i d f l u i d i z e d bed, Appendix 8.2,  these  Then  as shown i n  the f o l l o w i n g s t r a i g h t l i n e r e p r e s e n t s  the  p r e s s u r e drop f o r Z < Z : — in 9.x H  and  - p)  (pM  =  ±  Z e3  (p 3  - px)  the p o i n t of i n t e r s e c t i o n of t h i s  H = H max  H  (8.2.18)  s t r a i g h t l i n e with  satisfies  max  (p . - p.) M 1 K  =  A  Z max  e, 3  (p, - p, ) 3 1  (8.2.19)  The measured 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 f o r a typical  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  F i g u r e 3.10.  The  s t r a i g h t l i n e f o r the f l u i d i z e d bed r e g i o n  i s obtained by f i t t i n g  the b e s t l i n e through  drop data f o r Z «  Z  by the method of l e a s t  intercept of t h i s  s t r a i g h t l i n e w i t h the averaged  H = H  max  m a x  the p r e s s u r e squares.  The  value of  g i v e s the v a l u e of Z , from which the expanded ^ max' c  bed h e i g h t i s c a l c u l a t e d  Z max  +  as  8.7  (3.6)  150  60 55 o o  50 45 -  o  E 40 = 35 21 Q <  cr  30 25  cr LU hLU  o <  20 WATER - Q 5 m m GLASS BEADS  15  W  =.l200gm. = 1.59  10  L-b.o  =  cm/sec.  3 4 . 2 cm.  Visually, L  b  = 5 4 . 5 cm.  <Zmax>cm +8-7  -  ".-54.8cm.  .  max  5 10 15 20 25 30 35 DISTANCE FROM TAP 1, Z, in. FIGURE  3.10  40  TYPICAL PRESSURE DROP PROFILE IN LIQUID-SOLID FLUIDIZATION  45  The expanded bed h e i g h t so c a l c u l a t e d was found gene r a l l y to be i n good agreement w i t h the bed h e i g h t measured by d i r e c t o b s e r v a t i o n o f the bed boundary.  However, f o r the  p a r t i c u l a r case o f s m a l l g l a s s beads a t a l a r g e degree o f bed expansion, the observed p r e s s u r e drop d a t a near Z = Z d e v i a t e d c o n s i d e r a b l y from the i n i t i a l  straight line.  m a x  This  d e v i a t i o n i s b e l i e v e d to be caused by the n o n - u n i f o r m i t y o f l o n g i t u d i n a l 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 by s i z e o f the i m p e r f e c t l y s i z e d s o l i d s .  stratification  I n such cases the  expanded bed h e i g h t c a l c u l a t e d by the s t r a i g h t l i n e  inter-  s e c t i o n method was found to be s m a l l e r than the measured bed h e i g h t by v i s u a l e s t i m a t i o n o f the bed boundary; n e v e r t h e l e s s the former was used to c a l c u l a t e the s o l 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 e q u a t i o n  e  =  3  W /P AL 3  (3.1)  b  and a l s o from the s l o p e , S^, o f the b e s t s t r a i g h t l i n e through the p r e s s u r e drop d a t a . £  2  =  Then from e q u a t i o n 8 . 2 . 1 3 , s i n c e  ° '  E  3  =  S  I  (  PM~  P  1  ) / ( P  3"  P  1  )  (  3  ,  7  )  The v a l u e s o f s o l i d s holdup o b t a i n e d from equations 3.1 and 3.7 r e s p e c t i v e l y ,  were found t o be i n good agreement, and  an a r i t h m e t i c mean o f these two v a l u e s i s r e p o r t e d as the s o l i d s holdup.  152 The procedure of  a three-phase  f o r o b t a i n i n g the expanded bed h e i g h t  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  d e s c r i b e d above.  The observed  p r e s s u r e drop, as r e p r e s e n t e d  by the manometer r e a d i n g , i s p l o t t e d a g a i n s t the d i s t a n c e from tap 1.  As shown i n Appendix 8.2.1, the f o l l o w i n g s t r a i g h t  l i n e r e p r e s e n t s the p r e s s u r e drop data f o r Z « Z : max c  c  HI  H (P -P ) M  X  =  Z  [ e  3  ( p  t £  3  3~ l p  " 2  )  £  (Pi P _  (8.2.12)  ) ] 2  while f o r Z > Z , — max'  H  (p  M" l>  =  p  Z  max  - Z e  2  ( p  3- l p  )  ( £  max M- l> (p  p  =  £  )  ( p  l~ 2 p  ) ]  (P -P ) 1  (8.2.10)  2  The i n t e r s e c t i o n o f these two l i n e s  H  2- 2  satisfied  W ^ a - P l *  - 2 e  ( p  l- 2 p  ) ]  ( 8  '  2  The measured 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 f o r 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 F i g u r e 3.11. fitting  The l i n e f o r the f l u i d i z e d bed r e g i o n i s o b t a i n e d by the b e s t s t r a i g h t l i n e through  data f o r Z «  the p r e s s u r e drop  Z by the method o f l e a s t squares. max  Similarly  the l i n e f o r the r e g i o n above the bed i s o b t a i n e d by the b e s t s t r a i g h t l i n e through  fitting  the p r e s s u r e drop d a t a f o r  153  551  —i  5 FIGURE  3.11  1  1  1  1  1  1  10 15. 20 25 30 35 40 DISTANCE FROM TAP I, Z, in. TYPICAL PRESSURE DROP PROFILE IN THREE-PHASE FLUIDIZATION (arrows i n d i c a t e upper l i m i t o f bed l e v e l as v i s u a l l y observed)  r~—-—  45 50  154 by l e a s t squares. The p o i n t o f i n t e r s e c t i o n o f Z » .. Z max these two s t r a i g h t l i n e s determines Z from which the 'max' expanded bed h e i g h t i s c a l c u l a t e d :  + 8.7 Z max  L,b  (3.6)  The expanded bed h e i g h t from e q u a t i o n 3.6 was found to be i n good agreement w i t h the bed h e i g h t measured by d i r e c t v i s u a l o b s e r v a t i o n o f the bed boundary a t s m a l l gas flow r a t e s , when t h i s boundary c o u l d be c l e a r l y d e f i n e d .  However,  f o r h i g h e r gas flow r a t e s (> 4 cm/sec), the bed boundary was quite diffuse  and c o u l d no l o n g e r be l o c a t e d v i s u a l l y w i t h  any c o n f i d e n c e and c o n s i s t e n c y .  Under these  circumstances  the method o u t l i n e d above p r o v i d e s a meaningful the expanded bed h e i g h t , and was used throughout  d e f i n i t i o n to this  study  to o b t a i n expanded bed h e i g h t s w i t h a h i g h degree o f r e p r o d u c i b i l i t y and c o n f i d e n c e . The  s o l i d s holdup  was then o b t a i n e d  HI  i n the three-phase  f l u i d i z e d bed  from  =  W/p AL 3  (3.1)  b  Care was e x e r c i s e d i n a l l experiments  t o prevent e l u t r i a t i o n  or e j e c t i o n o f 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 r a t e s and the s i z e o f the p a r t i c l e s p r e s e n t i n  155 the column, d i d occur, as evidenced particles  i n the screen c a t c h - a l l  s t a t i c bed h e i g h t .  by the presence o f  and by the r e d u c t i o n i n  I f the weight o f p a r t i c l e s l o s t from the  column d u r i n g a p a r t i c u l a r r u n was d i s p r o p o r t i o n a t e l y l a r g e (>5%), t h a t r u n was d i s c a r d e d ; otherwise  the weight of  p a r t i c l e s i n the system was taken as the mean o f the weights o f p a r t i c l e s i n the system b e f o r e and a f t e r the r u n .  No  such problem was encountered f o r the l a r g e and heavy p a r t i c l e s . I t i s a l s o important  t o p o i n t o u t t h a t no measurable  f r i c t i o n a l p r e s s u r e drop was observed  when o p e r a t i n g  without  s o l i d s i n the column w i t h water as the t e s t l i q u i d , a t a l l the water flow r a t e s i n v e s t i g a t e d (2-39 cm/sec).  However,  w i t h 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 as the t e s t l i q u i d , f r i c t i o n a l p r e s s u r e drop i n the column without measurable f o r two o f the flow r a t e s s t u d i e d 18.84  cm/sec).  s o l i d s was  (13.8 2 and  In such i n s t a n c e s , the measured  pressure  drops f o r the l o n g i t u d i n a l p r o f i l e s were c o r r e c t e d by s u b t r a c t i n g the a p p r o p r i a t e f r i c t i o n a l p r e s s u r e drop from each o f the measured v a l u e s .  The c o r r e c t e d p r e s s u r e drop  r e a d i n g s were than used t o o b t a i n the expanded bed h e i g h t f o r determining  t h e s o l i d s holdup.  In the 20 mm glass-column  the expanded bed h e i g h t was  measured.by l o c a t i n g the bed boundary v i s u a l l y .  The s m a l l  c r o s s - s e c t i o n o f the tube employed f a c i l i t a t e d o b s e r v a t i o n of the bed boundary; n e v e r t h e l e s s , the upper range o f gas flow r a t e i n v e s t i g a t e d 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 o f d e f i n i n g the bed boundary s o l i d s holdup was  3.4.2  Gas  a t h i g h e r gas flow r a t e s .  The  c a l c u l a t e d from e q u a t i o n 3.1 as b e f o r e .  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 o f s t a t i c p r e s s u r e drop g r a d i e n t and the measurement o f liquid  l e v e l a f t e r i s o l a t i n g the s e c t i o n s by s h u t t i n g o f f  the v a l v e s , are d e s c r i b e d i n Appendix t h a t due to changes  8.2.  I t i s shown there  i n k i n e t i c energy of the stream and  f r i c t i o n a l p r e s s u r e l o s s e s , which Neal and Bankoff  [103]  estimated to be o n l y about 2.5% o f the t o t a l s t a t i c p r e s s u r e drop i n t h e i r own  g a s - l i q u i d system, the gas holdup o b t a i n e d  by s t a t i c p r e s s u r e drop measurements i s s u b j e c t to some error.  However, i f these l o s s e s are p r o p e r l y accounted f o r ,  the p r e s s u r e drop per u n i t l e n g t h i n g a s - l i q u i d flow i s very n e a r l y equal to the mean d e n s i t y o f the two-phase flow stream.  In such i n s t a n c e s , the gas holdup i s g i v e n by  e  2  =  "  H  ( p  M  _ p  l  ) / ( p  l" 2 P  )  Z  (8.2.17)  The gas holdups above and below the experimental s e c t i o n , are thus o b t a i n e d by measuring the p r e s s u r e drop w i t h the U-tube manometer p r o v i d e d i n each of these s e c t i o n s .  These  measured gas holdups are then s u i t a b l y c o r r e c t e d from the  157 midpoint p r e s s u r e o f the measuring s e c t i o n to a standard p r e s s u r e o f 760 mm o f mercury. at t h i s  A l l gas  holdups a r e r e p o r t e d  pressure. For the e s t i m a t i o n o f gas holdup i n a  f l u i d i z e d bed, is required.  a complete l o n g i t u d i n a l p r e s s u r e drop  I  =  [ e  profile  As shown i n Appendix 8.2, the s l o p e o f the  s t r a i g h t l i n e f o r the three-phase  S  three-phase  3  ( p  3" l p  " 2  )  e  ( p  l" 2 p  r e g i o n i s g i v e n by  ~  ) ]  (8.2.13)  (P  M~ 1 P  )  from which, on rearrangement,  £  2  =  where  [ £  3  ( p  3- l p  - I  }  S  ( p  M- l p  ) ]  , , (P x -P 2 ) 1  ( 3  '  8 )  i s o b t a i n e d from equation 3.1 as d i s c u s s e d i n the  p r e c e d i n g s e c t i o n , and Sj i s o b t a i n e d from the s l o p e o f the b e s t s t r a i g h t l i n e through by the method of l e a s t  the p r e s s u r e drop data f o r Z « %  m  a  K  squares.  A l t e r n a t e l y , i n f l u i d i z a t i o n one can u t i l i z e the f a c t t h a t p r e s s u r e drop a c r o s s the bed i s equal to the weight of H  the bed p e r u n i t area  max ^  Z  max  a n c  s a  m  e  a  s  u  r  e  °f ^  e  [98].  In three-phase  fluidization  p r e s s u r e drop a c r o s s the bed o f h e i g h t  ^ s a t i s f i e s equation 8.2.11.  Then by r e a r r a n g i n g  equation 8.2.11, the gas holdup i n t h e three-phase bed  i s g i v e n by  fluidized  158  - H  e 3 ( P 3 - P i > . Zmax  II  e2  max  (P -P2)  z max  1  The off  method o f m e a s u r i n g gas  the v a l v e s  each i s o l a t e d discussed  and  recording  t h e r e , t h e gas  that s e c t i o n .  the as  equations gas  760  experimental The  8.2.29 and  t o o b t a i n gas of  quick  d e p e n d i n g on  due  to  the  As  h y d r o s t a t i c head  a  above  corrections  given  8.2.3 0 were a p p l i e d r e s p e c t i v e l y t o below the  of mercury.  s e c t i o n , no  experimental sections at a  For  the  gas  s e c t i o n , so standard  h o l d u p above  such c o r r e c t i o n f a c t o r  s h u t - o f f measurements o f gas  s e c t i o n under s l u g flow  in  i n each s e c t i o n i s a t  the  holdups i n these  valve  level  i n A p p e n d i x 8.2.2.  suitable pressure  i n and  mm  quickly shutting  subsequent l i q u i d  collected  Therefore  collected  pressure  the  h o l d u p by  section i s described  d i f f e r e n t pressure  by  (3.9)  is  the  necessary.  holdups i n  this  c o n d i t i o n s were f o u n d t o be u n r e l i a b l e  short length of  the  s e c t i o n , and  were  therefore  discarded. As  shown i n A p p e n d i x  p h a s e f l u i d i z e d bed i s obtained  ni  e0  8.2,  r e g i o n by  t h e gas  holdup i n the  the q u i c k v a l v e  three-  s h u t - o f f method  from  n  = e_ + [e  n  2EC  "  £  2  ]  VLb  (8.2.33)  n  where e~  i s t h e gas  holdup i n the  two-phase g a s - l i q u i d  region  159 above the bed and i s measured independently  by a U-tube  manometer l o c a t e d near the top o f the experimental s e c t i o n . For low gas flow r a t e s it.was not p o s s i b l e to measure the l i q u i d l e v e l i n the t r a n s p a r e n t experimental  s e c t i o n . In  such i n s t a n c e s a mean o f gas holdups o b t a i n e d from 3.8 and 3.9 i s r e p o r t e d ; otherwise,  equations  i n a l l o t h e r cases, a  mean of gas holdups by the two p r e s s u r e drop methods and by the q u i c k v a l v e s h u t - o f f method i s r e p o r t e d . The gas holdup i n the 20 mm g l a s s column was measured mainly by o b s e r v i n g the p r e s s u r e drop a c r o s s the bed on a Utube manometer.  Then the r e d u c t i o n i n manometer r e a d i n g on  i n t r o d u c t i o n of the gas corresponds tween the t a p s .  to the gas f r a c t i o n be-  No attempt was made t o s p e c i f i c a l l y  the gas f r a c t i o n i n s i d e the three-phase  calculate  region; nevertheless  the gas holdup measured i n t h i s manner c o u l d r e v e a l whether the presence l i q u i d holdup  o f s o l i d p a r t i c l e s m o d i f i e d the two-phase gassignificantly.  160  CHAPTER 4 RESULTS AND DISCUSSION  In t h i s chapter, the mathematical Chapter  models d e r i v e d i n  2 are f i r s t compared w i t h 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 o b t a i n e d i n t h i s study are used  4.1  to e v a l u a t e the proposed  Comparison of proposed  mathematical  models.  models w i t h p r e v i o u s  work 4.1.1  Gas  holdup  i n gas-liquid  Zuber and F i n d l a y  flow  [39] d e r i v e d equation 2.26,  i n S e c t i o n 2.1.2, f o r two-phase g a s - l i q u i d a t i o n w i t h equation 2.24  v  2  =  =  gas-liquid  flow; i n combin-  i t can be w r i t t e n as  C (<j 0  1 +  j » 2  +  <a 2 >  Equation 4.1  presented  _ 2 - 2 l _  (  4  i  l  )  <a 2 >  i s q u i t e g e n e r a l and  i s a p p l i c a b l e t o a l l the  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 o b t a i n e d  indepen-  dently . The d i s t r i b u t i o n parameter, CQ, w i t h the h e l p o f equations  was  shown  theoretically,  2.27a-c, to vary between 1.0  and  161  1.5  f o r most cases of g a s - l i q u i d flow. [113]  Smissaert  Using the data o f  f o r a i r - w a t e r flow i n a 2 i n c h v e r t i c a l  p i p e , Zuber and F i n d l a y were a l s o a b l e to show e m p i r i c a l l y t h a t f o r the c h u r n - t u r b u l e n t bubbly CQ was  equal to 1.2.  and s l u g flow  regimes,  N i c k l i n e t a l . [ 1 1 4 ] used a d i f f e r e n t  l i n e of argument and a l s o found a v a l u e of 1.2  for C N  to  s a t i s f y a wide range of data f o r g a s - l i q u i d flow, i f the Reynolds number based c o n d u i t exceeded  on the g a s - l i q u i d f l u x through  the  8,000.  For d r i f t v e l o c i t y , Zuber and F i n d l a y proposed  v  =  2 j  V  a  (1 - a )  (2.31)  m  2  found to vary between 0 and  where the exponent m was depending on the bubble  size.  the l o c a l d r i f t v e l o c i t y was and bubbly  3  =  ( i . e . , m = 0 ) ; then  S~T  0.35  <a >  (4.2)  *  2  the s l u g flow regime and  <a„v .>  ag  0  —£_£i_  <a„> for  flow  mean d r i f t v e l o c i t y i s simply  <a„v„ . > 1  further noticed that  c o n s t a n t f o r both s l u g  flow i n a t u r b u l e n t stream  the weighted  for  I t was  3,  =  1.53  n  [—V'^ p  l  the c h u r n - t u r b u l e n t bubbly flow regime.  (4.3)  162  The p r e s e n t author has proposed v e l o c i t y i n the bubble earlier, yields  V  ~  2 jJ  V  ~  flow regime  "'T  ,  tanh  /-i,  1/3  (2.37)  [ 0 . 2 5 ( l / aX ) '  r r t  o  c  2  equation 2 . 1 0 f o r l a r g e  (r  = R), a p p l y i n g  Eotvos number, s i m p l i f i e s t o  = 0.35  2 j  Thus f o r the s l u g ocity  [76] which, as shown  the equation  and which f o r the s l u g flow regime  v  a model f o r d r i f t  ( 2 > 3 9 )  flow regime the two models f o r d r i f t  vel-  (equations 2 . 3 9 and 4 . 2 ) are i d e n t i c a l , so t h a t  equation 4 . 1 , w i t h CQ = 1 . 2 , reduces relationship  proposed  — — = 1.2 <a >  to the s l u g  flow  by N i c k l i n [ 1 9 ] :  (<j +j >) + 0 . 3 5 /gD~ 1  (4.4)  2  2  V a r i o u s models f o r p r e d i c t i n g the bubble Happel's The  the d r i f t v e l o c i t y i n  flow regime are shown i n F i g u r e 4 . 1 , along w i t h  [ 4 9 ] equation f o r sedimentation of s o l i d  following  remarks are based  on F i g u r e 4 . 1 :  (a) The d i s c r e p a n c y between the curves f o r bubble and s o l i d p a r t i c l e s a r i s e s tangential  spheres.  swarms  from the f a c t t h a t the  l i q u i d v e l o c i t y i s zero a t the s u r f a c e o f  163  FIGURE 4.1  PROPOSED MODELS FOR DRIFT VELOCITY OF BUBBLE SWARMS  164  LEGEND FOR FIGURE 4.1  1.  Sedimentation  2.  Equation 2.31 w i t h m = 3 f o r s m a l l bubbles (d^ < 0.5 mm)  3.  of s o l i d p a r t i c l e s by Happel's model.  obeying Stokes law [74].  Equation 2.31 w i t h m = 2 recommended f o r bubble  flow  regime by Bhaga [ 1 ] . 4.  E q u a t i o n 2.31 w i t h m = 1.5 f o r l a r g e r (1 < d  5.  b  bubbles  < 20 mm) [74].  Equation 2.31 w i t h m = 0 f o r churn t u r b u l e n t - b u b b l y flow and s l u g flow regimes [39].  6.  Equation 2.34 proposed Marrucci  7.  f o r bubble  flow regime by  f o r bubble  flow regime by  [75] .  Equation 2.37 proposed B h a t i a [76].  a s o l i d p a r t i c l e b u t i s not zero a t the s u r f a c e o f a bubble.  Consequently  the energy  dissipation i s  s m a l l e r and the d r i f t v e l o c i t y i s h i g h e r f o r the bubble (b)  swarm.  The model proposed tial  by M a r r u c c i  [75] based on poten-  flow shows a dependence o f d r i f t v e l o c i t y on  v o l u m e t r i c gas f r a c t i o n which i s v e r y s i m i l a r t o t h a t  (c)  of  the Z u b e r - F i n d l a y model w i t h the exponent m equal  to  1.5 .  The exponent m i n the Zuber-Findlay model  (equation  2.31) was r e p o r t e d to vary between 0 and 3, depending on the bubble to  s i z e , the l a r g e r v a l u e s corresponding  the s m a l l e r bubble  sizes.  The s l o p e o f the curve  r e p r e s e n t i n g equation 2.37 shows a g r a d u a l r e d u c t i o n w i t h i n c r e a s i n g gas holdup. s i z e and the gas holdup i n the l a t t e r  S i n c e the s t a b l e  bubble  are i n t e r r e l a t e d , i n c r e a s e s  accompanying i n c r e a s e s i n the former,  i t can be argued  a t l e a s t q u a l i t a t i v e l y t h a t the  p r e s e n t model has the v i r t u e of p r e d i c t i n g the c o r r e c t trend i n d r i f t v e l o c i t y o f a bubble  swarm over a wide  range of o p e r a t i n g v a r i a b l e s . In  order to t e s t the q u a n t i t a t i v e a p p l i c a b i l i t y o f  the p r e s e n t model, a comparison o f the p r e d i c t i o n s w i t h some o f the a v a i l a b l e l i t e r a t u r e data f o r bubble  columns  and f o r c o c u r r e n t g a s - l i q u i d flow i n v e r t i c a l p i p e s has been p u b l i s h e d by the p r e s e n t author  [76] and i s presented  166 i n Appendix 8.8. (a)  (b)  I t was found t h a t :  The model i s a p p l i c a b l e t o low v i s c o s i t y and comparatively  pure g a s - l i q u i d  For bubble  columns, the model s a t i s f i e d the data  o b t a i n e d i n s m a l l columns  systems.  (D _< 2 inch) , s y s t e m a t i c  b u l k c i r c u l a t i o n not being important i n such columns [59, 6 0 ] . The s y s t e m a t i c c i r c u l a t i o n found i n columns w i t h D ^ 4 i n c h [59, 60] would i n c r e a s e the bubble  c o n c e n t r a t i o n i n the upward moving  c o r e , thereby r e d u c i n g the average  central  gas holdup and  i n c r e a s i n g the v a l u e ' o f 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 (c)  (as opposed t o f l a t ) .  For c o c u r r e n t flow the p r e s e n t model i n c o n j u n c t i o n w i t h equation 2.20 f o r bubble diameter  [69] agrees  w e l l w i t h Hughmark's e m p i r i c a l c o r r e l a t i o n  [52], which  has been c o r r o b o r a t e d a g a i n s t experimental data by Dukler e t a l . [72] . In o r d e r to check the g e n e r a l v a l i d i t y o f the above model f o r determining the d r i f t v e l o c i t y of a bubble s y s t e m a t i c data f o r average bubble diameter  and  swarm,  gas-holdup  as a f u n c t i o n o f gas and l i q u i d v e l o c i t i e s a r e needed.  The  model, which s t r i c t l y speaking a p p l i e s o n l y l o c a l l y , should be supplemented by equation 4.1  to account  n o n - u n i f o r m i t i e s i n v e l o c i t i e s and holdups.  f o r any r a d i a l  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 g e n e r a l model f o r d e s c r i b -  i n g 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 w i t h the e m p i r i c a l c o r r e l a t i o n s based on data o f v a r i o u s i n v e s t i g a t o r s it  [14, 17, 79] can be attempted,  i s necessary t h a t a mathematical e x p r e s s i o n be o b t a i n e d  to p r e d i c t the wake volume f r a c t i o n .  As d e f i n e d  earlier,  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 g i v e n by  HI £  k  =  £  2  >  (  *  ( 4  5 )  where the r a t i o of wake volume to bubble volume i n the t h r e e phase f l u i d i z e d bed may be r e p r e s e n t e d by  <nr> - K> B  f  (  e  (2.H7)  )  B  f b e i n g a continuous f u n c t i o n o f bed voidage which u n i t y as the s o l i d s f r a c t i o n , £ 3 / f l u i d i z e d bed approaches zero.  approaches  i n the three-phase  The s i m p l e s t such f u n c t i o n  is  f(e)  =  (l-e ) 3  P  (4.6)  168 Combining equations 4.5, 2.117 and 4.6, the necessary mathematical  e x p r e s s i o n f o 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 g i v e n by  e  k  =  e  2  (  }  3  ( 1 _ e  ) P  ( 4  *  7 )  ^k " where (^—) may be estimated from the data o f Letan and "B  Kehat  [61] f o r a l i q u i d - l i q u i d system, as shown p r e v i o u s l y . In order t o e v a l u a t e the exponent p, simultaneous  measurements o f e^, are s c a r c e .  and  a r e needed.  Such measurements  N e v e r t h e l e s s Efremov and Vakhrushev [16]  presented an equation f o r e^, based on measurements o f e!}  1  and  i n beds o f g l a s s beads  (0.32 - 2.15 mm)  fluidized  by a c o c u r r e n t stream o f a i r and water: (JS) ft o  = 5.1  (e!') 4  U b  n  - 3.32 ( e j )  where  [1-tanh {40  x  i s the voidage  5 , 4 5  ->2  (el')  1 0  x  } ]  (2.128)  i n the l i q u i d - s o l i d f l u i d i z e d bed  b e f o r e the i n t r o d u c t i o n of gas and can be computed equation 2.46.  from  Tables 4.1 and 4.2 p r e s e n t the r a t i o o f  wake to bubble volume f o r two s i z e s o f g l a s s beads, f l u i d i z e d by a c o c u r r e n t stream o f a i r and water, as p r e d i c t e d by equations 4.7 ( f o r p=3), 2.128 and 1.13, r e s p e c t i v e l y , along w i t h some l i m i t e d  data o f Rigby and  169 TABLE 4.1 RATIO OF WAKE TO BUBBLE VOLUME IN THREE-PHASE FLUIDIZED BED PARTICLES - G l a s s beads (d =0.775 mm) p  p =2.67 gm/cm  3  3  D > 4 inches  Liquid Flux, <j > (cm/sec) x  2.61  4.35  (1)  Gas Flux, <  ^2  >  (1)  (2)  (3)  V°B  (4)  (cm/sec)  [x =0.0]  [x =0.0]  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  -  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  k  k  [x =1.0j k  [x =0.0] k  -  From g e n e r a l i z e d wake model, u s i n g e q u a t i o n 4.7 w i t h p = 3  (2)  Equation 2.128 by Efremov and Vakhrushev [16]  (3)  Equations 1.12 and 1.13 by 0stergaard [8]  (4)  From F i g u r e 2 o f Rigby and Capes [8 0]  (*)  Values i n b r a c k e t s a r e f o r x, =1.0  17 0 TABLE 4.2 RATIO OF WAKE TO BUBBLE VOLUME IN THREE-PHASE FLUIDIZED BED PARTICLES - G l a s s beads (d =2.0 mm) p  p^= 2.88 gm/cm  3  D = 2.0 inches Liquid Flux,  <J!>  Gas Flux,  (1)  (2)  (3)  V°B  (cm/sec)  (cm/sec)  "k^B [x =0.0]  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  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  8.51  k  [x =0.0] k  [x =1.0] k  (1) From g e n e r a l i z e d wake model, u s i n g e q u a t i o n 4.7 w i t h p=3. (2) E q u a t i o n 2.128 by Efremov and Vakhrushev (3) Equation 1.13 by 0 s t e r g a a r d [ 8 ] .  [16].  Capes [80].  A comparison of these v a l u e s r e v e a l t h a t :  (a) The values p r e d i c t e d by equation 4.7 w i t h p=3 show mode r a t e agreement w i t h the equation o f Efremov and Vakhrushev  [16] and the data o f Rigby and Capes [80] .  The wake was assumed t o be f r e e o f s o l i d s i n the g e n e r a l i z e d wake model i n order t o match the assumption i n h e r e n t i n the equation of Efremov and Vakhrushev. (b) The v a l u e s p r e d i c t e d by 0stergaard s 1  [8] equation a r e  g e n e r a l l y higher than those by the other two equations; t h i s d i s c r e p a n c y i s p a r t i a l l y due t o 0stergaard's assumption t h a t the c o n c e n t r a t i o n o f s o l i d  particles  i n the wake i s equal to the c o n c e n t r a t i o n i n the p a r t i c u l a t e phase. The agreement o f p r e d i c t i o n s by equations 4.7 w i t h those by equation 2.128 was found to improve by assuming d i f f e r e n t v a l u e s o f the exponent p f o r d i f f e r e n t o p e r a t i n g conditions.  However, a v a l u e o f 3 f o r the exponent gave a  rough agreement w i t h a l l a v a i l a b l e i n f o r m a t i o n , b u t i s not to be i n t e r p r e t e d as recommended v a l u e . behind bubbles  e i t h e r the b e s t o r the u n i v e r s a l l y  Only a s y s t e m a t i c i n v e s t i g a t i o n of wakes  i n three-phase  f l u i d i z e d beds, w i t h s i m u l t a n -  eous measurements of gas and s o l i d s holdups, a justifiable correlation.  could provide  N e v e r t h e l e s s , i n t h e absence o f  s u i t a b l e d a t a f o r wake volume f r a c t i o n , e q u a t i o n 4.7, w i t h p=3,  w i l l be used  i n the g e n e r a l i z e d wake model o f S e c t i o n  172 2.3 2 f o r the purpose of comparing i t s p r e d i c t i o n s o f and s o l i d s holdups w i t h experimental  data.  gas  I t w i l l be seen  t h a t i n many i n s t a n c e s the p r e d i c t i o n s 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 f u n c t i o n f ( e ) i n equation 2.117.  The more complex equation 2.128  w i l l be used  s i m u l t a n e o u s l y to p r o v i d e g u i d e l i n e s w i t h i n i t s range of applicability. Now,  a c c o r d i n g to the model proposed i n S e c t i o n 2.3.2,  the r i s e v e l o c i t y o f a bubble swarm i n a  three-phase  f l u i d i z e d bed i s g i v e n by  v  2  =  C"» <j,+j > ± — 2  +  e" ( l - e - e > - i i ±—— 2  £  where v ^  k  _ v™  i s c a l c u l a t e d from equation 2.107  and from equation 2.108a f o r D < 4 i n c h e s . of  the d i s t r i b u t i o n parameter i n three-phase  CQ',  (2.114)  ±  £  f o r D >_ 4 inches S i n c e the v a l u e f l u i d i z e d beds,  cannot be estimated a t p r e s e n t , t h e r e f o r e , i n a l l the  c a l c u l a t i o n s and d i s c u s s i o n s which f o l l o w , i t w i l l tacitly  assumed t h a t CQ  = 1.0  and  be  t h a t the e f f e c t s o f non-  uniform r a d i a l p r o f i l e s can be lumped i n t o 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 p a s t f o r g a s - l i q u i d flow  [19]•  The gas holdup i s then o b t a i n e d from  <j >/v 2  2  (2.94)  173 Tables 4.3-A /2 e  a n <  ^ 2'  and  4.3-B  p r e s e n t the v a l u e s of  r e s p e c t i v e l y , p r e d i c t e d by the p r e s e n t  v  a l i z e d wake model [equations 2.114, 2.107 0.775 mm and water  and  2.94]  generfor  g l a s s beads f l u i d i z e d by a c o c u r r e n t stream of a i r [80], along w i t h the values c a l c u l a t e d from  e m p i r i c a l c o r r e l a t i o n s presented i n S e c t i o n 2.3.2. estimate of bubble Rigby e t a l . [79] r  e  - 0.6  cm)  l e n g t h was (1 = 0.61  and was  [80].  An  o b t a i n e d from the data of  cm;  then from equation 2.125,  used f o r c a l c u l a t i o n s i n the p r e s e n t  model as w e l l as i n the c o r r e l a t i o n proposed Capes  Equation 2.107  was  r e l a t i v e v e l o c i t y of a bubble  used  by Rigby  for calculating  and the  swarm i n the model because  the e m p i r i c a l c o r r e l a t i o n s presented i n Tables 4.3-A were a l l based  the  and  on data obtained i n columns o f D > 4 i n c h .  On comparing the p r e d i c t i o n s from the model w i t h those  from  the c o r r e l a t i o n s i n Table 4.3-A, i t can be seen t h a t : (a) The p r e d i c t i o n s from the model are i n reasonable agreement w i t h the r e s u l t s o f Efremov and [16], who  Vakhrushev  were f o r c e d 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 u n c e r t a i n t y of l o c a t i n g the three-phase  boundary a t high-gas  r a t e s , e s p e c i a l l y f o r l a r g e bed (b) Although  -B  flow  expansions.  the c o r r e l a t i o n r e p o r t e d by M i c h e l s e n  0stergaard i s a l s o based on data f o r low  and  gas  v e l o c i t i e s , i t too i s found to g i v e r e s u l t s which are i n f a i r l y good agreement w i t h the p r e d i c t i o n s o f 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 .775 mm) P  P  = 2 .67 gm/cm  3  3  Liquid Flux, ^1* (cm/sec)  Gas Flux <j > (cm/sec)  CD  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  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  4.35  (2)  (3) c  e  2  in/,-, I I  2  / e  2  (4) c  e  in /  2  c  / e  II  2  (1) From g e n e r a l i z e d wake model w i t h x, = 0 (2) From e q u a t i o n o f Efremov and Vakhrushev [16] (3) From equations [14]  2.121 and 2.122, M i c h e l s e n and 0stergaard  (4) From e q u a t i o n 2.120, V a i l e t a l . [17]  175 TABLE 4.3-B RISE VELOCITY OF BUBBLE SWARMS IN THREE-PHASE FLUIDIZED BED (D > 4 INCHES)  PARTICLES - G l a s s beads  Liquid Flux,  Gas Flux, <  (cm/sec)  2.61  4 .35  h  >  (cm/sec)  (d = 0.775 mm)  (1) v , cm/sec 2  (2)  p  = 2.67 gm/cm  3  3  (3)  ^2  ^2  (4)  (5) ^2 [1=0.612 cm]  [r =0.6 cm] e  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  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 g e n e r a l i z e d wake model w i t h 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 e t a l . (5) From e q u a t i o n 2.123, Capes e t a l .  [79]  [17]  (c) The p r e d i c t i o n s from the model a r e c o n s i s t e n t l y h i g h e r than the r e s u l t s o f V a i l e t a l . [17].  This discrepancy  c o u l d be p a r t i a l l y due to the approximate method used by V a i l e t a l . t o measure the s o l i d s holdup and p a r t i a l l y due t o t h e i r method f o r measuring  the gas holdup  itself  (see Appendix 8.2). Table 4.3-B p r e s e n t s the r i s e v e l o c i t i e s o f bubble swarms c a l c u l a t e d from v a r i o u s e m p i r i c a l c o r r e l a t i o n s .  A  comparison o f these v a l u e s w i t h the p r e d i c t i o n s from the g e n e r a l i z e d wake model r e v e a l s  that:  (a) The o r i g i n a l 0stergaard c o r r e l a t i o n f o r e s t i m a t i n g the r i s e v e l o c i t y of a bubble swarm, e q u a t i o n 1.12, g i v e s v a l u e s which are f a r o u t o f l i n e from those g i v e n by the other c o r r e l a t i o n s as w e l l as from the model.  The  c o r r e l a t i o n proposed l a t e r by M i c h e l s e n and 0stergaard [14], on the o t h e r hand, g i v e s v a l u e s which a r e i n good agreement w i t h the p r e d i c t i o n s from the model. (b) The bubble r i s e v e l o c i t i e s c a l c u l a t e d from the c o r r e l a t i o n proposed b y . V a i l e t a l . [17] are found to be l a r g e r than those p r e d i c t e d by the model, though the two e x h i b i t s i m i l a r trends w i t h r e s p e c t t o i n c r e a s e i n gas velocity. (c) The v a l u e s c a l c u l a t e d 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 w i t h  those o f the model.  A t the s m a l l e r l i q u i d f l u x the  bubble v e l o c i t i e s a r e s m a l l e r , whereas a t the l a r g e r  177 liquid  f l u x the bubble v e l o c i t i e s are much l a r g e r ,  than p r e d i c t e d by the model. to  However, i t i s important  n o t i c e t h a t the c o r r e l a t i o n of Rigby e t a l . (equation  2.123) i s q u i t e s e n s i t i v e to the l e n g t h of bubbles i n the swarm.  S i n c e o n l y l i m i t e d data f o r bubble  are a v a i l a b l e 2.123  [7 9],  a fair  lengths  comparison of e q u a t i o n  w i t h the other c o r r e l a t i o n s as w e l l as w i t h  the  model i s not p o s s i b l e a t p r e s e n t . S i n c e the model proposed the 2 i n c h diameter an attempt  here i s used l a t e r to t e s t  column data obtained i n the p r e s e n t  study,  i s f i r s t made t o compare the p r e d i c t i o n s of the  model w i t h d a t a o b t a i n e d from 2 i n c h diameter earlier investigators.  Although  columns by  the bed voidage d a t a f o r  the 2 i n c h columns are a v a i l a b l e , no such data f o r gas h o l d up are r e p o r t e d . and T h e i s e n  T h e r e f o r e the bed voidage data o f 0 s t e r g a a r d  [18] , f o r 2 mm  g l a s s beads f l u i d i z e d by a  c o c u r r e n t stream of a i r and water i n a 2 i n c h column, are used as a b a s i s f o r o b t a i n i n g the v a l u e s of e'^/z^ from e m p i r i c a l c o r r e l a t i o n s .  where the  v e l o c i t y f o r the s l u g flow regime i s o b t a i n e d by to account  0.2  <j.  v  2  drift  modifying  for non-uniformities i n r a d i a l  p r o f i l e s , f o l l o w i n g the recommendation of N i c k l i n  =  ^  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,  equation 2.39  a n c  +  j > 9  +  0.35  vfqD  [19]:  (4.8)  17 8 The c a l c u l a t e d v a l u e s a r e presented i n T a b l e s 4.4-A and 4.4-B. A comparison o f v a r i o u s v a l u e s o f 4.4-A  ^  e^/t 1^  n  T a  ble  reveals that:  (a) The p r e d i c t i o n s from the model a r e i n f a i r l y good agreement w i t h the r e s u l t s of Efremov and Vakhrushev [16] f o r s m a l l 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 p r e d i c t i o n s from the model agree w i t h the r e s u l t s of V a i l e t a l . [17] o n l y i n t h e i r r e s p e c t i v e t r e n d s , but not i n a b s o l u t e v a l u e s . (c) The p r e d i c t i o n s from the model do not agree w i t h the values c a l c u l a t e d from the Michelsen-0stergaard [14] c o r r e l a t i o n , e i t h e r a b s o l u t e l y or i n trends d i s p l a y e d . 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 o f bubble swarms i n three-phase f l u i d i z e d beds (Table 4.4-B) r e v e a l s that: (a) The c o r r e l a t i o n proposed by 0stergaard  [ 8 ] , e q u a t i o n 1.12,  p r o v i d e s a poor estimate f o r the bubble r i s e  velocity,  i f one g i v e s any credence a t a l l t o the e m p i r i c a l c o r r e l a t i o n s o f r e f e r e n c e s [14] and [17].  The c o r r e l a -  t i o n proposed l a t e r by M i c h e l s e n and 0stergaard [14] g i v e s moderate agreement w i t h the p r e d i c t i o n s from the model. (b) The bubble r i s e v e l o c i t i e s c a l c u l a t e d from the c o r r e l a t i o n proposed by V a i l e t a l . [17] do n o t agree w i t h the p r e 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 Liquid Flux,  Gas Flux,  h (cm/sec)  <  (cm/sec)  >  (d = 2.0 mm)  (1) £  2 2 £  e  p, = 2.88 gm/cc  (2)  (3)  (4)  2 2  e '"/e " 2 2  2 2  e  fc  3.38  0.5 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0  0.471 0.485 0.484 0 .495 0.504 0.507 0 .515 0.532 0 .531 0.535  0.436 0.436 0.436 0.436 0.436 0 .436 0.436 0 .436 0 .436 0.436  0.216 0 .216 0 .218 0.220 0 .225 0 .232 0 .241 0 .253 0 .257 0.260  0.645 0 .532 0 .442 0.396 0 .366 0 .345 0.328 0.315 0 .304 0.294  8.51  0.5 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0  0 .714 0.679 0 .685 0.705 0.703 0.713 0 .720 0.728 0 .733 0.741  0 .560 0 .560 0 .560 0 .560 0 .560 0 .560 0.560 0 .560 0 .560 0 .560  0 .436 0.433 0.430 0.433 0 .440 0 .455 0 .472 0.475 0 .479 0.482  1.044 0.868 0.720 0.650 0 .597 0 .562 0.535 0.514 0.495 0 .480  (1) From g e n e r a l i z e d wake model w i t h x. = 0 (2) From e m p i r i c a l e q u a t i o n o f Efremov and Vakhrushev [16] (3) From e q u a t i o n 2.120, V a i l e t a l . [17] (4) From equations 2.121 and 2.122, M i c h e l s e n and 0 s t e r g a a r d [14]  180 TABLE 4.4-B RISE VELOCITY OF BUBBLE SWARMS IN THREE-PHASE FLUIDIZED BED PARTICLES - G l a s s beads  Liquid Flux,  Gas Flux,  — V  (d  (1)  2  (D = 2 INCHES) =2.0  _ V  mm)  (2)  2  p, = 2.88 gm/cc  _ V  (3)  2  _ V  (4)  2  (cm/sec  (cm/sec)  (cm/sec)  3.38  0.5 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0  60.84 61.80 63 .66 65.40 67 .00 68.36 69 .48 70.26 71.87 73 .54  28 .27 25.08 21.89 20 .03 18 .70 17 .68 16.84 16.13 15.52 14 .97  49.74 57.93 67 .47 73 .77 78 .59 82 .54 85.92 88.89 91.54 93 .94  100.2 116.7 134 .6 146.0 152 .0 154 .8 155.1 152.8 155.0 157 .1  8.51  0.5 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0  52.26 52.88 54.11 55.28 56.36 57.32 58 .21 59 .38 60.57 61.79  33.40 30.21 27 .02 25.16 23 .83 22.81 21.97 21.26 20.65 20 .10  35.46 41.17 47 .95 52.42 55.85 58.66 61.06 63.16 65.05 66 .75  106.1 119 .6 140.3 152.3 160 .0 162.2 162.7 167 .3 170 .8 174.2  (1) From g e n e r a l i z e d wake model w i t h x, = 0 (2) From e q u a t i o n 1.12, 0stergaard [8] (3) From equations  2.121 and 2.94, M i c h e l s e n and 0stergaard  [14] (4) From equations 2.118 and 2.94, V a i l e t a l [17]  181 Thus i t can now be t e n t a t i v e l y concluded t h a t the p r e s e n t model, when used w i t h a s u i t a b l e c o r r e l a t i o n f o r 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, p r o v i d e s an e f f e c t i v e method f o r e s t i m a t i n g the gas holdups and bubble r i s e v e l o c i t i e s i n three-phase ation, since  the p r e d i c t i o n s  fluidiz-  from the model g e n e r a l l y  agree  w i t h the a v a i l a b l e c o r r e l a t i o n s o f d a t a from columns o f D >_ 4 inches [14, 16] .  S i n c e the bubble dynamics  l i q u i d flow i n small columns from those i n l a r g e r ones  f o r gas-  (D < 4 i n c h ) d i f f e r markedly  [59, 60, 68], no s i n g l e  correlation  can be s u c c e s s f u l l y used f o r a l l column s i z e s u n l e s s these dynamics  are p r o p e r l y accounted f o r .  The p r e s e n t model does  so by a l l o w i n g a separate c o r r e l a t i o n f o r the r e l a t i v e v e l o c i t y o f bubble swarms t o be used f o r s m a l l diameter columns than f o r l a r g e  columns.  182 4.1.3  Voidage i n three-phase f l u i d i z e d beds The measurement o f voidage i n three-phase  fluidized  beds has been c a r r i e d o u t by v a r i o u s i n v e s t i g a t o r s , o f whom.0stergaard and co-workers and Vakhrushev  [8, 14, 18, 54] and Efremov  [16] a r e most noteworthy  because o f the  wide range o f p a r t i c l e s i z e s which they i n v e s t i g a t e d . Most o f these measurements were o b t a i n e d i n columns w i t h diameter D > _ 4 i n c h e s ; we note, however, t h a t 0stergaard and Theisen [18] r e p o r t e d on l i m i t e d data o b t a i n e d i n a 2 i n c h diameter column. The number o f t h e o r e t i c a l analyses attempted f o r p r e d i c t i n g the voidage i n a three-phase f l u i d i z e d bed has been l i 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  a n a l y s i s to s a t i s f y the r e p o r t e d paradox  [6, 9] o f bed  c o n t r a c t i o n i n three-phase f l u i d i z a t i o n .  However t h i s  model f a i l e d to s a t i s f y the e x t e n s i v e d a t a o f 0stergaard and Theisen [18] q u a n t i t a t i v e l y .  Efremov and Vakhrushev  [16] then proposed and d e r i v e d a model q u i t e s i m i l a r to the wake model b u t w i t h the assumption content o f bubble wakes was zero.  t h a t the p a r t i c l e  From simultaneous  measurements o f 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 c a l c u l a t e d equations 2.126 - 2.127.  from  The wake f r a c t i o n s so c a l c u l a t e d  were found to be adequately r e p r e s e n t e d by equation 2.128.  183 Efremov and Vakhrushev then t e s t e d the a p p l i c a b i l i t y of equations 2.126 - 2.128 f o r p r e d i c t i n g the data o f 0stergaard and  Theisen  [18], and r e p o r t e d a s a t i s f a c t o r y agreement.  M i c h e l s e n and 0stergaard  [14] , i n a l a t e r study o f t h r e e -  phase f l u i d i z a t i o n of 1, 3 and 6 mm g l a s s beads i n a 6 i n c h diameter column, r e p o r t e d data on bed voidage and gas holdup f o r a wider range of o p e r a t i n g v a r i a b l e s cm/sec and J 2 <  >  U  P to 15.0 cm/sec).  (<j-^> up t o 26.0  These d a t a have n o t  been t e s t e d h i t h e r t o by any o f the t h e o r e t i c a l a n a l y s e s . T h e r e f o r e the g e n e r a l i z e d wake model d e r i v e d i n S e c t i o n 2.3, which assumes v a r i o u s p o s s i b l e s o l i d s contents o f the wake w i t h consequent c i r c u l a t i o n o f s o l i d s i n the p a r t i c u l a t e phase, w i l l now be compared w i t h the data o f M i c h e l s e n and 0stergaard  [14] and the model of Efremov and Vakhrushev  [16] . I t i s important to note a t the o u t s e t t h a t the g e n e r a l i z e d wake model w i t h  = 0.0 i s , i n essence,  identical  to the model formulated by Efremov and Vakhrushev [16], d i f f e r i n g only i n the exact e x p r e s s i o n s used the r i s e v e l o c i t y o f a bubble i n a three-phase  to c a l c u l a t e  swarm and the wake f r a c t i o n  f l u i d i z e d bed.  From a comparison o f the  models w i t h r e s p e c t t o these two q u a n t i t i e s i n the p r e ceding s e c t i o n , i t was found t h a t (i) the r i s e v e l o c i t y o f a bubble  swarm c a l c u l a t e d  from the g e n e r a l i z e d wake model i s i n g e n e r a l s m a l l e r than p r e d i c t e d by the Efremov-Vakhrushev equations, and  184 (ii)  the 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  pre-  d i c t e d by the g e n e r a l i z e d wake model w i t h x^. = 0 shows a s c a t t e r e d agreement w i t h e q u a t i o n 2.128 of Efremov and Vakhrushev.  The f u l l  comparison  i s summarized i n Table 4.5. M i c h e l s e n and 0stergaard dent techniques  [14], who used  two indepen-  ( r e s i d e n c e time d i s t r i b u t i o n by t r a c e r s t u d i e s  and p r e s s u r e drop measurement) to measure the gas holdup i n a three-phase results  f l u i d i z e d bed, found  the techniques t o g i v e  that d i f f e r e d widely, e s p e c i a l l y  particles.  f o r beds o f l a r g e  Hence, s i n c e no agreement i s found  p r e d i c t e d v a l u e s o f r i s e v e l o c i t i e s o f bubble  f o r the swarms from  the g e n e r a l i z e d wake model w i t h those from the EfremovVakhrushev e q u a t i o n s , and s i n c e the measurements o f gas h o l d up by M i c h e l s e n and 0stergaard are u n c e r t a i n , a  comparison  between measured and p r e d i c t e d v a l u e s i s made f o r l i q u i d fraction,  e^, r a t h e r than f o r bed voidage,  e(=£^+£2)•  dictions  by the g e n e r a l i z e d wake model a r e compared  Prehere  w i t h those by the Efremov-Vakhrushev e q u a t i o n s , as w e l l as w i t h the experimental data of M i c h e l s e n and 0stergaard. Another  special  case of the g e n e r a l i z e d 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 , i n s i g n i f i c a n t and n e g l i g i b l e . s i m p l i f i e s to  £^., to be  I n t h a t case e q u a t i o n 2.106  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  \  \  Particle Size Large  Bed >v Expansion  High (e > 0.8)  Medium (0 .6<e<0.8)  Medium  (d >3mm) P adequate agreement a t a l l gas flow r a t e s  excellent agreement a t a l l gas flow rates  Small  (l<dp<3mm)  (d <lmm) p  poor agreement a t a l l gas flow r a t e s  worst agreement a t a l l gas flow r a t e s  adequate agreement up to  poor agreement a t a l l gas flow r a t e s  j  2  - 10 cm/  sec  Low (0.4<e<0.6)  favorable agreement a t a l l gas flow rates  favorable agreement a t a l l gas flow rates  adequate agreement up to j  2  sec  - 10 cm/  186 £  i£  -  I  <J!>  m  -  1  1/n  which, when s u b s t i t u t e d i n t o equation 2.91, voidage  as  e =  E q u a t i o n 4.9 2.3.1  g i v e s the bed  (1-e"') [  <j,> 1/n ^-ir, ] + e"«  i s i d e n t i c a l to equation 2.. 62 d e r i v e d i n S e c t i o n  f o r the g a s - f r e e model.  fluidized  (4.9)  bed,  Thus f o r a  three-phase  i f e^. i s s m a l l , the g e n e r a l i z e d wake model  approaches the g a s - f r e e model. Before any comparisons c o u l d be attempted necessary  to f i n d a c o r r e l a t i o n which c o u l d  r e p r e s e n t the bed voidage As shown i n Table 4.6,  for liquid-solid  i t was  satisfactorily fluidization.  the N e u z i l - H r d i n a c o r r e l a t i o n  (equation 2 . 5 1 ) p r e d i c t s 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 M i c h e l s e n and 0stergaard are outs i d e the recommended range o f a p p l i c a b i l i t y  of the  I t i s of i n t e r e s t to p o i n t out too t h a t the bed data f o r 3 and  6 mm  g l a s s beads  (ROp  >  1000)  former.  expansion  also  agree  w e l l w i t h the p r e d i c t i o n s o f Trupp  [87], thereby s u p p o r t i n g  the h y p o t h e s i s t h a t t u r b u l e n c e 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.  correlation  recommended by N e u z i l and  (equation 2.51)  The e m p i r i c a l Hrdina  187 TABLE 4.6 COMPARISON OF MEASURED AND PREDICTED BED VOIDAGES FOR LIQUID-SOLID  Particle Diameter (mm)  (p -2.67 a» ) cm  2.95  (p -2.45 S* ) cm  5.93 (p =2.63 SE» ) cm  (1) (2) (3) (4)  (1)  (2)  (3)  (4)  £  £  £  £  3.0  0 .575  0.487  0 .552  _  4.2  0.645  0.555  0.625  -  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  -  6.6  0 .570  0 .494  0 .579  0 .563  8.4  0.630  0.546  0.633  0 .626  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  10.0  0 .580  0 .495  0 .577  0 .547  14.0  0 .646  0.570  0 .653  0 .634  20 .0  0.776  0 .662  0 .746  0 .741  26 .0  0.833  0.739  0.822  0 .832  Liquid Flux <J1> (cm/sec)  1.25  FLUIDIZATION  Measurements by M i c h e l s e n and 0stergaard [14] E q u a t i o n 2.46 by Richardson and Zaki [2] E q u a t i o n 2.51 by N e u z i l and Hrdina [47] Dimensional c o r r e l a t i o n proposed by Trupp [87]  188 [47] w i l l , n e v e r t h e l e s s , be used to d e s c r i b e the bed expans i o n c h a r a c t e r i s t i c s o f a l l p a r t i c l e s i z e s 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 f o r c a l c u l a t i o n s  of e£* from the g e n e r a l i z e d wake model i t i s assumed t h a t (a) the r e l a t i v e v e l o c i t y o f bubble swarms i s r e p r e s e n t e d by equation 2.107 (since a l l the data r e p o r t e d i n r e f e r e n c e s 14 and 16 were obtained i n columns o f L\>4 in) , and (b) the voidage  i n the p a r t i c u l a t e phase i s r e p r e s e n t e d  by the N e u z i l - H r d i n a c o r r e l a t i o n  (equation 2.51).  I t i s worth n o t i n g t h a t s i n c e the voidage  i n the  p a r t i c u l a t e phase, f o r x^ = 0.0, i s g i v e n by  ^1 " 2 k / [ — — - ] V ^ l - e ^ ) <  'If  >  v  e  X  2  (2.106-a)  and s i n c e 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 g i v e n by  e  l  =  £  l f  ( 1 _ £  then f o r determining accurately  2" k e  )  +  e  k  e£' a c c u r a t e l y ,  (as normally  (2.91-a)  should be c a l c u l a b l e  > e^, and both e  2  and e^. are  very much s m a l l e r than u n i t y ) , and f o r determining  189 a c c u r a t e l y , the product v An overestimate o f of v of  2  2  £^ need be known a c c u r a t e l y .  would r e s u l t from an underestimate  (as i n 0stergaard's e q u a t i o n s ) , w h i l e an underestimate would c o n v e r s e l y r e s u l t from an overestimate of v  2  (as  i n the Efremov-Vakhrushev e q u a t i o n s ) . The p r e d i c t e d values of l i q u i d f r a c t i o n i n a t h r e e phase f l u i d i z e d bed c a l c u l a t e d from the g e n e r a l i z e d wake model f o r x^. = 0.0 and from the g a s - f r e e model f o r the l a r g e r p a r t i c l e s , a r e shown i n F i g u r e s 4.2-4.4, along w i t h the data of M i c h e l s e n and 0stergaard  [14] and the p r e d i c t e d v a l u e s  from the equations of Efremov and Vakhrushev 0stergaard  [8].  [16] and o f  Some r e s u l t s a r e a l s o presented i n Tables  4.7 and 4.8.  Beds o f 6_ mm p a r t i c l e s  (Figures 4.2a, b and Table 4.7)  The data o f M i c h e l s e n and 0stergaard show t h a t the l i q u i d holdup reduces g r a d u a l l y as the gas v e l o c i t y i s i n creased.  The p r e d i c t i o n s from the model f o r <j^> = 20 cm/sec <  show e x c e l l e n t agreement w i t h the data up t o J and then o n l y an agreement i n t r e n d f o r j  2  > 2  ~ 6 cm/sec,  > 8 cm/sec,  whereas the p r e d i c t i o n s from the Efremov-Vakhrushev  equations <  show o n l y an agreement i n t r e n d w i t h the data up to J 7 cm/sec, and no agreement f o r j  2  > 8 cm/sec.  > 2  =  The p r e d i c t i o n s  from 0stergaard's model a r e a t b e s t q u a l i t a t i v e up t o <J2> = 3 cm/sec.  The d i s c r e p a n c y between the p r e d i c t i o n s o f the  190  0.8  0.4 0  2  4  6  8  10  14  .12  16  < j > , cm/sec 2  FIGURE 4.2a  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; g e n e r a l i z e d wake model w i t h x^ = 0; Efremov-Vakhrushev equations 2.12 6 2.128, f o r J i = 20.0; 0stergaard s ' e q u a t i o n s , f o r j i = 20.0; g a s - f r e e model) ]  1  191  LU  <  9  o > O UJ DO  0  2  6  8  10  12  14  r  < j > , cm/sec 2  FIGURE  4.2b  BED VOIDAGE DATA OF MICHELSEN AND 0STERGAARD [14] FOR 6 MM GLASS BEADS ( H -j 1 =20.0; O -Jl= 10.0; g e n e r a l i z e d wake model w i t h x^=0; g a s - f r e e model)  TABLE COMPARISON  OF  MEASURED AND d  Gas Flux, < j > ? (cm/sec)  Measured  P  e  mm,  Predicted  (1)  1  5 .93  =  PREDICTED  / E  2  e  k  P3  GAS  2 .63  =  _  <j1> =  Predicted  k  /  0.0  0.730  1.0  0.720  0 .018  1 . 827  0 .644  0.011  2.0  0.710  0.040  1. 729  0 .625  0.018  IN THREE-PHASE  20.0 c m / s e c , D =  e  2  £  k /  £  _  _  1.282  0 .730  0.015  .12.95  1.135  0.716  0.030  8.80  BEDS  0.662  Predicted  (4)  e "' 1  2  0.744  FLUIDIZED  6.0 i n c h  Predicted  (3)  e "' 1  2  _  _  0 .662  LIQUID FRACTIONS  gm/cc -  (2)  e'" 1  2  AND  4.7  E  c  2  _  (5)  IM  1  e e  l  E  0 .741  0.741  2  0.615  0 .024  0 .73 4  0 .752  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 .5 5 1  0 .050  0.450  0.672  0.098  0.674  0.830  9.0  0.667  0.839  0.659  0.849  0.654  0 .854  0.648  0.862  0.628  0 .17 2  0 .893  0 . 550  0.054  0.422  0 .665  0.107  10.0  0 .617  0.190  0 .779  0 .551  0.058  0.416  0 .65.8  0.115  11.0  0.604  0 .200  0 .674  0 .554  0.062  0.409  0 .651  0.123  0 .214  0 .578  0 .558  0.066  0.403  0.644  0.131  0 .492  0 .563  0.070  0.397  0 .637  0.138  0 .416  0 .568  0.073  0.391  0.630  0.145  0 .349  0 .574  0.077  0 .386  0.624  0 .151  12.0 13.0 14.0 15.0  -  (1)  Measurements  (2)  From  -  -  by M i c h e l s e n  equations  2.126  and  (3)  From  g e n e r a l i z e d wake  (4)  From  equations  (5)  From  gas-free  1.10  and  0stergaard  2.128, Efremov  -  -  -  -  -  -  -  -  -  -  -  -  -  -  [14] and Vakhrushev  [16]  model  - 1.13,  0stergaard  model, equation  [8]  4.11, u s i n g  the values  1  o f e'l m e a s u r e d  by M i c h e l s e n  and  0stergaard  H vo to  193 g e n e r a l i z e d wake model and the Efremov-Vakhrushev equations a r i s e s , i n p a r t , from o v e r e s t i m a t i o n of v e s t i m a t i o n of £ )  ky  2  t  n  latter,  e  2  (due to under-  the estimates o f e^. by  these two methods being i n reasonable agreement w i t h each other. However, i f i t i s assumed t h a t the r o l e of  bubble  wakes i s i n s i g n i f i c a n t , the g a s - f r e e model can be used f o r 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 s u b s t i t u t i n g T r u p p s 1  dimensional equation f o r  e'"  =  (l-e™) [  X  Z  and the bed voidage  e  The  =  ej' +  0.36  i n t o equation  <j,> ^rp V„ (1-e*)  ]  2.60:  1/2.28  (4.11)  from  e' "  (1.3)  2  l i q u i d f r a c t i o n s c a l c u l a t e d from equation 4.11,  the gas holdup data as r e p o r t e d by M i c h e l s e n and [14], 4.2a.  are presented i n Table 4.7 These v a l u e s o o f l i q u i d  using  0stergaard  and a l s o shown i n F i g u r e  f r a c t i o n appear to be i n as  good agreement w i t h the experimental v a l u e s as those l a t e d from the g e n e r a l i z e d wake model.  The v a l u e s of bed  voidage c a l c u l a t e d from the g a s - f r e e model and  2.91)  are presented i n F i g u r e 4.2b  s m a l l percentage  calcu-  (equations  4.11  and a l s o show o n l y  d e v i a t i o n s from the experimental v a l u e s  [14].  194 Thus f o r f l u i d i z e d beds o f 6 mm p a r t i c l e s the r o l e o f bubble wakes appears to be i n s i g n i f i c a n t and the bed voidage can be w e l l represented by the simple g a s - f r e e model.  Beds of 3 mm p a r t i c l e s  (Figures 4.3a,b)  The data o f M i c h e l s e n and 0stergaard f r a c t i o n i n three-phase  [14] f o r l i q u i d  f l u i d i z e d beds show e x c e l l e n t  agreement with the p r e d i c t e d v a l u e s from the g e n e r a l i z e d wake model, p a r t i c u l a r l y i f the parameter x^. i s s u i t a b l y adjusted.  On the other hand, the o v e r a l l v o i d f r a c t i o n ,  especially at j ^ <  >  =  14.0 cm/sec, shows poor agreement w i t h  the p r e d i c t i o n s from the model.  T h i s d i s c r e p a n c y i s caused  by the p r e d i c t e d gas holdup being c o n s i d e r a b l y s m a l l e r a t the h i g h e r l i q u i d f l u x and c o n s i d e r a b l y l a r g e r a t the lower l i q u i d f l u x than the measured v a l u e s r e p o r t e d by M i c h e l s e n and 0stergaard. liquid  They observed  the bubble behaviour f o r  f l u x below 7 cm/sec to be markedly d i f f e r e n t  t h a t f o r higher l i q u i d  fluxes.  than  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 g e n e r a l i z e d wake model i t was assumed t h a t the r e l a t i v e v e l o c i t y f o r a l l gas f l u x e s i s w e l l r e p r e s e n t e d by equation 2.107 w i t h r  g  = 6 mm [ 7 9 ]  r  the d i s c r e p a n c y from the measured v a l u e s i s n o t e n t i r e l y unexpected.  To improve the p r e d i c t i o n s i t would be  necessary to o b t a i n a r e l a t i o n s h i p f o r d e s c r i b i n g the average diameter  o f bubbles  i n the swarm as a f u n c t i o n o f  195  0.8  0.41  0  .  .  2  4  _  i  6  _  . 8  « 10  f 12  . I 14 16  < j 2 > , cm /sec  FIGURE 4.3a  LIQUID FRACTION DATA OF MICHELSEN AND 0 S T E R G A A R D ' [14] FOR 3 MM GLASS BEADS ( p - j = 1 4 . 0 ; 8-^=6.6 g e n e r a l i z e d wake model; — ; gasrfree model) 1  196  X, = 0.6  h: 0  - : ° ^ o o - o o uJ 0.7  < 9  o > Q UJ 00  x =o k  0.5 0  8  10  12  14 16  <L> > cm/sec FIGURE 4.3b  BED VOIDAGE DATA OF MICHELSEN AND 0STERGAARD [14] FOR 3 MM GLASS BEADS ( O - J i = 1 4 . 0 ; H-j-^6.6; ' g e n e r a l i z e d wake model; gas-free model)  197 gas and l i q u i d f l u x e s and p a r t i c l e p r o p e r t i e s . However, i f i t i s assumed a g a i n t h a t the r o l e o f bubble wakes i s i n s i g n i f i c a n t , the g a s - f r e e model can be used f o r d e s c r i b i n g the bed behaviour.  Thus f o r the l i q u i d  flow r a t e of 14.0 cm/sec, the l i q u i d f r a c t i o n s  calculated  from e q u a t i o n 4.11, u s i n g the measured v a l u e s of gas holdup [14], are shown i n F i g u r e 4.3a.  These v a l u e s are found  to be i n good agreement with the experimental v a l u e s up to a gas flow r a t e o f 5.0 cm/sec, the agreement becoming poorer f o r l a r g e r gas flow r a t e s .  The values of bed voidage,  c a l c u l a t e d from e q u a t i o n 2.91 and presented i n F i g u r e 4.3b, e x h i b i t a s i m i l a r range of agreement w i t h the measured values.  However, f o r the l i q u i d flow r a t e o f 6.6  cm/sec,  the agreement o f c a l c u l a t e d v a l u e s of l i q u i d f r a c t i o n and bed voidage w i t h the corresponding experimental v a l u e s was poor a t almost a l l gas flow r a t e s . beds o f 3 mm  Therefore f o r f l u i d i z e d  g l a s s beads, the r o l e of bubble wakes  be c o n s i d e r e d t o t a l l y i n s i g n i f i c a n t .  cannot  N e v e r t h e l e s s , the  bed voidage p r e d i c t e d by the g a s - f r e e model can be used as a first  approximation.  Beds of 1 mm p a r t i c l e s  (Table 4.8 and F i g u r e 4.4)  The data of M i c h e l s e n and 0 s t e r g a a r d show a g r a d u a l reduction i n l i q u i d bed i s i n c r e a s e d .  f r a c t i o n as the gas f l u x through the For a v o l u m e t r i c l i q u i d f l u x o f 7.8  cm/sec,  TABLE  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 FOR T H R E E - P H A S E F L U I D I Z E D =  d  Gas Flux,  P  Measured e " 1 1  1 . 2 5 mm ,  P  2 . 67  =  3  (1) P  gm/cc, Liquid  Predicted k  /  2  Flux,  J1  (2)  >  = 7 . 8 cm/sec, D =6 . 0  Predicted  HI P  in  2  <  1  inch  (3)  Predicted  e /e'" k 2  2  BEDS  k  / e  / E  2  e'" l E  (cm/sec)  -  -  -  -  0.0  0 .780  1.0  0.710  0.027  2.225  0 .667  0.021  1.298  2.0  0 .680  0.045  1.539  0.635  0.034  1.026  3.0  0 .663  0 .060  0.869  0 .644  0 .045  0.826  4.0  0.650  0 .070  0 .416  0.667  0.054  0.667  5.0  0.640  0.079  0.181  0 .684  0.063  0.532  6.0  0 .630  0.088  0.075 .  0;693  0 .072  7.0  0 .625  0.094  0.030  0 .696  8.0  0.620  0 .100  0.012  9.0  0 .615  0.107  10.0  0.612  0.112  11.0  0 .609  0 .118  12.0  0 .608  0 .124  13 . 0  0 .605  0.128  14.0  0 .600  15.0  -  -  by M i c h e l s e n  0 .728  0.786  _  _  0.706  (4) £"• 2  _  0.021  4 .974  0 .610  0.034  0.706  .040  3.322  0 .570  0 .076  0.683  .057  2.615  0.537  0 .123  0.669  .072  2.202  0.506  0.173  0 .662  .087  1.926  0.477  0 .226  0.411  0.661  .101  0 .080  0.393  0 .645  .112  0 .700  0.087  0 .377  0 .630  .122  -  0 .005  0 .697  0 .094  0 .362  0.616  .132  0 .002  0.695  0.101  0 .348  0.602  .140  0.001  0 .693  0 .108  0.335 •  0.590  .148  0.000  0 .690  0 .114  0 .324  0 .577  .154  0.000  0 .688  0 .120  0.315  0 .564  .160  0 .000  0 .686  0 .126  0.307  0 .552  .166  0.000  0 .684  0 .132  0.298  0.540  .171  -  and  0stergaard  (1)  Measurements  (2)  Predicted  values  from  correlations  (3)  Predicted  values  from  generalized  (4)  Predicted  values  f