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UBC Theses and Dissertations

Gas absorption in cocurrent turbulent bubble flow Lamont, John Craig 1966

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GAS ABSORPTION IN C O C U R R E N T T U R B U L E N T B U B B L E FLOW by JOHN CRAIG L A M O N T B. A. Sc. , University of British Columbia,  I960  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E O F DOCTOR O F PHILOSOPHY in the Department of C H E M I C A L ENGINEERING  We accept this thesis as conforming to the required standard  T H E UNIVERSITY O F BRITISH C O L U M B I A  August,  1966  In presenting for  this thesis  i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s  an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h Columbia,, I agree  t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e study,  and  I f u r t h e r agree t h a t permission., f o r e x t e n s i v e c o p y i n g o f t h i s  t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s .  I t i s understood that  copying  or p u b l i c a t i o n of" t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n  permission.  Department o f The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  Columbia  The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION . FOR THE  DEGREE OF  DOCTOR OF PHILOSOPHY of  JOHN CRAIG LAMONT B.A.Sc,  The University of B r i t i s h Columbia, 1960  August 4, 1966, at 10:00 a.m. IN ROOM 207, CHEMICAL ENGINEERING BUILDING  COMMITTEE IN. CHARGE Chairman: I. McT, Cowan R. M. R„ Branion E„ G. Hauptman S. D. Cavers D„ A„ Ratkowsky J . S„ Forsyth D„ S„ Scott R„ W. Stewart External Examiner: I. A. Johnson McMaster University Hamilton, Ontario Research Supervisor:  D„ S, Scott  GAS ABSORPTION IN COCURRENT TURBULENT BUBBLE FLOW ABSTRACT Mass transfer rates have been measured f o r streams of CO2 bubbles of controlled frequency being.absorbed i n to water i n cocurrent pipeline flow.  Superficial liquid  Reynolds number varied from 1810 to 24000 . Mass transfer c o e f f i c i e n t s based on equivalent spherical areas were between 0.6 and 4.5 cm/min.  For 5/16-  and 5/8 inch I.D.  tubes oriented both horizontally and v e r t i c a l l y , the mass transfer c o e f f i c i e n t s were proportional to (Reynolds , .0.52 , ,, . -0.85 ,. , „ number) and (tube diameter) at high Reynolds J  number.  Bubble v e l o c i t i e s were measured for a l l test  sections and flow conditions.  Photographs of bubbles  in turbulent flow were obtained by a high speed f l a s h technique. The mass transfer results support a postulated mechanism of surface renewal by turbulent eddies which result from the mean flow of l i q u i d through the tube. Two t h e o r e t i c a l approaches have been described i n an attempt to r e l a t e the surface renewal rate to the pipe flow turbulence. A model based on.mixing length theory gives good agreement with the experimental r e s u l t s . the larger scales of motion dominate*  In this model  A second model was  based on the assumption that the very small scales dominate.  The flow and convective d i f f u s i o n equations  were solved for idealized viscous eddy c e l l s which represent the small motions.  The size, v e l o c i t y and mass  transfer rate of. these c e l l s were linked to the turbulent energy spectrum for both s o l i d / l i q u i d and gas/liquid  interfaces.  The p r e d i c t e d dependence of mass t r a n s f e r  coefficient  on Schmidt number .and energy d i s s i p a t i o n i s  i d e n t i c a l w i t h experimental  results for solid surfaces. .69  However, the Reynolds number dependence (Re than f o r the p r e s e n t experiments. eddy c e l l model may be v a l i d s u f f i c i e n t l y h i g h l y developed  ) i s higher  N e v e r t h e l e s s , the  f o r bubbles  and s o l i d s i n  turbulence*  GRADUATE STUDIES  Field  of Study-  Gas A b s o r p t i o n D, S S c o t t S. D. Cavers  S o l v e n t E x t r a c t i o n and Gas Absorption F l u i d and P a r t i c l e Dynamics  £  R„ M. R.  Branion  R. W.  Stewart  Turbulence Other S t u d i e s : I n d u s t r i a l K i n e t i c s and C a t a l y s i s Chemical E n g i n e e r i n g Thermodynamics  P. L  D. S. S c o t t e  Silveston  Mathematical O p e r a t i o n s i n Chemical E n g i n e e r i n g  N, E p s t e i n  Analogue Computers  E. V. Bohn  Programming  H. Dempster  Complex V a r i a b l e and . D i f f e r e n t i a l Equations  E. Macskasy  Fortram  PUBLICATION Lamont, J . C. and S c o t t , D. S., "Mass T r a n s f e r From Bubbles i n Cocurrent Eng,,  In P r e s s .  Flow", Can. J , Chem.  T h e s i s S u p e r v i s o r : Dr. D. S. Scott  ii  ABSTRACT  M a s s t r a n s f e r r a t e s h a v e b e e n m e a s u r e d f o r s t r e a m s of  CO^  b u b b l e s of c o n t r o l l e d f r e q u e n c y b e i n g a b s o r b e d into w a t e r i n c o c u r r e n t p i p e l i n e flow. 24000. •  S u p e r f i c i a l l i q u i d R e y n o l d s n u m b e r v a r i e d f r o m 1810 to  M a s s t r a n s f e r c o e f f i c i e n t s b a s e d on e q u i v a l e n t s p h e r i c a l a r e a s  w e r e b e t w e e n 0. 6 and 4. 5 c m / m i n .  F o r 5/16-  o r i e n t e d both h o r i z o n t a l l y and v e r t i c a l l y ,  a n d 5/8 i n c h I.D.  the m a s s t r a n s f e r  tubes  coefficients  0 5Z - 0 85 w e r e p r o p o r t i o n a l to ( R e y n o l d s n u m b e r ) ' a n d (tube d i a m e t e r ) ' at high Reynolds number.  Bubble velocities w e r e m e a s u r e d f o r all test  s e c t i o n s and flow conditions.  Photographs of bubbles in turbulent flow  were obtained by a high speed flash technique.  T h e m a s s t r a n s f e r r e s u l t s s u p p o r t a p o s t u l a t e d m e c h a n i s m of s u r f a c e r e n e w a l b y t u r b u l e n t e d d i e s w h i c h r e s u l t f r o m the m e a n f l o w o f l i q u i d t h r o u g h the tube.  Two  t h e o r e t i c a l approaches have been  described  i n an a t t e m p t to r e l a t e the s u r f a c e r e n e w a l r a t e to the p i p e f l o w t u r b u l e n c e .  A m o d e l b a s e d on m i x i n g l e n g t h t h e o r y g i v e s g o o d a g r e e m e n t the e x p e r i m e n t a l r e s u l t s . inate.  A  s e c o n d m o d e l was  scales dominate.  with  In t h i s m o d e l . t h e l a r g e r s c a l e s of m o t i o n d o m b a s e d o n t h e a s s u m p t i o n that the v e r y s m a l l  T h e flow and c o n v e c t i v e d i f f u s i o n equations w e r e s o l v e d  f o r i d e a l i z e d v i s c o u s e d d y c e l l s w h i c h r e p r e s e n t the s m a l l m o t i o n s ,  The  s i z e , v e l o c i t y a n d m a s s t r a n s f e r r a t e o f t h e s e c e l l s w e r e l i n k e d to the turbulent e n e r g y s p e c t r u m f o r both s o l i d / l i q u i d and g a s / l i q u i d i n t e r f a c e s .  iii  The predicted dependence of mass transfer coefficient on Schmidt number and energy dissipation is identical with experimental results for solid sur•69  faces.  However, the Reynolds number dependence (Re  for the present experiments.  ) is higher than  Nevertheless, the eddy cell model maybe valid  for bubbles and solids in sufficiently highly developed turbulence.  iv  T A B L E OF CONTENTS  Page INTRODUCTION  THEORY  -  M a s s T r a n s f e r in C o c u r r e n t F l o w / . T.  i"  -  Surface Renewal Models  5  Scope of. R e s e a r c h  8  -  M a s s T r a n s f e r Equations  . . .  9  S i m p l i f i e d Equations for L o w Gas /  '- - »  APPARATUS  -  L i q u i d Ratios  16  C o r r e c t i o n for Bubble V e l o c i t y  18  C o r r e c t i o n for P r e s s u r e D r o p  20  G e n e r a l Requirements  22  . .  L i q u i d C y c l e and T e s t Sections  . . . . .  23  Gas F l o w and Bubble N o z z l e s  . . . . .  27  A u x i l i a r y Equipment for F r e q u e n c y , Velocity, PROCEDURE  ;  -  and Photography  33  A b s o r p t i o n Runs  39  Sampling  41  A n a l y s i s for D i s s o l v e d C O ^  4 5  R E S U L T S A N D DISCUSSION  49 A. B U B B L E ' V E L O C I T Y B. A B S O R P T I O N R E S U L T S  49 . . . . . .  57  H o r i z o n t a l 5 / 1 6 - i n c h T e s t Section . .  57  Uncertainty in Reynolds Number Exponent  68  V  Page 'Comparison of a l l T e s t Sections  75  E v i d e n c e cf or ^Turbulent T r a n s f e r M e c h a n i s m  8 1  TURBULENT RENEWAL MODELS  -  —  84  A. T U R B U L E N T M I X I N G L E N G T H M O D E L  87  B. I D E A L I Z E D E D D Y C E L L M O D E L . ... .  90  Introduction  90  ....  The M o d e l  92  Idealized Viscous Eddy Cells  9 5  D i f f u s i o n into E d d y C e l l .  98  E n e r g y of the Motions  100  O v e r a l l T r a n s f e r Rate, k_  10 3  C. C O N C L U S I O N S . . . . . .  105  NOMENCLATURE  110  REFERENCES. ..  114  APPENDICES I II III  EQUIPMENT AND INSTRUMENT SPECIFICATIONS  117  B U B B L E VELOCITY DATA  12 3  C A L C U L A T I O N S F O R A B S O R P T I O N RUNS  IV V VI  .  129  D A T A A N D R E S U L T S F O R A B S O R P T I O N R U N S . . ...  14 6  CALCULATIONS FOR RENEWAL MODELS  154  SOLUTIONS FOR F L O W AND DIFFUSION F O R EDDY CELLS  158  vi  LIST O F  FIGURES  Page 1.  Differential Element for M a t e r i a l Balance  10  2.  The  17  3.  S c h e m a t i c D i a g r a m of A p p a r a t u s  2 4  4;  H o r i z o n t a l Test Sections  2 8  5.  V e r t i c a l Test Sections  2 9  6.  Bubble Nozzle A s s e m b l y  3 0  7.  I n v e r t e d Cup  8.  T r a c k i n g W i r e Method for Bubble Velocity  3 5  9-  O p t i c a l A r r a n g e m e n t for Bubble Photography  3 7  10. 11.  Function,  F,  (f)  Nozzles •  31  Concentration Profile A c r o s s Concentrations  Across  5/16  H o r i z o n t a l T u b e : Re  = s 10200 . = 7020 and  5/l6 H o r i z o n t a l Tube: Re S  4810  12.  V e l o c i t y R a t i o s f o r the H o r i z o n t a l T u b e s  13.  V e l o c i t y R a t i o s f o r the V e r t i c a l T u b e s  14.  V e l o c i t y R a t i o s i n V e r t i c a l F l o w at F i x e d R e y n o l d s N u m b e r  15.  B u b b l e s i n V e r t i c a l F l o w i n 5/16  16.  D e p e n d e n c e of A b s o r p t i o n F u n c t i o n o n T e s t L e n g t h  17.  D e p e n d e n c e of A b s o r p o t i o n  18.  3 D e p e n d e n c e of A b s o r p t i o n F u n c t i o n on N- L../.Q  19.  D e p e n d e n c e of A b s o r p t i o n F u n c t i o n on R  20.  E f f e c t of I n c l u d i n g F e e d N o z z l e sorption Results  21.  Absorption  Data  43  44 51 5 3 54  Tube  5 5  F u n c t i o n on B u b b l e F r e q u e n c y  R e p r e s e n t e d as N T U  . .  g  5 9 6 0 6 2  •. -  e  and E n t r a n c e  ........  .  64  Length in Ab- . 6 6 6 8  vii Page 22.  E f f e c t o f R e y n o l d s N u m b e r on M a s s T r a n s f e r e f f i c i e n t f o r H o r i z o n t a l 5/16  23.  Co-  Tube  69  P o s t u l a t e d I n t e r p r e t a t i o n of M a s s T r a n s f e r to I l l u s t r a t e the U n c e r t a i n t y i n the R e y n o l d s  Coefficients Number  Exponent 24. 25.  7 3  B u b b l e s i n H o r i z o n t a l F l o w i n 5/16 Comparison  Tube  7 4  of M a s s T r a n s f e r C o e f f i c i e n t s f o r a l l T e s t  Sections  7 6  26.  B u b b l e s i n H o r i z o n t a l F l o w i n 5/8  Tube  27.  F l u i d E d d y D e f l e c t e d b y an I n t e r f a c e  9 3  28.  A Small Eddy Superimposed  9 4  29.  Idealized Viscous Eddy Cell  9 6  30.  P o s s i b l e F l u i d M o t i o n s S i m i l a r to the E d d y C e l l  9 7  31.  M a s s T r a n s f e r Into I d e a l i z e d E d d y C e l l . . -  9 9  1-1  F e e d N o z z l e B l o c k f o r 5/8 T u b e s  119  1-2  F e e d Nozzle Fittings  1-3  5/8  1-4  Counter Circuit Details  121  1-5  Time Delay Circuit  122  III-1  T y p i c a l Data Sheet for A b s o r p t i o n Run  130  VI-1  Idealized Viscous Eddy C e l l  159  VI-2  E l e m e n t of C o n c e n t r a t i o n B o u n d a r y  VI-3  T y p i c a l N o d e of M e s h f o r F i n i t e D i f f e r e n c e s  oh a L a r g e One  ,  8 0  12 0  -Inch Tubing Connectors  120  Layer  162 166  viii  LIST O F T A B L E S Page 1.  S u m m a r y o f A b s o r p t i o n R e s u l t s f o r 5/16 H o r i z o n t a l Tube  2.  E f f e c t o f P r e s s u r e V a r i a t i o n im M a s s  63  Transfer  Calculations  3.  72  S u m m a r y of Absorption. Results for V e r t i c a l T u b e s a n d 5/8 H o r i z o n t a l T u b e  78  ix  ACKNOWLEDGEMENT  The  a u t h o r w i s h e s to e x p r e s s h i s t h a n k s to t h e f a c u l t y a n d  and s t a f f o f the C h e m i c a l E n g i n e e r i n g : D e p a r t m e n t , British Columbia. particular,  T h e U n i v e r s i t y of  T h e author; w i s h e s to t h a n k - M r . JS.  Rudischer, in  for his assistance and cooperation.  The  a u t h o r i s i n d e b t e d to M r . F . K. B o w e r s o f the E l e c t r i c a l  E n g i n e e r i n g D e p a r t m e n t f o r t h e d e s i g n o f the b u b b l e c o u n t e r c i r c u i t , a n d to the s t a f f o f t h e I n s t i t u t e of E a r t h S c i e n c e s f o r t h e i r a s s i s t a n c e a n d f o r the u s e o f t h e i r i n s t r u m e n t s .  T h a n k s i s e x t e n d e d t o D r . D; S. Scott, u n d e r w h o s e g u i d a n c e t h i s w o r k was u n d e r t a k e n .  F i n a n c i a l s u p p o r t f o r this r e s e a r c h was m o s t g r a t e f u l l y  received  f r o m the N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a a n d f r o m the B r i t i s h .Hydro and P o w e r Authority.  Columbia  1  INTRODUCTION M a s s T r a n s f e r in C o c u r r e n t F l o w C o c u r r e n t g a s / l i q u i d pipe flow has been extensively studied during the past five or six y e a r s .  T h i s type of flow occurs n a t u r a l l y in g a s - o i l  pipelines, p r o c e s s equipment such as condensers and r e b o i j e r s , steam generating equipment.  and i n  In addition c o c u r r e n t pipe flow provides an  efficient means of contacting a gas and l i q u i d for c a r r y i n g out p h y s i c a l absorption or c h e m i c a l r e a c t i o n .  Note that for a pure gas there is no  d r i v i n g force disadvantage for c o c u r r e n t contacting.  Heat t r a n s f e r and  p r e s s u r e drop are of p r i m a r y importance i n the f o r m e r situations, and are of considerable design importance in the l a t t e r .  Consequently heat  t r a n s f e r and p r e s s u r e drop have r e c e i v e d the most attention and are s t i l l being v e r y actively investigated. M a s s t r a n s f e r r e c e i v e d little attention until about 1962, the l i t e r a t u r e .  at which time only two papers (1,  2) were available i n  In 1963 Scott (3) and Dukler and Wicks (4) presented  thorough reviews of the l i t e r a t u r e on p r e s s u r e drop and heat and mass t r a n s f e r in cocurrent two-phase pipe flow.  Some papers on mass  transfer  which have appeared since 1963 w i l l subsequently be d i s c u s s e d . B y nature, two-phase flow exhibits a v a r i e t y of widely different flow patterns depending on the relative and absolute flow rates of the two phases.  M u c h effort has been expended just to chart the o c c u r r e n c e of  various flow patterns as a function of the flow v a r i a b l e s and to m e a s u r e the holdup of the phases (5,  6).  Because of the different flow patterns  2 that a r e p o s s i b l e no s i n g l e , c o m p r e h e n s i v e t r e a t m e n t  can d e s c r i b e and  e x p l a i n m o m e n t u m , heat o r m a s s t r a n s f e r for a l l two-phase flow s i t u a t i o n s . I n s t e a d i t i s n e c e s s a r y to c o n s i d e r a n u m b e r o f d i s t i n c t r e g i m e s ,  for which  the t r e a t m e n t i n c r e a s e s i n d i f f i c u l t y m o r e o r l e s s w i t h the c o m p l e x i t y of the i n t e r f a c i a l g e o m e t r y .  The  m e c h a n i s m for interphase mass transfer  i s m u c h d i f f e r e n t f o r e x a m p l e f o r d i s c r e t e gas b u b b l e s continuous l i q u i d stream, v e l o c i t y c o r e of gas.  moving in a  and f o r a m o v i n g l i q u i d annular f i l m with a h i g h -  T h e r e f o r e a c o m p l e t e k n o w l e d g e of m a s s t r a n s f e r f o r  t w o - p h a s e p i p e f l o w r e q u i r e s a k n o w l e d g e of e v e r y r e g i m e that m a y  be  encountered.  R e c e n t l y a n u m b e r of m a s s t r a n s f e r i n v e s t i g a t i o n s h a v e b e e n r e p o r t e d . H a y d u k (7) m e a s u r e d a n d c o r r e l a t e d l i q u i d - p h a s e c o n t r o l l e d m a s s t r a n s f e r i n h o r i z o n t a l pipes o v e r a wide range of flow r a t e s w h i c h gave bubble, plug, slug and annular flow p a t t e r n s . s t u d i e d to d e t e r m i n e  H a y d u k a l s o v a r i e d the s y s t e m s  the e f f e c t of t h e f l u i d p r o p e r t i e s .  In h i s w o r k m a s s  t r a n s f e r c o e f f i c i e n t s c o u l d not be o b t a i n e d b e c a u s e s u r f a c e a r e a s w e r e  un-  known.  For  H e n c e the r e s u l t s w e r e p r e s e n t e d i n t e r m s of t r a n s f e r u n i t s .  certain regimes,  the i n t e r f a c e g e o m e t r y i s s i m p l e e n o u g h t h a t an e s t i m a t e  o f s u r f a c e a r e a c a n be m a d e , w h i c h m a k e s p o s s i b l e a t h e o r e t i c a l a n a l y s i s of the p r o c e s s i f s o m e i n f o r m a t i o n about the h y d r o d y n a m i c s n e a r the i n t e r face c a n be i n f e r r e d .  T h u s e x p e r i m e n t a l m e a s u r e m e n t s of g a s - p h a s e  controlled mass transfer in horizontal,  annular f i l m flow have b e e n  t h e o r e t i c a l l y e x p l a i n e d w i t h s o m e s u c c e s s (2).  A l s o the l i q u i d - p h a s e c o n -  t r o l l e d t r a n s f e r f r o m l o n g s l u g s i n a v e r t i c a l gas l i f t h a s b e e n s t u d i e d (8)  and good a g r e e m e n t has b e e n found between t h e o r y a n d e x p e r i m e n t . K i n g ( 9 ) has r e c e n t l y m e a s u r e d m a s s t r a n s f e r i n h o r i z o n t a l f r o t h flow i n which m a n y s m a l l , n e a r l y s p h e r i c a l bubbles phase.  Interfacial areas,  efficients were estimated,  residence times  a r e d i s t r i b u t e d i n the l i q u i d and hence m a s s t r a n s f e r c o r  a n d s o m e c o n c l u s i o n s c o u l d be d r a w n a b o u t  the l i q u i d - p h a s e c o n t r o l l e d p r o c e s s .  A f l o w p a t t e r n w h i c h i s p e r h a p s t h e s i m p l e s t to a n a l y z e i s the type of  b u b b l e flow i n w h i c h s i n g l e gas bubbles  the p i p e d i a m e t e r bubbles  of a p p r o x i m a t e l y  o n e - t h i r d of  a r e s e p a r a t e d b y at l e a s t s e v e r a j . tube d i a m e t e r s .  s m a l l e n o u g h that the s h a p e i s a p p r o x i m a t e l y  spherical,  For  surface  a r e a c a n be o b t a i n e d f r o m the f r e q u e n c y and gas v o l u m e t r i c flow. F o r s p a c i n g l a r g e e n o u g h that i n t e r a c t i o n s a r e n e g l i g i b l e , t h e h y d r o d y n a m i c s of the l i q u i d s h o u l d a p p r o x i m a t e the r e l a t i v e l y w e l l known c a s e of s i n g l e phase flow i n a pipe. ing meaningful  H e n c e t h e r e should be a good p o s s i b i l i t y of o b t a i n -  l i q u i d - p h a s e m a s s t r a n s f e r c o e f f i c i e n t s a n d u s i n g t h e m to  d r a w c o n c l u s i o n s about the n a t u r e of the t r a n s f e r p r o c e s s .  Unfortunately  this bubble flow m a y be of little p r a c t i c a l i n t e r e s t f o r g a s - l i q u i d contacting b e c a u s e i t o c c u r s o n l y f o r v e r y l o w g a s to l i q u i d v o l u m e r a t i o s o f the o r d e r of 3 % o r less.  Nevertheless,  b u b b l e f l o w m a y o c c u r i n the l a t t e r  s e c t i o n of an a b s o r b e r  or condenser,  and its m a s s t r a n s f e r  behaviour  m u s t b e s t u d i e d i n o r d e r to o b t a i n a c o m p r e h e n s i v e k n o w l e d g e o f t w o - p h a s e pipeline mass transfer.  F u r t h e r m o r e , knowledge gained f r o m this flow  p a t t e r n m i g h t b e a p p l i c a b l e to h e l p i n t e r p r e t  mass transfer in certain  o t h e r r e g i m e s s u c h as the f r o t h f l o w m e n t i o n e d a b o v e . F i n a l l y a n a n a l y s i s  4 df the t r a n s f e r p r o c e s s f o r b u b b l e f l o w m a y y i e l d i n f o r m a t i o n o f f u n d a m e n t a l i n t e r e s t to the g e n e r a l f i e l d o f m a s s t r a n s f e r a c r o s s a f l u i d i n t e r face.  T h i s latter p o s s i b i l i t y w i l l be d i s c u s s e d f u r t h e r i n s o m e detail.  F i r s t i t is; n e c e s s a r y ' t o d i s c u s s t h e g e n e r a l f e a t u r e s p o s t u l a t e d f o r this bubble flow. T h e l i q u i d flows u n d e r c o n s i d e r a t i o n a r e h i g h enough that t u r b u l e n t f l o w w o u l d e x i s t e v e n i n t h e a b s e n c e o f g a s b u b b l e s .  The  flattened turbulent velocity profile should result in bubble velocities n e a r l y t h e s a m e as t h e l i q u i d v e l o c i t y o v e r the c e n t r a l r e g i o n o f the p i p e , except-in s o m e c a s e s of v e r t i c a l flow i n w h i c h the t e r m i n a l v e l o c i t y of the ;  bubble i s m u c h l a r g e r than the l i q u i d v e l o c i t y . r e l a t i v e to t h e b u b b l e s h o u l d b e s m a l l . to the f l u c t u a t i n g t u r b u l e n t v e l o c i t i e s ,  The mean liquid velocity  H o w e v e r the bubble i s e x p o s e d a n d i t i s p o s t u l a t e d that this  turbulent v e l o c i t y f i e l d p r e d o m i n a t e s o v e r the m e a n r e l a t i v e v e l o c i t y . i s f u r t h e r p o s t u l a t e d that i n this t u r b u l e n t situation, m a s s t r a n s f e r place b y surface renewal;  It  takes  i . e. , e d d y m o t i o n s b r i n g f r e s h f l u i d v e r y c l o s e  to t h e s u r f a c e s o t h a t m a s s i s t r a n s f e r r e d b y m o l e c u l a r  diffusion for a  s h o r t t i m e b e f o r e the f l u i d i s r e p l a c e d b y m o r e f r e s h f l u i d .  Now  i n this  b u b b l e flow, the t u r b u l e n t v e l o c i t y f i e l d s h o u l d b e d e t e r m i n e d  primarily  b y t h e f l o w o f l i q u i d t h r o u g h the p i p e at h i g h e n o u g h p i p e f l o w  superficial  ;  Reynolds numbers.  T h e r e f o r e the pipe d i a m e t e r  s h o u l d be the m e a n i n g f u l diameter  and l i q u i d flow rate  c o r r e l a t i n g p a r a m e t e r s i n s t e a d of the bubble  and relative velocity.  F o r b u b b l e f l o w H a y d u k (7) o b t a i n e d : the  r e s u l t t h a t m a s s t r a n s f e r i s p r o p o r t i o n e d to t h e m o l e c u l a r  diffusivity  5  r a i s e d to an e x p o n e n t of 0. 5, w h i c h i s c o n s i s t e n t w i t h the s u r f a c e r e n e w a l mechanism  (10).  In h o r i z o n t a l b u b b l e f l o w the b u b b l e s , q u a r t e r to o n e - h a l f of a p i p e d i a m e t e r , due to t h e i r b o u y a n c y .  The  which are approximately  one-  t r a v e l c l o s e to the u p p e r p i p e w a l l  t h i n f i l m of l i q u i d b e t w e e n the s t a t i o n a r y w a l l  and the m o v i n g b u b b l e s h o u l d p r o v i d e e f f i c i e n t m a s s t r a n s f e r i n a d d i t i o n to the a b o v e t u r b u l a n t r e n e w a l .  H o w e v e r i n v e r t i c a l flow this  m e c h a n i s m s h o u l d b e m u c h l e s s i m p o r t a n t b e c a u s e the b u b b l e s a m o r e c e n t r a l p o s i t i o n i n the p i p e .  second travel in  T h e r e f o r e a c o m p a r i s o n of m a s s t r a n s -  f e r r a t e s f o r h o r i z o n t a l and v e r t i c a l b u b b l e f l o w s h o u l d h e l p to c o n f i r m w h e t h e r the r a t e c o n t r o l l i n g m e c h a n i s m i s s u r f a c e r e n e w a l b y t u r b u l e n t eddies.  It i s i n t e r e s t i n g to note a l s o t h a t c o c u r r e n t b u b b l e f l o w of the t y p e s t u d i e d i n t h i s w o r k i s one o f the v e r y few p o s s i b l e w a y s to s t u d y m a s s transfer in a turbulent shear field,  a n d i h e n c e it r e p r e s e n t s a u s e f u l r e -  s e a r c h tool.  Surface Renewal Models An  a d d i t i o n a l t e s t of the m e c h a n i s m p o s t u l a t e d a b o v e w o u l d b e to  c o m p a r e ' t h e e x p e r i m e n t a l e f f e c t o f l i q u i d f l o w on t h e t r a n s f e r r a t e a g a i n s t a t h e o r e t i c a l p r e d i c t i o n f r o m a m o d e l b a s e d on t h i s m e c h a n i s m .  However,  t h e r e i s no t h e o r y a v a i l a b l e w h i c h s a t i s f a c t o r i l y r e l a t e s m a s s t r a n s f e r at a g a s / l i q u i d i n t e r f a c e to the s t r u c t u r e of the t u r b u l e n t f i e l d i n the n e a r b y liquid.  In D a n c k w e r t s ' m o d e l (10),  surface elements are r e p l a c e d by  6  f r e s h l i q u i d after s o m e r a n d o m t i m e d e t e r m i n e d by per  second.  This  a surface  r e n e w a l rate,  d e v i c e s a t i s f a c t o r i l y p r e d i c t s the w e l l - t e s t e d  p e n d e n c e of m a s s t r a n s f e r c o e f f i c i e n t on the d i f f u s i v i t y to the power for liquid-phase  de-  one-half  c o n t r o l l e d t r a n s f e r at the g a s / l i q u i d i n t e r f a c e ;  i . e. ,  a c c o r d i n g to the D a n c k w e r t s m o d e l ,  Y ^  7  D  k  =  L  (i)  h o w e v e r , i t i s not p o s s i b l e to p r e d i c t the v a l u e of^,,  c h a n g e s w i t h the i n t e n s i t y of the t u r b u l e n c e n e a r a s u r f a c e . r e n e w a l models.(11,  12,  13) c o n s i d e r c o m b i n e d f i l m and  p a r t i a l r e n e w a l s o r r e j u v e n a t i o n s / and  v  o r e v e n the w a y  different surface  M o r e elaborate  renewal transfer, age  distributions.  T h e s e m o d e l s p r e d i c t d i f f u s i v i t y d e p e n d e n c e , but do not p r o v i d e a n y f o r r e l a t i n g the t r a n s f e r to the t u r b u l e n c e l e v e l .  R e c e n t l y K i n g (14)  p r o p o s e d a r e n e w a l m o d e l in, w h i c h t r a n s f e r to s u r f a c e m o l e c u l a r diffusivity, p i r i c a l way t a l way  elementsis  f r o m the s u r f a c e .  A g a i n t h e r e i s no  to r e l a t e the e d d y d i f f u s i v i t y to the t u r b u l e n c e ,  p a r t i c u l a r l y that of C a l d e r b a n k and  to be  due  surface  dispersed  M o o - Y o u n g (15)  em-  fundamen-  correlations,  for t r a n s f e r  from  p a r t i c l e s u n d e r the i n f l u e n c e of t u r b u l e n c e .  c o r r e l a t i o n of C a l d e r b a n k and  i n t e r e s t to the  due.to  although some con-  clusions have been i n f e r r e d f r o m e x p e r i m e n t a l m a s s t r a n s f e r  The  has  and to a n e d d y d i f f u s i v i t y w h i c h v a r i e s i n s o m e  with distance  f i x e d s u r f a c e s and  basis  M o o - Y o u n g i s of p a r t i c u l a r  p r e s e n t w o r k s i n c e i t c o n s i d e r s the m a i n t r a n s f e r p r o c e s s  to t u r b u l e n t  e d d i e s of s i z e s s m a l l e r t h a n the d i m e n s i o n of the  (particle or fixed surface).  Further,, this c o r r e l a t i o n  empirically  relates the mass t r a n s f e r to the energy dissipated by the turbulence,, which is the fundamental p a r a m e t e r in determining the size and energy of the s m a l l motions, and which can be r e a d i l y estimated for a number of t u r bulent flow situations.  However this c o r r e l a t i o n did not fit data for gas  ;  bubbles, p o s s i b l y due to the different nature of the g a s / l i q u i d interface. A l t e r n a t e l y , the l a c k of agreement for gas bubbles m a y s i m p l y indicate that the n e c e s s a r y condition of turbulent velocities predominating was not realized.  The applicability of C a l d e r b a n k ' s c o r r e l a t i o n to a g a s / l i q u i d  interface is difficult to evaluate,, because no m o d e l was presented to d e s c r i b e the fluid motions or the way they actually a i d the t r a n s f e r of m a s s (or heat).  Nevertheless,  this c o r r e l a t i o n may be of some use for bubble  flow, in that the extreme case of a s m a l l bubble contaminated with a monolayer of surface active m a t e r i a l m a y be s i m i l a r to the s o l i d p a r t i c l e . The turbulent t r a n s f e r at g a s / l i q u i d free (clean) surfaces, l i q u i d and p a r t i a l l y - c o n t a m i n a t e d g a s / l i q u i d interfaces,  at l i q u i d -  and at s o l i d s u r -  faces can only be s a t i s f a c t o r i l y c o m p a r e d and understood when an i d e a l i z e d but p h y s i c a l l y r e a l i s t i c m o d e l of the v e l o c i t y f i e l d near the v a r i o u s i n t e r faces is available. recent papers (14,  S e v e r a l aspects of the p r o b l e m are c o n s i d e r e d in 16,  motions has e m e r g e d .  17,  18,  19) but no satisfactory link to the turbulent  T h i s a r e a of fluid mechanics needs a m u c h i m p r o v e d  understanding, because the mass transfer i s s i m p l e in p r i n c i p l e once the fluid velocity field is known i n detail.  A study of mass t r a n s f e r in c o -  current bubble flow m a y lead to some conclusions that w i l l be of general interest to the field of turbulent mass  transfer.  8  Scope of R e s e a r c h T h e e x p e r i m e n t a l w o r k o f t h i s t h e s i s i n v o l v e d the m e a s u r e m e n t o f m a s s t r a n s f e r rates f o r the type of c o c u r r e n t bubble flow d i s c u s s e d  above.  T h e p r i m a r y a i m s w e r e to e s t a b l i s h the m a g n i t u d e o f m a s s t r a n s f e r c o efficients,  a n d to t e s t the a p p l i c a b i l i t y o f the t u r b u l e n t r e n e w a l m e c h a n i s m .  The c h e m i c a l s y s t e m was not v a r i e d b e c a u s e the d i f f u s i v i t y dependence of m a s s t r a n s f e r h a s b e e n e s t a b l i s h e d b y H a y d u k (7) f o r b u b b l e f l o w . T h e s y s t e m p u r e c a r b o n dioxide - water was u s e d for its convenience* and b e c a u s e r e s i s t a n c e to m a s s t r a n s f e r i s i n the l i q u i d p h a s e s o that the s o l u t i o n o f g a s c a n b e t r e a t e d p u r e l y as a l i q u i d - p h a s e a b s o r p t i o n p r o c e s s . T w o tube d i a m e t e r s and a r a n g e of bubble d i a m e t e r s a n d t u r b u l e n t l i q u i d flow rates w e r e studied. was  Bubble flow in both h o r i z o n t a l and v e r t i c a l tubes  s t u d i e d i n o r d e r to d e t e r m i n e i f the t h i n f i l m b e t w e e n b u b b l e a n d tube  w a l l i n the h o r i z o n t a l c a s e i s a n i m p o r t a n t t r a n s f e r m e c h a n i s m . v e l o c i t i e s w e r e r e q u i r e d i n o r d e r to c a l c u l a t e m a s s t r a n s f e r  Bubble  coefficients,  and t h e r e f o r e s u c h v e l o c i t y d e t e r m i n a t i o n s w e r e m a d e f o r a l l flow conditions used.  T w o attempts w e r e m a d e to t h e o r e t i c a l l y r e l a t e the rate of m a s s t r a n s f e r to the t u r b u l e n t f i e l d b e h a v i o u r .  I n the f i r s t a p p r o a c h ,  the D a n -  c k w e r t s r e n e w a l m o d e l (10) w a s a p p l i e d , w i t h the d e p e n d e n c e o f the r e n e w a l r a t e , - ^ , o n the t u r b u l e n c e l e v e l e s t i m a t e d f r o m the m i x i n g l e n g t h c o n cept.  T h e s e c o n d a p p r o a c h c o n s i s t e d o f s e t t i n g up a n i d e a l i z e d m o d e l o f  the t u r b u l e n t m o t i o n s i n the i m m e d i a t e  v i c i n i t y o f the i n t e r f a c e , a n d  i n v e s t i g a t i n g the m a s s t r a n s f e r as a f u n c t i o n o f the t u r b u l e n c e .  9  THEORY  Mass Transfer  Equations  If t h e b u b b l e s i z e , bubbles, gas  o r frequency, i s known for u n i f o r m l y  sized  t h e n t h e a v e r a g e m a s s t r a n s f e r c o e f f i c i e n t c a n b e r e l a t e d to  a n d l i q u i d f l o w r a t e s a n d the l i q u i d p h a s e c o n c e n t r a t i o n b y m e a n s o f  a differential m a t e r i a l balance, andthis  equation c a n be i n t e g r a t e d  p r o v i d i n g c e r t a i n a s s u m p t i o n s a r e m a d e r e g a r d i n g the v e l o c i t y and shape of t h e b u b b l e s .  It i s c o n v e n i e n t to i n c o r p o r a t e t h e f o l l o w i n g s i m p l i f i -  c a t i o n s i n t o t h e m o d e l o f b u b b l e f l o w s h o w n di a g r a m a t i c a l l y i n F i g u r e 1.  . T h e bubbles a r e smooth spheres.  2.  • B u b b l e v e l o c i t y i s e q u a l t o t h e a v e r a g e v e l o c i t y o f the total v o l u m e t r i c  '  3.  flow,  that i s ,  '  I  P r e s s u r e d r o p a l o n g the t u b e i s n e g l i g i b l e r e l a t i v e to the a b s o l u t e p r e s s u r e  4.  1.  of the s y s t e m ,  B u b b l e s w i l l neither c o a l e s c e n o r b r e a k - u p within the tube;; that i s , t h e b u b b l e f r e q u e n c y , N  is  , remains con-  stant. 5.  L i q u i d f i l m r e s i s t a n c e to m a s s t r a n s f e r c o n t r o l s t h e t r a n s f e r rate.  6.  B u b b l e s a r e s u f f i c i e n t l y l a r g e that i n t e r n a l p r e s s u r e a function of diameter.  i s not  10 7.  T h e t e m p e r a t u r e is constant.  8.  A s i n g l e a v e r a g e v a l u e o f the m a s s t r a n s f e r c o e f f i c i e n t c a n b e a p p l i e d to the e n t i r e s u r f a c e o f a b u b b l e ,  and this  a v e r a g e c o e f f i c i e n t i s not s t r o n g l y d e p e n d e n t o n b u b b l e diameter.  Therefore,  a constant value c a n be a s s u m e d  o v e r t h e l e n g t h o f the a b s o r b e r .  This assumption may well  be q u e s t i o n a b l e , b u t t h e c o m p l e x i t y o f a n y m o r e c a t e d m o d e l i s not w a r r a n t e d  sophisti-  initially.  A s s u m p t i o n s 2 a n d 3 c a n be r e l a x e d b y i n c l u d i n g the v a r i a t i o n o f v e l o c i t y a n d p r e s s u r e , b u t i t i s c o n v e n i e n t to r e t a i n t h e s e r e s t r i c t i o n s a n d i n t r o d u c e v e l o c i t y a n d p r e s s u r e c o r r e c t i o n s at a l a t e r  F I G U R E 1.  DIFFERENTIAL ELEMENT BALANCE  FOR  initially  stage.  MATERIAL  11 T h e m a t e r i a l b a l a n c e i s a p p l i e d to an e l e m e n t of l e n g t h ,  dL,  of a b s o r b e r as s h o w n i n F i g u r e 1, a c r o s s w h i c h the d i f f e r e n c e i n c o n c e n t r a t i o n of d i s s o l v e d gas i s dq, w h e r e q i s e x p r e s s e d as the e q u i v a l e n t gas v o l u m e at the s y s t e m t e m p e r a t u r e a n d p r e s s u r e p e r u n i t v o l u m e o f l i q u i d . Gas  i s f e d to the a b s o r b e r at a v o l u m e t r i c r a t e Q_ , b a s e d o n the s y s t e m G o t e m p e r a t u r e and p r e s s u r e . L i q u i d i s f e d at a v o l u m e t r i c r a t e Q with a  L  c o n c e n t r a t i o n q^.  In u n i t t i m e , the gas v o l u m e a b s o r b e d i n the e l e m e n t  d L is therefore dq =  ^ N u m b e r of b u b b l e s ^ S u r f a c e a r e a / b u b b l e j ^ D r i v i n g f o r c e ]  (3)  In the a b o v e e x p r e s s i o n , the d r i v i n g f o r c e i s e x p r e s s e d i n v o l u m e t r i c terms,  and k  h a s the u s u a l u n i t s of cm;  /min.  " L  I  T h e b u b b l e s i z e i s d e t e r m i n e d b y the f r e q u e n c y , gas f e e d r a t e ,  and  the a m o u n t o f t r a n s f e r that h a s t a k e n p l a c e . T h u s , the v o l u m e o f a b u b b l e Q = „Si.  Q  r  i s g i v eB nu bbbyl e V o l u m e  where N_  B  =  =  - Q  G  Z2  L  (q -  q ) _ Q  the b u b b l e f r e q u e n c y .  If the b u b b l e s a r e s p h e r i c a l ,  then  Surface a r e a p e r bubble - S^ = 4 j ^ ~ : ^  ^  H>  ^  ^°/^  T h e t i m e a v e r a g e n u m b e r o f b u b b l e s i n the e l e m e n t d L i s d e t e r m i n e d the f r e q u e n c y a n d b u b b l e v e l o c i t y as N g  E q u a t i o n (2)  c a n be a p p l i e d and,  from  dL.  and Q  e x p r e s s e d as i n e q u a t i o n (4) to G  obtain,  ^  12  2 N u m b e r of bubbles i n d L  •=  7T 4  The  D  N  B  d  (6)  L  + q -  Q (1 L  '  Q  d r i v i n g f o r c e f o r t r a n s f e r i s the d i f f e r e n c e b e t w e e n the  concentration,  q#,  at the i n t e r f a c e , a n d the a v e r a g e s t r e a m  It i s c o n v e n i e n t to d e f i n e a p a r a m e t e r , total concentration  L-  i f a l l gas  —  ,  as the  concentration,  equivalent  fed were in solution^  ^  q  of equations (5),  Substitution  Xl  equilibrium  (  o  (6)  and (7)  into (3)  r e s u l t s i n the  7)  differential  equation,  If' e q u a t i o n (8) i s i n t e g r a t e d o v e r a l e n g t h L = L . - L . , t h e n the l e f t s i d e of the e q u a t i o n i s r e p r e s e n t a t i v e  of the t o t a l gas  d e t e r m i n e d b y the c o n c e n t r a t i o n s  q-  and  a b s o r b e d w i t h i n L,  q..  In o r d e r to p r e d i c t f u l l y the d e p e n d e n c e of the a m o u n t of o n the v a r i a b l e s i n e q u a t i o n ( 8 )  as  absorption  i t w o u l d be n e c e s s a r y to k n o w s p e c i f i c a l l y  the r e l a t i o n s h i p  k  L  =  Function  of (D,  N , fi  Q^,  J^yA  >  &)  T h i s r e l a t i o n s h i p i s u n k n o w n f o r b u b b l e flow, but a p r e d i c t i o n of the  inter-  d e p e n d e n c e of the v a r i a b l e s i s m a d e p o s s i b l e b y p o s t u l a t i n g that the f u n c t i o n has  the f o r m w h i c h i s k n o w n to a p p l y to m a n y o t h e r m a s s  trans-  13'  fer situations'  Sherwood N u m b e r = constant  (Reynolds  Number) (Schmidt  Number) (9)  F o r flow p a s t spheres,  the f o r m of e q u a t i o n (9) i s w e l l c o n f i r m e d b y  the F r o e s s l i n g e q u a t i o n (20) i n w h i c h S h e r w o o d a n d R e y n o l d s n u m b e r s b a s e d o n the s p h e r e d i a m e t e r In the p r e s e n t  are  a n d l i q u i d v e l o c i t y r e l a t i v e to the s p h e r e .  investigation, the m o s t p r o b a b l y m e a n i n g f u l  character-  i s t i c l e n g t h a n d v e l o c i t y a r e a s s u m e d to b e the tube d i a m e t e r  and  l i q u i d v e l o c i t y r e l a t i v e to t h e tube. T h e s e p a r a m e t e r s a r e m o r e  average closely  r e l a t e d to the state o f l i q u i d t u r b u l e n c e , a n d t h e r e f o r e t h e i r u s e i s c o n s i s t e n t w i t h the a s s u m p t i o n  t h a t the t u r b u l e n t v e l o c i t y f i e l d i s c o n t r o l l i n g  the m a s s t r a n s f e r r a t e as p o s t u l a t e d i n the I n t r o d u c t i o n . definitions of S h e r w o o d and R e y n o l d s n u m b e r s Sh  =  k  T  Hence,  logical  w o u l d be,  D/£>  (10)  T h e f r a c t i o n o f the tube o c c u p i e d b y l i q u i d ,  Q  Q  L  1  L +  Q  G  -  -q  '  is i n t r o d u c e d i n e q u a t i o n (11) to a d j u s t the l i q u i d v e l o c i t y for' the a v e r a g e decrease  in flow c r o s s - s e c t i o n .  definitions,  as  E q u a t i o n (9) c a n b e w r i t t e n , u s i n g t h e s e  14 w h e r e K^,  °^  and  *7T  are unknown constants.  If t h e p r e d o m i n a n t v e l o c i t i e s a f f e c t i n g m a s s t r a n s f e r a r e t h e t u r b u l e n t v e l o c i t i e s due to t u r b u l e n t l i q u i d f l o w t h r o u g h t h e tube, t h e n the m a s s t r a n s f e r c o e f f i c i e n t s h o u l d be u n a f f e c t e d b y the s p a c i n g b e t w e e n adjacent bubbles,  that i s , k  s h o u l d be independent  H e n c e e q u a t i o n (12) c a n b e u s e d to r e p l a c e k  of bubble  frequency.  i n the r r a t e r i a l b a l a n c e  (8), w i t h the r e s u l t a f t e r i n t e g r a t i n g a n d r e a r r a n g i n g , 3 -I 1/3 N  r y . dq 2/3 ^  -.=. K_ R e 2 s  (q* -q)(4- q)  where  K  2  Re^  =  / 3 4 ^  B  (13)  c  \2 /3  ( — )  =  S  L  •  K  j  s u p e r f i c i a l l i q u i d Reynolds n u m b e r b a s e d on l i q u i d o n l y f l o w i n g i n the tube  4  Q L  Sc  =  Schmidt number,  The unknown exponent  oC  c a n be e l i m i n a t e d f r o m the left side of  0  e q u a t i o n (13) b y m a k i n g the a p p r o x i m a t i o n t h a t (1 4v\ - q)  1  -Ad-  is unity.  For  the n o n - c o a l e s c i n g b u b b l e f l o w u n d e r c o n s i d e r a t i o n h e r e t h e v o l u m e t r i c fraction occupied by liquid,  e s t i m a t e d at 0.92.  i  } T"A,  would likely* exceed  a value  - q,  F o r a wide v a r i e t y of m a s s t r a n s f e r situations, the  m a s s t r a n s f e r c o e f f i c i e n t h a s b e e n f o u n d o r p r e d i c t e d to b e p r o p o r t i o n a l to a R e y n o l d s n u m b e r r a i s e d to p o w e r s i n the r a n g e 0. 5 to 1. 5 (21). oL  i s also i n this range,  If  t h e n the m a x i m u m e r r o r p o s s i b l e i n t h e a b o v e  15 approximation is only 5% at the highest possible gas to liquid ratios for bubble flow. It is convenient to combine the parameters j£. and q*  by expressing  concentration in dimensionless form as a fraction of the saturation value f  =  q/q*  (14)  Equation (13) can now be written, F (f) =  r~i '  Af (1 - f ) ( ^ - f ) 2/3 ^  = K q*  2 / 3  2  Aj - R eot-1 * " ^ ^ " I /Q 1  3  T  \ i/  3  N b L  1  f.  V  L  (15)  1  The above equation is completely in terms of dimensionless quantities. Any convenient  unit of concentration, such as normality or parts per  million, can be used to determine f Th<5  and £/q* in the left side of the equation.  left hand side of (15) is a measure of the amount of absorption  achieved in a section of the tube, and is analogous to the NTU packed tower absorbers.  used in  In the case of bubble flow, the interfacial area  per unit length of tube decreases along the tube.  Consequently,  the 2/3  normal NTU  is modified by the surface area correction, ( X. /q* - f)  The left hand side of (15) will be referred to hereafter as the "Absorption Function", Absorption Function  F(f) =[ f.  =  df  1  (i  - f)( £ / * q  :  (16)  f The integral F.(f) = C  df  (1 - f) (je/q* - f) 0  for values of Jt/q*  J  has been evaluated numerically  ^  o  from 0. 01 to 0. 1 and these results have been summar-  i z e d g r a p h i c a l l y i n F i g u r e 2.  T h e A b s o r p t i o n F u n c t i o n c a n be e v a l u a t e d  f r o m t h e s e c u r v e s as F ( f ) = F . (f .) -  '  J  F . ( f .) 1  (17)  i  E q u a t i o n (15) p r o v i d e s a c o r r e l a t i n g e q u a t i o n f o r t h e a b s o r p t i o n d a t a w h i c h s h o u l d b e s a t i s f a c t o r y p r o v i d e d a l l the a b o v e a s s u m p t i o n s a r e a p p r o x i m a t e l y a p p l i c a b l e to b u b b l e flowtested with this c o r r e l a t i o n .  T h e data w i l l be i n i t i a l l y  If a g o o d fit i s obtained,, i t w i l l i n d i c a t e  that t h e m a s s t r a n s f e r , c o e f f i c i e n t i s r e a s o n a b l y c o n s t a n t . T h i s c o n c l u s i o n would g r e a t l y s i m p l i f y the c a l c u l a t i o n of m a s s t r a n s f e r coefficients b y j u s t i f y i n g t h e i n t e g r a t i o n u s e d i n d e r i v i n g e q u a t i o n (15). g a s / l i q u i d s y s t e m at one t e m p e r a t u r e  A s o n l y one  i s c o n s i d e r e d i n the p r e s e n t w o r k  it i s i m p o s s i b l e to t e s t t h e e f f e c t o f S c h m i d t n u m b e r . . T h e r e f o r e e q u a t i o n (15) m u s t b e t e s t e d i n the f o r m ,  (18)  where  i n c l u d e s b o t h t h e S c h m i d t n u m b e r a n d q*, w h i c h i s i n d e p e n d e n t  of p r e s s u r e w h e r e H e n r y ' s L a w o f s o l u b i l i t y a p p l i e s .  S i m p l i f i e d Equations for L o w Gas / L i q u i d Ratios F o r l o w g a s to l i q u i d v o l u m e t r i c r a t i o s s o m e s i m p l i f i c a t i o n o f t h e a b o v e e q u a t i o n s i s p o s s i b l e i f the c o n c e n t r a t i o n d r i v i n g f o r c e , v a r i e s only over a n a r r o w range. p o i n t s i n the a b s o r b e r ,  This case occurs i f q*  q * - q,  y*y>  <1 at a l l  as i s the c a s e f o r C O ^ a b s o r p t i o n i n bubble  flow  w i t h a l o w c o n c e n t r a t i o n o f C O ^ i n the f e e d w a t e r . T h e u s e of a n a r i t h metic mean driving force q*  -  (q. +  q.)  / 2, t h e n i n t r o d u c e s a  18  neglible error.  The absorption function can therefore be simplified to  F (f)  V  ^  r  df  J  (1 - f)m  f.  1  which can be integrated analytically. The fraction of the tube occupied by liquid, —:—;—» r  only slightly for this case.  1+ J c - q  •  7  n  , also varies  Hence an arithmetic mean value of 1 + j£ - q  can be used with little error.  With this approximation and equations (14)  and (19), equation (8) can be to give 1/3 rearranged 7 1 + Jf and „ integrated f \ 4/ 3 q*V3 / 1 + Jf \\ Q* q* h7 m k TCI*  .-.' ~ S : i  >  V"  J3  (20  The experimental mass transfer coefficient for a single absorption run can be calculated from equation (20) subject to the foregoing simplifying assumptions, and without introducing any postulated functions such as equation (9) Correction for Bubble Velocity If the bubble velocity is not equal to the mean volumetric flow velocity,  A  V  =  B  —  ^  1.0, then the residence time of a bubble within an  T element of the absorber is inversely proportional to the velocity ratio, Hence a correction to the calculated mass transfer coefficient must be made if  is significantly different from unity.  However ^  is a function of  bubble diameter for a given flow rate and consequently varies along the absorber. quadratic,  If the velocity-diameter data can be fitted satisfactorily by a _ ~  W  o  +  W  l  ^'  +  W  2  ^  2  (21)  19  t h e n the v e l o c i t y r a t i o c a n be a p p l i e d to the m a t e r i a l b a l a n c e (8) a n d the r e s u l t i n g e q u a t i o n c a n be a n a l y t i c a l l y i n t e g r a t e d to g i v e the c o r r e c t e d m a s s transfer  coefficient k  e x p r e s s e d b y an e q u a t i o n a n a l o g o u s  to e q u a t i o n (20).  V  The bubble s p h e r i c a l  d i a m e t e r ; i s s i m p l y r e l a t e d to the b u b b l e  volume  w h i c h c a n be e x p r e s s e d i n t e r m s of the c o n c e n t r a t i o n at a g i v e n l o c a t i o n i n the  absorber. d =  volume)J  (bubble  3  =  (22)  ^  s u b s t i t u t i o n of the r e l a t i o n (14) i n e q u a t i o n (22) y i e l d s  (23)  T h e r e f o r e e q u a t i o n (21) c a n be e x p r e s s e d  /4  =/6( I f  =  w h e r e the p r i m e d  To  (-£  +  w 0  as  -)i4 f  T  'U \%  "+W2 (Jl- »r *- - V f  (24)  1  c o n s t a n t s a r e d e f i n e d b y e q u a t i o n s (21) a n d (23).  a p p l y the v e l o c i t y r a t i o to the m a t e r i a l b a l a n c e , the t i m e  n u m b e r of bubbles  given by  i n t r o d u c e s the f a c t o r  e q u a t i o n (6) i s m u l t i p l i e d b y  .  average  This  o n the r i g h t s i d e of e q u a t i o n (8) w h i c h  can  b e r e a r r a n g e d to g i v e  Q  k  *L,  = r  ** q * L 3, 8 D ^ N J J L  (f)  I  (1 f  -" (\- T  3  I  F o r the l o w gas to l i q u i d r a t i o s e q u a t i o n (25) c a n be w r i t t e n  df  (25)  20  Q  %  4/iii  f)  D" N  (l-fj \ /m  df 8  B  » L  (26) which i s readily integrated.  T h e r e s u l t i s c o n v e n i e n t l y e x p r e s s e d as a r  v e l o c i t y c o r r e c t i o n f a c t o r C^. to b e a p p l i e d to the c o e f f i c i e n t c a l c u l a t e d b y e q u a t i o n (20) V l  w h e r e C, V  T  -  w  w. o  2 wf)ci-fy <£  m  WL  F  Correction for Pressure The  2  (28)  Cf) (1-f)  m  Drop  e f f e c t o f v a r i a b l e p r e s s u r e a l o n g the t u b e c a n be e x p l i c i t l y  a l l o w e d f o r i n e q u a t i o n s (8), (13) o r (20).  The quantities Q  and q c a n be  d e f i n e d as v a l u e s o f a v e r a g e g a s f l o w a n d v o l u m e t r i c c o n c e n t r a t i o n at a n y p o i n t e x p r e s s e d at the t e m p e r a t u r e o f t h e s y s t e m a n d the i n l e t p r e s s u r e P o . W i t h t h i s d e f i n i t i o n , the e q u i l i b r i u m i n t e r f a c e c o n c e n t r a t i o n q * v a r i e s w i t h the p r e s s u r e P, a n d i f H e n r y ' s L a w o f s o l u b i l i t y h o l d s q * = H i s a Henry's Law P = Po  constant.  + p L where p  HP Po  where  F o r a l i n e a r p r e s s u r e d r o p along the tube ,  dP  The  dL  mass transfer driving force is  therefore a*  - a  =  H(Po  + p L)  -  q  (29)  Po  The  s u r f a c e a r e a p e r bubble given  b y e q u a t i o n (5) m u s t a l s o b e  21 corrected for pressure . A3  .  Y£  /Q  G o  -  Q .(q -  ^ P  L  ^  o  ( 3 0 )  Substitution of these modified relations into equation (3) produces the variable pressure equation analogous to (8)  The square bracket on the right side of (21) corresponds to the concentration driving force and therefore use of a mean value for the second term f P J q* »  q.  • is justified for low gas to liquid ratios with  This approximation removes the variable q from the right side  and permits integration of equation (31). used for 1  q.  Similarly a mean value can be  An expression for calculating the mass transfer co-  efficient is obtained by substituting equation (14) into (31), rearranging: k  =  P  . L  u/ 4  r  , </. ~"  where C = — — ; <• f P 4p  3  v  .  q* \\ q*  n  3« 8 D N „ ' B  ,  H  / im  integrating and  /j df  C p  (p,> • p L ) ^ :  - ( p „ . p LJ)*] -  I . ( f  P  ' / i ) . (33) n  The termsCp represent the effect of pressure drop on the concentration driving force and the surface area.  22  APPARATUS  General The  Requirements l o w gas to l i q u i d r a t i o s e n c o u n t e r e d i n b u b b l e f l o w r e s t r i c t  the d i s s o l v e d gas (ppm)  for CO^  c o n c e n t r a t i o n s to l e s s t h a n about 40 p a r t s p e r  at one  a t m o s p h e r e (atm).  o p e r a t i o n at p r e s s u r e s up to 3 a t m  The  designed for  to i n c r e a s e the s o l u b i l i t y f o r e a s e of  a n a l y s i s . O p e r a t i o n at the h i g h e r p r e s s u r e s was l i q u i d f l o w s to k e e p the p r e s s u r e  a p p a r a t u s was  million  also r e q u i r e d for high  d r o p s m a l l r e l a t i v e to the a b s o l u t e  pressure.  P r e l i m i n a r y e x p e r i m e n t s i n d i c a t e d t h a t a t e s t s e c t i o n l e n g t h of the o r d e r of twelve  feet was  r e q u i r e d to a c h i e v e a s a t i s f a c t o r y a m o u n t o f a b s o r p o r t i o n .  at h i g h l i q u i d f l o w s .  However,  a b s o r p t i o n of the b u b b l e s was  complete i n m u c h s h o r t e r test lengths f o r l o w e r flows. tubes w i t h a n u m b e r of s a m p l e taps w e r e r e q u i r e d .  essentially  Consequently long  In o r d e r to e n s u r e  that the gas p h a s e p r e s e n t e d no r e s i s t a n c e to m a s s t r a n s f e r s - d i s s o l v e d a i r h a d to b e r e m o v e d f r o m the l i q u i d f e e d . A l o w l e v e l of d i s s o l v e d a i r was  c o n v e n i e n t l y a c h i e v e d b y c o n t i n u o u s l y c i r c u l a t i n g the w a t e r t h r o u g h a  v a c u u m s t r i p p e r w h i c h a l s o r e m o v e d m o s t of the CO_  a b s o r b e d i n the t e s t  section.  F o r the b u b b l e f l o w c o n s i d e r e d i n t h i s w o r k , the gas m u s t b e d u c e d into the t e s t s e c t i o n i n s u c h a w a y i n s i z e and s p a c i n g r e s u l t s .  that a s t r e a m o f b u b b l e s  intro-  uniform  F e e d n o z z l e s w e r e d e s i g n e d to a c h i e v e t h i s  u n i f o r m i t y and to g i v e s o m e c o n t r o l o v e r the b u b b l e s i z e f o r any g i v e n  gas  and l i q u i d flows.  T h e f r e q u e n c y of bubble f o r m a t i o n i s an e s s e n t i a l  v a r i a b l e f o r the e v a l u a t i o n of bubble s i z e and m a s s t r a n s f e r c o e f f i c i e n t s . A d i g i t a l c o u n t e r w a s d e s i g n e d to m e a s u r e f r e q u e n c y .  Additional equipment  was r e q u i r e d to p h o t o g r a p h b u b b l e s a n d m e a s u r e t h e i r v e l o c i t y i n t h e t e s t section.  L i q u i d Cycle and T e s t Section A schematic  d i a g r a m o f the a p p a r a t u s i s s h o w n w i t h t y p i c a l o p e r a t i n g  c o n d i t i o n s i n F i g u r e 3.  F u r t h e r details and specifications of c e r t a i n  p i e c e s of equipment a r e i n c l u d e d i n Appendix.I.  W a t e r l e a v i n g the test  s e c t i o n e n t e r e d a s t o r a g e tank and was p r e h e a t e d  t o 48 ° C b y d i r e c t c o n t a c t  with s t e a m i n an i n - l i n e s t e a m m i x e r . necessary flows.  A s m a l l c i r c u l a t i o n pump was  i n t h e h e a t i n g l o o p to a v o i d h a m m e r i n g i n the m i x e r at l o w s t e a m  T h e s t o r a g e t e m p e r a t u r e w a s c o n t r o l l e d to w i t h i n 1° C b y a F e n w a l l  t h e r m o s w i t c h o p e r a t i n g a s o l e n o i d v a l v e w h i c h t r i m m e d about 3 0 % o f t h e steam load.  A c y c l o n e w a s p r o v i d e d i n t h e s t e a m l i n e to p r e v e n t  f r o m e n t e r i n g the s y s t e m .  F l o w of p r e h e a t e d  condensate  w a t e r to t h e v a c u u m  stripper  tank w a s c o n t r o l l e d b y a s o l e n o i d v a l v e a c t u a t e d t h r o u g h a r e l a y b y a d i p ^ w i r e c o n t a c t i n g the m e r c u r y s u r f a c e i n the d r y s i d e o f a l e v e l - i n d i c a t i n g manometer,  as d e s c r i b e d b y H a y d u k (7). .  T h e w a t e r w a s s p r a y e d i n t o the  s t r i p p e r w h e r e s o m e f l a s h i n g o c c u r r e d to a s s i s t s t r i p p i n g o f d i s s o l v e d gases.  T y p i c a l c o n d i t i o n s i n the s t r i p p e r w e r e 4 7 ° C a n d 29 i n c h e s of  m e r c u r y vacuum.  T h e v a c u u m was m a i n t a i n e d  by a water-jet ejector.  A d d i t i o n a l heat f o r s t r i p p i n g was p r o v i d e d b y the f e e d p u m p w h i c h r e c y c l e d about 3 g p m t h r o u g h the b a c k - p r e s s u r e  c o n t r o l v a l v e a n d into t h e t o p o f t h e  FEED NOZZLE 5 ppm C 0 20 °C  ,  O  2  0  TEST  SECTION  30 psig SAMPLE VAPOUR TO EJECTOR  WATER ROTAMETER  4  DIFF'L FLOW REGULATOR  t CO,  1 COOUNG WATER SOLENOID VALVE  F I G U R E 3.  SCHEMATIC DIAGRAM OF APPARATUS  ^ 7 0 ppm C 0  2  tank.  A t y p i c a l c o n c e n t r a t i o n o f d i s s o l v e d C O ^ i n the s t r i p p e d f e e d w a s  5 ppm, a l t h o u g h c o n c e n t r a t i o n s  as h i g h as 16 p p m o c c u r r e d f o r h i g h l i q u i d  t h r o u g h p u t s a n d s t r i p p e r i n l e t c o n c e n t r a t i o n s (2 to 3 g p m a n d 100 p p m ) .  The  s t r i p p i n g tank w a s l o c a t e d s i x f e e t a b o v e t h e f e e d p u m p i n o r d e r to  e n s u r e that a s u f f i c i e n t l i q u i d h e a d w a s m a i n t a i n e d  f o r good o p e r a t i o n of  the p u m p w h i c h w a s h a n d l i n g w a t e r n e a r i t s b o i l i n g p o i n t .  A turbine  p u m p was c h o s e n f o r t h i s p u r p o s e to p r o v i d e the t o t a l h e a d o f a p p r o x i m a t e l y 50 p s i r e q u i r e d f o r o p e r a t i o n of the a b s o r b e r pump suction under vacuum.  Pressure  b y the a b o v e m e n t i o n e d b a c k p r e s s u r e the s t r i p p e r .  at 3 a t m w i t h t h e  at the p u m p o u t l e t w a s r e g u l a t e d  c o n t r o l v a l v e o n the r e c y c l e b a c k t o  T h e f e e d w a t e r w a s c o o l e d i n h e a t e x c h a n g e r s to 20 +_ 0. 2 ° C .  M a n u a l v a l v e s c o n t r o l l e d the w a t e r f l o w t h r o u g h w e i g h t - c a l i b r a t e d r o t a m e t e r s to the t e s t s e c t i o n .  Pressure  i n the t e s t s e c t i o n w a s m a i n t a i n e d  c o n t r o l v a l v e f o r the h i g h p r e s s u r e  with a b a c k - p r e s s u r e  runs and with an e l e v a t e d h e a d tank  for runs near a t m o s p h e r i c p r e s s u r e .  T h e o v e r f l o w f r o m the h e a d tank was  c o n f i n e d to a C O ^ a t m o s p h e r e to a v o i d a e r a t i o n of the w a t e r . the l i q u i d c y c l e w e r e 3/8-,  A l l lines i n  1/2- a n d 3/4- i n c h c o p p e r w i t h b r a s s  fittings..  A c i d f l u x w a s u s e d on s o l d e r j o i n t s a n d T e f l o n tape o n p i p e t h r e a d s to prevent  grease f r o m contaminating  The supported  the  system.  test sections w e r e a l l c o n s t r u c t e d of p r e c i s i o n - b o r e glass tubing between a P e r s p e x feed n o z z l e and connector  blocks, which  w e r e a t t a c h e d to a r i g i d f r a m e . T h e two t u b i n g s i z e s u s e d w e r e 5/16  (0. 313 in) and 5/8 was  inside diameter  (ID).  critical in forming a uniform stream  tee was  u s e d w i t h l i q u i d t h r o u g h the run,  o r r i g h t i n the l i q u i d s t r e a m , the b u b b l e s  The  f o r m of the f e e d n o z z l e  of b u b b l e .  If a u n i f o r m - d i a m e t e  and b u b b l e s  f o r m e d i n the  branch  w o u l d be t o r n a p a r t b y the h i g h  t u r b u l e n c e i n the r e g i o n of the tee o r b u b b l e n o z z l e at h i g h l i q u i d f e e d r a t e s Consequently  a t a p e r e d f e e d n o z z l e (12°  approximately  t o t a l angle) was  designed  with  a t e n - f o l d r e d u c t i o n i n c r o s s s e c t i o n so that b u b b l e s  were  f o r m e d i n a r e g i o n of l o w v e l o c i t y and a c c e l e r a t e d into the t e s t s e c t i o n . The  g e n e r a l f e a t u r e s of the f e e d n o z z l e c a n be s e e n i n F i g u r e 3,  d e t a i l s f o r the 5/8d i x I.  The  i n c h n o z z l e a r e g i v e n i n F i g u r e 1-1  s a m e l i q u i d f e e d n o z z l e was  v e r t i c a l test sections. The  of  Appen-  u s e d f o r b o t h h o r i z o n t a l and  c e n t r e l i n e of the t a p e r was  that the u p p e r s u r f a c e of the b o r e was  and 1-2  and  inclined  such  p a r a l l e l to the t e s t s e c t i o n t u b i n g  i n o r d e r to a v o i d a gas p o c k e t w h e n i n u s e h o r i z o n t a l l y . l o w w a t e r f e e d r a t e s i n h o r i z o n t a l flow, b u b b l e s  However,  at  t e n d e d to c o a l e s c e b e f o r e  t h e y m o v e d h o r i z o n t a l l y out of the f e e d n o z z l e . T h e r e f o r e n o z z l e l i n e r s as s h o w n i n F i g u r e 1-2 flow velocity.  The  w e r e u s e d f o r t h e s e c o n d i t i o n s to i n c r e a s e the  w a t e r f l o w was  i n t r o d u c e d i n t o the n o z z l e s t h r o u g h  1 0 - i n c h l e n g t h of 1 1 / 2 - i n c h p i p e to a v o i d s e v e r e e d d i e s . was  A  a  seal block  p r o v i d e d i n the b a s e of the f e e d n o z z l e to p e r m i t c h a n g i n g  and  ad-  j u s t i n g the b u b b l e n o z z l e s .  The  P e r s p e x tubing connector b l o c k s p r o v i d e d a smooth,  b o r e t h r o u g h the t e s t s e c t i o n . T h e  continuous  t u b i n g ends w e r e g r o u n d s q u a r e  to l e n g t h on a l a p p i n g w h e e l and s e a t e d a g a i n s t a s q u a r e  and  s h o u l d e r i n the  connector O-rings  block.  operated  T h i s s y s t e m of g l a s s t u b i n g s e a l e d into the b l o c k s s a t i s f a c t o r i l y at p r e s s u r e s o f 30 p s i g .  The  with  connector  b l o c k s c o n t a i n e d s a m p l e taps t h r o u g h w h i c h 1 / 8 - i n c h O D t u b i n g s a m p l e p r o b e s c o u l d be i n s e r t e d to a n y r a d i a l p o s i t i o n . T h e p r o b e s e a l s w e r e p r o v i d e d b y 1 /8-inch ID O-rings  i n 1/8 N P T b y 1/8 O D  c o m p r e s s i o n fittings  b o r e d t h r o u g h to a c c e p t the t u b i n g . T h e d e t a i l s o f the 5 / 8 - i n c h  connector  b l o c k s a r e s h o w n i n F i g u r e 1-3.  D e t a i l s o f the l o c a t i o n o f s a m p l e t a p s a r e s h o w n f o r a l l t e s t s e c t i o n s i n F i g u r e s 4 arid:5.  T h e s e f i g u r e s a l s o s h o w t y p i c a l a r r a n g e m e n t s of the  pressure measuring lines.  A m e r c u r y m a n o m e t e r was u s e d to m e a s u r e  p r e s s u r e d r o p a n d the o p e r a t i n g p r e s s u r e w h e n t h i s was not t o o h i g h .  A  h y d r a u l i c a l l y c a l i b r a t e d B o u r d o n gauge was u s e d f o r h i g h o p e r a t i n g pressure.  A l l m a n o m e t e r a n d gauge l i n e s w e r e w a t e r - f i l l e d .  taps s e r v e d as p r e s s u r e  Gas  F l o w and Bubble  The sample  taps.  Nozzles  C a r b o n dioxide flowed f r o m a c y l i n d e r and p r e s s u r e - r e d u c i n g through a rotameter  valve  a n d M o o r e c o n s t a n t - d i f f e r e n t i a l f l o w r e g u l a t o r to  the n o z z l e as s h o w n i n F i g u r e 3.  A s h u t - o f f v a l v e a n d b y p a s s to a n  a t m o s p h e r i c s o a p - b u b b l e f l o w m e t e r was p r o v i d e d f o r the a c c u r a t e  deter-  m i n a t i o n of gas f l o w r a t e f o r e a c h r u n .  A n u m b e r of designs of bubble n o z z l e tips w e r e r e q u i r e d f o r the d i f f e r e n t test s e c t i o n s . F o r the 5/16-inch a b s o r b e r , m a t i o n was l e s s than 6 m m  b u b b l e s i z e at f o r -  equivalent s p h e r i c a l diameter,  and a s i m p l e  14.00  NOTE: ALL DIMENSIONS IN FECT, EXCEPT AS NOTEO  5/16 - INCH TUBE  14.29  5/8-INCH  F I G U R E 4.  TUBE  H O R I Z O N T A L TEST SECTIONS  oo  5/16 - INCH TUBE FIGURE  5/8 - INCH TUBE 5.  VERTICAL TEST  SECTIONS  30  NOZZLE ORIFICE PERSPEX NOZZLE TIP  BRASS INSERT SEAL TUBE, 1/2 O . D . COPPER TUBING S.S. CAPILLARY 0.014 I . D . X 5 INCH C0 FEED 2  F I G U R E 6.  BUBBLE NOZZLE ASSEMBLY  | DRAIN c i r c u l a r o r i f i c e as s h o w n i n F i g u r e 6 was s a t i s f a c t o r y f o r b o t h v e r t i c a l and h o r i z o n t a l test s e c t i o n s . were used. as 9 m m orifice,  O r i f i c e d i a m e t e r s o f 1/32 to 7/32  inch  F o r the 5 / 8 - i n c h a b s o r b e r , b u b b l e d i a m e t e r s w e r e as l a r g e  a n d i t was i m p o s s i b l e to f o r m b u b b l e s o f t h i s s i z e w i t h a s i m p l e s i n c e a g a s - l i q u i d i n t e r f a c e c o u l d not b e m a i n t a i n e d a c r o s s the  l a r g e o r i f i c e . . I n v e r t e d c u p n o z z l e s s h o w n i n F i g u r e 7 p e r m i t t e d the gas to r o l l o v e r the l i p o f the c u p i n u n i f o r m b u b b l e s of t h e r e q u i r e d s i z e at f r e q u e n c i e s as h i g h as 4 0 0 / m i n . us ed.  C u p s i z e s of 1/4 to 1/2 i n c h I D w e r e  N O Z Z L E  H O R I Z O N T A L  D E T A I L  F L O W  F I G U R E 7.  V E R T I C A L  INVERTED CUP  NOZZLES  F L O W  The bubble nozzle assembly, t h r o u g h the s e a l b l o c k , nozzle assembly  was  w h i c h e n t e r e d the l i q u i d feed  i s s h o w n i n F i g u r e 6.  a d o p t e d to e l i m i n a t e  The  d e s i g n of the  irregularities  i n the  frequency which resulted f r o m slight pressure  fluctuations  s e c t i o n ( «< 0. 2 p s i a t a p r e s s u r e  A  o f 30 p s i g ) .  nozzle bubble  bubble  i n the  test  5 - i n c h l e n g t h of  stain-  less steel capillary tubing provided approximately 2 psi pressure  drop  to s t e a d y out the f l o w of gas to the n o z z l e t i p .  The c a p i l l a r y  s o l d e r e d into a b r a s s i n s e r t w h i c h was p r e s s e d  into the b a s e of the  tip This  to l e a v e  tubing  was  nozzle  a v e r y s m a l l v o l u m e i m m e d i a t e l y b e l o w the o r i f i c e i n the t i p .  s m a l l v o l u m e e n s u r e d that the p r e s s u r e  b e n e a t h the o r i f i c e  q u i c k l y a d j u s t e d to a n e w l e v e l i n r e s p o n s e to a s l i g h t p r e s s u r e  very variation  3 in the test section, conditions.  e v e n at C Q  The nozzle tip,  f l o w s as l o w as  insert,  w i t h a t u r n of T e f l o n tape w h e n  of bubbles  nozzle materials  / m i n at  and seal tube w e r e  nozzle  sealed  together  assembling.  Two additional difficulties were stream  10 c m  encountered  in producing a uniform  f r o m the p l a i n c i r c u l a r o r i f i c e s . and treatments,  the f r e q u e n c y  f l o w s v a r i e d e r r a t i c a l l y b y as m u c h as  30%.  For  a variety  at f i x e d w a t e r  and C O ^  T h i s behaviour is  to r e s u l t f r o m v a r i a b l e w e t t i n g as t r a c e i m p u r i t i e s a d s o r b e d o n nozzle tip.  Finally Perspex,  w a s f o u n d to be a l m o s t was  due to w a t e r  believed the  washed after machining in p e t r o l e u m  e n t i r e l y f r e e of t h i s p r o b l e m .  The  expelled some water,  s m a l l volume flooded.  The  ether,  second difficulty  f l o w i n g d o w n the w e t t e d w a l l of the n o z z l e o r i f i c e  b u i l d i n g up b e n e a t h it u n t i l the  of  gas  p r o d u c i n g one o r two i r r e g u l a r b u b b l e s .  and  flow then This  f l o o d i n g was e l i m i n a t e d b y r e n d e r i n g the w a l l of the n o z z l e o r i f i c e n o n w e t t i n g w i t h a t h i n c o a t i n g o f p a r a f f i n w a x c a r e f u l l y a p p l i e d to the l o w e r p a r t o f the w a l l , l e a v i n g the u p p e r p a r t a n d s h a r p e d g e c l e a n .  W i t h the  above techniques, v e r y u n i f o r m bubble s t r e a m s w e r e p r o d u c e d with s p h e r i cal  d i a m e t e r s f r o m 0. 25 to 0. 6 c m .  a n d f r e q u e n c i e s f r o m 100 to 1 5 0 0 / m i n .  F r e q u e n c i e s w e r e t y p i c a l l y s t e a d y to w i t h i n 2%.  Auxiliary Equipment  for Bubble Frequency,  V e l o c i t y and  Photography  F i g u r e 1^4 s h o w s the c i r c u i t u s e d to d e t e r m i n e b u b b l e f r e q u e n c y .  A  s m a l l l i g h t a n d p h o t o d i o d e w e r e m o u n t e d on. o p p o s i t e s i d e s of the a b s o r p t i o n tube n e a r the f e e d n o z z l e . A b u b b l e p a s s i n g t h i s d e t e c t o r g e n e r a t e d a voltage d i s t u r b a n c e . T h e s i g n a l a m p l i f i e r s e c t i o n a m p l i f i e d this s i g n a l and put  out a p u l s e o n l y i f the i n i t i a l d i s t u r b a n c e h a d a c c e p t a b l e m a g n i t u d e a n d  r a t e of change,  i n o r d e r to a v o i d r e s p o n d i n g to n o i s e .  the  c i r c u i t w h i c h p u l s e d the b i n a r y s w i t c h e s a n d r e m a i n e d  monostable  The pulse triggered  i n a c t i v e f o r an a d j u s t a b l e t i m e i n t e r v a l . T h i s d e a d t i m e p r e v e n t e d m u l t i p l e c o u n t i n g of a s i n g l e b u b b l e due to a n i r r e g u l a r i n i t i a l d i s t u r b a n c e .  Two  b i n a r y s w i t c h e s a n d an a m p l i f i e r w e r e u s e d to d i v i d e the c o u n t r a t e b y a f a c t o r of two o r f o u r ,  a n d a m p l i f y the output so that a m e c h a n i c a l t a l l y  c o u l d e a s i l y h a n d l e the c o u n t i n g . T h e f r e q u e n c y m e t e r c i r c u i t g e n e r a t e d a c o n t i n u o u s v o l t a g e w h i c h i n d i c a t e d the f r e q u e n c y .  Two velocity.  i n d e p e n d e n t t e c h n i q u e s w e r e d e v e l o p e d f o r m e a s u r i n g the b u b b l e F o r a s i n g l e b u b b l e i n the a b s o r p t i o n tube,two photo  diodes; i n  s e r i e s w e r e p l a c e d s e v e r a l f e e t a p a r t on the tube a n d c o n n e c t e d to the  34 counter circuit.  A single bubble passing the two detectors in turn  generated two pulses in the monastable circuit.  The time interval between  the two pulses was measured with a digital interval-timer and used to determine the bubble velocity.  For this technique,, bubble size was deter-  mined by a positive-displacement injector connected directly to the bubble nozzle. A micrometer feed permitted the formation of a single bubble and the accurate measurement of the volume of a number of slowly formed bubbles. The second technique was developed for bubbles flowing in streams as in the absorption work, in order to ascertain if bubble velocity was affected by flow irregularities due to adjacent bubbles. This possibility was of particular interest in the vertical flows at low liquid rates where the bubbles rise through the wakes of preceeding bubbles. For these measurements the tracking wire method illustrated in Figure 8  was developed.  A soft, rubber-covered wire, painted with alternate black and white bands, ran around two 6-inch diameter pulleys and travelled parallel and close to the absorption tube.  The head pulley was driven by a variable-speed motor,  and tripped a microswitch once every revolution.  A digital interval-timer  connected to the microswitch permitted accurate determination of the speed of the wire.  To measure bubble velocity, the speed was adjusted  until the bands on the wire were moving synchronously with the bubble stream.  This matching of velocity was possible for steady velocities  up to 4 ft/sec to the extent that the relative positions of bubbles and wire bands remained within 1/2 inch over an 8-foot length.  For higher velocities  F I G U R E 8. T R A C K I N G W I R E M E T H O D BUBBLE VELOCITY  FOR  36 the single bubble method had to be used. For the tracking wire method, bubble size was determined by metering a continuous gas feed.and counting the bubble frequency.  Nitrogen bubbles were used for both techniques to  avoid bubble size variation due to absorption during a velocity measurement. In order to obtain bubble photographs which were not blurred at the experimental velocities which were as high as 10 ft/sec, an exposure time was required at least an order of magnitude faster than the typical flash duration of one millisecond obtained with common electronic flash units.  A  flash duration of approximately 30 microseconds was achieved by reducing the capacitance of a.Braun Hobby F60 electronic flash from 350 to 12 microforads. It was necessary to synchronize the flash with the bubble motion as bubble spacing was generally so large that bubbles would only rarely be in the field of view if the flash was fired at random. This synchronization was achieved by the variable-time delay circuit, shown in Figure 1-5, initiated through the bubble counter by a bubble passing the photo diode located 6 inches upstream from the camera. The circuit became active when the camera shutter was opened, and triggered the electronic flash after initiation and a pre-set delay. The delay, was pre-set using bubble velocity data and delay circuit calibration data.  Delay times from 10  milliseconds to 2 seconds were available. The optical arrangement is shown in Figure 9-  A water-filled,  Per-  spex viewing cell was fitted on the tube in order to reduce optical distortion which results from the curvature of the water surface inside the flow  37  ELECTRONIC FLASH  F I G U R E 9-  OPTICAL ARRANGEMENT FOR BUBBLE PHOTOGRAPHY  38 tube.  R e f r a c t i o n b y the g l a s s tube w a l l s t i l l c a u s e d s i g n i f i c a n t d i s t o r t i o n  n e a r the s i d e s of the tube. i n F i g u r e 9 to g i v e two  Two  mutually  t a n e o u s l y on one p h o t o g r a p h .  f r o n t s u r f a c e m i r r o r s w e r e p l a c e d as p e r p e n d i c u l a r v i e w s of the b u b b l e  I l l u m i n a t i o n was  accentuate  screens  distortion.  to p r o v i d e  W e r e r u l e d w i t h a d i a g o n a l g r i d to  the i m a g e of the tube and to p r o v i d e  bubble s u r f a c e  simul-  p r o v i d e d w i t h the f l a s h  p l a c e d b e h i n d the f l o w tube a n d d i r e c t e d at t r a n s l u c e n t s c r e e n s a diffuse back light. The  shown  a q u a l i t a t i v e i n d i c a t i o n of  39  PROCEEDURE  Absorption The  Runs apparatus  w a s t h o r o u g h l y f l u s h e d out w i t h s e r v i c e w a t e r b e f o r e  absorption runs commenced. to t h e i r o p e r a t i n g l e v e l ,  T o b e g i n a r u n , t h e two t a n k s w e r e f i l l e d  w h e n e a c h c o n t a i n e d about 10 g a l l o n s .  The  feed p u m p and e j e c t o r w e r e s t a r t e d . T h e f e e d w a t e r flow v a l v e was o p e n e d to s e t a p p r o x i m a t e l y t h e d e s i r e d f l o w r a t e o n t h e r o t a m e t e r , taneously, the test s e c t i o n b a c k - p r e s s u r e level.  and s i m u l -  w a s a d j u s t e d to the d e s i r e d  T h e l e v e l c o n t r o l v a l v e s w e r e a d j u s t e d to g i v e s a t i s f a c t o r y c o n t r o l .  P r i o r to t h i s s t a r t u p the s t e a m v a l v e at the b o t t o m o f t h e c y c l o n e w a s o p e n e d to b l o w d o w n c o n d e n s a t e f r o m t h e l i n e .  The heater  circulation  p u m p w a s n o w s t a r t e d a n d t h e s t e a m v a l v e s a d j u s t e d to a d m i t s t e a m to t h e mixer.  D u r i n g the heating-up  p e r i o d , feed water flow was f r e q u e n t l y  a d j u s t e d a n d c o o l i n g w a t e r to the h e a t e x c h a n g e r s w e r e g r a d u a l l y i n c r e a s e d . Approximately  30 m i n u t e s w a s r e q u i r e d f o r t h e v a c u u m ^ t e m p e r a t u r e s a n d  f l o w to b e c o m e  satisfactorily  steady.  B e f o r e s t a r t i n g t h e l i q u i d c y c l e , the b u b b l e n o z z l e w a s p r e p a r e d a n d a s s e m b l e d as p r e v i o u s l y d e s c r i b e d , a n d i n s t a l l e d i n the l i q u i d f e e d n o z z l e . C a r b o n d i o x i d e f l o w to the n o z z l e w a s g e n e r a l l y s t a r t e d b e f o r e  admitting  w a t e r to t h e f e e d n o z z l e i n o r d e r to p r e v e n t f l o o d i n g t h e b u b b l e n o z z l e a s s e m b l y . W h e n t h e l i q u i d c y c l e w a s steady,  CO_ f l o w w a s a d j u s t e d to g i v e  the d e s i r e d f r e q u e n c y .  A t t h i s p o i n t c a r e was t a k e n to o b s e r v e the b u b b l e  flow f o r u n i f o r m i t y and p o s s i b l e c o a l e s c e n c e r e s u l t i n g f r o m too c l o s e bubble s p a c i n g . T h e f r e q u e n c y r e c o r d e r was p a r t i c u l a r l y u s e f u l as a guide to u n i f o r m i t y f o r h i g h flow v e l o c i t i e s w h e r e o b s e r v a t i o n was d i f f i c u l t , i r r e g u l a r i t i e s i n the b u b b l e In s e t t i n g the f r e q u e n c y ,  stream produced  as a n y  a r a g g e d t r a c e on the c h a r t .  the b u b b l e n o z z l e m i g h t b e p u s h e d into the b u b b l e  s t r e a m o r w i t h d r a w n to v a r y the f r e q u e n c y o r to a c h i e v e the m o s t u n i f o r m bubble  stream  The  f o r the p r e v a i l i n g f l o w s .  apparatus  was o p e r a t e d f o r an a d d i t i o n a l 10 m i n u t e s w i t h  bubble flow before s a m p l i n g  steady  c o m m e n c e d so t h a t the c o n c e n t r a t i o n of CO^  i n the f e e d w a t e r c o u l d r e a c h s t e a d y s t a t e . e a c h s a m p l e t a p o f i n t e r e s t b y the p r o c e d u r e  S a m p l e s w e r e then d r a w n d e s c r i b e d below.  from  Two  feed-  w a t e r s a m p l e s w e r e s l o w l y d r a w n f r o m u p s t r e a m , o f the r o t a m e t e r s  at the  beginning and end of each set of test s e c t i o n s a m p l e s . was a c c u r a t e l y d e t e r m i n e d  T h e bubble  frequency  o n c e d u r i n g the r u n b y r e c o r d i n g the t a l l y  o v e r a 6 0 - s e c i n t e r v a l . V a r i a t i o n s o f l e s s t h a n 2 % i n the f r e q u e n c y  counts  were  r e a d i l y d e t e c t e d on the r e c o r d e r t r a c e . P r e s s u r e r e a d i n g s w e r e m a d e at t h i s s t a g e , b u t no s i g n i f i c a n t d i f f e r e n c e i n p r e s s u r e d r o p w a s d e t e c t a b l e , f o r a g i v e n w a t e r r a t e , w h e t h e r b u b b l e s w e r e p r e s e n t o r not.  W h e n s a m p l i n g was was  complete,  the s h u t - o f f v a l v e i n t h e C O ^  feed line  c l o s e d a n d the b u b b l e m e t e r b y p a s s v a l v e was t h r o t t l e d to h o l d the  nozzle feed pressure,  P  i n F i g u r e 3, at the s a m e l e v e l as d u r i n g the r u n .  T h i s p r e s s u r e s l i g h t l y a f f e c t e d the f l o w t h r o u g h t h e M o o r e f l o w r e g u l a t o r .  The  f l o w f o r e a c h r u n w a s thus d i r e c t l y d e t e r m i n e d b y m e a s u r i n g  the t i m e  3 f o r a s o a p b u b b l e to b e d i s p l a c e d 25, 50 o r 200 c m rate.  d e p e n d i n g o n the f l o w  F r e q u e n t c h e c k s w e r e m a d e to e n s u r e that t h e g a s f e e d s y s t e m  was  t i g h t r i g h t u p to t h e n o z z l e t i p ,  and that shutoff v a l v e s s e a l e d p o s i t i v e l y , 3 s i n c e f l o w r a t e s as l o w as 10 c m / m i n w e r e u s e d . The  C O ^ f l o w w a s a g a i n d i r e c t e d to the n o z z l e a n d a n e w f r e q u e n c y  m a i n t a i n e d f o r 10 m i n u t e s b e f o r e s a m p l i n g .  F r o m two to f i v e f r e q u e n c i e s  w e r e g e n e r a l l y u s e d f o r a f i x e d w a t e r flow.  If i t w a s n e c e s s a r y t o c h a n g e  n o z z l e t i p s f o r a n e w f r e q u e n c y t h i s c o u l d be done t h r o u g h t h e s e a l b l o c k q u i c k l y e n o u g h t h a t o n l y a b r i e f u p s e t to the l i q u i d c y c l e o c c u r r e d . shutting down the apparatus,  Before  a s u r f a c e t e n s i o n s a m p l e was always taken i n  g l a s s w a r e w h i c h h a d b e e n c l e a n e d i n c h r o m i c a c i d . S u r f a c e t e n s i o n was determined with a weight-calibrated,  DuNouy  ring tensionmeter.  Sampling A l l s a m p l e s w e r e d r a w n s l o w l y into i n v e r t e d , had p r e v i o u s l y been flushed with nitrogen.  100-ml p i p e t t e s w h i c h  T h e details of s a m p l i n g  s o m e w h a t d i f f e r e n t f o r the h o r i z o n t a l a n d v e r t i c a l t e s t s e c t i o n s .  were  In the  h o r i z o n t a l c a s e s , the p r o b e t i p w a s c o n f i n e d to t h e l o w e r h a l f o f the tube c r o s s - s e c t i o n s o that s a m p l i n g o f the g a s p h a s e d i d n o t o c c u r at s a m p l e 3 f l o w s as h i g h a s 50 c m 4 minutes  /min.  S a m p l e s w e r e d r a w n o v e r a p e r i o d o f 2 to  f r o m o n l y one s a m p l e t a p at a t i m e a f t e r a d e q u a t e l y p u r g i n g  the s a m p l e l i n e .  S i n c e the c o m p l e t e  f r e q u e n c y m i g h t t a k e 20 m i n u t e s ,  s a m p l i n g s e q u e n c e f o r one b u b b l e  two f e e d w a t e r  samples were  required  as a. c h e c k o n the c o n s t a n c y of the f e e d c o n c e n t r a t i o n . T h e  concentration  p r o f i l e a c r o s s the tube was t e s t e d f o r the 5 / 1 6 - i n c h h o r i z o n t a l tube. detailed t r a v e r s e , extending could  be obtained,  as f a r a c r o s s the tube as g a s - f r e e  i s s h o w n i n F i g u r e 10 f o r Re^=  10200.  The  A  samples concentration  of s a m p l e s d r a w n w i t h t h e p r o b e f l u s h w i t h t h e tube w a l l o r w i t h d r a w n into the w a l l a r e w i t h i n 2 % o f the c o n c e n t r a t i o n n e a r m i d - s t r e a m .  It i s s u p p o s e d  i n the p r e s e n t w o r k that t h e t i m e a v e r a g e c o n c e n t r a t i o n a c r o s s t h e e n t i r e c r o s s - s e c t i o n i s n o t s i g n i f i c a n t l y h i g h e r t h a n the m i d - s t r e a m v a l u e , e f f i c i e n t e d d y m i x i n g o v e r the c e n t r a l c o r e o f the t u r b u l e n t flow.  due to  The use  of the c o n c e n t r a t i o n o b t a i n e d w i t h p r o b e s f l u s h w i t h the w a l l i s t h e r e f o r e s a t i s f a c t o r y f o r R e ^ ^ ^ l O , 000.  However,  e v e n at t h e s e h i g h R e y n o l d s  n u m b e r s , the p o s s i b i l i t y c a n n o t b e d i s c o u n t e d that c o n c e n t r a t i o n i n the u p p e r h a l f o f the t u b e i s c o n s i d e r a b l y h i g h e r t h a n i n the l o w e r h a l f due to s o m e r e g u l a r p a t t e r n i n the f l o w o f w a t e r r e l a t i v e to the b u b b l e .  F i g u r e 11 s h o w s c o n c e n t r a t i o n s  at t h r e e r a d i a l p o s i t i o n s f o r the s a m e  tube at l o w e r R e y n o l d s n u m b e r s , w h e r e the r a d i a l m i x i n g i s s l o w e r a n d the c o n c e n t r a t i o n g r a d i e n t w i t h t u b e l e n g t h i s s t e e p e r t h a n i n the a b o v e The  case.  s l o p e s of the l i n e s i n F i g u r e 11 r e f l e c t c h a n g e s i n f e e d c o n c e n t r a t i o n  d u r i n g the s a m p l i n g s e q u e n c e . F o r t h e s e c a s e s concentration gradient exists. Therefore  a significant radial  f o r ReS-7020 the p r o b e was  about o n e - t h i r d o f a tube d i a m e t e r into the s t r e a m f o r e a c h s a m p l e .  pushed It i s  e s t i m a t e d that the t r u e a v e r a g e c o n c e n t r a t i o n s h o u l d b e w i t h i n 3 % o f the s a m p l e v a l u e e x c e p t p e r h a p s f o r v e r y s m a l l b u b b l e s at l o w R e  .  Probes  u p s t r e a m of the one i n u s e w e r e a l w a y s w i t h d r a w n to a v o i d a n y d i s t u r b a n c e  s  TYPICAL BUBBLE POSITION  15-  ic  Re = 10200 N = 936 / min SAMPLED AT 10-ft TAP FEED C0NC. STEADY, 6 ppm SAMPLE F L O W » 50 cmVmin g  TUBE CENTRELINE /  10  CONCENTRATION PROFILE  5  < I- u, CO (Q  5 UJ  m IU O  0C CL  I  3  TUBE WALL  / / / ' / / / / / / / / /  7  >  /  / / / / / / / / / / / / / / / / / / / / / /  NUMBERS INDICATE SAMPLING SEQUENCE  O  03  -5 54 F I G U R E 10.  1  55  56 57 SAMPLE CONCENTRATION, p p m  CONCENTRATION TUBE:  1  58  P R O F I L E A C R O S S 5/16 H O R I Z O N T A L  R e = 10200 s  OJ  44  4^  R e « 7020  2.5%  s  42L-  40  SYMBOL  PROBE  O  FULLY  7  HALF-WAY  < £ 38}  ui o z, o o  •  POSITION WITHDRAWN  NEAR  TO  TUBE  TUBE  t-  <L  36}  V 6%  Ul  a.  2 34[ Re « 4800 N > 520/min 32  30  6-ft SAMPLE L .  L  I  2 SAMPLING  F I G U R E 11.  TAP  = 50 cmVmin  FLOW  L  I  3  4  SEQUENCE  C O N C E N T R A T I O N S A C R O S S 5/16 H O R I Z O N T A L T U B E : Re = 7020 a n d 4810  s  of the bubble flow. F o r the 5 / 8 - i n c h h o r i z o n t a l tube, drawn with the probe  a l l samples were  pushed about o n e - t h i r d tube diameter into the  stream.  F o r the v e r t i c a l test sections it was found that gas was withdrawn with 3  the samples unless the sample rate was held to less than 15 c m with the probes completely withdrawn f r o m the s t r e a m . this s a m p l i n g e r r o r ,  / m i n even  In o r d e r to prevent  valves on each sample line were r e p l a c e d with lengths 3  of c a p i l l a r y tubing s i z e d to r e s t r i c t the flow to 10 c m  / m i n . . At this  sample rate, approximately 15 minutes was r e q u i r e d to purge the line and accumulate a s a m p l e . taneously.  Hence samples were collected f r o m a l l taps s i m u l -  In this sampling p r o c e d u r e ,  the sample concentrations  must  be somewhat lower than the c r o s s - s e c t i o n average because the probes were always withdrawn f r o m the s t r e a m . gible for R e = = » 1 0 , 200,  T h i s e r r o r is estimated to be n e g l i -  and to be within 6% for lower Reynolds n u m b e r s ,  s• based on the .concentration gradients o b s e r v e d for h o r i z o n t a l flow as a p e s s i m i s t i c estimate since no average a s y m m e t r y exists in the v e r t i c a l flow case. A n a l y s i s for D i s s o l v e d C O ^ The method of analysis used in the present work is based on the standard carbonate determination (22), C O ^ " to H C O ^ converts the H C O  in which acid is u s e d to convert  at the f i r s t equivalence point, pH 8.4.  Additional acid  to C O _ at the second equivalence point, p H 3 . 8.  acid titres r e q u i r e d for the two steps are equal for a pure sample, but unequal for mixtures of carbonate-bicarbonate,  The  carbonate or  carbonate  46 and e x c e s s h y d r o x i d e .  F o r these mixtures,  the c a r b o n a t e c a n be  d e t e r m i n e d f r o m the r e l a t i v e a m o u n t s o f t i t r e i n the two s t e p s , o r i f the a m o u n t o f s e c o n d s p e c i e s i s known, f r o m the t i t r e f o r o n l y the f i r s t s t e p . To  a p p l y t h i s m e t h o d f o r d i s s o l v e d CO^, t h e s a m p l e  a m o u n t o f N a O H s o l u t i o n to e n s u r e that p H > Z> CO  to C O  The O H  s u i t a b l e for the above a n a l y s i s , and p r e v e n t s  desorption during storage. contamination the C O ^  8.4.  i s a d d e d to a k n o w n  CO  losses by  H o w e v e r , the b a s i c s o l u t i o n i s s u s c e p t i b l e to  b y a t m o s p h e r i c CO^.  S i n c e t h e a m o u n t o f N a O H i s known,  c a n b e d e t e r m i n e d b y the s i n g l e t i t r a t i o n o f C O ^  equivalence;  converts the  i . e. , t h e e q u i v a l e n t s o f O O ^  to the p H 8 . 4  c o n v e r t e d to H C O ^  are given  b y the d i f f e r e n c e i n e q u i v a l e n t s b e t w e e n the N a O H b l a n k d e t e r m i n e d b y b l a n k titrations  a n d the a c i d t i t r e .  M o l e s GO_  =  J Equivalents i n ] I  F o r the C O ^  Hence  —  N a O H blank HCO^  c r e s o l r e d - t h y m o l blue,  /  I acid titre  d e v e l o p e d b y S i m p s o n (2,3) w a s u s e d .  f r o m p i n k - v i o l e t to y e l l o w - o r a n g e , 8.4.  "  (34)  A  c o i n c i d e d w i t h the i n f l e c t i o n p o i n t at * HCO^  H e n c e the e n d point i n t i t r a t i o n m u s t be  a p p r o a c h e d d r o p w i s e o v e r a p e r i o d o f 2 to 3 m i n i n o r d e r to a v o i d shooting. CO^  pH  that the m o s t r a p i d c o l o u r c h a n g e ,  T h e t i t r a t i o n i n v o l v e s the n o n - i o n i c r e a c t i o n C O ^  which r e q u i r e s a finite time.  /  e q u i v a l e n c e point, the m i x e d i n d i c a t o r ,  c u r v e obtained f o r the t i t r a t i o n c o n f i r m e d  pH  f Equivalents i n  T h e i n i t i a l d i s s o c i a t i o n of d i s s o l v e d m o l e c u l a r  over-  C O ^ to f o r m  i n b a s i c s o l u t i o n i s e s t i m a t e d to r e q u i r e as l o n g as 4 h r s f o r c o m -  p l e t i o n (24).  T o t e s t the i m p o r t a n c e o f s t o r a g e t i m e ,  a s e r i e s of  47 i d e n t i c a l s a m p l e s o f CC>  2  s o l u t i o n w e r e a d d e d to N a O H b l a n k s ,  and t i t r a t e d a f t e r s t o r a g e t i m e s  stoppered,  r a n g i n g f r o m 20 m i n u t e s to 5 h o u r s .  No  m e a s u r a b l e c h a n g e o c c u r r e d a f t e r the f i r s t h o u r .  B e c a u s e t h e c o n c e n t r a t i o n s o f d i s s o l v e d CO^, i n the p r e s e n t w o r k w e r e low,  s p e c i a l p r e c a u t i o n s w e r e r e q u i r e d to p r e v e n t s i g n i f i c a n t  f r o m atmospheric expected  CO.,-  contamination  N a O H b l a n k s of 5, 10 o r 15 m l , d e p e n d i n g o n  s a m p l e c o n c e n t r a t i o n , w e r e m e a s u r e d into 2 5 0 - m l e r l e n m e y e r  w h i c h w e r e t h e n f l u s h e d w i t h N,, a n d t i g h t l y s e a l e d w i t h r u b b e r  flasks  stoppers.  S a m p l e s f o r a n a l y s i s w e r e c o l l e c t e d into i n v e r t e d p i p e t t e s w h i c h w e r e f l u s h e d with N^  before filling.  T h e 1 0 0 - m l s a m p l e s w e r e a d d e d b e n e a t h the s u r f a c e  of the N a O H b l a n k s i n the f l a s k s w h i c h w e r e a g a i n f l u s h e d a n d s t o p p e r e d f o r s t o r a g e t i l l the e n d o f the a b s o r p t i o n r u n . a g a i n f l u s h e d w i t h N^of o t h e r w i s e  D u r i n g t i t r a t i o n the f l a s k s w e r e  T o t e s t the n e c e s s i t y o f t h i s p r o c e d u r e ,  identical samples were prepared  these precautions.  two s e r i e s  and a n a l y z e d with and without  A c o n s i s t e n t d i f f e r e n c e o f about 1.5 p p m C O ^ r e s u l t e d  (an e r r o r o f 5 % at a t y p i c a l s a m p l e c o n c e n t r a t i o n o f 30 p p m ) .  Standard  s o l u t i o n s (N/50) w e r e p r e p a r e d f r o m r e a g e n t g r a d e N a O H  and H C l a n d b o i l e d ,  d i s t i l l e d water. T h e N a O H s o l u t i o n was s t a n d a r d i z e d  and c h e c k e d p e r i o d i c a l l y w i t h p o t a s s i u m  a c i d phthalate,  Acid normality  a g r e e d w i t h the b a s e w i t h i n o n e d r o p i n 25 m l .  The  a n a l y t i c a l m e t h o d c h e c k e d w i t h i n 1 % ( l p p m CO., e q u i v a l e n t at  a c o n c e n t r a t i o n o f 130 p p m ) a g a i n s t a s t a n d a r d N a ^ C O ^ s o l u t i o n . A s a c h e c k o n the a n a l y s i s at l o w C O  c o n c e n t r a t i o n s , a s t o c k s o l u t i o n o f 75  48  p p m C O ^ was p r e p a r e d . 100-,  S a m p l e s w e r e t a k e n b y the a b o v e p r o c e d u r e  25-, a n d 10- m l p i p e t t e s .  The samples  w e r e m a d e u p to t h e n o r m a l  volume b y adding dilution water which had been thoroughly degassed. diluted samples  t h e n c o r r e s p o n d e d to s a m p l e s  initial concentration. 2.5%  f o r the 2 5 - m l , a n d 7. 5 % f o r the 10- m l a l i q u o t , These  The  w i t h 1/4 a n d 1/10 o f the  T h e m a x i m u m e r r o r s i n the d i l u t e s a m p l e s  d e t e r m i n a t i o n s to b e c o r r e c t .  using  assuming  were  the 1 0 0 - m l  e r r o r s c o r r e s p o n d to o n e d r o p o f  reagent.  A u s e f u l c h a r a c t e r i s t i c o f the a n a l y s i s i s that the s h a r p n e s s o f c o l o u r change i m p r o v e s  as the C O ^ c o n c e n t r a t i o n  the a n a l y s i s i s r e l i a b l e to w i t h i n at the 5 p p m l e v e l .  Hence  1 p p m at 70 p p m a n d to w i t h i n 0. 5 p p m  T y p i c a l r e p r o d u c i b i l i t y of the s a m p l i n g a n d a n a l y t i c a l  p r o c e d u r e can be seen i n F i g u r e w e l l w i t h i n 0. 5 p p m .  of the s a m p l e i s l o w e r e d .  10, w h e r e t h e s c a t t e r o f the r e s u l t s i s  49  RESULTS  A. B U B B L E  AND  DISCUSSION  VELOCITY  B u b b l e v e l o c i t i e s w e r e m e a s u r e d f o r a n u m b e r of bubble at f i x e d l e v e l s of s u p e r f i c i a l l i q u i d R e y n o l d s n u m b e r , niques d e s c r i b e d under "Apparatus. " the r a t i o , ^ The  diameters  Re^, u s i n g t h e t e c h -  V e l o c i t y d a t a w e r e e x p r e s s e d as  , o f b u b b l e v e l o c i t y to m e a n t o t a l v o l u m e t r i c f l o w v e l o c i t y .  v a l u e s f o r a l l i n d i v i d u a l d e t e r m i n a t i o n s of v e l o c i t y a r e l i s t e d i n T a b l e  II- 1 o f A p p e n d i x II. T h e r e s u l t s for v a r y i n g bubble s i z e s ( e x p r e s s e d as e q u i v a l e n t s p h e r i c a l d i a m e t e r s ) at f i x e d R e  w e r e s m o o t h e d and u s e d to p r e p a r e s  cross  plots of ^ )  versus Re  f o r fixed bubble  diameters.  The  resulting  c u r v e s w e r e s m o o t h e d w h e r e n e c e s s a r y a n d a r e s h o w n i n F i g u r e s 12(a) a n d (b).  S m o o t h e d data forj@  v e r s u s bubble d i a m e t e r  c u r v e s at e a c h R e y n o l d s n u m b e r  these  r e q u i r e d f o r a b s o r p t i o n r e s u l t s , and fitted  w i t h p a r a b o l a s i n o r d e r to e v a l u a t e the c o n s t a n t s w  (21) f o r t h e v e l o c i t y c o r r e c t i o n .  were taken f r o m  , ,w, and w i n e q u a t i o n o ' 1 2  The constants calculated  in this way a r e  l i s t e d i n T a b l e II-2. the p a r a b o l a s g i v e a g r e e m e n t w i t h a l l i n d i v i d u a l v e l o c i t y d e t e r m i n a t i o n s to w i t h i n +_ 2%, e x c e p t f o r c e r t a i n l i m i t e d r e g i o n s o f u n c e r t a i n t y as p o i n t e d out b e l o w significance,  T h e p a r a b o l a s a r e not i n t e n d e d to h a v e a n y f u n d a m e n t a l  but a r e s i m p l y an a p p r o x i m a t e a n a l y t i c a l r e p r e s e n t a t i o n o f t h e  data f o r u s e with the m a s s t r a n s f e r  equations. However,  the v e l o c i t y c u r v e s w a r r a n t  comment.  further  V e l o c i t y c u r v e s f o r the 5 / l 6 - i n c h ID, h o r i z o n t a l  s o m e f e a t u r e s of  tube a r e s h o w n i n  50 F i g u r e 12(a).  T h e r a n g e o f b u b b l e s i z e s i n t h e a b s o r p t i o n r u n s e x t e n d e d to  s l i g h t l y s m a l l e r s i z e s t h a n i n the v e l o c i t y d e t e r m i n a t i o n s . was e x t r a p o l a t e d f o r i n t e r m e d i a t e R e  H e n c e , the d a t a  to a b u b b l e d i a m e t e r of 0.20  cm,  s w h e r e a s the l o w e s t - d i a m e t e r The R  m e a s u r e m e n t w a s f o r 0. 2 6 - c m b u b b l e s .  e x t r a p o l a t e d v a l u e s a r e s h o w n b y the d a s h e d c u r v e i n F i g u r e 12(a). 2500, an a b r u p t d r o p i n  e g  meters.  occurs  For  f o r the s m a l l e r b u b b l e d i a -  It i s s u p p o s e d that t h i s d r o p r e f l e c t s the t r a n s i t i o n f r o m t u r b u l e n t  to e s s e n t i a l l y l a m i n a r  l i q u i d flow.  p a r a b o l i c shape of l a m i n a r  flow,  A s the v e l o c i t y p r o f i l e a p p r o a c h e s the s m a l l b u b b l e s t r a v e l l i n g n e a r the t o p o f the  tube s h o u l d b e i n c o n t a c t w i t h f l u i d of l o w e r v e l o c i t y f o r the f l a t t e r t u r b u l e n t p r o f i l e .  r a t i o o n the a v e r a g e t h a n  V e l o c i t i e s i n this r e g i o n a l s o v a r y  errati-  c a l l y as m u c h as 5%, s u p p o s e d l y due to t h e i n s t a b i l i t y of the l i q u i d f l o w . Near R e  g  = 15000, t h e c u r v e s  rise sharply.  This increase i s attributed  to a t h i c k e n i n g o f the l i q u i d f i l m b e t w e e n b u b b l e a n d t u b e w a l l as b u b b l e velocity increases, and g u i d e (25).  a c c o r d i n g to the t h e o r y of l u b r i c a t i o n b e t w e e n a s l i p p e r  E l l i s (26) o b s e r v e d a s i m i l a r i n c r e a s e i n v e l o c i t y r a t i o f o r  s o l i d c a p s u l e s h e a v i e r t h a n w a t e r , as the c a p s u l e s l i f t e d o f f t h e b o t t o m o f the tube.  T h e i n c r e a s e d f i l m t h i c k n e s s s h o u l d r e d u c e the d r a g f o r c e at t h e t o p  of the b u b b l e , a n d f o r c e t h e b u b b l e d o w n i n t o the h i g h e r v e l o c i t y r e g i o n o f the flow c r o s s - s e c t i o n .  B u b b l e p h o t o g r a p h s c a n not p r o v i d e q u a n t i t a t i v e i n f o r m a t i o n  about t h i s f i l m t h i c k n e s s , b e c a u s e i n the s i d e e l e v a t i o n v i e w t h e r e i s m u c h d i s t o r t i o n c l o s e to the w a l l at the t o p o f the tube. (e) s h o w that f o r R e  H o w e v e r , F i g u r e 24 (d) a n d  = 18300, the b u b b l e s a r e t r a v e l l i n g w e l l a w a y f r o m the s u p p e r w a l l o f the tube.  51  (a)  2  5/i6-iN§H  Re  F I G U R E 12.  | 6  5 s  TUBE  4  2  (b) 5/8-INCH TUBE V E L O C I T Y RATIOS FOR T H E H O R I Z O N T A L TUBES  V e l o c i t y c u r v e s f o r the 5 / 8 - i n c h , h o r i z o n t a l tube F i g u r e 12 (b).  The  d a s h e d c u r v e ( d = 0. 4 cm)  v e l o c i t y d a t a f o r b u b b l e s l a r g e r t h a n 0. 5 c m  are shown i n  i s an e x t r a p o l a t i o n  diameter.  from  In c o n t r a s t to the  5 / 1 6 - i n c h tube r e s u l t s , t h e s e c u r v e s s h o w a h i g h v e l o c i t y r a t i o f o r l a r g e b u b b l e s at l o w R e y n o l d s n u m b e r , Re  7 500.  s  No  a n d a d e c r e a s e to a m i n i m u m r a t i o at  e x p l a n a t i o n can be p r o v i d e d f o r this d i f f e r e n c e in  c u r v e s f o r the two h o r i z o n t a l t u b e s .  The  d a t a i n T a b l e II-1 i n d i c a t e w h e t h e r  m e a s u r e m e n t s w e r e m a d e f o r s i n g l e b u b b l e s o r s t r e a m s of b u b b l e s .  The  v e l o c i t y r a t i o s w e r e i n d e p e n d e n t of b u b b l e s p a c i n g f o r b o t h h o r i z o n t a l t u b e s for  f l o w s r a n g i n g f r o m a s i n g l e b u b b l e to a s t r e a m of b u b b l e s s e p a r a t e d b y  as l i t t l e as 6  cm.  V e l o c i t y c u r v e s f o r the 5 / 1 6 - i n c h , 13 (a).  The  v e r t i c a l tube a r e s h o w n i n F i g u r e  c u r v e s f o r s m a l l b u b b l e d i a m e t e r c r o s s o v e r the c u r v e s f o r  l a r g e r d i a m e t e r s as a r e s u l t of m a x i m a i n t h e / ^ —rr s h o w n i n F i g u r e 14 f o r two R e y n o l d s n u m b e r s . b u b b l e d i a m e t e r s at l o w R e  d i a m e t e r c u r v e s as  T h e m a x i m a s h i f t to l o w e r  , c a u s i n g the c u r v e s i n F i g u r e 13 (a) to c r o s s . s  L a r g e b u b b l e s , r i s i n g f a s t e r t h a n the m e a n l i q u i d v e l o c i t y due to b o u y a n c y , o c c u p y m u c h o f the tube c r o s s - s e c t i o n . H e n c e the b a c k f l o w a r o u n d the b u b b l e s c o n f i n e d to a s m a l l p o r t i o n o f the c r o s s - s e c t i o n , motion.  g r e a t l y r e t a r d s the b u b b l e  S o m e w h a t s m a l l e r b u b b l e s r e s t r i c t the b a c k f l o w to a l e s s e r e x t e n t  and h a v e a h i g h e r v e l o c i t y .  M u c h s m a l l e r b u b b l e s r e s t r i c t the b a c k f l o w  v e r y l i t t l e , but t r a v e l e c c e n t r i c a l l y i n the tube, p r e s u m a b l y  approaching  c l o s e e n o u g h to the tube w a l l to b e r e t a r d e d b y the m o r e s l o w l y m o v i n g l i q u i d .  (a)  10  F I G U R E 13.  2  R e $  5/16-INCH  5  TUBE  10  2  (b) 5 / 8 - I N C H TUBE V E L O C I T Y RATIOS FOR THE V E R T I C A L T U B E S  54  -  1.6  °  SYMBOL  BUBBLE SPACING  •  > 16 cm  V  6 to 16 cm  O  2.7 to 6 cm  \  > ^ r - FITTED  / OQ.  PARABOLAS  >  1.4  ^  \°  **o  s  ' o \ 1.2 -  /  p>0  1.0  FIGURE 1  .30  14. i  V E L O C I T Y R A T I O S IN V E R T I C A L F L O W  AT  FIXED REYNOLDS l i I  i  .40  BUBBLE  NUMBER I I  .50  .60  i  i  .70  DIAMETER , d , cm  T h e s e s m a l l b u b b l e s a r e o b s e r v e d to r i s e i n an i r r e g u l a r s p i r a l . p r e f e r r e d eccentric position is c o n f i r m e d by all photographs  The  showing p e r -  p e n d i c u l a r v i e w s o f the s m a l l b u b b l e s , s u c h as F i g u r e s 15 (a, b a n d d).  The effect of bubble  s p a c i n g on v e l o c i t y was e x p e c t e d to b e g r e a t e s t  f o r the v e r t i c a l t u b e s , s i n c e at c l o s e s p a c i n g the b u b b l e s r i s e t h r o u g h the w a k e o f p r e c e e d i n g b u b b l e s . V e l o c i t y m e a s u r e m e n t s f o r the 5/16 tube w e r e o b t a i n e d b y the t r a c k i n g w i r e m e t h o d f o r Re "is^. s  bubble method for h i g h e r Re , g  8150, a n d b y the s i n g l e  A t l i q u i d f l o w R e y n o l d s n u m b e r s o f 1430 a n d  1950 the v e l o c i t y o f c l o s e l y s p a c e d b u b b l e s i s a p p r o x i m a t e l y 1 0 % l o w e r t h a n the v e l o c i t y of w i d e l y s e p a r a t e d o r s i n g l e b u b b l e s o f the s a m e d i a m e t e r .  The  = 5250 = 0. 30  cm  = 11200 = 0. 30  cm  = 11200 = 0. 55  cm  = 18300 = 0. 30  cm  = 18300 = 0. 55  FIGURE  15.  BUBBLES  I N V E R T I C A L F L O W I N 5/16  TUBE  cm  56  v e l o c i t y r a t i o s f o r two R e y n o l d s n u m b e r s a r e s h o w n i n - F i g u r e 14. Re  For  5230 the v e l o c i t y r a t i o i s i n d e p e n d e n t o f b u b b l e s p a c i n g .  s  The  e f f e c t of s p a c i n g i s m o s t p r o n o u n c e d f o r l i q u i d f l o w s w h i c h a r e t o o l o w to sustain turbulence,  s i n c e the w a k e b e h i n d t h e b u b b l e s  the o t h e r w i s e l a m i n a r f l o w .  significantly disturbs  A t h i g h e r l i q u i d f l o w s , the w a k e m a k e s l i t t l e  d i f f e r e n c e to t h e i n d e p e n d e n t l y t u r b u l e n t s t r e a m . T h e r e d u c t i o n i n r e l a t i v e v e l o c i t y between bubbles  a n d l i q u i d at c l o s e s p a c i n g due to w a k e d i s t u r b a n c e s  i s c o n s i s t e n t w i t h the m o d e r a t e i n c r e a s e i n d r a g c o e f f i c i e n t due to f r e e s t r e a m , t u r b u l e n c e r e p o r t e d b y T o r o b i n a n d G a u v i n (27). was l e s s t h a n 10 c m f o r a l l a b s o r p t i o n r u n s w i t h R e  Bubble spacing  =s  5230.  H e n c e the  p a r a b o l a s f i t t e d to the c l o s e l y s p a c e d b u b b l e v e l o c i t i e s a r e a d e q u a t e f o r t h e v e l o c i t y c o r r e c t i o n to m a s s t r a n s f e r .  The  v e l o c i t y c u r v e s f o r the 5 / 8 - i n c h ,  v e r t i c a l tube,  13 (b) h a v e t h e s a m e g e n e r a l f e a t u r e s as F i g u r e 13 (a). curves.  shown i n F i g u r e  The curves  cross  The velocity ratios are  c o n s i d e r a b l y h i g h e r at a g i v e n R e y n o l d s n u m b e r f o r t h e 5/8 tube,  s i n c e the  m e a n l i q u i d v e l o c i t y i s o n l y h a l f as m u c h f o r t h e l a r g e tube as f o r t h e s m a l l tube at the s a m e R e y n o l d s n u m b e r , w h e r e a s r e l a t i v e v e l o c i t i e s b e t w e e n b u b b l e a n d l i q u i d s h o u l d be o f the s a m e o r d e r f o r t h e two t u b e s . of b u b b l e s p a c i n g w a s n o t i n v e s t i g a t e d f o r t h i s tube,  T h e effect  s i n c e the 5/16 tube  r e s u l t s i n d i c a t e d that the e f f e c t was of l i t t l e s i g n i f i c a n c e w i t h r e s p e c t to t h e velocity correction applied i nmass transfer  coefficients.  57 B. A B S O R P T I O N  RESULTS  Horizontal,  5/16-inch  The  5/16-inch,  Test Section h o r i z o n t a l tube w a s m o s t e x t e n s i v e l y s t u d i e d f o r gas  absorption because a wider The  r a n g e o f the v a r i a b l e s w a s p o s s i b l e f o r t h i s tube.  a b s o r p t i o n f u n c t i o n c o r r e l a t i n g e q u a t i o n was t e s t e d u s i n g r e s u l t s  t h i s tube,  from  and m a s s t r a n s f e r c o e f f i c i e n t s w e r e also c a l c u l a t e d and c o r r e l a t e d  f r o m the s a m e d a t a .  S u p e r f i c i a l l i q u i d R e y n o l d s n u m b e r s b e t w e e n 1810 a n d 22400 w e r e s t u d i e d i n the 5/16, h o r i z o n t a l tube. A b o v e t h i s r a n g e b u b b l e s w e r e u n s t a b l e i n the f e e d n o z z l e , a l t h o u g h t h e y r e m a i n e d s t a b l e i n the t e s t s e c t i o n at s o m e what h i g h e r R e . g  above a t m o s p h e r i c  T h e range of o p e r a t i n g p r e s s u r e s was v a r i e d f r o m to 3 a t m a b s o l u t e .  slightly  A b s o r p t i o n d a t a f o r the m a i n c o r r e l -  a t i o n s w e r e o b t a i n e d u s i n g t h e 2 - f t a n d s u b s e q u e n t s a m p l e t a p s ( S e e F i g u r e 4) in o r d e r to o m i t the e f f e c t o f a b s o r p t i o n i n the f e e d n o z z l e a n d tube B u b b l e d i a m e t e r s b e y o n d the 2 - f t t a p w e r e b e t w e e n 0. 22 a n d 0. 54 ( d / D = 0.28 to 0.69).  Bubble frequencies ranged  from  entrance. cm  160 to 1250 /min,  a l t h o u g h the w i d e s t r a n g e at a s i n g l e l i q u i d f l o w r a t e was 280 to 1 0 8 0 / m i n . A l l a b s o r p t i o n r e s u l t s w e r e o b t a i n e d at 20. 0+^0.2 ° C .  In o r d e r to t e s t the a p p l i c a b i l i t y o f the s i m p l i f y i n g a s s u m p t i o n s l e a d i n g to e q u a t i o n (18), the a b s o r p t i o n f u n c t i o n values,. F ^ (f), w e r e c a l c u l a t e d the e x p e r i m e n t a l d a t a u s i n g e q u a t i o n (19)w e r e l e s s t h a n 0. 03 f o r a l l data.  from  G a s to l i q u i d v o l u m e t r i c r a t i o s  T h e m e a n v a l u e of p r e s s u r e o v e r the t e s t  s e c t i o n was u s e d i n t h i s c a l c u l a t i o n .  The equilibrium interface concentration  58 q * was  calculated using a Henry's Law  f r a c t i o n of C O with raw  at 20 ° C (28).  experimental  c o n s t a n t of 1.42  x 10  3  atm/mole-  C o m p l e t e d e t a i l s of the c a l c u l a t i o n s t a r t i n g  data a r e p r e s e n t e d  s a m p l e s w e r e d r a w n f r o m m o r e t h a n two  i n A p p e n d i x III (a). F o r m o s t r u n s s a m p l e t a p s i n the t e s t s e c t i o n ,  p e r m i t t i n g c a l c u l a t i o n of a b s o r p t i o n r e s u l t s f o r a r a n g e of t e s t l e n g t h s b u b b l e s i z e s w i t h a g i v e n gas  and l i q u i d f l o w and n o z z l e t i p .  the c o n c e n t r a t i o n di f f e r e n c e b e t w e e n a p a i r of s a m p l e s was n o r m a l a n a l y t i c a l e r r o r s i n the s a m p l e c o n c e n t r a t i o n s e r r o r i n the d i f f e r e n c e .  and  Occasionally, so s m a l l that  made a large potential  T h e s e r e s u l t s w e r e r e j e c t e d i f the u n c e r t a i n t y  g r e a t e r than a p p r o x i m a t e l y  10%  of the d i f f e r e n c e . T h e  experimental  was  results  f o r a l l d a t a p o i n t s a r e t a b u l a t e d in A p p e n d i x I V (a). The  p r o p o r t i o n a l i t y of the a b s o r p t i o n f u n c t i o n and t e s t l e n g t h p r e d i c t e d  b y e q u a t i o n (18) is w e l l c o n f i r m e d result,  it m a y  b y F i g u r e 16 f o r t y p i c a l r u n s .  F r o m this  be i n f e r r e d that the m a s s t r a n s f e r c o e f f i c i e n t a v e r a g e d o v e r  the e n t i r e b u b b l e s u r f a c e i s at l e a s t a p p r o x i m a t e l y o v e r the s i z e r a n g e s t u d i e d .  The  i n d e p e n d e n t of b u b b l e s i z e  c a l c u l a t i o n of m a s s t r a n s f e r c o e f f i c i e n t s  c a n t h e r e f o r e be g r e a t l y s i m p l i f i e d b e c a u s e the i n t e g r a t i o n u s e d i n d e r i v i n g equations  (18) and (20) i s j u s t i f i e d i n s t e a d of a d i f f e r e n t i a l a p p r o a c h .  In o r d e r to t e s t the e f f e c t of b u b b l e f r e q u e n c y r e s u l t s w e r e r e q u i r e d f o r a w i d e r a n g e of N l i q u i d flow,  Q  L  .  The  widest  r a n g e of N  i n c l u d e d the f e e d n o z z l e and e n t r a n c e  B  B was  length.  on the a b s o r p t i o n f u n c t i o n ,  at c o n s t a n t t e s t l e n g t h  and  available for results which Hence these results,  tabulated  in A p p e n d i x I V (b) w e r e u s e d , w i t h the a s s u m p t i o n that f o r a f i x e d l e n g t h , the e f f e c t of b u b b l e f r e q u e n c y  s h o u l d not be  c h a n g e d b y i n c l u d i n g a 2. 17-ft  TEST F I G U R E 16.  LENGTH , ft  DEPENDENCE O F ABSORPTION ON  TEST  FUNCTION  LENGTH  entrance  s e c t i o n i n t e s t l e n g t h s of 6. 17 and  10. 17 ft. T h e  e f f e c t of b u b b l e  frequency  s h o w n i n F i g u r e 17 i s not as w e l l d e f i n e d as the e f f e c t o f t e s t  l e n g t h , b e c a u s e of the l i m i t e d r a n g e of f r e q u e n c i e s and the on f r e q u e n c y .  The  frequency  weak dependence  d e p e n d e n c e i n d i c a t e d b y F i g u r e 17 i s a p p r o x i -  m a t e l y the 1/3 p o w e r p r e d i c t e d b y e q u a t i o n  (18).  a l s o r e p r e s e n t a r a n g e of o p e r a t i n g p r e s s u r e s ,  The  r e s u l t s i n F i g u r e 17  and i n d i c a t e that the  a b s o r p t i o n f u n c t i o n a d e q u a t e l y t a k e s into a c c o u n t the o p e r a t i n g p r e s s u r e this i s r e a s o n a b l y pressure  c o n s t a n t a l o n g the t e s t s e c t i o n (/^  L  ai  s=::  6 % of m e a n  at R e = 7020). s  It f o l l o w s f r o m F i g u r e s N  p " *  where  16 and  17 that the d i m e n s i o n l e s s  group'.  3 . / Q. . is s u i t a b l e to c o r r e l a t e the a b s o r p t i o n at a g i v e n l i q u i d f l o w . T h e  ' 7020  o  0.4  SLOPE  = 0 35  0.3— —  •  SLOPE  = 0.29  b. o  0.2—  SYMBOL  o • 1  3 00  I  1  1  2280 mm Hg  10 7 15  o 1  MEAN OPERATING PRESSURE  RUN  1  1790 919 1  1  500 BUBBLE FREQUENCY , min" 1  F I G U R E 17. D E P E N D E N C E O F A B S O R P T I O N F U N C T I O N O N B U B B L E FREQUENCY  1  1000  1  1  1  61 c o r r e l a t i o n i s s h o w n f o r a t y p i c a l r u n i n F i g u r e 18.  M o s t of the r u n s  p l o t t e d as i n F i g u r e 18 g i v e a s l o p e b e t w e e n 0. 32 a n d 0. 345  when  as p r e d i c t e d b y 3  e q u a t i o n (18).  D a t a f o r the r e m a i n i n g r u n s c o v e r a n a r r o w r a n g e of N L D  a n d t h e r e f o r e do not s i g n i f i c a n t l y d e t e r m i n e  the s l o p e , a l t h o u g h t h e s e  /Q  results  a l s o t e n d to c o n f i r m e q u a t i o n (18). The  d e p e n d e n c e of the a b s o r p t i o n f u n c t i o n on l i q u i d R e y n o l d s n u m b e r 3  was  t e s t e d at a v a l u e o f N ^ L  6 /QL  = 10  .  I n t e r c e p t s at t h i s v a l u e w e r e  o b t a i n e d f r o m c o r r e l a t i o n s s u c h as F i g u r e 18. adequately determine  F o r r u n s w h i c h d i d not  the s l o p e as m e n t i o n e d above, the d a t a w e r e f i t t e d w i t h  the b e s t l i n e of 1 /3 s l o p e as the b e s t e s t i m a t e of the 10^ i n t e r c e p t . Re^  exponent,  10^ i n t e r c e p t a n d 9 5 %  The  confidence limits are listed in Table 1  f o r a l l r u n s i n the h o r i z o n t a l , 5/16 tube;. The intercepts are plotted against Re i n F i g u r e 19 w h i c h i n d i c a t e s a R e exponent ( -1 i n e q u a t i o n (18)) of s s -0. 54.  The  s c a t t e r ( 9 5 % l i m i t s ) of F i g u r e 19 i s +_  11%.  A l l individual  d e t e r m i n a t i o n s of a b s o r p t i o n f u n c t i o n f o r the p r e s e n t w o r k (176 v a l u e s ) a r e r e p r e s e n t e d by F (f) = 0.228 R e " ' 0  The  Re  s  , S h S c  5 4  (N  L /Q .) 3  T  +  18%  (36)  e x p o n e n t o b t a i n e d i m p l i e s t h a t e q u a t i o n (9) c a n be w r i t t e n &  Re  s  0.46 , that is k  The  L  R  e  °s  4 6  (37)  c o r r e l a t i n g e q u a t i o n (18) r e p r e s e n t s gas a b s o r p t i o n i n b u b b l e  f l o w i n t h i s tube q u i t e w e l l , w i t h o u t the r e f i n e m e n t of b u b b l e v e l o c i t y c o r r e c t i o n s . The  exponent  *  0. 46 i s o n l y v e r y s l i g h t l y o u t s i d e the p o s t u l a t e d r a n g e  F I G U R E 18. D E P E N D E N C E O F A B S O R P T I O N F U N C T I O N O N T H E D I M E N S I O N L E S S G R O U P , N ^ L /Q  63 TABLE  1. S U M M A R Y  O F A B S O R P T I O N R E S U L T S F O R 5/16 H O R I Z O N T A L  TUBE  A b s o r ation F u n c t i o n RUN  j  Re  s  Slope  Int. @10  : 1/3  0.  k  LV  (cm/min)  P*  + 2 <7-"  %  Mean  +  2<T-  o  %  Mean  % Mean f  Pressure (mmHg)  0 1  881  10. 0  0 2  884  .1. 30  33. 0  0 3  879  1. 36  6. 6  1 1  890  12. 0  1. 60  6. 2  0 7  900  . 304  21. 9  1. 82  23. 1  1 0  888  0. 344  . 2 54  4. 2  2. 03  5. 3  1 3  898  4810  . 334  . 252  11. 0  2. 05  9. 2  1 3  901  9  5880  . 324  . 229  9- 5  2. 35  7. 7  2  1  910  7  7  7020  . 336  . 199  6. 0  2. 29  6. 5  2 9  1790  10  4  7020  . 343  . 203  19. 0  2. 38  13.  2 3  2280  15  8  7020  . 319  , 193  7. 6  2. 26  9-  5 8  919  27  18  7510  . 327  . 197  15. 0  2. 53  14. 6  6.0  923  26  12  8760  . 325  . 170  9. 5  2. 56  6. 3  3 3  2310  4  6  10200  . 344  . 158  14. 9  2. 70  15. 2^  4.4  2310  14  7  10200  . 302  . 161  17. 8  2. 81  15. 0 •13. 7  4 4'  2290  24  8  10200  . 332  . 151  9. 3  2. 62  8.  4.4  2280  41  12  13200  . 329  . 133  9- 7  3. 13  .7. 5  6 8  2280  25  15  14200  . 341  . 127  14. 6  3. 20  11. 3  8 2  8  7  15300  . 336  . 118  13. 1  3. 19  15. 0  8 9  9  9  18300  . 331  . 123  18. 6  4. 10  16.  40  12  18300  . 342  . 103  16. 0  3. 57  15.  39  9  22400  . 339  . 103  12. 6  4. 44  .1. 10  32  2  1810  18  3  I960  1/3  . 352  13. 4  .1. 02  31  6  2350  1/3  . 328  29. 0  19  2  2604  1/3  . 314  6. 8  17  4  3170  1/3  . 307  30  7  3517  1/3  16  5  4810  29  6  28  AP  379  o v e r the l e n g t h i n u s e f o r a g i v e n r u n  D a t a f i t t e d t o b e s t l i n e o f 1/3 s l o p e  *\  3  7  °J  a  12  $  2240 2280  1  2280  7J  12 6  2230  12. 6  18 2  2190  F I G U R E 19. D E P E N D E N C E G F A B S O R P T I O N F U N C T I O N O N LIQUID R E Y N O L D S N U M B E R  SUPERFICIAL  65 0. 5 —=r  <*^- 1.5,  a n d so the e l i m i n a t i o n of.(1 + % - q)  1-oC  from  equation  (13) i s j u s t i f i e d .  The  e f f e c t o f i n c l u d i n g the f e e d n o z z l e a n d 2 - f t e n t r a n c e  o f the t e s t  s e c t i o n i n the a b s o r p t i o n r e s u l t s i s s h o w n i n F i g u r e 20 f o r the s a m e r u n as i n F i g u r e 18.  absorption  T h e l i n e in; F i g u r e 20 i s the l e a s t s q u a r e s fit o f the d a t a  i n F i g u r e 18 w h e r e t h e e n t r a n c e l e n g t h i s o m i t t e d .  The results with  i n c l u d e d f a l l b e l o w the r e s u l t s w i t h e n t r a n c e omitted', •}. •  and the d i f f e r e n c e is  3  g r e a t e s t (15 to 2 0 % ) at l o w N _ L f e e d n o z z l e a n d tube e n t r a n c e .  entrance  /Q, . w h e r e the t e s t l e n g t h c o n t a i n s o n l y the T h e d a t a o f f i g u r e 20 a r e f r o m A p p e n d i x IV(.b).  It i s i n t e r e s t i n g to c o m p a r e the p r e s e n t r e s u l t s f o r a h o r i z o n t a l tube w i t h H a y d u k ' s r e s u l t s (7) f o r b u b b l e flow, all v a r i a b l e s n e c e s s a r y less,  f o r c a l c u l a t i o n of the a b s o r p t i o n f u n c t i o n . N e v e r t h e -  one c o m p a r i s o n i s p o s s i b l e .  of t r a n s f e r u n i t s , N T U , of N T U  versus  a l t h o u g h H a y d u k d i d not m e a s u r e  H a y d u k p r e s e n t e d h i s r e s u l t s as n u m b e r  i n a g i v e n t e s t l e n g t h , a n d f o u n d that a l o g - l o g p l o t  gas f e e d f l o w g a v e a s t r a i g h t l i n e o f s l o p e 2/3 f o r b u b b l e f l o w  at a c o n s t a n t l i q u i d f e e d . F o r the p r e s e n t d i l u t e s y s t e m , the a p p l i c a b l e definition of N T U i s  NTU  However,  change i n c o n c e n t r a t i o n log mean driving force  (38)  f o r the p r e s e n t w o r k , the b u l k c o n c e n t r a t i o n i s a l w a y s l e s s  3 % of the e q u i l i b r i u m v a l u e . m a t e l y the e q u i l i b r i u m v a l u e .  than  H e n c e the l o g m e a n d r i v i n g f o r c e i s a p p r o x i -  F I G U R E 20. E F F E C T O F I N C L U D I N G F E E D N O Z Z L E A N D LENGTH  IN A B S O R P T I O N  RESULTS  ENTRANCE  67  F o r t h i s r e p r e s e n t a t i o n of the r e s u l t s the f e e d n o z z l e i s i n c l u d e d , s i n c e the gas f e e d f l o w QQ  c h a n g e s s i g n i f i c a n t l y b e f o r e the 2 - f t tap i s r e a c h e d .  Q  NTU  v a l u e s f o r one l i q u i d f l o w h a v e b e e n c a l c u l a t e d b y e q u a t i o n (39)  from  the r e s u l t s i n A p p e n d i x I V (b), a n d a r e s h o w n i n F i g u r e 21.  The  NTU  r e s u l t s f a l l on a s t r a i g h t l i n e f o r e a c h  a b s o r b e r length,  but the s l o p e i s c l o s e to u n i t y , w h e r e a s H a y d u k s r e s u l t s h a v e a s l o p e of 1  2/3. was  T h i s d i f f e r e n c e i s e x p l a i n e d b y the f a c t that H a y d u k ' s b u b b l e almost  c o n s t a n t w i t h v a r y i n g gas flow.  H e n c e bubble  surface area  a p p r o x i m a t e l y p r o p o r t i o n a l to gas f l o w to the p o w e r 2/3. of F i g u r e 21, since bubble  frequency  In the r e s u l t s  s u r f a c e a r e a i s a p p r o x i m a t e l y p r o p o r t i o n a l to the gas s i z e i s a l m o s t c o n s t a n t w i t h v a r y i n g gas  flow,  flow.  In a f u n d a m e n t a l s t u d y of m a s s t r a n s f e r , the m a s s t r a n s f e r  coefficient  is of m o r e i n t e r e s t t h a n the i n t e g r a t e d , d e s i g n o r i e n t e d f u n c t i o n s s u c h NTU  or absorption function.  The  was  m a s s t r a n s f e r c o e f f i c i e n t was  as  calculated  f o r e a t h a b s o r p t i o n m e a s u r e m e n t ( i . e. , f o r e a c h p a i r of s a m p l e s ) b y  equation  (20) a n d c o r r e c t e d f o r v a r i a b l e v e l o c i t y i n the tube b y e q u a t i o n s (27)  and  (28).  The  constants w  , w. o  and w 1  i n e q u a t i o n (28) w e r e o b t a i n e d f r o m  the  2  v e l o c i t y data a l r e a d y presented,  a n d a r e t a b u l a t e d i n T a b l e II-2.  The  c a l c u l a t e d v a l u e s of m a s s t r a n s f e r c o e f f i c i e n t c o r r e c t e d f o r v e l o c i t y , k ! a r e l i s t e d i n A p p e n d i x I V (a) f o r a l l d a t a p o i n t s . T h e  , J-1  V  m e a n v a l u e s of k  J_i V f o r e a c h r u n at c o n s t a n t R e confidence l i m i t s on k J_i  The  m e a n v a l u e s of k  V  J_i V  s  a r e l i s t e d i n T a b l e 1 a l o n g w i t h the  for e a c h run,  95%  e x p r e s s e d as % of the m e a n v a l u e .  a r e p l o t t e d i n F i g u r e 22 w h i c h h a s  a s l o p e ( i . e. ,  68  .02  Re = 7020 s  Z  .01 SLOPE »  0.99  SLOPE =  0.96  .005  J  CO.  FEED  RATE, Q  L  , cm*/min  r  F I G U R E 21. A B S O R P T I O N D A T A R E P R E S E N T E D A S  R e y n o l d s n u m b e r e x p o n e n t ) o f 0. 52 a n d a s c a t t e r o f + 1 3 % . i n d i v i d u a l v a l u e s of k  100  a r e c o r r e l a t e d against Re  • XJ V  s  NTU  If the 176  , the s c a t t e r i s 1 8 %  These values are represented by k  The  „ = 0. 023 R e LV s  0. 52  T  slight increase  cm/min  +  (40)  18%  i n Reynolds number dependence indicated b y equation  (40) o v e r that i n d i c a t e d b y (39) i s due to t h e u s e o f v e l o c i t y c o r r e c t i o n f a c t o r s which slightly decrease k  at l o w R e  • J-i  and slightly increase S  k ;  at h i g h R e . JLJ  S  H o w e v e r the s c a t t e r i n t h e r e s u l t s i s n o t r e d u c e d b y t h e v e l o c i t y c o r r e c t i o n .  Uncertainty  in Reynolds Number  Exponent  F r o m t h e l e a s t s q u a r e s a n a l y s i s o f the d a t a i n F i g u r e  22, i t i s p o s s i b l e  to e s t i m a t e the u n c e r t a i n t y i n the s l o p e , as d e s c r i b e d i n A p p e n d i x III (b). This analysis indicates 9 5 %  c o n f i d e n c e l i m i t s on the s l o p e as 0. 52 +_ 0. 04.  H o w e v e r , t h i s r a n g e o f c o n f i d e n c e i s b a s e d on the a s s u m p t i o n  that the r e s u l t s  a r e f u n d a m e n t a l l y l i n e a r w h e n p l o t t e d on l o g - l o g c o o r d i n a t e s .  In fact,  the d a t a i n F i g u r e 22 m i g h t be f i t t e d w i t h l e s s v a r i a n c e b y a l i n e w i t h s l i g h t c u r v a t u r e s i n c e the m e a n v a l u e s of k  a b o v e the l i n e a r b e s t f i t .  „ f o r 2600 LV  T  ^  Re  s  7020 a l l l i e  It i s i m p o r t a n t to c o n s i d e r the p o s s i b i l i t y of a  m o r e c o m p l e x r e l a t i o n s h i p i f a n y i n f e r e n c e s about the f u n d a m e n t a l p r o c e s s a r e to b e d r a w n f r o m the R e y n o l d s n u m b e r e x p o n e n t .  Some possible causes  of a m o r e c o m p l e x r e l a t i o n s h i p w i l l s u b s e q u e n t l y be q u a l i t a t i v e l y d i s c u s s e d . H o w e v e r f i r s t the a m o u n t of s c a t t e r due to e x p e r i m e n t a l e r r o r m u s t be considered."  In A p p e n d i x III (c) i t i s s h o w n t h a t the r a n d o m e r r o r i n c a l c u l a t i n g k  ,_ LV  T  i s a l m o s t e n t i r e l y due to u n c e r t a i n t y i n f. and f., a n d t h a t r a n d o m i j  error  c a n a c c o u n t f o r a s c a t t e r of + 1 3 % ( 9 5 % c o n f i d e n c e ) f o r i n d i v i d u a l d e t e r m i n a t i o n s of k  .  F o r the a v e r a g e  v a l u e s p l o t t e d i n F i g u r e 22,  the r a n d o m  L V s c a t t e r s h o u l d be c o n s i d e r a b l y l e s s t h a n 13%. l i m i t s on k  LV  experimental  with  r u n s ( R u n s 30 and 31) w h i c h h a v e s i g n i f i c a n t v a r i a t i o n  with bubble diameter.  F o r t h e s e r u n s at R e  s  m a s s t r a n s f e r coefficient i s lower for s m a l l bubbles, an e r r o n e o u s  confidence  f o r e a c h r u n , l i s t e d i n T a b l e 1, a r e of the o r d e r of 1 3 %  the e x c e p t i o n of two i n k, LV  The  = 2350 a n d 3517,  s u p p o s e d l y due to  (low) s a m p l e c ' o n c e n t r a t i o n f o r the s m a l l b u b b l e s  with slow r a d i a l mixing.  1  the  in a  stream  H e n c e the s c a t t e r of r e s u l t s f r o m the l i n e a r f i t i n  71 F i g u r e 22 ( 1 3 % f o r m e a n v a l u e s a n d 1 8 % f o r i n d i v i d u a l appears  determinations)  to b e s o m e w h a t h i g h e r t h a n c a n b e a t t r i b u t e d s o l e l y to r a n d o m  Two  a d d i t i o n a l f a c t o r s w e r e c o n s i d e r e d as p o s s i b l e c a u s e s  in F i g u r e 22. T h e u s e of m e a n t e s t s e c t i o n p r e s s u r e w a s  error.  of the s c a t t e r  c o m p a r e d with  v a r i a b l e p r e s s u r e f o r R u n 39, w h i c h h a d the h i g h e s t p r e s s u r e d r o p ( 1 8 % of the m e a n p r e s s u r e o v e r the 12-ft t e s t l e n g t h ) .  T a b l e 2 s h o w s t h a t the  m a s s t r a n s f e r c o e f f i c i e n t s c a l c u l a t e d b y e q u a t i o n s (20) a n d (32) d i f f e r e d b y l e s s t h a n 3. 1 % f o r t h i s e x t r e m e c a s e . F o r R e  ""^ 15000, the v a l u e s d i f f e r b y s  l e s s t h a n 1%.  T h e s u r f a c e t e n s i o n of w a t e r s a m p l e s f o r a l l r u n s was i n t h e  r a n g e 72.8 +  0. 6 d y n e / c m at 20 ° C . - H e n c e the p o s s i b i l i t y of s u r f a c e t e n s i o n  e f f e c t s c a u s i n g s c a t t e r c a n b e r u l e d out. In t h e ' I n t r o d u c t i o n " , i t w a s p o s t u l a t e d t h a t k  • J-i V  i s p r o p o r t i o n a l to  oC Re .  H e n c e a l i n e a r p l o t i s e x p e c t e d f o r l o g ^--^y  versus log Re ,  provided  s o m e p o w e r l a w m e c h a n i s m p e r s i s t s o v e r the e n t i r e r a n g e o f R e .  However  g  g  g  if s u r f a c e r e n e w a l b y l i q u i d f l o w t u r b u l e n c e i s the c o n t r o l l i n g m e c h a n i s m , t h e n at R e  =  2100 the r e s u l t s s h o u l d d r o p b e l o w the p o w e r l a w c u r v e ,  s  as  ;  i n d i c a t e d at the l o w e n d o f the c u r v e i n F i g u r e 23, i n w h i c h i t i s s u p p o s e d t h a t o n l y the c e n t r a l p o r t i o n o f the d a t a f o l l o w s a s i m p l e p o w e r l a w .  A l l results  h a v e b e e n b a s e d on the e q u i v a l e n t s p h e r i c a l s u r f a c e a r e a . T h e p h o t o g r a p h s i n F i g u r e 24 s h o w that f o r a l l b u b b l e  s i z e s used and for Re  bubbles  e x c e p t f o r s o m e f l a t t e n i n g a g a i n s t the  are approximately  tube w a l l . bubble  spheres,  H o w e v e r for Re  u p to 10200, t h e  = 18300, s i g n i f i c a n t d i s t o r t i o n s o c c u r i n the s s h a p e . T h e i n c r e a s e i n s u r f a c e a r e a as d i s t o r t i o n d e v e l o p s r e s u l t s  72  T A B L E 2.  E F F E C T OF TRANSFER  PRESSURE  V A R I A T I O N IN MASS  CALCULATION  R e s u l t s o f R u n #39, w i t h o u t v e l o c i t y c o r r e c t i o n R e = 22400 P r e s s u r e D r o p = 1 8 % of m e a n p r e s s u r e in test s e c t i o n  Mass T r a n s f e r Coefficient, k  Sample Taps i  C a l c u l a t e d with m e a n pressure by E q ' n (20)  j  (cm/min) L Calculated with v a r i a b l e p r e s s u r e b y e q u a t i o n (32)  :  Difference %  6  10  4. 02  3.96  -1. 5  2  6  3. 56  3.45  -3. 1  2  10  3. 79  3. 70  -2. 3  10  14  3. 58  3. 59  +0. 3  6  10  4. 28  4.22  -1.4  2  6  3. 54  3.43  -3. 1  6  14  3. 93  3.91  -0. 6  2  10  3.91  3. 82  -2. 3  2  14  3. 80  3. 75  -1. 3  r  5.0  TURBULENT RENEWAL MECHANISM NOT E F F E C T I V E  SURFACE AREA INCREASING  E u  2.0  1.0  0.5 10  1 1 1  10*  Re. F I G U R E 23. P O S T U L A T E D I N T E R P R E T A T I O N O F M A S S T R A N S F E R C O E F F I C I E N T S TO I L L U S T R A T E T H E U N C E R T A I N T Y IN THE REYNOLDS NUMBER EXPONENT  74  (d) R e d  s  = 18300 =0.45 c m  F I G U R E 24. B U B B L E S  (e) R e d  s  = 18300 = 0. 34 c m  (f) R e d  I N H O R I Z O N T A L F L O W I N 5/16  s  = 22400 =0.44  TUBE  in c a l c u l a t e d c o e f f i c i e n t s that a r e too h i g h ,  e v e n i f the p o w e r l a w  mechanism  s t i l l a p p l i e s f o r the t r a n s f e r p e r u n i t a r e a . H e n c e the d a t a r i s e a b o v e the p o w e r l a w c u r v e as i n d i c a t e d at the h i g h e n d of the c u r v e i n F i g u r e  The  o b j e c t i v e of t h i s r a t h e r s p e c u l a t i v e  23.  i n t e r p r e t a t i o n of t h e d a t a  i n F i g u r e 23 is to e s t i m a t e the s i g n i f i c a n c e of the R e y n o l d s n u m b e r e x p o n e n t 0. 52 i n d i c a t e d b y the p o w e r l a w fit of F i g u r e 22. curve  The  c e n t r a l p a r t of the  i n F i g u r e 23 s h o u l d b e f r e e of the a b o v e c r i t i c i s m s ,  and has  been  f i t t e d w i t h a s t r a i g h t l i n e of s l o p e 0. 40 to g i v e a l o w e s t e s t i m a t e of the p o w e r l a w e x p o n e n t . T h i s e x p o n e n t c a n o n l y be s t a t e d w i t h c o n f i d e n c e as b e t w e e n 0. 40 a n d 0. 55.  The  somewhere  p h y s i c a l s i g n i f i c a n c e of t h i s e x p o n e n t w i l l b e  c o n s i d e r e d in a later section.  The  a b o v e d i s c u s s i o n e m p h a s i z e s one  of  the f u n d a m e n t a l d i f f i c u l t i e s of t w o - p h a s e flow: n a m e l y that m a n y f l o w p a t t e r n s c a n be m a i n t a i n e d  o v e r o n l y a n a r r o w r a n g e of the v a r i a b l e s , so t h a t it i s  d i f f i c u l t to e s t a b l i s h  The No  b u b b l e d i a m e t e r h a s not b e e n c o n s i d e r e d i n the a b o v e d i s c u s s i o n .  c o r r e l a t i o n h a s b e e n f o u n d b e t w e e n b u b b l e s i z e and k  of the l o w R e k  LV  and  r e l i a b l y the e f f e c t of the v a r i a b l e s i n a g i v e n r e g i m e .  f o r two  s  r a n g e n o t e d a b o v e . A l s o at Re  i n Run  40) d i f f e r b y 14%.  exception  = 18300, the m e a n v a l u e s of  r u n s w i t h q u i t e d i f f e r e n t b u b b l e s i z e s (0. 385 to .251  . 554 to .371  bubble  s  , w i t h the  This difference may  i n Run>9,  be due to  diameter.  C o m p a r i s o n of a l l T e s t S e c t i o n s A v e r a g e v a l u e s of the m a s s t r a n s f e r c o e f f i c i e n t c o r r e c t e d f o r v e l o c i t y a r e p l o t t e d i n F i g u r e 25 f o r the h o r i z o n t a l 5/8  tube a n d f o r b o t h  vertical  F I G U R E 2 5. C O M P A R I S O N O F M A S S T R A N S F E R C O E F F I C I E N T S FOR A L L TEST SECTIONS  77 tubes.  T h e s e c o e f f i c i e n t s w e r e c a l c u l a t e d b y e q u a t i o n s (20), (27) a n d (28),  and a r e l i s t e d i n A p p e n d i x I V (a). s u m m a r i z e d i n T a b l e 3. 5/16  results,  is also  Average values for each run are  E q u a t i o n (40), w h i c h r e p r e s e n t s the h o r i z o n t a l ,  p l o t t e d i n F i g u r e 25.  In T a b l e 3, the a b s o r p o t i o n  function c o r r e l a t i o n s are also s u m m a r i z e d for e ach run. s e c t i o n s s t u d i e d , the a b s o r p t i o n to 1/3  on the g r o u p N  f u n c t i o n was  B  L  3  f u n c t i o n c o r r e l a t i o n h a s an e x p o n e n t c l o s e  /Q. ,. as p r e d i c t e d b y e q u a t i o n (18). L  not c o r r e l a t e d w i t h R e  s  f o r the d a t a of T a b l e 3.  w e r e i n the s a m e r a n g e f o r b o t h 5/16 diameters  F o r a l l test  tubes.  w e r e b e t w e e n 0. 32 a n d 0. 98 cm;  F o r the 5/8  Absorption Bubble  tubes,  i . e. , b u b b l e to tube  diameters  bubble diameter  r a t i o s w e r e i n a p p r o x i m a t e l y the s a m e r a n g e f o r a l l t e s t s e c t i o n s s t u d i e d .  In the ' I n t r o d u c t i o n " i t was  p o i n t e d out t h a t i n h o r i z o n t a l  bubble  flow, the t h i n l i q u i d f i l m b e t w e e n b u b b l e a n d tube w a l l m i g h t p r o v i d e e f f i c i e n t mass transfer.  If t h i s t h i n f i l m t r a n s f e r i s an i m p o r t a n t m e c h a n i s m ,  different surface average  then  m a s s t r a n s f e r c o e f f i c i e n t s s h o u l d be o b t a i n e d f o r  h o r i z o n t a l a n d v e r t i c a l f l o w s , b e c a u s e the f i l m s h o u l d be m u c h t h i c k e r i n the v e r t i c a l c a s e w h e r e b o u y a n c y d o e s not act to f o r c e the b u b b l e the top of the tube.  The  r e s u l t s f o r the h o r i z o n t a l a n d v e r t i c a l ,  tubes a r e i n e x c e l l e n t a g r e e m e n t f o r R e  >  against 5/16  8000 w h e r e t u r b u l e n c e i s w e l l  developed and where r e l a t i v e v e l o c i t y between bubble  a n d l i q u i d due to  b o u y a n c y i n the v e r t i c a l tube i s m u c h s m a l l e r t h a n the f l o w a n d t u r b u l e n c e velocities. lent r e n e w a l  T h i s a g r e e m e n t at h i g h R e  i s a s t r o n g i n d i c a t i o n that t u r b u -  i s the c o n t r o l l i n g m e c h a n i s m ,  a l t h o u g h e v e n i n the v e r t i c a l  TABLE  3. S U M M A R Y AND  RUN  Re  N  OF ABSORPTION  H O R I Z O N T A L 5/8  F(f)  versus  H o r i z Dntal,  N^ L  Intercept @  10  6  VERTICAL  TUBES  TUBE  s Slope  RESULTS FOR  3  /Q  L  %Y  / •V  p*  n) + 2<r M e(acnm / m i +2 o~ % o f Mean  %  %  Mean Pressure (mm  Hg)  5/8 T u b e  56  12  3140  0. 330  0. 231  17.  0. 62  18. 3  960  55  12  5090  .358  . 290  22.  1. 12  17. 7  1070  54  20  7630  . 324  .239  15, 6  1. 43  15. 7  0. 3  2320  57  16  11600  . 329  . 174  11. 3  1. 63  12. 9  .4  2320  58  10  16800  . 346  . 153  13. 7  2. 12  13. 3  1.2  Vertical,  2260  5/16 T u b e  51  8  3180  . 364  . 321  16. 7  2. 25  16. 9  10. 0  934  50  8  52 50  . 335  . 224  7.9  2. 37  14. 8  22. 0  948  49  18  8150  . 322  . 150  14. 4  2. 50  14. 7  9- 5  2300  48  12  11200  . 335  . 142  6.9  3. 09  10. 8  2270  46  15  15300  . 329  . 105  11, 1  3. 20  11.9  13. 0  2290  47  13  18300  . 327  . 102  11.8  3. 70  13. 1  15. 6  2200  Vertical,  5/8  7. 1  tube  38  4  1930  . 293  . 430  13. 2  2. 15  14. 5  14, 6  918  37  6  4070  . 321  . 323  12. 9  2. 13  12. 7  8. 8  1550  36  8  6620  . 282  . 194  20. 5  1. 86  25. 2  8. 3  2330  34  16  10200  . 336  . 161  14. 7  1. 92  12. 5  8. 4  2320  35  12  15300  . 329  . 126  11.7  2. 12  11.4  8. 8  2320  A P o v e r the length i n u s e f o r a given r u n  tube, the t h i n f i l m m a y b e i m p o r t a n t f o r t r a n s f e r  b e c a u s e the b u b b l e s  t r a v e l s o m e w h a t e c c e n t r i c a l l y as s h o w n i n F i g u r e 15. w i l l s u b s e q u e n t l y be c o n s i d e r e d f u r t h e r .  This possibility  D i s t o r t i o n of the bubbles i n F i g u r e  15 i s n o t s e v e r e w i t h the e x c e p t i o n o f t h e l a r g e b u b b l e at R e c o r r e l a t i o n has b e e n found between k  = 18300. • No  and bubble s i z e f o r any o f the  test sections.  For  are  b o t h v e r t i c a l t u b e s , the m a s s t r a n s f e r  c o e f f i c i e n t s at l o w R e  a s y m p t o t i c to a v a l u e o f a p p r o x i m a t e l y 2.2 c m / m i n .  s h o u l d be the m a s s t r a n s f e r  s  T h i s asymptote  coefficient for bubbles r i s i n g i n a c o l u m n of  s t a g n a n t l i q u i d , w h e r e the b a c k f l o w a r o u n d t h e b u b b l e s i s h i n d e r e d b y t h e w a l l e f f e c t i n the tube. N o m a s s t r a n s f e r r e s t r i c t e d case. similar,  H o w e v e r , i f the b u b b l e a n d tube d i a m e t e r s a r e n o t too  t h e n t h e r e s u l t f o r b u b b l e s r i s i n g i n an i n f i n i t e m e d i u m s h o u l d be  approximately applicable. Johnson  data has been found for this  F o r the u n r e s t r i c t e d  (30) h a v e o b t a i n e d m a s s t r a n s f e r  for CO^ bubbles  r i s e of bubbles,  c o e f f i c i e n t s c l o s e to 2. 0 c m / m i n  in water over a range of equivalent s p h e r i c a l  f r o m 0.4 to 0. 85 c m .  H e n c e the low Re  B o w m a n and  diameters  asymptote for both tubes i s i n good  a g r e e m e n t w i t h r e s u l t s f o r an i n f i n i t e s t a g n a n t medium.* T h e m a s s c o e f f i c i e n t s r i s e a b o v e t h i s a s y m p t o t e at a l o w e r v a l u e o f R e tube t h a n f o r t h e l a r g e tube, b e c a u s e  g  transfer  f o r the s m a l l  r e l a t i v e t e r m i n a l v e l o c i t i e s a r e o f the  s a m e o r d e r f o r t h e two t u b e s , b u t the f l o w v e l o c i t y i s h i g h e r f o r the s m a l l tube at a g i v e n R e . g  predominates  It i s s u p p o s e d that t h e f r e e l y r i s i n g b u b b l e  regime  u n l e s s t h e r e l a t i v e b u b b l e to l i q u i d v e l o c i t y i s s m a l l e r t h a n  80  F I G U R E 26.  B U B B L E S IN H O R I Z O N T A L F L O W IN 5/8  TUBE  81 the  turbulent velocities.  T h e c u r v e f o r the 5/8 v e r t i c a l tube i n F i g u r e 2 5 i s d r a w n a s y m p t o t i c to the h o r i z o n t a l , tubes.  5/8 tube c u r v e , i n a g r e e m e n t w i t h the c u r v e s f o r the  T h i s asymptote  i s l o w e r t h a n the 5/16  c u r v e b y a f a c t o r o f 0. 57.  a p o w e r l a w r e l a t e s the m a s s t r a n s f e r c o e f f i c i e n t a n d tube d i a m e t e r , these asymptotes  indicate k  T h e m a s s t r a n s f e r c o e f f i c i e n t s f o r the 5/8, b e l o w the p o w e r l a w c u r v e f o r R e  -0  (Tube diameter)  LJ  <T  5000.  5/16  "  85  If  then (41)  h o r i z o n t a l tube f a l l  Photographs  o f b u b b l e s i n 5/8  h o r i z o n t a l tube a r e s h o w n i n F i g u r e 26. T h e l a r g e s t b u b b l e s a r e g r e a t l y flattened by bouyancy  a n d the s h e a r f l o w at h i g h R e  ( F i g u r e 26 (a), (d) a n d s  (f)), b u t the b o t t o m o f the b u b b l e s , e x p o s e d to the t u r b u l e n t s t r e a m q u i t e s m o o t h e v e n at the h i g h e s t R e g  remains  T h e l i q u i d f i l m b e t w e e n tube a n d  b u b b l e i s r i p p l e d n e a r the r e a r o f the b u b b l e . F o r the s m a l l e r b u b b l e s , d i s t o r t i o n f r o m s p h e r i c a l shape is s m a l l . Evidence for Turbulent T r a n s f e r One  Mechanism  o f the m a i n o b j e c t i v e s o f the p r e s e n t w o r k was  to e s t a b l i s h  w h e t h e r the m a s s t r a n s f e r i s c o n t r o l l e d b y the t u r b u l e n t v e l o c i t y f i e l d r e s u l t i n g f r o m the l i q u i d flow, o r b y s o m e o t h e r v e l o c i t y f i e l d s u c h as e x i s t s i n the t h i n f i l m b e t w e e n b u b b l e a n d tube w a l l f o r h o r i z o n t a l f l o w . The  e x p e r i m e n t a l r e s u l t s p r e s e n t e d in: F i g u r e 25 a p p e a r to be  w i t h the t u r b u l e n t r e n e w a l m e c h a n i s m a l t h o u g h t h e r e i s no  consistent  satisfactory  t h e o r e t i c a l e x p l a n a t i o n of the R e y n o l d s n u m b e r e x p o n e n t o f a p p r o x i m a t e l y  82 1/2,  o r the tube d i a m e t e r d e p e n d e n c e .  H o w e v e r , t h e s e r e s u l t s do  c o n c l u s i v e l y e l i m i n a t e the p o s s i b i l i t y that the t h i n f i l m b e t w e e n the tube w a l l i s important.  not  bubble and  H e n c e t h i s f i l m r e g i o n w i l l be g i v e n f u r t h e r c o n -  sideration.  The  p l a n v i e w s of b u b b l e s in; F i g u r e s 24  and 26  s h o w that the t h i n  f i l m has  r i p p l e s n e a r the r e a r of the b u b b l e , e s p e c i a l l y f o r the l a r g e r  bubbles,  and h e n c e t h i s r e g i o n s h o u l d p r o v i d e e f f i c i e n t m a s s t r a n s f e r .  estimate  of the t r a n s f e r r a t e w o u l d be u s e f u l , but i s d i f f i c u l t to m a k e b e c a u s e  no i n f o r m a t i o n i s a v a i l a b l e on the f l o w p a t t e r n a n d  An  f i l m thickness. There is  a s t r i k i n g s i m i l a r i t y b e t w e e n the p l a n v i e w s i n p h o t o g r a p h s of l a r g e b u b b l e s i n h o r i z o n t a l flow, gas l i f t (8).  The  a n d the p h o t o g r a p h of a T a y l o r b u b b l e i n a n a r r o w ,  f l o w a n d s u r f a c e b e h a v i o u r i n the t h i n f i l m i n h o r i z o n t a l  b u b b l e f l o w a p p e a r s to be s i m i l a r to the T a y l o r b u b b l e i n a n a r r o w , gas l i f t ,  vertical  vertical  i n c l u d i n g s t r e t c h i n g of the s u r f a c e n e a r the f r o n t of the b u b b l e ,  c a p i l l a r y w a v e s n e a r the r e a r , a n d  e d d i e s i n the w a k e at the r e a r .  However  the l a c k o f a x i a l s y m m e t r y and the c o m p l e x s h a p e of the f i l m i n h o r i z o n t a l f l o w m a k e s c o m p a r i s o n of the t r a n s f e r r a t e s d o u b t f u l . F r o m the gas m a s s t r a n s f e r data of v a n H e u v e n and B e e k (8), :  t r a n s f e r c o e f f i c i e n t i s e s t i m a t e d to be water. The  m o s t i n h o r i z o n t a l flow. to be at l e a s t 10  Therefore  the s u r f a c e a v e r a g e m a s s  approximately.  t h i n f i l m c a n a c c o u n t f o r a b o u t 1/3  lift  1 c m / m i n for CO^  —  of the b u b b l e s u r f a c e at  the t r a n s f e r r a t e i n the f i l m w o u l d h a v e  t i m e s as h i g h as f o r the T a y l o r b u b b l e , to a c c o u n t f o r the  present experimental  t r a n s f e r r a t e s . H e n c e i t s e e m s u n l i k e l y that t h i s  film  c a n p r o v i d e the m a i n t r a n s f e r m e c h a n i s m , higher velocity greatly reduces f r o m f r o n t to r e a r of the  The  even i f s m a l l e r bubble size  and  the e f f e c t i v e s u r f a c e age of f l u i d f l o w i n g  bubble.  r e s u l t w i t h m o s t s i g n i f i c a n c e to t h i s q u e s t i o n of m e c h a n i s m  i s the e x c e l l e n t c o i n c i d e n c e o f the t r a n s f e r c o e f f i c i e n t s f o r the h o r i z o n t a l and v e r t i c a l 5/16  tubes  at h i g h R e  s h o w n i n F i g u r e 25.  If the t h i n f i l m  s b e t w e e n b u b b l e a n d tube w a l l was f o r the two  controlling mas  t r a n s f e r , t h e n the r e s u l t s  t u b e s s h o u l d be the s a m e o n l y i f the f i l m s h a p e a n d f l o w p a t t e r n  w e r e the s a m e . C o m p a r i s o n of F i g u r e s for example,  15(b) w i t h 24 (b) o r 15(c) w i t h 24 (c),  s h o w s t h a t the f i l m s a r e q u i t e d i f f e r e n t i n s p i t e of the e c c e n t r i c  p o s i t i o n p r e v i o u s l y n o t e d f o r v e r t i c a l flow. In c o n c l u s i o n , the p r e s e n t w o r k p r o v i d e s g o o d e v i d e n c e t h a t m a s s t r a n s f e r i n bubble flow i s p r e d o m i n a n t l y  due to s u r f a c e r e n e w a l b y l i q u i d  f l o w t u r b u l e n c e f o r a l l tubes s t u d i e d at h i g h R e y n o l d s n u m b e r . t h i n f i l m b e t w e e n b u b b l e a n d t u b e p r o b a b l y i s of s e c o n d a r y  F l o w i n the  importance for  t r a n s f e r i n h o r i z o n t a l b u b b l e flow, a l t h o u g h t h i s m e c h a n i s m m a y important  at l o w R e  .  F o r v e r t i c a l b u b b l e f l o w at l o w R e  s  be  most  the t r a n s f e r s  m e c h a n i s m i s s i m i l a r to the m e c h a n i s m of b u b b l e s  r i s i n g through  l i q u i d . F o r the t u r b u l e n t r e n e w a l t r a n s p o r t at h i g h Re  s  a stagnant  , the m a s s t r a n s f e r  c o e f f i c i e n t b a s e d on e q u i v a l e n t s p h e r i c a l s u r f a c e a r e a i s p r o p o r t i o n a l to Re  r a i s e d to a p o w e r of a p p r o x i m a t e l y  1/2  Increase in bubble surface  a r e a due to s m a l l s c a l e d i s t o r t i o n s p r o b a b l y c o n t r i b u t e s a s m a l l p a r t o f this R e y n o l d s n u m b e r  dependence.  84  TURBULENT RENEWAL  MODELS  T h e p r e c e d i n g r e s u l t s s u p p o r t the p o s t u l a t e d m e c h a n i s m of s u r f a c e r e n e w a l b y t u r b u l e n t e d d i e s w h i c h o r i g i n a t e i n the s h e a r f l o w of the l i q u i d . The  s a t i s f a c t o r y c o r r e l a t i o n of r e s u l t s with l i q u i d s u p e r i f i c i a l Reynolds  n u m b e r , i n s t e a d of s o m e m o r e c o m p l i c a t e d f u n c t i o n of v e l o c i t y s u c h s l i p v e l o c i t y o r s o m e bubble mechanism.  Reynolds number,  as  is c o n s i s t e n t with this  H o w e v e r i t w o u l d be v e r y u s e f u l to s h o w t h a t the e x p o n e n t of  a p p r o x i m a t e l y 0. 5 on the R e y n o l d s n u m b e r i s a l s o c o n s i s t e n t w i t h the turbulent renewal.  *  S o m e m o d e l i s r e q u i r e d w h i c h l i n k s m a s s t r a n s f e r at  the i n t e r f a c e to the e n e r g y a n d s i z e of the t u r b u l e n t m o t i o n s .  Unfortunately  no s a t i s f a c t o r y t h e o r y h a s b e e n f o u n d to a p p l y to t h i s p r o b l e m .  In the  I n t r o d u c t i o n i t was  n o t e d t h a t s o m e a s p e c t s of t h i s p r o b l e m h a v e r e c e n t l y  been studied.  s e m i - e m p i r i c a l m o d e l of C a l d e r b a n k  The  and' M o o  Young  (15) s e e m s m o s t p e r t i n e n t b e c a u s e i t u t i l i z e s the t u r b u l e n t e n e r g y t i o n , Q,  dissipa-  , w h i c h i s a f u n d a m e n t a l p a r a m e t e r o f the s t a t e o f t u r b u l e n c e .  A p p e n d i x V (a), S approximately  £  f o r p i p e f l o w of a s i n g l e p h a s e i s s h o w n to be  In  given  by  = . | G R e  Calderbank  2  -  7  5  ^  /  f  D  o b t a i n e d the r e s u l t k  (41) C  for transfer  from  l_i s o l i d s u r f a c e s , that i s , data f r o m  /<j_  s o l i d s o v e r a 10  Re  '^ .  - f o l d r a n g e of  T h i s c o r r e l a t i o n fitted t r a n s f e r The  e x p o n e n t of 0. 69 i s s i g n i f i c a n t l y h i g h e r t h a n r e s u l t s f o r the p r e s e n t w o r k .  85 It h a s b e e n p o i n t e d out i n t h e I n t r o d u c t i o n  t h a t c o m p a r i s o n of r e s u l t s f o r a  g a s / l i q u i d i n t e r f a c e w i t h C a l d e r b a n k ' s r e s u l t f o r s o l i d s i s of d o u b t f u l s i g n i f i c a n c e s i n c e he d i d n o t p o s t u l a t e a n y d e t a i l s o f the f l u i d m o t i o n w h i c h w o u l d p e r m i t a n e s t i m a t e of the e f f e c t of s u r f a c e type.  However, i f it i s  p o s t u l a t e d that s u r f a c e t e n s i o n h a s no e f f e c t o n t h e t r a n s f e r r a t e i n t h i s turbulent situation, a d i m e n s i o n a l r e s u l t is obtained for a gas/liquid surface also.  that  S o m e p r o p e r t i e s of a w e l l d e v e l o p e d t u r -  b u l e n t f i e l d a r e r e q u i r e d to o b t a i n t h i s d i m e n s i o n a l r e s u l t .  The  e n e r g y o f the t u r b u l e n c e  o r i g i n a t e s f r o m the m e a n flow  energy,  w h i c h i s c o n v e r t e d i n t o the l a r g e r s c a l e s o f t u r b u l e n t m o t i o n . T h e s e l a r g e s c a l e s i n t e r a c t and d e g e n e r a t e into s m a l l e r  and s m a l l e r  scales,  thereby  t r a n s f e r r i n g e n e r g y d o w n to s c a l e s o f m o t i o n w h i c h u l t i m a t e l y d i s s i p a t e the e n e r g y b y v i s c o u s  stresses.  F o r well developed turbulence  there is  a w i d e r a n g e o f s c a l e s w h i c h a r e l a r g e e n o u g h that i n e r t i a l f o r c e s dominate and negligible d i s s i p a t i o n o c c u r s .  A c r o s s these s c a l e s of  m o t i o n , t h e r e i s a f l o w of e n e r g y to the v i s c o u s p a t e d at a r a t e  £  m o t i o n to s m a l l e r  s c a l e s where it i s d i s s i -  , A n i m p o r t a n t f e a t u r e o f the c o n v e r s i o n s c a l e s i s the t e n d e n c y t o w a r d i s o t r o p y :  lose directional preference  of l a r g e scale the s m a l l  and are t e r m e d l o c a l l y 'isotropic.  p a r a m e t e r s that c a n i n f l u e n c e t h e s t r u c t u r e o f the t u r b u l e n c e  T h e only  H e n c e the  s t r u c t u r e of t h i s e q u i l i b r i u m r a n g e (32) i s a u n i v e r s a l f u n c t i o n o f for a l l turbulent fields where l o c a l isotropy occurs.  scales  in this range  a r e t h e k i n e m a t i c v i s c o s i t y a n d the r a t e o f e n e r g y d i s s i p a t i o n .  1/"  pre-  £  and  F o r the s u b -  r a n g e of s c a l e s of m o t i o n w h i c h a r e l o c a l l y i s o t r o p i c , but w h i c h a r e s t i l l  86 so l a r g e that v i s c o u s  dissipation is negligible, inertial forces predominate.  In t h i s i n e r t i a l s u b r a n g e , the s t r u c t u r e of the t u r b u l e n c e the s i n g l e p a r a m e t e r ,  £  .  range is d e t e r m i n e d by both  is determined  by  H o w e v e r , the extent of the i n e r t i a l s u b £  and  IT  •  If i t i s p o s t u l a t e d that m a s s t r a n s f e r r e s u l t s f r o m s c a l e s of m o t i o n s m a l l e r t h a n the b u b b l e and that t h e s e s c a l e s a r e l o c a l l y i s o t r o p i c , k  c a n d e p e n d on  £  , If  and  J©  i f s u r f a c e t e n s i o n and  d i a m e t e r a r e not i m p o r t a n t v a r i a b l e s .  Hence  . :  k^  OC  Therefore,  £  m i g h t a l s o a p p l y to b u b b l e s i f  w o r k , the d e p e n d e n c e on R e y n o l d s n u m b e r (that i s , on . 69  bubble  dimerisionally  l o c a l l y i s o t r o p i c s c a l e s m u c h s m a l l e r t h a n the b u b b l e e x i s t .  l o w e r than Re  then  & l/a  w h i c h c o r r e s p o n d s to C  c o u l d r e s u l t b e c a u s e p i p e f l o w at Re  •  In the  present  ) is significantly  T h i s l a c k of a g r e e m e n t  10 000 i s not w e l l e n o u g h d e v e l o p e d ;  to h a v e a l o c a l l y i s o t r o p i c r a n g e .  L e v i c h (33) a l s o c o n s i d e r e d suspended in a turbulent stream.  t r a n s f e r f r o m p a r t i c l e s and He  considered  bubbles  that a r e l a t i v e v e l o c i t y  b e t w e e n b u b b l e and f l u i d a r i s e s as a r e s u l t of the d e n s i t y d i f f e r e n c e b e t w e e n b u b b l e a n d f l u i d w h e r e the f l u i d i s a c c e l e r a t i n g due He  to t u r b u l e n c e  motions.  e s t i m a t e s t h i s r e l a t i v e v e l o c i t y and u s e s i t to e s t i m a t e m a s s t r a n s f e r  f r o m r e l a t i o n s t h a t a p p l y to s t e a d y f l o w p a s t a b u b b l e . T h i s t h e o r y k  _~ 0  0  XJ  Re'  • "75  for bubbles in a turbulent stream,  of C a l d e r b a n k ' s f o r s o l i d s , bubbles.  In L e v i c h ' s  predicts  a r e s u l t c l o s e to that  a n d to the a b o v e d i m e n s i o n a l r e s u l t f o r  theory,  i t i s m a d e c l e a r that the p o s t u l a t e d v e l o c i t y  87 f i e l d c l o s e to the b u b b l e s u r f a c e i s b a s i c a l l y the s a m e as i n s t e a d y f l o w p a s t a sphere, although t h i s relative v e l o c i t y a r i s e s f r o m turbulent motions. a l t e r n a t e v i e w i s p o s t u l a t e d i n the p r e s e n t w o r k ;  The  i . e. , that the f l u i d s u r r o u n -  d i n g the b u b b l e h a s o n the a v e r a g e a n e g l i b l e v e l o c i t y r e l a t i v e to t h e b u b b l e , but that e d d i e s m u c h s m a l l e r t h a n the b u b b l e r e p e a t e d l y b r i n g f r e s h f l u i d c l o s e to the i n t e r f a c e .  A r e a l i s t i c m o d e l o f e d d y m o t i o n s n e a r a gas / l i q u i d i n t e r f a c e h a s r e c e n t l y b e e n p r e s e n t e d b y D a v i e s (34), w h o s e m a i n o b j e c t i v e h a s b e e n to a s c e r t a i n the e f f e c t o f s u r f a c e c o n t a m i n a t i o n o n the h y d r o d y n a m i c s , by mass transfer,  f o r an e d d y o f a g i v e n c h a r a c t e r i s t i c v e l o c i t y .  and t h e r e -  However,  t h i s c h a r a c t e r i s t i c v e l o c i t y m u s t b e e m p i r i c a l l y r e l a t e d to the t u r b u l a n c e Reynolds number by mass transfer results, l i t t l e d i r e c t u s e f o r the p r e s e n t p u r p o s e .  and t h e r e f o r e the t h e o r y i s of  Furthermore,  Davies' model  not c o n s i d e r t h e r e l a t i v e e f f e c t i v e n e s s o f e d d i e s o f d i f f e r e n t s i z e s  does  which  m a y be p r e s e n t n e a r the i n t e r f a c e .  In the p r e s e n t w o r k , two a t t e m p t s a r e m a d e to t h e o r e t i c a l l y r e l a t e the p o s t u l a t e d e d d y  A. T U R B U L E N T As  r e n e w a l to the s u p e r f i c i a l l i q u i d R e y n o l d s  MIXING L E N G T H  d e s c r i b e d above,  number.  MODEL  t h e m e a n v e l o c i t y o f f l u i d c l o s e to t h e b u b b l e i s  s u p p o s e d to b e n e g l i g i b l e r e l a t i v e to the b u b b l e .  M a s s transfer is supposed  to o c c u r m a i n l y due to e d d i e s s m a l l e r t h a n the b u b b l e r e n e w i n g p o r t i o n s o f the b u b b l e s u r f a c e .  If a c h a r a c t e r i s t i c t i m e f o r the e d d y m o t i o n s c a n  be d e f i n e d , t h e n t h e m a s s t r a n s f e r s h o u l d b e r e l a t e d to t h i s t i m e b y  Danckwerts' surface renewal model (10).  In order to define such a  characteristic time it is necessary to suppose that the turbulent velocity field can be characterized by some mean velocity and length scale of motion. The root mean square (, r m s' ) fluctuating velocity length 2-  Uj  , and the mixing  are chosen to define a characteristic time,  I  U  (43)  The mean effective residence time of fluid at the interface should be proportional to  provided the eddy motions maintain kinematic similarity  near the interface as the mean flow Reynolds number changes. The mean renewal rate,  =  1  Therefore from Danckwerts (10),  l"tf  (44) F r o m elementary mixing length theory (35) for a shear flow,  MJ ^  *L u Oi t\)  i  (45)  For flow in pipes a general relation for the velocity gradient follows from the universal velocity profile,  ,46,  JLU, - 0.4-0(11P«fV*'/\ duh'  Therefore, at a fixed radial position, since Re c<r U . „, AV  2  ^  RjvVJ  1  Further, for smooth tubes, in the present range of Re, J approximately.  (47) r\&  With the relations (44) and (47), this results in  89  (48)  T h i s R e y n o l d s n u m b e r d e p e n d e n c e i s i n g o o d a g r e e m e n t w i t h the p r e s e n t experimental result,  ^Ly  ' *  If i t i s f u r t h e r a s s u m e d that the m e a n s u r f a c e age i s of the s a m e o r d e r "t> , t h e n an e s t i m a t e of the m a g n i t u d e of k  as the c h a r a c t e r i s t i c t i m e be m a d e f r o m t h i s t h e o r y .  C o n s i d e r Re  tube w a l l and c e n t r e l i n e i n the 5/16 A p p e n d i x V (d), a n d r e s u l t s i n k  =  101)00, a n d a p o i n t h a l f w a y b e t w e e n  - i n c h tube. ^  can  This estimate is made in  • 2 c m / m i n , w h i c h i s w e l l w i t h i n the  range of e x p e r i m e n t a l m a s s t r a n s f e r c o e f f i c i e n t s .  This encouraging  a g r e e m e n t w i t h the e x p e r i m e n t a l R e y n o l d s n u m b e r  d e p e n d e n c e m u s t be c o n s i d e r e d w i t h c a u t i o n b e c a u s e the m o d e l h a s of s h o r t c o m i n g s  that m a y  s e r i o u s l y affect its applicability.  The  a number  eddy s t r u c t u r e  n e a r the i n t e r f a c e m u s t be s o m e w h a t a l t e r e d ( f r o m the s i n g l e p h a s e f l o w s i t u a t i o n ) b y the c l o s e p r o x i m i t y of a b a r r i e r w h i c h u l t i m a t e l y e l i m i n a t e s one c o m p o n e n t of v e l o c i t y .  The  s t r u c t u r e cannot s t r i c t l y m a i n t a i n k i n e m a t i c  s i m i l a r i t y , b e c a u s e the r a n g e of e d d y s i z e s p r e s e n t i n c r e a s e s w i t h Re.  No  c o n s i d e r a t i o n i s g i v e n to the a n i s o t r o p y of s c a l e s o f m o t i o n as l a r g e as the m i x i n g l e n g t h . In the r e g i o n n e a r the b u b b l e s ,  w h i c h i s of i n t e r e s t h e r e ,  t y p i c a l p i p e s h e a r f l o w i s d i s t u r b e d , a n d so m a y  the  generate a quite d i f f e r e n t  e d d y s t r u c t u r e t h a n that c o n s i d e r e d a b o v e . F i n a l l y , the u s e of an e f f e c t i v e eddy l e n g t h w h i c h has p r o v e n u s e f u l f o r t u r b u l e n t d i f f u s i o n down a c o n c e n t r a t i o n g r a d i e n t i s not n e c e s s a r i l y a p p l i c a b l e i n the p r e s e n t c o n t e x t ,  where  90 it i s p o s s i b l e that the s m a l l e s t s c a l e of m o t i o n w h i c h r e a c h e s t h e i n t e r f a c e m a y h a v e a l a r g e i n f l u e n c e o n the m a s s t r a n s f e r as a r e s u l t o f the  importance  of m o l e c u l a r d i f f u s i o n i n the i n t e r f a c i a l zone. N e v e r t h e l e s s the a g r e e m e n t m a y h a v e s o m e s i g n i f i c a n c e , p a r t i c u l a r l y i f the l a r g e r s c a l e s o f m o t i o n are r e s p o n s i b l e for the m a s s t r a n s f e r .  This model also provides  a n e s t i m a t e o f the e f f e c t o f tube d i a m e t e r  on the m a s s t r a n s f e r c o e f f i c i e n t .  Equations  (44), (45) a n d (46) c a n b e v \/i  w r i t t e n i n the f o r m  J  \ OX Jf(l'  ''L w h e r e R e = D U ^ y / if.  Therefore  (49)  the m a s s t r a n s f e r c o e f f i c i e n t  should  be i n v e r s e l y p r o p o r t i o n a l to tube d i a m e t e r f o r a f i x e d R e a n d r a d i a l p o s i t i o n . Experimentally,  k  L  oC  D  85 '  o v e r the l i m i t e d r a n g e ( t w o - f o l d ) o f tube  d i a m e t e r that w a s i n v e s t i g a t e d .  B.  IDEALIZED EDDY C E L L  MODEL  Introduction It i s p r o p o s e d to c o n s i d e r i n d e t a i l the w a y a n i d e a l i z e d e d d y m o t i o n c l o s e to t h e s u r f a c e o p e r a t e s t o t r a n s p o r t s o l u t e o r h e a t f r o m the i n t o t h e b u l k o f the f l u i d .  T h e t e r m m a s s t r a n s f e r w i l l be used,  i s t h e m a i n c o n c e r n i n the p r e s e n t  work.  since this  H o w e v e r the d i s c u s s i o n  a p p l y as w e l l to h e a t t r a n s f e r , at l e a s t f o r s o l i d s u r f a c e s . m a s s t r a n s f e r were c o r r e l a t e d together  surface  should  Both heat and  f o r s o l i d s b y C a l d e r b a n k (15).  s o m e p h y s i c a l l y r e a l i s t i c m o t i o n s c a n be v i s u a l i z e d , t h e n i t m a y b e  If  possible  91 to q u a n t i t a t i v e l y d e s c r i b e how  the s i z e a n d e n e r g y of the m o t i o n s a f f e c t the  m a s s t r a n s f e r . T h i s m o d e l s h o u l d t h e n p r o v i d e a d i r e c t l i n k w i t h the h y d r o d y n a m i c s of the t u r b u l e n t f l u i d s i n c e the s i z e and e n e r g y of the s m a l l m o t i o n s a r e e n t i r e l y d e t e r m i n e d b y the e n e r g y d i s s i p a t i o n subrange exists.  The  parameter  £  £  p r o v i d e d an i n e r t i a l  c a n be r e a d i l y e s t i m a t e d o r m e a s u r e d ,  and i t s d e p e n d e n c e on R e y n o l d s n u m b e r o f p i p e f l o w i s k n o w n ( A p p e n d i x V (a)).  T h i s p o s s i b i l i t y of a l i n k to the h y d r o d y n a m i c s i s the m a i n i n c e n t i v e to d e v e l o p the p r e s e n t m o d e l .  M o d e l s of the D a n c k w e r t s type (10) a r e to s o m e  extent p h y s i c a l l y u n r e a l i n v i s u a l i z i n g a f l u i d e l e m e n t i n s t a n t a n e o u s l y a r r i v i n g at the s u r f a c e , r e s i d i n g t h e r e a . c e r t a i n , t i m e , replaced by f r e s h fluid.  Nevertheless  d e s c r i b e the e f f e c t of m o l e c u l a r  and t h e n b e i n g i n s t a n t a n e o u s l y  t h e s e m o d e l s , s e e m to s a t i s f a c t o r i l y  d i f f u s i o n at l e a s t f o r g a s / l i q u i d i n t e r f a c e s ,  a n d p e r h a p s f o r s o l i d / l i q u i d i n t e r f a c e s (12,  36).  H o w e v e r it i s v e r y difficult  to a p p l y t h e s e m o d e l s to p r e d i c t the e f f e c t of the R e y n o l d s ' n u m b e r on t u r bulent m a s s t r a n s f e r , p a r t i c u l a r l y i f s o l i d p a r t i c l e s , l i q u i d drops, b u b b l e s w i t h c l e a n and c o n t a m i n a t e d s u r f a c e s a r e a l l c o n s i d e r e d . o b j e c t i v e of the p r e s e n t m o d e l i s to c o n s i d e r how f o r the two The  e x t r e m e t y p e s of i n t e r f a c e ;  and A  second  the t r a n s f e r p r o c e s s  solid/liquid,  and c l e a n g a s / l i q u i d .  m o d e l p r o p o s e d h e r e m i g h t g i v e s o m e u s e f u l i n f o r m a t i o n about the  cess,  differs  pro-  e v e n w i t h a g r o s s o v e r s i m p l i f i c a t i o n of the r e a l t u r b u l e n t f l u i d ,  p r o v i d e d the m o s t e s s e n t i a l f e a t u r e s a r e r e t a i n e d .  In the s i t u a t i o n u n d e r c o n s i d e r a t i o n , the t u r b u l e n t v e l o c i t y f l u c t u a t i o n s p r e d o m i n a t e o v e r a n y r e l a t i v e v e l o c i t y of the a d j a c e n t f l u i d p a s t the b u b b l e  92 or particle.  T h e r e a r e motions of s i z e s c a l e s c o n s i d e r a b l y s m a l l e r than the  d i m e n s i o n of the p a r t i c l e ,  a n d f o r s u c h s c a l e s , the s u r f a c e a p p e a r s f l a t .  It  i s a l s o s u p p o s e d that a l l s c a l e s o f m o t i o n s m a l l e r t h a n t h e b u b b l e a r e i n the l o c a l l y isotropic e q u i l i b r i u m range p r e v i o u s l y discussed. is not f u l f i l l e d i n the p r e s e n t e x p e r i m e n t a l ideal case m a y  This latter condition  w o r k , b u t e x a m i n a t i o n o f the  give u s e f u l i n f o r m a t i o n for m o r e g e n e r a l turbulent r e n e w a l  situations.  The  Model The  f l u i d m o t i o n s w h i c h a r e e f f e c t i v e f o r m a s s t r a n s f e r a c r o s s the  i n t e r f a c e m u s t have the f o r m of u p w e l l i n g s  o r e d d i e s ( F i g u r e 27) i n w h i c h  f l u i d e l e m e n t s t r a v e l t o w a r d t h e s u r f a c e due t o t u r b u l e n c e f o r c e s pressure,  or viscous coupling)..  ( inertial,  T h e s e e l e m e n t s a r e d e f l e c t e d b y the  s u r f a c e a n d f l o w a l o n g the s u r f a c e a n d s u b s e q u e n t l y  p l u n g e b a c k i n t o the b o d y  of t h e f l u i d . T h i s m o t i o n b r i n g s f r e s h f l u i d v e r y c l o s e to the s u r f a c e s o that h e a t o r m a s s i s t r a n s f e r r e d to i t b y m o l e c u l a r  diffusion.  The fundamental  d i f f e r e n c e i n b e h a v i o u r b e t w e e n t h e two e x t r e m e s u r f a c e t y p e s ,  s o l i d and  free fluid (clean gas/liquid),  i s the f l o w p a t t e r n w i t h i n the u p w e l l i n g ,  c u l a r l y c l o s e to the s u r f a c e ,  due to t h e d i f f e r e n t b o u n d a r y c o n d i t i o n s ,  V*  0 and  Therefore  parti-  it m i g h t be  p o s s i b l e to p r e d i c t d i f f e r e n c e s i n m a s s t r a n s f e r b e h a v i o u r f o r the two s u r faces f r o m a g r e a t l y s i m p l i f i e d flow m o d e l . T h e l i q u i d / l i q u i d c a s e has s o m e behaviour between these extremes, the two l i q u i d p h a s e s .  d e p e n d i n g o n the r e l a t i v e v i s c o s i t i e s o f  93  GAS, LIQUID OR SOLID SURFACE  F I G U R E 27.  FLUID EDDY  D E F L E C T E D BY AN  INTERFACE  INTERFACE  A s e c o n d d i f f e r e n c e i n flow p a t t e r n i s p o s s i b l e b e c a u s e the r e s t r a i n t to v e l o c i t y  IL  at the g a s / l i q u i d i n t e r f a c e c a n o n l y b e p r o v i d e d b y s u r f a c e  t e n s i o n f o r c e s a f t e r the s u r f a c e h a s y i e l d e d to p r o v i d e  sufficient  T h i s l a t t e r effect should be s m a l l f o r a high s u r f a c e t e n s i o n water,  curvature.  s u c h as c l e a n  a n d f o r s m a l l s c a l e s o f m o t i o n b e c a u s e the c u r v a t u r e i s v e r y l a r g e  even f o r a s m a l l amplitude,  i f t h e s c a l e o f the d e f o r m a t i o n  is small.  t h e o r y of D a v i e s (34) s u r f a c e t e n s i o n a n d s u r f a c e d e f o r m a t i o n portant role,  In the  p l a y an i m -  as t h e y do i n L e v i c h ' s t h e o r y (37), w h i c h f o r m s p a r t o f t h e  basis for Davies' theory.  H o w e v e r Davies theory is c o n c e r n e d with  eddies  at the s u r f a c e o f a s t i r r e d v e s s e l , a n d s c a l e s of m o t i o n a r e o f the o r d e r of s e v e r a l m i l l i m e t r e s . F o r t h e v e r y m u c h s m a l l e r s c a l e s of the e q u i l i b r i u m range of turbulence,  the s u r f a c e s h o u l d b e v e r y n e a r l y f l a t .  Furthermore,  a c l e a n s u r f a c e s h o u l d p r o v i d e no r e s t r a i n i n g f o r c e a g a i n s t the s e p a r a t i o n o r  • 94 a p p r o a c h of s u r f a c e e l e m e n t s o f f l u i d .  H e n c e i t i s p o s t u l a t e d that  surface  t e n s i o n i s n o t a n i m p o r t a n t p a r a m e t e r i n the t r a n s f e r p r o c e s s u n d e r  consider-  ation.  If we s u p p o s e that a n e d d y as d e s c r i b e d a b o v e h a s s u p e r i m p o s e d o n it a s i m i l a r m o t i o n o n a m u c h s m a l l e r of t h i s s m a l l e r  s c a l e ( F i g u r e 28),  t h e n i n the v i c i n i t y  eddy, the m a s s t r a n s f e r s h o u l d be m a i n l y c o n t r o l l e d b y t h e  small scale motion if its energy is sufficiently large. This same argument s h o u l d a p p l y down to the s m a l l e s t s c a l e s that e x i s t . H e n c e i t a p p e a r s that the s m a l l e s t s c a l e s i n the f i e l d m a y c o n t r o l the o v e r a l l t r a n s f e r r a t e . If t h i s i s the c a s e t h e n a g r e a t l y s i m p l i f i e d a n a l y s i s of t h e p r o c e s s m a y be p o s s i b l e , for these s m a l l e s t motions a r e p r e d o m i n a n t l y v i s c o u s can  b e s u p e r i m p o s e d to c o m p l i c a t e  the m a s s t r a n s f e r p r o c e s s ,  t h e i r flow pattern.  a n d no s m a l l e r  T o apply this view of  the f o l l o w i n g s t e p s w i l l b e t a k e n :  SURFACE  F I G U R E 28.  A S M A L L EDDY SUPERIMPOSED ON A EDDY  eddies  LARGE  1)  R e p r e s e n t the a b o v e s m a l l e d d i e s b y g r e a t l y i d e a l i z e d v i s c o u s e d d y c e l l s f o r w h i c h the f l o w p a t t e r n s a n d m a s s t r a n s f e r c a n be  evaluated.  2) Diffusivity the l o c a l t r a n s f e r r a t e of s o l u t e a c r o s s the i n t e r f a c e f o r the s o l i d and f r e e f l u i d s u r f a c e s .  3)  ' F r o m the t h e o r y  of the e n e r g y of s m a l l s c a l e t u r b u l e n c e ,  and the m a s s t r a n s f e r - e n e r g y d e p e n d e n c e of the e d d y c e l l s , d e t e r m i n e i f the s m a l l , v i s c o u s s c a l e s of m o t i o n h a v e s u f f i c i e n t e n e r g y to c o n t r o l .  4)  D e p e n d i n g on the o u t c o m e of the above step,  d e t e r m i n e how  o v e r a l l m a s s t r a n s f e r d e p e n d s on the g o v e r n i n g s i z e and e n e r g y of m o t i o n ;  Idealized Viscous  that i s  and  the  parameters for 'V  Eddy Cells  A v e r y g r e a t l y i d e a l i z e d e d d y m o t i o n i s r e q u i r e d to m a k e the f l u i d f l o w and c o n v e c t i v e d i f f u s i o n e q u a t i o n s  t r a c t a b l e . The  m a i n f e a t u r e s of the  u p w e l l i n g f l o w d e s c r i b e d a b o v e m i g h t be w e l l r e p r e s e n t e d b y the t w o - d i m e n s i o n a l m o t i o n s h o w n i n F i g u r e 29, amplitude  A  w h e r e a s i n u s o i d a l s h e a r i n g m o t i o n of  e x i s t s at an a r b i t r a r y d i s t a n c e  of s y m m e t r y i t i s o n l y n e c e s s a r y  a, b e n e a t h the s u r f a c e .  Because  to c o n s i d e r the r e g i o n b e t w e e n a d j a c e n t  f l o w and d o w n f l o w s t a g n a t i o n p o i n t s .  up-  This distance is a r b i t r a r i l y supposed  96  F I G U R E 29.  to be a a l s o , wide,  I D E A L I Z E D VISCOUS E D D Y  giving a s q u a r e flow cell,  shallow cells or v e r y narrow,  CELL  s i n c e i t s e e m s u n l i k e l y that v e r y  deep c e l l s would o c c u r .  No  consider-  a t i o n i s g i v e n to the m e c h a n i s m o f d r i v i n g s u c h a m o t i o n f r o m the b o d y of the fluid.  T h e m o t i o n i n one e d d y c e l l m i g h t c l o s e l y r e p r e s e n t a r o t a t i n g e l e m e n t  of f l u i d n e a r the s u r f a c e as i n F i g u r e 30 (a).  A l t e r n a t e l y it might  represent  a j e t of f l u i d d i r e c t e d t o w a r d the i n t e r f a c e , a n d f o r c e d to f l o w b a c k i n t o the b o d y of the f l u i d b e c a u s e o f c o n t i n u i t y c o n s i d e r a t i o n s ,  as i n F i g u r e 30 (b).  F u r t h e r m o r e i t i s not s u p p o s e d that a l l of the e n e r g y of the t u r b u l e n c e i s i n the  f o r m o f t h e s e m o t i o n s that a r e u s e f u l f o r m a s s t r a n s f e r , b u t o n l y s o m e  s i g n i f i c a n t p o r t i o n o f the t o t a l e n e r g y .  97  (q) ROTATING ELEMENT  (b) JET  F I G U R E 30. P O S S I B L E E D D Y M O T I O N S S I M I L A R T O EDDY C E L L  The  THE  f l o w w i t h i n the c e l l i s t r e a t e d as v i s c o u s f o r e a s e o f s o l u t i o n  of the f l o w p r o b l e m ,  a n d i n k e e p i n g w i t h the p o s t u l a t e that the  (viscous) m o t i o n s a r e m o s t i m p o r t a n t f o r the m a s s t r a n s f e r .  smallest In o r d e r t o  f u r t h e r f a c i l i t a t e t h e s o l u t i o n o f the f l o w a n d d i f f u s i o n e q u a t i o n s f o r t h e e d d y c e l l , t h e m o t i o n i s t r e a t e d as t i m e w i s e s t e a d y .  This simplification is a  r e a s o n a b l e a p p r o x i m a t i o n i f the r e a l e d d y m o t i o n s e x i s t f o r l o n g e n o u g h t h a t t h e i r m a s s t r a n s f e r b e h a v i o u r a p p r o a c h e s t h e b e h a v i o u r of the i d e a l , cells. The  F o r the s o l i d s u r f a c e ,  the s u r f a c e b o u n d a r y c o n d i t i o n i s I T =  steady ii — © .  f r e e f l u i d s u r f a c e i s s u p p o s e d to r e m a i n f l a t a n d to e x e r t no f o r c e i n t h e  p l a n e o f the s u r f a c e ,  as d i s c u s s e d  above.  B e c a u s e o f the v e r y m u c h l o w e r  v i s c o s i t y o f g a s t h a n l i q u i d , no m o m e n t u m i s t r a n s f e r r e d a c r o s s face.  Therefore  t h e s u r f a c e b o u n d a r y c o n d i t i o n s a r e il - o a n d  for the f r e e s u r f a c e .  the i n t e r / ^ t y "" ^  In A p p e n d i x V I (a), t h e f l o w e q u a t i o n s a r e s o l v e d f o r  98 the two t y p e s o f s u r f a c e . T h e i n e r t i a l t e r m of t h e s e e q u a t i o n s i s n e g l e c t e d f o r v i s c o u s flow, a n d t h e v e l o c i t y c o m p o n e n t s a r e r e p r e s e n t e d b y a s t r e a m  function  ^IJT , H e n c e the N a v i e r S t o k e s e q u a t i o n s r e d u c e to the b i h a r m o n i c e q u a t i o n , and c l o s e d solutions a r e r e a d i l y obtained. F o r the s o l i d  surface,  f . *fl((' •  f) 4lkt (% fr)  ° «*« £  (50) F o r the f r e e f l u i d i n t e r f a c e ,  1f*»*[. M « The  I f ccsh (If)-,  m  j  c o n s t a n t s i n e q u a t i o n s (50) a n d (51) a r e a p p l i c a b l e o n l y f o r a s q u a r e  eddy  cell.  D i f f u s i o n into E d d y C e l l It i s n e c e s s a r y to a p p r o x i m a t e state c o n v e c t i v e d i f f u s i o n equation eddy c e l l s .  The  a s o l u t i o n to the t w o - d i m e n s i o n a l , £ x '  "™"  ^  S  ^  ^  e  steady idealized  a r e a v a i l a b l e f r o m the a b o v e s t r e a m f u n c t i o n s o l u t i o n s .  F o r thin concentration boundary l a y e r s (Pe = ^  i s large), a solution can  be a p p r o x i m a t e d b y a s s u m i n g that t h e c o n c e n t r a t i o n p r o f i l e i s a f i x e d f u n c t i o n of the b o u n d a r y l a y e r t h i c k n e s s , a n d n e g l e c t i n g m o l e c u l a r d i f f u s i o n i n t h e d i r e c t i o n p a r a l l e l to the s u r f a c e , transfer f r o m spheres.  as done b y B o w m a n et a l (38) f o r m a s s  The solution method is d e s c r i b e d in Appendix  VI (b)  F o r b o u n d a r y l a y e r t h i c k n e s s l e s s t h a n about o n e - t e n t h of the c e l l s i z e , s o l u t i o n f o r the t o t a l m a s s t r a n s f e r i s d i r e c t l y o b t a i n e d . T h e r e s u l t s a r e  t  a  99 e x p r e s s e d i n t e r m s of S h e r w o o d a n d P e c l e t n u m b e r s the s o l i d  For  surface,  0. 9/5 F o r the f r e e f l u i d  L  k  where k  d e f i n e d f o r the c e l l .  fi  '/3  (52)  surface,  O  CU  = local mass  transfer  'fe  (53)  c o e f f i c i e n t f o r the s u r f a c e  of the e d d y c e l l .  F o r v e r y l o w P e c l e t n u m b e r w h e r e the c o n c e n t r a t i o n b o u n d a r y l a y e r is deep,  an a p p r o x i m a t e m a s s  s o l u t i o n of f i n i t e d i f f e r e n c e A p p e n d i x V I (c).  Pe.  rate was c a l c u l a t e d u s i n g an  diffusion equations.  It w a s n o t  at t h e s a m e P e ,  transfer  The results  This method is d e s c r i b e d i n  p o s s i b l e to c o m p a r e  b e c a u s e the f i n i t e d i f f e r e n c e  r e s u l t s b y the two  equations  f o r d i f f u s i o n into the c e l l are  iterative  are unstable  techniques at h i g h  summarized in Figure  31.  FINITE DIFF. V FLUID 20 ESTIMATE o SOLID ^ io  FREE FLU CO  l<  F I G U R E 31.  SOLID SURFACE 100 Pe • oA/i) MASS TRANSFER INTO IDEALIZED EDDY  100 C E L L  100 T h e f o r m of t h e s e two c u r v e s i n d i c a t e s that the a b s o l u t e v a l u e of P e f o r the e d d y c e l l s s h o u l d h a v e an i m p o r t a n t e f f e c t o n w h e t h e r the two t y p e s of s u r f a c e w i l l h a v e s i m i l a r o r d i f f e r e n t m a s s t r a n s f e r b e h a v i o u r . . It i s e x p e c t e d that P e  ^  100,  at l e a s t f o r s o l u t e d i f f u s i n g t h r o u g h l i q u i d  \J9*S tO  cm  /sec),  and t h e r e f o r e i t i s s u p p o s e d that the s i m p l e p o w e r l o w p o r t i o n of the c u r v e s i s of i m p o r t a n c e . A n e s t i m a t e of the s i z e o f P e w i l l s u b s e q u e n t l y be m a d e .  E n e r g y of the M o t i o n s In o r d e r to a p p l y the r e s u l t s o f F i g u r e 31 to a t u r b u l e n t f l u i d c o m p o s e d of a r a n g e o f s c a l e s o f m o t i o n , the r e l a t i v e e n e r g y of d i f f e r e n t s i z e d m u s t be c o m p a r e d .  motions  T h e e n e r g y s p e c t r u m of the t u r b u l e n c e p r o v i d e s a m e a n s  of m a k i n g t h i s c o m p a r i s o n .  The  e n e r g y s p e c t r u m i s b a s e d on a F o u r i e r  d e c o m p o s i t i o n of the t u r b u l e n t v e l o c i t y f i e l d , i n w h i c h v e l o c i t i e s a r e r e p r e s e n t e d b y the s u p e r p o s i t i o n of s i n u s o i d a l s h e a r i n g m o t i o n s o f d i f f e r e n t w a v e l e n g t h s . T h e e n e r g y s p e c t r u m i n d i c a t e s how  the t u r b u l e n t  energy is  d i s t r i b u t e d o v e r the v a r i o u s w a v e n u m b e r s o f the F o u r i e r c o m p o n e n t s .  The  f u n c t i o n E (n) i s the k i n e t i c e n e r g y p e r u n i t m a s s o f f l u i d p e r u n i t i n c r e m e n t of w a v e n u m b e r .  C e r t a i n f e a t u r e s of the s p e c t r u m h a v e b e e n w e l l e s t a b l i s h e d  t h e o r e t i c a l l y a n d e x p e r i m e n t a l l y (32).  In the e q u i l i b r i u m r a n g e , the e n e r g y  s p e c t r u m i s a u n i v e r s a l f u n c t i o n o f Q j "\f occurs.  and  n  where local isotropy  In the i n e r t i a ! s u b r a n g e ,  (54)  V i s c o u s d i s s i p a t i o n of e n e r g y i s the m a i n f e a t u r e at the s m a l l e s t w a v e n u m b e r s a n d c a u s e s the s p e c t r u m to f a l l o f f m o r e s t e e p l y t h a n n'  '*  beyond  a value  101 of a p p r o x i m a t e l y 1 /3  where  a  T h e s i z e o f the i d e a l i z e d e d d y c e l l c o r r e s p o n d s t o a w a v e n u m b e r  In t h e c e l l shown, the m o t i o n c o r r e s p o n d s o n l y to w a v e n u m b e r s w h i c h l i e i n a p l a n e p a r a l l e l to the s u r f a c e , f r a c t i o n o f the t o t a l e n e r g y .  a n d s o r e p r e s e n t s o n l y an i n f i n i t e s i m a l  H o w e v e r i f an a n a l y s i s w e r e ma'de f o r s i m i l a r  m o t i o n s f o r a l l w a v e n u m b e r s w h i c h m a k e an a n g l e w i t h i n s a y 3 0 ° o f the s u r f a c e , i t s e e m s p r o b a b l e that the m a s s t r a n s f e r b e h a v i o u r w o u l d b e n o t s i g n i f i c a n t l y d i f f e r e n t f r o m the p a r a l l e l c a s e . T h i s r a n g e of w a v e n u m b e r d i r e c t i o n s w o u l d r e p r e s e n t about h a l f o f t h e t o t a l t u r b u l e n t e n e r g y . a m p l i t u d e o f the s h e a r i n g m o t i o n  The  i n the i d e a l i z e d c e l l s i s p r o p o r t i o n a l to t h e  s q u a r e r o o t of t h e e n e r g y o f m o t i o n s o f the s i z e r a n g e u n d e r  consideration.  If e d d y m o t i o n s a r e c o n s i d e r e d w h i c h h a v e w a v e n u m b e r s w i t h i n t h e range  A  %  ,  t h e n the e n e r g y i n t h e s e m o t i o n s i s E (n)  Afft  • To compare  the m a s s t r a n s f e r c o n t r i b u t i o n due to d i f f e r e n t s i z e d m o t i o n s , the w a v e i n c r e m e n t s m u s t be a c o n s t a n t f r a c t i o n of the w a v e n u m b e r ; w h e r e the i n c r e m e n t  /(Lit  w i t h i n the i n c r e m e n t A  frftoc *Y\.dy  that i s ,  i s c o n s t a n t a c r o s s the s p e c t r u m . i s p r o p o r t i o n a l to  fl> E (n).  number  The energy  T h e r e f o r e the  a m p l i t u d e A of t h e e d d y c e l l v e l o c i t y i s p r o p o r t i o n a l to lA'n  £  (Tt)  F o r the i n e r t i a l s u b r a n g e t h i s r e s u l t s i n  A °<- t' * ' k  %  (56)  F o r an e d d y m o t i o n o f s c a l e a, L e v i c h (39) i n d i c a t e s that t h e c h a r a c t e r i s t i c velocity  i s o f the s a m e o r d e r as ( c  ^  a  )  >  a  r e s u l t e q u i v a l e n t to e q u a t i o n  102 (56), but w h i c h s h o w s that the c o n s t a n t  of p r o p o r t i o n a l i t y i s of the o r d e r  of  unity.  A p p r o x i m a t e absolute values  of the e d d y c e l l P e c l e t n u m b e r c a n  e s t i m a t e d f r o m the a b o v e result..  In the p r e s e n t  p i p e flow, the  be  inertial  s u b r a n g e i s not d e v e l o p e d , but the s i z e a n d v e l o c i t y of the s m a l l s c a l e m o t i o n s s h o u l d be o f the s a m e o r d e r  as g i v e n b y e q u a t i o n s (55) and (56).  V (b) a n d (e) i t i s e s t i m a t e d that at R e  (.a  ^.0  3 cm)  has  an a m p l i t u d e A  b y e q u a t i o n (53), k  2  ^  In A p p e n d i x  = 10,000, a s c a l e of w a v e n u m b e r 1/3 ' 10 c m / s e c ,  and  Pe ^  20000, w h i c h  c m / m i n f o r the f r e e f l u i d s u r f a c e .  i n the s a m e r a n g e as the e x p e r i m e n t a l v a l u e s ,  n a  gives  This result is  w h i c h i n d i c a t e s that m o t i o n s of  t h i s s i z e c o u l d m a k e a s i g n i f i c a n t c o n t r i b u t i o n to m a s s t r a n s f e r i f t h e y r e a c h e d the i n t e r f a c e . The  a b o v e d i s c u s s i o n of the e n e r g y  b u l k f l u i d f a r f r o m any  surface,  w h e r e a s the p r e s e n t  i m m e d i a t e l y a d j a c e n t to the s u r f a c e . m o t i o n of s i z e  a  of t u r b u l e n t m o t i o n s a p p l i e s to the r e g i o n of i n t e r e s t i s  H o w e v e r i t s e e m s r e a s o n a b l e that a  w i l l not be m u c h a f f e c t e d b y a s u r f a c e u n l e s s  approximately a distance a  f r o m the s u r f a c e .  it is within  F u r t h e r m o r e , t h e way  the m o t i o n  w i l l be a f f e c t e d b y the s u r f a c e o v e r the d i s t a n c e a m u s t be m o r e o r l e s s as d e s c r i b e d b y the e d d y c e l l m o d e l .  H e n c e i t i s s u p p o s e d that the b u l k t u r b u l e n t  e n e r g y s p e c t r u m c a n be a p p l i e d to the e d d y c e l l m o d e l a d j a c e n t to the  The  i n i t i a l p o s t u l a t e that the m a s s t r a n s f e r i s c o n t r o l l e d b y  s c a l e s of m o t i o n c a n now  be t e s t e d .  The  surface.  the.smallest  local mass transfer coefficient k  w i l l be e s t i m a t e d as a f u n c t i o n of c e l l , s i z e o v e r the i n e r t i a l s u b r a n g e to  see  103 i f w a v e n u m b e r s c l o s e to 1/3  nj  predominate.  The  viscous treatment is  hot a p p l i c a b l e f o r t h e s e i n e r t i a l m o t i o n s , - b u t i f t h e i r m a s s t r a n s f e r b e h a v i o u r i s not too d i f f e r e n t f r o m v i s c o u s to i n d i c a t e w h e t h e r the s m a l l e r and (56) and the d e f i n i t i o n of Pe,  F o r the f r e e f l u i d s u r f a c e ,  These expressions smaller  eddies,  t h e n the a n a l y s i s s h o u l d be  adequate  scales are c o n t r o l l i n g . F r o m equations f o r the s o l i d  surface  a p p l y i n g e q u a t i o n (53),  s h o w that i n the p r e s e n t m o d e l of t u r b u l e n t t r a n s f e r ,  s c a l e s ( h i g h e r n) a r e m o s t e f f e c t i v e . H o w e v e r the e x p o n e n t s on  a r e s o l o w i n e q u a t i o n s (57) and (58) that the i n e r t i a l s u b r a n g e c a n not ignored  (52)  as was  initially postulated.  In f a c t the i n e r t i a l m o t i o n s a r e  to t h i n the c o n c e n t r a t i o n b o u n d a r y l a y e r m o r e t h a n i n the v i s c o u s t h e r e f o r e the l a r g e r s c a l e s m a y and (58) i n d i c a t e .  The  f o r e , i s that m a s s t r a n s f e r i s not due but i s due  n be  expected  case,  be of m o r e i m p o r t a n c e t h a n e q u a t i o n s  o n l y c o n c l u s i o n p o s s i b l e f r o m the p r e s e n t  the  and (57)  model  there-  to a n y n a r r o w r a n g e of s c a l e s of m o t i o n ,  to s c a l e s w h i c h e x t e n d f r o m the s m a l l e s t v i s c o u s m o t i o n s w e l l  into the i n e r t i a l m o t i o n s , w i t h o n l y a m i n o r p r e f e r e n c e  O v e r a l l T r a n s f e r Rate,  f o r the s m a l l e r  scales.  k  J_i To  d e t e r m i n e the o v e r a l l t r a n s f e r c o e f f i c i e n t , i t i s n e c e s s a r y to  up the c o n t r i b u t i o n s due  to s c a l e s of a l l s i z e s . It h a s  m o t i o n s of e a c h s c a l e b e h a v e i n d e p e n d e n t l y . a s s u m p t i o n i s e x p e c t e d to be  As  add  b e e n a s s u m e d that  previously discussed  s a t i s f a c t o r y f o r the s m a l l , v i s c o u s  this  scales.  104  H o w e v e r , i t h a s b e e n f o u n d that the l a r g e r , i n e r t i a l s c a l e s a r e o f s o m e i m p o r t a n c e to t r a n s f e r . T h e r e f o r e  the a s s u m p t i o n of i n d e p e n d e n c e i s o f d o u b t -  ful v a l i d i t y f o r the l a r g e s c a l e s . F u r t h e r m o r e , the v i s c o u s m o d e l i s not a p p l i c a b l e to t h e i n e r t i a l m o t i o n s . In s p i t e o f t h e s e f a i l u r e s o f t h e m o d e l , the i n t e g r a t i o n w i l l b e c a r r i e d out, of s m a l l , haviour  viscous motions.  motions.  p r o p o r t i o n a l i t i e s i n e q u a t i o n s (57) a n d (58) h o l d i f i n c r e m e n t s  A?? Qt- Tl Jl)  of  It i s a l s o p o s s i b l e that t h e m a s s t r a n s f e r b e -  o f the i n e r t i a l m o t i o n s i s n o t t o o d i f f e r e n t f r o m t h e v i s c o u s  The  n,  s i n c e a l a r g e p o r t i o n of the r a n g e c o n s i s t s  a  r  e  considered.  In o r d e r to i n t e g r a t e w i t h r e s p e c t to t h e v a r i a b l e  e q u a t i o n s (57) a n d (58) m u s t b e c o n v e r t e d to the b a s i s o f e q u a l i n c r e m e n t s  <L 7t\ , T h a t i s , k  (n) = k  /n e x p r e s s e s t h e c o n t r i b u t i o n to m a s s t r a n s f e r  Li JL/ coefficient p e r unit wave number.  A n u m b e r of functions have b e e n p r o p o s e d f o r the energy, s p e c t r u m i n the d i s s i p a t i o n r a n g e . T h e K o v a s z n a y s p e c t r u m (32) w i l l be u s e d f o r the present purpose, ,  ,  /  ^  -5/3  fl  0,6 a /  _  ") ^  +l\ "j ^ \  (59) T h i s f u n c t i o n i s i d e n t i c a l w i t h e q u a t i o n (54) i n t h e i n e r t i a l s u b r a n g e , a n d d r o p s t o z e r o at a v a l u e n r e c a l l i n g that  P  °**  VtL  Q  = 1.47 £  (in)  the f o l l o w i n g r e l a t i o n i s o b t a i n e d  U s i n g e q u a t i o n s (52) a n d (59), a n d and  ft/  f o r the s o l i d  - "^£J " ~^~"~gf ) surface:  (60)  105 The upper limit of integration is n  fi  ,  In establishing the lower limit,  it is supposed that all scales smaller than the dimension of the particle or bubble are involved. This dimension corresponds to a wave number  ng  The overall mass transfer coefficient for the solid surface is therefore m/  (6i)  The integrand in equation (61) can be expanded by the binomial theorem and integrated. If at least 3 octaves of scales exist in the inertial subrange below the particle or bubble dimension, then  n' ^ B  JL . n. , 20  °  an d '  the lower limit can be omitted from the integrated equation with only a small error.  In Appendix V (f) it is shown that equation (61) can be integrated and  simplified to (62) Similarly for the fluid surface,  k *Je*J&\'*ll-6%"  y 4n, (63)  L  which results in  K  «-(nf/£>)'  %  (Of)  <h (64)  The results of the viscous eddy cell model, equations (62) and (64) are consistent with the dimensional requirement previously derived. this sense, no new information has been obtained from the model.  In  However,  an estimate of the constants of proportionality in equations (62) and (64) also  106 r e s u l t s f r o m the m o d e l ,  s i n c e the c o n s t a n t s i n a l l the r e l a t i o n s i n v o l v e d a r e  of the o r d e r o f u n i t y . C a l d e r b a n k  1  s empirical result k  =  0. 2  S c ^ fe^f^ ^  f o r s o l i d s i n a t u r b u l e n t f l u i d i s at l e a s t w i t h i n an o r d e r of m a g n i t u d e o f the p r e s e n t t h e o r e t i c a l r e s u l t .  T h e eddy c e l l m o d e l also p r e d i c t s f r o m  first  p r i n c i p l e s t h e s a m e S c h m i d t n u m b e r d e p e n d e n c e as f o u n d b y C a l d e r b a n k f o r the s o l i d i n t e r f a c e , a n d as f o u n d b y H a y d u c k f o r t h e gas / l i q u i d i n t e r f a c e . F r o m e q u a t i o n s (62) a n d (64), t h e m a s s t r a n s f e r r a t e s f o r s o l i d p a r t i c l e s and u n c o n t a m i n a t e d bubbles only.  s h o u l d d i f f e r as a r e s u l t o f t h e S c h m i d t n u m b e r  O f m o r e interest perhaps,  i s t h e r e s u l t that b u b b l e s  i n a well  t u r b u l e n t f i e l d s h o u l d have the s a m e m a s s t r a n s f e r rate (aside f r o m f a c t o r of o r d e r u n i t y ) w h e t h e r c l e a n o r c o n t a m i n a t e d ,  developed a constant  b e c a u s e Sc i s u n c h a n g e d  b y the p r e s e n c e  of contamination.  types of bubbles  w o u l d i n d i c a t e that d a m p i n g o f the e d d i e s b y t h e s o l i d o r  contaminated  A l a r g e d i f f e r e n c e i n r a t e s f o r t h e s e two  s u r f a c e was an i m p o r t a n t f a c t o r .  T h e p r e s e n t e x p e r i m e n t a l r e s u l t that k l o w e r t h a n the d e p e n d e n c e o n £ ' ~  Li  ( i . e. R e  ^  Re'  is significantly  S  ) p r e d i c t e d b y e q u a t i o n (64).  T h i s l a c k of a g r e e m e n t d o e s not n e c e s s a r i l y c o n t r a d i c t e q u a t i o n (64),  since  the e x p e r i m e n t a l f l o w s d i d n o t h a v e a w e l l d e v e l o p e d i n e r t i a l s u b r a n g e as the eddy c e l l m o d e l r e q u i r e s .  C.  CONCLUSIONS T h e t u r b u l e n t m i x i n g l e n g t h m o d e l p r o v i d e s b e t t e r a g r e e m e n t w i t h the  p r e s e n t e x p e r i m e n t a l r e s u l t s than the v i s c o u s eddy c e l l m o d e l . T h e e q u i l i brium  r a n g e o f t u r b u l e n t m o t i o n s was not d e v e l o p e d i n t h e p r e s e n t e x p e r i m e n -  107 t a l c o n d i t i o n s . H e n c e i t i s r e a s o n a b l e that t h e m i x i n g l e n g t h a n d v e l o c i t y , w h i c h a r e m a i n l y d e t e r m i n e d b y the l a r g e s c a l e s o f m o t i o n , be  appropriate  The  should  to d e t e r m i n e the r e n e w a l r a t e .  idealized, viscous  eddy c e l l m o d e l m a y b e applicable for bubbles  o r p a r t i c l e s s u s p e n d e d i n a f l u i d w h i c h i s s u f f i c i e n t l y t u r b u l e n t so that a s i g n i f i c a n t e q u i l i b r i u m r a n g e of s c a l e s s m a l l e r t h a n the b u b b l e e x i s t s . m a s s t r a n s f e r c o r r e l a t i o n s p r e d i c t e d b y the e d d y c e l l m o d e l f o r s o l i d  The surfaces  are i n r e a s o n a b l e a g r e e m e n t with the e x p e r i m e n t a l r e s u l t s c o r r e l a t e d b y Calderbank.  H e n c e the m o d e l i s o f s o m e i n t e r e s t i n that i t i n t r o d u c e s  a clearly  d e f i n e d f l o w a n d d i f f u s i o n p r o c e s s as t h e b a s i s o f the t u r b u l e n t m a s s t r a n s f e r . It i s n o t p o s s i b l e to m a k e a n y f u r t h e r c o n c l u s i o n s  about the g e n e r a l  validity  of t h i s m o d e l .  A b a s i c d i f f e r e n c e b e t w e e n the a b o v e m o d e l s s h o u l d b e r e c o g n i z e d .  The  m i x i n g length m o d e l i s b a s e d on the l a r g e r s c a l e s , w h e r e a s the i d e a l i z e d e d d y c e l l m o d e l i s b a s e d o n the s m a l l e r  s c a l e s of m o t i o n that o c c u r .  Calder-  bank's c o r r e l a t i o n i s a l s o b a s e d o n l a r g e s c a l e s (but i n i n e r t i a l s u b r a n g e ) , a l t h o u g h h i s c o r r e l a t i o n h a s i d e n t i c a l f o r m to e q u a t i o n (62).  His correlation  i s b a s e d on t h e s t r u c t u r e f u n c t i o n  •(Ul(s)-  tt/(X+V)*  °°  G  ^  B  w h i c h i s d o m i n a t e d b y s c a l e s c l o s e to the s i z e d  *  (65)  T h i s question of the  B. s i z e o f m o t i o n s that a r e i m p o r t a n t , i s a p o s s i b l e a r e a f o r f u r t h e r C a r e f u l m e a s u r e m e n t s o f the v e l o c i t y f i e l d c l o s e t o f l u i d a n d s o l i d  research. surfaces  that a r e m o v i n g w i t h t h e m e a n f l u i d v e l o c i t i e s s h o u l d be the n e x t s t e p i n  108 studying this question. of t u r b u l e n c e , fer.  S u c h m e a s u r e m e n t s w o u l d f i l l a gap i n the k n o w l e d g e  and s h o u l d be of v a l u e to o t h e r f i e l d s i n a d d i t i o n to m a s s t r a n s -  A l t e r n a t e l y , s o m e o p t i c a l m e a n s m i g h t be d e v i s e d to s t u d y the s t r u c t u r e  of f l u c t u a t i o n s i n m a s s t r a n s f e r r a t e on a t u r b u l e n t s u r f a c e u n d e r g o i n g m a s s transfer.  F o r e x a m p l e , the t o t a l s o l u t e  i n a l a y e r b e n e a t h the s u r f a c e i s a  m e a s u r e of c o n c e n t r a t i o n b o u n d a r y l a y e r t h i c k n e s s , and t h e r e f o r e t r a n s f e r r a t e m i g h t be e s t i m a t e d b y the a b s o r p t i o n of l i g h t b y s o m e c o m p o n e n t i n solution.  F u r t h e r w o r k m i g h t a l s o be done w i t h the p r e s e n t m o d e l to c o n s i d e r the e f f e c t s of s u r f a c e d i s t o r t i o n , the e d d i e s .  i n e r t i a l f o r c e s , and the u n s t e a d y n a t u r e  of  E x p e r i m e n t s s i m i l a r to the p r e s e n t work, u s i n g l a r g e s o l i d s p h e r e s  o r c o n t a m i n a t e d b u b b l e s m i g h t g i v e m o r e i n f o r m a t i o n about the v a l i d i t y of the model.  It i s i m p o s s i b l e to d i s t i n g u i s h b e t w e e n the v a r i o u s m o d e l s  presented  i n t h i s w o r k on the b a s i s of the m a g n i t u d e of the p r e d i c t e d t r a n s f e r r a t e s . The  estimates  model,  of t r a n s f e r c o e f f i c i e n t for the m i x i n g l e n g t h and eddy c e l l  and f o r the t h i n f i l m r e g i o n b e t w e e n b u b b l e and tube w e r e a l l of the  o r d e r of 1 to 2. c m / m i n , c l o s e to the p r e s e n t e x p e r i m e n t a l t r a n s f e r c o e f f i c i e n t f o r CO,, mately 2 cm/min.  The  result.  The  bubbles r i s i n g i n stagnant water is a l s o a p p r o x i -  i m p l i c a t i o n of t h e s e v a l u e s i s that the e x a c t  of the h y d r o d y n a m i c s i s of o n l y m i n o r i m p o r t a n c e i n d e t e r m i n i n g f e r r a t e s i n t w o - p h a s e flow,  mass  nature  mass trans-  and that f o r d e s i g n p u r p o s e s , the p r e d i c t i o n of  i n t e r f a c i a l a r e a m i g h t be the m o s t f r u i t f u l a r e a of r e s e a r c h .  Further  research  into d e t a i l e d m e c h a n i s m s  c a n o n l y be j u s t i f i e d i f i t l e a d s to f u n d a m e n t a l  u n d e r s t a n d i n g t h a t i s of v a l u e i n s o m e o t h e r c o n t e x t s u c h as, f o r e x a m p l e selectivity i n competing gas/liquid reactions.  110  NOMENCLATURE  N. B. T y p i c a l d i m e n s i o n s a r e s h o w n w e r e  applicable  A  v e l o c i t y amplitude in i d e a l i z e d eddy cell,  a  width of i d e a l i z e d eddy c e l l ,  C  dimensionless concentration;  0^  factor representing  C  c o r r e c t i o n factor for bubble v e l o c i t y  v  D  tube d i a m e t e r ,  cm/sec  cm e. g. , q / q *  a correction for variable  pressure  cm 2  diffusivity of C O ^ i n water, c m /sec bubble diameter, c m turbulent e n e r g y s p e c t r u m (3-dimensional) function,  E(n)  cm  /sec  F(f)  Absorption  function  F  s o l u t e c r o s s i n g a p l a n e n o r m a l to s u r f a c e  f  d i m e n s i o n l e s s concentration, q/q  in boundary layer  t  f  f r i c t i o n factor for p r e s s u r e drop in pipe flow  H  Henry's Law  K, (  k  K^, K  3  dimensionless  constant constants  liquid phase controlled m a s s transfer coefficient, cm/min; k  , coefficient corrected L  coefficient corrected k  T  L  for bubble velocity; k  , i-tlr  V  for variable  pressure  m a s s t r a n s f e r c o e f f i c i e n t for an i d e a l i z e d eddy c e l l ;  k (n), L  c o n t r i b u t i o n to o v e r a l l c o e f f i c i e n t , o n the b a s i s of u n i t w a v e number.  Ill  length of test section, ultimate  o r p o s i t i o n along absorption  final concentration  i f a l l f e e d gas w e r e  tube, ft  dissolved  bubble frequency, m i n n u m b e r of t r a n s f e r units wavenumber,  cm  ^; n „ w a v e n u m b e r o f s c a l e t h e s i z e o f  B the bubble', n ,, w a v e n u m b e r d n  o  , wavenumber  range of m a i n dissipation,  at w h i c h E ( n ) = o i n K o v a s z n a y  p r e s s u r e at gas f e e d n o z z l e ,  mm  m e a n p r e s s u r e i n test section,  Hg  mm  Hg  p r e s s u r e at a p a r t i c u l a r l o c a t i o n i n t e s t s e c t i o n , p r e s s u r e gradient  i n test section,  mm  concentration  temp  and p r e s s u r e y c m  tube r a d i u s ,  a.A/J@  /mm  o f d i s s o l v e d gas, ( c m /  Po+pL  Hg/ft  P e c l e t number f o r idealize d eddy cell, v o l u m e t r i c flow rate, c m  spectrum  3  gas at t e s t s e c t i o n  3 liquid  cm  d i m e n s i o n l e s s positioni i n eddy c e l l , y / a r a d i a l distance  f r o m tube c e n t r e l i n e ,  cm  Reynolds n u m b e r f o r single p h a s e pipe flow s u p e r f i c i a l liquid pipe flow Reynolds surface  number  a r e a of a s p h e r i c a l bubble of equivalent  surface renewal rate, s e c Schmidt number, ^ ^  3 volume,  cm  112  Sh  Sherwood number, k  ^  (length)  jj&  c h a r a c t e r i s t i c t i m e of t u r b u l e n t e d d i e s  U  t i m e m e a n p i p e f l o w v e l o c i t y at a g i v e n r a d i a l p o s i t i o n , pi-P  e  flow v e l o c i t y averaged  over cross  section  v e l o c i t y n o r m a l to s u r f a c e i n e d d y c e l l , IP  rms  V  v e l o c i t y of b u b b l e ,  V^,  m e a n v e l o c i t y of t o t a l v o l u m e t r i c flow,  iP*  v e l o c i t y p a r a l l e l to s u r f a c e i n e d d y c e l l  w  , w,, w/ i l  o  , etc. , coefficients f o r v e l o c i t y ratio  Z  m i x i n g length,  A  polynomials  Re  b u b b l e to m e a n f l o w v e l o c i t y r a t i o  '  ft/sec  cm  u n k n o w n e x p o n e n t on  1  cm/sec  ft/sec  coordinates i n eddy c e l l  ft- f  cm/sec  fluctuating velocity i n turbulent field,  x, y  ^  /V^  u n k n o w n e x p o n e n t s o n Sc  ^  t h i c k n e s s of c o n c e n t r a t i o n b o u n d a r y l a y e r ,  £  rate of e n e r g y d i s s i p a t i o n by t u r b u l e n c e p e r unit m a s s , 2 cm  cm  3 /sec  d i m e n s i o n l e s s p o s i t i o n i n the b o u n d a r y l a y e r , yl>l  cm/sec  liquid viscosity,  y/ $  gm/cm-sec 2  jj  liquid kinematic viscosity,  cm  /sec  ^  d i m e n s i o n l e s s p o s i t i o n i n eddy c e l l , x / a  ^  l i q u i d density,  gm/cm  d i m e n s i o n l e s s t h i c k n e s s of c o n c e n t r a t i o n b o u n d a r y l a y e r ^/a; also s t a n d a r d deviation i n s t a t i s t i c a l context  113  r  dimenionsless  s t r e a m function f o r eddy c e l l ^  s t r e a m function for eddy cell,  j/^/aA  2 c m /sec  S u p e r s c r i p t *; p r o p e r t y e v a l u a t e d at the i n t e r f a c e Subscripts G  gas  L  liquid  B  bubble  o  at the g a s f e e d l o c a t i o n ( e x c e p t f o r n^)  i,  j : inlet and outlet s a m p l e tap l o c a t i o n code n u m b e r s ;  tensor subscripts m  m e a n value  Equalities =  equals approximately  equals  of the s a m e o r d e r o f m a g n i t u d e as cO  p r o p o r t i o n a l to  also general  114  REFERENCES  1.  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J . , C h e m . E n g . S c i , , 18, 377 (1963)  C. J . , H e u s s , J . M. , a n d W i l k e ,  C, R. , A . I. C h . E . J . , 11, 866  ( 1965) 10.  D a n c k w e r t s , P. V. , Ind. E n g . C h e m . , 43, 1460 (1951)  11.  Toor,  12.  H a r r i o t , P., C h e m . E n g . S c i . , 17, 149 (1962)  13.  Perlmutter,  14.  King,  15.  Calderbank,  16.  H a r r i o t , P. , A . I. C h . E . J . , _8, 93 (1962)  17. 18.  *  H. L . , M a r c h e l l o ,  D. D. , C h e m . E n g . S c i . , 16, 287 (1961)  C J . , I. E . C  Ruckenstein, Middleman,  J . M. , A . I. Ch. E . J . , 4, 97 (1958)  P.H.,  Fund.,  5_, 1 (1966)  M o o - Y o u n g , M. B. , C h e m . E n g . S c i . , 16, 39 (1961)  E . , C h e m . Eng.Sci.-,  2j_, 113 (1965)  S. , A . I. C h . E . J . , 11_, 750 (1965)  115 19-  Chan, W i n g - C h e n g ,  20.  Sherwood,  T. K. , P i g f o r d ,  McGraw-Hill, 21.  Davies,  Ph.D. T h e s i s , U n i v e r s i t y of Minnesota,  1965.  R. L . , p. 72, A b s o r p t i o n a n d E x t r a c t i o n ,  N. Y. , 1952.  J . T. ,  A d v . i n C h e m . E n g . , 4, p. 1, A c a d e m i c P r e s s , N.Y-.,  1963. 22.  Hamilton, p. 169,  L . F . , S i m p s o n , S. G. , Q u a n t i t a t i v e C h e m i c a l  10th E d . , M a c M i l l a n ,  Analysis,,  N. Y. , 1952.  23.  S i m p s o n , S., I. E . C.,l_6, 709 (1924)  24.  H u g h e s , R.R.,  G i l l i l a n d , E.R.»  Ch. E n g . P r o g . S y m p . S e r i e s ,  5L  101 (1955) 25.  Prandtl,  L . , F l u i d D y n a m i c s , p.153,  B l a c k i e and Son Ltd. ,  L o n d o n , 1952. 26.  Ellis,  H.S., Can, J . C h e m . E n g . , 42_, 69 (1964)  27.  T o r o b i n , L . B. , G a u v i n ,  28.  P e r r y , J , H. , ed. , C h e m i c a l McGraw-Hill,  W. H. , C a n . J . C h e m . E n g . _38, 189 (1960) E n g i n e e r s ' H a n d b o o k , III, p. 674,  N.Y., 1950.  29.  V o l k , W. , A p p l i e d S t a t i s t i c s f o r E n g i n e e r s ,  30.  Bowman,  31.  Bennett^, C. A . , F r a n k l i n ,  CW.,  McGraw-Hill,  N.Y. , 1958  J o h n s o n , A . I . C a n . J. C h e m . Eng., 40, 139 (1962) N. L . , S t a t i s t i c a l A n a l y s i s i n t h e  C h e m i c a l I n d u s t r y , p. 227, J . W i l e y , N.Y. , 1954. 32.  Hinze,  J . Q. , T u r b u l e n c e ,  33.  L e v i c h , W. G. , P h y s i c o c h e m i c a l H y d r o d y n a m i c s , p. 176 a n d p. 470, Prentice Hall,  p. 184,. M c G r a w - H i l l ,  Englewood Cliffs,  N. Y. , 1959-  N.J. , 1962.  34.  Davies,  J . T. , P r o c . R o y . S o c . A, 290, 515 (1966)  35.  Sherwood,  T. K. P i g f o r d ,  36.  Haurathy,  T, J . ,  37.  L e v i c h , W. G. , Op. C i t . , p. 689-  38.  B o w m a n , C W. , et. a l . , C a n . J . C h e m . E n g . , 39, 9 (1961)  R. L . , Op. C i t . , p. 37.  A . I. C h . E . J . , 2, 359 (1956)  116  39-  L e v i c h , W. G.  40.  Irving,  J . ,Mullineaux,, N. , M a t h e m a t i c s i n P h y s i c s a n d  e e r i n g , p. 40, 41.  Op. C i t . , p. 24.  Davidson,  Engin-  A c a d e m i c P r e s s , N. Y. , 1959•  J . F. , Cullen,. E . J . , T r a n s . F a r a d a y S o c . , 53,  51 (1957)  117  APPENDIX I  EQUIPMENT AND INSTRUMENT Heater  Circulation  SPECIFICATIONS  Pump  E a s t e r n Industries Company,  Model E - l ,  T y p e 100, c e n t r i f u g a l ,  c a p a c i t y a p p r o x i m a t e l y : 5 g p m at h e a d of 5 p s i .  Heater  C o n t r o l Switch F e n w a l M o d e l 17150  Water Jet Ejector S c h u t t e a n d K o e r t i n g , M o d e l 0, c a p a c i t y a p p r o x i m a t e l y 1. 5 s c f h at 26 i n H g v a c u u m w i t h w a t e r p r e s s u r e o f 60 p s i g .  Storage and Stripper  Tanks  G l a s s - l i n e d , m i l d s t e e l d o m e s t i c hot w a t e r tanks, c a p a c i t y a p p r o x i m a t e l y 30 g a l  Feed  Pump Pumps and Power (Vancouver)  mechanical  P a r a m o u n t T u r b i n e P u m p , M o d e l 5241,  seals suitable for vacuum,  c a p a c i t y 5 g p m at a t o t a l h e a d o f  50 p s i , m o t o r 1 hp, 1750 r p m  C0  2  Flow Regulator Moore  P r o d u c t s Co. , c o n s t a n t - d i f f e r e n t i a l f l o w r e g u l a t o r ,  Model  63BU-L  118  Digital Interval T i m e r T r a n s i s t o r S p e c i a l t i e s Inc.,  Plainview,  N. Y. , A p t i m e t e r  Model  361.  Tensiometer C e n t r a l S c i e n t i f i c Co. , D u N o u y r i n g i n t e r f a c i a l t e n s i o n b a l a n c e , 10403, C e n c o b u l l e t i n  Liquid Feed  101.  Nozzles  D e t a i l s of the 5/8 f e e d n o z z l e was  approximately  - i n c h f e e d n o z z l e a r e s h o w n i n F i g u r e 1-1.  u s e d f o r b o t h h o r i z o n t a l and v e r t i c a l r u n s .  of r u n s w i t h the 5/16,  h o r i z o n t a l t e s t s e c t i o n , a n o z z l e was  g e o m e t r i c a l l y s i m i l a r to the 5/8  b u b b l e n o z z l e to f i r s t s a m p l e tap was h i g h e s t J^g  Model  ( R u n s 39 a n d 40),  nozzle. The  approximately  This  F o r the m a j o r i t y used which  was  distance f r o m  0. 17 f t . F o r r u n s at  a l a r g e r d i a m e t e r was  r e q u i r e d i n the b u b b l e  • <S forming and 0.42  zone.  H e n c e a n o z z l e s i m i l a r to F i g u r e I-1,  ft b e t w e e n b u b b l e n o z z l e  5 /16 f e e d n o z z l e was  «  with a 12° total angle  and f i r s t s a m p l e tap, was  u s e d f o r a l l 5 /16 v e r t i c a l  runs.  used. T h i s  second  F I G U R E 1-1.  FEED NOZZLE BLOCK  FOR  5/8  TUBES  120 NOZZLt  SS.AL  F I G U R E  1 - 2 .  F I G U R E  1-3.  BLOCK  F E E D  LIQ.VIO  N O Z Z L E  5/8 - I N C H  FttD  PIPE  FITTINGS  T U B I N G  C O N N E C T O R S  COMPONENTS C, C  2  TEST  + 9'  SECTION-  VARIABLE, APPROX. 0.25/xf "  "  a  TRANSISTOR  b  PHOTO DIODE  c  DIODE  0.01  "  100 kXl  2 N404 IN2175  IN34A  :600Xl RECORDER  • / lE £ 00l  800 kQ, 800k& —vw—w-  100/i/if  A  W  ^niiokft^s  c  w -  40/xf; •TALLY  3.3k||33kil  |l50k&  |lOk&  |?.2 Meg&  -6'  FREQUENCY METER CIRCUIT FIGURE 1-4.  BINARY SWITCHES 8r AMPLIFIER  MONOSTABLE CIRCUIT  SIGNAL AMPLIFIER  DETECTOR  COUNTER CIRCUIT DETAILS ro  100  500kif2  a - TRANSISTOR  2N404  R,-  b - TRANSISTOR  2NI377  C,- STEP VARIABLE  c-DIODE  IN34A  ZERO -  0.25/JLf 0,50^L£f I.O/Xf 2.0/If  F I G U R E 1-5.  TIME  DELAY  CIRCUIT  5.0/JLf  ro  123  A P P E N D I X II  BUBBLE VELOCITY  DATA  V e l o c i t y d a t a w e r e o b t a i n e d b y the two m e t h o d s "Apparatus".  described in  F o r the c o m p u t e r outputs u s e d i n T a b l e s II-1 (c) a n d (d),  the c o l u m n h e a d s h a v e the f o l l o w i n g  meaning:  RE  • =  Re s  NB  =  N_  DB  =  d = equivalent s p h e r i c a l bubble diameter,  DRAT  =  r a t i o o f b u b b l e to tube d i a m e t e r =  QG/Q  =  gas v o l u m e t r i c flow r a t i o = Q _ / ( Q _ + Q_ )  DELT  =  bubble spacing, c m  V=  =  m e a n v e l o c i t y o f t o t a l v o l u m e t r i c flow,  =  bubble velocity ratio  B  , m i n ~*  vj  V^  BETA = For  =• V _  B  cm  d/D  Lr  JU  /N_  B ft/sec  T a b l e s II-1 (a) a n d (b) c o l u m n h e a d s a r e s t a n d a r d n o m e n c l a t u r e .  the h e a d i n g " M e t h o d " : T d e n o t e s t r a c k i n g bubbles;  TABLE  S denotes v e l o c i t y of a single bubble.  II-1. E X P E R I M E N T A L V E L O C I T Y  (a) V e r t i c a l , 5/8 Re  4070  DATA  Tube d(cm)  s  1930  w i r e m e t h o d for a s t r e a m of  Method  2. 50  0.89  2. 59  .70  S  2,96  .42  S  . .90  S  1.72  '  ' S  1.86  .70  S  1.87  .42  S  Under  Re  d(cm)  s  6620  10200  15400  17800  (b) H o r i z o n t a l , 1690  I960  2200  2600  3170  Method  1.46  .91  S  1.57  .70  S  1. 51  .42  S  1.35  .91  S  1.42  .70  S~  1.30  .42  S  1.30  .96  S  1. 33  .73  S  1.26  .42  S  1.28  .96  S  1.31  .73  S  0.519  S  5/16 T u b e 0.988 .92  .. 372  .84  .291  S  S  1.002  .519  S  .930  .372  S  .807  .291  S  .993  .372  S  .88  .291  S  1.022  .519  S  1.000  .372  S  .955  .291  S  1.03  .519  S  1. 00  . 372  •'• S  • 952  .291  S  4240  1.036  .519  S  3840  1.005  .329  T  .947  .261  T  • 996  .415  T  1.037  .505  T  1.022  .519  S  1.00,  .327  S  • 957  .291  S  Re  d(cm)  s  Method  5880  1. 05 1. 016 . 963  . 519 . 372 . 291  S S S  7020  0. 980 . 980 1, 06 1.01 . 952  0. 285 . 338 . 519 . 372 • 291  T T S S S  10200  1. 017 1. 022 1. 084 1, 058 1. 02 • 994 1. 07 1.05  333 413 538 519 372 291 49 39  T T T S S s s s  13200  1. 083 1. 045 1. 002  519 372 291  S S S  15300  1. 11 1. 07 1. 032 1, 11 1. 10  519 372 291 49 39  S S S S S  18300  1. 14 1. 12  48 39  S S  22400  1.15 1. 14  48 39  S S  26300  1. 19 1. 17  47 39  S s  126  (c) H o r i z o n t a l , 5/8 Tube VELOCITY BY TRACKING WIRE TUBE HORIZONTAL, 5/8 1.0. RE 1879. 1879. 1879. 2942. 2942. 2942. 2942. 2942. 2942. 2942. 2942. 2942. 5085. , 5065. 5085. 5085. 5085. 5085. 5085.  5085. 5085. 7625. 7625. 7625. 7625. 12668. 12668. 12668. 12668. 16271. 16271. 16271. 16271. 16271.  NB  Ob  94. 28. 150. 98. 182. 162. 164. 51. 184. 57. 156. 92. 172. 308. 304. 31. 230. 170. ?CC. 2 78. 236. 172. 316. 264. 452. 49. 466. 280. 640. 130. 596. 19C. 230. 750.  0.918 0.879 C.755 C.92C 0.963 C.771 1.23d 1.C91 1.263 0.914 1.025 0.495 0.952 1.C22 0.776 1.098 1.297 1.269 0.749 1.C34 C.509 1.260 C.779 0.9 V2 0.533 1.C91 0.815 1.038 C.561 1.15? 0.919 0.738 1.011 0.563  VELOCITY OF SINGLE BUBBLE, RE 1879. 1879. 2942. 2942. 2942. 5085. 5085. 5085. 5085. 5085. 5085. 7625. 7625. 7625. 7625. 12668. 12668. 12668. 12668. 16271. 16271. 16271. 16271.  NB  OR AT  QG/Q  OELT  0.578 0.553 0.475 0. 579 0.606 0.485 0 . 778 0.686 0.794 0.575 0.645 0. 311 0. 599 0.643 0.488 0.691 0.816 0. 790 0.471 0.650 0. 320 0. 793 0.490 0.624 0. 335 0.686 0. 513 0.653 0.353 0. 724 0.57 8 0.464 0.636 0.354  0.026 0.007 0.023 0.018 0.037 0.017 0.068 0.015 0.080 0.010 0.0*8 0.00 3  7.60 23. 06 4.42 12.06 6. 75 6.95 8.30 24.49 7.44 19. 69 8.08 10. 59 11. 44 6. 74 6.22 65. 04 9. 72 12 .80 9.45 7.49 7.38 18.56 8.70 11. 16 5.89 101. 44 10. 52 17. 48 7.24 51. 64 10. 67 33. 16 27. 76 8.24  0.02O  0.043 0.019 0.006 0.064 0.045 o.oti 0.040 0.004 0.030 0.013 0.023 0.006 0.003 0.014 0.017 0.0U6 0.008 0.019 0.003 0.010 0.006  V 0.401 0.393 0.399 0.622 0.634 0.622 0.656 0.620 0.664 0.617 0.635 0.612 1.077 1.103 1.076 I .062 1.128 1. 106 1.068 I.100 1.060 1.633 1.605 1 .620 1.593 2.640 2.667 2.675 2.647 3.407 3.445 3.389 3.413 3.398  BETA 0.975 0.898 0.90b 1.039 1.058 0.991 1. 135 1.100 1.126 0.994 1.084 0.869 0.993 1.028 0.960 1.038 1.083 1.075 0.967 1.034 0.897 1.068 0.936 0 . 994 0.913 1.029 1.005 1.000 0.957 1.077 1.008 1.016 1.022 0.994  TUBE HORIZONTAL, 5/8 I .0.  DB  DRAT  0.875 C.6B8 0.875 C.6bl U075 0.866 0.681 0.87C 1.083 0.681 0.567 o.e70 1.058 C.681 0.556 C.848 1.036 0.670 C.556 0.e48 1 .C36 C.681 C.546  0.550 0.433 0.550 0.428 0.676 0.54 5 C.428 0. 54 7 0.681 0.428 0 . 356 0. 547 0.665 0.428 0. 3 50 0.533 0.651 0 . 421 0. 350 0.533 0.65 I 0.428 0. 343  OG/0  OELT  V 0.393 0.393 0.614 0.614 0.614 1 .062 1.062 1.062 1.062 1.062 1.062 1.593 1.593 1.593 1.593 2.646 2.646 2.646 2.646 3.398 3.398 3.398 3.398  BETA 0.943 0.896 1.044 0 . 954 1.145 0.988 0.908 0.987 1.038 0.92 9 0.877 0.925 1.007 0.914 0.907 0.976 0.993 0.979 0.928 1.009 1.015 1.009 0.955  (d) V e r t i c a l ,  5/16  Tube  VELOCITY BY TRACKING MIRE. RE NB OB 1428. 1428. 1428. 1428. 1428. 1428. 1428. 1428. 1428. 1428. 1428. 1945. 1945. 1945. 1945. 1945. 1945. 1945. 1945. 1945. 1945. 1945. 3157. 3157. 3157. 3157. 1157. 3157. 3157. 3157. 3157. 3157. 3157. 3157. 3157. 5234. 5234. 5234. 5234. 5234. 5234. 5234. 5234. 5234. 5234. 5234. S234. 52*4. 5234. 5234. 5234. 8151. 8151. 8151. 8151. 8151. 8151. 8151. 8151. 8151. 8151. 8151.  368. 592. 124. 712. 336. 122. 800. 494. 306. 375. 570. 832. 476. 264. 78. 732. 318. 105. 764. 360. 410. 570. 600. 380. 188. 72. 680. 288. 112. 980. 448. 200. 680. 375. 570. 692. 612. 876. 324. 118. 908. 468. 180. 1216. 240. 908. 375. 820. 570. 985. 1150. 450. 150. 1209. 1160. 596. 164. 908. 815. 1220. 405. 990.  0.584 0.619 0.615 0.502 0.475 0.449 0.469 0.424 0.412 0.387 0.274 0.612 0.S56 0.533 0.533 0.503 0.467 0.443 0.440 0.403 0.376 0.274 0.579 0.547 0.526 0.527 0.519 0.483 0.463 0.453 0.409 0.409 0.386 0.367 0.283 0.587 0.612 0.614 0.521 0.525 0.533 0.492 0.470 0.495 0.431 0.439 0.363 0.360 0.271 0.279 0.265 0.568 0.527 0.589 0.598 0.507 0.467 0.434 0.376 0.405 0.356 0.277  VELOCITY OF SINGLE BUBBLE. RE NB OB 11224. 11224. 11224. 11224. 15305. 15305. 15305. 18306. 18306. 18306.  0.4?2 0.344 0.384 0.419 0.494 0.344 0.429 0.489 C.344 0.437  TUBE VERTICAL . 5/16 1.0. ORAT OG/0 DELT 0.735 0.778 0.774 0.631 0.597 0.564 0.590 0.534 0.519 0.487 0.344 0.770 0.699 0.671 0.671 0.633 0.587 0.558 0.553 0.506 0.473 0.344 0.728 0.688 0.662 0.663 0.653 0.607 0.582 0.570 6.514 0.515 0.486 0.462 0.356 0.739 0.770 0.773 0.655 0.661 0.670 0.619 0.591 . 0.623 0.542 0.552 0.457 0.478 0.340 0.351 0.333 0.714 0.663 0.741 0.752 0.638 0.587 0.546 0.474 0.510 0.448 0.348  0.066 0.119 0.027 0.080 0.034 0.011 0.074 0.035 0.020 0.021 0.011 0.119 0.055 0.028 0.008 0.062 0.022 0.006 0.044 0.016 0.015 0.008 0.048 0.026 0.012 0.005 0.040 0.014 0.005 0.038 0.013 0.006 0.017 0.008 0.006 0.036 0.036 0.051 0.012 0.004 0.035 0.015 0.005 0.038 0.005 0.020 0.005 0.012 0.003 0.006 0.006 0.014 0.004 0.040 0.040 0.013 0.003 0.012 0.007 0.014 0.003 0.004  4.76 2.91 13.71 2.83 6.22 17.91 2.69 O.Sl 7.29 5.89 3.82 2.70 4.91 9.03 32.45 3.38 8.07 25.57 3.47 7.36 6.26 3.88 5.50 8.87 18.11 47.62 5.12 12.53 32.23 3.71 8.07 17.63 5.12 8.96 5.21 7.39 8.25 5.75 15.83 44.11 5.75 11.47 30.12 4.47 21.73 6.03 13.32 6.20 7.58 4.39 3.76 16.71 50.41 6.30 6.52 12.79 47.09 8.65 8.95 6.20 17.65 6.67  TUBE VERTICAL. 5/16 1.0. ORAT OG/0 DELT 0.594 0.432 0.483 0.527 0.621 0.432 0.540 0.615 0.432 0.549  V 0.639 0.677 0.613 0.648 0.617 0.603 0.644 0.618 0.609 0.609 0.603 0.922 0.859 0.835 0.819 0.866 0.831 0.817 0.849 0.825 0.824 0.819 1.385 1.354 1.334 1.324 1.373 1.337 1.325 1.371 1.336 1.326 1.341 1.329 1.326 2.266 2.266 2.302 2.212 2.195 2.264 2.217 2.196 2.270 2.196 2.229 2.196 2.211 2.192 2.197 2.197 3.451 3.416 3.546 3.546 3.448 3.413 3.446 3.428 3.450 3.414 3.415  V 4.676 4.676 4.676 4.676 6.376 6.376 6.376 7.627 7.627 7.627  BETA 1.499 1.392 1.515 1.699 1.850 1.981 1.826 1.969 2.002 1.983 1.972 1.330 1.486 1.560 1.689 1.564 1.688 1.796 1.704 1.753 1.700 1.475 1.303 1.361 1.395 1.415 1.386 1.475 1.489 1.450 1.478 1.453 1.419 1.382 1.224 1.233 1.218 1.196 1.268 1.296 1.261 1.323 1.349 1.310 1.297 1.341 1.244 1.257 1.077 1.074 1.074 1.191 1.210 1.174 1.165 1.208 1.237 1.245 1.162 1.198 1.144 1.057  BETA 1.178 1.135 1.163 1.193 1.180 1.180 1.190 1.186 1.196 1.198  T A B L E II-2. V E L O C I T Y C O N S T A N T S F O R A L L T E S T  Re  w  s  Tube  o  w  1  W  SECTIONS  2  V e r t i c a l , 5 /8 I n c h I. D.  15300 10200 6620  0. 962  4070 1930  1. 477 4. 045  0. 662 0. 977  Tube Horizontal,  0.. 984 2.,170 1.. 901 1.,514 -3..342  -0. 659 -1. 554 -1. 505 -1. 383 1. 805  0., 528  -0. 395 -0. 544  5/16 I n c h I. D.  22400 18300  1. 001 0. 931  15300  0. 732  14200  0. 757  0.689 1., 464 1., 246  13200  0. 784  1.. 010  -0. 818  10200 8760  0..350 0., 361  0. 000 0. 026  7510 7020  0. 0. 0. 0.  5880  889 873 847 852  -1. 432 -1. 141  0.430  -0. 074 -0. 050  0. 845  0.438  -0. 048  4810  0.840  0., 472  ~0, 124  3517  0.778  0.,832  -0. 648  3170  0. 720 0. 562  2350  0. 313  1., 129 1.,920 3., 076  -1. 023  2604 I960  0. 034  3.. 798  -3. 359 -3. 737  1810  0.019  3.. 825  -3. 746  0., 134 o.. 036  -0.. 049 0. 069  0.. 103 0,.313  0. 064 -0. 030  0.433  -0. 046  -9. 211 -5. 786 -4. 143  Tube Horizontal,  0.937  11600  0. 895 0.821  5085 3140 Tube  -2. 014  5/8 I n c h I. D.  16800 7625  0.455  0. 729 0. 675 V e r t i c a l , 5/16 Inch, I. D.  3140  -0.361  8.. 196  5250  0. 043  8150  0. 261  5,, 429 3.. 991  11200  0. 606  2.486  15300  0. 866  1,.590  -2. 677 -1. 948  18300  0. 988  0,. 980  -1, 161  129  A P P E N D I X III CALCULATIONS  (a) C A L C U L A T I O N S A N D D A T A F O R A B S O R P T I O N  RUNS  A t y p i c a l calculation for an absorption m e a s u r e m e n t w i l l be w o r k e d through in detail,  s t a r t i n g f r o m the r a w d a t a s h o w n i n F i g u r e III- 1,  is a t y p i c a l a b s o r p t i o n r u n data sheet.  which  I n c l u d e d w i t h the c a l c u l a t i o n s a r e  the c o r r e s p o n d i n g a r i t h m e t i c s t a t e m e n t s  f o r u s e i n the F o r t r a n I V p r o g r a m .  A c o m p l e t e l i s t of w o r d d e f i n i t i o n s f o r t h i s p r o g r a m p r e c e d e s  the p r o g r a m  listing.j E x a m p l e : R u n 28,  D a t a P o i n t 28106, 5/16 T u b e  See F i g u r e III- 1 f o r r a w e x p e r i m e n t a l Water Flow .  .  data  - S m a l l r o t a m e t e r @ 0. 60 g p m o n r o t .  scale  M a s s f l o w r a t e = 4.90 l b / m i n f r o m w e i g h t c a l i b r a t i o n 5  01 c p = 0.0101  Re  4 ( M a s s flow)  TT  s  D  4  cm  sec  ^4.90 /minW454 lb  g  m  /  "7T (.794 c m ) (.0101 g m / c m sec)(60 s e c / m i n ) = 1200 (4.90) = 5880  RE  = 1200.  * QLW  Superficial water velocity  4.90  (30.5)  62.4  (.495) (60)  2  0. 500 (4. 90) = 2. 45 f t / s e c VEL  = 0. 500 * Q L W  M  m  10612  i  *  \VJ'!S}  S  ^  «  ? 7 z  -  / j r 3»  TS; (  SE s ^ 4 L  :i<  J f  %  9  l  \ 1 1  C, U_ C c  CV  li / z  _  C  x  -  l Ifif IC J  C Cf F  r t  • •  £  .61 Z,SS 4. GJ t>%1*  /  *  io  • sr  .T  7.90  to c>  n i*  s 5  FIGURE III-1  Pj  &  ivj)  2 S  ZoO'C,  ts*c\  A £ ^w)  3Z.S  /o3D  SO  • < I * <h.o? J O J I . 71  IS  S O  I ,1 H .lo  j . ss 4-. oo 4 . HO 10.ot /0 ,10  -  . 2 0  /,?-T  ia s ss s.  6JUL  * Z.t f -3 /  f , jf6>  S.LZ  J j )  ///•  T%iJZZ c o '  S~,99  S  ^  7-i'.V  77/r* ;  . A "  • C ,  S 7  (  ^7~T7  *  SO  / .00 • 20 ,  t 10..6%  S. oo £. 'JTX  T Y P I C A L D A T A S H E E T FOR ABSORPTION RUN: RUN 28  i  95 ~±  3 ^ 7  131  Test Section P r e s s u r e P r e s s u r e i s m e a s u r e d with a m a n o m e t e r with w a t e r - f i l l e d l i n e s . Therefore  1 cm  difference in Hg  ^13. 6 x 10 = 9. 27 m m Pressure  h e i g h t i s e q u i v a l e n t to (13. 6 - 1)  Hg  r e l a t i v e to a t m o s p h e r i c  i s m e a s u r e d a g a i n s t an o p e n  h e a d tank at tube l e v e l ( F i g u r e 4 (a) ) . '.  Manometer reads  - F o r high pressures,  directly a gauge was  u s e d . G a u g e was  to r e a d d i r e c t l y i n p s i r e l a t i v e to a t m o s p h e r i c  calibrated  at the tube l e v e l ,  allowing for w a t e r - f i l l e d lines. - F o r v e r t i c a l t e s t s e c t i o n s , s t a t i c h e a d b e t w e e n h e a d tank l o c a t i o n and p o s i t i o n i n tube was  allowed for  I F o r Run  28,  P^  = p r e s s u r e at 2 - f t tap. - 159 m m  Hg  above  atmospheric  = 754 + 159 = 913 m m The .  .  l a s t s a m p l e t a p u s e d f o r t h i s r u n i s at 6 f t . M e a n p r e s s u r e i s c a l c u l a t e d at 3-ft p o s i t i o n : P  E q u i l i b r i u m CO  =  p 2  '  ~[z  ( P  2  " 14  *  P  =  9  1  "  3  3  3  /  1  2  =  910...mm H g  Concentration  c*  F r o m P e r r y (28), H e n r y ' s L a w  constant @ 3  = 1.42 CO^  Hg  x 10  20°C  atm/mole-fraction  p a r t i a l p r e s s u r e i n t e s t s e c t i o n == t ePs t s-e c18 t i omnm p r eHsgs u r e ~  M o l e f r a c t i o n CO_  Ct  at eg'm  =  (P 1420  - 18 ) y 7 6 0 /  atm  atm  / mole-fraction  CO,, v a  P  o r  pressure  132 w i noi n n ^ u n , o\ m o l e s CO2 / nccc M o l a r c o n c e n t r a t i o n = . 926 (10 ) ( P - 18) ; - ;, -0555 m o l e sol'n V 8 I 3 = 5.13(10" ) gm - moles C 0 / cm  m o l e sol'n - T — cm J J  2  q*  = e q u i l i b r i u m c o n c e n t r a t i o n i n equiv. = (5.13)(10- )(P-  18) g " " " ;  8  1  1  1 6  cm°  -J  ( I  = 0.940  8 )  .  •  9  4  0  =  mm  Hg  \  K gm-mole  /  293 K 293° Hg P mm  ° 2 solution  cm  ("flu")  gm  C  F o r R u n 28, P - 910 m m  =  f62360 ^  P  q*  gas v o l u m e at P a n d 2 0 ° C  0.921  Hg  cm CO 3  z  3  j  cm  sol'n  C* = concentration in parts per million by mass =  5.13(10- )(5-  (44JELU1  i8J^2oj£_  8  cm->  \  _£HL_\ o ) 6  (1  mole/ \  gm  /  = 2. 26 ( P - 18) p p m = 2.26 (910 - 18) = 2020 p p m C S T A R = 2. 263 * ( P B T S  - 18. )  Q S T A R = . 4153 * C S T A R / P B T S  Sample  Concentration  j  F o r i d e n t i c a l N a O H and H C l reagent  c o n c e n t r a t i o n s (0.02N),  e q u a t i o n (34) b e c o m e s Moles CO  = N o r m a l i t y x D i f f e r e n c e b e t w e e n blank and t i t r e 3  /  = (.02) ( D i f f e r e n c e , c m ) / 1000 S a m p l e v o l u m e = 100 c m .  .  3  Concentration in p p m 2(10" )(difference) moles CO? , „ „ / . -,^ 6 ^ (44 g m / m o l e C O g ) (10 ) 5  =  w i f t  5  100 g m s o l ' n = 8.8 ( d i f f e r e n c e ) , p p m  /  ,  133 VOL  I = d i f f e r e n c e between blank and t i t r e f o r s a m p l e f r o m tap i  a f  /  = fractional concentration, 7 q*  F I ( K ) = 8.8 V O L Ij C S T A R F o r . d e t e r m i n a t i o n 28106 [rJ  *  = 8.8 ( V O L l y C  = 1030/min), the c o n c e n t r a t i o n s  (N = 10 3 0 / m i n ) , f r o m the 2 - f t a n d 6 - f t taps a r e u s e d . T h e r e f o r e B  and  i = 2,  j = 6. S u b s c r i p t " o " d e n o t e s the l i q u i d e n t e r i n g f e e d n o z z l e F o r 28106,  = 8.8 (2.45) j 2020 = 0.0107  f = f  f. = f. = 8. 8 (5.90) /2020 = . 0257 f Ultimate  / • /  = 8. 8 (. 19)/ 2020 = 0. 00083 J[ /q  Concentration,  G a s f e e d m e t e r e d at a t m o s p h e r i c p r e s s u r e (754 f o r R u n and m e t e r t e m p e r a t u r e ( 2 5 ° C f o r R u n QQ  Q  - g-  as  28)  f e e d at t e s t s e c t i o n t e m p a n d  Mete r V o l  /  Mete r Sec  ^  28)  pressure  50 32.5  x  Q G O T (K) = V O L  ' \ 910  * PMET  / \  298  * 17580.^(SEC  * PBTS*  TMET)  If a l l gas f e d w e r e d i s s o l v e d , t h i s w o u l d p r o d u c e a c o n c e n t r a t i o n q e q u i v a l e n t i n a d d i t i o n to the f e e d w a t e r c o n c e n t r a t i o n 3 Q cm CO2 / m i n q equiv = — TT-T —-. —. r rr (QLW / m i n ) (454 § / l b ) ( l . O c m sol'n/gm) G  l b  =  7  5  ,  m  2  (4.90) (454)  .  =  0.0338  6  cm c  m  3  3  CQ  3  ,  n  S Q l  fo  +  q, e q u i v a*  "L =  . 00\083  +  •  0  3  3  =  8  0.0375  •921  )  E L Q S T (K) = F E E D + . 0 0 2 2 * Q G O T (K) / ( Q L W  * QSTAR)  F r o m equation ( 2 3 )  Bubble Diameter  let xx =  6 Q  <±*j J  L  N  B  xx is constant f o r a given bubble f r e q u e n c y and l i q u i d flow 3  6  =  xx  foLW /minV454 lb  TT =  XX  867  /  l b ) (l.  Q  c  m  f R  m  )  q*  (r^/min) = 3. 81  cm  3  Q L W * Q S T A R / E N B ( K )  = X X * * .  DIAU(K)  m  ( 4 . 9 0 ) (. 9 2 1 ) ^ 1 0 3 0  = 867. *  X X I 3  g  333  = XX*(< ELQST(K)  - FI(KJ)) **  . 333  d. - b u b b l e d i a m e t e r at u p s t r e a m tap, i  -  1  n/  3  1/3 For  28106,  d. =  J3.  81 ( . 0 3 7 5  -  . 0107)j  B u b b l e v o l u m e at f e e d p o i n t = Q G  d  0  =  6  -  6  D I A O (K) = ( l . 9 0 9 * Q G O T ( K )  (75.2)  j  ENB  = l e n g t h i n ft b e t w e e n t a p s i a n d j  cm  B = 0. 519  (1030)^  L e n g t h of T e s t S e c t i o n DL  N  1/3  Qr-^ N1,0 B  /  =0.468  (K)J * * . 3 3 3  cm  135  If the t a p s o c c u r  at p o s i t i o n s o f i n t e g r a l n u m b e r s o f feet,  then D L = j - i If t a p s a r e n o t at p o s i t i o n s o f i n t e g r a l n u m b e r s o f f e e t ^e. g. , f e e d s a m p l e , E Y E  o r t a p 11 i n F i g u r e 5(a) ^, t h e n t h e v a l u e  o r E J i s s e t e q u a l to a c o n s t a n t b y a d e c i s i o n s u c h as  s t a t e m e n t 59 i n t h e a t t a c h e d F O R T R A N  program.  F o r 28106, D L = 6. 00 - 2. 00 = 4. 00 f t . Absorption Function  F (f) i s c a l c u l a t e d b y e q u a t i o n (19)  Mean driving force,  ( l - f ) m = D F B = 1 - (f. + f. ) j  2  = 1. - ( . 0107 + . 0 2 5 7 ) ^ 2 = .964 DFB  = (z.  - FI(K) - F J ( K ) ) / 2 .  •f  J  df FINT =  3  r  1  = 3 F I N T = 3. * ( ( E L Q S T ( K )  '/si  3  y  (. 0375 - .0107) - ( . 0375-.0257) -  FI(K)J  ** .333  - ( E L Q S T ( K )  A b s o r p t i o n function, F(f) = F I N T / ( l - f ) m = FFN  ( K ) = FINT /  Bubble Spacing velocity.  =  j  =0.212  - F J ( K ) ) * * . 333) 0.219  DFB  T h i s c a l c u l a t i o n i s b a s e d o n the s u p e r f i c i a l l i q u i d No c o r r e c t i o n is made for $  Bubble spacing = V E L j (.500 Q L W  ^  B N, ft/sec)(30.5  C m  / f t ) (60 s e c / m i n )  N, B /min 914. 6 Q L W / N  B 914. 6 (4. 90)  1  1030 = 4. 35 c m  136 DELT  (K) = 914. 6 * Q L W / E N B(K)  Velocity Correction  E q u a t i o n (28) c a n b e w r i t t e n p a r t l y i n t e r m s  of  bubble d i a m e t e r , u s i n g equation (23), with  6  L  Q  7T C  v  =  W  -  Q  <1* N  xx  B  _3 w 2 F  2  (f)(l-f)m  (xx)^>  F 2  f f ), (1 - f) v  ,  ,  m  B u t f r o m e q u a t i o n s ( 2 1 ) , ( 2 3) a n d ( 2 4 )  W  =  |  Also  2  w  C  0  =  v  Re  .'.  f  (  F (f)  C  For  W  s  w  XX  )  J  =  F I N T . /  . •  +  5880,  =  .845,  w . {di 2  f r o m Table  -  d  2  v  +  2  )  "  a  w +  f  i')(  x x  FINT  1  II-2 ;  M.439)(.468  +  i  (xx)''3  =.439,  W|  w /xx 1  (l-f)m  2 (FINT)  =  =.845  3  w^' =  j  w  2  =  f t  -.0486  - . 356 ) 2  +  ( . 2 1 2 ) ( 3 . 8 1 ) '5  (-.0486)  (.0257  - .0107)  =  1.02  .212 V C F 2 = 1. 5 * x * ( D I A U ( K ) * * 2 - D I A D ( K ) * * 2 / ( F I N T  *  V C F 3 = Y * x x 13 * * 2 * ( F J ( K ) - F I ( K ) ) ^ F I N T V C F (K) = W Mass  Transfer k  ^  L  + VCF2 + V C F3  Coefficient  i s g i v e n b y e q u a t i o n (20) w h i c h c a n be w r i t t e n  =  Q  ^  3.8D  2  h  K  N ISLq** B  F  ^  *  L  2<" 2  xxl3)  )^  137  where  * / ^ v r  J ) ' = r e c i p r o c a l of l i q u i d v o l u m e t r i c /TTV : f r a c t i o n i n tube = RLFM  {-—j-  R L F M = . 921  +  .(-  . 0375 -  R L F M = Q S T A R * (1, /QSTAR Q  +  0 1 0 7  '  0  2  5  7  ) =  1. 018  +. E L Q S T (K) - (FI(K) + F J ( K ) )/2)  *  L  = constant f o r a given r u n at f i x e d Q  3 . 8D (j Z  -  *  r  [ (QLW  %  l b /  m i n ) (454  l b ) (1. 0  g m /  3. 8 (.794 ) c m  ^ * ^  2  = .417 J454 (4. 90)J  *  ^  C  m  7  -----  (.921 )  J  gm  3  cm / 2  = 12800  . mm  *  Y Y = Q S T A R ** . 6667 B K L = . 4171 * (454. * QLW) ** 1. 333/ Y Y cm / . € 'mmmi n-3) 3  (BKL ( N  g  / min)  ( D L ft) (30. 5  1/3  b  ( R L F M ) ( F j (f) ) C m  12800 (1. 018) (-219)  /ft) , _,  =  '  .  2.32 c m / m m  (1030) "••••(4) (30. 5) C K L (K) = B K L * F F N (K) * R L F M / (( ENB'(K) ** . 333) * 30.48 * D L ( K ) ) with v e l o c i t y c o r r e c t i o n , k  =  T  C  LV k  LV  k v  = 1. 02 (2. 32) = 2. 36  C m  L  /min  3  D i m e n s i o n l e s s Group, N B B G R (K) = N  L  3  L / Q , L  /QL =  N  B / m i n [ ( P L ft)(30. 5 ^ f 0 ] C  (QLW 62.4(1030) 4 4^0  /min) (454  3  =  8 4 0  '  0 0  °  C  m  /lb)  '  BGR  (K) = 62. 42 * E N B ( K ) * D L (K) * * 3/ Q L W  WORD DEFINITIONS F O R ABSORPTION  PROGRAM  TMET  t e m p e r a t u r e ( K ) at s o a p b u b b l e m e t e r  PMET  b a r o m e t e r r e a d i n g ( m m Hg)  QLW  water flow rate (lb/min)  PBTS  m e a n p r e s s u r e i n test section, P  NRUN  Run number  N  number of individual determinations  CSTAR  E q u i l i b r i u m c o n c e n t r a t i o n ( p p m CO-,)  QSTAJl  E q u i l i b r i u m c o n c e n t r a t i o n , <y  BKL  constant f o r calculating k  R U N (K)  ( m m Hg)  i n the r u n  2. • «/ (cm / min J  L  2  )  identification number for each determination  e. g.  Run Number  28  F i r s t f r e q u e n c y i n R u n 28  RUN(K) =  Sixth d e t e r m i n a t i o n f o r the first  I (K), J (K) =  frequency  l o c a t i o n code n u m b e r s f o r u p s t r e a m a n d d o w n s t r e a m taps, respectively.  F I (K), F J ( K ) , F E E D = c o n c e n t r a t i o n s at t a p s i a n d j, a n d i n the l i q u i d f e e d VEL  =  V O L (I), V O L ( J ) ,  S u p e r f i c i a l liquid velocity (ft/sec) V O L (F) = Difference between blank and titre f o r s a m p l e s i,  j and feed water ( c m )  RE  Re  E N B (K)  N  B  (min  )  139 W, X, Y Q G O T (K)  c o n s t a n t s /tJo ; ^ / =  AA)^ r e s p e c t i v e l y f r o m equation('21.')  3 ( c m /min)  Q_ Go  SEC  T i m e for bubble m e t e r m e a s u r e m e n t (sec)  VOL  V o l u m e i n bubble m e t e r ( c m )  E L Q S T (K) =  F(f)  F F N (K) B G R (K)  N  B  L  3  /  Q  L  DFB  mean driving force,  C K L (K)  k  C K L I (K)  k  D L (K)  test length,  D E L T (K)  =  L LV  (1 - f ) m  ( cm/min) (cm  /min) L (ft)  bubble s p a c i n g without v e l o c i t y c o r r e c t i o n (cm)  D I A O ( K ) , D I A U ( K ) , D I A D (K) = b u b b l e s p h e r i c a l d i a m e t e r s ( c m ) YY  if *)  XX  constant for calculating bubble d i a m e t e r  FINT RLFM  r e c i p r o c a l of l i q u i d v o l u m e f r a c t i o n i n tube  VCF  V e l o c i t y c o r r e c t i o n factor,  RUNFIT  n a m e o f s u b r o u t i n e f o r l e a s t s q u a r e s f i t t i n g ( A p p e n d i x I I I (b))  F O R T R A N IV P R O G R A M  FOR ABSORPTION  Cy  RESULTS  T h e p r o g r a m s h o w n i s f o r 5/16 h o r i z o n t a l  tube  T h e s e t t i n g o f l e n g t h s f o r the c o d e n u m b e r s i a n d j w i l l b e d i f f e r e n t for each test section,  a c c o r d i n g to the d i m e n s i o n s i n F i g u r e s 4 a n d 5.  0 < t l H F T C TDRHSV >C TAHULATIUH OF PAT A AND RESULTS FOR ABSORPTION RUNS • c TIIRE HORIZONTAL, 5 / 1 6 INCH 1 . 0 . 1 < DIMENSION R U N I 5 0 ) , 1 ( 5 0 ) , J 1 5 0 I . F H H I 5 U I , F K 5 0 ) , QG0TI50) 2 4 0IMENSI0N F F N I 5 0 ) , B G R I 5 0 ) , C K L I 5 U ) , t)L 1 5 0 ) , D E L T I 5 0 ) , O I A 0 I 5 0 ) DIMENSION 0IAUI50),DIAD(50). FJ150I, ELOST(50),CKL1150),VCF(50) 3 4 <» 12 FORMAT ( 2 1 1 0 , 4 F 1 0 . 3 ) 5 < CALL SKIP TO (1) READ 12, NRUN, N , PMET, TMET, QLW, PBTS ft> 10 11 IF ( N - 0 ) 1 6 0 , 1 6 0 , 1 1 12 <• 11 RE * 1200.2»QLW  i  > c  13 14 15 16 17 20  4 4  21  4  22 23 24 25 26 27 30 31 32 33 34 35 36 37 40 41 42 43 44 45 46 47 50 51 52 53 54 55 56 57 60 61 62 63  > 21.0  4  t  14 c  > c  4  t 4340  4  4> 4  4•  52 > 56 > < i  <> • • » *• 4• <  58 59 45 860 880 895  • • • • 4•  50 760 80 9'i 75 77  > • • « • • •  78 81 82 85 C 87 89  t  97  i  4  c 65 4 145 66 4 115 67 4 106 70 4 71 4 201  ni li 4  204  74 4 75 * 76 4 212 77 4 ICO i 220 102 4 103 4 242 104 1 105 4 250 107 4 110 4 111 « 112 4 113 4 114 4  r \>  loO  (  600.  f o r  5/8  tubes)  READ 2 1 0 , R , W , X , Y FORMAT ( F 1 0 . 0 . 3 F 1 0 . 5 ) CSTAR o 2 . 2 6 3 * (PBTS - 1 8 . ) CSTAR = , 4 1 5 3 * C S T A R / P B T S YY=USTAR*«.6667 BKL • = . 4 1 7 1 * 1 4 5 4 . * U L W I * * 1 . 3 3 3 / Y Y VEL  ( (  . 1044  f o r  5/8  tubes  )  . 1254  f o r  5/8  tubes  )  = .5001  • QLW  00 97 K• « l.li READ 4 3 , RUN IK) , K K ) , J I K ) , E N B ( K ) , V 0 L 1 , V O L J , V O L F , V O L , SEC FORMAT ( F 8 . 0 , 2 1 5 , F 7 . 0 , 3 F 8 . 2 , F 6 . 0 , F 8 . l l El • JIK) QGOTIK) «VUL*PME T * l 7 5 8 0 . / ( S E C * P B T S * T M E T ) XX =359.8*CSTAR*QLW/(PaTS*ENBIK)1 XX13°XX**.333 F J ( K ) » 8 . 8 * VOLJ /CSTAR FEED =8.8 * VOLF /CSTAR ELQST(K) » FEED • .00220*QGOT1Kl/(QLW*OSTAR) IF ( I ( K ) - l ) 5 0 , 4 5 , 50 EYE » - 0 . 1 7 F I ( K ) » 8 . 8 * VOLF /CSTAR OFB = ( 2 . - FEEO - F J ( K ) l / 2 . D I A U ( K ) = (XX * ( E L Q S T I K l - F E E D ) I * * . 3 3 3 GO TO 75 EYE « K M FIIK)°b.8*V0LI/CSTAR OFO = ( 2 . - F I ( K I - F J I K ) ) / 2 . D1AUIK)»IXX » I E L Q S T ( K ) - F I I K ) ) ) * * . 3 33 DL(K)=EJ-EYE BUR(K)»62.42*ENBIK)*DL<KI**3/0LW FINT «3.*(<ELQST(K)-FI(K))**.333 -1ELOSTIK)-FJ1 K))**.3331 R L F M - Q S T A R * ( 1 . / Q S T A R • ELQSTIK) - 1 F 1 < K ) * F J I K ) ) / 2 . 1 FFN(K) - F I N T / O F B CKLIK)»BKL*FFN(K)*RLFM/(IEN8IK)**.333)*30.48*DLIK)) OELTIK)«914.6*QLW/ENB(K) (229.  3  f o r  5/8  tubes)  DIAOIK) = t 1 . 9 0 9 * Q G 0 T ( K ) / E N B I K ) ) • * . 3 3 3 DIADIK) »IXX * ( E L Q S T ( K ) - F J ( K ) ) ) * * . 3 3 3 VCF2»1.5*X*IDIAU(KI**2-DIA0(K)**2)/IFINT»XX13) VCF3»Y*XX13**2*<FJ(K|-FI(K))/FINT VCF(KI=W+VCF2*VCF3 CKL1(K)=CKL<K)*VCFIKI CONTINUE FORMAT I F B . 0 , 2 F 1 0 . 5 , 2 F 8 . 2 , 4 F 8 . J ) FORMAT ( F 8 . 0 , 2 I 4 , F 8 . 0 , F 9 . 4 , F U . O , F 7 . 2 , 2 F 8 . 3 , F 9 . 4 ) FORMAT (44H TUBE HORIZONTAL, 5 / 1 6 INCH INSIDE DIAMETER //) PRINT 2 0 1 , N R U N , R E , P B T S , C S T A R FORMAT IIX.BHRUN NO. 1 3 , 8H RE « F 7 . 0 , 14H PRESSURE • F 7 . 0 1. 15HMM.HG. C« » F 7 . 0 . 3HPPM ) PRINT 204.QLW,VEL FORMAT!IX,28HWATER TEMP.=20.0 C , FLOW " F 6 . 2 . 1 6 H L B / K I N VEL." 1 F 7 . 4 , 7 H FT/SEC I PRINT 106 PRINT 212 FORMAT ( 4 X . 3 H R U N . 4 X . ! . H I . 3 X . 1 H J . 4 X . 2 H N B . 6 X . 4 H F I F ) , 7 X . 3 H B G R . 4 X •> 16HLENGTH,4X,2HKL,SX,3HKLl,5X,4HL/0* )  DO 220  K'l.li  PRINT U S , R U N ( K ) , I I K ) , J ( K ) , F N B I K » , F F N I K I , B G R ( K I , D L ( K I . C K L I K I , C K L 1 ( I K ) , E L O S T t K) PRINT 242 FORMAT ( / / / 4 X . 3 H R U N . 7 X . 2 H F I , 8 X . 2 H F J . 6 X . 3 H Q G 0 . 4 X , 5 H D E L T A , 4 X , 3HVCF. 16X.2H01,6X,2HDJ,6X,2H00 ) DO 250 K»l,N PRINT 1 4 5 , R U N ( K ) , F I I K ) , F J ( K ) , Q G O T I K ) , D E L T ( K ) , V C F I K ) , 0 1 A U I K 1 , DIADIK 1),DIAOIK) CALL RUNFIT I B G R . F F N . C K L . C K L 1 . A . B . N . N R U N ) CALL SKIP TO ( I ) GO TO 10 CONTINUE STUP END  F o r the 5/8 t u b e s ,  c e r t a i n c o n s t a n t s i n t h e p r o g r a m w i l l be  changed.  The^se c o n s t a n t s a r e i n d i c a t e d i n t y p e f o l l o w i n g the s t a t e m e n t i n w h i c h t h e y a r e to be u s e d . (b) L E A S T S Q U A R E S  ANALYSIS  T o f i t n p a i r s o f v a l u e s (X, ^ ) to the l i n e a r r e l a t i o n s h i p = a  +  bx  the f o l l o w i n g e x p r e s s i o n s a r e u s e d :  •LV-(f»i> where  ^  =  and a l l s u m m a t i o n s a.  "  Rvalues  m e a n of a l l are  X>  b  V a r i a n c e o f r e s u l t s f r o m the f i t t e d l i n e i s g i v e n b y ^  ~  £  l  c  a  «/fc)r:-fc-£*ig.  -  ,  S I G Z X  ;  9 5 % c o n f i d e n c e l i m i t s o n the i n t e r c e p t a a r e c a l c u l a t e d ,  ( + T<) 2  a  Upper limit A H  =  Lower limit A L  = a - 2  Confidence  a  +  2  l i m i t s on the s l o p e b a r e a l s o c a l c u l a t e d a c c o r d i n g to  B e n n e t a n d F r a n k l i n (31) : 95% limits = b +  where  ;/  "H-Z  y  0.OS  t ^  =  i  i  °  -  o  S  v a l u e o f the t d i s t r i b u t i o n f o r n - 2 d e g r e e s of f r e e d o m and  95%  c o n f i d e n c e lfcmits, p r o v i d e d i n t a b l e s .  F o r computing,  D I F T 9 5 '=• S I G Z X j e.g.,  , _•  set  -  —  v  (SQRT (SXX - X M  ~ \U  = DIFT95  * S X ))  f o r the d a t a i n F i g u r e 22, n=23 D I F T 9 5 - 0. 0190 f -«  *  n  r  = 2. 08 f r o m  T a b l e III of R e f e r e n c e (31)  b = 0.520 .  .  9 5 % c o n f i d e n c e l i m i t s on t h e s l o p e = b  +  (2.08) (.019) = . 520 +  .04  M e a n v a l u e a n d v a r i a n c e a r e a l s o c a l c u l a t e d f o r the i n d i v i d u a l determinations of k  T  L SKL = SSKL =  k  L  and k Ly  I (k ) • L  £ ( k j .  % - /  143 0 * $1BFTC RUNFIT SURROUTINE RUNFIT ( BGR, FFN,CKL,CKL1,A,B,N,NRUN) 1 * * C LEAST MEAN SQUARES PROGRAM FOR Z = A + B*X DIMENSION B G R < 5 0 ) , F F N ( 5 0 ) , C K L I 5 0 ) , C K L 1 I 5 0 ) 2 * SX= 0. 3 * 4 * SZ = 0. SXX = 0. 5 6 * SXZ = 0. 7 * SZZ = 0. 10 * SKL = 0 . SSKL = 0. 11 * SKL1=0. 12 * 13 * S S K L 1 = 0. 14 * DQ 50 K=1,N X = AL0G10(BGR(K ) ) - 6 . 15 16 * SX = SX + X 17 * Z SZ= ALOG = S7TO•tFFN Z IK) ) 20 * 21 * SXX =sxx+x*x 22 * sxz =sxz + x*z 23 * 48 SZZ = SZZ + z*z SKL1=SKL1+CKLI(K) 24 * SSKL 1 = SSKL 1+CKL K K )**2 25 26 * • SSKL=SSKL+CKL(K)**2 SKL=SKL + C K L ( K ) 27 * 50 EN = N 31 * 51 XM = SX/EN 32 * 52 ZM = SZ/EN 33 * 53 34 * 54 CKLM = SKL/EN SMXX=SXX-SX*XM 35 * 60 SMXZ=SXZ-SX*ZM 36 * 61 B=SMXZ/SMXX 37 * 62 A=ZM-B*XM 40 * 63 SMZZ=SZZ-SZ*ZM 41 * 64 SMEE=SMZZ-B*SMXZ 65 42 RSQ0=1.-SMEE/SMZZ 43 * 68 44 * 69 CONST = 10.0 ** A 45 SIGZX = SQRT ( ( S Z Z — ( A * S Z + B * S X Z ) ) / ( E N - 2 . ) ) 70 46 * 71 AH = A + S I G Z X * 2 . 47 * 72 AL = A - S I G Z X * 2 . HLIM = 10.0**AH 50 * 73 BLIM = 10.0**AL 51 * 74 DIFT95 = SIGZX/(SQRT(SXX-XM*SX ) ) 52 CKL1M= S K L I / E N 53 * VAR=(SSKL-SKL**2/EN)/(EN-1.) 54 VAR1=(SSKL1-SKL1**2/EN)/(EN-1.) 55 C A L L S K I P TO ( 1 ) 56 * PRINT 126, NRUN, N 57 * 125 FORMAT (IX,8HRUN NO. 1 3 , 1 7 , 13HDATA POINTS /) 60 * 126 PRINT 1 3 1 , XM, ZMt CONST, B, CKLM 61 * 130 FORMAT ( 2 E 1 6 . 6 , 2 F 1 3 . 7 , F 1 4 . 3 /) 62 * 131 P R I N T 1 6 1 , A L , A, AH, B L I M , CONST, H L I M 63 * 160 64 161 FORMAT ( 6 F 1 2 . 5 . /) .. FORMAT ( 3 E 1 6 . 6 /) 6 5 * 141 PRINT 1 4 1 , SMEE, RSOD, S I G Z X 66 * 143 67 * 150 70 * 151 71 * 72 * 165 ... 73 74 * 200 75 76 * 201 77 * 100 *  PRINT 1 5 1 , SX, S Z , SXX, S X Z , SZZ FORMAT ( 5 E 1 6 . 6 ) PRINT 165 , 0 I F T 9 5 FORMAT (/26H +/- .95 CONF. L I M . = T * F 1 0 . 7 PRINT.200,CKLM,VAR . FORMAT <///37H NOT CORRECTED FOR V E L O C I T Y , KLM PRINT 201,CKL1M,VAR1 FORMAT ( 3 7 H CORRECTED FOR V E L O C I T Y . KLIM = RETURN END  ) ,F20.3,F20.8/) .F20.3.F20.8//)  144 (c) E S T I M A T E O F RANDOM ERROR IN MASS T R A N S F E R C O E F F I C I E N T A l l quantities in equations (19) and (20) for calculating k •  n.  to within 1% or better with the exception of the integral C, is reliable to within 2% except for R V e  fJ I  are reliable I f a  a  2500.  r  The largest uncertainty in these equations is due to uncertainties in f. and f. in the integral when f. - f. is small. i  j  j  6  i  As pointed out in "Results  and Discussion" for the 5/16, horizontal tube, the approximate criterion for, rejecting data was that uncertainty in f. - f. was greater than 10% of f. J  J•  1  i  In order to estimate the effect of this uncertainty on k , it is assumed that one concentration f. is exact and all uncertainty is in f..  This is done since  J  1  the maximum uncertainty of 10% of the difference occurs in this worst case.  The statistical theory of random errors (29) estimates the variance in a derived function  —>//( X j ,  - - - ^ ) due to random error in the i n -  dependent variables "X.. based on the variance of each variable:  + A-^X* where  and  ^  *  (III(c)-l)  . are variances of each quantity. F o r the  integral in equation (20),  if/?-*)*)  ' (V •  J  S  ^  The maximum uncertainty of 0. 1 (f. - f. ) is taken to be 2 uncertainty would be within 10% of (f. - f. ) in 9 5% of all cases.  3  i  ( I I I ( C )  - ' 2  ; i . e. , the  The  c o n t r i b u t i o n of f. 3 to the v a r i a n c e i n  ^  J  4 I  -77-  —,/  /  is  (HI(c)-3)  E q u a t i o n (III(c)-l) s h o w s that i f the s t a n d a r d d e v i a t i o n f o r a n y q u a n t i t y i s s e v e r a l t i m e s l a r g e r t h a n f o r the o t h e r q u a n t i t i e s , t h e n t h i s q u a n t i t y a l m o s t entirely determines  &g  »  T h e r e f o r e i n the p r e s e n t c a s e , e q u a t i o n  (III(c)-3) a b o v e g i v e s the t o t a l v a r i a n c e i n  f o r e q u a t i o n (20)  (III(c)-4) The  % l i m i t s f o r the 9 5 % c o n f i d e n c e l e v e l i s g i v e n b y  T h e l i m i t s g i v e n b y (III(c)-5) d e p e n d on the v a l u e s of f^, f^ a n d are widest for 1  and f  c l o s e to  i s one of the w o r s t c o n d i t i o n s :  /  ^^*  •  D a t a p o i n t # 2 5105  0. 0163,  f. = 0. 0118,  ^/^* f r o m Run  and 25  f. = 0.0142  S u b s t i t u t i o n of t h e s e v a l u e s i n e q u a t i o n I I I ( c ) - 5 g i v e s 95% Limits  ( . 0 0 2 4 ) — — ^ 2 ^ 1 — ^ ^ ^  1±  _  +  _I3%_  F o r the m a j o r i t y of d a t a p o i n t s h o w e v e r the m a x i m u m u n c e r t a i n t y s h o u l d be m u c h l e s s t h a n +  _  1 3 % s i n c e the d i f f e r e n c e , f.  j  -f , is  i n doubt b y the  x  m a x i m u m v a l u e of 1 0 % o n l y w h e n the d i f f e r e n c e , f. - f., i s s m a l l . J  1  APPENDIX D A T A AND  RESULTS  FOR  IV  ABSORPTION  RUNS  In the f o l l o w i n g t a b u l a t i o n s , t h e c o l u m n h e a d i n g s h a v e t h e f o l l o w i n g definitions: RE  s u p e r f i c i a l Reynolds number,  VEL  superficial liquid velocity (ft/sec)  RUN  identification  I. J  l o c a t i o n code n u m b e r s i and j f o r s a m p l e taps  NB  bubble frequency,  F(F)  absorption function, F(f)  BGR  N  LENGTH  t e s t s e c t i o n l e n g t h (ft)  KL  mass transfer coefficient, k  L /Q  Re  g  number  N  (min  ^)  3  B  L  (cm/min)  •L KL1  corrected for velocity (cm/min)  f  k L  V  L/Q* FI, F J  inlet and outlet s a m p l e concentrations,  J  gas f e e d flow,  DELTA  bubble s p a c i n g without v e l o c i t y c o r r e c t i o n (cm)  VCF  velocity c o r r e c t i o n factor, C DJ, DO =  G  . (cm  1  QGO  DI,  Q  3  f. a n d f.  /min)  equivalent s p h e r i c a l bubble diameter  u p s t r e a m and  d o w n s t r e a m t a p s a n d at f e e d n o z z l e Note:  The  code number,  1=1,  i s u s e d to denote that t h e gas f e e d  n o z z l e i s the i n l e t of the test length,  s i n c e no s a m p l e taps  were  l o c a t e d at t h e 1 - ft p o s i t i o n o f a n y t e s t s e c t i o n . S e e A p p e n d i x IV(b), w h e r e the e n t r a n c e  length is included.  147  (o)  RESULTS  USED  F O RMAIN  CORRELATIONS  '  (UN N O . 4 RE • 1 0 1 6 6 . MATER T E M P . " 2 0 . 0 C . FLCW TUBE H O R I Z O N T A L * 9 / 1 6 I N C H RUN 4103. 410*. 4101. 4203. 4303. 4403.  I 6 2 2 6 6 6  J 10 6 10 10 10 10  NB 1020. 1020. 1020. 712. 660. 530.  HORIZONTAL,  PRESSURE • 2314.PM.HG. = 8 . 4 7 LU/MIN VEL." 4.2356 INSIDE DIAMETER rif l 0.1235 0.1231 0.2464 0.1288 0.1032 C.C091  BGK 461004. 401004. 3646668. 37J547. 3112(1-7. 249975.  LENGTH 4.00 4.00 B.00 4.00 4.00 4.00  KL 2.664 2.667 2.666 3.016 2.568 2.383  <\1H NCI. 7 HE • 7021. P-USSURfc • 1792 . H f . H C . MATER T E M P . " 2 0 . 0 C , ^LOH 5 . C 5 Lb/I-IH VL:L. • 2 . 9 2 5 6 l U B t HOR I Z U N T A L * 5 / l u I N C . - i N S I O f r DIAMfcTfcK  NB IC 76. 1C76. 1C76. 8/5. 89B. 88b. 52B.  Flf 1 C.190J 0 . 183I 0.369/ 0 . 1 6 «4 .1649 . 3281 .1410  PGR 7347R3* 7347H5. 3876278. 597325. 613231. 41)512 I B . 360364.  LENGTH 4.00 4.00 8.00 4.00 4.00  KL 1 2.696 2.770 2.7J3 3.033 2.594 2.411  RUN  104.  1 10 6 6 10 6 6 10  J 14 10 14 14 10 14 14  NH 1C44. 1C72. 1050. 866. 866. 848. 576.  FIF ) C.C827 C.CQ02 0 . 1629 0.068? C.C661 0.1548 0.0669  HCR 327110. 3 3 5 6 R 1. 2631921. i 7 I J 39 . 271339. 21235B9. 100475.  LF'NTiTH 4.00 4.00 8.00 4.00 4.00 K. 00 4.00  L/0* RUN 0.0204 4103. 0.0204 4104. 0.0204 4105. 0.0155 4209. 0.0125 " 4101. 0.0100 4403.  Fl 0.01186 0.00567 0.00567 0.005-48 0.00720 0.00567  FJ 0.016C1 0.01 IB6 0.01601 0.01279 0.C0974 0.00762  Fl 0.01710 0.C0767 0.00767 0.01436 0.00669 fl. 0 0 6 6 9 0.00699  FJ 0.02247 0.01710 0.02247 C.01063 0.01436  QCO 66.00 66.00 66.00 49.56 40.61 32.25  11.74 14.62  VCF 1.012 1.039 1.029 1.004 1.010 1.012  DI . 386 0.463 0.463 0.3/4 0.381 0 . 303  VCF 0.968 1.007 0.987 0.971  ni 0.328 0.425 0.425 0.332 0.424 0.4*5 0.332  0.310 0.316 0. 110 0.206 0.306 0.313  DO 0*498 0.498 0.496 0.493 0.491} 0.488  OJ 0.226 0 . 326 0.226 0.238 0.329  DU 0.468 0.466 0.466 0.473 0.469  0.236  0.472  OJ 0.308 0.*63 0.307 0.255 0 . 107 0.2*7 0.26b  DO 0.512 0.507 0.311 0.4ol 0.461 0.464 0.473  C» • 4013.PPM FT/SEC  KL I 2.380 2.363 2.372 2.200 .295 .24b .240  KL 2.459 2.348 2.403 2.265 2 . 2 78 2.270 2.307  *UN NO. 8 R€ - 1 5 3 0 3 . PRESSURE • 2 2 7 6 . M h . H G . WATER T E K P . « 2 0 . 0 C t FLOW • 1 2 . 7 5 L B / K | N VfcL." 6.3763 TUBE HOR I Z O N T A L , 5 / 1 6 I M C H INSlOt DI A M E T E H  a8 1 0 5 . 8106. 8204. 6205. 8206. B303.  TUBE  C» • 5196.PPM FT/SEC  :  RUN 7104. 7104. 7106. 72C4. 7203. 7206. 7303.  5/16  L/C* 0.0231 0.0231 0.0251 .0211 0.0211 0.0211  o.oiii  RUN 7104. 7103. T106. 7204. " 7203. T206. 7303.  0.0116/  57.74 5 7 . 74 57.74 46.47 48.47 48.47 29.11  " FJ C.01231 0.C1010 U.01231 0.00816 0.00713 u.OOfllO 0.00594  73.12 73.12 73.12 44.28 44.28 44.28 31.88  OCO 50.51 50.51 50.51 25.55 25.55 25.55 36.47 36.47 36.47  DFLTA 4.97 6.11 5. 6.03 10.13  i.ooe  .969 0.971  TTsT" ~uT4TT"  C* > 5110.PPM FT/SEC  KL 3.046 2.936 2.998 2.669 3.375 3.053 2.999  KL 1 3.241 Ja 192 3.225 2 . 7 74 3.596 3.211 3.126  L/Q* 0.0153 0.0153 0.0153 0.0096 0.0096 0.0096 0.0070  RUN 8104. B103. 8106. 8204. 8203. 6206. 6303.  *r  0.01030 0.U0761 0.00761 0.00715 0.00525 6.605^5" 0.00308  "oco  DELIA 11.17 10.88 11.11 13.47 13.47 13. 20.25  VCF 1.064 1.067 1.076 1 . 0 32 1.065 1.052 1.042  01 0.366 0.419 0.422 0 . J07 0.372 0.375 0.326  .<UN N O . 9 RE • 1 8 3 0 3 . PRESSUKE • 2277.MM.HC. C» • 5112.PPH WATER T E M P . " 2 0 . 0 C . FLOW • 1 5 . 2 5 L b / M I \ VEL 7 . e > 2 6 5 FT / S E C TUBE H O R I Z O N T A L * 5 / 1 6 INCH I N S l O t D I A M E T E R RUN 9104. 9105. 9106. 9204. 9203. 9206. 9307. 9 308. 9309.  [ 10 6 6 10 6 6 10 6 6  J 14 10 14 14 10 14 14 IC 14  NH ICOO. 1000. 1000. 504. 504. 504. 728. 72B. 72H.  F IF 1 C.C062 0.0792 0 . 1 6 53 0.0645 0.0640 0.1335 0.0690 0.C594 0.1284  DOR 261959. 261959. 2U95675. 132028. 142026. 1056220. 190706. 190706. 1125651.  LENGTH 4.00 4.00 8.00 4.00 4~.00" 8.00 4.00 4.00 8.00  "tUN N O . 10 RE • 7021. PHLSSURE • 2279.MM.HG. h AT Eft T E M P . - 2 0 . 0 C * FLOW > 5.115 L L / M l N VEL. • 2.92S6 TUBE H 0 R I 2 C N T A L . 5 / 1 6 I N C H I N S I O h D I A M E T E R RUN 10103. 10205. 10206. 10207.  1 2 6 2 2  J 6 10 6 10  NB 1056. 684. 684. 684.  FIFI 0.1654 0.1774 0.1489 0.3261  BGR 721127. 46 7094. 46 7 0 9 4 . 3716749.  LENGTH 4.00 4.00 4.00 8.00  KL 1 4.422 4.149 4.265 4.4*16  KL 4 , 081 3 . 754 3. 917 4 . .132 .BOB 3. ,970 3. ,632 3 . , 129 3. ,360  ""i.iio 4 . 3->2  3.948 3.460 3.7U4  N8 1170. 1170. 1170. 828. 628. 828.  FIFI 0.1277 C.1180 0.244R 0.1322 6.1153 0.2474  BGK 351831. 551831. 4414649. _31C527._ 390527. 3124213.  LENGTH 4.00  4.00  KL 2.177 2.664 2.247 2.455  KL 2.638 2.441 2.536 •JjObl 2.670 2.860 3.111  NO. 13 RE * 7021. PRESSURE • 9I9.HH.HG. WATER T E M P . " 2 0 . 0 C . FLU* 5.H5 LB/HIN VEL." 2.9256 TUBE H O R I Z O N T A L . 5 / 1 6 I N C H I V S K H ' C I A N E T F K RUN 15107. 15108. 15109. 15304. 15305. 15106. 13403. 15303.  1 6 2 2 6 2 2 6 6  J 10 6 10 10 6 10 10 10  NB ICOO* 1CU0. 1000. 688. 686. 688. 462. 278.  FIFI 0.1669 0.1743 0.340B 0.1423 C.1528 O.2950 0.1324 0.1213  8GR 6B28H5. 662R65. 5463084. 469825. 469825. 37586C2. 313493. 189B42.  LENGTH 4.00 4.00 6.00 4.00 4.00 8.00 4.00 4.O0'  KL 2.229 2.344 2.207 2.131 2.320 2.215 2.279 2.471  RUN 9104. 9105. 9106. 4204. 9205. 9206. 9307. 9308. 9309.  FI 0.00660 0.00503 0.0O303 fl.0OJ7"6 0.00275 0.00273 0.00465 0.00356 0.0035b  FJ 0.00809 0.00680 0.006C9 0 . 0 0 4 19 C.00370 0.00439 0.00534 0.00465 0.0CS14  RUN 10103. 10205. 10206. 10207.  Fl 0.00906 0.01238 0.00597 0.00597  FJ 0.02029 0.0166C  Fl 0.01322 0.00616 0.C06I6 0.00911 0.00445  FJ C.01849 0.01322 0.01649 0.01250 .00911  S.O044S 0.0C556  6.6US6  DELTA 13.95 13.95 13.95 2*.67 27.67 27.67 19.16 19.16 19.16  VCF 1.083 1. 105 1.094 1.066 1. 108 1.096 1.087 1.106 1.096  DI 0.317 0.377 0.377 0.323 0.384 0.184 0 . 323 0 . 4 73 0.373  VCF 1.029 0.975 1.015 0.993  0.477 0.350 0.443 0.443  OJ 0.251 0.317 0.251 0.2*6 0.323 0.256 0.264 0.323 0.264  DO 0.459 0.459 0.459 0.459 0.459 0.459 0.458 0.438 0.456  C* • 5117.PPM F T / SEC  HUN N O . 14 RE - 1 0 1 6 6 . PRLSSUKE • 2289.NM.HG. HATER T E M P . " 2 0 . 0 C . FLCW • 8 . 4 7 LB/MIN V E L . " 4./23S8 TUBE H O R I Z O N T A L . 5 / 1 6 I N C H I N S I D F D I A M E T E R RUN 14104. 14105. 14106. 14204. 14205. 14206.  L/C* 0.0094 0. 0094 0.0094 0.0051 0.0051 0.0051 0.0066 0.0066 0.0066  K U 2.241 2.597 2.282 2.439  L/C* 0.0333 0.0186 0.0IB6' 0.0186  0.61238 0.01660  CELTA 5.07 42.44 42.44  0.3A8 0.239 0.350 0.239  DO 0.522 0.491 0.491 0.491  DJ 0.331 0.408 0.333 0.261 0.369 0 . 2 8 1 6 0.262  Db 0.512 0.512 0.512 0.482 0.482 T 4 8 2 0.465  C* • 5139.PPh FT/SEC  KL 1 2.689 2.562 2.617 3.062 2.758 .910 3.103  L/0»' 0.0247 0.0248 0.0248  RUN 14104. 14105. 14106. JSsSlSi. 1 * 2 0 4 . 0.0152 14205. 0.0152 0.0089  U246. 14303.  0.00753  uco  8 2 . 14 62.14 8 2 . 14 48.33 46.33  CELIA 6.62 6.62  27.61 —ffrrr-  VCF 1.019 1.045 1.032 1.004 I.U33 1.017 0.998  0 I 0.407 0.478 0.478 0.369 0.446 0.446 0.332  C* > 2039.PPM FT/SEC  KL! 2.179 2.37B 2.278 2.106 2.356 2.231 -2.206 2.383  L/C 0.0264 0.0264 0.0264 0.0L92 0.0192 0.0192 0.0122 0.0076  RUN 15107. 15109. 15109. 15304. 15305. 15306. 15403. 13303.  Fl 0.0I7S7 0.00798 0 . 0 0 796 0.01295 0.00626 0.00626 0.00665 0.00565  FJ 0.02287 0.01757 0.02287 0.01662 C.012-15 0.01662 0.011CI 0.00699  OGO 60.48 60.46 60.4B 4 2 . 36 42.36 42.3R 25.65 14.39  DELTA 5.35 5.35 5.35 7.78  i.n  7 . 78 11.58 19.23  VCF 0.977 1.014 0.996 0.979 l.Oll 0.998 0.968 0.965  01 0.346 0.442 0.442  0.350 0.445 0.445 0.324 0.320  DJ 0.233 0.346 0.255 0.261 6 . Ho 0.261 0.229 0.217  DO 0.467 0.487 0.487 0.490 0.490 0.490 0.474 0.463  148  RUN N O . 16 RE • 4813. PRESSURE > 898.MM.HG. WAFER T E M P . - 2 0 . 0 C , FLOW • 4.01 L6/H1N VEL.2.0054 TUBE H O R I Z O N T A L , 5 / 1 6 I N C H I N S I O E D I A M E T E R RUN 16105. 16205. 16303. 16405. 16503.  1 2 2 2 2 2  J 6 6 6 6 6  NB 880. 702. 552. 377. 258.  F (F) 0.2378 0.22/5 0.2135 0.1792 0. I5B9  BGR 676682. 619353. 549919, 375578. 257027.  LENCTH 4.00 4.00 4.00 4.00 4.00  C* * 1991.PPM FT/SEC  KL 2.022 2.079 2.10T 2.002 2.0U  KL 1 2.022 2.074 2.0'16 1.977 1.984  HUN N O . 17 RE • 3169. PRESSURE 900.MK.HG. WATER T E M P . « 2 0 . 0 C , FLCW > 2.64 LB/HIN VEL.1.3203 T U B E H O R I Z O N T A L , 5 / ) 6 I N C H I N S I DC D I A M E T E R RUN 17105. 17205. 17303. 17405.  I 2 2 2 2  J 6 6 6 6  NB 496. 367. 234. 396.  F(F1 C.30S9 0.2457 0.2110 C.2515  BGR 750553. 555349. 354092. 599232.  LENGTH 4.00 4.00 4.00 4.00  RUN 18105. 18201. 18J03.  240. 202. 174.  MF1 C.2865 0.2671 0.2869  BGR 586203. 495071. 42644 7.  LENGTH 4.00  308. 308.  '.UN N O . 24 RE • 1 0 1 6 6 . « 4 T E R TEMP.-20.0 C, FLCW fUflE H O R I Z O N T A L . 5 / 1 6 INCH RUN . 24103. 24203. 24304. 24305. 24306. 24404. 24405. 24406.  I 6 6 2 6 2 2 6 2  J 10 10 6 10  to  6 IC 10  NB 1012. 636. 612. 612. 612. 976. 976. 9 76.  F IF) 0.2539 C.2666  BGR 567015. 667015.  LENGTH 4.00 4.00  BGR 477310. 299970. 286650. 268650. 2309201.. 4C0331. 460331. 3682647.  KLl 1.005 0.965 1.090  I 10 6 2 6 2 2 6 10 6 10 6 2 6 2 2  J 14 10 6 14 10 14 10 14 14 14 10 6 14 10 14  NB 1246. 1246. 1246. 1246. !246. 124o. *)12. 912. 912. 624. 624. 624. 624. 624. 624.  FIF J 0.1125 0.0956 0.C964 0.2081 0 . 1919 C.3043 0.0696 0.0870 0 . 1765 0.CT39 0.0669 0.0690 0.1428 0.1379 0.2117  BGR 42C052. 420052. 420052. 3360419. 3360419. 1 1341414. 307454. 307454. 245<>>632. 210363. 210163. 210361. 1662906. 1662906. 56/9809.  KLl 1.329 1.392  KL 3.543 3.017 3.053 3.280 3.035 3.204 3.134 3.040 3.087 2.929 2.738 2.745 2.833 2.742 2.804  •WH N O . 26 HE * 8761. PRESSURf 2306.MM.HG. » A T E R T E M P . - 2 0 . 0 C. FLCW * 7.30 LD/KIN VEL.3.6507 TUBE H O R I Z O N T A L , 5 / 1 6 INCH I N S I D E DIAMETER  RUN 26107. 2610B. 26109. 26207. 26206. 26209. 26104. 26305. 26306. 26404. 26405. 26406.  I 6 2 2 6 2 2 6 2 2 6 2 2  J 10 6 10 10 6 10 10 6 10 10 6 10  N8 740. 740. 74 0 . 1110. 1110. 1110. 658. 858. 859. 543. 543. 543.  FIF> 0.1347 0.1179 0.2524 0 . 139 3 0.1155 0.2746 0. M72 0.130 3 0.2674 C.121B 0.1161 0.2378  BGR 404960. 404960. 3239683. 607441. 607441. 4659525. 469535. 469535. 3756282. 297151. 297153. 2377227.  LENGTH 4.00 4.00 6.00 4.00 4.00  a.oo  4.00 4.00 6.00 4.00 4.00 8.00  KL 2.652 2.332 2.492 2.410 2.361 2.385 2.575 2.460 2.517 2.652 2.535 2.594  QL.0 54.70 42.66 30 . 8 9 19.68 13.19  DELTA 4.17 5.22 6.64 9.73 14.22  VCF 1.000 0.997 0.995 0.988 0.987  DI 0 . 436 0.431 0.425 0.404 0.401  OJ 0.316 0.307 0.298 0.281 0.280  00 0.492 0.4HB 0.47S 0.464 0.461  L/Q* 0.0257 0.0190 0.0122 0.0266  RUN 17105. 17205. 17303. 17405.  0.01049 0.00895 0.00617 0.01058  0.02249 0.01653 0.01080 0.02182  26.99 19.09 11.57 27.52  0.02110 0.017T4 0.01558  QGO 16.00 11.17 1 1 . OZ  VCF 0.975 0.973  0.402 0.387 0 . 380 0.4". 2  0.239 0.242 0.23i 0.298  0.470 0.463  L/Q* 0.0244 0.0204 0.0174  RUN 18105. 18203. 16303.  0.Q1CCI 0.00676 0.00732  DELTA 6.21 6.57  VCF 0.694 0.B72  0.42R 0.422 0.422  0.263 0.259 0 . 2 3B  0.503 0.500 0.495  L/Q* 0.O238 0.0236  RUN 19102. 19202.  0.00936 0.00936  0.01971 C.020C2  CGO 20.67 20.67  FI 0.010B2 0.0C712 0 . C034«» 0.00721 0.00349 0.00438 0.00982 0.0043B  FJ 0.01537 0.00967 0.00721 0 . 0 0 9 79 0.00979 0.00982 0.01394 0.01394  UCU 6 8 . 11 4 b . 60 43.60 43.60 43.60 61.92 61 . 9 2 61.92  VCF 0.989 0.987  0.434 0.434  0.285 0.276  0.504 0.504  01 0 . 398 0.402 0.474 0.402 0.474 0.461 0 . 390 0.461  OJ 0.31b 0.328 0.402 0 . 3 JO 0 . 130 0.390 0 . 3 13 0 . 3L 3  OU 0.505 0.506 0.51b U . 51 5 0 . 5 15 0.49b 0.49 5 0.4«5  C* M26.PPM FT/SEC  LENGTH KL 4.00 2.7B5 4.00 2 . 5 54 4.00 2.498 4.00 2.471 8.00 2.465 4.00 2.439 4.00 2.704 0.00 . 2.571  LENGTH 4.00 4.00 4.00 8.00 8.00 12.00 4.00 4.00 8.00 4.00 4.00 4.00 8.00 8.00 12.00  FJ 0.02545 0.02059 0.0156C 0.01C43 0.00734  C* • 1973.PPM FT/SEC  *UN N O . 25 RE • 1 4 2 2 2 . PRESSUKE • 2238.MM.HG. ..•ITER T E M P . " 2 0 . C C FLOW - U . C 5 L B / M I N VEL.* 5.9262 IUBE H O R I Z O N T A L . 5 / 1 6 INCH I N S I D E DIAMETER RUN 25105. 25106. 25107. 25108: 25109. 25110. 25204. 25205. 25206. 25304. 25305. 25306. 25307. 25306. 25309.  FI 0.01140 0.00941 0.00698 0.00535 0.00389  C* I960.PPM FT/SEC  1.343 1.410  PRESSUK.E • 2283.MM.HG. 8.47 L8/MIN V E L . » 4.2358 I N S I D E OIAMETER Ff F| 0.1207 0 . 1013 0.O9T6 0.0966 0.1943 0.1109 0.1235 0.2343  KL 1 I.748 1.545 1.532 1.586  1.124 I . ion 1.250  OIK N O . 19 RE * 2604. PRESSURE > B90.HH.HG. WATER T E H P . - 2 0 . 0 C , FLOW • 2.17 L8/MIN VEL. • 1.0852 TUBE H O R I Z O N T A L , S / 1 6 I N C H I N S I D E D I A M E T E R  RUN 16105. 16205. 16303. 16405. 16503.  C* • 1996.PPM FT/SEC  KL 1.792 1.587 1.560 1.591  KUN N O . 18 RE • 1956. PRtSSURE 864.MM.HG. t.4TFR T E M P . . 2 0 . 0 C , FLCW • l . o 3 IB/MIN V E L . « 0.8152 TUBE H O R I Z O N T A L . 5 / 1 6 INCH I N S I O E DIAMETER  L/Q* 0.0341 0.0269 0.0202 0.0131 0.0091  KLl 2.827 2.599 2.60 7 2.516 2.561 2.534 2.739 2.636  L/Q* 0.0201 0.0131 0.0130 0.0130 0.0130 0.0183 0.0183 0.0183  RUN 24103. 24203. 24304. 24305. 24 3 0 6 . 24404. 24405. 24406.  CELTA 7.6S 12.1" 12.66  bt  12. 66 7.94 7.94 7.94  VCF 1.01 * 1.018 1.043 1.01H 1.031 1.039 1.013 1.025  C* 5024.PPM FT/SEC  KLl 3.621 3.1H9 3.297 3.405 3.243 3.366 3.343 3.154 3.249 3.126 2.973 3.009 3.049 2.991 3.036  L/Q* 0.0162 0.0162 0.0162 0. 0162 0.0162 0.0162 0.0136 0.0136 0.0136 0.0124 0.0124 0.0124 0.0124 0.0124 0.0124  RUN 25105. 25106. 25107. 29108. 25109. 25110. 25204. 25205. 25206. 25304. 25305. 25306. 25307. 25306. 25309.  0.01182 0.00876 0.00447 0.00676 0.00447 0.00447 0.00694 0.00960 0.00694 0.00/66 0.00575 0.00301 0.00575 0.00301 0.00301  0.01419 0.01182 0.00676 0.61419 0.0118? 0.01419 0.00960 0 . 01139 0.01139 0.0C96C 0.CU786 0.00575 0.0096C 0.00768 0.00960  or-o  73.29 73.29 73.29 73.29 73.29 73.29 59.55 59.55 59.55 56.42 56.42 56.42 56.42 56.42 56.42  CELTA B . 70 6.70 6.70 8.70 8.70 8.70 11.88 11.88 11.86 17.37 17.37 17.37 17.17 17.37 17.37  VCF 1.022 1.057 1.080 1.030 1.06B 1.051 1.067 1.038 1.05? 1.067 1.0R6 1.096 1.076 1.09) 1.083  0 . 124 0.386 0.449 0.386 0.449 0.449 0.412 0.347 9.412 0.411 0.468 0.524 0.466 0.524 0.524  0.251 0.324 0.366 0.251 0 . 324 0.251 0.347 0.264 0.264 0.350 0.411 ' 0.468 0.350 0.411 0.350  0.463 0.483 0.463 0.483 0.483 0.463 0.500 0.500 0.500 O.S57" 0.557 0.557 0.557 0.557 0.557  C* 5178.PPM FT/SEC  KLl 2.677 2.426 2.552 2.462 2.484 2.472 2.604 2.566 2.585 2.645 2.612 2.628  L/Q* 0.0193 0.0193 0.0193 0.0323 0.0323 0.0323 0.0226 0.0228 0.0228 0.0124 0.0124 0.0124  RUN 26107. 26108. 26109. 26207. 26206. 26209. 26304. 26305. 26306. 264 0 4 . 26405. 26406.  FI 0.01054 0.00459 0.00459 0.01649 0.00646 0.00646 0.01241 0.00498 0.00498 0.00743 0.00323 0.00323  FJ 0.01504 0.01054 0.01504 0.02362 0.01649 0.0236? 0.017*4 0.01241 0.01759 0.01020 0.00743 0.01020  QGO 56.68 S6.68 56.68 96.51 96.51 96.51 67.02 67.02 67.02 35.68 35.66 35.88  CELTA 9.02 9.02 9.02 6.01 6.01 6.01 7.76 7.76 7.78 12.30 12.30 12.30  VCF 1.009 1.041 1.024 1.021 1 . 0 52 1.036 1.011 1.043 1.027 0.997 1.030 1.013  DI 0.411 0.469 0.489 0.438 0.516 0.516 0.415 0.497 0.49 7 0.379 0.465 0.465  OJ 0.323 0.411 0.123 0 . 359 0.438 0.359 0 . 129 0.415 0.329 0.290 0.379 0.290  00 0.527 0.527 0.527 0.550 0.550 0.550 0.531 0.531 0.531 0.502 0.502 0.502  149  ..UN N O . 27 RE • 7513. .ITER T E M P . . 2 0 . 0 C. FLOW HIDE H O R I Z O N T A L . 5 / 1 6 INCH  RUN 2110.. 27105. 27106. 27205.  J  I 6 2 2  NB 1030. 1C30. 1030. 816. 616. 616. 816. 616. 816.  10 6 10  i7206. 27207. 27208. 27209. 27210. 27305. 27306. 27307. 27306. 27309. 27310. 27*0*.  10 10 10  574. 5*4. 574. 574. 574. 574. 274. 274. 274.  27.e;. 27.06.  PKESSURE • 923.MM.HG. • 6.26 LB/HIN VEL.' 3.1306 INSIOE OIAMETER  0.1519 0.0706 0.0812 0.3021 0.2211 0.1499 0.1407 0.0710 0.069T 0.2904 0.2209 0.0683  BGR 657305. 657305. 525a<>36. 520736. 520738. 65092. 65092. 4165907. 1757492. 366304. 366304. 4578R. 45786. 2930430. 1236275. 21857.  O.OttOB C l 190  174856.  FIF> 0.1S30 0.169$ 0.3521 0.150*  <UN N O . 26 RE 5661. HATER T E M P . - 2 0 . 0 C , FLOW TUBE H O R I Z O N T A L , 5 / 1 6 INCH  RUN  26104. 28105. 28106. 28204. 28205. 28206. 28304. 28305. 28306.  I 4 2 2 4 2 2 4 2 2  J  6 4 6 6 4 6 6 4 6  NB 1030. 1030. 1030. 720. 720. 720. 317. 357. 357.  1 4 2 2 2  J 6 4 6 6  FIF1 0.1122 0.1076 0.2196 0.097b 0.0910 0.1087 0.0651 0.0771 0.1622  NB 548. 548. 548. 357.  407. 407. 407. 704. 184. I M . 164.  F (F 1 C.1069 0.0937 0.2005 0.1815  RUN  1 4 2 2 4 2 2  J 6 4 6 6 4 6  FIFI 0.1216 0.1176 0.239U C.1436 C.0B65 0.1192 0.2057  NB 187. 187. 1B7. 346. 346. 346.  HUN NQ. 32 RE * 18)2. WATER T E M P . - 2 0 . 0 C , FLCW TUBE H O R I Z O N T A L * 5 / 1 6 INCH  RUN 32103. 32203.  1 2 2  J 4 4  NB 293. 160.  MIN N O . 39 RE - 2 2 3 B 4 . ..ATER T E M P . . 2 0 . 0 C » FLCW I'IHE H O R I Z O N T A L . 5 / 1 6 INCH  RUN 39109. 39106. 39108. 39204. 39209. 39206. 39207. 39206. 39209.  I 6 2 2 10 6 2 6 2 2  J 10 6 10 14 10 6 14 10 14  NB 1260. 1260. 1260. 1012. 1012. 1012. 1012. 1012. 1012.  BGR 104968. 104968. 839740. 73375. 73375. 567003. 36382. 36382. 291056.  LENGTH 2.00 2.00 4.00 2.00 2.00 4.00 2.00 2.00 4.00  BGR 68242. 68242. 545934. 355654.  LENGTH 2.00 2.00 4.00 4.00  BGR 69365. 69345. 554920. 1I99B3. 3135.: 31339. 250673.  ' PRESSURE . 1.96 LB/HIN INSIDfc DIAMETER  FIFI 0.0992 0.1142 0.2293 0.1363 0.1735 0.3096  HOR 47643. 47643. 361144. 88152. B6L52. 705219.  BGR 96696. 52912.  .566  .Q«2 "2.340 2.711  KLl 2.377 2.347 2.362 2.323 2.218 2.271 2.526 2.350 2.438  KLl 2.141 1.926 2.034 2.102  LENGTH 2.00 2.00  KLl 1.747 1.736 1.741 1.775 1.9.4 2.249 1.917  0.127O 0.1262 0.1640  BGR 269699. 269899. 2159163. 216773. 216773. 216T73. 1734183. 1734185. 5842874.  LENGTH 4.00 4.00 8.00 4.00 4.00 4.00 6.00 8.00 12.00  27309. 27310. 2 7404. •J7405. 2 7406.  0.00636 0. 00636 0.01104 0.01053 C.00494 0.00804 0.00444 0.00494 0.00804 0.OO43B 0.003C9 0.0O309  FJ 0.02492 0.01826 0.U24S2 0.01912 0.01431 0.01431 0.011C4 0.01912 0.01912 0.01396 0.01C53 0.U101.3 0.00804 0.01396 0.01396 0.00571 0.00438 0.00571  QGO 71.50 71.50 71.50 55. Zl 55.21 55.21 55.21 55.21 55.21 38.17 38.17 38.17  DELTA 5.56 5.56 7.02 7.02 7.02 7.02 7.02 7.02 9.97 9.97 9.97  0.990 0.963 1.023 1.013  38.1/ 38.17 3 8 . 17 16.84 1 6 . 84 16.84  9.97 9.97 9.97 20.90 20.90 20.90  1.032 1.002 0.993 1.006 1.026 1.015  l)J  VCF  0.986 1.027  t.OOf  0.990 1.026 1.017 1.035  l.UOQ  0.376 0.471 ( 1 . 4 Tt 0.375 0.466 0.417 0.466 0.466 0.417 0.363 0.459 0.412 0.45 V 0.4S9 0.412 0.398 0.443 0.443  1.275 0.376 0.275 0.2*4 0.375 0 . 375 0.417 0.204 0.284 0.262 0.163 Q. 363 0.412 0.<!62 0.262 0.339 0.398 0 . 339  0.510 U.V 10 0.106 0.506 0.506 " 0.506 0.506 0.503 0.503 0.503 0.50J 0.503 0.103 0.490 0.490 0.490  L/Q* 0.0375 0.0375 0.0375 0.0259 0.0259 0.0299 0.0130 0.0130 0.0130  RUN 2aio«. 28109. 26106. 21204. 26209. 28206. 2630.. 26305. 26306.  0.01909 0.01066 .01068 0.MSJ6 0 . 0 0 7 76 0.00776 0.00719 0.00.36 0.00.36  FJ 0.02572 0.019C9 0.02572  b.olVsi  0.01330 0.01783 0.0095C 0.00719 0.00950  QI.O 75.20 75.20 7!.. 20 51.24 51.24 51.24 24.61 24.61 24.81  DELTA 4.35 4.35 4.35 6.22 6.22 12.55 12.55 12.55  VCF 1.006 1.029 1.017 1.005 1.027 1.016 L.COO 1.024 1.011  01 0 . 4 12 0.467 0.467 0.410 0 . 462 0.462 0.399 0.455 0.455  UJ 0.355 0.412 0.355 0.353 0.410 0.353 0 . 336 0.399 0 . 3 36  UO 0.519 0.519 0.519 0.514 0.514 0.514 0.5LO 0.510 0.510  L/Q* 0.0282 0.0282 0.0282 0.0179  RUN 29104. 29105. 29106. 29206.  .01427 .00619 0.00C19 0.00573  .01951 .01427 0.dl9«l 0.01299  RUN 30104. 30105. 30106. 30203. 30 3 0 4 . 50365. 1C306.  FI 0.01833 0.01126 0.01126 0.01743 .00903 0.60541 0.00941  0.02347 0.01633 C.C234 7 0.02896 C.01073_ 6.0090T' .01C71  OGO 45.60 45.80 6.69 10.27  VCF 1.010 1.032 . 0 20 .013  0.371 0.434 0.311  0.543 0.543 0.5*3  L/3« 0.0301 0.0301 0.0301 _<M>4 7 1 0. 0T2T 0.0127" O.U127  Fj  OGI] 34.86 34.66 34.66 5 5 . 75 13.67 15.ST 13.87  3.81 14.56 14.56 14.96  VCF 0.999 1.020 1.009 1.016 0.962  l.olo  0.999  01 0.406 0.47T 0.477 0.462 0.362 0.454 0.454  OJ 0.337 0.408 0.337 0.392 0.295 0.362 0.299  DO 0.547 0.547 0.547 0.533 0.524 "0.924 0.524  C. . 1946.PPM FT/SEC  KL 1.034 1.461 1.248 1.210 1.552 1.381  KLl 0.988 1.470 1.229 1.165 1.569 1.367  L/U. 0 .0183 0 .0163 0 .0183 0 .0391 0 .0351 0. . 0 3 9 1  RUN '1104. •1105. .1106. '1205. '1206. 31207.  FI 0.01310 0.OC799 C.00799 0.02430 0.01364 0.01364  FJ 0.01545 0.01310 0.01543 0.02958 C.024)0 0.02956  OUO 14.07. 14.07 14.07 27.79 27.79 27.79  RUN 32103. 32203.  FI .01496 1.00942  0.02613 0.01487  OGO 23.40 11.85  RUN 19109. •9106. 391C8.  FI 0.01379 0.01043 0.01043 0.00948 0.00)14 0.00473 0.00714 0.C0.73 0.00473  CELTA 9.59 9.59 9.5V 5.1H 5 . IB 5.18  DJ 0.289 0.153 0.289 0.293 0.366 0.293  00 0.524 0.524 0.524 0.536 0.536 0.536  Dl .456 .426  .368 .339  0.933 0.921  01 0.511 0.558 0.556 0.427 0.463 0.530 0.463 0.530 0.530  DJ 0.459 0.511 0.459 0.3H0 0.427 0.461 0.360 0.427 0.380  00 0.599 0.599 0.599 0.971 0.571 0.571 0.571 0.971 0.571  VCF 0.956 1.006 0.965 0.961 1.011 0.990  Dl 0 . 15 3 0.442 0.442 0.366 0.460 0.460  VCF 0.960 0.933  C* • 1953.PPH FT/SEC  KL 1.145 1.178  PRESSURE 2187.MM.HG. • 18.69 LH/MIN V.EL.* 9 . 3 2 6 9 I N S I O E 01AHETF.K  FIFI 0.0696 0.C619 0.1311 0.0979' 0.0692 0.0570  "7TJ7J8T  FI 0.01626 0.00776 O.U0776 0.01431 0.00636 0.01104  C. . 1969.PPH FT/SEC  1.612 f.226 1.919  LENGTH 2.00 2.00 4.00 2.00 2.00 4.00  BUN 2T164. 27105. 27106. 27205. 27206. 27207. .'7208. ' 7209. 27210. 27305. 27306. 27307.  C* . 1996.PPH FT/SEC  KL 1.749 1.701 1.725  879.MM.HG. VEL.. 0.9802  L/Q* 0.0292 0.0292 0.0292 0.0229 0.0229 0.0229 0.0229 0.0229 0.0229 0.0160 0.0160 0.0160 0.0160 0.0160 0.0160 0.0078 0.0076 0.0078  C* 2019.PPM FT/SEC  KL 2.120 1.867 1.993 2.071  PKESSURE • B81.MM.HG. • 1.51 LB/MIN VEL.0.7552 INSIOE DIAMETER  FIFI 0.1713 0.1453  2.621 2.544 2 . 583 2.330 2.453 2.259 2.647 2.391 2.306 2.586 2.515 2.5)3 2.537 2.560  KL 2.363 2.262 2.322 2.311 2.160 2.235 2.527 2.296 2.411  PRESSURE » B6B.MM.HG. . 2.93 IB/HIN VEL.. 1.4653 1NSIUE OIAMETER  i:MN N O . 31 RE • 2352. NATER T E H P . . 2 0 . 0 C t FLOW TUBE H O R I Z O N T A L . 3 / 1 6 INCH  31104. 31109. 31106. 11209. 31206. 11207.  4.00 2.00 2.00 8.00 6.00 4.00  "2.390 2.223 2.559 2.372 2.310 2.631 2 . 4 79 2.500 2.458 2.555 2.567 3.064 2.2*0 2.672  PRESSURE - . 901.HH.HG. . 4.01 LB/HIN VEL.. 2.0054 I N S I D E DIAMETER  1UN N O . 30 RE . 3517. WATER 7 E H P . . 2 0 . 0 C i FLCW TUBE H O R I 2 0 N T A L . 5 / 1 6 INCH  RUN 30104. 10109. 30106. 30203. 30304. 30309. 30306.  2.652 2.477 2.564  2.00 4.00  KLl  KL  4.00 4.00 8.00 4.00  4.00 2.00 2.00 6.U0 6.00 2.00  FT/SEC  PRESSURE 910.MH.hC. > 4 . 9 0 Lfl/MIN VEL.* 2.4505 INSIOE DIAMETER  tUN NO. 29 RE > 4613. WATER T E H P . - 2 0 . 0 C . FLCW IU8E HOHIIONTAL. 5/16 INCH  RUN 29104. 29103. 29106. 29206.  LENGTH  ' • iO*t."*  C  KL 4.021 3.364 3.792 3.379 4.285 3.940 3 . 9 32 3.912 3.801  KLl 1.099 1.098  L/Q* 0.0302 0.0196  on  C« > 4908.PPM FT/SEC  KLl 4.666 4.173 4.411 4.120 4.973 4.137 4.946 4.955 4.410  L/0« 0.0249 0.0249 0.0249 0.0147 0.0147 0.0147 0.0147 0.0147 0.0147  "WS04. <»203. •1206. 19207. (7208. 19209.  FJ 0.01683 0.01375 0.01683 O.OllOj 0.0094b 0.00714 C.01103 0.00946 0.01103  QGn 141.11 141.31 141.11 9B.41 98.41 98.41 98.41 98.41 98.41  DELTA 13.34 13.54 13.54 16.86 16.86 1 6 . R6 16.66 16.86 16.86  VCF 1.165 1.172 1.168 1.131 1.161 1.166 1.156 1.164 1.160  150 KU-N N O . 4 0 Rt • 18303. PRESSURE " 2229.MH.HO. WATER T E P P . - 2 0 . 0 C , FLCW « 1 5 . 2 5 L B / P I N VEL.* 7.6265 TUOE H O R I Z O N T A L . 5 / 1 6 I N C H I N S I l ' E D I A M E T E R  RUN 40104. 40109. 40106. 40107. 4 0108. 40109. 4S204. 41)205. 40206. 40207. 40208. 4 0209.  1 10 6 2 6 2 2 10 6 2 6 t 2  J 14 10 6 14 10 14 14  to  6 14 10 14  RUN 41104. 41105. 41106. 4 HOT. 41108. 41109. 41204. 41205. 41206. 41207. M208. 41209.  NB 1200. 1200. 1200. 1200. 1200. 1200. 746. 746. 746. 746. 746. 746.  NB  1044. 1044. 1044. 1044. 1044. 1044. 632. 632. 632. 632. 632. 632.  FIFI 0.0739 C.0724 0.0675 0.1462 0.1398 0.2136 0.0523 0.0699 0.0523 0. 1222 0. 1222 0.1745  FCFI  0. 1032 0.0906 C.0940 0.1937 0. 1845 0.2B75 0.0914 0.07B6 C.0810 0.1699 0.1596 0.2508  BGR 314351. 314351. 314351. 2514810. 2514810. 6487483. 195422. 195422. 195422. 1563373. 1563373. 5276385.  LENGTH 4.00 4.00 4.00 6.00 6.00 12.00 4.00 4.00 4.00 8.00 8.00 12.00  BGR 379150. 379150. 379150. 3033ZQ3. 3033203. 10237061. 229524. 229524. 229524. IH36192. 1636192. 6197 148.  4.00 6.00 6.00  8.00 12.00  KL 3.311 3.253 3.043 3.262 3.148 3.202 2.734 3.662 2.745 3.196 3.203 3.047  .131 .756 .874 .944 2.815 2.920 3.264 2.614 2.907 3.039  C« • 5003.PPH FT/SEC  KL 1 3.732 3.700 3.483 3.716 3.591 3.636 3.059 4. 141 3.130 3.600 3.635 3.443  1/Q* 0.0231 0.0231 0.0231 0.0231 0.0231 0.0231 0.0123 0.0123 0.0123 0.0123 0.0123 0.0123  KL 1 3.290 2.957 .124 .123 3.040 3.123 3.3H4 2.992 3.143 3.1H8 .068 .173  L/0» 0,0248 0.0248 0.0248 0.0248 0.0248 0.0248 0.0134 0.0134 0.0134 0.0134  HORIZONTAL,  Fl  40104. 40105. 40106. 4C107. 40108. 401Q9. 40204. 40205. 40206. 40207. 40206. 40209.  0.01398 0.01050 0.00653 0.01050 0.00653 0.00653 0.00795 0.00584 0.00387 0.00584 0.00387 0.00367  RUN 41104. 41105. 41106. 41107. 41108. 41109. 41?C5.  41206. 41207. 41208. 41209.  5/8  Fl 0.01653 0.01232 0.00662 0.01232 0.00662 0.00662 0.00943 0 . 0 0 7 1 •> 0.004C0 0.00715 0.00400 0.00400  r j 0.01660 0.01398 0.01050 0.01680 0.01398 0.01680 0.00918 0.00795 0.00584 0.00918 0.C0795 0.00918  FJ 0.02002 0.01653 0.01232 0.02CC2 0.01653 6.O2002 0.00943 0.00715 0.01129 0.00943 0.01129  SCO 134.92 134.92 134.92 134.92 134.92 134.92 69.45 69. 45 69.45 69.45 69.45 69.45  QtiO  109.44 109.44 109.44 109 .44 109.44 109.44 5 7 . 13 57.13 57.13 S 7 . 13 5 7 . 13 57.13  DELTA 11.62 11.62 11.62 11 . 6 2 11.62 11.62 18.70 18.70 18.70 18.70 IB. 70 18.70  VCF 1.12 7 1.137 1. 1 4 4 1.132 1.141 1.136 1.119 1.131 1.140 1.126 1.135 1.130  DELTA 9.64 9.64 9.64 9.64 9.64 9.64 15.92 15.92 15.92 15.92 15.92 15.92  VCF 1.051 1.073 1.08 7 1.061 1.080 1.064 1.037 1.06 3 1.081 1.049 1.072 1.059  01 0.413 0.474 0.537 0.474 0.537 0.537 0.3b2 0.445 0.509 0.445 0. 5 0 9 0.509  nj 0. 344 0.413 0.474 0.344 0.413 0. 144 0. 309 .382 .445 .309 .382 . 309  DO 0.565 0.565 0.565 0.585 0.585 0.5B5  VCF 0.921 0.938 0.930 0-937 .895 .908 0.924 0.942 C.902 0.917 0.932 0.911 0.926 0.919 0.901 0.919 0.937 0.910 0.927 0.919  DT 0.722 0.812 0.812 0-812 0.574 0.644 0.743 0.829 0.644 0.743 .829 0. 743 0.829 0.829 0.617 0.716 0.603 0.716 0.803 0.803  OJ 0.639 0. 722 0.639 0-713 0.501 0.5 74 0.644 0.743 0.501 0-574 0.644 0.501 0.574 0.501 0.525 0.617 .716 0.525 0.617 0.525  00 0.886 0.886 0.886 0-886 .906 0.906 0.906 0.906 0.906 0-906 0.906 0.906 0.906 0.906 0.876 0-876 .876 0.876 0.876 0.876  Dt 0.454 0.506 0.554 0.506 0.554 0.554 0.415 0.474 0.518 0.474 0.518 0.518  DJ 0.401 0.454 0. 5 0 6 0.401 0.454 0.401 0.371 0.415 0.474 0.371 0.415 0.371  DU 0.599 0.399 0.599 0.599 0.599 0.599 0.563 0.563 0.563 0.563 0.563 0.563  0.557 0.557 0.557 0.557 0.557  TUBE  RUh N O . 3 4 R E- 7 6 2 5 PRESSURE - 2323.MH.HG. C »- 5 2 1 6 . P P * kATER TEMP.-20.0 C, FLC* - 12.70 LB/MIN VEL.• 1.5926 FT/SEC TUBE HORIZONTAL* 5 / 6INCH INSIDE DIAMETER RUN 54101. 34102. 54103. 54 Itl4. 54201. 34202. 54203. 54204. 34205. 343C*. 34207. 54208. 54209. 54210. S43C1.  333. 333. 333. 333-  243. 245. 245. 245. 245. 245. S45. 245. 245. 245. 157. 157. 157. 157. 157. 1ST.  34302.  54303. 54304. 54305. 34306.  WATER T E P P . " 2 0 . C TUBE B O R U C N T A L . RUN 55101. 55102. 55103. 55202. 53203. 552C4. 53203. 53206. 352C7. 55208. 55209. 55210.  1 6 3 3 9 6 3 9 6 3 6 3 3  J 9 6 9 12 9 , 6 1« 12 9 14 12 14  C. 5/8  WATCH T E H TUBE H O I I C N T A L , RUN 56101. 56102. 56103. 561114. 36103. 56106. 56201. 36202. 562C3. 36204. 56205. 562C6.  3 3  J 12 9 6 12 9 12 12 9 6 1? 9 12  3/B  BGR 44190. 44190.  KL 1.464 1.558 1.511 1.722 a.oo 1.673 2.29 1.225 3.00 1.710 3.00 1.504 3.00 1.419 5.29 .467 tl.PQ 6.00 1.607 8.29 1.524 9.00 1.479 11.29 1.519 3.00 1.592 3.00 , 1.697 3.00 1.516 6.00 1.645 6.00 1.606 9.00 1.602  44190. 14461. 32512. 32512. 32512. 178260. 260100260100. 666040. 877837. 1732878. 20835. 20835. 20835. 166676. 166676. 562532.  PRESSURE  FIFI 0.090B 0.0779 C.1686 C.0831 0.0810 0.0629 0.1641 C.1641 0.1439 0.2450 0.2269 0.307B  LENGTH 3.00 3.00 6.00  353523.  -  FLCW - 8 . 4 7 L B / M I N INCH I N S I D E DIAMETER  NB 184. 184. 184. 112. 112. 112. 112. 112. 112. 112. 112. 112.  RUM N O  FIFI C.0813 0.0862 0.1675 0-0953 0.0644 0.0617 0.0860 C.0753 0.1262 0.14.77 .1613 C.2121 0.2229 0.2872 0.0693 0-0738 • 0656 0.1431 C.1396 0.2089  1065.MM.HG.  BGR 36612. 36612. 292895. 22283. 222B3. 22285. 122187. 178264. 17B2B4. 470243. 601708. 1187791.  3140. PRESSURE • FLOW .23 LB/MIN INCH INSIDE DIAHE1ER NB 119. 119. 119. 119. 119. 119. 97. 97. 97. 97. 97. 97.  FIFI 0.0704 C.0766 0.0B53 0.1470 0.1619 C.2322  c.oeso  0.0622 0.0B14 0.1431 0.1496 0.2264  BGR 38347. 38347. 38347. 306777. 906777. 1033372. 31258. 31258. 3123B. 250062. 25C062. 643959.  VEL.-  LENGTH 3.00 3.00 6.00 3.00 3.00 3.00 5.29 6.00 6.00 8.29 9.00 11.29  1. 0 6 2 1  Kl 1.170 1.009 1.090 1.255 1.226 0.955 1.404 1.240 1.090 1.339 1.145 1.237  96l.WM.HG. VEL." 0.6538  LENGTH 3.00 3.00 3.00 6.00 6.00 9.00 3.00 3.00 3.00 6.00 6.00 9.00  KL 0.552 0.604 0.676 0.576 0.640 0.611 0.694 0.522 0.6B6 0.60B 0.604 0.634  KL 1 1.348 1.462 1.405 1.615 1.496 1.112 1.579 1.417 1.279 1-346 1.498 1.368 1.369 1.395 1.435 1.559 1.420 1.497 1.489 1.471  L/Q« 3.0244 3.0244 0.0244 0.0244 0.0193 0.0193 0.O193 0.0193 0.0193 Q.Q193 .0193 0.0193 0.0193 0.0193 0.0115 0.0113 0.0115 0.0115 0.0115 0.0115  RUN 54101. 54102. 54103. 54104. 54201. 54202. 54203. 542C4. 54205. 54206. 54207. 54206. 542C9. 54210. 54301. 543C2, 54303. 54304. 54305. 54 306.  L/0» 0.0333 0.0333 0.0333 0.0200 0.0200 0.0200 0.0200 0.0200 0.0200 0.0200 0.0200 0.0240  .00358 0.00590 0.00358 0.00358  FJ C.01598 0.01220 0.01598 0-01267 C.01635 0.01483 0.01296 0.C0957 0.0163S 0.01483 .01296 0.01635 0.01483 0.01635 C.00930 C00791 0.0059C .00930 .00791 .00930  OGO 121.35 121.35 121.33 121 . .33 95.45 95.45 95.45 95.43 95.45 95.43 , 95.45 95.45 95.45 95.45 55.35 SS.35  55.33 55.35 35.35 55.35  8.75 8-75 11.89 11.89 11.69 11.89 11.89 11.89 11.89 11.69 11.69 11.69 18.55 18.55 18.55 18.35 18.55 18.55  RUN 55101. 55102. 55103. 552C2. 35203. 55204. 55205. 55206. 55207. 55208. 552C9. 55210.  Fl 0.01697 0.01151 0.01131 0.01363 0.C1C47 0.00739 0.01363 0.01047 0.0C739 0.01047 0.00739 0.00739  FJ 0.02206 0.01697 0.022C6 0.01604 0.01363 0.01047 C.01772 0.01604 0.01363 0.01772 0.01604 0.01772  OGO 10B.79 108.79 108.79 64.01 64.01 64.01 64.01 64.01 64.01 64.01 64.01 64.01  DELTA 10.56 10.56 10.56 17.34 17.34 17.34 17.34 17.34 17.34 17.34 17.34 17.34  VCF 0.959 0.983 0.970 0.927 0.955 0.980 .912 0.941 0.966 0.926 0.952 0.937  DI 0.845 0.930 0.930 0.726 0.832 0.914 .726 0.832 0.914 0.832 0.914 0.914  DJ 0.746 0.845 0.746 0.621 0.728 0.632 .516 0.621 0.728 0.516 0.621 0.516  00 1.041 1.041 1.041 1.029 1.029 1.029 .029 1.029 1.029 1.029 1.029 1.029  RUN 56101. 56102. 56109. 5.104. 51105. 56106. 36201. 56102. 36203. 3620*. 56205. 36206.  Fl 0.02566 0.02082 0.0139S 0.02082 0.01398 0.01398 0.02012 0.01678 0.01146 0.0167B 0.01146 0.011*6  FJ 0.02565 0.02586 0.02082 0.02965 0.02586 0.02965 0.02371 0.02012 0.0167B 0.02971 0.02012 0.02371  OGO B3.46 85.46 B5.46 83.46 B5.46 A3.46 64.87 64.87 64.87 64.87 6 4 . 67 64.B7  DELTA 10.08 10.OB 10.08 10.08 10.08 1 0 . OB 12.36 12.36 12.36 12.36 12.36 12.36  VCF 0.983 1.010 1.041 0.997 1.027 1.013 0.970 1.000 1.029 0.983 1.017 1.000  DI 0.808 0.890 0.981 0.890 0.981 0.961 0.784 0.856 0.949 0.856 0.949 0.949  DJ 0.734 0.608 0.B90 0.734 O.BOB 0.734 0.690 0.784 0.B56 0.690 0.7B4 0.990  DO 1.111 1.111 1.111 1.111 1.111 1.111 1.085 1.083 1.085 1.085 1.085 1.0BJ  Fl  0.01220 0.0O709 C.C0709 0-00709 .01483 .01296 0.00937 0.00574 0.01296 C.CC937 .00 5 74 0.00957 0.00574 C.00574 0.00791  Q.QQ»Q  FT/SEC  KL1 1.123 0.992 1.058 1.163 1.171 0.936 1.261 1.167 1.053 1.241 1.090 1.160  C • 213*.PPM FT/SEC  KL 1 0.543 0.610 0.703 0.376 0.637 0.619 0.674 0.522 0.706 0.398 0.614 0.634  L/Q" 0.0409 0.0409 0.0409 0.0409 0.0409 0.0409 0.0313 0.0313 0.0313 0.0313 0.0313 0.0313  151  RUN N O . 57 RE • 1 1 6 4 0 . PRESSURE • 2321.UN.HG. WATER T E M P . - 2 0 . C C . FLOW • 1 9 . 4 0 L B / M I N VEL.• 2.4328 TUBE H O R I 2 0 N T A L . 9 / 8 INCH I N S I O E O I A M E T E R  RUN 97101. 97102. 97103. 97104. 971C9. 97106. 57107. 97108. 371C9. 57110. 97201. 97202. 972C3. 97204. 97203. 972C6.  I 12 9 6 3 9 6 3 6 3 3 9 6 3 6 3 3  J 14 12 9 6 14 12 9 14 12 14 14 9 6 14 9 14  NB 348. 348. 346. 348. 346. 348. 348. 348. 348. 348. 298. 298. 298. 298. 298. 298.  F IF I C.0422 0.0488 0.0612 C.0963 C.0909' 0.1100 C.U79 0.1921 C.1663 0.2084 C.0799 0.0902 0.0909 0.1100 0.1010 0.1809  8GR 13446. 30232. 30232. 30232. 169796. 241899. 241899. 637916. 816261. 1611323. 122688. 22413. 22413. 472939. 179306. 1194603.  LENGTH 2.29 3.00 3.00 3.00 3.29 6.00 6.00 6.29 9.00 11.29 5.29 3.00 3.00 8.29 6.00 11.29  KL 1.717 1.919 1.910 1.762 1.604 1.719 1.836 1.719 1.730 1.728 1.592 1.721 1.748 1.613 1.739 1.649  RUIN N O . 58 RE • 1 6 6 1 1 . PRESSURE 2265.MM.HG. • ATER T E M P . - 2 0 . 0 C » FLCW • 2 8 . 0 0 L B / M I N VEL.3.5112 T U B E H C R I 2 C N T A L * 9 / 8 INCk INSIOE DIAMETER RUN 36101. 98102. 961C3. 58104. 96109. 96106. 962C1. 96204. 96209. 5!2J16j_  1 9 6 3  I  3 3 9 6 3 3  J 14 9 6 14 9 14 14 14 9 14  NB 374. 374. 374. 374. 374. 374. 240. 240. 240. 240.  FIFI C.0810 0. . 0 4 4 8 C.0372 0,. 1 2 4 7 C.C620 0 .1629 0,. 0 6 3 9 0.. 1 0 2 9 c. 0 6 6 6 0,L 1 A 2 J L  BGR 123423. 22911. 22311. 474006. 160091. 1199828. 79203. 304818. 113966. 769943.  LENG7H 3.29 3.00 3.00 8.79 6.00 11.29 9.29 6.29 6.00 11.79  C> • 9212.PPM FT/SfcC  Kll 1.630 1.493 1.844 1.719 1.929 1.649 1.782 1.643 1.672 1.663 1.463 1.641 1.683 1.528 1.662 1.969  L/Q» 0.0214 0.0214 0.0214 0.0214 0.0214 0.0214 0.0214 0.0214 0.0214 0.0214 0.0123 0.0121 0.0123 0.0123 0.0123 0.0123  RUN 46109. 46106. 46107. 46108. 46109. 46201. 46202. 46203. 46204. 46205. 46301. 46302. 46303. 46304. 46309.  NB 1064. 1064. 1064. 1064. 1064. 814. 814. 814. 814. 814. 632. 632. 632. 632. 632.  RUN N O . 47 RE . WA7ER 7 E M P . - 2 0 . 0 C , TUBE V E R T I C A L . 5 / 1 6 RUN 67101. 47102. 47103. 47104. 47103. 47201. 47202. 47203. 47204. 47209. 47301. 4 7302. 47303.  4 4 2 6 2 2  10 6 4 10 6 10  FIFI 0.0339 0.0403 0.0797 0.0756 0.1919 0.0349 0.03S1 0.0644 0.0699 0.1343 0.0316 0.0289 0.0946 0.0601 0.1147  KL 2.279 2.223 1.890 2.256 2.037 2.140 2.126 2.134 1.909 2.011  FIF) 0.0330 0.0412 0.0663 0.0763 0.1426 0.0342 0.0349 0.0696 0.0687 0.1384 0.0614  780. 780.  0.0639 0.1234  FIF) 0.0393 0.0S60 0.1137 0.1112 0.2246 0 . 1689 0. 0307 0. 0437 0 . 0917 0. 0944 0 . 1860  LENGTH 2.00 2.00 4.00 4.00 . 8.00 2.00 2.00 4.00 4.00 8.00 2.00 2.00 4.00  KLl 2,300 2.237 1.883 2..2S ^ 2.070 2.178 2.144 2.195 1.936 2.033  L/Q* 0.0197 0.0197 0.0197 0.0197 0.0197 0.0197 0.0109 0.0109 0.0109 Q,QL0_9_  BGR 61205. 61203. 489640. 489640. 1692934. 36744. 36744. 293939. 293933. 2331636.  KLl 3.107 3.922 3.310 3.314  _J.312  1.3J0 3.351 1.063 1.341 3.202 3.278 2.958 2.813 3.UB 2.965  I/O* 0.0169 0.0169 0.0169 0.0169 0.0169 0.0112 0.0112 0.0112 0.0112 0.0112 0.007B 0.0078 0.0078 0.0378 0.0078  KL 2.866 3.398 2.728 3.142 2.933 3.132 3.176 3.202 3.164 3.163 3.139  KLl 3.438 4.039 3.233 3.749 3.491 3.738 3.701 3.764 3.759 3.762 3.690  L/0" 0.0126 0.0126 0.0126 0.0126 0.0126 0.0079 0.0073 .0079 .0079 .0079 • 0047  .00  3.269 3.224  3.893 3.791  0.0047 0.0047  LENGTH 2.00 2.00 4.00 4.00 6.00 2.00 2.00 4.00 4.00 6.00 6 . 0 0 2  FJ C.01569 0.01425 0.01231 C.00941 0.01569 0.01425 C.01231 C.01569 C.01425 0.01569 0.00944 0.00753 C.00598 0.00944 0.00753 0.00944  OGO 150.29 150.29 150.29 190.29 190.29 150.29 150.29 150.29 150.29 150.29 84.07 84.07 84.07 84.07 64.07 64.07  CELTA 12.78 12.78 12.78 12.76 12.78 12.76 12.76 12.76 12.78 12.76 17.24 17.24 17.24 IT.24 17.24 17.24  VCF 0.949 0.996 1.965 1.976 0.953 0.961 1.970 1.958 0.966 0.963 0.943 0.954 .963 1.947 0.958 0.952  .686 .74 3 0.615 0.881 0.743 0.819 0.881 _0.B15 0.681 0.861 0.662 0.72 7 0.793 0.727  OJ 0.637 0.666 .743 0.815 0.637 0.686 0.743 _ 0 . 6 37 0.686 0.637 0.558 0.662 0 . 727 0.558 0.662 0.558  0.936 _0-_?_3ft_ 0.938 0.938 0.936 0.936 0.938 0.938 0.936 0.936 0.854 0.654 0.854 0.854 0.654 0.654  FI 0.01075 O.0C87C C.00678 0.00870 0.0067B 0.00678 0.00613 n.004«fl C.CC39G C.00396  FJ 0.01376 0.01075 0.0C87C 0.01376 C.01079 0.01376 O.CC775 C.00775 0.00613 0.00775  QGO 160.16 180.16 160.16 160.18 160.18 180.18 99.80 99.60 99.80 99.80  RUN 46109. 46106. 46107. 46108. 46109. 46201. 46202. 46203. 46204. 46209. 46301. 46302. 46303. 46304. 46903.  FI 0.00772 0.00386 0.00918 0.00586 0.00986 0.00527 0.00406 0.00634 0.00406 0.00406 0.00383 0.00310 0.00439 0.00310 0.00310  FJ 0.00918 0.00772 0.01175 0.00918 0.01173 0.00694 0.00927 0.00799 0.00634 0.00799 0.00499 0.00383 0.00963 0.00439 0.00363  RUN 47101. 47102. 47103. 47104. 47105. 47201. 47202. 47203. 47204. 47209. 47301. 47302. 47301.  0.00474 0.00737 0.00474 0.00474 0.00399 0.00314 0.00472 0.00314 0.00114 0.00313 0.00212 0.00212  RLN 981C1. 98102. 98103. 981C4. 96109. 96106. 98201. 9A2C4. 98209. 98206.  CELTA 17.17 17.17 17.17 17.17 17.17 17.17 26.75 26.75 26.75 26.75  VCF 1.011 1.015 1.018 1.012 1.016 1.014 1.008 1.010 1.014 1.011  01 0.814 0.872 0 . 920 0.B72 0.920 0.920 0.769 0.825 0.869 0.869  OJ 0.709 0.814 0.672 0.709 0.614 0.709 0.670 0.670 0.769 0.670  DO 0.973 0.973 0.973 0.973 0.973 0.973 0.926 0.926 0.926 0.926  OGO 78.93 78.93 78.93 78.93 78.93 90.85 90.89 90.09 90.89 90.89 39.60 39.60 39.60 35.60 13.60  DELTA 10.96 10.96 10.96 10.96 10.96 14.33 14.33 14.33 14.33 14.33 18.46 16.46 18.46 16.46 ' 18.46  VCF 1.191 1.187 1.191 1.168 1.190 1.192 1.190 1.189 1.191 1.190 1.191 1.192 1.189 1.191 1.189  01 0.446 0.479 0.421 0.479 0.479 0.422 0.449 0.999 0.449 0.449 0.401 0.429 0.379 0.429 0.429  OJ 0.421 0.446 0.968 0.421 0.368 0.393 0.422 0.345 0.395 0.345 0.375 0.401 0.329 0.375 0.329  OD 0.322 0.922 0.522 0.522 0.522 0.493 0.493 0.493 0.493 0.493 0.476 0.476 0.476 0.476 0.476  FJ 0.00737 0.00623 0.00912 0.0073T 0.00912 0.00472 0.00399 0.00388 0.00472 0.00588 0.00383  UGO 69.26 69.28 69.28 69.28 69.28 39.82 39.82 39.62 39.82 39.62 24.37  DELTA 9.06 9.06 9.06 9.06 9.06 12.73 12.75 12.75 12.79 12.79 17.89  VCF 1.194 1.169 1.193 1.189 1.186 1.191 1.175 1.188 1.182 .168  Dl 0.371 0.396 0.348 0.398 0.398 0.341 0.367 0.316 0.367 0.367 0.293  OJ 0.348 0.171 0.304 0.346 0.304 0.316 0.341 0.264 0.316 0.264 0.242  00 0.442 0.442 0.442 0.442 0.442 0.411 0.411 0.411 0.411 0.411 0.391  0.00189  24.37  17.89  1.176  0.346  0.242  0.391  TUBE  C. . 4940.PPM FT/SEC  LENGTH 2.00 2.00 4.00 4.00 B.00 2.00 2.00 4.00 4.00 8.00 4.00  2271.MM.HG. VEL." 4.6739  FI 0.01429 C01231 C.00941 0.00623 0.01231 0.00941 C.00623 0.00941 0.00623 C.00623 C.00753 0.00556 0.004C9 0.00996  C» • 5199.PPM FT/SEC  KL 2.610 2.966 2.779 2.789 2_.784_ 2.794 2.819 2.977 2.804 2.691 2.792 2.482 2.373 2.617 2.493  2201.NM.HG. VEL.. 7.6269  BGR 90427. 90427. 403417. 403417. 3227339. 39623. 39623. 286986. 286984. 2292668. 204326. 204328. 1634626.  11222. PRESSURE FLOW • 9 . 3 3 LB/MIN INCH INSIOE DIAMETER  N8 1146. 1146. 1146. 1146. 1146. 1146. 688. 688. 688. 688. 688.  2289.MM.HG. V E L . « 6.3763  BGR 41672. 41672. 313377. 333377. 2667013. 31681. 31881. 295046. 299046. 2040369. 24793. 24733. 196021. 198021. 1964166.  18303. PRESSURE FLOW - 1 5 . 2 3 L B / N I N INCH INSIOE DIAMETER  NB 1940. 1940. 1940. 1940. 1940. 1094. 1094. 1094. 1094. 1094. 780.  HUN N O . 48 RE • WATER T E M P . " 2 0 . 0 C , TUBE V E R T I C A L . 9 / 1 6 RUN 4B101. 48102. 48101. 48104. 48103. 48106. 48201. 48202. 48203. 48204. 46209. 48206. "  15303. PRESSURE • FLOW • 1 2 . 7 3 L t t / H I N INCH INSIOE DIAMETER  97103. 97104. 57109. 97106. 571C7. 57106. 57109. 57110. 57201. 972C2. 97203. 57204. 572C9. 97206.  C* 5085.PPM FT/SEC  VERTICAL. S/16 RUN N O . 46 RE • WATER T E M P . . 2 0 . 0 C . TUBE V E R T I C A L . 9 / 1 6  RLN 97101.  FI  C» • 5099.PPM FT/SEC  KL 2.629 2.666 2.693 2.546  KLl 3.089 3.137 1.132 3.123  L/Q* 0.0222 0.0222 0.0222 0.0222  2.670 2.843 2.492 2.569 2.649 2.607 . 6 3 8 1 .  3,. 0 9 6 3.. 3 3 0 2..897 2,. 9 2 9 3,.114 3,.021 0 6 2  0.0222 0.0124 0.0124 0.0124 0.0124 0.0124 0.0124  RUN 48101. 46102. 46103. 46104. 48109. 48106. 48201. 48202. 48201. 48204. 46205. 46206.  FI 0.01063 0.00736 0.01120 0.00796 0.00796 0.01063 0.00618 0.00459 0.00771 0.00439 0.00499 0.00618  FJ 0.01320 0.01063 0.01723 O.01320 0.01723 0.01721 0.00773 0.00616 0.00984 0.00771 0.00984 0.009B4  OGO 80.67  ao.sy  80.67 60.67 80.67 60.67 44.32 44.12 44.32 44.32 44.32 44.32  DELTA 7.47 7.47 7.47 7.47 7.47 7.47 12.43 12.43 12.43 12.43 12.43 12.43  VCF 1.176 1.164 1.132 1.160 1.166 1.160 1.171 1.161 1.142 1.176 1.199 1.192  01 0.424 0.459 0.390 0.459 0.459 0.424 0.409 0.441 0.372 0.441 0.441 0.409  DJ 0.. 3 9 0 0.. 4 2 4 0.. 3 2 0 0.. 3 9 0 0.. 3 2 0 0.. 3 2 0 0.. 3 7 2 0.. 4 0 9 0.. 3 0 5 D. . 3 7 2 0.. 3 0 5 0., 3 0 5  DO 0.513 0.513 0.513 0.513 0.511 0.513 0.498 0.498 0.498 0.498 0.498 0.498  152  RUN N O . 49 RE . 8149. PRESSURE WATER T E M P . - 2 0 . 0 C . FLOW • 6 . 7 9 LB/HlN T U B E V E R T I C A L , 5 / 1 6 I N C H I N S I O E 0 1 A M E TER  2299.MH.HG. VEL.B 3.3937  5 0 R E 5 2 " * 5 ' PRESSURE WATER T E M P . - 2 0 . 0 C , FLOW 4.37 LB/HlN TUBE V E R T I C A L . 5 / 1 6 I N C H I N S I D E D I A M E T E R  RUN 50101. 50102. 50103. 50104. 50105. 50201. 50202. 5020).  Nfl 740. 740. 740. 740. 740. 429. 429. 429.  2 2  FIFI 0.0913 0.1042 0.1955 0.2018 •0.293D .0837 .0797 .1634  RUN 51101. 51102. 51103. 51104. SU03. 31201. 51202. 51204.  NB 462. 462. 462. 462. 462. 268. 268. 266.  FIFI 0.1367 0.1243 0.3084 0.2609 .4446 0.0980 0.1234 0.2213  948.MM.HG. C. 2 1 0 5 PPH V E L . - 2 1854 F l / W r T / " C  BGR 64560. 64560. 676479. 676479. 2263115.  LENGTH 2.00 2.00  49022. 49022. 392173.  RUN N O . 31 KE • 3181. PRESSURE WATER T E M P . - 2 0 . 0 C , FLOW 2 . 6 5 LB/MIN TUBE V E R T I C A L . 5 / 1 6 INCH I N S I D E DIAMETER  C* 5162.PPH FT/SEC  KL 1.632 2.100 1.966 2.016 1.955 2.003 1.913 1.959  934.MH.HG. VEL.1.3253  BGR 6705B. 87058. 696466. 6964A6, 2350572. 50501. 50501. 404011.  LENGTH 2.00 2.00 4.00 4.00 6.00 2.00 2.00  KL 1 2.249 2.706 2.476 2.156 2.187 2.361 2.408 2.384  RUN 34105. 34106. 34108. 34204. 34306. 34307. ' 34306. 34 3 0 9 . 34310.  591. 591. 391. 316. 360. 380. 380. 380. 360.  34404. 34405. 34406. 34407. 34408. 34409.  3p-Qt 258. 258. 258. 256. 258. 259,  mi*-  FIFI 0.0722 0.0640 0.1362 0-0555 0.0366 0.0530 0.0538 0.0896 0.1067 C.1433  KL 1.653 1.515 1.852 1.564 786 1.411 1.784 1.597  KL 1 2.358 2.215 2.118 2.267 2.1.8 1.934 2.596 2.265  0.0374 0.0460 0.0483 0.0834 0.0943 - 0.1317  2322.HM.HG. VEL.- 2.1316  BGR 58590. 58590. 468723. 31526. 20219. 37672. 37672. 224376. 301378. 83B257. 13728. 25578. 2S57B. 152339. 204620. __6__U?*__  RUN 35104. 35105. 35106. 35107. 33108. 35109. 33205. 35206. 35208. 35303. 35306. 35308.  J 11 9 6 11 9 11  NB 357. 337. 357. 357. 357. 357. 714. 714. 423. 423. 423.  FIFI 0.0318 0.0334 0.0370 0.0652 0.0704 0.1022 0.0427 0.0507 0.0934 0.0371 0.0409 0.0779  !w.l! ; -» ...  BCR 12713. 23688. 23668. 141083. 189501. 527080. 47375. 47375. 379002. 28067. 28067. 224535.  0 4 * " 6 , PRESSURE c WATER T E M P . . 2 0 . 0 C , FLCW • 1 1 . 0 2 L B / P I N TUBE V E R T I C A L . 5 / 8 INCH I N S I D E D I A M E T E R  RUN 36106. 36203. 36306. 36307. 36309. 36407. 3640B. 36410.  1 3 3 3  6  3 3 6 3  J  6  6  6 11 11 6 11 11  NB 566. 322. 199. 199. 199. 336. 336. 336.  FIFI 0.1135 0.C968 0.C654 0.1120 0.1773 0.0784 0.1277 0.2060  L/U. 0.0442 0.0442 0.0442 0.0442 0.0442 0.0230 0.0230 0.0230  5/8  KL 1.577 1.401 1.489 1.488 1.142 1.344 1.368 1.254 1.356 W294  KL 1 2.156 1.960 2.058 2.0*3 1.620 1.912 1.939 1.781 1.925 W837  L/Q* 0.0179 0.0179 0.0179 0.0111 0.0205 0.0205 0.0205 0.0205 0.O205 0.0205  2. 3.00 3.00 5.44 6.00 8.44  1.323 1.323 1.390 1.323 1.357 _L_347  1.860 1.877 1.978 1.869 1.927 1.908  0.0126 0.0126 0.0126 0.0126 0.0126 0.0126_  LENGTH 2.44 3.00 3.00 S.44 6.00 6.44  KL 1.725 1.474 1.633 1.586 .553 1.603 1.494 1.776 1.635 1.542 1.701 1.622  F| 0.01785 0.01208 0.01208 0.02162 0.01785 0.01012 0.00748 0.00748  FJ 0.02162 0.01785 .02162 .02655 .02655 .01213 .01012 .01213  UGO 46.65 *8.65 46.65 48.65 48.65 24.03 24.03 24.03  DELTA~ 5.40 5.40 5.40 5.40 5.40 .32 .32 .32  VCF 1.228 1.289 1.260 1.069 .119  win  DI 0.369 0.426 0.426 0.319 0.369 0.341  1.259 1.217  • J 0.319 0.369 0.319 0.209 0.209 0.286 0 . 341 0.286  00 0.501 0.501 0.501 0.501 0.501 0 . 4 75 0.475 0.475  RUN 51101. 51102. 51103. 51104. 51105. 51201. 51202. 51204.  F l " 1.02810 .01906 .03513 .01906 .02810 .01622 '.01087 .01067  FJ 0.03515 0.02810 0.04288 0.03515 0.04288 0.01910 0.01622 0.01910  OGO 47.50 47.50 47.50 47.50 47.50 24.16 2 4 . 16 24.16  DELTA 5.25  VCF 1.426 1.463 1.143 1.444 1.211 .371 .455 .418  DI 0.420 0.487 0 . 346 0.487 _0.j»20_ 6.377 0.456 0.458  RUN 34105. 34106. 34108. 34204. 34306. 34307. 34308. 3*i?09,  Fl 0.01089 0.00830 0.00830 0.C0744 0.01214 0.00979 0.00699 0.00979  0.01316 0-01089 .01318 .00866 .01354 .01214 0.00979 •01?S4  QGO 97.32 9 T t » 97.32 54.68 126.45 126.45 126.45 126.45  DELTA 6.60 6.60 6.60 12.26 10.26 10.26 10.26  VCF 1.367 1.396 1.362 1.373 1.419 1.422 1.417  DI .546 .606.. 0.606 0.553 0.671 0.729 .786 .729  34310. .34311. 34404. 34405. 34406. 34407. 34408. 34409.  0.00699 0.00699 0.00619 0.00664 0.00315 0.00684 0.00515 0.00515  0.01214 0.01354 0.00911 0.00619 0.00684 0.00911 0.00819 0.00911  126.43 126.45 71.32 71.32 71.32 71.32 71.32 71.32  10.26 10.26 15.11 15.11 15.11 15.11 15.11 15.11  1.420 1.419 1.406 1.418 1.423 1.413 1.421 1.416  0.768 0.768 0.619 0.676 0 . 7 36 0.676 0.736 0.736  DJ 0.346 0.420 0.182 0._«tb 3.182 0 . J13 0.377 3.313  DO 0.5B1 0.581 0.581 0.581 0.581 0.557 0.557 0.557  TUBE  C* 5214.PPM FT/SEC  3.00 6.00 3.00 2.44 3.00 3.00 5.44 6.00 8.44  V!.. " " E " 1 5 " ° PRESSURE • 2319.HM.HG. WATER T E M P . . 2 0 . 0 C , flOW . 2 5 . 4 0 LB/MIN VEL."" 3 . 1 8 5 2 IU6E V f R I I C A L , 5 / 6 INCH I N S I O E DIAMETER  RUN 50101. 50102. 50103. 50104. 50105. 50201. 50202. 50203.  C* 2073.PPM FT/SEC  VERTICAL. RUK N O . 34 RE « 1 0 2 0 7 . PRESSURE WATER T E M P . - 2 0 . 0 C . FLOW - 1 7 . 0 0 L B / M I N TUBE V E R T I C A L , 5 / 8 INCH I N S I D t DIAMETER  L/Q* 0.0285 0.0285 0.0285 .0285 .02B5 0.0150 0.0150 0.0150  OJ 0.478  _P_- 5 4 6 0.478 0.489 0.631 0.671 0.729 .631  0.671 0.631 0.572 0.619 0.676 0.572 0.619 0.572  DO 0.680 _ .0.680 0 . 6 BO 0.690 0.860 0.860 0.860 0.660 0.860 0.808 0.808.  o.soe  0.808 0.606 0.808  C. 5207.PPM PPM FT/SEC  KL 1 2.284 1.959 2.173 2.105 2.066 2.129 1.917 2.326 2.132 2.014 2.241 2.127  L/0» 0.0117 0.0117 0.0117 0.0117 0.0117 0.0117 0.0117 0.0117 0.0117 0.0084 0.0084 0.00B4  RUN 35104. 35103. 35106. 35107. 35108. 35109.  FI 0.0D636 0.00551 0.00419 0.00331 0.00419 0.00419  F J 0.00744 0.00656 0.00SS1 0.00744 0.00656 0.00744  35205. 35206. 35208. 35305. 35306. 35)08.  0.00622 0.00450 0.00450 0.00448 0.00336 0.00338  0.00744 0.00622 0.00744 O.C0532 0.00448 0.00532  OCD 99.71 99.71 99.71 99.71 99.71 99.71 102.37 102.37 102.37 69.29 69.29 69.29  DELTA 16.31 16.31 16.31 16.31 16.31 16.31 8.16 B.16 8.16 11.77 13.77 13.77  VCF 1.324 1.329 1.331 1.327 1.330 .328 1.297 1.310 1.304 1.306 1.317 1.312  01 0.665 0.707 0.754 0.707 0.754 0.754  DJ 0.624 0.665 0.7U7 0.624 0.665 0.624  0.541 0.592 0.592 0.575 0.624 0.624  0.498 0.541 0.496 0.630 0.575 0.530  DO  tt.Sir  0.811 0.811 0.811 0.811 0.811 0.650 0.650 0.650 0.679 0.679 0.679  2332.MM.HG. C- 5237.PPM VEL - 1 3 8 1 9 f t / « i " / S E C  BGR 89620. 49245. 30434. 1B1265, 677195. 51386. 306055. 1143405.  LENGTH 3.00 3.00 3.00  5.44  8.44 3.00 5.44 8.44  KL 1.401 1.451 1.151 1 .086 L.109 1.167 1.044 1.088  KL 1 2.193 2.267 1.617  L/0" 0.0240 0.0132 0.0128  1.742 1.833 1.646 1.712  0.0128 0.0128 0.0253 0.0253 0.0253  RUN 36106. 36203; 36306. 36307. 36309. 36407. 36408. 36410.  Fl 0.01070 0.00622 0.00553 0.00770 0.00553 0.01005 0.01433 0.01005  FJ 0.01603 0.00916 0.00770 0.01032 0.01032 0.01433 0.01944 0.01944  OGO 97.56 50.37 50.97 50.97 50.97 102.47 102.47 102.47  CELTA 4.31 7.85 12.70 12.70 12.70 7.52 7.52 7.52  VCF 1.565 1.562 1.5T9 1.566 1.370 1.571 1.576 1.574  DI 0.587 0.577 0.6B7 0.610 0.667 0.740 0.663 0.740  DJ 0.495 0.481 0.610 .0.479 0 . 4 79 0.663 0.536 0.338  00 0.683 0.669 0.788 0.7S8 0.768 0.835 0.835 0.835  153  HUM N O . 37 RE 4071. PRESSURE h A T E R T E M P . - 2 0 . 0 Ct FLOW • 6.78 LB/MIN TUBE V E R T I C A L . 5 / 8 I N C H I N S I D E D I A M E T E R  RUN 37104. 37105. 37106. 37204. 37205. 37206.  I 6 3 3 6 3 3  J 9 6 9 9 6 9  NB 309. 309. 309. 152. 152. 152.  FIFI 0.1424 0.1294 0 . 2 717 0.1235 0.1089 0.2322  1551.MM,HGVEL." 0.8502  BGR 76810. 76810. 614478. 37783. 37783. 302267.  RUM N O . 38 RE • 1927. PRESSURE WATER T E M P . - 2 0 . C C * FLOW 3.21 LB/MIN TUBE V E R T I C A L * 5 / 8 I N C H I N S I D E O I A M E T E R  RUN 38104. 38105. 38106. 36205.  (b)  I 6 3 3 1  J 9 6 9 6  RESULTS  BGR 129157. 129157. 1033255. 63003.  F(F) 0.2218 0.2246 0.4457 0.2066  NB 246. 246. 246. 120.  INCLUDING  THE  LENGTH 3.00 3.00 6.00 3.00 3.00 6.00  O « 3469.PPM FT/SEC  KL 1.140 1.043 1.092 1.245 1.100 1.172  918.MH.HG. VEL.• 0.4025  LENGTH 3.00 3.00 6.00 3.00  FEED  1 1 1 1 t 1 1 1 1 1 1  J 10 6 10 6 10 6 10 6 10 6  NB 1076. 10 7 6 . 864. 888. 528. 328. 355. 135. 256. 256.  FIFI 0.4492 0.2600 0.4062 0.2433 0.3443 C.2036 0.2936 0.1944 0.2603 0.1676  KlJh N C . 10 RE 7021. M&TER T E H P . - 2 0 . 0 C . FLOW TUBE H O R I Z O N T A L , 5 / 1 6 I N C H RUN 10101. 10201. 10202. ..10*04.  I 1 1 1 I  J 6 10 6 10  NB 1056. 684. 684. 684.  RUN NO. 15 RE » WATER T E M P . - 2 0 . 0 C * TUBE H C R I 2 C N T A L * 5 / 1 6 RUN 13101. 15103. 15201'. 15301,_ 15302. 15.401. 15402. 15501. 13302.  J 10 6 6 J.O.. 6 10 6 10 6  7021, FLOW INCH  NS ICOO. 1000. ICOO. 6B8. 688. 442. 462. 218. 278.  RUN N O . 27 RE • 7513. WATER T E M P . - 2 0 . 0 C , FLOW TUBE H O R I Z O N T A L , 5 / 1 6 INCH  RUN 27101. 27102. 27101. 2.7.107, 27201. 77202. 27203. 2 7204. 27101. 27302. 27303. 27304. 27401. 27402. 27403.  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  J 10 6 2 4 10 6 4 2 10 6 4 2 6 4 2  NB 1030. 1030. 1030. 1Q0S. 616. 816. 816. 816. 574. 574. 574. 574. 274. 274. 274.  BGR 12076554. 2696719. 9697158. 2225545. 5926041. 1323297. 1984365. 889717. 2873232. 641598.  LENGTH 10.17 6 . 17 10.17 6.17 10.17 6.17 10.17 6 . 17 10.17 6.17  KL1 2.137 1.998 2.067 2.379  BGR LENGTH 2646594. 6.17 IC.t 7 7676917. 1714271. 6.17 76 7 6 9 1 , 7 . . . - I 0 . l _ 7  BGR 11223563. 2506244. 2506244. 7721811. 1724296. 5183286. 1157885. 3120150. 696716.  LENGTH 10.17 6.17 6.17 10.17 6.17 10.17 6.17 10.17 6.17  KL 2.110 2.2)9 2.199 2._)81  KL 2.263 2.216 2.218 2.1.83 2.207 2.316 2.340 2.117 2.217  PRESSURE • 923.KM.HG. » 6.26 LB/PIN VEL.3.1306 INSIOE OIAMETER  FIF 1 0.4217 0.2393 0.0701 0.1487 0.3668 0.2167 0.1461 0.0649 C.3S42 0.2046 0.1336 0.0640 0.1726 0.1044 0.0536  8GR 10603127. 2412360. 104946. 726647. 855S594. 1911151. 589994. 63142. 6020384. 1344364. 415020. 58484. 64I73S. 198111. 27918.  LENGTH 10.17 6.17 2.17 4.17 10.17 6.17 4.17 2.17 10.17 6.17 4.17 2.17 6.17 4.17 2.17  FI 0.02308 0.01492 0.01492 0.01048 0.00698 0.00698  FJ 0.02917 0 . 0 2 308 0.C2917 0.01294 0.01048 0.01294  KL 2.423 2.275 1.902 2.112 2.270 2.216 2.713 1.894 2.455 2.341 2.265 2.068 2.514 2.250 2.223  QGQ 96.01 96.01 96.01 37.74 37.74 37.74  10.23 10.23 10.23  VCF 1.892 1.875 1.864 1.883 1.890 1.886  DI 0.605"' 0.715 0.715 0.534 0.653 0.653  0.463 0.605 0.485 0.400 0.534 0.400  00 0 8 .4 0 0 8 .< t 0 0 8 .4 0 0 7 .6 0 0 7 .8 0 0. 780  L/U* 0.0520 6.0520 0.0520 0.0224  RUN 38104. 38105. 38106. 36205.  Fl 0.04213 0.02839 0.02839 0.01292  FJ 0., 0 4 9 0 4 0,.04213 0..04904 0..01944  QGU 68.49 68.49 68.49 28.75  DELTA 2.99 2.99 2.99 6.13  VCF 3.013 7.757 2.884 2.637  DI 0.470 0.627 0.627 0.588  DJ 0.316 0.470 0.316 0.401  DD 0.810 0.810 0.810 '0.771  LENGTH  C* • 4015.PPM FT/SEC  KL 2.303 2.203 2.237 2.192 2.225 2.171 2.161 2.359 2.111 2.265  PRESSURE 919.MM.HG. 5 . 8 5 LB/MIN VEL.2.9256 INSIDE OIAMETER FIFI C.4276 0.2557 0.2536 0.1658 C.221B 0 . 3409 0.2088 0.2887 0.1675  RUN 37104. 37105. 37106. 37204. 37205. 37206.  NOZZLE a TUBE E N T R A N C E HORIZONTAL, 8/16 TUBE  PRESSURE • 2279.MM.HG. 5 . 8 5 LB/MIN VEL.• 2.9236 I N S I O E OIAMETER FIFI 6.2463 0.3772 0.224) g.40_lj  L/Q» 0.0356 0.0356 0.0356 0.0147 0.0147 0.0147  C- • 2037.PPM FT/SEC  KL 0.709 0.723 0.717 0.839  RUN N O . T RE • 7021. PRESSURE • 1792.MP..HC. WATER T E M P . - 2 0 . 0 C , PLOW • 5.85 LB/MIN VEL." 2.9256 1UBE H O R I Z O N T A L , 5 / 1 6 I N C H I N S I D E C I A M E T E R RUN 7101. 7102. 7201. 7202, 7301. 7302. 7401. 7*02. 7601. 7602.  KL1 2.158 1.956 2.057 2.343 2.079 2.211  KL 1 2.294 2.238 2.236 2.210 2.221 2.209 2.154 2.388 2.121 2.291  L/U« 0.0251 0.0251 0.0211 0.0211 0.0131 0.0131 0.00H6 O.DU06 0.0060 0.0060  RUN 7101. 7102. 7201. 7202. 7101. 7302. 7401. 7402. 7601. 7602.  0.00175 0.00175 0.00153 0.00153 0.00132 0.00132 0.00110 0.00110 0.C0077 0.00077  0.0224 7 0.01710 0.01861 C O 1436 C.01162 0.00859 0.C0T67 0.00636 0.00537 0.00438  OGO 57.74 57.74 48.47 46.47 2 9 . U 29.11 16.65 18.63 „ 13.06 13.06  DELTA 4.97 4.97 6.19 6.03 10.13 10.13 15.OT 15.07 20.90 20.90  VCF 0.996 1.666 1.017 0.999 1.018 0.997 1.012 0.995 1.012  DI 0.46B 0.466 6.475" 0.471 0.472 0.472 0.46* _0i465 0.460" 0.460  DJ 0.226 0.32B 6.239" 0.33U 0.236 O.J32 0..34 0.312 6.232" 0 . 113  DO 0.466 0.466_ 6 . 4 75 0.471 0 . 4 ti 0.472 0.465 0.465 "0.450 0.460  C* « 5117.PPM FT/SEC  KL1 2.190 2.252 2.253 _2.388  L/C* 0 . 0 3 33 0.0186 0.0186 0.0186  RUN 10101. 10201. 10202. 10204.  Fl 0.00167 0.00144 C.00144 0.00144  FJ 0.02C29 0.O162C .01238 .01660  OGO 7B.34  RUN 15101. 13103. 15201. 15301. 15302. 15401. 13402. 15501. 13502.  FI 0.00160 0.00173 0.00173 C.00194 0.00194 0.00173 0.00171 0.00173 0.00173  F J 0.02287 C.01737 0.91746 0.01662 0.01295 0.CI1C1 0.00885 0.00699 0.00565  OGO 60.48 60.48 60.48 42.38 42.38 25.65 25.65 14.39 14.39  RUN 27101. 27102. 21103. 27107. 27201. 2 7202. 27203. 27204. 27301. 21302. 27303. 27304. 27401. 27402. 27403.  Fl 0.00193 0.00193 C.00193 0.00193 0.00180 C.OOIAO C.00180 O.OOlflC C.00146 C.00146 0.00146 0.00146 C.00142 0.00142 0.00142  FJ 0.02492 0.01826 C.00778 0.01323 0.01912 0.C1431 0.01104 0.00636 0.01396  OGO 71.50 71.50 71.50 71.50 55.21 56.21 55.21 55.21 38.17 3 8 . 17 38.17 38.17 16.64 1 6 . B4 16.84  VCF 1.038 1.00.6 , .025 .00)  01 0.521 0._4?1. 0.491 0.491  .254 . 350 .239  CELTA 5.33 5.35 5.35 7.78 7.78 11.58 11.58 19.25 19.25  VCF 1.005 1.021 1.023 1.007" 1.025 0.997 1.016 0.993 1,011.  01 0.487 0.487 0.4B7 0.490 0.490 0.473 0.473 0.462 0.4.2  DJ 0.252 0.346 0 . 147 " 0.261" ' 0 . 350 0.229 0.324 0.217 0.320  DELTA 5.56 5.56 5.56 5.70 7.02 7.02 7.02 7.02 9.97 9 . 9 7" 9.97 9.97 20.90 20.90 20.90  VCF 1.015 1.035 1.051 1.046 1.016 1.034 1.042 1.052 1.011 1.031 1.041 1.050 1.024 1.035 1.044  UU 0 5 .2 2 04. 91 04.91 U4.91  C* 2 0 3 9 ,, P P M FT/SEC  KL I 2.273 2.288 2.270 2.200 2.261 2.310 2.378 2.300 2.247  L/0* O.0263 0.0264 0.0264 0.0192 0.0192 0.0122 0.0122 0.0076 0.0076  00 0.487 0.467 0.487 0.490 0.49U 0 . 4 74 0.474 0.463 0.463  C4 • 2046.PPM FT/SEC  KL1 2.459 2.354 2.004 2.209 2.306 2.291 2.306 1.992 2.482 2.414 2.357 2.192 2.574 2.3)0 2.321  L/0* 0.0292 0.0292 0.0292 0.0292 0.0229 0.0229 0.0229 0.022V 0.0160 0.0160 0.0160 0.0160 0.0078 0 . 0 0 78 0.0078  6.01053 C.C0804 0.00494 0.00571 0.00436 C.C03C9  01 0.510 0.510 0.510 0.514 " 0.506 0.506 0.506 0.506 0.503 "0.503" "" 0.503 0.501 0.490 0.490 0.490  DJ 0.275 0.376 0.471 0 . 4 30 0.264 0.375 0.417 0.466 0.262 0.163 0.412 0.459 0 . 3 39 0.398 0.443  on 0.510 0.510 0.510 0.514 0.506 0.506 0.506 0.506 0.*D3 ' 6.V03 0.303 0.503 0.490 0.490 0.490  154  APPENDIX V C A L C U L A T I O N S FOR R E N E W A L MODELS (a) Turbulent Energy Dissipation F r o m the Blasius equation,  - XA  • 160 Re  A P =  D  2  L G  175  2  • ifeo Re  _  f  D  JL L  3  Per unit length of pipe, the rate of energy input into the fluid is u ( T D /4 2  ) ( A  P )  For highly developed turbulence in pipe flow, the energy is almost entirely dissipated by the turbulence. .  . Energy input rate =  turbulent dissipation rate,  rate of dissipation per unit mass of fluid UAP h  •._  .16  Re ' 1  f  B  f  3  7 5  M  U  2  '  .16  Re '  71  "  f V  2  7 5  #  3  4 '  .  (b) Wavenumber n^ of Dissipation Scale n  d  - ( ^fyl)^  e. g. for Re = 10, 000:  = 0.63  N  6  9  / D from (VIa-1)  Assume that the dissipation wave number is at least  of the same order as given by { / ^ ) turbulence.  Re*  which applies for well developed  For the 5/l6-inch I. D. tube (D= 0.795 cm), d  =  . 63 (10000)"  6 9  7 9 5 = 287 c m "  which corresponds to a scale of  T  —r—— 287-  1  = .011  cm  (c) S c h m i d t N u m b e r f o r CO_  At2o°c,  yW. H  - Water  - = -oi  e  j,0  m  cm-sec  CO^H^.0  = 1. 68 x 10  cm  /sec  from  d a t a p r e s e n t e d b y H a y d u k (7) b a s e d on a review by Davidson  S  C  ~~  =  .- -  JL.  e.g.,  CD  (1-68) (10  f r o m Mixing Length  Re  C u l l e n (41)  =  5  9  5  5  ./ ^ (d) E s t i m a t e k  &  = 10000, 5/16  3  )  Model  I. D.  tube  C o n s i d e r a p o i n t h a l f w a y f r o m tube w a l l to c e n t r e l i n e . R  -  ^j^y  T' ~ 1  -  D/4  = 0. 199  cm  126 c m / s e c f o r w a t e r at 2 0 ° C  f' = 0. 0076 f o r a s m o o t h tube. A p p l y e q u a t i o n s (44), (45) a n d  1  dU dr =  k.  126V.  0.4 |/ ^ " f T "  0076/2 —  n  =  0. 199  = [ ( ! • 68 ( 1 0 " ) ( 9 8 ) ] 5  • = (e) E s t i m a t e E d d y C e l l P e  (46)  2.4  and k  -.040  -1  o  98  sec  cm/sec  cm/min  1  1_ e. g. , Re  = 10000. C o n s i d e r that s i z e a n d a m p l i t u d e  of the s m a l l e s t s c a l e  is at l e a s t of the s a m e o r d e r as g i v e n b y the r e l a t i o n s that a p p l y to an e q u i l i b r i u m r a n g e of t u r b u l e n c e . C o n s i d e r a s c a l e at a p p r o x i m a t e l y the l o w e n d of the i n e r t i a l s u b r a n g e : i.e.,  a  3  ( T./  N  ) /-'.05  cm.  156 F r o m equation (56 ) From(VIa-l),  £  the velocity characteristic of a scale a is **** ( € a) .16 (10000) '. 2  -  (..Ql)  75  3  =  39200  'A  2 3 cm /sec  .795) A  *s  ( €  a)  10..5 cm/sec (.03) (10.5)  aA  Pe  (1. 68) (10  For a free fluid interface,  )  equation ( 53 )  a  Sh  18, 700  45 Pe  'A  ~ 61. 5  (.61.5 ) (1. 68) (10~ ) 5  = 0.035 c m / s e c ^ 2 c m / m i n  0. 05 (f) Integration for Overall Transfer Coefficient, k  Kovasznay Spectrum (32) applies through inertial subrange (ISR) and the viscous dissipation range,  C 73  E (n) =  n  Experimentally, E (n) = 0.45 6  \  n  n in the ISR  n  \ d)  '/2  E (n) r . 4 5 r £  F r o m equation (60), k  (Vf-1)  1.27  CxC = Also  n  1  /  n  I -  • 6^  c<  Tr~  +/3 n  (Vf-2)  n  dn (Vf-3)  is defined by  j1  £'*  157  n  The  = 1. 47  ~ -  fr  = 1.47  %  i n t e g r a l c a n be e x p a n d e d b y m e a n s of the b i n o m i a l t h e o r e m :  ;  (  -- l  n  .6)  VJ  i)  —rr n  n  6  i /,  a ) \ ^ h  - -——/  —  n  81  I  E q u a t i o n (V(f)-3) b e c o m e s rn  k  L  cx  3  n n.  The t e r m s  I £'6  dn  9  i n the s u m m a t i o n a r e a l l n e g a t i v e , and the s e r i e s  (by the r a t i o t e s t ) f o r a l l  n < n  At n = n  b y c o m p a r i s o n w i t h the s e r i e s  which converges for a l l  , the s e r i e s a l s o c o n v e r g e s  1  ti  converges  4  k  4.  I" /  4  k  n  If at l e a s t 3 o c t a v e s o f s c a l e s e x i s t i n the ISR b e l o w the b u b b l e d i m e n s i o n . 'then  n„ B  — — 20 o  and the l o w e r l i m i t c a n be o m i t t e d i n ( V ( f ) - 4 )  n  with only a s m a l l e r r o r . k  Substituting N  =-1.47  fr  {f/jt*)  as the u p p e r  (2. 24 - . 22 - . 043  L  -. 017  =  CxC  2  - . 0088 -  )  limit.  158  A P P E N D I X VI SOLUTIONS FOR FLOW FOR  EDDY  AND  DIFFUSION  CELL  (a) F l o w i n the V i s c o u s E d d y C e l l  ^  F o r slow, v i s c o u s f l o w that i s t i m e steady, and the t i m e d e r i v a t i v e  .  H  —  the i n e r t i a l termy.  c a n b e d r o p p e d f r o m the  —xr  Navier-Stokes  equations. . ' . —  " y ~ "  Define a s t r e a m  " V^Vr  = 0  function  d e s c r i b e s t h e flow.  y/ (x, y) s u c h t h a t  *T = —fj  and  F o r two d i m e n s i o n a l m o t i o n , the f l o w e q u a t i o n s  (j = -  y ~  become  0  T h i s i s the b i h a r m o n i c  d i f f e r e n t i a l equation,  and a s o l u t i o n is g i v e n b y  I r v i n g a n d M u l l i n e u x (40); 1^/  =  |^E c o s h p y + B p y s i n h p y + C s i n h p y + D p y c o s h p y j cos p x  F o r the e d d y c e l l a d j a c e n t to a s o l i d s u r f a c e The  ( F i g u r e VI-1):  following boundary conditions must be satisfied: At x =  ~ Zn  3  ,  ~ Zn  ,  etc.;  W 7  =  0  f r o m eq'n • ( V I a - 1 )  T h e b o u n d a r y o f the c e l l i s d e f i n e d b y a s t r e a m l i n e p a s s i n g x=  (VIa-1)  ~ — 2n  ,  ~ r ~ ~ » Therefore  Zn-  W~  1  0  through  a l l a r o u n d the b o u n d a r y ;  except  159 on y = a.  For  y = 0 ,  For  y = o ;  u  0 for all  =  -  —  >  x =  o  . '.  For y = a . '.  V =  ; B  .,.!•  =  =  Cpa  ;  -  0  for all x  y sinh  E  =  o, o b t a i n  D =  )  =  x  D i f f e r e n t i a t i n g (VIa-1) and setting C  E  o for all x  p a  , sinh p a + p a cosh p a  For  y = a ;  U =  -  c  d  = - A  sin  n x  x  n  , and  _ C  =  n a sinh  A / ( n a s i n h n a) — <n a  sinh n a + n a c o s h n a  F I G U R E VI-1. I D E A L I Z E D V I S C O U S E D D Y  sinh n a - n a cosh n a n a sinh n a  CELL  160  Note that  a=7f/n  . " .  t  n a = 7f  t  Define C  =  C/Aa ;  The constants C square c r o s s section. = Aa  and B  Equation  (C  B'  + B  — a  = B/Aa can now be evaluated for the c e l l of  (VIa-1) becomes, y ) sinh —• y - C a  — y cosh — y | cos - — x a a a 1  .(VIa-20 1  I  where B ' = - 0. 09351 and C " = - 0. 1236 F o r the free fluid (clean gas/liquid) surface, are the same as for the solid, on the surface,  y = o.  a l l boundary conditions  except for the velocity boundary conditions  F o r the free fluid at y = o ; u =  —- = 0 for a l l x, * y  but  V 4 o for this case .  Applying a l l the boundary conditions to equation E = 0  ,  C = -  (VIa-1),  obtain  B = 0 D (cosh na + na sinh na ) cosh n a  and D =  A / (n sinh na) cosh n a f n a sinh n a - n a cosh n a cosh n a sinh n a  Equation (VIa-1) becomes for the free fluid surface = where  Aa D  D  ' I T  1T  — y cosh — a a  = 0.02822,  y + C  =  C  '  TT  sinh— a  1cosIT— x  y  a  (VIa-3)  -0.1l6;&  (b) Convective Diffusion - P r o f i l e S i m i l a r i t y Solution The following analysis follows c l o s e l y the technique used by Bowman and Johnson (38) for transfer f r o m s p h e r e s .  161  F o r the e d d y c e l l ,  the c o n v e c t i v e d i f f u s i o n e q u a t i o n m u s t be  ? c  solved:  C =  F o r the two d i m e n s i o n a l c e l l ,  0  (VIb-1)  i and j t a k e v a l u e s 1 o r Z  v; =v  •= x  X  X  2  =  y  A s s u m e m o l e c u l a r d i f f u s i o n i n the x - d i r e c t i o n i s n e g l i g i b l e ; i.e. , n e g l e c t  >  -  2  w. r. t. o t h e r t e r m s i n ( V I b - 1 ) .  X  Let  the con-  c e n t r a t i o n b o u n d a r y l a y e r be of t h i c k n e s s _t(x) at a n y p o s i t i o n x.  Assume  that the c o n c e n t r a t i o n i n the b o u n d a r y l a y e r i s s o m e f i x e d f u n c t i o n , which satisfies all n e c e s s a r y boundary conditions:  c =  (0  q  c = o  (ii)  1  @  @  y =  h y =  (iii)  =  0  @  (iv)  = 0  @  y = 0  j  y = o  C o n d i t i o n (iv) a r i s e s f r o m e q u a t i o n ( V I b - 1 ) e v a l u a t e d at y = o, w h e r e C = 1 a n d u = 0 x  T h e p o l y n o m i a l "C = 1 - ~ ^ w h e r e 7|  =  +  ~ TJ  ( y / £ ), s a t i s f i e s t h e s e b o u n d a r y c o n d i t i o n s .  The total solute c r o s s i n g plane A A  i n F i g u r e VI-2 i s  r  rA (y)  C(y) l/"(y) d y = J A  (VIb-2)  2  J  f-  & Y•r= d  C  162  Between the pair of planes separated b y d x ( s i n c e there is no l a t e r a l diffusion), the mass t r a n s f e r a c r o s s the interface equals the difference in solute c r o s s i n g the two planes. ro —  -  Tjid-c  fdC  t  ' y - o  x + d x  the m a t e r i a l balance in the boundary layer becomes,  c  = J 5  Equation ( V I b - 3 ) can be put i n dimensionless form-by defining r ^  =  x/a,  CT~=  £/a  F I G U R E : VI-2 .  and f =  ^ / a A  E L E M E N T OF CONCENTRATION BOUNDARY LAYER  (VIb-3)  =y/a  163  W A * .  [  _ where Pe  a  =  -  i  3  Ir  3  "  4  )  2-^""  VA.«o  c o s h TTr  (VIb-5)  (VIa-3),  -'.117 s i n h ^Tr) tosjr^/  (VIb-6)  \f  2  I  2<r-  B  &  2^<T~\  (.0282 7/-r  C =  I  A  F o r the f r e e f l u i d s u r f a c e , f r o m e q u a t i o n  (j? d  V  JO  2£\ =  (  —  1)1  . 0282 T r c o s h f r - . 117  F o r only m o d e r a t e l y thin B.L. sinh(7Tr)  If*  =*  ( 0~~^.  sinhTfr  cosT^dr  0.2),  and c o s h  (fr  ) =^ 1  o c^dC Equations  =  (VIb-4, -5 a n d  l a y e r t h i c k n e s s s cT~( *>") .  0. 104 <T^cos  (VIb-7)  -7) g i v e a d i f f e r e n t i a l e q u a t i o n f o r the b o u n d a r y T h i s c o u l d be s o l v e d and  0~ (?.) u s e d to e v a l u a t e .  m a s s t r a n s f e r r a t e s f r o m the s l o p e of c o n c e n t r a t i o n p r o ffiill ee I 1  However a m o r e direct approach avoids  o  Define F = F ( ^ ) F Then  = -  t\)  ^  e v a l u a t i o n of  *  f  d C  -/  O" ( \ )-  ' (VIb-8)  1  = F^, is p r o p o r t i o n a l to the t o t a l s o l u t e c r o s s i n g  the i n t e r f a c e b e t w e e n a d j a c e n t s t a g n a t i o n p o i n t s . F r o m ( V I b - 7 and  0"~' =  -8), . 104  F r o m ( V I b - 4 -5 and  F " c o s -jry  - 8),  (VIb-9)  164  ~) (- F)  > *  2 P e ^ ^  o »  F d F  a  3(.  =  104)  2  J  3/2 (VIb-10)  cos  Pe 1/2  E q u a t i o n ( V I b - 1 0 ) i s r e a d i l y s o l v e d to g i v e - V12 F_ T  =  0. 445  Pe  r -  Solute t r a n s f e r r e d =  aA  J,  1-  a A  solute t r a n s f e r r e d (area)(driving  :  Sh =  I_  a  P  Sh = 0. 445  Pe  d C =0.  R  F.  the s a m e p r o c e d u r e i s f o l l o w e d ,  r =  F  (VIb-11)  e q u a t i o n ( V I a - 2 ) f o r the s t r e a m f u n c t i o n .  F  A  1/2  F o r the s o l i d s u r f a c e ,  results in  Pe  F„  F„ =  AFT  = aA  /  force)  a  P  dC  185  (f  .For  cos  using  a thin b o u n d a r y l a y e r , this  7T^  (VIb-12)  1/3 and  Sh  = 0.815  (VIb-13)  Pe  (c) C o n v e c t i v e D i f f u s i o n - F i n i t e D i f f e r e n c e The  convective  d i f f u s i o n (continuity) equation  • c a n be  Approximation  «4^-  -i5v c=o 2  r e n d e r e d dimensionless by defining dimensionless quantities:  165  v =  y / A  ,  u =  n  X =  x/a  ,  Y =  y/a  Pe  = aA/_£) 3  obtain V  IA  +•  c  ' ^  T.  »U  3  c  " ^  X  r;  —  y  o (VIc-1)  Set up c e n t r a l f i n i t e d i f f e r e n c e f o r m f o r t h i s e q u a t i o n , node shown in F i g u r e  C =  2  Pe  Y  u s i n g the t y p i c a l  VI-3.  E x p r e s s the C. i n T a y l o r s s e r i e s about C i  o  and  s o l v e to o b t a i n  central differences for _> C  _  C  2  V  2  C i  C  - C  2 C  r  ~>  3  +  C  C_  =  2 +  >  U s i n g these r e l a t i o n s and ( V i c - 1 ) s o l v e f o r four concentrations  2  C . - 2 C 2"  "  C =  +  C  4  C . ±-  -  i n t e r m s o f the  surrounding  C. ,  2 (C^  +  Cj  +  °  2T(:..C +  C )  2  • 4 (1 +  4  - VP:e|'x(C  r  Gj-UPe |y  (C^-Cj  X?) (VIc-2)  2  where  "Jf =  ( ex  /  £y  )  B o u n d a r y c o n d i t i o n s a r e C = 1 on Y = o ^ C ^  > For  =  X  =  o  on  1 X = 2—  and x =  o on Y = 1 f o r 1  •<  3 —2  X  , by  3/2  symmetry  as an a p p r o x i m a t i o n o n l y .  Y  ~  X  1  on Y = 1, two  boundary conditions were tried:  166  The  finite d i f f e r e n c e equations f o r these b o u n d a r i e s  the F O R T R A N p r o g r a m ,  a r e shown i n  i d e n t i f i e d b y the n o d a l c o d e n u m b e r s w h i c h d e f i n e  t h e i r l o c a t i o n i n the e d d y c e l l .  This system  of f i n i t e d i f f e r e n c e e q u a t i o n s i s s t a b l e o n l y i f  x  .  .  Pe.  u  and  Pe  y  f o r a p r a c t i c a l m e s h s i z e ( s a y 20 x 20) t h e t e c h n i q u e i s l i m i t e d to l o w T h e c a s e s o f P e - 40 a n d P e - 10 w e r e e v a l u a t e d f o r t h e f l u i d a n d  s o l i d s u r f a c e s o n a 20 x 20 s q u a r e a r r a y .  V  and U were evaluated  from  the a p p r o p r i a t e s t r e a m f u n c t i o n s ( V I a - 2 a n d -3).  The  condition  V PE  x  •H  i m p l i e s that b a c k d i f f u s i o n i n  the u p s t r e a m d i r e c t i o n c a n o c c u r i n s p i t e o f t h e f o r w a r d c o n v e c t i o n . o c c u r s i n t h e p r e s e n t s o l u t i o n i f o n l y the b o u n d a r y c o n d i t i o n A.  r  Co  2c  —  *  = ois Y  I  C,  F I G U R E VI-3.  TYPICAL NODE O F MESH FOR DIFFERENCES  This  FINITE  167 u s e d f o r the i n c o m i n g s t r e a m .  In t h i s c a s e , the w h o l e c e l l w o u l d b e c o m e  s a t u r a t e d a f t e r an i n f i n i t e n u m b e r of tterations.. H o w e v e r a f t e r a r a p i d i n i t i a l c h a n g e i n the m a s s t r a n s f e r r a t e , the c h a n g e i s v e r y s l o w .  For  P e = 40 a n d the f l u i d s u r f a c e ,  N u m b e r of I t e r a t i o n s Sherwood  The  No.  alternate boundary  20  60  280  3. 7  3. 0  2.9  c o n d i t i o n C = o f o r —-  <C  the i n c o m i n g s t r e a m to z e r o c o n c e n t r a t i o n , was  X  1, w h i c h  forces  t r i e d . F o r the a b o v e c a s e  t h i s g a v e the i d e n t i c a l r e s u l t f o r 60 i t e r a t i o n s (Sh -m 3. 0).  The  r e s u l t s of  all cases evaluated were:  Surface  Pe  Sh  Fluid  40  3.0  10  2. 06  40  2. 52  10  2. 08  Solid  The  F o r t r a n p r o g r a m f o r the s o l u t i o n f o l l o w s .  S h w e r e e v a l u a t e d f r o m the  M a s s transfer rate  f o r the s u r f a c e  and  increments  e v a l u a t e d o v e r the w h o l e s u r f a c e .  The approximate  i t e r a t i v e s o l u t i o n s o b t a i n e d h e r e a r e i n t e n d e d m a i n l y as e s t i m a t e o f the P e v a l u e at w h i c h S h d e p a r t s s i g n i f i c a n t l y  the p o w e r l a w o f e q u a t i o n s (52) a n d (53). factory for Pe  40.  These  an from  r e l a t i o n s a p p e a r to be s a t i s •  0 < 1 < 2 <  i1 i 7 i 10 < 3 4  5 6 l  11  l  12 13 14 17 20 22 23  < * < i 4 < 4  * I BF TC R E L A X I T E R A T I V E S O L U T I O N F O R F R E E F L U I D I N T E R F A C E AT LOW C DIMENSION CI 2 8 , 2 8 1 , V P ( 2 6 , 2 6 I . U P I 2 6 . 2 6 1 , M I 2 6 , 2 6 1 1NU =22 JND = 22 XX=l./20. XY=l./20.  C  25 12  40 41 42 43 44 45 46 47 50 51 52 53 54 57 60 66 67 70 76 77 100 101 102 1C3 104  i  i  1  2 • 3  F 4 5  F 6  iFF  14 7 150 151 15 2 160 161 162 163  SURFACE  i  114 115 116 117  126 127 132 133 134 135 136 137 140 141  Ml I N D , J ) = 2 CO 14 I = 3 , I N D M 1 Md.JNDI = 3  14  105 i 106 1C7 < 110 * 1114 112 F 1 113  120 121 122 123 124 125  NUMBERS  Ml I , 2) = 4 M(IND,21=5 M(IND,JND>=6 INPUT F O r l l N I T I A L CONCENTRATIONS C DO 20 J = 2, JND C(2,J»=100. 4 20 DO 21 1=3,IND < < DO 2 1 J = 2 , J N D 1• 2 1 ClI,J)=0. I N P U T FOR V E L O C I T Y M A T R I C E S CASE 2 FLUID F C DO 3 5 1=2,IND < EYE= I Y = XY»(EYE-2.) < PI=3.1416 4 PY=3.1416*Y 1 FVY=(-.2775»C0SH(PY) • .27B5»Y*SINHIPY))*PE * FUY=(.2785«Y*COSHIPY> - . 3 6 6 2 * S I NH 1 P Y ) ) * P E < DO 3 5 J = 2 , J N D EJ =J * X=.5+XX*IEJ-2.> PX=3.1416*X 4 VP I I , J ) = F V Y * C O S ( P X ) UP!I»J)=FUY*SIN(PX) 4• 3 5 DO 6 2 I=2,INU,2 < PRINT 69,I V P ( I ,J),J=2,JND,2) *F 6 2 < PRINT 107 DO 6 3 1=2,IND,2 PRINT 69, l U P I I . J ) , J=2,JND,2) • 61 FORMAT I12F10.3) •<F 6 4 C A L L S K I P TO I 1) I T E R A T I V E PROCEDURE F C TURN=0. t 4B TR = 0 . Z= C ..t. 50 CO 8 0 1=3,IND i DO 8 0 J = 2 , J N 0  i  24 26 1 27 4  30 31 33 34 35  G=l. PE = 4 0 . INDM1=IND-1 JNDM1=JND-1 I N P U T FOR N O D A L C O D E DO 2 5 I = 3 , I N D M 1 CO 2 5 J=3,JNDK1 MI I . J ) = 1 CO 12 J = 3 , J N D M 1  Pt  F F  73 80  F  100 101 102 105  F  107  •  F 110 •  104  F  115  K=M( I , J 1 C1=C1 I . J t l ) C2=CI1+1,J) C3=C(I,J-1I C4=Cll-l,J) GO TO I 1 , 2 , 3 , 4 , 5 , M , K W=(2.»ICI+C3)+2.*G*(C2+C4)-VP<I,J)*XX*(C1-C31-UP(I,JI*5*XY«(C2-C4) 1l/I4.*(l.+GI) GO TO 78 W=(C1+L3+2.*G*C4)/I2.»ll.+G)) GO TO 78 W=(4.*C3+2.*G*(C2+C4) UP!I,J)*G*XY»(C2-C4)1/(4.*(1.+G)) GO TO 78 W=(4.*Cl+2.*G*(C2iC4)-UPII.JI*G*XY*(C2-C4))/(4.«(1.+G)) GO TO 78 W=(Cl+G*C4l/ll.+GI GC TO 78 W=(C3+G*C4)/(l.+G) Z =Z*IC(11 Jl-W)**2 C(I.J)=W TR=TR+1. TURN=TURN*1. RF S= SORT 1 Z ) PRINT 100,TURN,RES FORMAT I / F 2 0 . 0 . F 2 0 . 1 / ) I F I T R - 5 . ) 50. 102.'JO DO 105 1=2, I M i l PRINT PRINT FORMAT 0 0 110 PKI NT FORMAT GO TO STOP END  1 0 4 , (C 1 1 , J ) , J = 2,12 ) 107 (/) 1=2,IND 104, ICII.J) , J=12,22 111F10.2) 48  )  ••  

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