liquid circulation, bubble size distributions, and solids movement in two- and three-phase bubble...

8/3/2019 Liquid Circulation, Bubble Size Distributions, And Solids Movement in Two- And Three-phase Bubble Columns

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Pergamon

S0009-2509(96)00029-2 0099-2509/96 $15.00 + 0 . 0 0

L i q u i d c i r c u l a t i o n , b u b b l e s i z e d i s t r i b u t i o n s , a n d s o l i d s m o v e m e n t i n

t w o - a n d t h r e e - p h a s e b u b b l e c o l u m n s

S . G r e v s k o t t B . H . S a n n ~e s M . P . D u d u k o v i d * K . W . H j a r b o t H . F . S v e n d s e n

D e p a r t m e n t o f C h e m i c a l E n g in e e ri n g , U n i v e rs i ty o f T r o n d h e i m , N - 7 03 4 T R O N D H E I M , N o r w a y

A b s t r a c t - O n e t w o - p h a s e b u b b le c o l u m n a n d t w o t h r e e -p h a s e s l u rr y r e a c to r s h a v e b ee n e x p e r i m e n t a l lyc h a r a c t e r i s e d wi t h s p e c i a l e m p h a s i s o n b u b b l e s iz e d i s t r i b u t io n , l i q m d c i r c u l a ti o n a n d s o l i d s m o v e m e n t .Th e m e a s u r e m e n t s i n t h e t wo p h a s e b u b b l e c o l u m n we r e b a s e d o n a f i v e p o i n t c o n d u c t i v i t y p r o b e m e t h o da n d o n a s t e a d y s t a t e h e a t t r a c e r t e c h n i q u e . Fo r t h e s o l i d s m o v e m e n t , t h e CARPT t e c h n i q u e wa s u s e d .

By n u m e r i c a l s i m u l a t i o n s u s i n g a t wo fl u id m o d e l , n e w m o d e l s f o r b u b b l e s i ze d i s t r i b u t i o n a n d s o l id sp r e s e n c e h a v e b e e n t e s t e d . Th e n e w b u b b l e s i ze m o d e l is fo u n d t o im p r o v e t h e s i ze d i s t r ib u t i o n p r e d i c t i o n sc o m p a r e d t o p r i o r m o d e l s , b u t . i s s t il l n o t s a t is f a c to r y . Te m p e r a t u r e p r o f il e s we r e we l l p r e d i c t e d b y t h em o d e l . Fo r t h e s o l id s m o v e m e n t , t wo c i r c u l a ti o n c e ll s we re fo u n d e x p e r i m e n t a l l y i n t h e r e a c t o r s t e s t e d .Th i s h a s a ls o b e e n v e r i fi e d b y n u m e r i c a l si m u l a t io n s . Th e f o r m a t i o n o f s e c o n d a r y c e l l s t r u c t u r e s o fm a g n i t u d e a p p r o x i m a t e l y e q u a l to t h e e o h u n n d i a m e t e r , a r e i n d i c a t e d b y t h e e x p e r i m e n t a l l y d e t e r m i n e d

Re y n o l d s s t r e s s p a t t e r n s ' .

I N T R O D U C T I O N

S l u r r y b u b b l e c o l u m n s a r e p r e s e n t l y b e i n g u s e d f o r a w i d e r a n g e o f c a t a l y t i c r e a c t i o n s i n b o t h t h e b i o - a n d

p e t r o c h e m i c a l i n d u s t r y . T h e f lo w p a t t e r n s o f l i q u i d , g a s , a n d s o li d s in t h e se r e a c t o r s a r e v e r y c o m p l e x , a n d

t h e b a s i c p r i n c i p l e s g o v e r n i n g t b e f lo w p h e n o m e n a a r e n o t y e t fu l ly u n d e r s t o o d .

C o m p u t a t i o n a l f lu i d d y n a m i c s is an i m p o r t a n t t o o l in t h e i n v e s ti g a t io n o f m u l t i d i m e n s i o n a l t w o - p h a s e

f lo w . P h e n o m e n o l o g i c a l m o d e l s h a v e b e e n w i d e l y u s e d t o d e s c r i b e su c h fl ow s , b u t t h e s e m o d e l s a r e n o t

w e l l s u i t e d f o r s c a l e -u p a n d d e s ig n i u g e n e r a l, s i n c e th e y r e q u i r e p r i o r k n o w l e d g e o f t h e f l o w s t r u c t u r e . B y

i n t r o d u c i n g m o r e f u n d a m e n t a l m o d e l s f o r t h e i n t e r n a l f lo w v a ri a b l e s , a to o l t h a t c a n b e u s e d f o r a v a r i e t y o f

m u l t i - p h a s e s y s t e m s a n d c o n d i t i o n s m a y b e o b t a in e d .

R e c e n t l y , s e v e r a l p a p e r s t h a t d e s c r i b e a d v a n c e d m o d e l s fo r tw o p h a s e f lo w s i n r e a c t o r s h a v e a p p e a r e d .

A m o n g t h e se a r e S o k o li c h in a n d E i g e n b e r g e r ( 1 9 94 ) w h o u s e a d y n a m i c E u l e r - E u l e r f o r m u l a t i o n w i t h n oc o n s i d e r a t i o n o f t u r b u l e n c e . L a p i n a n d L i i b b e r t ( 1 9 9 4 a) u se a n E u l e r - L a g r a n g i a n a p p r o a c h w h e r e t h e y t r a c k

t h e m o t i o n o f i n d i v i d u a l b u b b l e s o r b u b b l e c l u s te r s i n a n E u l e r i a n l i q u i d . R e c e n t l y , t h e y e x t e n d e d t h e i r

m o d e l t o i n c l n d e 3 d i m e n s i o n a l e f f e ct s (L a p i n a n d L i i b b e r t 1 9 9 4 b) . H i l l m e r e t . a l. ( 1 9 9 4 ) u s e d a n E u l e r - E u l e r

a p p r o a c h t o m o d e l s lu r r y b u b b l e c o l u m n s b y t r e a t i n g t h e l i q u id a n d s o l id p h a s es a s a p s e u d o - h o m o g e n e o u s

p h a s e w i t h a x i a l l y v a r y i n g d e n s i t y a n d v i s c o s i t y f ie l ds b a s e d o n a s e d i m e n t a t i o n - d i s p e r s i o n m o d e l .

A f u n d a m e n t a l t i m e a v e r a g e d m o d e l f o r t h e p r e d i c t i o n o f l o c a l f lo w s t r u c t u r e s i n b u b b l e c o l u m n s h a s b e e n

d e v e l o p e d d u r i n g t h e l a s t y e a r s ( T o r v i k a n d S v e n d s e n 1 9 9 0 , S v e n d s e n e t . a l . 1 9 92 , J a k o b s e n e t . a l. 1 9 9 3) . T h e

m o d e l i s s h o w n c a p a b l e o f p r e d i c t i n g l i q u i d p h a s e v e l o c i t y pr o f il e s an d v o i d p ro f i le s fo r l i m i t e d v a r i a t i o n s i n

g a s - l i q u i d s y s t e m s , s u p e r f i c i a l g a s v e l o c i t y a n d c o l u m n d i a m e t e r .

L i t e r a t u r e s h o w s t h a t a n a c c u r a t e d e s c r ip t i o n o f t h e in t e r -p h a s e m o m e n t u m e x c h a n g e t e r m s i s i m p o r t a n t

t o t h e a c c u r a c y o f t h e f u n d a m e n t a l m o d e l s . I n b u b b l e c o l u m n s t h e t r a n s v e r s a l a n d d r a g f o r c e s a r e o f

p a r t i c u l a r i n t e r e s t, s i n ce t h e y h a v e g r e a t i n f lu e n c e o n t h e v o i d p ro f il e s a n d t h e g a s h o l d u p .

T h e c o r r ec t d e t e r m i n a t i o n o f b u b b l e s iz e d i s t ri b u t io n s is im p o r t a n t t o t h e m o d e l l i n g o f b u b b l e c o l u m n s

a s it i s a n i n t e g r a l p a r t o f t h e p a r a m e t e r c o m p l e x d e t e r m i n i n g c i r c u l a t i o n a n d v o i d f r a c t i o n p r o f il e s . T h e

m o v e m e n t o f t h e i n d i v i d u a l b u b b l e , i n p a r t i c u l a r l a t er a l ly , d e p e n d s o n i t s s iz e a n d i n t e r a c t s w i t h t h e t u r b u l e n t

e d d y s t r u c t u r e s o f t h e c o l u m n t h r o u g h m o m e n t u m t r a n s fe r a n d t h e b u b b l e b r e a k u p a n d c o a l e sc e n c e pr o c es s es .

P r e v i o u s l y ( J a k o b s e n e t . a l . 1 9 9 3) , a s i m p l e m o d e l w h e r e th e l o c a l b u b b l e s i z e w a s a s s u m e d p r o p o r t i o n a l t o

t h e t u r b u l e n t l e n g t h s c a le , w a s u se d . T h i s m o d e l h as b e en f o u n d u n s a t i s f a c t o r y , a n d a n e w m o d e l h a s b e e n

d e v e l o p e d d e p e n d i n g d i r e c t l y on t u r b u l e n t k i n e t i c e n e r g y . T h e m o d e l h a s b e e n te s t e d f o r s e v e r a l g a s - l i q u i d

s y s t e m s t h r o u g h b u b b l e - s i z e d i s t r i b u t i o n s a n d v o i d f r a c t i o n p ro f i le m e a s u r e m e n t s p e r f o r m e d u s i n g a f iv e

p o i n t c o n d u c t i v i t y p r o b e m e t h o d ( B u c h h o tz a n d S t e i n e m a n n 1 9 84 ).

T h e s o l i d s c o n c e n t r a t i o n p r o fi le s i n s l u r r y b u b b l e c o l u m n s h a v e t r a d i t i o n a l l y b e e n s i m u l a t e d u s i n g s ed i -

m e n t a t i o n - d i s p e r s i o n m o d e l s . T h e s e m o d e l s ar e l i m i te d in t he i r a b il i ty to r e p ro d u c e e x p e r i m e n t a l d a t a

( S a x e n a a n d T h i m m a p u r a m 1 9 92 ), b u t a r e a b l e t o 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 a b o u t t h e a x i a l c o n -

c e n t r a t i o n p r o f il e s o f t h e s o l id s . T h e r a t e o f r e a c t i o n t a k i n g p l a c e o n a s u s p e n d e d c a t a l y s t w i l l d e p e n d o nt h e c o m p o s i t i o n o f, a n d t h e m a s s t r a n s f e r f r o m , t h e f l u id p h a s e i n c o n t a c t w i t h t h e p a r t i c l e . I t is t h e r e f o r e

i m p o r t a n t t o b e a b l e t o p r e d i c t , n o t o n l y t h e s o li d s c o n c e n t r a t i o n p r o fi le s , b u t a l so t h e m o v e m e n t o f t h e

s o l i d p a r t i c l e s .

* D e p a r t m e n t o f C h e m i c a l E n g i n e e r i n g , W a s h i n g t o n U n i v er s it y , S a in t L o u i s, M i s s o u ri , U S A

? S I N T E F D i v i si o n f o r A p p l i e d C h e m i s t r y , 70 3 4 T R O N D H E I M , N o r w a y

1704 S. GREVSKOTr t al .

The two fluid model is presently being extended to include a suspended solid phase.

For three phase model verification, experimental results from the CARPT (Computer Automated Ra-

dioactive Particle Tracking) facility at Washington University have been used. The CA RP T technique has

previously been used to map liquid movement in bubble columns, using a fully wettable solid radioactive

particle with density close to the liquid density. In the mapping of solids movement a particle closely match-

ing the size, shape, and density of the solids was used. Since the similar ity between the t racer partic le and

the solids used was very good, it is expected that the solids movement is well determined with this technique,and possibly better than for liquid movement.

M O D E L L I N G

The mathematical model used for both the two and three phase systems is a two fluid model, with the two

fluid phases trea ted in an Euler ian frame of reference. Tim e averaged conservation equations in cylindrical

coordinates of pressure, velocities and volume fraction of both phases are solved. In addi tion a k - e model for

turbul ence is used (Launder and Spalding 1974), with source terms of bubbl e generated turbulence (Svendsen

et. al 1992). We assume equal pressure for both fluid phases, no mass exchange between the phases and a

spatial averaging larger tha n the scale of the dispersed phases. The inter-phase moment um exchange terms

modelled between the fluid phases are steady inter -facial drag, added mass force and lift force. The lift force

is only considered in the rad ial direction, since the drag force is domina ting in the axial direction. If the void

fraction is not small compared to the tota l volume considered, the inter-phase momen tum exchange termshave to be mult ipli ed by the liquid fraction (Johansen 1990), and this is used in the model. The drag force

formula tion used is given as• 3 CD

F ~ = " ~ l o ~ g p z - - ~ u l - u g l ( u l , i - u g , i ) (1)

where i can mean bo th the axial and radial direction. For the drag coefficient, a curve has been fitted to

experi mental data for bubble movement in pure water (Gaud in 1957), giving

5.645C D - 1 . 0 ( 2 )

~go + 2.385

where E o is the EStvSs number, dependent on the surface tension. The bubble size will have an important

effect on the drag force through the drag coefficient, and therefore on the circu lation patt ern in general. The

new model still assumes that the bubble size is proportional to the turbulent length scale (Jakebsen 1993),but in addition the Sau ter Mean Diameter coefficient depends on the local turbu lent kinetic energy, giving:

k 3 / 2ds : CS MD , CS MD ----a r k a~ (3)

The coefficients al and a2 are to be considered empirical parameters. The added (virtual) mass force is

expressed by (Auton 198l):- D u L i D u g , i

F ~ = c ~ lc ~ g p zf y( D t D t ) (4)

where the added mass coefficient f v is set to 0.2. The transversal lift force can be expressed to include

non-l inear turbu len t drag effects in the radial direction (Jakobsen 1993), giving

F r ( 4 I ~ I , tC D C r / ~ ,1--~ ~ ( u , , ~ - - u , , z ) ~ - - - ~ ( u z , ~ ) (5)

where C~ is an empi rical coefficient. The f L in equat ion (5) is the conventional lift force coefficient and set

equal to 0.5 (Thomas et.al. 1983).

As a first approach to the three phase model, the two fluid model developed was used to determine the

flow pat terns. The viscosity and density of the liquid phase were modified to account for the presence of

solids. The drag coefficient was modified from equation (2) to include a formula tion for the drag on small

bubbles, giving5.645

1 o 2 , d ~ > 2 . 0 r a mCv = ~ + .385 - (6)

8 25"(1 -c~g) , d~ <2.0r am

It has been assumed tha t the solids are uniformly suspended in the column. Accordingly, the density of

the slurry, flrn, was adjusted to1 cw,~ Cw, l

- - - + ( 7 )

P,~ P~ Pl

Two- and three-phase bubble column s 1705

T h e v i s c o s i t y w a s c a l c u l a t e d a c c o r d i n g t o a n e m p i r i c a l c o r r e l a t i o n ( T h o m a s 1 9 65 )

P r e l - - 1 + 2 . 5 a s + 1 0 . 0 5 c ~ + 0 . 0 0 2 7 3 e x p ( 1 6 . 6 c t , ) ( 8 )

w h e r e ct~ i s t h e v o l u m e f ra c t i o n o f s o l id s . T h e b o u n d a r y c o n d i t i o n s f o r v e lo c i t i e s a n d v o i d fr a c t i o n a t t h e

i n l e t a r e s p e c i fi e d in a c c o r d a n c e w i t h t h e e x p e r i m e n t s t h e y a r e t o b e c o m p a r e d w i t h . T h e k i n e t i c t u r b u l e n t

e n e r g y a t t h e i n l e t i s a s s u m e d t o b e g e n e r a t e d m a i n l y b y th e g a s b u b b l e s e n t e r i n g t h e r e a c t o r , a n d t h e k i n e t i ce n e r g y i n t h e g a s i t s e l f , g i v i n g :

1 A ~ K p g , u gk ei,~ = ~ A , A g ( a + ~ ) ~ - [ u g (9 )

w h e r e K a s c a le s t h e k i n e t i c e n e r g y g e n e r a t e d b y t h e b u b b l e s . T h e p r o fi le s ar e a s s u m e d f l a t a t t h e i n l e t, a n d

n o d i f fu s i o n is a l l o w e d a t t h e b o u n d a r y . A t t h e o u t l e t , t h e p r e s s u r e i s s e t e q u a l t o t h e a m b i e n t p r e s s u r e, a n d

t h e o u t l e t m a s s f l o w s b e c o m e a p a r t o f t h e o v e r a l l s o l u t i o n . T h e r a d i a l g r a d i e n t s o f a l l v a r i a b l e s a r e s e t t o

z e r o a t t h e s y m m e t r y a x i s, e x c e p t f o r r a d i a l v e l o c it i e s, w h i c h a r e s e t to z e r o t h e m s e l v 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 e n s u r e z e r o fl u x o v e r th e s y m m e t r y a x is . A t t h e w a l l a s i n g l e - p h a s e l o g a r i t h m i c w a l l f u n c t i o n i s

u s e d .

T h e p a r t i a l d i f f er e n t i a l e q u a t i o n s a r e d i s e r e t iz e d u s i n g a f i n i te v o l u m e t e c h n i q u e w i t h s t a g g e r e d g r i d a n d

u p w i n d d i f f e re n c i n g i n a t w o - d i m e n s i o n a l c y l i n d r i c a l p o l a r m e s h w i t h 1 5 r a d i a l a n d 3 0 a x ia l g r id c e l ls . A

f u r t h e r e x p a n s i o n o f t h e g r i d w a s f o u n d n o t t o c h a n g e t h e r e s u l ts . T h e f i n it e v o l u m e e q u a t i o n s a r e s o l v e di t e r a ti v e l y u si n g t h e S I M P L E S T a n d I P S A a l g o r i t h m s ( S p a l d in g 1 9 77 ) a s i m p l e m e n t e d i n t h e P H O E N I C S

c o d e .

E X P E R I M E N T A L W O R K

T w o p h a s e e x p e r i m e n t s

T h e g a s - l i q u i d e x p e r i m e n t s w e r e p e r f o r m e d i n a c o l u m n w i t h in n e r d i a m e t e r 0 . 2 8 8 m a n d h e i g h t 4 .2 5 m .

T h e s y s t e m u s e d w a s a i r / w a t e r , a n d t h e s u p e r f ic i a l v e l o c i t y w a s v a r i e d in t h e r a n g e 0 . 0 2 - 0 . 18 m / s f o r

g a s , a n d i n t h e r a n g e 0 . 0 0 6 - 0 . 02 m / s f o r l i q u i d . T h e t e m p e r a t u r e v a r i a t i o n r a n g e w a s 1 2 0 C - 7 00 C .

T h e g a s d i s t r i b u t o r w a s a p e r f o r a t e d p l a t e w i t h 25 0 , 1 n a m d i a m e t e r h o l e s. T h e l i q u i d e n t e r e d t h r o u g h 1 9

h o l e s o f d i a m e t e r 2 8 r n m . B u b b l e s iz e d i s tr i b u t i o n s w e r e m e a s u r e d a t p o s i t io n s 0 . 3 m a n d 2 . 0 m a b o v e t h e

g a s in l e t. S t e a d y s t a t e h e a t t r a c e r e x p e r i m e n t s w e r e p e r f o r m e d b y a d d i n g s t e a m t h r o u g h a s y s te m o f t h r e e

c o n c e n t r i c r i n g s. T h i s p r o v i d e d a r a d i a l l y e v e n l y d i s t r i b u t e d i n f lu x o f e n e r g y a t a w e l l d e f i n e d a x i a l p o s i t io n .A p o s i t i o n 2 m a b o v e t h e g a s i n l e t w a s u se d i n t h e e x p e r i m e n t s . T e m p e r a t u r e s w e r e m e a s u r e d a t f iv e t o s ix

a x i a l p o s i t i o n s a n d f iv e r a d i a l p o s i t i o n s .

T h r e e p h a s e e x p e r i m e n t s

T h e m o v e m e n t o f 1 1 0 -1 8 0 # m g l a ss b e a d s ( d e n s i t y 2 9 5 0 k g / m 3 ) i n a n a i r - w a t e r s y s t e m w a s s t u d i e d i n 0 . 1 4

m a n d 0 . 2 6 m i n n e r d i a m e t e r c o l u m n s . T h e s t a t i c h e i g h t s o f t h e s u s p e n s i o n s w e r e 0 . 99 m a n d 1 .3 4 m i n th e

t w o c o l u m n s , r e s p e c t i v e l y . W a t e r a n d g l a s s b e a d s w e r e b a t c h f e d , a n d t h e s u p e r f i c ia l g a s v e l o c i t y w a s v a r i e d

f r o m 0 . 0 5 m / s t o 0 .1 4 m / s . T h e g a s d i s t r i b u t o r w a s a p e r f o r a t e d p l a t e w i t h a p o r o s i t y o f 0 . 05 % . S o l i d s

l o a d i n g w a s v a r i e d f ro m 7 w t - % t o 2 0 w t - % . A r a d i o a c t i v e 1 5 0 # m S c a n d i u m p a r t i c l e ( d e n s i ty 2 8 9 0 k g / m 3

w a s u s e d a s t r a c e r . I t s a c t i v i t y w a s a p p r o x i m a t e l y 6 0 0 # C u .

C o l u m nd i a m e t e rS t a t i che i gh tSol idsl oad i ng

0.26m 0.14m

1.33m 0.97mSuperf ic ia l

gas veloci ty0 .05 m / s 0 .02 m / s0 .08 m / s 0 .08 m / s0 .11 m/ s0 .14 m / s 0 .14 m / s0 .05 m / s 0 .05 m / s0 .08 m / s 0 .08 m / s0 .11 m/ s0 .14 m / s 0 .14 m / s0 .05 m / s 0 .05 m / s0 .08 m / s 0 .08 m / s0 .11 m/ s

0 .14 m / s 0 .14 m / s

T ab l e 1 : L i s t o f a l l r uns pe r f o r med f o r t he s l u r r y sys t em

1706 S. GREVSKOTr t a l .

The C A R P T f a c i l i t y

Up to 32 scintill ation detectors con taini ng NaI crystals were placed around the column. During each sampling

interval the number of hits for each detector was registered and transferred to a computer.

Intensity vs. distance data were established for each detector by in-situ calibrations performed ahead of

the experiment. When one r un of 5-7 hours was finished, the instantaneous positions were calculated from

the pre-established intensity vs. distance curves and the intensity data for each detector through a linear

regression scheme (Devanathan 1990).The column was divided into calculation cells for treat ment of the data. The i nstantaneous velocities

were calculated from the inst antaneous positions as

U ( i m , J m , ]era) = r( n + 1) -- r( n) (10)At

and assigned to the calculation cell ( i r a , j r a , k i n ) contain ing the mid poin t of the two positions. Tim e (ensem-

ble) averaged velocities were calculated by su mmi ng up all velocities assigned to a certa in cell and div iding

by the number of velocities in that cell.

The number of registrations in each cell divided by the cell volume gave the occurrence density of the

particle in the cell.

Stagnant regions of the column can be found by examining the in stant aneous velocities of the particle. If

the instantaneous velocity was small in one cell compared to other cells, this cell was considered stagnant for

the current registration. Tim e (ensemble) averaging gives the percentage of stagnant registrations in each

cell. For the 0.26 m diamete r column the trans ition velocity from stagnan t to active status of a registration

was set to 1 m/s . The transitio n velocity was chosen so as to give approxi mately the same numb er of

stagna nt and active registrations.

The fluct uating velocities were calculated as

Ufluct(i, j, k) = Uinst (i, j , k) - uav( i, j, k) (11)

and the Reynolds stresses (both shear and normal) were calculated as

u ~ u t ( i , j , k ) = m=l NVEL(i, j , k ) s , t = r , O , z ( 1 2 )

The turbulent kinetic energy was calculated as

1 (u-7-U-7(i, , k) + ~ ( i , j, k) + u-7~(i , j, k)) (13)kturb (i, j , k) =

R E S U L T S A N D D I S C U S S I O N

T w o p h a s e c o n d i t i o n s

The new bubble size distri but )n model was tested by comparing with experimental data. An example is

given in Figure 1 at inle t gas and liquid superficial velocities of 11.0 cm/s and 1.0 cm/s respectively. The

simulated results show how changes in the two parameters al and a2 affect the size distribution, and we

concentrat e on the value of these since the turb ulen t intensity levels have previously been found to be in good

agreement with experiment al (Svendsen et. al. 1992). It is found that a 1 jus t acts as a scaling parameter.This is not an obvious dependency, since the interactions between bubble size and level of turbulent energy

are complex. It is seen that a increase in the exponent a2 gives an decrease in bubble size. This was expected

since the t urb ulent kinetic energy is usually < 1 m2/s 2.

The choice of parameter values also affects the change in radial bubble size profile with superficial gas

velocity. An increase in the exponential parameter a~ increases the shift in predicted bubble diameter with

superficial gas velocity. In Figure 2 is shown the predicted and experimental radial bubble size profiles as

function of gas superficial velocity. For all velocities the magni tudes of the predictions are qui te good. The

predic tions also show the correct increase in bubble size with increasing superficial velocity. However, the

experimental profiles are flatter than the predicted ones. It would be possible also to make the predicted

profiles flatt er by adj ust ing the parameters al and a~. However, the shift in bubble size with superficial gas

velocities would then become to small. Thi s indicates that the present modification of the bubb le size modelstill is too simple, and t hat it may be difficult to account for the changes in bubb le size with superficial gas

velocity with a model based on ly on the local flow properties and that coalescence/breakup based po pulat ion

balance models are needed.

For the heat tracer experiments the inlet values for gas- and liquid superficial velocity were 10.0 cm/s

and 1.2 cm/s respectively. In Figure 3 are shown experimental and predicted t emper atur e profiles at dif-

Tw o- and thr e e -phase bubble c o lumns

S A U T E R M E A N D I A M E T E R I m]

. . .. . 2 i i i . . . . . .

0.C06 , . . .%

0 . 006 " - . . ~ o . ,

0 .004 !S o l i d l i n e : a ! = 0 0 6 , a 2 = 0 . 2 5

a l = 0 . 0 6 , ~ 2 = 0 . 2 0

/:-:/:ill::: :i=o.os, a 2=o ~o

o . 0 0 2 c i r c l e s : E x p e r i m e n t a l d a t a

\o . o 0 0 i I

o.oo o.05 o. lo 0 . 1 5

F i g u r e 1 : T h e S a u t e r M e a n D i a m e t e r , a s f u n c t i o n o f r a d i a l p o s i t i o n , s y s t e m a i r / w a t e r , U g , s u p er fi ci a I = 1 1 . 0 c m / s ,

'U l ,supe r fic ia I = 1 . 0 , ax ia l po s i t ion 2 . 0 m abov e in le t

S A U T E R M E A N D I A M E T E R ( ml

o . o 1 2 ] I I

O . O L O ~ -

) ~30 " - ' . *q . - o0 .0 06 ~ ~

0 . 0 0 4 [ ~

H o l i d l i n e : s u p e r f i c i a l g a s v e l o c i t y = 0 , 0 8m/ s

. . . . . . . . . . : ~ u p e r f i c i a l g a s v e l o c i t y = 0 . 1 1 m/ s. . . : s u p e r f i c i a l g a s v e l o c i t y = 0 . 1 6 m/ s

0.802 ~ Squares: Experimenta l data 0.08 m/sc i r c l e s : E x p e r i m e n t a l d a t a O . l l m / s

i D i ~ o n d s : E x p e r i m e n t a l d a t a 0 1 6 /

o . o 0 o [o . ~ o o c 5 o . I o

F i g u r e 2 : T h e S a u t e r M e a n D i a m e t e r a s f u n c t i o n o f r a d i a l p o s i t i o n f o r t h r e e s u p e r f i ci a l g a s v e l o c i t ie s a t 2 . 0 i n a b o v ei n l e t . S y s t e m a i r / w a t e r , a l = 0 . 0 6 a n d a s = 0 . 2 5

f e re n t v a l u e s o f t h e b u b b l e s iz e m o d e l p a r a m e t e r s a l a n d a 2 . G e n e r a l l y t h e p r e d i c ti o n s a g r ee w e l l w i t h t h e

e x p e r i m e n t a l d a t a , a l t h o u g h t h e t e m p e r a t u r e d e c re a s e b e l o w t h e s t e a m f e e d p o i n t i s p r e d i c te d s t e e p e r t h a n

e x p e r i m e n t a l l y f o u n d . T h i s i n d i c a t e s t h a t t h e r e ci r c u la t i o n o f l i q u i d i n t h e b u b b l e c o l u m n i s w e l l m o d e l l e d , a

f a c t t h a t i s s u p p o r t e d b y e a r li e r s i m u l a t i o n s o f l i q u i d v e l o c i t y p r o f il e s ( J a k o b s e n e t . a l 1 9 9 3 ) . I t i s n o t e w o r t h yt h a t t h e r e l a t iv e l y l a r g e s h if t in p r e d i ct e d b u b b l e s iz e , ca u s e d b y c h a n g i n g t h e p a r a m e t e r s i n t h e m o d e l , d o e s

n o t i n f l u e n c e t h e c i r c u l a t i o n p r o f i le s to a l a r g e e x t e n t . T h i s i s s o m e w h a t s u r p r i s i n g , a s a s h i f t t o w a r d s l a r g e r

b u b b l e s i n t h e c e n tr e o f t h e c o l u m n w o u l d b e e x p e c t e d t o g iv e h i g h e r c ir c u l a t io n r a te s . T h e e x p l a n a t i o n

i s t h a t t h e v o i d f r a c t i o n p r o f i l e s b e c o m e f la t t e r ( n o t s h o w n ) , s u c h t h a t t h e v o l u m e t r i c g a s v e l o c i t i e s i n t h e

c e n tr e s e c ti o n r e m a i n a l m o s t u n c h a n g e d .

I n F i g u r e 4 t h e e x p e r i m e n t a l a n d p r e d i c t ed r a d i a l t e m p e r a t u r e p r o fi le s a t a p o s i t i o n 0 . 3 m a b o v e t h e g a s

d i s t r i b u t o r a r e s h o w n . T h e r e s u lt s a r e c h a r a c t e r i s ti c a l s o f o r t h e p o s i t i o n s f u r t h e r u p . T h e g e n e r a l t r e n d

i s t h a t t h e p r e d i c t i o n s s e e m t o o v e r e s t im a t e t h e r a d i a l t e m p e r a t u r e v a r i a t io n s a n d t h i s i n d i c a t e s t h a t t h e

r a d i a l m i x i n g i s s o m e w h a t u n d e r e s t i m a t e d . H o w e v e r , is s h o u l d b e n o te d t h a t t h e v a r ia t i o n s in te m p e r a t u r e

o b s e r v e d e x p e r i m e n t a l l y a r e s m a l l , a n d t h a t t h e e x p e r i m e n t a l a c c u r a c y i s i n t h e r a n g e o f +O. I ° C. C h a n g i n g

t h e b u b b l e s i z e d i s t r i b u t i o n h a s n e g l i g i b l e e f fe c t o n t h e t r e n d i n t h e r a d i a l t e m p e r a t u r e p r o f i l e s g i v e n a b o v e .

T h r e e phase experiments

S o l i d s m o v e m e n t :

T w o c i r c u l a t io n c e ll s w er e fo u n d f o r a l l e x p e r i m e n t s , o n e l o w e r ce ll t h a t w a s a p p r o x i m a t e l y o n e d i a m e t e r

h i g h , a n d o n e u p p e r c e ll e x t e n d i n g t o t h e t o p o f t h e c o l u m n . I n t h e l o w e r c e ll th e s o l i d s w e r e a s c e n d i n g a t t h e

w a l l a n d d e s c e n d i n g i n t h e c e n tr e . T h e u p p e r c e ll s h o w e d t h e o p p o s i t e c i r c u l a ti o n p a t t e r n . F i g u r e 6 s h o w s

L I Q U I D T E M P E R A T U R E ( C)

S . GREVS KOTT et a l .

s o l i d l i n e : a l = 0 . 0 6 , a 2 = 0 . 2 5

. . . . . . . . . . : a l = 0 . 0 6 , a 2 = 0 . 2 0

. . . _ . _ . : a l = 0 , 0 5 , a 2 = 0 . 2 0

c i r c l e s : E X p e r i m e n t a l d a t a

I I i2 ~ 4

Ax ~ P o s iTio t~ l

F i g u r e 3 : A x i a l t e m p e r a t u r e 2 .8 c m f r o m c e n t r e o f c o l u m n

L I Q U I D T ~ p E ~ R E ( C)

/ / ~ ~ . . S ol i d l i ne : a 1= 0 , 06, a 2 =0 . 25. * " . . . . . . . . . . : a l=0, 06, a2=0,2 0

- _ . . . . . . . : a l = O . 0 5 , a 2 = 0 . 2 0

i . C i r c l e s : E x p e r i m e n t a l d a t a

0 ,0 5 0 .1 0

~I~ POSlTIOS {m~

0 , 1 5

F i g u r e 4 : R a d i a l t e m p e r a t u r e a t 0 . 3 m a b o v e i n l e t ( b )

t h e m e a s u r e d f l o w fi e ld s f or t w o d i ff e re n t e x p e r i m e n t s . T h e l e ft p l o t i s f o r t h e 0 . 1 4 m d i a m e t e r c o l u m n , a n d

t h e m i d d l e p l o t i s f or th e 0 . 2 6 m d i a m e t e r c o l u m n . I n b o t h e x p e r i m e n t s t h e s o l i d s l o a d i n g w a s 7 % - w t a n d

t h e g a s s u p e r f i c i a l v e l o c i t y w a s 0 . 0 8 m / s . T h i s f lo w p a t t e r n i s o b s e r v e d f o r l o w g a s s u p e r f i c i a l v e l o c i t i e s a l s o

i n t w o p h a s e b u b b l e c o l u m n s , w h e r e a s f o r h i g h g a s su p e r f ic i a l v e l o c it i e s t h e s m a l l e r c e ll at t h e b o t t o m o f

t h e c o l u m n d i s a p p e a r s , l e a v i n g o n l y o n e c e ll e x t e n d i n g t o t h e to p ( D e v a n a t h a n e t .a l . 1 9 9 0 , D u d u k o v i ~ e t. a l .

1 9 9 1 ) . F o r t h e t h r e e p h a s e s y s t e m t h e i n t e r fa c e o f t h e t w o c i r c u l a t i o n c e l l s w a s a t a n a n g l e , a s o p p o s e d t o

t h e i n t e r fa c e o f li q u i d c i rc u l a t io n c e ll s in b u b b l e c o l u m n s , w h e r e t h e i n t e rf a c e i s a p p r o x i m a t e l y h o r i z o n t a l .

T h e m a x i m u m a x i a l s o l i d s v e l o c i t y w a s f o u n d t o in c r ea s e w i t h c o l u m n d i a m e t e r a n d w i t h g a s su p e r f ic i a l

v e l o c it y , a n d t o d e c re a s e w i t h s o l i d s lo a d i n g . T h i s d e c r e a se w a s m o s t p r o n o u n c e d a t t h e lo w e r g a s su p e r f ic i a l

v e l o c i t i e s .

T h e o c c u r r e n c e d e n s i t y w a s f o u n d t o d e c r e a se w i t h h e i g h t a b o v e t h e d i s t r i b u t o r a s c a n b e s e e n i n F i g u r e 5 .

A s f a r a s t h e o c c u r r e n c e d e n s i t y c a n b e u s e d a s a r e l a t i v e m e a s u r e f o r s o l i d s c o n c e n t r a t i o n , t h i s i n d i c a t e s a

d e c r ea s e i n s o l i d s c o n c e n t r a t i o n q u a l i t a t i v e l y in a c c o r d a n c e w i t h t h e s e d i m e n t a t i o n - d i s p e r s i o n m o d e l s . F o r

h i g h s o l id s l o a d i n g , a p e a k in th e o c c u r re n c e d e n s i t y a t th e t o p w a s s ee n . T h i s p e a k w a s m o s t n o t i c e a b l e

a t h i g h g a s s u p e r f ic i a l v e l o c i t ie s a n d m a y b e c a u s e d b y t h e h i g h g a s f r a c t io n s a n d l o w s o l i d s r e t e n ti o n s i n

t h e d i s e n g a g e m e n t z o n e , t h u s c h a n n e l l i n g th e so l i d s d i r e c tl y b e l o w t h i s z o n e . T h e o c c u r re n c e d e n s i t y w a s

h i g h e r a r o u n d p o s i t i o n r/R = 0 . 7 t h a n a t t h e c e n t re o r a l o n g t h e w a l l s . T h i s c o i n c i d e s w i t h t h e p o s i t i o n o f

f l o w r ev e r s a l i n s l u r r y r e a ct o r s. F o r 1 4 % a n d 2 0 % s o l i d s l o a d i n g , a r e g i o n w i t h o u t o c c u r r e n c e s w a s f o u n d a tt h e g a s d i s t r i b u t o r . T h e s i z e o f t h i s r e g io n i n c r e a s e d w i t h s o li d s l o a d i n g a n d w i t h g a s s u p e r f i c ia l v e l o c i t y .

a u s t a b o v e t h i s r e g io n a p e a k i n o c c u rr e n ce d e n s i t y w a s f o u n d , b u t n o st a g n a n t z o n e a t t h i s p o i n t c o u l d b e

v i s u a l l y o b s e r v ed . A n e x p l a n a t i o n f o r t h i s o b s e r v a t i o n h a s n o t y e t b e e n fo u n d . T h e o c c u r r e n c e d e n s i t y w a s

m o r e u n i f o r m t h r o u g h o u t t h e c o l u m n fo r h i g h e r s o l id s l o a d i n g s e x c ep t f o r t h e p e a k n e a r t h e b o t t o m .

Two- and three-phase bubble columns 1709

r = 0 3 5 c m

. . . . r = 2 4 c m

- - - r = 4 5 c m

r - 6 . 6 c m

t 5 O / r = 0 . 6 5 c=m

. . . . r = 4 . 5 5 c m

- - - r = 8 4 5 cr n

1r=72.35 can

0 2 0 4 0 6 0 8 0 1 0 0 2 0 4 0 6 0 8 0 1 0 0 ? 2 0 1 4 0 1 6 0

axi a l p o si t i o n [cm] axi a l p o si t i c~ [ c m ]

Figure 5: Axial occurrence density profiles for the 0.14 m (left) and 0.26 m diameter columns with 7 %-wt solids and0.08 m/s gas superficial velocity.

There were no regions that were considered stagnant. A comparison of the ins tant aneous velocities

in the two col umns showed larger velocities in the large column when solids loading and gas superficialvelocities were equal. When the t ransit ion velocity for the larger column was used for the sma ller one, most

registrations were considered stagnant.

The Reynolds stresses varied both axially and radially in a way that suggests the existence of secondary

circulat ion cells in the column. These cells were each approximately one diameter high as can be seen in

Figure 8, and evenly distrib uted along the vertical direction. These "cells" may be caused by recurring

transient cells in these specific positions. This conclusion is still subject to further experimental studies of

long duration to provide better statistics.

W avele t t~l tering

The gam ma radiat ion from the tracer particle fluctuates randomly. This causes the particle to be detected up

to 5-10 mm away from the true position. This, in turn , causes the inst antaneous velocities to be calculated

as larger than they really are. Through the averaging procedure this evens out for parameters calculatedfrom time averaging of one fluctuating or instan taneous velocity, and thus does not affect the results. For

correlations of two or more fluc tuat ing velocities the "spurious" effects, however, do not average out. For

the s tagnancy, which is determined from the size of the instantaneous velocity, these "spurious" effects are

also not averaged out.

A mathematical filter based on wavelets has recently been developed at Washington University (De-

galeesan 1995). By this technique the inst antane ous positions of the tracer particle is adjusted based on

knowledge of the noise characteristics in the signal due to the fluctuations of the radioactivity. The "spuri-

ous" effects have been shown to almost completely disappear after the filtering procedure, and the f luct uati ng

velocities are reduced. The filter uses routines from the WavBox 4@ software (Taswell 1995).

Figu re 7 shows that the me an velocities are not significantly changed by the filtering, as expected. The

turbulent kineti c energy level is seen to be significantly reduced, but the profiles are retained . Simil ar results

are obt ained for all Reynolds stresses, and for the stagnancy as welt.

Numer ica l s imula t ions for three phases

Figure 6 shows measured and simulated flow fields for the 0.26m diameter column with 7%-wt solids and

0.08m/s superficial gas velocity. The middle plot in the figure clearly shows the two measured circulation

cells as menti oned earlier. The right plot in the figure shows tha t the model predicts the two circulati on

cells observed experimenta lly, and also positions the cells in good agreement with the measured da ta.

Note that the measured flow field is for the solids, while the simulated flow field is for the slurry and does

not allow for relative velocity between the solids and the liquid. The measured velocities vary more with

axial position than the simulated velocities, but they show the same trend. The predicted slurry velocities

are higher than the meas ured solids velocities in the up-flow region. This is reasonable considering the

difference in densi ties and relative velocities between solid and liquid. Similarly, in the down-flow region

near the bottom the solids velocities are larger than the slurry velocities. The shift in angle of the interface

between the two circulation cells observed in experiments when going from two phase to three phase flow,can also be seen in Figure 6.

The changes in the flow due to changes in viscosity as described in equa tion 8 are negligible for small

solids loadings (e.g. 7%-wt). The viscosity changes due to higher solids loadings (e.g. 20%-wt.) clearly affect

the flow in the column.

1 7 l O S . G R E V S K O I T e t a l .

l f t t , , - , H ,

t t t t , , . ' ~ 1

~ t t , . . . U j

t t t h . , * ~ 1

I ~ l m , , 11

f l ~ l l , . . , i I

l l t ~ I , , . i ; I

f l t l ~ , . , l i j

8 0 b t r ' " ' r ~

, ~ t f t . ' i l l

t t f h , ' ~ l l

I t t h . . . . l j

l t t , , , . , d

t t h , . ' , l l

~ 4 1 , , . . . , ~

l f l t l , ' , ~ 1 1

~ l t f t , ' , l J

t t t t l , ' ~ z l

I t t h . . , ~ Z I

4 0 ~ll~,.,~

~ t f f h , ' I l l

t t t h . , ~ d

l t l ~ , , . , , i I

I t t h ' " I l l

2 0 L ' " " , ,~

0 " ~ - ~

r a d i u s [ c rn ]

' t t t f , I

t t t , . . . . . . t

t m , , i : ! 1

I lh '

t t t t " ~ 1~ o t t t t , , ' . ' . ' . ~

t T t t t , , ; ' 1t t t , , , , i t lt t t , , . i ' . ' . It t t , , . . ; t l

~ ° ° t t t t t , . . ' . ~ 1t t t t , , , i , It f t t t , , " 1t t t h , , . . i l

8 ° r t t h . . . .

t t m , , , I

t f t t t , , . i i

t r Y S t , , I

6 0 r t ' t t t t , I

t t ~ t t l , I

t t ~ l t , . I

4 0 ? , , ~ 1 ~ I , . I

2 0 T ~ ' , . , , , , - ~

o l Or a d i u s [ c r n ]

% , . ' %

, , , , , ,

% * . " ' ~ 4

i f J , i ,

% * . " 4

m , r ,

~ . ' * 4

H l ~ r ,

~ . " 4

% ~ . * ' 4

H , , / ,

H l l , ,

~ +. "V

[ 7 7 t ~ .

l l J H ' " . . . .

, \ \ \ \ . . ~ r .

F i g u r e 6 : M e a s u r e d a v e r a g e f l o w f i e l d s i n 0 . 1 4 m ( l e f t ) a n d 0 . 2 6 m ( m i d d l e ) d i a m e t e r c o l u m n s w i t h 7 % - w t s o l i d s a n d0 . 08 m /s g as supe r f i c i a l ve loc i ty . S im u la t ed ( r i g h t ) ave rage fl ow fi eld fo r t he 0 . 26m d iam ete r co lum n w i th 7 % -w t

so l ids and 0 . 08m /s gas supe r f i c i a l ve loc i ty . T he sca l e o f t he ve loc i ty vec to r s i s t he sam e in a l l t h r ee f i gu res .

~0 . 2 5

~ 0 . 2

~ 0 . 1 5

:2 o.1

. . . . . U n f l ll e re d d a ta

- - R l te md d a ta

-0.1 -0 .05 0 0 .05 0 .1 -0 .1 ~) .05 O 0,05 0 ,1R a d i a l p o s i t i o n {m] Radia l posiUon mJ

F i g u r e 7 : F i l t e r e d a n d u n f i l t e r e d r e s u l t s f o r a x i a l m e a n v e l o c i t y a n d t u r b u l e n t k i n e t i c e n e r g y

T a k i n g i n t o a c c o u n t t h e s i m p l i f i c a ti o n s m a d e i n t h e m o d e l l i n g of t h e s l u r r y m o v e m e n t , t h e a g r e e m e n tb e t w e e n p r e d i c t e d a n d m e a s u r e d v e l o c i t ie s i s c o n s i d e r e d s a t i s f a c to r y .

Tw o- and three-phase bubble colum ns 1711

~go 6O

0 0- 5 0 5 - 1 0 0 1 0

r a d i u s [ c m ] r a d i u s [ c m ]

Fi g u r e 8 : Re y n o l d s s t r e s s e s f o r t h e 0 . 1 4 m ( l e ft ) a n d t h e 0 .2 6 m d i a m e t e r c o l u m n w i t h 7 % - wt s ol id s l o a d i n g a n d 0 .0 8m / s g a s s u p e r f i c i a l v e l o c i t y .

Further modelling

A f u l l t h r e e p h a s e m o d e l i s b e i n g d e v e l o p e d b a s e d o n t h e t w o f l u id m o d e l . T h e s o l i d s w i l l b e i n c l u d e d a s

a s u b - p h a s e o f t h e l i q u i d p h a s e . T h e r e w i l l b e n o d i r e c t i n t e r a c t i o n s b e t w e e n s o l i d s a n d g a s . T h e s o l i d s

w i l l i n t e r a c t w i t h t h e l i q u i d a n d t h r o u g h t h i s b e a f f ec t e d b y t h e p re s e n c e o f t h e b u b b l i n g g a s . T h i s w i l l b e

a c c o m p l i s h e d b y u s i n g a n a l g e b r a i c s l ip m o d e l f o r t h e i n t e r a c t i o n b e t w e e n s o l id s a n d l i q u i d , i n s t e a d o f a f u l l

i n t e r - p h a s e s l ip a l g o r i t h m ( I P S A ) . T h e g a s w i ll i n t e r a c t w i t h a s l u r r y p h a s e t h a t i s d e n s e r a n d m o r e v i s c o u s

t h a n t h e p u r e l i q u i d p h a s e, a n d r e l a t i o n s s i m i l a r t o e q u a t i o n s 7 a n d 8 w i l l b e u s e d l o c a l l y t o d e t e r m i n e t h e

d e n s i t y a n d v i s c o s i t y o f t h e s l u r r y i n t e r a c t i n g w i t h t h e g a s.

C O N C L U S I O N S

O n e t w o - p h a s e b u b b l e c o l u m n a n d t w o t h r e e -p h a s e s l u rr y r e a c t o rs h a v e b e e n e x p e r i m e n t a l l y c h a r a c t e r is e d

w i t h s p e c i a l e m p h a s i s o n b u b b l e s i z e d i s t r i b u t i o n , l i q u i d c i r c u l a t i o n a n d s o l i d s m o v e m e n t .

T h e d a t a h a v e b e e n c o m p a r e d w i t h s i m u l a t i o n s b as e d o n a tw o fl u id m o d e l u si n g a n e w s u b - m o d e l f o r

b u b b l e s i ze , a n d a l s o t a k i n g i n t o a c c o u n t t h e p r e s e n c e o f s o l id s .

T h e n e w b u b b l e s i z e m o d e l , b a s e d o n t h e b u b b l e i n d u c e d t u r b u l e n t l e n g t h s c a le a n d t h e l o c a l t u r b u l e n t

k i n e t i c e n e r g y l e v e l, i s f o u n d t o g i v e i m p r o v e d p r e d i c t i o n s c o m p a r e d t o t h e m o d e l b a s e d o n l e n g t h s c a l e a l o n e .

B u b b l e s i z e s i n t h e s a m e r a n g e a s t h e e x p e r i m e n t a l o n e s a r e o b t a i n e d i n t h e h e t e r o g e n e o u s f lo w re g i o n , a n d

a c o r r e c t s h i f t i n b u b b l e s iz e w i t h s u p e r f i c ia l g a s v e l o c i t y is fo u n d . T h e r a d i a l s i ze p r o f il e s a re , h o w e v e r , t o o

s t e e p . I t is c o n c l u d e d t h a t i n o r d e r t o o b t a i n a s a t i s f a c t o r y d e s c r i p t i o n o f b u b b l e s i ze d i s t r i b u t i o n s i n b u b b l e

c o l u m n s , a m o d e l b as e d o n l o c a l c o n d i t io n s a l o n e i s i n a d e q u a t e . T h u s , a p o p u l a t i o n b a l a n c e b a s e d m o d e l

t a k i n g i n to a c c o u n t c o a le s c e n ce a n d b r e a k u p m e c h a n i s m s m u s t b e i m p l e m e n t e d .

I n t h e h e t e r o g e n e o u s f l o w r e g i o n , c b a n g e s in b u b b l e s iz e d i s t r i b u t i o n s a r e f o u n d t o h a v e o n l y m o d e r a t e

e f fe c t o n l iq u i d c i r c u l a t i o n r a te s . T h e s e a r e t h e r e f o r e w e l l p r e d i c t e d w i t h o u t a n a c c u r a t e d e t e r m i n a t i o n o f

171 2 S. GREVSKOq'r t a l .

t h e b u b b l e s i z e d i s t r i b u t i o n .

I n t h e t h r e e p h a s e f l o w e x p e r i m e n t s , t h e e x i s t en c e o f tw o p r i m a r y s o l i d s c i rc u l a t i o n c e l ls h a s b e e n e s -

t a b l i s h e d f o r t h e c o l u m n s i z e s us e d . A n a l y s i s o f th e R e y n o l d s st r e ss p a t t e r n s i n d i c a t e s t h e e x i s t e n c e o f a

s e c o n d a r y c e l l s t r u c t u r e e x t e n d i n g f r o m t h e b o t t o m o f t h e u p p e r p r i m a r y c i r c u l a t i o n c e ll a n d a l m o s t t o t h e

t o p o f t h e d i s p e r s i o n l a y e r . T h e c e l l s i z e i n t h i s r e g i o n i s i n t h e o r d e r o f t h e c o l u m n d i a m e t e r a n d a t a n a n g l e

w i t h t h e h o r i z o n t a l .

N u m e r i c a l s i m u l a t i o n s g i v e f lo w s t r u c t u r e s v e r y s im i l a r t o t h e o n e s o b s e r v e d e x p e r i m e n t a l l y a n d t h e t w op r i m a r y c i r c u l a t i o n c e l ls a r e p r e d i c t e d i n c o r r ec t p o s i t i o n s , in d i c a t i n g t h a t t h e s i m p l i f i e d a p p r o a c h u s e d , m a y

b e v i a b l e .

N O M E N C L A T U R E

L a t i n l e t t e r s

i , j , k

N V E L

a l ~ a 2

] ¢ e i n

s u b - a n d s u p e r s c r i p t s

d r a g c o e f f ic i e n t r r a d i a l

S a u t e r M e a n D i a m e t e r o f b u b b l e z a x i a l

a d d e d m a s s c o e ff ic i e nt t t u r b u l e n t

l i f t f o r c e c o e f f ic i e n t T t r a n s v e r s a l

v e l o c i t y c o m p o n e n t D d r a g

v e l o c i t y v e c t o r m m i d p o i n t o r m i x t u r e ( i . e , s l u r r y )

t r a n s v e r s a l d r a g c o e f fi c i e n t l l i q u i dp a r t i c l e r e l a x a t i o n t i m e s s o l i d s

p a r t i c l e R e y n o l d s n u m b e r g g a s

w e i g h t f r a c t i o n r e l r e l a t i v e

i n d i c e s f o r c a l c u l a t i o n c e l l s f l u c t f l u c t u a t i n g

t i m e i n d ic e ; 1 s t t o l a s t r e g i s t r a t i o n i n s t i n s t a n t a n e o u s

t i m e b e t w e e n r e g i s t r a t i o n s a v a v e r a g e

# v e l o c i t i e s i n a c a l c u l a t i o n c e l l t u r b t u r b u l e n t

p o s i t i o n v e c t o r

E S t v 6 s n u m b e r

s c a l i n g fa c t o r f o r b u b b l e g e n e r a t e d k i n e t i c e n e rg y

e m p i r i c a l c o e f fi c ie n t s o f t h e b u b b l e s iz e m o d e l

i n l e t t u r b u l e n t k i n e t i c e n e r g y

G r e e k l e t t e r s

a v o l u m e f r a c t i o n p l a m i n a r v i s c o s it y

p d e n s i t y rL L a g r a n g i a n t i m e s c a l e

A C K N O W L E D G E M E N T S

O n e o f t h e a u t h o r s ( M . P . D u d u k o v i ~ ) i s g r a t e f u l f o r t h e D e p a r t m e n t o f E n e r g y c o n t r a c t s a n d g r a n t s ( D O E -

F C 2 2 - 95 P C 9 5 0 5 1 a n d D E - P S 2 2 - 9 5 P C 9 5 2 0 0 ) a n d t o t h e i n d u s t r i a l s u p p o r t e r s o f t h e C h e m i c a l R e a c t i o n

E n g i n e e r i n g L a b o r a t o r y ( C R E L ) t h a t m a d e h i s c o n t r i b u t i o n p o s si b l e . T w o o f t h e a u t h o r s ( B . H . S a n n m s a n d

S . G r e v s k o t t ) a r e g r a t e f u l fo r t h e f i n a n c ia l s u p p o r t o f S T A T O I L .

R E F E R E N C E S

A u t o n , T . R . , 1 98 1, T h e d y n a m i c s o f b u b b l e s , d r o p s a n d p a r t i c l e s in m o t i o n i n l i q u id s , P h . D . T h e s i s , U n i -

v e r s i t y o f C a m b r i d g e , C a m b r i d g e , U K .

B u c h h o l z , J . , a n d S t e i n e m a n n , R . , 1 98 4, A p p l i c a t i o n o f a n e l e c t r i c a l c o n d u c t i v i t y m i c r o - p r o b e f o r t h e c h a r -

a c t e r i s a t i o n o f b u b b l e b e h a v i o u r i n g a s - l i q u i d b u b b l e f l ow . Par t . Charac t . , 1 .

D e g a l e e s a n , S . a n d D u d u k o v i d , M . P ., 1 9 95 , A p p l i c a t i o n o f W a v e l e t s f o r f i lt e r i ng C A R P T D a t a f o r B u b b l e

C o l u m n s , S u b m i t t e d t o Exp . i n F l u i d s .

D e v a n a t h a n , N . , 19 90 , I n v e s t i g a t i o n o f L i q u i d H y d r o d y n a m i c s i n B u b b l e C o l u m n s v i a a C o m p u t e r A u t o -

m a t e d R a d i o a c t i v e P a r ti c l e T ra c k i n g ( C A R P T ) F a c il i ty , D . S c . T h e s i s , W a s h i n g t o n U n i v e r s i t y , S a i n t L o u i s ,

M i s s o u r i , U S A .

D e v a n a t h a n , N . , M o s l e m i a n , D . a n d D u d u k o v i d , M . P . , 19 90 , F l o w m a p p i n g i n b u b b l e c o l u m n s u s i n g C A R P T ,

Chem. Eng . Sc i . , 4 5 , 2285-2291 .

D u d u k o v i ~ , M . P ., D e v a n a t h a n , N . a n d H o l u b , R ., 1 99 1, M u l t i p h a s e r e a c t o rs : M o d e l s a n d e x p e r i m e n t a l v e r -

i f i c a t i o n , Revue de l ' I n s t i t u t e Fran~a i s du Pg t ro l e , 4 6 , 4 3 9 -4 6 5 .

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Gaudin, A.M., 1957, Flo ta t ion , 2nd ed., McGraw-Hill, New York.

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Johansen, S.T., 1990, On the modelling of disperse two-phase flows, Dr. Techn. Thesis , The Norwegian In-stitute of Technology, Trondheim, Norway.

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bubble columns, Chem. Eng . Sc i . , 49, 3661-3674.

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in app l ied Mechan ics and Eng ineer ing , 3,269-289.

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