This article was downloaded by: [Columbia University]On: 06 October 2014, At: 09:02Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Solvent Extraction and Ion ExchangePublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lsei20

DROPLET COALESCENCE AND BREAKAGE RATES INLIQUID EXTRACTION COLUMNSH.R.C. Pratt aa Department of Chemical Engineering , University of Melbourne , Parkville, Victoria, 3052,AUSTRALIAPublished online: 29 Mar 2007.

To cite this article: H.R.C. Pratt (1984) DROPLET COALESCENCE AND BREAKAGE RATES IN LIQUID EXTRACTION COLUMNS,Solvent Extraction and Ion Exchange, 2:4-5, 521-551

To link to this article: http://dx.doi.org/10.1080/07366298408918462

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

SOLVENT EXTRACTION AND I O N EXCHANGE, 2(465), 521-551 0 9 8 4 )

DROPLET COALESCENCE AND BREAKAGE RATES

I N LIQUID EXTRACTION COLUMNS

H.R.C. P r a t t

Department o f Chemical Engineer ing V ~ ~ i v e r s i t y o f Me'l bourne

P a r k v i l l e , V i c t o r i a , 3052 AUSTRALIA

ABSTRACT

A review i s g iven o f recen t work on t h e measurement o f d r o p l e t coalescence and breakage r a t e s i n a packed and a pulsed p l a t e e x t r a c t i o n column us ing a newly developed c o l o r i m e t r i c technique. The r e s u l t s , which were i n t e r p r e t e d i n terms o f second o rde r coalescence and f i r s t o rde r breakage r a t e constants, showed t h a t t h e d r o p l e t i n t e r a c t i o n r a t e s a r e cons iderab ly lower i n t h e pu lsed column. The r a t e constants can a l s o be used t o p r e d i c t accu ra te l y t h e steady s t a t e d r o p l e t s i z e d i s t r i b u t i o n , and t o s tudy t h e o r e t i c a l l y t h e e f f e c t o f d r o p l e t coalescence and breakage on mass t r a n s f e r r a t e . De f i c ienc ies i n t h e a v a i l a b l e mass t r a n s f e r c o e f f i c i e n t data f o r d rop le ts , bo th i n d i v i d u a l and i n "swarms",are p o i n t e d ou t .

INTRODUCTION

Dev ia t ions o f l i q u i d e x t r a c t i o n columns f rom simple

p lug f l o w behaviour r e s u l t f rom two causes, v i z 521

Copyright @ 1984 by Marcel Dekker, Inc. 07364299/84/0204-0521$3.50/0

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

522 PRATT

( i ) a x i a l d i s p e r s i o n o f one o r both phases, and ( i i ) , po l y -

d i s p e r s i v i t y o f t h e d r o p l e t phase. Two models, termed

" d i f f u s i o n " (1,Z) and "backf low" ( 3 ) r e s p e c t i v e l y , have been

devised t o account f o r t h e former, and methods o f scale-up

us ing these a r e a v a i l a b l e (4 ) .

P o l y d i s p e r s i v i t y , an e n t i r e l y d i f f e r e n t e f f e c t , r e l a t e s

t o t h e range o f d r o p l e t s i z e s present i n a d i spe rs ion , each

w i t h d i f f e r i n g sur face areas, mass t r a n s f e r . c o e f f i c i e n t s and

v e l o c i t i e s . I n consequence t h e sma l le r drop1 e t s approach

e q u i l i b r i u m more r a p i d l y than t h e l a r g e r ones, l ead ing t o a

l o s s i n performance (5 ) ; Rod ( 6 ) termed t h i s e f f e c t , r a t h e r

l oose ly , " forward mixing". T h i s performance l o s s would be

expected t o be reduced by repeated coalescence and breakage,

which would tend t o average t h e d r o p l e t concent ra t ions l a t e r a l l y .

Theore t i ca l computations conf i rmed t h a t t h i s i s so (7), a1 though

i n t h e absence o f exper imental data t h e coalescence r a t e was

expressed i n terms o f an assumed "coalescence h e i g h t " .

Exper imental measurements o f i n t e r - d r o p l e t coalescence

r a t e s a r e few i n number, and almost e n t i r e l y conf ined t o

a g i t a t e d vesse ls a t low d ispersed phase holdup. The

t h e o r e t i c a l mode l l i ng o f d ispersed systems i n terms o f

cont inuous popu la t i on balances has been descr ibed b y Valentas

e t a1 (8) and Ba jpa i e t a1 ( 9 ) . Theore t i ca l expressions f o r

coalescence and breakage r a t e constants have been obta ined

by Coula log lou and Tav la r ides (10). and Sovova (11) compared

these w i t h exper imental data f o r a s t i r r e d tank.

Hamil t o n and P r a t t (12) r e c e n t l y prov ided a s o l u t i o n t o

t h e problem o f measuring d r o p l e t coalescence ra tes , i n t h e

form o f a novel c o l o r i m e t r i c techn ique (12) . Th is i nvo lved

p r e l i m i n a r y s i z e e q u i l i b r a t i o n o f two equal streams o f methyl

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE 523

i sobuty l ketone ( M I B K ) d r o p l e t s c o n t a i n i n g r e s p e c t i v e l y

d i t h i z o n e (green) and n i c k e l d i (e thy l xan tha te ) ( y e l l o w ) i n

a v e r t i c a l l y p a r t i t i o n e d s e c t i o n o f column. These were then

al lowed t o ming le and e n t e r a second, u n p a r t i t i o n e d column

sect ion, when coalescence o f green w i t h y e l l o w d r o p l e t s gave

r i s e t o red d rop le ts , t h e p ropor t i ons and s i zes o f which were

determined by co lou r photography.

The method has been app l i ed t o a packed (12-14) and t o a

pulsed p e r f o r a t e d p l a t e column ( l 5 , l 6 ) . I n both cases t h e

r e s u l t s were i n t e r p r e t e d i n terms o f coalescence and breakage

r a t e constants, us ing a novel d i s c r e t e popu la t i on balance

model. A b r i e f account o f t h e method i s g i ven below,

together w i t h a comparison o f t h e r e s u l t s f o r t h e two cases.

EXPERIMENTAL

Equipment

I n t h e case of t h e packed column t h e s i z e e q u i l i b r a t i o n

s e c t i o n comprised a 1.0m l e n g t h o f 72.45m diameter p r e c i s i o n

bore tube, d i v i d e d i n t o quadrants by means o f l o n g i t u d i n a l

SS b a f f l e s . Th is was surmounted by a photographic window,

a 0.3m long coalescence sect ion, a second window and a 0.5m

long i n t e r f a c e sec t ion . Both s i z e e q u i l i b r a t i o n and

coalescence sect ions were packed w i t h 12.5 x 2mm t h i c k ceramic

Raschig r i n g s . The so lven t d i s t r i b u t o r was d i v i d e d i n t o f o u r

sec t ions each prov ided w i t h 7 x 4.5mm nozzles, oppos i te p a i r s

o f which were f e d w i t h one o f t h e so lven ts . The general

arrangement o f t h e equipment i s shown i n F ig . 1, and f u l l

d e t a i l s a r e g iven i n r e f s . 12-14.

The pulsed column was s i m i l a r (15.16) except t h a t t h e

lower window was omi t ted and t h e upper p l a t e s were c a r r i e d on

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

Equilibration Section E Rotameter (0.3-3.0 L/min) Coalescence Section F Solvent Pumps (0.25hp) Interface Section G Water Pump (0.11 hp) Solvent Stock Tanks (80L) H Activated Carbon Columns Solvent Receiving Tank (160L) J Sand Filter Water Stock Tank ( t60L) K Interface Control Valve Rotameters (0.2-1.5L/min) L Photographic Window

FIGURE 1. Arrangement of equipment f o r packed column.

an ex tens ion o f t h e support r o d f o r t h e p l a t e s i n t h e lower

sec t ion , t o avo id coalescence between t h e two sec t ions . The

s i z e e q u i l i b r a t i o n sec t ion , which was mounted d i r e c t l y on t o

t h e c y l i n d e r o f a pu lse pump o f v a r i a b l e speed and s t roke , was

prov ided w i t h 16 p la tes , and t h e coalescence s e c t i o n w i t h one

t o e i g h t . These were o f 1.5751nn t h i c k SS d r i l l e d w i t h 3.2mm

holes t o g i v e 21.7% f r e e space, and were spaced 50m apar t .

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE 525

M a t e r i a l s

The phases cons i s ted o f commercial MIBK and de ion i zed

water, m u t u a l l y sa tu ra ted be fo re use. The spent s o l v e n t was

regenerated f o r reuse by passage through two beds o f a c t i v a t e d

carbon. The concen t ra t i ons o f d i t h i z o n e and n i c k e l d i ( e t h y l

xanthate) were 0.05 and 0.10 g / l r e s p e c t i v e l y f o r t h e packed

column, and 0.075 and 0.17 g / l f o r t h e pu lsed column; t h e

s t o i c h i o m e t r i c r a t i o s were between 3 and 4, w i t h t h e n i c k e l

reagent i n excess.

Photographic Techniques

Photographs were taken on 35mm c o l o u r s l i d e f i l m us ing

s u i t a b l e i l l u m i n a t i o n and exposure (14.16). The s l i d e s were

p r o j e c t e d on t o a screen us ing s u i t a b l e m a g n i f i c a t i o n s as

determined by sca les on t h e windows. Around 150 d r o p l e t s pe r

s l i d e were counted f o r t h e packed column and 200-250 f o r t h e

pu lsed column w i t h i n randomly se lec ted areas, and t h e i r

d iameters o r s i z e s o f major and minor axes, dl and d2, were

recorded w i t h t h e i r co lou rs . For o b l a t e d r o p l e t s t h e 2 1/ e f f e c t i v e diameters were taken as (dl d2) 3.

RESULTS

Runs were c a r r i e d o u t w i t h the packed column a t f l o w r a t e s

corresponding t o c a l c u l a t e d d ispersed phase holdup va lues (1 7 ) o f 4.8%, 10.6% and 17.0% r e s p e c t i v e l y . I n each s e r i e s f o u r

o r f i v e packing he ights , f rom 5 t o 30cm, were used. Coalescence

i n the lower window was n e g l i g i b l e throughout.

For t h e pu lsed column, pu lse f requencies o f 60, 90 and

120 min-' were used, w i t h a f i x e d ampl i tude o f 1.4cm. For

each f requency, 2, 4 and 8 p l a t e s were used w i t h t o t a l

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

526 PRATT

throughputs, Qc+Qd, o f 10,15,30 and, f o r t h e two h igher f r e -

quencies,45cm3s-1, a l l w i t h a so lvent /water r a t i o o f u n i t y .

V isua l i n s p e c t i o n i n d i c a t e d t h a t t h e t h r e e f requenc ies

corresponded t o t h e m i x e r - s e t t l e r , t r a n s i t i o n and emulsion

reg ions o f opera t ion r e s p e c t i v e l y . Holdup values were

c a l c u l a t e d f rom Thornton's c o r r e l a t i o n (18) f o r t h e emulsion

reg ion, and from t h e f o l l o w i n g r e l a t i o n f o r t h e m i x e r - s e t t l e r

r e g i o n

Pre l iminary Smoothing o f Data

For each case, t h e d r o p l e t s were grouped i n t o d i s c r e t e

s i z e ranges jl t o jM based on i n t e r v a l s o f d iameter o f 1 / 2 3, i . e . , so t h a t t h e mean d r o p l e t volume o f any g i ven

s i z e i s t w i c e t h a t o f t h e s i z e below. The s i z e ranges used

a re g i ven i n Table 1; t h e raw data a re tabu la ted on t h i s

bas is i n r e f s . 13 and 15.

The packed column data were found t o be b e t t e r f i t t e d

w i t h t h e ac tua l packing h e i g h t reduced by 2.5cm, suggest ing

t h a t t h e f i r s t two courses o f r i n g s were i n e f f e c t i v e i n

promot ing coalescence. The data w e r e s m o ~ t h e d b y f i t t i n g t o

t h e f o l l o w i n g express ion

where h ' = (h-2.5). The r e s u l t i n g values o f A and n were

used t o c a l c u l a t e smoothed curves o f f . vs h ' . Typ ica l J

p l o t s , f o r t h e 10.6% holdup case, a r e shown i n F ig . 2.

The pu lsed column data were smoothed by f i t t i n g t o t h e

express ion

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE

Column I t ype

Packed

Pulsed

TABLE 1 Drop le t S ize Ranges

Pulse frequency

(mi n-')

No. o f i n t e r v a l s

Mean diameter (mm) o f s i z e

4.520

1.793

1.423

where

A t y p i c a l p l o t o f B vs d . i s shown i n F ig . 3. The J

minimum i n B c o r r e c t l y r e f l e c t s t h e observa t ion t h a t t h e

coalescence r a t e f o r t h i s column passed through a minimum

a t an in te rmed ia te va lue o f d.. J

DEVELOPMENT OF MODEL

Bas is o f Method

The model i s based on two s e t s o f p o p u l a t i o n balances

f o r each s i z e i n t e r v a l , one i n v o l v i n g t h e number concen t ra t i on

and t h e o t h e r t h e r a t e o f r e d d r o p l e t fo rmat ion. The

assumptions i n v o l v e d a r e as f o l l o w s :

1. The d r o p l e t breakage and coalescence processes a r e

m u t u a l l y independent.

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

FIGURE 2. Smoothing of f. values for packed column (10.6% holdup). J

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE

KEY:

[I] f = 60 cycleslmin

A f = 90 cycleslmin

X f = 120 cycleslmin

- quadratic approximation

0.0 I I I 0.4 0.8 1.2 1.6 2.0

Droplet Diameter, d. (mm) I

FIGURE 3. V a r i a t i o n o f constant B i n eqn. ( 3 ) w i t h d r o p l e t diameter (Qc+Qd = 3 0 c m ' s - ~ ) .

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

530 PRATT

2. Only coalescences between d r o p l e t s i n t h e same, and i n

ad jacen t s i z e ranges a r e considered.

3. Coalescence r a t e s can be expressed i n terms o f second

o rde r r a t e constants, k. ; thus, f o r two equal s i z e d J

drop1 e t s o f s i z e j, forming one o f s i z e j + 1,

and f o r two ad jacent -s ized d r o p l e t s

For t h e pulsed p l a t e column, -dn. /d t i s rep laced by J

-hnj /ht .

4. The r a t e constants f o r ad jacent coalescences can be

expressed as f o l l o w s

5. O f coalescences o f d r o p l e t s i n ad jacent s i z e i n t e r v a l s

j and j-1, a f r a c t i o n p ( g e n e r a l l y taken as 0.5) o f t h e

d r o p l e t s formed e n t e r s i z e j + l and t h e remainder s t a y

i n s i z e j.

6. Breakages o f d r o p l e t s o f s i z e j i n t o p a i r s o f s i z e j -1

can be expressed i n terms o f f i r s t o rde r breakage r a t e

constants , K., as f o l l o w s J

7. I n t h e pu lsed column, as observed, t h e d r o p l e t s undergo

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE 531

coalescence and breakage o n l y on passage through t h e

p l a t e s and n o t w i t h i n t h e i n t e r v e n i n g spaces.

I t should be noted t h a t t h e dimensions o f k., K . and J J

n . d i f f e r accord ing t o t h e type o f column and method o f J

operat ion. Thus, f o r t h e packed column t h e r a t e s a r e

expressed per u n i t v o i d volume, and t h e d r i v i n g fo rce , n . J '

as number o f d r o p l e t s pe r u n i t v o i d volume. For t h e pu lsed

column, however, t h e r a t e s a r e expressed pe r u n i t c ross

sect ion, w i t h n . as number o f d r o p l e t s pe r u n i t area f o r J

m i x e r - s e t t l e r , and pe r u n i t volume f o r emulsion opera t ion

r e s p e c t i v e l y . The values o f n , according t o e i t h e r j

d e f i n i t i o n , a r e ob ta inab le d i r e c t l y f rom t h e holdup, xd, and t h e

exper iemental d r o p l e t s i z e d i s t r i b u t i o n data .

Popula t ion Balance Equat ions

The d r o p l e t number balance comprises f o u r terms, as

f o l l o w s , f o r each s i z e i n t e r v a l j

(a) The number o f d r o p l e t s e n t e r i n g s i z e j f rom below due

t o coalescences o f ( i ) , two d r o p l e t s o f s i z e j-1 and

( i i ) , d r o p l e t s o f s i z e j and j-1, one-ha l f o f which g i v e

s i z e j.

( b ) The number l e a v i n g s i z e j by coalescence o f ( i ) , two o f

s i z e j, ( i i ) , s i z e j w i t h j+l, and ( i i i ) , s i z e j w i t h

j-1, one-ha l f o f which g i v e s i z e j+l.

( c ) The number e n t e r i n g s i z e j f rom j-1 by breakage.

(d) The number l e a v i n g s i z e j by breakage t o j-1.

The value o f dn. /d t can be r e l a t e d t o dn./dh i f i t i s J J

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

532 PRATT

assumed t h a t a l l d r o p l e t s have t h e same v e l o c i t y i r (=U /E x ) d P d

r e l a t i v e t o s t a t i o n a r y coord inates. On t h i s bas is t h e number

balance takes t h e form:

where A . = ( U /E x ) dn./dh f o r t h e packed column and J d P d J

sbn./AN f o r t h e pu lsed column, w i t h s = f f o r t h e m i x e r - s e t t l e r J

and Ud/xd f o r t h e emulsion reg ion .

The r e d d r o p l e t balance, which takes i n t o account t h e

var ious p o s s i b i l i t i e s a r i s i n g f rom t h e t h r e e co lou rs o f d r o p l e t s

present, i s o f t h e form

R R k +cR I( t c R k Aj = cj, j-2kj-2tcj, j- l j-1 j,j j j,j+l jtl

where nR = (Udnj/c'pd) d f / d h f o r t h e packed column, and J

s a ( n . f .)/AN f o r t h e pu lsed column. J J

The c o e f f i c i e n t s Ci,j and B i n eqn. (9) con ta in t h e n., i ,j R~

e t c . t oge the r w i t h p and numerical c o e f f i c i e n t s ; t h e Ci,j

and BR i n eqn. (10) c o n t a i n a l s o t h e f . , fj-l, e t c . These 1 ,j J

equat ions can be expressed i n m a t r i x form as f o l l o w s :

C k + B K Z A - - - - - (11

R R C k + B K = A R - - - - - (12)

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE 533

R where C and C a r e quadr id iagonal mat r ices o f t h e C. . and R - - R 1 *J

C . . c o e f f i c i e n t s and B, B b id iagona l ma t r i ces o.f t h e 1 , J R - -

Bi . and B . . c o e f f i c i e n t s . r J 1 *J

Eq. (11) and (12) can be f u r t h e r condensed as a p a r t i t i o n e d

m a t r i x equat ion, v i z

where

T

Before s o l v i n g eqn. (13)

t h e end s izes, i . e . j = 0 and

i t i s necessary t o cons ider

M+1, s ince t h e numbers o f

d r o p l e t s i n these ranges, a l though smal l , were trot i napprec iab le

and i t i s e s s e n t i a l t o ensure t h a t d r o p l e t s a r e n o t l o s t t o t h e

system. Consequently, eqns. (9 ) and (10) were w r i t t e n f o r

these i n t e r v a l s sub jec t t o t h e c o n d i t i o n s ( i ) n = 0 f o r j

j < 0 and >M+1, ( i i ) KO = 0, and ( i i i ) kM+l = 0, and

inco rpo ra ted i n t o eqn. (13) .

S o l u t i o n o f Popula t ion Balance Equat ions

To so lve eqn. (13) , i t was assumed t h a t A = 0, s ince t h e - - s i z e d i s t r i b u t i o n s o f t h e d r o p l e t s e n t e r i n g t h e coalescence

sec t ions o f t h e columns were e f f e c t i v e l y a t t h e steady s t a t e . R Also, A was eva luated a t t h e sma l les t p o s s i b l e he igh t , t o -

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

534 PRATT

minimise t h e p o s s i b i l i t y o f m u l t i p l e coalescences, i . e . , o f

r e d w i t h o t h e r co lours . For t h e packed column, t h e r e f o r e , R A was eva luated a t h ' = 2.5cm ( i . e . , f o r h = 5.0cm) and f o r

t h e pu lsed column, ove r t h e range N = 0 t o 1.

S o l u t i o n o f eqn. (13) by d i r e c t m a t r i x i n v e r s i o n was

i n i t i a l l y found n o t t o be poss ib le s ince A was s i n g u l a r (13,

14). However i t was l a t e r found t h a t t h i s was a r e s u l t

o f t h e assumption t h a t p = 0.50, and t h a t s o l u t i o n s were

o b t a i n a b l e f o r o t h e r values o f p, e.g., 3.4 and 0.6 (15,16).

Using the p r e f e r r e d va lue o f p = 0.50, eqn. (13) t h e r e f o r e

c o n s t i t u t e s and underdetermined s e t o f equat ions. However,

t h e s o l u t i o n can a l t e r n a t i v e l y be formula ted as an

o p t i m i z a t i o n problem, g i v i n g a s o l u t i o n v e c t o r which

minimises a r e s i d u a l vec to r , E, de f i ned by

i n terms o f an appropr ia te norm. O f t h e general c l a s s o f

norms, t h e f o l l o w i n g a r e t h e most commonly used

Using t h e l2 norm, s u b s t i t u t i o n o f eqn. (14) i n t o

(16) g ives

As a * , t h e o b j e c t i v e f u n c t i o n , i s q u a d r a t i c i n x, t h e l a t t e r - can be obta ined by q u a d r a t i c programming. So lu t i ons were

obta ined i n t h i s way f o r both column types us ing an a v a i l a b l e

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE 535

mu1 t i -pu rpose o p t i m i z a t i o n program (MPOS). Garg a l s o

at tempted t o o b t a i n s o l u t i o n s f o r the pulsed column i n terms

o f t h e l1 norm, o b t a i n i n g r e s u l t s s i m i l a r t o those obta ined

w i t h MPOS.

The de r i ved values o f the r a t e c o e f f i c i e n t s were c o r r e l a t e d

i n terms o f d r o p l e t s i z e and holdup by expressions o f t h e

Values o f a, b a n d c a r e g i v e n i n r e f s . 13-16. P l o t s o f

t y p i c a l data f o r t h e pu lsed column are shown i n F ig . 4.

APPLICATIONS OF RATE CONSTANTS

D r o p l e t S ize D i s t r i b u t i o n

The change i n d r o p l e t number concen t ra t i on w i t h h e i g h t

o r stage number i s g i ven by

n . . = n . . +A. /s J .1 J , 1 - 1 J (19)

where s u b s c r i p t i r e f e r s t o stage number o r h e i g h t increment.

Hence, s t a r t i n g f rom a g i ven s i z e d i s t r i b u t i o n , eqn. (19) can

be used r e c u r s i v e l y w i t h eqn. ( 9 ) t o determine the change i n

s i z e d i s t r i b u t i o n w i t h stage number o r he igh t . I n t h i s way,

s t a r t i n g w i t h monosized d r o p l e t s o f t h e Sauter mean diameter,

t h e number o f stages requ i red t o a t t a i n a steady s t a t e s i z e

d i s t r i b u t i o n was c a l c u l a t e d f o r t h e pu lsed column a t a l l

t h r e e pu lse frequences. The r e s u l t s agreed w e l l w i t h t h e

measured d i s t r i b u t i o n , as shown i n F ig . 5 f o r a frequency

o f 90 m i n - l .

As an a l t e r n a t i v e , a Monte Car lo random s e l e c t i o n method

was used f o r t h e same purpose. Th is i nvo lved expressing

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

PRATT

FIGURE 4.

0.3 0.4 0.5 0.6 0.8 Droplet Diameter. di (mm)

C o r r e l a t i o n o f d r o p l e t coalescence and breakage r a t e constants f o r pu lsed column ( f = 120 m i n - I ) .

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE 537

N = O 100

KEY: - Predicted Size Distribution

r --- Exptl. Steady Size Oistrib

Size Interval j

FIGURE 5. P red ic ted a o ~ r o a c h o f d r o o l e t s i z e d i s t r i b u t i o n t o steady s t a t e i n pulsed' column ( f = 90 min-1, Qc+Qd = 15cm3s-I).

t h e t o t a l d r o p l e t i n t e r a c t i o n s , i .e . , coalescences o f equal

and o f ad jacent s izes, and breakages, pe r u n i t v o i d volume

o r c ross -sec t ion i n h e i g h t ~ h , as f o l l o w s

A g iven i n t e r a c t i o n type was f i r s t se lec ted us ing random

numbers a l l o c a t e d i n p r o p o r t i o n t o t h e t h r e e summation terms

i n eqn. (20); t h e drop s i z e invo lved was then assessed from

f u r t h e r random numbers a l l o c a t e d amongst t h e j s izes . Th is

method gave r e s u l t s f o r t h e pu lsed column i d e n t i c a l w i t h those

obta ined by t h e prev ious method. Values o f d32 f o r t h e

packed column c a l c u l a t e d i n t h i s way a l s o agreed w e l l w i t h t h e

exper imental values .

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

538 PRATT

Coalescence Rate

The r a t e o f r e d d r o p l e t f o rmat ion can be c a l c u l a t e d by R s i m i l a r methods. Thus, f rom t h e d e f i n i t i o n o f A . J

T h i s represents a s e r i e s o f M simultaneous l i n e a r equat ions

which were so lved f o r t h e f . by Newton's method. Typ ica l J

r e s u l t s a re shown i n F ig . 6 f o r t h e pulsed column. ' The

Monte Car lo method can a l s o be used f o r t h i s purpose, u s i n g

random numbers t o assess t h e c o l o u r o f each se lec ted d r o p l e t .

Th is method, a p p l i e d t o t h e packed column, gave s i m i l a r

agreement w i t h t h e exper imental data (14 ) .

E f f e c t on Mass T rans fe r

The e f f e c t o f p o l y d i s p e r s i v i t y on mass t r a n s f e r r a t e was

i n v e s t i g a t e d t h e o r e t i c a l l y f o r the e x t r a c t i o n o f 5% w/v

aqueous a c e t i c a c i d w i t h MIBK. A stepwise procedure was used

f o r t h i s purpose, s t a r t i n g f rom t h e so lven t i n l e t and assuming

t h e steady s t a t e s i z e d i s t r i b u t i o n throughout w i t h p l u g f l o w

o f d ispersed phase.

The procedure cons is ted i n c a l c u l a t i n g t h e change i n

concen t ra t i on o f each d r o p l e t over h e i g h t i n t e r v a l s ah, taken

as 0.lOcm i n t h e case o f t h e packed column, and s t o r i n g t h e i r

r e s u l t i n g concent ra t ions; i n a l l , 1500-2000 d r o p l e t s were

considered, w i t h one s torage l o c a t i o n per d r o p l e t . The mean

dispersed and cont inuous phase composi t ions were then ca lcu la ted ,

assuming e i t h e r p l u g o r backmixed f l o w o f t h e l a t t e r . Fo l lowing

t h i s , Monte Car lo procedures were used as descr ibed e a r l i e r t o

modi fy t h e s to red values o f t h e d r o p l e t concen t ra t i ons i n

accordance w i t h changes due t o coalescence and breakage.

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

540 PRATT

Four cases were considered, v i z a p o l y d i s p e r s i o n ( i ) , w i t h

no d r o p l e t i n t e r a c t i o n s , ( i i ) , w i t h measured i n t e r a c t i o n ra tes ,

and ( i i i ) w i t h " i n f i n i t e " i n t e r a c t i o n s ; and ( i v ) , a mono-

d i spe rs ion o f t h e same Sauter mean diameter. The mass

t r a n s f e r c o e f f i c i e n t s f o r t h e d r o p l e t phase were c a l c u l a t e d

f rom t h e c o r r e l a t i o n o f Rozen and Bezzubova (20) f o r "medium

s ized" d rop le ts , and f o r t h e cont inuous phase f rom t h a t o f

Weber (21) except f o r t h e sma l les t s ize , f o r which G r i f f i t h ' s

formula no. 1 (22) was used. The r e s u l t s f o r 0.1% a c e t i c

a c i d i n t h e r a f f i n a t e , 10% holdup and p lug f l o w showed t h a t

packing he igh ts o f 251, 243, 222 and 228cm r e s p e c t i v e l y were

requ i red f o r t h e f o u r cases. The values f o r t h e backmixed

case were 63-69% h igher . The o p e r a t i n g diagram f o r case ( i i ) ,

w i t h a 1;0% r a f f i n a t e , i s shown i n Fig.7.

For t h e pu lsed column, the h e i g h t increment f o r mass

t r a n s f e r was taken as t h e p l a t e spacing, and d r o p l e t coalescence

and breakage were assumed t o occur o n l y du r ing passage through

t h e p l a t e s . Values o f kc f o r t h i s case were obta ined from

Thorsen and Ter jesen 's c o r r e l a t i o n (23) f o r Re>50, and from

G t i f f i t h ' s formula no. 1 (22) f o r s m a l l e r s izes. The e f f e c t

o f p o l y d i s p e r s i v i t y was found t o be s u r p r i s i n g l y smal l f o r t h i s

column; thus, t h e d i f f e r e n c e s i n numbers o f p l a t e s between

cases ( i ) and ( i i i ) were o n l y 2.6-4.6% f o r p l u g f l ow , and 1.4-

2.6% f o r backmixed f low, so t h a t d r o p l e t i n t e r a c t i o n s apparen t l y

have a n e g l i g i b l e e f f e c t on performance.

DISCUSSION

The present technique c l e a r l y s a t i s f i e s t h e need f o r a

d i r e c t method o f measurement o f d r o p l e t coalescence r a t e s .

Apart f rom t h e c o l o u r r e a c t i o n i t s e l f , t h e p r e l i m i n a r y d r o p l e t

s i z e e q u i l i b r a t i o n , a l though probably n o t e s s e n t i a l , simp1 i f i e s

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE 541

FIGURE 7. Pred ic ted o p e r a t i n g diagram f o r po lyd ispe rse system: e x t r a c t i o n o f a c e t i c a c i d i n a packed column ( p l u g f low, 10% holdup, a l l o w i n g f o r d r o p l e t coalescence and breakage).

cons iderab ly t h e i n t e r p r e t a t i o n o f t h e data.

The use o f a d i s c r e t e r a t h e r than a cont inuous s i z e

d i s t r i b u t i o n model t o represent t h e data a l s o leads t o

cons iderab le s i m p l i f i c a t i o n . An a l t e r n a t i v e d i s c r e t e model

has r e c e n t l y been proposed by J i r i c n y e t a1 (24,25), b u t t h i s

uses s i z e i n t e r v a l s based on simple m u l t i p l e s o f t h e volume

o f t h e sma l les t drop; i t t h e r e f o r e l a c k s t h e s i m p l i c i t y o f

t h e present model i n which d r o p l e t s pass t o an ad jacent s i z e

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

54 2 PRATT

range on coalescence o r breakage. I t i s t r u e t h a t some o f

t h e present assumptions, e s p e c i a l l y nos.. 2,4 and 5, appear

r e s t r i c t i v e , b u t t h e agreement o f t h e p r e d i c t e d and e x p e r i -

mental s i z e d i s t r i b u t i o n s suggest t h a t t h e accuracy o f t h e

11:odel i s adequate f o r most purposes.

One o t h e r assumption, t h a t a l l d r o p l e t s move w i t h t h e

same mean v e l o c i t y i r r e s p e c t i v e o f s i ze , does however r e q u i r e

f u r t h e r comment. Some j u s t i f i c a t i o n f o r t h i s assumption i s

prov ided by t h e analogous case o f the h indered s e t t l i n g o f

s o l i d suspensions, i n which a l l p a r t i c l e s appear t o depos i t

a t t h e same average ra te . On t h i s bas is i t would be

expected t h a t f l u c t u a t i o n s would occur i n t h e v e l o c i t i e s o f

i n d i v i d u a l d r o p l e t s w i t h i n swanns, l ead ing t o an apparent

d i s t r i b u t i o n o f res idence t imes even though t h e mean v e l o c i t y

may be constant. Such behaviour would account f o r t h e

apprec iab le d ispersed phase backmixing c o e f f i c i e n t s repor ted

f o r , e.g. packed columns (32), f o r which t h e r e i s evidence t h a t

i n f a c t backmixing does n o t occur i n t h i s phase (33 ) .

I t i s o f i n t e r e s t t o compare t h e d r o p l e t i n t e r a c t i o n r a t e s

a t t h e steady s t a t e f o r t h e two types o f columns used; thus,

t h e values summarized i n Table 2 were c a l c u l a t e d f rom t h e

smoothed values o f t h e r a t e constants. The small i n t e r a c t i o n

r a t e f o r t h e pulsed, as compared w i t h t h e packed column i s due

t o t h e occurrence o f i n t e r a c t i o n s o n l y w i t h i n and immediately

ad jacent t o the p l a t e s i n t h e former case, b u t throughout t h e

vo id space i n t h e l a t t e r , f o r which t h e mean d r o p l e t l i f e t i m e

was o n l y 0.67 sec.

There i s .a need f o r s i m i l a r coalescence data f o r o t h e r

so l ven t systems, t o a s c e r t a i n t h e e f f e c t o f phys ica l p r o p e r t i e s

on t h e r a t e c o e f f i c i e n t s . Un fo r tuna te l y , t e s t s w i t h o t h e r

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE

TABLE 2 D r o p l e t I n t e r a c t i o n Rates

Basis: 10% holdup; 2,000 d r o p l e t s

column i Type O f

Packed

Pulsed

Pulse

f requency (min-') .

- Coal gual

923

39

147

141 -

No. o f i n t e r a c t i o n s i n

so lvents , e.g. kerosene, to luene and b u t y l aceta te , showed t h a t

t h e present c o l o u r r e a c t i o n i s t o o s low t o be o f use w i t h them,

presumably because they are i n s u f f i c i e n t l y p o l a r . However, some

guidance i n t h i s regard can be prov ided by dimensional

ana lys i s . Thus, assuming d e n s i t y d i f f e r e n c e and i n t e r f a c i a l -

t ens ion t o be t h e c o n t r o l l i n g phys ica l p r o p e r t i e s t h e f o l l o w i n g

r e l a t i o n s a r e obta ined p xz F ~ ( L ~ ) o r F ~ ( K ~ ) = const . (w9-) (22)

where t h e holdup term, xd, accounts f o r t h e e f f e c t o f f l o w

r a t e s and t h e groups on t h e l e f t hand s ide a r e as f o l l o w s Column Type Operat ion Fhk,) F2(Kj)

Packed *

Pulsed M i x e r - s e t t l e r kj4A(9-

Y

Emulsion 4 7 5 k, A! g

4 @P

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

544 PRATT

Values of t h e exponents p and q i n eqn. (22) can be obta ined

from b and c i n eqn. ( l a ) , as g i ven i n r e f s . 13-16.

S i m i l a r data a r e a l s o r e q u i r e d f o r o t h e r types o f e x t r a c t o r ;

however, m o d i f i c a t i o n s t o t h e method would be r e q u i r e d f o r those

contactors , e.g., w i t h r o t a r y a g i t a t o r s , wh ich cannot be

p a r t i t i o n e d v e r t i c a l l y . I n such cases i t would perhaps be

p r a c t i c a b l e t o omi t t h e s i z e e q u i l i b r a t i o n stage, and t o use

non-zero values o f A . ob ta ined from t h e s i z e d i s t r i b u t i o n data J

i n eqn. ( 9 ) .

F i n a l l y , t h e u n s a t i s f a c t o r y s t a t e o f knowledge o f d r o p l e t

mass t r a n s f e r c o e f f i c i e n t s r e q u i r e s mention. Th is i s c l e a r l y

shown by the comparison i n Figs. 8 and 9 o f t h e kc and kd

values c a l c u l a t e d f o r t h e MIBK-acetic acid-water system from

var ious t h e o r e t i c a l and exper imental sources f o r i s o l a t e d

d r o p l e t s i n " s i n g l e f i l e * " columns. These i n t u r n may w e l l

d i f f e r w ide ly f rom those f o r d r o p l e t "swarms" i n r e a l

ex t rac to rs , i n which the e f f e c t o f d r o p l e t v e l o c i t y under

hindered s e t t l i n g cond i t i ons may be expected t o d i f f e r f rom

t h a t f o r i s o l a t e d d r o p l e t s .

CONCLUSIONS

(1) A novel c o l o r i m e t r i c method has been developed f o r t h e

d i r e c t measurement o f i n t e r - d r o p l e t coalescence r a t e s i n

packed and pu lsed p l a t e e x t r a c t i o n columns.

(2 ) The data have been i n t e r p r e t e d s a t i s f a c t o r i l y i n terms

o f second o rde r coalescence and f i r s t o r d e r breakage

r a t e constants us ing a d i s c r e t e popu la t i on balance model

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE 545

Weber (21) e-• Thorsen and Terjeren (23) -A Garner et al. (26) - - - Calderbank and Moo Young, Eqn.1 (27)

0-0 Griffith (22) Calderbank and Moo Young, Eqn. 3 (27)

FIGURE 8. Comparison o f p red ic ted k values f o r MIBK d r o p l e t s (MIBK-acetic ac id-water &stem).

'13 based on a 2 progress ion o f mean d r o p l e t diameters

The de r i ved r a t e constants can be used t o p r e d i c t w i t h

acceptable accuracy t h e steady s t a t e s i z e d i s t r i b u t i o n

and t h e coalescence r a t e .

A t h e o r e t i c a l study o f mass t r a n s f e r i n a po lyd isperse

system, assuming mass t r a n s f e r c o e f f i c i e n t s based on

values f o r i s o l a t e d d rop le ts , has shown t h a t t h e e f f e c t

o f d r o p l e t coalescence and breakage i s r e l a t i v e l y smal l ,

e s p e c i a l l y f o r t h e pu lsed column.

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

546 PRATT

M Rozen and Bezzubova (20) - -- Skelland and Wellek, Eqn. 9 (31) 0-0 Kronig and Brink (28) --- Handlos and Baron (30) h - A Newman (29)

FIGURE 9. Comparison o f p r e d i c t e d k va lues f o r MIBK d r o p l e t s (MIBK-acetic acid-water s jstem).

(5) Fur the r data o f s i m i l a r t ype a r e requ i red f o r o t h e r

systems, t o determine t h e e f f e c t o f phys ica l p r o p e r t i e s ,

and f o r o t h e r e x t r a c t o r types.

(6 ) There i s an u rgen t need f o r data on area mass t r a n s f e r

c o e f f i c i e n t s f o r t h e d r o p l e t swarms present i n r e a l

e x t r a c t o r s .

NOTATION

B = constant i n eqn. ( 3 )

R B i ,B i , j = elements rep resen t ing breakage terms i n ith row

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE

and jth column of matrix A (eqns. 9,lO)

= elements representing coalescence terms in i t h

row and jth column of matrix A (eqns. 9,lO)

= mean diameter of jth droplet s i ze range, cm.

= mean volume-surface ( i .e. Sauter mean) droplet

diameter, cm.

2 -1 = longitudinal d i f fus iv i ty , cm s

= pulse frequency, min-l

= number f rac t ion of red droplets in s i ze range j

= height of packing, cm

= f i r s t order breakage r a t e constant fo r droplets of

s i ze j, s - l (packed column; pulsed column, mixer-

s e t t l e r operat ion), cm s-' (pulsed column, emulsion

operation)

= second order coalescence r a t e constant fo r droplets 3 -1 of s i ze j, cm s (packed column), cm2s-l and cm 4

- 1 s (pulsed column, mixer-settler and emu1 sion operation respect ively) .

= second order coalescence r a t e constant for droplets

of s ize j with j-1 (dimensions as fo r k . ) . J

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

P RATT

= stage h e i g h t (pu lsed column), cm

= no. o f d r o p l e t s i z e i n t e r v a l s

= no. o f stages (pu lsed column)

= number concen t ra t i on o f d r o p l e t s o f s i z e j, cm - 3

(packed columns; pu l sed column, emulsion

operat ion) , cm-*(pul sed column, m i x e r - s e t t l e r

opera t ion )

3 -1 Q,>Qd = vo lumet r i c f l o w r a t e o f phase, cm s

Red, Reid = d r o p l e t Reynolds number, dj ir/vc,dj ir/vd

respec t i ve1 y

= ( f o r pul'sed column), f o r U /x f o r mixer - d d s e t t l e r o r emulsion opera t ion r e s p e c t i v e l y

(eqns. 9 and l o ) , s-' o r cm s-'

= t ime, s.

= d r o p l e t con tac t t ime, s.

= s u p e r f i c i a l v e l o c i t y o f cont inuous o r d ispersed

phase r e s p e c t i v e l y , cm s-'

= mean d r o p l e t v e l o c i t y r e l a t i v e t o s t a t i o n a r y

coordinates, cm s-'

= f r a c t i o n a l holdup o f d ispersed phase

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE 549

Greek L e t t e r s

a = backmixing r a t i o ( i .e. r a t i o o f backf low t o n e t

forward f l o w )

A . A . R = r a t e o f change o f number concen t ra t i on o f t o t a l

J . J and of red d r o p l e t s r e s p e c t i v e l y , s - I

(packed column), s - I (pu lsed column)

V C

Subsc r ip ts

= r e s i d u a l vec to r

= f r a c t i o n a l voidage o f packing

= t o t a l number o f d r o p l e t i n t e r a c t i o n s pe r u n i t

v o i d volume o f packing,

= k inemat ic v i s c o s i t y o f cont inuous phase, 2 - 1

cm s

= cont inuous phase

= d ispersed phase

= s tage number

= d r o p l e t s i z e range

REFERENCES

C.A. S le i che r , A.1.Ch.E. J n l . , - 5, 145 (1959).

T. Miyauchi and T. Verrneulen, I.E.C. Fund, - 2, 133 (1963).

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

. 550 PRATT

C.A. S le i che r , A.1.Cti.E. J n l . , 5, 529 (1960).

H.R.C. P r a t t and M.O. Garg, I.E.C. Proc. Des. and Develop.

20, 439 (1981). - R.B. Olney, A.1.Ch.E. J n l . , 10, 827 (1964).

V. Rod, B r i t . Chem. Eng., 11, 483 (1966).

R.H. Chartres and W.J. Korchinsky, Trans. I n s t . Chem. Eng.,

53, 247 (1975). - K.J. Valentas and N.R. Amundsen, I.E.C. Fund, - 5, 533 (1966)

R.K. Ba jpa i . D. Ramkrishna and A. Prokop, Chem. Eng. Sc i . ,

31, 913 (1976). - C.A. Coula log lou and L.L. Tav lar ides, Chem. Eng. Sc i . ,

32, 1289, (1977). - H. Sovova, Chem. Eng. Sci., - 36, 1567 (1981).

J.A. Hami l ton and H.R.C. P r a t t , Proc. I n t e r n a t . Solv.

Extn. Conf. 1980 (ISEC '801, Session 3, Paper no. 80-19.

J.A. Hamilton, Droplet Coalescence and Breakage i n a

Packed Liquid Extraction C o l m , Ph.D. Thesis,

U n i v e r s i t y o f Melbourne (1981).

J.A. Hamil ton and H.R.C. P r a t t , A.1.Ch.E. Jn l . , 30, 442

(1984).

M.O. Garg, Measurement and Modelling of Droplet

Coalescence and Breakage i n a EILctsed Plate Liquid

Extraction Column, Ph.D. Thesis, U n i v e r s i t y o f Melbourne

(1982).

M.O. Garg and H.R.C. P r a t t , A.1.Ch.E. Jn l . , 30, 432

(1984).

R. Gayler, N.W. Roberts and H.R.C. P r a t t , Trans. I n s t .

Chem. Eng., 31, 57 (1953).

J.D. Thornton, Trans. I n s t . Chem. Eng., 35, 316 (1957).

I. Barroda le and F.D.K. Roberts, Corn. o f t h e A.C.M.,

17, 319 (1974). - A.M. Rozen and A.I. Bezzubova, Foundat ions Chem. Eng.

(Consul tants Bureau Trans.), - 2, 715 (1968).

Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014

DROPLET COALESCENCE 551

M.E. Weber, I.E.C. Fund, - 14, 365 (1975).

R.M. G r i f f i t h , Chem. Eng. Sci., 12, 198 (1960).

G. Thorsen and S.G. Terjesen, Chem. Eng. Sci., 17, 137

(1962).

V. J i r i c n y , M. K ra tky and J. Prochazka, Chem. Eng. Sci.,

34, 1141 (1979). - H. Sovova and J. Prochazka, Chem. Eng. Sci., 36, 163

(1981).

F.H. Garner, A. Foord and M. Tayeban, J. Appl. Chem.,

9, 315 (1959). - P.H. Calderbank and M.B. Moo-Young, Chem. Eng. Sc i . ,

16, 39 (1961).

R. Kron ig and J.C. B r ink , Appl. Sc i . Res., g, 142

(1950).

A.B; Newman, Trans. A.I. Chem. E., 27, 310 (1931).

A.E. Handlos and T. Baron, A.1.Ch.E. J n l . , - 10, 491

(1964).

A.H.P. Ske l land and R.M. Wellek, A.1.Ch.E. J n l . , g, 491 (1964).

T. Vermeulen, J.S. Moon, A. Hennico and T. Miyauchi,

Chem. Eng. Prog., 62, 95 (1966).

R. Gayler and H.R.C. P r a t t , Trans. I n s t . Chem. Eng.,

35, 267 (1957). -

Received by Editor

October 18, 1983 Dow

nloa

ded

by [

Col

umbi

a U

nive

rsity

] at

09:

02 0

6 O

ctob

er 2

014