DROPLET COALESCENCE AND BREAKAGE RATES IN LIQUID EXTRACTION COLUMNS
Transcript of DROPLET COALESCENCE AND BREAKAGE RATES IN LIQUID EXTRACTION COLUMNS
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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
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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
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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
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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
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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 .
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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
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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
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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.
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FIGURE 2. Smoothing of f. values for packed column (10.6% holdup). J
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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 - ~ ) .
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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
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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
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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)
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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 -
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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
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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
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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 ) .
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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 .
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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.
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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Received by Editor
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