Influence of Reaction Reversibility on Continuous-Flow Extraction by

Chapter 5

Influence of Reaction Reversibility on Continuous-Flow Extraction by Emulsion Liquid Membranes D. L. Reed , A. L. Bunge, and R. D. Noble Liquid Membranes Downloaded from pubs.acs.org by UNIV OF CALIFORNIA SAN DIEGO on 06/03/15. For personal use only.

1

13,

2

Colorado School of Mines, Golden, CO 80401 National Bureau of Standards, Center for Chemical Engineering, Boulder, CO 80303 1

2

This paper examines theoretically the continuous flow extraction by emulsion globules in which the transferring solute reacts with an internal reagent. The reversible reaction model is used to predict performance. These results are compared with advancing front calculations which assume an irreversible reaction. A simple criterion which indicates the importance of reaction reversibility on performance is described. Calculations show that assuming an irreversible reaction can lead to serious underdesign when low solute concentrations are required. For low solute concentrations an exact analytical solution to the reversible reaction problem is possible. For moderate solute concentrations, we have developed an easy parameter adjustment of the advancing front model which reasonably approximates expected extraction rates. E m u l s i o n l i q u i d membranes (ELM) a r e double e m u l s i o n s formed by mixi n g two i m m i s c i b l e phases and t h e n d i s p e r s i n g t h e r e s u l t i n g emuls i o n i n another c o n t i n u o u s phase under a g i t a t i o n . Proposed a p p l i c a t i o n s f o r e m u l s i o n l i q u i d membranes have i n c l u d e d s e l e c t i v e r e c o v ery of m e t a l i o n s ( 1 - 1 2 ) , s e p a r a t i o n of hydrocarbons (13-16), r e moval o f t r a c e o r g a n i c contaminants ( 1 7 - 2 7 ) , and e n c a p s u l a t i o n o f r e a c t i v e enzymes o r whole c e l l s ( 2 8 - 3 6 ) . N e a r l y a l l o f the l a r g e number of e x p e r i m e n t a l and t h e o r e t i c a l s t u d i e s r e p o r t e d have been performed i n b a t c h mode. W h i l e t h e i n d u s t r i a l s i g n i f i c a n c e of a c o n t i n u o u s s t e a d y - s t a t e o p e r a t i o n i s o b v i o u s , o n l y H a t t o n and coworkers have examined f l o w c o n f i g u r a t i o n s b o t h t h e o r e t i c a l l y and e x p e r i m e n t a l l y (27,37,38). They cons i d e r e d t h e s i t u a t i o n when a s o l u t e (A) d i f f u s e s t h r o u g h t h e o i l phase membrane and then r e a c t s w i t h a r e a g e n t (B) t r a p p e d i n t h e i n t e r n a l d r o p l e t s t o produce a p r o d u c t (P) a c c o r d i n g t o ,

To whom correspondence should be sent

3

0097-6156/87/0347-0062$06.00/0 © 1987 American Chemical Society

Liquid Membranes Downloaded from pubs.acs.org by UNIV OF CALIFORNIA SAN DIEGO on 06/03/15. For personal use only.

5.

REED ET AL.

Influence of Reaction Reversibility

63

The product i s i n s o l u b l e i n t h e membrane phase and t r a p p e d i n t h e i n t e r n a l d r o p l e t s a l o n g w i t h any u n r e a c t e d r e a g e n t . T h i s scheme p e r m i t s a c c u m u l a t i o n o f r e a c t e d s o l u t e w i t h i n t h e i n t e r n a l phase d r o p l e t s a t s i g n i f i c a n t l y l a r g e r concentrations than the o r i g i n a l solute concentration i n the feed. Examples o f such an ELM system a r e t h e e x t r a c t i o n o f o r g a n i c bases by a c i d i f i e d i n t e r n a l d r o p l e t s o r t h e e x t r a c t i o n o f a c i d s by basic d r o p l e t s . Experimental r e s u l t s f o r a continuous i n t e r n a l r e c y c l e r e a c t o r have been r e p o r t e d f o r ammonia e x t r a c t i o n u s i n g a s u l f u r i c a c i d i n t e r n a l phase ( 2 7 ) . To d e s i g n l a r g e s c a l e e x t r a c t o r s from bench s c a l e d a t a r e q u i r e s p r e d i c t i v e models which d e s c r i b e t h e p h y s i c o c h e m i c a l p r o c e s s es o p e r a t i n g i n an ELM e x t r a c t i o n u n i t c o r r e c t l y . Many o f t h e models proposed t o d e s c r i b e ELM e x t r a c t i o n s have l i m i t e d u t i l i t y s i n c e s e v e r a l parameters cannot be determined i n d e p e n d e n t l y . An i m p o r t a n t e x c e p t i o n i s t h e a d v a n c i n g f r o n t model p r e s e n t e d by Ho e t a l . (26) and H a t t o n e t a l . ( 2 7 ) . They c o n s i d e r t h e o p e r a t i o n a s a d i f f u s i o n - c o n t r o l l e d e x t r a c t i o n c o u p l e d w i t h an i n s t a n t a n e o u s r e a c t i o n when Κ i s i n f i n i t e . The r e s u l t i n g s p e c i e s c o n t i n u i t y e q u a t i o n i s solved with a perturbation technique producing several a l g e b r a i c e x p r e s s i o n s which depend o n l y on i n d e p e n d e n t l y m e a s u r a b l e - p h y s i c a l and o p e r a t i n g p a r a m e t e r s . H a t t o n and coworkers use t h e z e r o - o r d e r s o l u t i o n of t h e advancing f r o n t theory t o analyze t h e i r continuous ammonia e x t r a c t i o n e x p e r i m e n t s and t o p r e d i c t m u l t i s t a g e p e r f o r mance o f m i x e r - s e t t l e r t r a i n s and cascade m i x e r s (27,37,38). R e c e n t l y , Bunge and Noble (39) have extended t h e approach o f Ho e t a l . (26) t o i n c l u d e r e v e r s i b i l i t y o f r e a c t i o n 1. Batch e x t r a c t i o n s and c a l c u l a t i o n s from t h i s r e v e r s i b l e r e a c t i o n model de monstrate that r e a c t i o n r e v e r s i b i l i t y s i g n i f i c a n t l y a f f e c t s e x t r a c t i o n performance i n some cases (39,40). I n t h i s paper, we extend these batch e x t r a c t i o n c a l c u l a t i o n s t o a continuous s t i r r e d - t a n k e x t r a c t o r . We show t h a t a s i n g l e , d i m e n s i o n l e s s parameter can be used t o a s s e s s t h e l i k e l y c o n t r i b u t i o n o f r e v e r s i b i l i t y f o r a g i v e n set of conditions. Continuous Stirred-Tank

Extractor

Design

F i g u r e 1 diagrams a c o n t i n u o u s s t i r r e d - t a n k e x t r a c t o r and i n d i c a t e s the p e r t i n e n t d e s i g n p a r a m e t e r s . I n a w e l l - s t i r r e d , s t e a d y - s t a t e e x t r a c t o r , t h e b u l k phase c o n c e n t r a t i o n o f s o l u t e i s u n i f o r m throughout t h e tank and e q u a l t o t h e c o n s t a n t o u t l e t c o n c e n t r a t i o n , Cj[b« The bulk phase f e e d e n t e r s t h e e x t r a c t o r a t a c o n c e n t r a t i o n C^B

a

n

d

a

volumetric

f l o w r a t e , v ^ . The e m u l s i o n phase i s made

from an i n t e r n a l phase s o l u t i o n o f r e a g e n t , C ^ i , and a s o l u t e f r e e membrane phase. The volume f r a c t i o n o f membrane phase t o t o t a l e m u l s i o n i s denoted as f . I f some o f t h e spent e m u l s i o n i s r e c y c l e d t h r o u g h t h e e m u l s i f i e r , t h e average membrane phase concen t r a t i o n o f s o l u t e f e d t o t h e e x t r a c t o r , C ^ , w i l l be n o n z e r o . The v o l u m e t r i c f l o w r a t e o f t h e e m u l s i o n phase i s v . The volume o f b u l k and e m u l s i o n phases i n t h e e x t r a c t o r a r e and V , r e s p e c t i v e ly. For steady-state operation of the s t i r r e d extractor i n Figure 1, t h e f l o w r a t i o o f f e e d t o e m u l s i o n , ν ^ / ν , w i l l u s u a l l y e q u a l the volume r a t i o , V^/V . m

e

e

θ

e


64

LIQUID MHMBRANHS: THEORY A N D APPLICATIONS

Membrane Phase

Internal Phase Cm

Bulk Phase Feed

Membrane Phase

Emulsion

v

CÀb. b

Spent Internal Phase

Emulsifier

® v .v ν Ab b

(

u

Extractor Figure 1

v

Extracted Bulk Phase

b Settler

Schematic diagram of a c o n t i n u o u s f l o w u s i n g e m u l s i o n l i q u i d membranes

extraction

REED ET AL.

5.


65

Four d e s i g n parameters can be v a r i e d i n d e p e n d e n t l y t o a c h i e v e a d e s i r e d degree o f s o l u t e e x t r a c t i o n : the f r a c t i o n o f t h e emul s i o n which i s membrane phase ( f ) , t h e i n i t i a l c o n c e n t r a t i o n o f m

reagent i n the i n t e r n a l phase ( C g i ) , t h e f e e d r a t i o o f bulk phase t o e m u l s i o n phase ( v ^ / v ) , and the r e s i d e n c e time o f the e m u l s i o n (t = V /v ). That i s , t h e e x t e n t o f g l o b u l e u t i l i z a t i o n depends on the mass o f r e a g e n t a v a i l a b l e and the exposure time o f the emul s i o n t o the b u l k phase s o l u t i o n . These same f o u r c r i t e r i a f u l l y s p e c i f y t h e d e s i g n even f o r an a l t e r n a t e e x t r a c t o r c o n f i g u r a t i o n , such as H a t t o n s c o n t i n u o u s i n t e r n a l r e c y c l e m i x e r , which p e r m i t s b/ e * b e e q u i v a l e n t l y t * t ^ = V^/v^. We show l a t e r t h a t e


e

e

e

1

v

y

v

/ v

o

r

e

v a r y i n g the hold-up r a t i o (Vt,/V ) w h i l e keeping f , Cg^, v ^ / v and t c o n s t a n t does n o t a f f e c t the amount o f e x t r a c t i o n . e

m

e

e

E m u l s i o n L i q u i d Membrane Models To e s t i m a t e the e x t e n t o f g l o b u l e u t i l i z a t i o n f o r a g i v e n exposure time r e q u i r e s a m a t h e m a t i c a l d e s c r i p t i o n o f a d i f f u s i o n - c o n t r o l l e d e x t r a c t i o n c o u p l e d w i t h c h e m i c a l r e a c t i o n . Two approaches, t h e r e v e r s i b l e r e a c t i o n model ( 3 9 ) , and the a d v a n c i n g f r o n t model ( 2 7 ) , w i l l be d e s c r i b e d and compared. B a s i c assumptions i n b o t h models i n c l u d e : (1) membrane/bulk and membrane/internal phases a r e i m m i s c i b l e , (2) l o c a l phase e q u i l i b r i u m between membrane and i n t e r n a l phases, (3) no i n t e r n a l c i r c u l a t i o n i n the g l o b u l e , (4) u n i f o r m g l o b u l e s i z e , (5) mass t r a n s f e r i s c o n t r o l l e d by g l o b u l e d i f f u s i o n , ( 6 ) i n t e r n a l d r o p l e t s are s o l u t e s i n k s w i t h f i n i t e c a p a c i t y , (7) r e a c t i o n o f s o l u t e i n t h e i n t e r n a l phase i s i n s t a n t a n e o u s , (8) no c o a l e s c e n c e and r e d i s t r i b u t i o n of g l o b u l e s , and (9) a w e l l - m i x e d tank w i t h an e x p o n e n t i a l r e s i dence t i m e d i s t r i b u t i o n o f e m u l s i o n g l o b u l e s . Of t h e s e , assumption 8 may seem the most d a r i n g . I f c o a l e s cence and r e d i s t r i b u t i o n were e x t e n s i v e , t h e e m u l s i o n g l o b u l e s would approach the l i m i t o f u n i f o r m s o l u t e c o n c e n t r a t i o n and d i f f u s i v e r e s i s t a n c e s w i t h i n the g l o b u l e would not c o n t r i b u t e t o e x t r a c t i o n r a t e s . However, e x p e r i m e n t a l o b s e r v a t i o n s i n d i c a t e t h a t once formed, g l o b u l e s tend t o r e t a i n t h e i r i d e n t i t i e s . When g l o b u l e s do c o a l e s c e and b r e a k , u n d e s i r a b l e l e a k a g e o f i n t e r n a l drop l e t s i n t o the b u l k phase r e s u l t s . C o n s e q u e n t l y , p r e f e r r e d g l o b u l e f o r m u l a t i o n s m i n i m i z e breakage and r e c o a l e s c e n c e , t h e r e b y a s s u r i n g a p p l i c a b i l i t y o f assumption 8. The s o l e d i f f e r e n c e between the a d v a n c i n g f r o n t and r e v e r s i b l e r e a c t i o n approach i s t h e assumed s i z e o f the e q u i l i b r i u m c o n s t a n t f o r r e a c t i o n 1. Advancing f r o n t models assume t h a t r e a c t i o n 1 i s i r r e v e r s i b l e ; Κ i s i n f i n i t e l y l a r g e . F i n i t e values f o r Κ are a s sumed i n the r e v e r s i b l e r e a c t i o n t h e o r y . R e v e r s i b l e R e a c t i o n Model. I n an i d e a l , c o n t i n u o u s - s t i r r e d e x t r a c t o r , t h e s o l u t e c o n c e n t r a t i o n o f t h e b u l k phase i s u n i f o r m and equal t o the o u t l e t c o n c e n t r a t i o n , C ^ . The s o l u t e c o n c e n t r a t i o n i n the membrane p o r t i o n o f the g l o b u l e , C ^ , i s g i v e n as (39-m):

66

LIQUID MEMBRANES: THEORY AND APPLICATIONS

D


^Am 3t

eff

3_ 3r

~P~

/ . ^Am\ I 8r I r

t - 0

C

A m

-

cim

r = R

C

A m

=

K

=

0

m b

ΛΜ I f

/ ^ A i I I 3t

+

(R > r > 0) Clb

( t Ϊ 0)

^ P l ) 3t

(2)

(3) (it)

3C = 0

r

9r

(for a l l t )

(5)

where R i s t h e g l o b u l e r a d i u s , K b i s t h e s o l u t e p a r t i t i o n c o e f f i c i e n t between t h e membrane and bulk phases and f i s t h e volume f r a c t i o n o f t h e g l o b u l e o c c u p i e d by t h e membrane phase. For a com p o s i t e medium l i k e _ a n e m u l s i o n g l o b u l e , t h e mean e f f e c t i v e d i f f u s i o n c o e f f i c i e n t , D f f , based on t h e membrane d r i v i n g f o r c e , i n c l u d e s d i f f u s i o n o f b o t h r e a c t e d and u n r e a c t e d s o l u t e t h r o u g h t h e i n t e r n a l phase i n a d d i t i o n t o s o l u t e d i f f u s i o n through t h e membrane phase (24, 25, 3 9 ) . The i n i t i a l c o n d i t i o n , E q u a t i o n 3, assumes t h a t t h e membrane c o n c e n t r a t i o n i s ^ u n i f o r m w i t h p o s i t i o n a t C^ . £or a f r e s h e m u l s i o n f e e d , C ^ i s z e r o ; nonzero v a l u e s f o r C^ a r i s e when p r e v i o u s l y c o n t a c t e d e m u l s i o n i s t h o r o u g h l y mixed and r e i n t r o d u c e d . Nonzero v a l u e s a l s o o c c u r whenever t h e s o l u t e i s s o l u b l e i n t h e membrane phase and t h e membrane phase i s r e u s e d as shown i n F i g u r e 1. The c o n c e n t r a t i o n s o f s o l u t e A and p r o d u c t Ρ i n t h e i n t e r n a l phase, C ^ and Cp^, a r e r e s t r i c t e d by r e a g e n t c o n s e r v a t i o n and phase and r e a c t i o n e q u i l i b r i a : m

m

e

m

m

c

Ai

c

/K

( 6 )

- Am mi :

π 'Pi

i

B i

= "

m

(7)

1+K C " Ai A

where C g i i s t h e i n i t i a l c o n c e n t r a t i o n o f reagent Β i n t h e i n t e r n a l phase d r o p l e t s , and K ± i s t h e s o l u t e p a r t i t i o n c o e f f i c i e n t between t h e membrane and i n t e r n a l phases, assumed t o be con s t a n t . I f e q u i l i b r i u m between t h e membrane and i n t e r n a l phase i s a c h i e v e d i n s t a n t a n e o u s l y , and u s i n g t h e f o l l o w i n g d e f i n i t i o n s , m

η

=

r R

D

;

τ

eff,NR R*

t

( 8 )

C

Am C° Κ . Ab mb

oî 1

= f — m v. b

(9)

Κ

K

mb

(10)

5.


REED ET AL.

ν

+

σ

Κ (11)

2

m

= Κ C Liquid Membranes Downloaded from pubs.acs.org by UNIV OF CALIFORNIA SAN DIEGO on 06/03/15. For personal use only.

67

K

D

K

v

K

b

mi

(12)

Bi

C

(13)

/ K

mb A b m i

(14)

/ D

eff eff,NR

Equations

2-5 become:

d^m

λ

9

/

^^m

2

(15) 1 • (σ /q 2

Φ

1

- 0

3η

2

x

)[1 + σ /(1 + oS 4> ) ] 3

m

1 > η £ 0

(16)

τ 2 0

(17)

for a l l τ

(18)

The p l u s s i g n on a\ and σ d i s t i n g u i s h e s t h e s e d e f i n i t i o n s from a and σ p r e s e n t e d i n p r e v i o u s papers on b a t c h e x t r a c t i o n (39-41), which a r e based on the volume r a t i o ( V / V ^ ) r a t h e r than the r a t i o of v o l u m e t r i c f l o w r a t e s ( v / v ) . The d i s t i n c t i o n between b a t c h and c o n t i n u o u s e x t r a c t o r s i s o n l y i m p o r t a n t when t h e r a t i o o f v o l u m e t r i c f l o w r a t e s ( v / v ^ ) d i f f e r from t h e e x t r a c t o r volume r a t i o (Vg/V^). The d e f i n i t i o n s f o r σ and a\ as w e l l as σ and σ become i d e n t i c a l f o r o p e r a t i o n s r e s t r i c t e d t o v /Vfc = v /v . The r a t i o o f _ t h e mean e f f e c t i v e d i f f u s i o n c o e f f i c i e n t when r e a c t i o n s o c c u r , D f f , t o the d i f f u s i o n c o e f f i c i e n t when t h e r e i s no r e a c t i o n , D f f , i s estimated u s i n g the J e f f e r s o n - W i t z e l l S i b b e t t e q u a t i o n (42,43) f o r a composite media and c o r r e c t i n g f o r s o l u t e r e a c t i o n i n t h e i n t e r n a l phase: 2

x

2

e

e

b

e

χ

2

e

2

e

b

e

e

| N R

ι ψ

I,

D

eff,

R πι (19)

λ = eff,NR


68

D

D

D

eff,k = m

τ 4(U2p)

1

l . 2

2 (D. Ύ. /K . ) D ι k rai m (DiY /K .) - D mi m

F,k


t

ρ = 0.403 (1 - f )

R

Y

ml

D

m

D

Fk +

m

2 p D

, k=R

, k=R K

mi

D [ n

J

(20)

or

NR (21) (22)

0.5

2

or NR

Fk

l in j D

i k mi " '

and

(1+σϊφ )

/ K

/ K

•1/3

ÏJ

Ύ. = 1 •

_ J L _ 4(1+2p)

Vk D

K k

1

J

γ

(23)

= 1 N R

ιη

E q u a t i o n 19 d e t e r m i n e s the mean v a l u e o f the r e a c t i o n - i n f l u e n c e d t o n o n r e a c t i o n d i f f u s i o n c o e f f i c i e n t r a t i o between t h e i n l e t and max imum membrane c o n c e n t r a t i o n s . D e r i v a t i o n and d i s c u s s i o n of Equa t i o n s 19 t h r o u g h 23 a r e d e t a i l e d e l s e w h e r e (39,40). Advancing F r o n t Model. The a d v a n c i n g f r o n t model f o l l o w s a s i m i l a r approach e x c e p t t h a t s o l u t e d i f f u s i o n o n l y o c c u r s t h r o u g h the f u l l y reacted outer s h e l l . The n o - r e a c t i o n e f f e c t i v e d i f f u s i o n c o e f f i c i e n t , Deff,NR» a p p l i e s i n t h i s c a s e . The s o l u t e c o n c e n t r a t i o n i s zero a t the d i m e n s i o n l e s s l o c a t i o n of the r e a c t i o n f r o n t , χ, which moves from the g l o b u l e s u r f a c e (χ=1) toward the g l o b u l e c e n t e r . H a t t o n e t a l . (27) p r e s e n t e d the z e r o - o r d e r o r pseudosteadys t a t e s o l u t i o n f o r the a d v a n c i n g f r o n t t h e o r y , d e s c r i b e d h e r e i n our n o m e n c l a t u r e : φ =1 ra η T

^m 3n

1

[ ψ*

1-χ

]

(21)

(25)

1-χ

n=1 σΪ

J

σ

3

(1- 3χ

2

•

2χ ) 3

(26)

τ which a p p l i e s when C^m ata

3

»

σ?

(σΐ

i s z e r o and

+ σΪ)

the

condition (27)

i s met. E q u a t i o n 27 s t a t e s i n d i m e n s i o n l e s s form t h a t the c a p a c i t y of the g l o b u l e f o r a r e a c t i o n - b a s e d e x t r a c t i o n , measured by the r e a g e n t c o n c e n t r a t i o n and i n t e r n a l phase volume, f a r exceeds the c a p a c i t y of the g l o b u l e f o r e x t r a c t i o n by s o l u b i l i t y a l o n e . This r e s t r i c t i o n can be r e l a x e d by i n c l u d i n g a d d i t i o n a l terms i n t h e p e r t u r b a t i o n s e r i e s (26).

5.

REED ET AL.

Continuous


69

Stirred-Tank Extraction

The o v e r a l l s o l u t e t r a n s f e r r a t e , m^, must be r e l a t e d t o t h e s o l u t e f l u x i n t o the globules present a s : 3C 2

l^R D


Ab

eff

Am r-R

m 3r

dn(t)

(28)

where N j i s t h e t o t a l number o f g l o b u l e s i n t h e e x t r a c t o r , and n ( t ) i s t h e number o f g l o b u l e s w i t h r e s i d e n c e t i m e s l a r g e r t h a n t . The diameter o f a l l g l o b u l e s i s assumed t o be 2R. For a w e l l - m i x e d t a n k , t h e g l o b u l e r e s i d e n c e t i m e d i s t r i b u t i o n i s d e s c r i b e d as -t/t

n(t)

(29)

w i t h t , t h e average e m u l s i o n r e s i d e n c e , e q u a l t o t h e r a t i o of t h e volume o f e m u l s i o n i n t h e t a n k , V , t o t h e e m u l s i o n v o l u m e t r i c f l o w r a t e , v . A f t e r an o v e r a l l mass b a l a n c e on t h e e m u l s i o n , E q u a t i o n 28 i n d i m e n s i o n l e s s form becomes: e

e

e

3φ

-τ/τ. dx

3η where τ

θ

(30)

η=1

represents the dimensionless emulsion residence time:

D V eff,NR e (3D R ν e and t h e d i m e n s i o n l e s s i n l e t c o n c e n t r a t i o n , σ{ - Κ K b C^b/ mi· The q u a n t i t y (σ„ - σ°) r e p r e s e n t s t h e d i m e n s i o n l e s s c a p a c i t y o f the e x t r a c t o r a t a g i v e n r e s i d e n c e t i m e . E q u a t i o n 30 i s used t o p r e d i c t t h e i n l e t s o l u t e c o n c e n t r a t i o n which can be handled f o r a g i v e n s e t o f d e s i g n parameters: o* σ /σΐ, σ , and τ which i ο c o r r e s p o n d r e s p e c t i v e l y t o v / v , f , Cgi» C ^ , and V / v . The n o n l i n e a r i t y i n E q u a t i o n 15 d i s a p p e a r s when σ? i s s m a l l , p e r m i t t i n g an a n a l y t i c a l s o l u t i o n f o r t h e s u r f a c e f l u x : D

eff,NR

fc

e

2

K

m

f

e

> Χ

3η

' η=1

? Σ exp η=1

b

Γ η -— L

σ

χ

2

2

3

θ

m

π

+ σ

2

; 2

e

e

1

σΪ τ 1

(32)

(1 + σ )ϋ 3

S u b s t i t u t i n g E q u a t i o n 32 i n t o E q u a t i o n 30 produces the f o l l o w i n g algebraic solution: -1 σΐ - 6 a\ oS τ I θ

n=1

(33)

1 + Oj + σ

2

(1 + σ ) 3

E q u a t i o n 33 r e p r e s e n t s an e x a c t s o l u t i o n whenever t h e q u a n t i t y (1 + o j ) i s a p p r o x i m a t e l y one.


70

The a d v a n c i n g f r o n t c a l c u l a t i o n s r e q u i r e s u b s t i t u t i o n o f Equa t i o n s 25 and 26 i n t o E q u a t i o n 30 t o y i e l d (27): 1


i 0-

3σ^σ

X

j

3

α,

exp

ο,

2

3

(1-3χ +2χ )

(34)

dX

6σ! a j x

which i s n u m e r i c a l l y i n t e g r a t e d . The n o - r e a c t i o n d i f f u s i o n c o e f f i c i e n t i s a p p r o p r i a t e i n t h e a d v a n c i n g f r o n t t h e o r y and c o n s e q u e n t l y E q u a t i o n 34 i s w r i t t e n f o r λ - 1. A c c o r d i n g t o E q u a t i o n 34 t h e g l o b u l e c a p a c i t y f o r e x t r a c t i o n i s z e r o when t h e r e a g e n t c o n c e n t r a t i o n , σ , i s zero. T h i s i s c o n s i s t e n t w i t h t h e r e s t r i c t i o n o f Equa t i o n 27 which l i m i t s e x t r a c t i o n t o t h e s o l u t e which r e a c t s . The a d v a n c i n g f r o n t and r e v e r s i b l e r e a c t i o n models b o t h p r e d i c t t h a t t h e amount o f e x t r a c t i o n depends on t h e e m u l s i o n r e s i dence time and t h e r a t i o of e m u l s i o n and b u l k phase f e e d r a t e s , v / v b , as measured by t h e d i m e n s i o n l e s s group σ}. No indepen dent dependence on t h e hold-up r a t i o , ν / ν ^ , i s i n d i c a t e d . For e x t r a c t o r d e s i g n s which p e r m i t V / V t o d i f f e r from v /Vfc, model c a l c u l a t i o n s p r e d i c t t h a t t h e amount o f e x t r a c t i o n s h o u l d be i n d e pendent o f t h e hold-up r a t i o i f v /Vfc, x and Cj[b a r e f i x e d . T h i s was observed i n e x p e r i m e n t s w i t h t h e i n t e r n a l r e f l u x r e a c t o r d e s c r i b e d by H a t t o n and coworkers ( 2 7 ) . The maximum i n l e t c o n c e n t r a t i o n o f s o l u t e which can be reduced t o a r e q u i r e d o u t l e t c o n c e n t r a t i o n can be determined from an o v e r a l l s o l u t e b a l a n c e a t e q u i l i b r i u m c o n d i t i o n s . Assuming t h a t o n l y 3

e

Θ

e

b

e

e

e

f r e s h e m u l s i o n i s used ( t h a t i s , C^m = 0) and i n c l u d i n g r e a c t i o n reversibility: (35)

o î ) = σ°(σΐ • σΐ) + σ,σΐσΐ/(1 • σ!) RR

,max

The assumption o f an i r r e v e r s i b l e r e a c t i o n r e q u i r e s t h a t t h e max imum c o n c e n t r a t i o n o f r e a c t e d s o l u t e i n t h e i n t e r n a l phase e q u a l the o r i g i n a l c o n c e n t r a t i o n i r r e v e r s i b l e and C^m treated i s :

^\max

"

σ

V

=

σ

i s

*

o f r e a g e n t , Cgi · When r e a c t i o n 1 i s

z e r o , t h e maximum o\ which can be

( σ ι

+

°

2 )

+

σ 3

°

2

( 3 6 )

I f t h e r e q u i r e m e n t o f E q u a t i o n 27 i s met, then t h e g l o b u l e c a p a c i t y for extraction, ( a i - σ?), e q u a l s t h e c a p a c i t y f o r r e a c t i o n a l o n e , α^ο^· Comparing E q u a t i o n s 35 and 36, we see t h a t t h e d e v i a t i o n o f (1+σ;)/σ£ from 1 measures t h e a b i l i t y o f a g i v e n b u l k phase s o l u t e concentration t o d r i v e r e a c t i o n 1 t o completion. Consequently, the s i m p l e r a d v a n c i n g f r o n t approach i s s u f f i c i e n t when t h e d i m e n s i o n l e s s c o n c e n t r a t i o n o f s o l u t e i n t h e e x t r a c t o r i s l a r g e . One c o m p l i c a t i o n i s t h a t a t l a r g e o j t h e c o n t r i b u t i o n o f phase e q u i l i b r i u m t o t h e e x t e n t of e x t r a c t i o n i n c r e a s e s , meaning t h a t t h e pseudo> m a x


5.

REED ET AL.

influence of Reaction Reversibility

71

s t e a d y s t a t e r e s t r i c t i o n , E q u a t i o n 27, cannot be met. For s m a l l v a l u e s of o j , r e a c t i o n r e v e r s i b i l i t y needs t o be c o n s i d e r e d . S i n c e t h e n o r m a l i z e d a d v a n c i n g f r o n t model i s adequate except f o r s m a l l σ£ v a l u e s , a l e g i t i m a t e q u e s t i o n i s whether of 1.0 or l e s s i s l i k e l y to occur. T a b l e I shows r e a c t i o n e q u i l i b r i u m c o n s t a n t s f o r s e v e r a l a c i d i c and b a s i c compounds and c o n c e n t r a t i o n s c o r r e s p o n d i n g t o o j = 1. Values of o j when s o l u t e c o n c e n t r a t i o n s are 1 ppm are a l s o t a b u l a t e d . C l e a r l y , removal of t r a c e o r g a n i c a c i d s or bases w i t h s m a l l r e a c t i o n c o n s t a n t s can l e a d t o s m a l l σ£ v a l u e s . An a l t e r n a t e way of e v a l u a t i n g e x t r a c t o r performance i s t o measure the e x t e n t t o w h i c h t h e e m u l s i o n c a p a c i t y i s u t i l i z e d . We d e f i n e t h e f r a c t i o n a l u t i l i z a t i o n of the e m u l s i o n a s :

(37)

For example, b r i e f exposure t i m e s w i l l use o n l y a s m a l l f r a c t i o n of the emulsion c a p a c i t y f o r e x t r a c t i o n . Long e m u l s i o n r e s i d e n c e times w i l l a l l o w n e a r l y complete e q u i l i b r a t i o n w i t h t h e b u l k phase s o l u t i o n and f o r c e F t o approach 1 a s y m p t o t i c a l l y . e

R e s u l t s and

Discussion

R e s u l t s of sample c a l c u l a t i o n s a r e shown i n F i g u r e s 2 through 5. These r e s u l t s are based on t y p i c a l o p e r a t i n g and p h y s i c a l p a r a m e t e r s : a f r e s h e m u l s i o n f e e d ( c j ^ » 0 ) , a g l o b u l e membrane f r a c t i o n ( f ) of 0.6 and a v a l u e of 1.0 f o r both d i s t r i b u t i o n c o e f f i c i e n t s (K t> and K ± ). m

m

m

F i g u r e 2 r e p o r t s the d i m e n s i o n l e s s e x t r a c t o r c a p a c i t y f o r v a r i o u s v a l u e s of the d i m e n s i o n l e s s s o l u t e c o n c e n t r a t i o n l e a v i n g the e x t r a c t o r ( o j ) . The mass of reagent a v a i l a b l e f o r r e a c t i o n i s c o n s t a n t and s p e c i f i e d by σ of 2000 and v ^ / v of 9.0. The s o l i d c u r v e s denote p r e d i c t i o n s of the r e v e r s i b l e r e a c t i o n t h e o r y g i v e n i n E q u a t i o n 30; t h e broken c u r v e s a r e p r e d i c t i o n s of t h e ad v a n c i n g f r o n t model, E q u a t i o n 34. For a g i v e n d i m e n s i o n l e s s o u t l e t c o n c e n t r a t i o n , i n c r e a s i n g t h e average r e s i d e n c e time f o r t h e emul s i o n g l o b u l e s i n c r e a s e s the s o l u t e f e e d c o n c e n t r a t i o n w h i c h can be treated. 3

e

E x t r a c t o r Design. Data s u c h as t h o s e shown i n F i g u r e 2 a r e r e quired f o r extractor design. Except f o r the e m u l s i o n r e s i d e n c e t i m e , o p e r a t i n g parameters a r e f i x e d by f a c t o r s o t h e r than t h e de s i r e d amount of e x t r a c t i o n . For example, t h e f r a c t i o n of membrane phase i n t h e e m u l s i o n and the r e a g e n t c o n c e n t r a t i o n w i l l g e n e r a l l y be c o n t r o l l e d by e m u l s i o n d u r a b i l i t y and the c o n c e n t r a t i o n of r e a c t e d s o l u t e d e s i r e d . The r a t i o of f e e d - t o - e m u l s i o n f l o w r a t e w i l l depend on m i x i n g c o n s i d e r a t i o n s . S m a l l e r r a t i o s of v ^ / v increase the amount of s o l u t e w h i c h can be e x t r a c t e d f o r a g i v e n volume of b u l k phase f e e d . However, as t h e volume r a t i o of b u l k phase t o e m u l s i o n d e c r e a s e s , t h o r o u g h m i x i n g becomes more d i f f i c u l t . A low e r l i m i t of Vfc/v e q u a l t o 0.35 c o r r e s p o n d s t o u n i f o r m c l o s e - p a c k e d e

e


72

spheres of e m u l s i o n s u r r o u n d e d by b u l k s o l u t i o n ( i . e . , f r a c t i o n of v o i d s between e m u l s i o n g l o b u l e s i s 0.26). R e a l i s t i c a l l y , v ^ / v v a l u e s of l e s s than 1.0 a r e p r o b a b l y not p r a c t i c a l . To complete the d e s i g n r e q u i r e s s p e c i f i c a t i o n of the e x t r a c t o r s i z e , which i s determined from the x r e q u i r e d t o r e a c h a g i v e n amount of e x t r a c t i o n . The i n f o r m a t i o n r e q u i r e d a r e the d i m e n s i o n l e s s s o l u t e c o n c e n t r a t i o n of the b u l k phase f e e d σ£, and the d i m e n s i o n l e s s s o l u t e c o n c e n t r a t i o n w h i c h must be r e a c h e d , o j . The x needed t o e x t r a c t (σ£ - σ£) can be r e a d from the (σ£ - σ£) curve f o r the s p e c i f i e d o j . U s i n g the c o n d i t i o n s i n F i g u r e 2 and assuming σ£ and σ£ t o be r e s p e c t i v e l y 50 t o 10, the advanc i n g f r o n t and r e v e r s i b l e r e a c t i o n models b o t h p r e d i c t a τ of about 3. T h i s r e s u l t c o r r e s p o n d s t o 12.5 minutes f o r t y p i c a l v a l u e s of the average g l o b u l e diameter (1.0 mm) and the e f f e c t i v e d i f f u s i o n c o e f f i c i e n t (10~ m /s). A c c o r d i n g t o the r e v e r s i b l e r e a c t i o n p r e d i c t i o n , a σ£ of 20 and o j of 1 can be r e a c h e d w i t h one e x t r a c t o r s i z e d t o g i v e a x of 25 ( t y p i c a l l y about 100 m i n u t e s ) . The a d v a n c i n g f r o n t c u r v e p r e d i c t s a much s m a l l e r τ c o u l d a c c o m p l i s h t h i s same s e p a r a t i o n . By c o n t r a s t , the a d v a n c i n g f r o n t model c u r v e i n d i c a t e s t h a t the e x t r a c t i o n (σ£ - o j ) of (200-100) i s not p o s s i b l e i n a s i n g l e s t a g e ; the r e v e r s i b l e r e a c t i o n model shows a τ of 100 s h o u l d work. I f the ( o j - σ£) curve does not r e a c h the v a l u e f o r the de s i r e d (σ£-σ£), t h e n the g l o b u l e c a p a c i t y i s i n s u f f i c i e n t t o a c h i e v e the r e q u i r e d s e p a r a t i o n i n a s i n g l e e x t r a c t i o n u n i t . I f f r e s h e m u l s i o n i s d e l i v e r e d t o a l l e x t r a c t o r s of a m u l t i s t a g e scheme, commonly c a l l e d a c r o s s - f l o w c o n f i g u r a t i o n , t h e n t h e i n f o r m a t i o n i n F i g u r e 2 remains a p p l i c a b l e . I n an a l t e r n a t e scheme f o r the σζ of 20 and o j of 1, one e x t r a c t o r c o u l d reduce the f e e d t o a σ; of 10 ( r e q u i r i n g a τ of about 0.12 or 0.5 m i n u t e s ) . T h i s becomes t h e f e e d t o a second e x t r a c t o r w h i c h needs a x of a p p r o x i m a t e l y 2.5 (about 11 m i n u t e s ) t o complete the e x t r a c t i o n t o (o\ - σ£) = ( 1 0 - 1 ) . In t h i s example, the t o t a l e x t r a c t o r volume f o r t h e two-stage scheme i s c o n s i d e r a b l y l e s s t h a n the one s t a g e d e s i g n . However, some of the c a p i t a l s a v i n g s would be o f f s e t by the a d d i t i o n a l pumps, p i p i n g and t h e second b u l k phase-emulsion s e t t l e r which would be r e q u i r e d . A c o u n t e r c u r r e n t , c a s c a d e - t y p e c o n f i g u r a t i o n would i n c r e a s e the e m u l s i o n c a p a c i t y f o r s o l u t e e x t r a c t i o n . I n t h i s s i t u a t i o n , the e m u l s i o n from the second ( s m a l l o j ) s t a g e becomes the emul s i o n f e e d t o the f i r s t ( l a r g e σ?) s t a g e . To determine the r e s i dence time of the h i g h c o n c e n t r a t i o n s t a g e r e q u i r e s g e n e r a t i o n of curves l i k e those i n F i g u r e 2, but c a l c u l a t e d f o r the average s o l ute c o n c e n t r a t i o n of the e m u l s i o n l e a v i n g t h e low c o n c e n t r a t i o n s t a g e . The average s o l u t e membrane c o n c e n t r a t i o n from the d i l u t e s t a g e depends on the mass of s o l u t e e x t r a c t e d a c c o r d i n g t o e


e

e

θ

9

2

e

θ

Θ

θ

e

_o

m

•

σ

(38)

3

1 +

For a s e r i e s of c o u n t e r c u r r e n t e x t r a c t o r s , the i n t e r s t a g e membrane s o l u t e c o n c e n t r a t i o n can be c a l c u l a t e d from a s l i g h t l y m o d i f i e d E q u a t i o n 38.


5.

REED ET AL.


73

R e v e r s i b i l i t y E f f e c t s . F i g u r e 2 demonstrates t h a t d i f f e r e n c e s be tween a d v a n c i n g f r o n t and r e v e r s i b l e r e a c t i o n model p r e d i c t i o n s a r e s i g n i f i c a n t when o j i s l e s s than 1 o r when σϊ i s g r e a t e r t h a n 10. When o j i s s m a l l , as measured by t h e d e v i a t i o n o f (1+o5)/oJ from 1, t h e s o l u t e c o n c e n t r a t i o n i s t o o s m a l l t o f o r c e t h e reagent t o react completely. R e a c t i o n r e v e r s i b i l i t y causes t h e g l o b u l e e x t r a c t i o n c a p a c i t y t o depend on t h e o u t l e t s o l u t e c o n c e n t r a t i o n . T a b l e I I summarizes t h e maximum e x t r a c t i o n c a p a c i t i e s which c o r r e s p o n d t o t h e p l a t e a u v a l u e s i n F i g u r e 2. These numbers a r e c a l c u l a t e d u s i n g E q u a t i o n s 35, 36 and E q u a t i o n 36 r e s t r i c t e d by E q u a t i o n 27. When r e a c t i o n s a r e r e v e r s i b l e , t h e g l o b u l e c a p a c i t y f o r e x t r a c t i o n i n c r e a s e s as o j i n c r e a s e s . I f reaction 1 i s i r r e v e r s i b l e , a l l o f t h e r e a g e n t w i l l r e a c t independent o f s o l u t e concentration. Small d i f f e r e n c e s i n t h e a d v a n c i n g f r o n t ' s t o t a l c a p a c i t y a t d i f f e r e n t a j v a l u e s a r i s e from t h e s o l u b i l i t y o f u n r e a c t e d s o l u t e s o l u b i l i t y i n t h e membrane and i n t e r n a l phases. When i s s m a l l , r e a c t i o n r e v e r s i b i l i t y becomes s i g n i f i c a n t , making some e x t r a c t i o n s i m p o s s i b l e t o a c c o m p l i s h i n a s i n g l e u n i t . As αζ i n c r e a s e s t o 10 and (1+oJ)/oJ becomes more n e a r l y e q u a l t o 1, t h e d i f f e r e n c e s between t h e a d v a n c i n g f r o n t and r e v e r s i ble r e a c t i o n r e s u l t s disappear. But when o j i s 100 t h e p r e d i c t e d r e v e r s i b l e r e a c t i o n c a p a c i t y exceeds v a l u e s c a l c u l a t e d by t h e ad v a n c i n g f r o n t model. T h i s d i f f e r e n c e i s a consequence o f t h e r e s t r i c t i o n f o r t h e p s e u d o s t e a d y - s t a t e s o l u t i o n , E q u a t i o n 27, w h i c h c o n s i d e r s s o l u t e r e a c t i o n as t h e s o l e c o n t r i b u t o r t o e x t r a c t i o n capacity. When r e a g e n t c o m p l e t e l y r e a c t s and s o l u t e s o l u b i l i t y i n t h e membrane and i n t e r n a l phases i s i n c o n s e q u e n t i a l , t h e g l o b u l e capac i t y i s 88.88. T a b l e I I compares t h i s r e a c t i o n - o n l y number t o t h e t o t a l c a p a c i t y i n c l u d i n g s o l u t e s o l u b i l i t y . How c l o s e l y c o n d i t i o n 27 i s s a t i s f i e d w i l l determine t h e s i z e o f t h e d e v i a t i o n s i n t h e s e two numbers. When s o l u b i l i t y does c o n t r i b u t e , p r e d i c t i o n s o f e x t r a c t o r performance based on t h e p s e u d o s t e a d y - s t a t e a d v a n c i n g f r o n t approach w i l l tend t o be c o n s e r v a t i v e . T h i s u n d e r p r e d i c t i o n of e x t r a c t i o n performance can be removed by i n c l u d i n g a d d i t i o n a l terms i n t h e p e r t u r b a t i o n s o l u t i o n d e s c r i b e d by Ho e t a l . ( 2 6 ) . A s i m p l e a l t e r n a t e procedure i s d e s c r i b e d n e x t . F i g u r e 3 shows t h e f r a c t i o n a l u t i l i z a t i o n o f e m u l s i o n , F , as a f u n c t i o n o f e m u l s i o n r e s i d e n c e time f o r t h e same s e t o f c o n d i t i o n s as i n F i g u r e 2. To be c o n s i s t e n t w i t h t h e p s e u d o s t e a d y - s t a t e s o l u t i o n o f t h e a d v a n c i n g f r o n t t h e o r y , we have t a k e n ( o i , °°)AF °2°3 ? advancing f r o n t c a l c u l a t i o n s . C a l c u l a t i o n s made a t f l o w r a t e r a t i o s ( v ^ / v ) o f 9 and 19. Both models p r e d i c t t h a t F i s independent o f t h e f l o w r a t e r a t i o and g i v e t h e same v a l u e s when (1+a5)/oJ * 1. N o r m a l i z i n g t h e p s e u d o s t e a d y - s t a t e a d v a n c i n g f r o n t s o l u t i o n by t h e r e a g e n t c a p a c i t y e l i m i n a t e s t h e d i f f e r e n c e s a r i s i n g from s o l u t e s o l u b i l i t y which were observed i n F i g u r e 2. When (1+oJ)/oJ d e v i a t e s from 1, t h e r e v e r s i b l e r e a c t i o n model p r e d i c t s a f a s t e r e m u l s i o n u t i l i z a t i o n because t h e c a p a c i t y of t h e e m u l s i o n i s much s m a l l e r , as a l r e a d y noted i n T a b l e I I . F i g u r e 4 demonstrates t h e e f f e c t o f v a r i o u s c o n c e n t r a t i o n s o f i n t e r n a l phase r e a g e n t and b u l k phase s o l u t e (σ and o j ) a t f i x e d r a t i o s o f σ /σ5 ( 2 , 20, 200, and 2000). F o r t h i s c o m b i n a t i o n o f e

m a x

=

or

t

n

e

e

e

3

3


74

Table I. Example Concentration for Typical Solutes κ (25°C)


Component

°4° (for Cjk-I ppm)

(ίοτσ ° -1) 4

Phenol

11,000

8.6 ppm

0.12

Anffine

39,450

2.4 ppm

0.42

6,300

17.0 ppm

0.06

530,000

0.26 ppm

&81

o-cresd m-ritrophenol σ «2000

V

3

Figure 2

-

v

» "9

K =1.0

*m=0e

bm

Dimensionless extractor τ

6

θ

(f -°- » K - 1 . 0 , m

Table II.

m b

capacity

as a function o f

v /v =9) b

e

Maximum Globule Capacity, (c^ mrf-crf) 1

Reversible Reaction Total

Advancing Front Total

Reaction Only

0.1

8.09

88.89

88.88

1.0

44.56

88.99

ease

10

81.92

89.99

88.88

100

99.12

99.99

88.88

Ο3-2ΟΟΟ

V e» v

9

'm-O-e

REED ET AL.



J 1(f

10~

3

10"

2

1

I

L

10°

10

1

10

2

e F r a c t i o n a l u t i l i z a t i o n o f emulsion g l o b u l e s as f u n c t i o n o f x a t f i x e d σ /σ; r a t i o s o f 2, 20, 200 and 2000. For σ£>10 a d v a n c i n g f r o n t and reversible reaction predictions coincide (f -0.6 Kmb= -°. b e - 9 ) . T

Figure

4

-

e

3

m

1

v

/ v


76

c o n d i t i o n s , the e f f e c t of changing σ£ from 1 t o 100 i s n e g l i g i b l e when σ /σ2 i s 2.0 and s m a l l when o / a J i s l a r g e r t h a n 20. For oj e q u a l t o 10 or l a r g e r , the a d v a n c i n g f r o n t and r e v e r s i b l e r e a c t i o n models p r e d i c t i d e n t i c a l c u r v e s which depend o n l y on the r a t i o of o / o J . In t h i s s i t u a t i o n , e x t r a c t i o n c u r v e s l i k e t h o s e i n F i g u r e 2 can be developed by m o d i f y i n g the pseudosteadys t a t e advancing f r o n t c a l c u l a t i o n s l i g h t l y : 3

3


3

r i ο Ισ„ - a°J

C^'max " ^ R R g

=

Γ*

22

χ

exp

ο*

σ,

2

(1-3χ +2χ')

dX (39)

This i s an a p p e a l i n g r e s u l t s i n c e i t o b v i a t e s the n u m e r i c a l s o l u t i o n of the n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n . The approach of E q u a t i o n 39 i s not g e n e r a l l y s u c c e s s f u l when aj i s l e s s than 10. As e x p e c t e d , the d i f f e r e n c e s between the ad v a n c i n g f r o n t and r e v e r s i b l e r e a c t i o n p r e d i c t i o n s of F as a f u n c t i o n of τ grow l a r g e r as becomes s m a l l e r . Figure 5 i l l u s t r a t e s t h i s phenomenon f o r o / o J of 200. I f the a d v a n c i n g f r o n t c a l c u l a t i o n i s used as d e s c r i b e d i n E q u a t i o n 39, the d e s i g n w i l l always be c o n s e r v a t i v e . For example, the r e s i d e n c e time p r e d i c t e d by the a d v a n c i n g f r o n t f o r aj of 0.1 and F of 0.5 i s n e a r l y one o r d e r of magnitude l a r g e r than t h a t p r e d i c t e d by r e v e r s i b l e r e a c t i o n . However, when oJ i s 0.1, (1 + σ?) i s n e a r l y e q u a l t o 1 and E q u a t i o n 33 w i l l a p p l y . C o n s e q u e n t l y , s i m p l i f i e d approaches f o r c a l c u l a t i n g e x t r a c t o r c a p a c i t y e x i s t e x c e p t when oj i s n e i t h e r l a r g e nor s m a l l ( t h a t i s , b o t h (1 + σ?) and (1 + o j ) / oj are not a p p r o x i m a t e l y l ) . Next, we c o n s i d e r the case when n e i t h e r of these two l i m i t i n g c o n d i t i o n s are met. D i f f e r e n c e s between the a d v a n c i n g f r o n t and r e v e r s i b l e r e a c t i o n c u r v e s i n F i g u r e s 4 and 5 a r i s e because the a d v a n c i n g f r o n t t h e o r y assumes t h a t a l l of the t r a p p e d reagent r e a c t s . In f a c t , the reagent can o n l y r e a c t u n t i l the e q u i l i b r i u m c o n c e n t r a t i o n of r e a g e n t i s r e a c h e d . One way t o improve the a d v a n c i n g f r o n t p r e d i c t i o n when ai i s s m a l l but (1 + σ°) i s not c l o s e t o 1 i s t o a d j u s t the c o n c e n t r a t i o n of i n t e r n a l reagent ( t h r o u g h the d i m e n s i o n l e s s parameter σ ) t o the amount which a c t u a l l y can r e a c t . The a d j u s t ment of σ i s determined by e q u a t i n g the amount of s o l u t e which i s e x t r a c t e d by r e a c t i o n f o r the a d v a n c i n g f r o n t and r e v e r s i b l e r e a c t i o n t h e o r i e s (see E q u a t i o n s 35 and 36) t o o b t a i n : e

θ

3

e

3

3

°3 a d i ' a a j

=

1

^ 1 +

{ k 0 )

σ°

T a b l e I I I shows v a l u e s of o , j j f o r v a r i o u s aj when the r a t i o σ /σΐ i s f i x e d a t 200 and at and q\ are 0.0667 and 0.0W, respectively. When oj becomes s m a l l , the c o n t r i b u t i o n of r e a c t i o n t o the e x t r a c t i o n d e c r e a s e s and e x t r a c t i o n by phase s o l u b i l i t y becomes p r o p o r t i o n a l l y more i m p o r t a n t . The r i g h t hand column i n T a b l e I I I g i v e s the f r a c t i o n of t o t a l e x t r a c t e d s o l u t e which would be removed by r e a c t i o n o n l y . 3

3

a (


5.

REED ET AL.

Figure

5


-

77

F r a c t i o n a l u t i l i z a t i o n o f emulsion globules as a f u n c t i o n o f x f o r o /o% o f 200 and o% o f 0.1, 1.0, 10 and l a r g e r . For σ£>10 a d v a n c i n g f r o n t and r e v e r s i b l e r e a c t i o n p r e d i c t i o n s c o i n c i d e (f =0.6, K = 1 . 0 , v / v » 9 ) 3

e

ra

Table III.

mb

b

e

Adjusted Reagent Concentration and Fraction of Solute Extracted by Reaction with Reagent °2°3,odj °"3,odj

2

10

2000

1818

0.986

1

200

100

0.976

0.5

100

33.3

0.964

0.1

20

1.818

0.879

0.00198

0.073

0.01 σ /σ?=200 3

Influence of Reaction Reversibility on Continuous-Flow Extraction by

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