Some Observations About the Application of Fick's First Law for Membrane Separation of Multicompo

8/6/2019 Some Observations About the Application of Fick's First Law for Membrane Separation of Multicompo

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EL S EV I ER Journal of Membrane Science 135 (1997) 147-159

joum uoMEMBR A N ESC I ENCE

S o m e o b s e r v a t i o n s a b o u t t h e a p p l i c a t i o n o f F i c k ' s f i r s t l a wf o r m e m b r a n e s e p a r a t i o n o f m u l t i c o m p o n e n t m i x t u r e s

H . Denny Kamaruddin, William J. Koros*Dep artm ent of Chemica l Engineering, The University of Texas at Austin, Austin, T X 78712, USA

Received 10 October 1996; received in revised form 8 April 1997; accepted 10 April 1997

A b s t r a c t

Accord ing t o F i ck ' s f i r s t l aw o f d i f fu s ion , permeat i on f l ux o f a permean t t h rough a po re- f r ee dense po lymer i c membranewi th r espec t t o a f i xed f r ame o f r e f e rence i s equa l t o t he sum o f t he bu lk and d i f fu s iona l f lux . Under i dea l cond i t i ons and l owd o w n s t r e a m ( p e r m e a t e ) p r e s s u r e , m e m b r a n e p e r m e a b i l i t y , PA, of componen t A i s equa l t o t he p roduct o f t he mob i l i t ycoef f i c i en t , DA, and t he so lub i l i t y coef f i c i en t , SA. T h e d e f i n it io n o f m e m b r a n e p e r m e a b i l i ty a s s t a te d a b o v e c a n b e d e r i v e dus ing F i c k ' s f i r st l aw o f d i f fu s ion when t he f r ame o f r e f e rence (bu lk f l ux ) t e rm i s neg l i g ib l e . W hen ana lyz ing t he t r anspor t o fperm ean t t h rough po re- f r ee dense po lym er i c membra ne , t he bu lk f l ux con t r i bu t i on i s u sua l l y assum ed t o be neg l i g ib l e whenthe so rp t i on l eve l s o f t he componen t s a r e smal l . However , t h i s assumpt ion can be e r roneous i n t he case o f mu l t i componen tmix tu res when t he perm eat i on f l ux o f one o f t he perm ean t i s much h igher t han t he o thers . Two exa mp les wi l l be d i scu ssed t odem ons t r a t e t ha t neg l ec t i ng t he bu lk f l ux con t r i bu t i on migh t l ead r esearcher s t o i nco r rec t conc lus ions on t he m ater i a l sc i enceaspec t s o f t he mem brane separa t i on . The f i rs t exam ple i s C02 /CI - I4 gas separa t i on us ing a dense po re- f r ee g l assy po lym er i cm e m b r a n e a n d t h e s e c o n d e x a m p l e i s th e r e m o v a l o f t r a c e s o f p h e n o l fr o m w a t e r u si n g a p o l y e t h e r - b l o c k - p o l y a m i d ep o l y m e r .K e y w o r d s : CO2/CH 4 gas separa t i on ; Pervapora t i on o f VOCs; Membrane t r anspor t ; F i ck ' s f i r s t l aw o f d i f fu s ion ; So lu t i on -d i f fu s ion

1 . I n t r o d u c t i o n

I t i s w e l l k n o w n t h a t th e t r a n s p o r t p r o c e s s t h r o u g h ap o r e - f r e e p o l y m e r i c m e d i u m c a n b e d e s c r i b e d b yF i c k ' s f ir s t l a w o f d i f f u si o n . W h e n d e s c r i b i n g t h et r a n s p o r t p r o c e s s , i t is n e c e s s a r y t o s p e c i f y t h e f r a m eo f r e f e r e n c e o f t h e t r a n s p o r t p r o c e s s [ 1 ]. T h e r e a r e t w oc o m m o n f r a m e s o f r e f e re n c e : f i x ed a n d m o v i n g . I n t hem o v i n g f r a m e o f re f e r e n ce , t h e m a s s a v e r a g e b u l kv e l o c i t y i s t y p i c a l l y u s e d a s t h e f r a m e o f r e f e r e n c e f o r

*Corresponding Author.0376-7388/97/$17.00 1997 Elsevier Science B.V. All rights reserved.P I I S 0 3 7 6 - 7 3 8 8 ( 9 7 ) 0 0 1 4 2 - 7

t h e d i f f u s i o n a l v e l o c i t y . I n t h i s c a s e , f l u x o f t h ep e r m e a t i n g c o m p o n e n t i s c a u s e d b y t h e d if f u si o n a lv e l o c i t y r e l a t iv e t o t h e b u l k v e l o c i t y . I n t h e c a s e o ff i x e d f r a m e o f r e f e r e n c e t r a n s p o r t , t h e m e m b r a n e c a nb e u s e d a s t h e fr a m e o f r e f e r e n c e b e c a u s e t h e m e m -b r a n e i s s t a t i o n a r y a t s t e a d y s t a t e . F o r t h i s s t a t i cr e f e r e n c e f r a m e , t h e p e n e t r a n t t r a n s p o r t p r o c e s s i s ac o m b i n a t i o n o f d i f f u s io n a l a n d b u l k ( c o n v e c t i v e ) fl ux .W h e n d e s c r i b in g m e m b r a n e - r e l a t e d t ra n s p o r t p r o -c e s s e s , i t i s m o r e c o n v e n i e n t t o u s e t h e f i x e d f ra m eo f r e f e re n c e b e c a u s e t h e e x p e r i m e n t a l p e r m e a t i o n r a tei s m e a s u r e d w i t h r e s p e c t t o a f i x e d f r a m e o f r e f e r e n c e ,t h a t i s , t h e m e m b r a n e .


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14 8 H.D. K amaruddin, W.J. Koros/Jou rnal of Membrane Science 135 (1997) 147-159W h e n d e s c r i b i n g b i n a r y m i x t u r e p e r m e a t i o n , t h e

s y s t e m c o m p r i s e s a t e r n a r y m i x t u r e o f t h e p o l y m e ran d t w o p e rmean t s . E q s . ( 1 ) an d (2 ) s h o w n b e l o w a r et r an s p o r t eq u a t io n s fo r t he t w o p e rm ean t s an d E q . ( 3 )i s t h e t r an s p o r t eq u a t i o n fo r t h e p o l y mer .

&CAna = (ha +n B d - np )cda - - pDAm dx (1 )d U Bn B : (HA ~- nB @ r ip )b iB - - pOBm ~ ( 2 )

D d~vpn p = (na + n B + np)Wp -- p pm ~ ( 3 )A a n d B r e f e r t o t h e p e r m e a t i n g c o m p o n e n t s a n d pr e f e r s t o t h e p o l y mer . Dim i s t h e e f f ec t i v e b i n a ryd i f fu s i v i t y fo r d i f fu s i o n o f i i n t h e t e rn a ry mi x t u re[ 1 , t h a t i s, p o l y m er , co m p o n en t A an d B . n i i s th e m as sf l ux ( g / cm 2 s ) o f p e rm ean t i w i t h r e s p ec t t o a f i x edf r am e o f r e f e r en ce an d ~vi i s t h e co n cen t r a t i o n o fp e r m e a n t i i n t h e m e m b r a n e ( g / g ) . E q s . ( 1 ) - ( 3 ) c a nb e e x t e n d e d t o d e s c r i b e m u l t i c o m p o n e n t m i x t u r e s o fn . Fo r t h e s ak e o f s i mp l i c i t y , o n l y b i n a ry mi x t u rep e rmea t i o n w i l l b e d i s cu s s ed i n t h i s p ap e r .E q s . ( 1 ) - (3 ) a r e w r i t t en i n m as s f l u x u n it ( g / cm 2 s );t h ey can a l s o b e w r i tt en i n t e rms o f g a s mo l a r f l u x u n i t( cm 3 (S T P ) / c m 2 s ). E q . ( 4 ) i s t h e t r an s p o r t eq u a t i o nf o r c o m p o n e n t A i n t e r m o f g a s m o l a r f lu x .

N ~ VAC A _ Da md _ ~A (4 )NA = (NA + NB + pl 2 2 4 0 0W h e n d e s c r ib i n g p e r m e a t i o n t h r o u g h a p o r e - f r e e p o l y -m e r i c m e d i u m , i t i s m o r e c o n v e n i e n t t o u s e e q u a ti o n sw i t h mas s f l u x u n i t i n s t ead o f t h o s e u s i n g g as mo l a rf l ux u n it b ecau s e t h e m o l a r v o l u m e o f t h e s o rb edp en e t r an t s , l ? i , mi g h t n o t b e r ead i l y av a i l ab l e , e s p e -c i a l l y i n t h e ca s e o f g a s eo u s p e rmean t s .

T h e p o l y m e r m a s s f lu x , np , i s eq u a l t o ze ro a t s t ead ys t a t e s i n ce t h e memb ran e i s s t a t i o n a ry . T h e re fo re , t h ed i f fu s i o n a l mas s f l u x o f t h e p o l y m er is eq u a l i n v a l u ean d o p p o s i t e i n t h e d i r ec t io n t o t h e f l o w w i t h r e s p ec t t oi ts b u l k ( co n v ec t i v e ) mas s f lu x . T h e t o t a l ma s s f l u x o fA , n A, i s co m p r i s ed o f th e m as s f l u x r e s u l t i n g f ro m t h eb u l k mo t i o n o f t h e p e rmean t s an d mas s f l u x r e s u l t i n gf ro m d i f fu s i o n r e l a t i v e t o th e b u l k f lu x , an d i s g i v en b yE q . ( 5 ) . T h e s ame d e f i n i t i o n a l s o ap p l i e s t o co mp o -n e n t B a n d p . T h e s c h e m a t i c o f t h e t r a n s p o r t p ro c e s sfo r t h e t h r ee s p ec i e s i n t h e t h i ck n es s d i r ec t i o n o f am em b ran e o f t h i ck n es s { i s il l u s tr a t ed i n F ig . 1 .

~, = 1 O)AI - LOBI

O)AI

~t

~ x

p Dpm dO)p t dx

( n , + % + % ) ~ p

p DA m d f O A / d x

9 f

(Op2 ~ 1

Permeatenp = 0

1,rT = {n A + r iB) 0 )A - pDkmd(~A/dx

II=n B = (=?A + n e ) ~ - p D ~ d ( O ~ / d x

(0 ~ = o ~e ~ ~ 0( l o w d o w n s t r e a m p r e s s u r e )

F ig . 1 . I l l u s t r a t i on o f t he t ranspo r t p r ocess o f b ina r y m ix tu r em e m b r a n e p e r m e a t i o n .

RA = n bu lk + n d i f f ( 5 )n ~ ulk ---- ( h a + n B + r i p )W e ( 5 a )n d i f f - d ~ A=--D[-)Am ~ ( 5 b )

P a u l a n d E b r a - L i m a [ 2 - 7 ] r e c o g n i z e d t h e i m p o r -t an ce o f th e b u l k f l u x co n t r i b u t i o n i n s ingle c o m p o n e n tp e r m e a t i o n i n a h i g h l y s w o l l e n m e m b r a n e . I n p u r eco mp o n en t p e rmea t i o n , t h e b u l k f l u x i s a f u n c t i o n o ft h e s o rp t i o n l ev e l o f t h e p en e t r an t ; h o w ev e r , i n t h e ca s eo f b i n a ry m i x t u re s , b u l k f l u x i s a f u n c t i o n o f b o t h t h es o rp t i o n l ev e l an d f l u x o f b o th m o b i l e c o m p o n e n t s .W h e n t h e m a s s f lu x o f o n e c o m p o n e n t i s m u c h h i g h e rt h an t h e o t h e r, o mi t t i n g t h e f r am e o f r e f e r en c e t e rmcan b e e r ro n eo u s w h en d es c r i b i n g t h e t r an s p o r t o f th es l o w e r p e r m e a t i n g c o m p o n e n t . T h e r e f o r e , i t is n e c e s -s a ry t o d i s t i n g u i s h b e t w een t h e p e rmeab i l i t y ca l cu -l a t ed b as ed o n t h e t o t a l mas s f l u x an d p e rmeab i l i t yca l cu l a t ed b as ed o n t h e d i f fu s i o n f lu x . T h e l a t e r p a r t o ft h i s p ap e r w i l l ex p l o re t h e d i f f e r en ce , w h i ch i s n o tu s u a l l y r eco g n i zed , b e t w een t h e di f fusion-based andth e o b served m e m b r a n e p e r m e a b i l i t y .

2. D i ffusion-based permea bi l i t ies o f binarymi x tures

The bu lk f lux con t r ibu t ions in Eqs . (1 ) and (2 ) areo f t en n eg l ec t ed b as ed o n t h e a s s u mp t i o n t h a t t h eamo u n t o f s o rp t i o n o f t h e p en e t r an t s , WA and ~B,


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H.D. Kamaruddin, W.J . Kor os/Jo urna l o f Mem brane Science 135 (1997) 147-1 59 149

a re n eg l i g i b l e . I n s o m e cas es t h i s a s s u m p t i o n i s r ea -s o n ab l e , s u ch a s t h e s o rp t i o n o f s i m p l e g as es , t h a t i s ,H 2 , H e , 0 2 an d N 2 . T h i s a s s u m p t i o n h as co n v en t i o n -a l ly b e e n c a s u a l l y e x t e n d e d t o m o s t g a s e s a n d s o l v e n tss o t h a t b u l k f l u x co n t r i b u t i o n s a r e a l m o s t a l w ay sas s u m ed t o b e n eg l i g i b l e . T ran s p o r t eq u a t i o n s d i s -cu s s ed b e l o w a re o b t a i n ed u s i n g t h i s co n v en t i o n a la s s u m p t i o n t h a t b u l k f l u x i s m i n i m a l w i t h r e s p ec t t othe d i f fus ional f lux .n bulk = (nA q- r iB)idA ,~ 0 (6 )T h e r e f o r e t r a n s p o r t e q u a t i o n o f c o m p o n e n t A b e c o m e s

do3 An A ~ , ndAff = - - p D A r n ~ ( 7 )

T h e t o t a l f lu x o f A , ha, in Eq . (7 ) i s the mass f luxres u l t i n g f ro m dif fus ion t r an s p o r t. T h e a b o v e e q u a t i o nc a n a l s o b e w r i t t e n i n t e r m s o f g a s m o l a r f l u x , NA( c m 3 ( S T P ) / c m 2 s ), w h i c h i s o b t a i n e d f r o m E q . ( 4 )w i t h t h e a s s u m p t i o n o f m i n i m a l b u l k f lu x .N d i f f d C A" ~ - - D A m - - ~ ( 8 )

t h e d i f fu s i o n co e f f i c i en t s o f E q s . ( 7 ) an d (8 ) h av e t h es am e v a l u e an d u n i t s. I n t h e cas e o f m i n i m a l b u l k f l u x ,e q u a t io n s w i t h m o l a r g a s f l ux u n it a r e c o m m o n l y u s e dt o d e s c r i b e t r a n s p o r t t h r o u g h a p o r e - f r e e p o l y m e r i cm e d i u m b e c a u s e m o l a r v o l u m e s o f p e n e t r a n t s a n dp o l y m e r a r e n o t r eq u i r ed (E q . (4 ) ). T h e co n cen t r a t i o no f A i n s id e t h e m e m b r a n e , C A ( c m 3 ( S T P ) / c m 3 ) , i s af u n c t io n o f t h e u p s t r e a m p a r t ia l p r e s s u r e o f A . T h es o l u b i l i t y ( s o rp t i o n ) co e f f i c i en t o f t h e m em b ran e canb e o b t a i n e d f ro m t h e s o rp t i o n i s o t h e rm . T h e s o l u b i l it yco e f f i c i en t v a l u e i s t h e s l o p e o f t h e s o rp t i o n i s o t h e rm ,SA = CA/PA. In t h e cas e o f an i d ea l m i x t u re , t h ep a r t i a l p r e s s u re can b e u s ed t o d e t e rm i n e t h e s o l u b i l i t yco e f f i c i en t b u t fu g ac i t y h as t o b e u t i li zed w h en t h e g asb eh a v es n o n - i d ea l l y . Wh en t h e f eed s o l u t i o n i s a l iq u i dm i x t u re , t h e p a r t i a l p r e s s u re o r fu g ac i t y t e rm d o es n o tr e f e r t o t h e s t a ti c p r e s s u re . I n s t ead , t h e p a r t i a l p r e s s u reo r f u g a c i t y t e r m r e f e r s t o t h e e q u i l i b r iu m v a p o r p r e s -s u re o f t h e l iq u i d f eed m i x t u re . A s s u m i n g i d ea l i t y an dsubs t i tu t ing the def in i t ion of the so lub i l i ty coeff i c ien t ,SA , in to Eq . (8 )NAdi f d p a= - - D A m S A - ~ ( 9 )

T h e fo l l o w i n g a s s u m p t i o n s w ere u s ed i n E q . (9 ) :1 . T h e d e n s i ty o f th e m e m b r a n e i s a s s u m e d c o n s t a n t

t h r o u g h o u t t h e m e m b r a n e t h i c k n e s s .2 . D i f fu s i o n an d s o l u b i l it y co e f f i c i en t s o f t h e p e r -

m ea t i n g s p ec i e s a r e co n s t an t .T h e d e n s i t y o f t h e m e m b r a n e v a r i e s i n t h e t h i c k -n es s d i r ec t i o n d u e t o d i f f e r en t l ev e l s o f s o rp t i o n ,b u t an av e rag e d en s i t y can o f t en b e u s ed t o ar e a s o n a b l e a p p r o x i m a t i o n . T h e a v e r a g e d e n s i t y c a nb e e s t i m a t e d a s t h e m e a n o f t h e f e e d a n d p e r m e a t es i d e d en si t y . A m o re co m p l ex ex p res s i o n o f d en s i t ycan a l s o b e u s ed , b u t i t w i l l o n l y co m p l i ca t e t h ep i c t u re w i t h o u t ad d i n g i n s i g h t o n t h e p ro b l emb e i n g c o n s i d e r e d h e r e . S i n c e t h e f e e d a n d p e r m e a t es i d e d en s i t y d o n o t v a ry t o o m u ch i n t h e cas eo f l o w t o m o d e r a t e l y s w e l l i n g s y s t e m s , u s i n g t h es i m p l e a v e r a g e d e n s i t y v a l u e w i l l n o t c a u s e m u c he r ro r . O n t h e o t h e r h an d , t h e a s s u m p t i o n o f aco n s t an t d i f fu s i o n co e f f i c i en t can b e e r ro n eo u sb ecau s e d i f fu s i o n co e f f i c i en t s can b e a s t ro n g fu n c -t i o n o f t h e lo ca l p e rm e an t co n cen t r a t i o n s [8 -1 0 ] .T h e re fo re , t h e d i f fu s i o n co e f f i c i en t i n t h e eq u a t i o ni s ac t u a l l y t h e av e rag e d i f fu s i o n co e f f i c i en t o f t h ep e r m e a n t i n t h e m e m b r a n e o f t h ic k n e s s ( . K e e p i n gt h i s i n m i n d an d u s i n g t h e a s s u m p t i o n s s t a t ed ab o v ean d i n t eg ra t i n g E q . (9 ) w i t h t h e fo l l o w i n g b o u n d a ryco n d i t i o n s :X ~ 0 ; P A = P a lx ~ - ~ ; PA = PA2E q . (1 0 ) i s o b t a i n edDArnSA NAdiffd- ( 1 0 )P A l - - P A 2w h ere t h e av e rag e e f f ec t i v e d i f fu s io n co e f f i c i en t o f A ,Dim, i s def ined as fo l lows

CA1 D ( c ) d cbArn = dCa2 ( 1 1 )CAI -- CA2Perm eab i l i t y i s d e f i n ed a s t h e t h i ck n es s an d p re s s u re

n o rm a l i zed f l u x ; t h e re fo re , t h e p ro d u c t o f t h e s o l u b i -l it y a n d d i f f u s io n c o e f f i c i e n t i s th e m e m b r a n e p e r m e -ab i li ty . S i n ce t h e d e f i n it i o n o f m e m b ran e p e rm eab i l i t yas s t a t ed i n E q . (1 0 ) w as d e r i v ed u n d e r t h e l i m i t i n gco n d i t i o n o f ze ro b u l k m as s f l u x , i t i s t h e re fo re m o rea p p r o p r i a t e to r e f e r t o th e m e m b r a n e p e r m e a b i l i t y inE q s . ( 1 0 ) an d (1 2 ) a s a ' d i f fu s i o n -b as ed ' m em b ran e


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15 0 H.D. Kamaruddin, W .J. Koros/Journal of Membrane Science 135 (1997) 147-159p erm eab i l i t y , o r PA*:

N di f f dP A* - - - - D A m SA ( 1 2 )P A l - - P A 2T h i s w i l l d i f f e r en t i a t e t h e d i f fu s i o n -b as ed p e rm eab i l -i t y f r o m t h e e x p e r i m e n t a l l y o b s e r v e d m e m b r a n e p e r -m eab i l i t y , PA , w h i c h i s b a s e d o n b o t h t h e b u l k a n dd i f fu s i o n a l f lu x . T h e re fo re , i n t h e l i m i t wh en b u l k f l u xi s z e ro , t h e d i f f u s i o n -b a s e d m e m b r a n e p e r m e a b i l it y i seq u a l t o t h e p ro d u c t o f a k i n e t i c f ac to r , t h e av e rag em o b i l i t y (d i f fu s i o n ) co e f f i c i en t (Dim) a n d a t h e r m o -d y n am i c f ac t o r , t h e av e rag e s o l u b i l i t y co e f f i c i en t(Si re ). T h e t e rm o f s o l u t i o n -d i f fu s i o n t r an s p o r tm e c h a n i s m c o m e s f r o m t h e d e f i n i t i o n o f d i f f u s i o n -b as ed p e rm e ab i l i t y , t h a t i s, E q . (1 2 ). In t h e cas e o f ani d e a l m i x t u r e a n d p e r m e a t e p r e s s u r e a p p r o a c h i n gze ro , t h e i d ea l m em b ran e s e l ec t i v i t y o f A r e l a t i v e toB i s t h e r a t i o o f t h e d i f fu s i o n -b as ed p e rm eab i l i t i e s

P A * O a m S A (13)OgA/B -- PB* DBm SBD A m/ )B in = mo bi l i ty se lec t iv i tySA - - so lubi l i ty se lect iv i tySBId ea l m em b ran e s e l ec t i v i t y can a l s o b e t ak en t o b eeq u a l t o t h e p ro d u c t o f th e m o b i l i t y an d s o l u b i l i tys e l ec t i v i t i e s . T h e m o b i l i t y s e l ec t i v i t y i s a m eas u re o ft h e r e la t iv e e a s e w i t h w h i c h t h e c o m p o n e n t s p e r f o r md i f fu s i v e j u m p s wh i l e s o l u b i l i t y s e l ec t i v i ty i s a m ea-s u re o f th e r e l a t i v e co n d en s i b i l i ty o f t h e co m p o n en t sa n d a f f i n i ty o f t h e p o l y m e r t o t h e c o m p o n e n t s . U s i n gt h e m o b i l i t y an d s o l u b i l i ty a s p ec t s o f th e co m p o n e n t st o e x p l a i n p e r m e a t i o n p r o p e r t i e s o f m e m b r a n e s y s -t em s h as b een i n s t ru m en t a l i n b r i n g i n g ab o u t ab e t t e r u n d e r s t an d i n g o f t r an s p o r t i n p o l y m er i cm a t e r i a l s . I t is t h e re fo re i m p era t i v e t o u s e t h e co r r ec tp e rm eab i l i t y eq u a t i o n t o ca l cu l a t e t h e d i f fu s i o nco e f f i c i en t .

3 . P e r m e a b i l i t i e s w i t h r es p e c t t o f i x e d f r a m e o fr e f e r e n c e

T h e m a s s f lu x o f A a n d B c a n b e o b t a i n e d b yi n t eg ra t in g E q s . (1 ) an d (2 ) w i th t h e fo l l o wi n g b o u n d -

a ry c o n d i t io n s :X ~ 0~ ~a)A = 03A1 ~ ~-OB = 0.)BIX = ~; 03A : 03A2 ~ 03B : ~0B2w h e n p e r m e a t e ( d o w n s t re a m ) p r e s s u re a p p r o a c h e s t ozero, O)A2 and wn2 -~ 0. In the calcu la t ion , the bu lk f luxco n t r i b u t i o n w i l l b e fu l l y acco u n t ed fo r . T h e a s s u m p -t i o n s o f co n s t an t d i f fu s i o n co e f f i c i en t s an d d en s i t y w i l lb e u s ed a s d es c r i b ed p rev i o u s l y .

pDamln Fl -~2( l+l / r ) ] p[~amln 1--wAl(l+l/r)nAE = Ll -wA'( l+l / r )J(1 + l / r ) (1 + l / r )

(14)pDBmln [1-we2( l+r) l p/)Bmln [ ~ ]n J = L 1 - ~ . , ( l + r ) ]~

(1 + r) (1 + r)0 5 )(16)

q-nB)WB -- (1 q- r)WB gq-bulk __ (h A ~ avg1 IB nBIn ea r l i e r p a rag rap h s , i t was s t a t ed t h a t wh en s o rp t i o nl ev e l s a re s m a l l , n eg l ec t i n g t h e f r am e o f r e f e ren ce(b u l k f l u x ) s eem s t o b e a r eas o n ab l e e s t i m a t i o n .Ho w ev er , E q . (1 8 ) s h o w s t h a t i f t h e m as s f l u x o fc o m p o n e n t A i s m u c h h i g h e r t h a n B ( ' r ' i s h i gh ) , t h e nt h e f r ac t i o n o f b u l k f l u x fo r B can s t il l b e s i g n if i can te v e n t h o u g h ~ v g v a l u e i s s m a l l . L o o k i n g b a c k a tE q . (2 ) , wh en wB i s s m a l l , t h e b u l k f l u x o f B m i g h tb e s m a l l c o m p a r e d t o t h e t o t a l m a s s f l u x o f A a n d B ,( h a -~ - r i B ) . Ho w ev er , t h e b u l k f lu x o f B m i g h t s t il l b es ign i f ican t relative to t h e d i f fu s i o n a l f l u x o f B . T h e re -fo re , t h e b u l k f l u x o f B can n o t b e n e g l ec t ed .

On t h e o t h e r h an d , E q . (1 7 ) s h o ws t h e i m p ac t o f' r ' o n c o m p o n e n t A to b e t h e o p p o s it e o f c o m p o n e n tB. The value of I - I~ L~ app roac hes WA g wh en ' r ' i s

(17)

( 1 8 )

nAw h e r e r = - - ; ( n A > r i B )n Bf ract io n o f bu lk f lux cont r ibu t ion , 1 - I~ lk andh e

i--[bulk o f each c o m p o n en t p e rm ea t i n g t h ro u g h a m e m -B ,b ran e o f t h ick n es s : i s th e r a t i o o f b u l k m as s f l u x to t h eto ta l mass f lux as shown in Eqs . (17) and (18) . Sinceco n cen t r a t i o n g rad i en t s ex i s t i n th e d i r ec t i o n o f m em -b ran e t h i ck n es s , av e rag e co n cen t r a t i o n s , WA g an dav gWB , s h o u l d b e u s ed i n e s t i m a t i n g t h e ap p ro x i m a t ef r ac t i o n o f b u l k f l u x co n t r ib u t i o n s .


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H .D . K a m a r u d d in ,W . J .K o r o s / J o u r n a lo f M e m b r a neS c i en c e 1 3 5 ( 1 9 9 7 ) 1 4 7 - 1 5 9 151l a r g e . Th i s p i c tu r e i s ac tu a l ly q u i t e l o g ica l wh eno n e c o m p o n e n t h a s a m u c h h i g h e r f l u x t h a n t h eo t h e r c o m p o n e n t , t h e s l o w e r c o m p o n e n t i s ' s w e p t 'a lo n g b y th e f a s t e r co m p o n en t . Ob v io u s ly , t h e f lu xo f t h e s lo wer co m p o n en t i s t o o sm a l l t o i n f lu en ceth e b u lk f lu x o f t h e f a s t e r co m p o n en t . I n ex t r em ecases w h e r e th e l ev e l o f so r p t io n o f B an d th e v a lu eo f ' r ' a r e h ig h , t h e b u lk f lu x can ac tu a l ly d o m in a teth e to t a l f l ux o f B . W h i l e t h e se i s su es a r e c l ea r , wed o n o t b e l i ev e th a t t h ey h av e b een p o in t ed o u tp r ev io u s ly , an d th ey can b e r a th e r im p o r t an t i nprac t ica l s i tua t ions .

Av e r ag e co n cen t r a t io n s f o r ea ch co m p o n en t , WA gav gand oJB , c an b e o b ta in ed b y f i r s t d e t e r m in in g th eco n cen t r a t io n p ro f i l e s i n s id e th e m em b r an e , WA (X) a n d0JB (x) , and averaged as fo l lows:

feo OJA X )d XH o a x

1 WA1 (19)- - ( 1 + 1 / r ) I n [ l / ( 1 - - W A I ( 1 + I / r ) ) ]an d

1 O . ) B lf0 Cdx ( l + r ) l n [ 1 / ( 1 - W e l ( 1 + r ) ) ]

(20)De ta i l ed m an ip u la t io n o f Eq s . ( 1 9 ) an d ( 2 0 ) i s d i s -c u s s e d i n t he A p p e n d i x A .

Ex p r ess io n s f o r b u lk f lu x co n t r ib u tio n s a r e o b ta in edby subst i tu t ing Eqs. (19) and (20) to Eqs. (17) and( 1 8 )

H ul k 1 WA I(1 + 1 / r ) (21)A = l n [ 1 / ( 1 - OJAI( i + l / r ) ) ]I - I b u l k 1 ~ m ( 1 + r )8 = ln [1 / (1 - wal(1 + r ) ) ] (22)W h en th e f r ac t io n o f t h e b u lk f lu x wi th r e sp ec t t oth e to t a l m ass f lu x i s m in im a l , t h e co n cen t r a t io np r o f i l e i n s id e th e m em b r an e ap p r o ach es l i n ea r i ty ;th e r e f o r e , t h e av e r ag e co n cen t r a t io n in s id e th e m em -av gbran e, 0:~ g and we , c an b e e s t im a ted to b e th e s im p leav e r ag e o f t h e f eed an d p e r m ea te s id e co n cen t r a t io n s .Subst i tu t ing these va lues to Eqs. (17) and (18) g ives as im p le r m e th o d o f e s t im a t in g th e b u lk f lu x co n t r ib u -] ~ b u l ktion , ~ ~i

H b u l k ( ~ ) O d A 1+ ~ A 2 ~ ( 1 + ! ) 0 3 A Ia ~ 1 + 2 ~ - ( 2 3 )H ul k ~ B 1 71-~ ')B 2 ~ (1 + ~ 03B1 (24)B ~ ( 1 + r ) 2 r )~')A2 = C'0B2 ~ 0

Clear ly , the er ro r o f Eqs. (23) and (24) can besign if ican t i f the f rac t ion of bu lk f lux is s ign i f ican t .Th e r i ~ u lk ob ta ined f r om Eqs. (23) and (24) i s 3 .5%lo wer th an th a t o f Eq s . ( 2 1 ) an d ( 2 2 ) w h enH b u l k = 0 .1 an d 2 0 % lo wer wh en I -Ibulk = 0.5. Th ere -fore , the H bulk ca lcu la ted using Eqs. (23) a nd (24)sh o u ld o n ly b e u sed a s an ap p r o x im a t io n .

W i th th e v a lu e o f t h e f r ac t io n o f t h e b u lk f lu x inh an d , t h e d i f f u s io n - b ased p e r m eab i l i t y o f th e p e r -m ean t can th en b e ca l cu la t ed b ased o n d i f f u s io n a lm ass f lu x wh ich r ep r e sen t s t h e t r u e d i f f u s io n a lt ranspor t .

( ) a ) A I ( I + I / F ) P APA* = l - - I ~ ulk P A = l n [ 1 / ( l _ ~ A l ( l + l / r ) ) ](25)

( ) wm (1 + F ) P ,P B. = 1 - I I bulk P n = l n [ 1 / ( l _ o . ) m ( l + r ) ) ](26)

PA a n d P B a r e t h e e x p e r i m e n t a l l y o b s e r v e d m e m b r a n ep e r m e a b i l i ty o f c o m p o n e n t A a n d B . T h e e x p e r i m e n -t a l ly o b s e r v e d m e m b r a n e p e r m e a b i l i t y f o r c o m p o n e n ti , P i , c an b e o b ta in ed b y n o r m a l i z in g th e o b se r v edd i f f u s io n a l m ass f lu x , n . w i th th e m em b r an e th i ck n essan d d r iv in g f o r ce ( p a r ti a l p r e ssu r e o r f u g ac i ty d i f f e r -e n c e a c r o s s t h e m e m b r a n e ) , A p i .

2 2 4 0 0 n i fPi - - (27)M, ApiI f t h e o b s e r v e d p e r m eab i l i t y ( Eq . ( 2 7 ) ) , is ca l cu la t edb ased o n th e to t a l ( b u lk an d d i f f u s io n ) m ass f lu x , an d i su sed in s t ead o f t h e d i f f u s io n - b ased p e r m eab i l i t y( Eq s . ( 2 5 ) an d ( 2 6 ) ) t o ca l cu la t e t h e av e r ag e d i f f u s io nco e f f i c i en t s , t h e v a lu e o b ta in ed wi l l o v e r e s t im a te th eac tu a l v a lu e . Th e ex ten t o f t h e e r r o r d ep en d s o n th ef r ac t io n o f b u lk f lu x co n t r ib u t io n , t h e h ig h e r t h e b u lkf lu x co n t r ib u t io n th e h ig h e r t h e in accu r acy .

Th e d i f f u s io n - b ased p e r m eab i l i t y i s t h e ap p r o p r i a t ep r o p e r ty o f t h e m a te r i a l s in ce i t i s d ep en d en t o n ly o nthe mater ia l p roper ty , tha t i s , d i f fusion and so lub i l i ty .Th e im p o r t an c e o f t h e d i s ti n c t io n b e tween th e ca l cu -


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152 H.D. Kamaruddin, W.J. Koros/Journal of Membrane Science 135 (1997) 147-159l a t ed p e r m eab i l i t y b ased o n th e d i f f u s io n a l f l u x an dovera l l f lux d is t inc t ion wi l l be i l lus t ra ted wi th twom em b r an e sep a r a t io n ex am p les . Th e f i r s t ex am p le i sth e g as sep a r a t io n o f C O 2 / C H 4 u s i n g 6 F D A - T A D P Op o ly p y r r o lo n e m em b r an e ; t h e seco n d i s t h e r em o v a l o fp h en o l f r o m w a te r ( p e r v ap o r a t io n ) u s in g a p o ly e th e r -b l o c k - p o l y a m i d e ( P e b a x ) m e m b r a n e .

4 . C O 2 / C H 4 s e p a r a t io n u s i n g 6 F D A - T A D P Op o l y p y r r o l o n e sTh e d a ta u sed in th i s ex am p le a r e c i t ed f r o m R ef .

[ 1 1 ] an d th e m ix ed - g as d a t a a r e su m m ar ized inTab le 1 . 6 FDA- TA DPO p o ly p y r r o lo n e i s a r i g idsem i - l ad d e r p o ly m er . I t s o p en s t r u c tu r e an d ch a inr ig id i ty g iv e i t d e s i r ab le m em b r an e p r o p e r t i e s , t h a tis , h igh perm eabi l i ty and s e lec t iv i ty . I t s un iqu e proper -t ies enab le i t to res is t CO2 p las t ic iza t ion bet ter thanm o s t o th e r p o ly m e r s . Du e to th e u n iq u e p r o p e r t i e s o f6 F D A - T A D P O p o l y p y r r o lo n e , it is a n i d e al e x a m p l eto d em o n s t r a t e t h e im p o r t an ce o f t h e b u lk f lu x inm em b r an e sep a r a t io n o f b in a r y m ix tu r e s .

Th e so r p t io n i so th e r m s o f b o th g ases can b e m o d -e l ed wi th a h ig h d eg r ee o f accu r acy u s in g th e d u a l -m o d e m o d e l . T h e d u a l - m o d e s o r p ti o n m o d e l a s s u m e s ,f o r co n v en ien ce , t h a t g l a ssy p o ly m er s h av e two d i s -t i n c t t y p es o f so r p t io n en v i r o n m en t s , f r ee v o lu m e( d e f ec t s ) an d d en se m a t r ix [ 1 2 - 1 5 ] . Th e f r ee v o lu m e( d e f ec t s ) r eg io n ex i s ts b ecau s e o f t h e in ab i l i t y o f t h ep o ly m er ch a in s to p ack p e r f ec t ly a t t em p er a tu r e slo wer th an th e i r g l a ss t r an s i t i o n t em p er a tu r e (Tg). Ine f f ec t , g l a s sy p o ly m er s h av e n o t r each e d eq u i l i b r iu mp ack in g co n d i t io n s ; h o wev e r , t h e s t at e o f t h e p o ly m eri s m e ta s t ab le b ecau se th e r e l ax a t io n t im e o f t h e p o ly -m er ch a in s i s ex t r em e ly lo n g . Th e p o p u la t io n o f t h eco m p o n en t s so r b ed in th e f r ee - v o lu m e s i te s i s r e f e r r ed

to a s Lan g m u i r ' s p o p u la t io n wh i l e t h o se o ccu p y in gth e d en se m a t r ix a r e re f e r r ed to a s Hen r y ' s p o p u la tio n .T h e d u a l - m o d e e q u a t i o n f o r s i n g l e c o m p o n e n tsorp t ion is sh own in Eq . (28) ; the equat ion s fo rb in a r y m ix tu r e co m p o n en t s a r e sh o wn in Eq s . ( 2 9 )and (30) .

C--HAtbAPA~ MA (28)033 = kDAPA~ 1 + bAPAJ 2 2 4 0 0 p

Ctta'bAPA_ .~ MA (29)~3A = kDAFAq 1 + bAPA + bAP BJ 2 2 4 0 0 pCI4,'bBp8 ~ MBcob = ko sps q 1 + bAPA q- bAps J 22400 p (30)

k o i i s t h e ' Hen r y ' s l aw co n s t an t ' wh ich ch a r ac t e r i ze sth e so r p t io n in th e d en se r eg io n o f t h e p o ly m er m a t r ix ,Pi i s the par t ia l p ressure o f component , b , and is am easu r e o f t h e a f f in i ty o f t h e p en e t r an t t o t h e Lan g -m u i r p ack in g ' d e f ec t ' s i t e s an d Ct4' i s t h e L a n g m u i rcap ac i ty co n s t an t . Th e d en s i ty t e r m , p , r e f e r s t o t h ed en s i ty o f t h e sy s t em , th e p o ly m er an d p e r m ean t s .Ho wev e r , i n t h e ca se o f n o n - swe l l in g ( lo w l ev e l o fso r p t io n ) t h e d en s i ty o f t h e sy s t em can b e ap p r o x i -m a ted to b e th e p o ly m er d en s i ty w i th o u t cau s in gm a jo r e r r o r . W h en d esc r ib in g m ix tu r e s wh ich ex h ib i tn o n - id ea l b eh av io r , f u g ac i ty in s t ead o f p a r t i a l p r e ssu r eshould be used . S ince CO2/CI-I4 gas mix ture s are non-id ea l m ix tu r e s , t h e r e f o r e f u g ac i ty h as to b e u t i l i z edwh e n d esc r ib in g th e p e r m ean t s ' t r an sp o r t t h r o u g h th em e m b r a n e . T h e f u g a c i ty o f p u r e a n d m i x e d C O 2 / C H 4can b e ca l cu la t ed u s in g th e v i r i a l eq u a t io n o f s t a t e[ 1 9 ]. F ig . 2 sh o ws th e p u r e an d m ix e d ( 5 0 / 5 0C O2 /C H4 ) C O2 an d C H4 so r p t io n i so th e r m s a s af u n c t io n o f f u g ac ity . Th e p e r cen tag e d ec r ease o f so r p -t io n i so th e r m o f m ix e d C H4 i s m u ch l a r g e r t h an C O2w h e n c o m p a r e d t o t h e i r p u r e c o m p o n e n t s o r p t i o ni so th e rm s . T h i s i s b ecau se in t h e ca se o f m ix ed g as

T a b l e 1P e r m e a t i o n p r o p e rt i es o f t h e s e p a r a ti o n o f 5 0 / 5 0 C O 2 / C H 4 u s i n g 6 F D A - T A D P O p o l y p y r r o lo n e m e m b r a n e [ 1 1]

~ . t~ , b i l e ~ C ~ i l e I "1" u lk l - [ bu lkF u g a c i t y P c o 2 Pcr~ co2 nc o2 : x 109 r 11co2 l l c o ,( p s i a ) ( B a r r e r) ( B a r r e r) ( g / g ) ( g / g ) ( g c m / c m 2 s )1 3 9. 3 3 0 . 0 0 . 5 7 3 0 . 0 1 1 0 0 . 0 0 0 8 2 . 1 8 1 4 3 , 4 0 . 0 0 5 6 0 . 0 5 8 34 0 9 . 1 2 2 . 8 0 . 5 0 8 0 . 0 2 7 3 0 . 0 0 2 4 5 . 1 3 1 2 1 . 9 0 , 0 1 3 8 0 . 1 5 6 55 3 2 . 2 2 0 . 9 0 . 5 0 6 0 . 0 3 5 4 0 . 0 0 3 2 6 . 3 1 1 1 2 . 6 0 . 0 1 8 0 0 . 1 9 5 56 8 2 . 8 1 9 . 9 0 . 5 3 0 0 . 0 4 5 9 0 . 0 0 4 2 8 . 0 2 1 0 7. 8 0 . 0 2 3 4 0 . 2 4 1 37 6 7 . 8 1 8 .7 0 . 5 2 8 0 . 0 5 2 2 0 . 0 0 4 9 8 . 6 5 9 6 . 6 0 . 0 2 6 6 0 . 2 6 3 8B a r re r : 1 0 - 1 c m 3 ( S T P ) c m / c m 2 s c m H g .


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H.D. Kamaruddin, W.J. Koros/Journal of Membrane Science 135 (1997) 147-159 153

AgB

g= =o

0.140.120.100.080.060.040.020.00

0

. . . . i . . . . i . . . . i . . . . i . . . . i . . . . i . . . .: 6FDA-TADPO polypyrrolone5050 CO2/CH4, 35"C. . . . pure gas sorption .- . .

mixed gas sorption ." " Carbon dioxide

. . . . . . . . . . . . Methane. . . . . . . . . , ' , ' 7 0 ; 7 . . . . . . . . . . . . . . . . . . . . . .1 O0 200 300 400 SO0 600 700

g u g a c i t y ( p s i a )Fig. 2 . Pure and mixed (50/50 CO2/CH 4 mol % ) C O 2 and C H 4sorption isotherms.

s o r p t io n , C H 4 h as t o co mp e te wi th C O 2 for avai lab leLan g m u i r ' s s i te s an d CO2 h as a h ig h e r a f f i n it y fo r t h ef ree v o lu me ' d e fec t ' s i t es t h an CH4 .

Th e t ran sp o r t o f p en e t ran t s t h ro u g h ru b b e ry an dglassy po lymer i s s l igh t ly d i fferen t . As expla ined inthe prev ious paragraph , sorbed popula t ions in g lassyp o l y m e r s a re m a d e o f L a n g m u i r ' s a n d H e n r y ' s p op u -la t ions . Koros e t a l . [16] d iscussed the par t ia l imm o-b i l i za t i o n t h eo ry fo r g as mix tu re s wh ich su g g es t edth a t o n ly a f rac t i o n o f Lan g m u i r ' s p o p u l a t io n i s ab l e t op e r fo rm th e d i f fu s iv e j u mp . I t is a s su m ed t h a t Hen ry ' spopula t ion i s com ple te ly mobi le , tha t i s, they are ab leto fu l ly par t ic ipa te in per form ing the d i ffus ive jum p.Therefore , in g lassy po lymers , the mobi le species aremad e o f Hen ry ' s p o p u l a t i o n an d a f rac t i o n o f Lan g -mu i r ' s p o p u l a ti o n . W h en d i scu ss in g t ran sp o rt o f p e r -mean t s t h ro u g h g l a s sy memb ran e p o ly mers u s in gFic k 's fi rst law of dif fusio n equa tion, ~M and cob referto the concent ra t ion of the mo bi le species . I f the en t iresorbed popula t ion i s used , i t wi l l cause a s ign i f ican terror in the ca lcu la ted value . A deta i led descr ip t ion ofthe par t ia l im mo bi l iza t ion theory i s d i scussed in Ref .[ 1 6] . Th e v a lu e o f t h e d i f fu s io n co e f f i c ien t s o f H en ry ' san d Lan g mu i r ' s p o p u l a t i o n can b e o b t a in ed u s in gEqs. (31) and (32) . The cons tan t 'F ' in Eqs. (31)and (32) i s the ra t io of d i ffus ion coeff ic ien t o f Lang -mu i r ' s t o He n ry ' s p o p u l a ti o n s .

* ( F c2K c2 .'~Pc% = kDco2DDco2 1 -+ 1 + b c o 2 f c o 2 -~- bCHffCH 4(31)

Table 2Fugacity-based dual-mode and partial immobilization parametersof 6FDA-TADP O polypyrrolone for CO2 and CH4 at 35C

CO2 CthkD (cm3(STP)/cm3 atm) 1.526 0.327CH' (cm3(STP)/cm3) 34.084 22.838b (atm q) 1.023 0.160Do (cm2/s) 1.196e-7 1.12e-8F (Dt4/DD) 0.084 0.026

P* : kDcu4ODcH4 ( 1 + FcH 4K CH 4CH4 1 bco ~fc o2 + bCH4fCH4.](32)

Th e v a lu es o f t h e d u a l -mo d e an d p a r t i al imm o b i l iza -t ion parameters are summarized in Table 2 .

Th e mo b i l e co n cen t ra t io n o f CO2 an d CH4 i s sh o w nin Eqs. (33) and (34) . As d iscussed in the prev iousparagraph , the par t ia l imm obi l iza t ion theory requi resmo b i l e co n cen t ra t i o n t o b e u sed wh en d esc r i b in gpermeant t ransport in a g lassy po lymer. Thereforew ~bile an d w~B bile term s h ave to b e us ed in tran spo rtequat ions prev iously d iscussed , Eqs . (1 )-(26) .

mobi l e kDc%fC02MC02Wc2 - 224 00 p

mobile~JCH4 --

FC02 KCO2 . ') 1 -~ 1 + bc o~fco2 + bcn4fcn~ / (33)kDc.,fcH4MCH~

22400 pFcmKcH ~ -~ (34) 1 -~ 1 + bco~fco2 -4- bc nf fcr h//

Fig . 3 shows CO2 and CH4 mobi le concent ra t ions ofthe pure and 50/50 mol% CO2/CI-I4 . The i so thermsare calculated using Eqs. (33) and (34). Fig. 3 showsthat the sorp t ion i so therms of the mobi le species aremu ch l o wer t h an t h e so rp t i o n i so th e rm o f t h e t o t a lsorbed species . The concen t ra t ion of bo th m obi le CO2an d C H 4 a r e o n ly s l i g h t l y l o wer i n mix ed -g as t h anp u re g as . Th i s i s b ecau se mo s t o f t h e mo b i l e sp ec i e sa re d u e t o H en ry ' s p o p u l a t i o n an d o n ly a sma l l f rac ti o no f Lan g mu i r ' s p o p u l a t i o n i s mo b i l e . Th e d u a l -mo d eth eo ry a s su mes H en ry ' s p o p u l a t io n t o b e t h e same inp u re an d m ix ed -g as ca se.

Fig . 4 shows the exp erimen ta l an d pre d ic tedC O 2 / C H 4 mixed-gas ideal membrane se lec t iv i ty .


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154 H.D. Kamaruddin, W.J. Koros/Journal of Membrane Science 135 (1997) 147-1590 . 1 0 . . . . i . . . . i . . . . i . . . . i . . . .

6 F D A - T A D P O p o l y p y r r o l o n e C O z -t o ta l ( p ur e)3 5 " CA 0 . 0 8

g o.o6~ . - A " " "~ C O 2 " m o b i l e

/ . . " ( p u r e a n d m i x e d )" 0.04 / .~-.~"

O 0.02 / , a , - ' ' ' ' ' ' ~ ' " C H 4 - t o t a l ( p u r e )o . oo , ~ . . . . . . . . . . . . . . . . . . . . . . ~ - ~ o - ~ e ' ~ ; . ~ e d ) -. . . . i . . . . J . . . . i . , 4 . . i . . . .

1 0 0 2 0 0 3 0 0 4 0 0 5 0 0F u g a c i t y ( p s i a )

F i g . 3 . P u r e a n d m i x e d ( 5 0 / 5 0 C02/CH4 e e l % ) m o b i l e C O 2 a n dCH4 sorption isotherms.

14 x l 0 " a12 x l 0 " a1 0 x l 0 "a

8 x l O " a

6 x l O " aO4 x l 0 "a

~ , 2 x l O " a

i i i i iCarbond i o x i d e- , . - . _ . . - , - - . , - - - - , . , - - , - . - . , . . . . ; - - - .

C a r b o n d i o x i d e M e t h a n eo m i x e d - g a s z x m i x e d - g a s p u r e g a s p u r e g a s. . . . t h e o r e t i c a l . . . . t h e o r e t i c a l

I0 6 0 0

M e t h a n e- 'C A - . . . . . . . / r . . . . ~ r - - - - - ~ - - - ~ - -I I I I 1

1 0 0 200 3 0 0 400 500F u g a c l t y ( p s i a )

Fig. 5. Pure, mixed and predicted CO2/C H 4 mobility andsolubility coefficients of 6FDA-TADPO polypyrrolone.

~ " 8 0>1;eOl 7 0~ 6 0E~ s o"oE~ 4 o,o0=" 30000 20

. . . . i . . . . i . . . . i . . . . i . . . . i , . ,6 F D A - T A D P O p o l y p y r r o l o n e5 0 1 5 0 C O 2 / C H , , 3 5 " C

" " . . . . .. . . . . . . . . . . . . . . . .

o E x p e r i m e n t a l- - - N e g l e c t i n g t h eb u l k f l u x c o n t r i b u t i o n- - C a l c u l a t e d

. . . . i . . . . i . . . . i . . . . i , i i . J . . . .1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0

C H 4 F e e d F u g a c l t y ( p s l a )

Fig. 4. Comparison of the experimental (solid circle) andcalculated (solid line) observed membrane selectivities of 50/5 0CO2/CH4 mol%. The calculated selectivity was based on dual-mode and partial immobil ization parameters obtained from pureCO2 and CI-I4 sorption and permeation data. The dotted linerepresents the predicted selectivity when the bulk flux contributionis neglected.

Th e so l id c ir c l e s r ep r e sen t th e ex p e r im en ta l d a t a an dth e so l id l i n e i s t h e ca l cu la t ed v a lu e p r ed ic t ed u s in gpure component sorp t ion and permeat ion data . Fi r s t ,t h e m o b i l e co n cen t r a t io n o f C O2 an d C H 4 i s ca l cu la t edu s in g eq u a t io n s Eq s . ( 3 3 ) an d ( 3 4 ) . Th en th e m assf lu x es o f C O2 an d CH4 can b e ca l cu la t ed u s in gEqs. (14) and (15) . The ca lcu la t ion wi l l requ ire i te ra-t i o n s b ecau se th e v a lu es o f n co2 , ncH4 an d ' r ' a r eu n k n o wn . Th e r e f o r e , t h e ' r ' v a lu e o b ta in ed f r o mEq s . ( 3 1 ) an d ( 3 2 ) ( p a r ti a l im m o b i l i za t io n th eo r y )

can b e u sed a s an in i t i a l e s t im a te . Th e ca l cu la t io n swer e p e r f o r m e d u s in g a sp r ead sh ee t p r o g r am an d i t isac tu a l ly q u i t e s t r a ig h tf o r war d . Th e ca l cu la t ed v a lu eso f t h e o b se r v ed m em b r an e se l ec t iv i ty a r e i n g o o dag r eem en t w i th th e ex p e r im en ta l d a t a . F ig . 4 a l soshows tha t s ign i f ican t e r rors can ar ise i f d i f fusion-b ased p e r m eab i l i t i e s ( n eg lec t in g b u lk f lu x co n t r ib u -t io n ) a r e u sed to ca l cu la t e t h e o b se r v ed m em b r an eselec t iv i ty as i l lus t ra ted by the do t ted l ine . I t over -p r ed ic t s th e ex p e r im e n ta l ly o b se r v e d m e m b r an e se l ec -tivity.

F ig . 5 sh o w s th a t t h e p u r e , m ix ed - g as an d th eth eo r e t i ca l v a lu e o f d i f f u s io n ( m o b i l i t y ) co e f f i c i en to f C O2 an d CH 4 are ap p r o x im a te ly eq u a l wh en th eb u lk f lu x co n t r ib u t io n s a r e n o t a s su m ed to b e n eg l i -g ib l e. Th e d i f f u s io n co e f f i c i en ts o f t h e m ix ed - g as w er eca lcu la t ed b y u s in g Eq s . ( 1 4 ) an d ( 1 5 ). Th e r e f o r e , t h elo wer v a lu e o f t h e ex p e r im en ta l ly o b se r v ed se l ec t iv i ty(Fig . 4 ) in the mixed -gas case i s no t due to s l igh tm em b r an e p l a s t i c i za t io n cau sed b y th e p r e sen ce o fC O2 as o r ig in a l ly th o u g h t . Th i s p h en o m en o n can b ef u l ly ex p la in ed b y t ak in g in to acco u n t t h e b u lk f lu xco n t r ib u t io n . Pu r e C O2 p e r m ea t io n d a t a [ 1 1 ] a l soco n f i r m ed th e ab sen ce o f p l a s t i c i za t io n b y C O2 .

F ig . 6 sh o ws th e co m p ar i so n o f t h e f r ac t io n o f b u lkm ass f lu x o f C O2 in th e p u r e - an d m ix ed - g as ca se . T h ecu r v es wer e ca l cu la t ed u s in g Eq . ( 2 1 ) . As d i scu ssedea r l ie r , t h e b u lk m ass f lu x co n t rib u t io n o f t h e co m p o -n en t w i th th e h ig h e r m ass f lu x in a b in a r y m ix tu r em e m b r a n e p e r m e a t i o n c a n b e e s t i m a t e d t o b e a p p r o x i -m a te ly eq u a l t o i t s av e r ag e so r p t io n l ev e l wh en th e


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H.D. Kamaruddin, W.J. Koros/Journal of Membrane Science 135 (1997) 147-159 1550 . 1 0

~ 0.088

. 0 . 0 6

m 0 . 0 4"6

0 . 0 2.m

0 . 0 0

. . . . i . . . . i . . . . i . . . . i . . . . i . . . . i , , .

6 F D A - T A D P O polypyrrolone3 5 " CC a r b o n d i o x i d eo mixed-gas5 0 1 5 0 C O 2 / C H 4

- - e - - p u r e g a s . . e. -

. w- o. o o '

l . O - ' ' ~o o . r " o "

~ ' , " , ' ~ . i . . . . i . . . . i . . . . i . . . . i . . . . q . . . .1 0 0 Z O 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0

Carbon dioxide Fugacl ty (paid)Fig. 6. Bulk flux contribution of C O 2 in a 50/50 C O 2 / C H 4 mixed-gas experiments as a function of its fugacity.

0 . 3 0go.2s

t~ 0.20, . r 0 . 1 5

' 3 o . ~ o

0.05

0 . 0 0

. . . . ~ . . . . i . . . . , . . . . i . . . . i . . . . . . . - - ' "6 F D A ' T A D P O 5 . 1 ~ o l y P y r r l n e . o . -. o o -

s O0 M e t h a n e

- - o - - m m x e o - g a s " 5 0 1 5 0 C O ~ I C H 4

, - e " " - - = - . p u r e ga se

~-.-.-rQ-~.T-t~-,-9-r-,--,-r-?n-~e~. -,l-.'T-,'-.-T-:-l'T-T-7-.'

0 tOO 200 300 400 500 600Methane Fugaci ty ( p s l a )

Fig. 7. Bulk flux contribution of CH4 in a 50/50 C O 2 / C H 4 mixed-gas experiments as a function of its fugacity.

v a l u e o f ' r ' i s l a rg e (E q . ( 2 3 ) ) . O n t h e o t h e r h an d , a ss h o w n i n F i g . 7 , t h e f r ac t i o n o f t h e b u l k mas s f l u x o fC H 4 i n t h e m i x e d - g a s c a s e i s m u c h h i g h e r t h a n t h ep u r e - g a s c a s e . T h e f r a c t io n o f b u l k m a s s f l u x o f C H 4 i nt h e mi x ed -g as ca s e w as ca l cu l a t ed u s i n g E q . ( 2 2 ) . I nt h e m i x e d - g a s p e r m e a t i o n , C H 4 m o l e c u l e s w e r e' s w e p t ' a l o n g b y t h e m u c h f a s t e r C O 2 m o l e c u l e s .T h e r e f o r e , a c o n s i d e r a b le p o r t i o n o f t he m a s s f lu xi s a re s u l t o f b u l k f l u x w h i ch i f n o t acco u n t ed fo r canl ead r e s ea r ch e r s t o i n co r r ec t l y co n c l u d e t h a t l o w ermi x ed -g as s e l ec t i v i t y is d u e t o t h e l o s s o f t h e mo b i l i t yse lec t iv i ty , tha t i s , as a resu l t o f p las t i c i za t ion .

In o rd e r t o u n d e r s t an d t h e ma t e r i a l s s c i en ce a s p ec to f t h e p o l y m e r - p e n e t r a n t i n t e r a c t i o n , t h e b u l k m a s sf l u x co n t r i b u t i o n h as t o b e s u b t r ac t ed f ro m t h e t o t a l

m as s f l u x . T h i s i s b ecau s e t h e mo b i l i t y an d s o l u b i l i tyo f t h e p e n e t r a n t s i n t h e m e m b r a n e c a n o n l y b ed es c r i b ed u s i n g t h e d i f fu s i o n -b as ed p e rmeab i l i t i e s .I n t h i s e x a m p l e , t a k i n g i n t o a c c o u n t t h e f r a m e o fr e f e r en ce en ab l e s u s t o co m e t o a d if f e r en t co n c l u s i o na b o u t t h e p e r m e a t i o n p r o p e r t i e s o f t h e m e m b r a n e .

5 . R e m o v a l o f p h e n o l f r o m w a t e r u s i n g ap e r v a p o r a t i o n p r o c e s s

In t h e p r ev i o u s ex amp l e , t h e b u l k f l u x co n t r i b u t i o nh as a n eg a t i v e i mp a c t o n t h e s ep a ra t i o n . H o w e v e r , t h eb u l k f l u x co n t r i b u t i o n can b e p o s i t i v e o r n eg a t i v e fo rt h e s ep a ra t i o n p e r fo rm ed . I t d ep en d s o n t h e d es i r edco m p o n en t t o b e s ep a ra t ed , i t is p o s i t i v e i f t h e f l u x o ft h e c o m p o n e n t t o b e s e p a r a t e d is e n h a n c e d b y t h e b u l kf lu x m o r e t h a n t h e o t h e r c o m p o n e n t a n d v i c e v e r s a . I nt h e c a s e o f t h e r e m o v a l o f p h e n o l f r o m w a t e r , t he f l u xo f p h e n o l i s e n h a n c e d m o r e t h a n t h e f lu x o f w a t e r b yt h e b u l k f l u x co n t r i b u t i o n . T h e re fo re , t h e b u l k f l u xco n t r i b u t i o n ac t u a l l y b en e f i t s t h e s ep a ra t i o n b ecau s ep h e n o l i s t he d e s i r e d c o m p o n e n t t o b e s e p a r a te d .

T h e d a t a u ti l iz e d i n t h is e x a m p l e a r e s u m m a r i z e d i nT ab l e 3 an d t h ey a r e o b t a i n ed f ro m Ref . [ 1 7 ] . T h er e m o v a l o f p h e n o l f r o m w a t e r w a s d o n e u s i n g ap e rv ap o ra t i o n p ro ces s u t i l i z i n g a p o l y e t h e r -b l o ck -p o l y a m i d e ( P e b a x ( A t o c h e m ) ) m e m b r a n e . T h em e m b r a n e s e l e c t i v i t e l y r e m o v e s p h e n o l f r o m w a t e r .The overa l l pervapora t ion separa t ion fac to r , /~per~ap ,c a n b e t h o u g h t o f a s e q u a l t o t h e p r o d u c t o f e v a p o r a -t i o n s ep a ra t i o n f ac t o r , f levap , an d m em b ra n e s ep a ra t i o nfac to r , f lr ,~m [18] . Th e eva por a t ion se para t ion fac to r ,f levap , i s a m eas ure o f the re la t ive v o la t i l i ty o f thec o m p o n e n t s w h i l e t h e m e m b r a n e s e p a r a t i o n f a c t o r ,

Table 3Removal of phenol from water using a polyether-block-polyamidemembrane [16]

, . ,mobile bulkPhenol ~ p h e n o l n p h e n o l r ] - [p h e n o lFeed conc. (ppm) (g/g) (g/ m2h) ( n U ~ o / n p h e n o l )31 0.00105 0.4 434.5 0.2548 0.00200 0.6 290.7 0.3385 0.00330 1.1 161.5 0.30

153 0.00600 1.9 93.3 0.32324 0.01300 4.1 44.1 0.34585 0.02300 7.5 24.6 0.34994 0.03500 12 .7 14.9 0.32


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156 H.D. Kamaruddin, W.J. Koros/Journal of Membrane Science 135 (1997) 147-159/~mem, is a measure of the relative solubility anddiffusivity of the permeants in the membrane. Atlow permeate pressure, the membrane separation fac-tor, /~mem,is approximately equal to the ideal mem-brane selectivity

Pebax membrane is selective toward phenol overwater, that is, phenol has a higher permeability thanwater In this example, phenol has a lower mass fluxthan water because phenol has a much smaller drivingforce than water due to low phenol concentration inthe feed solution Since water has a higher mass fluxthan phenol, the mass flux of water has a positiveeffect on phenol bulk mass flux, much more than theeffect of phenol mass flux on water The linearity ofconcentration of phenol as a function of phenol feedconcentration suggests the sorbed population is mostlyHenry' s (Fig. 8). Therefore, it is assumed that all of thesorbed phenol in the polymer is mobile The summaryof the data is shown in Table 3 and Fig. 8. The fractionof phenol bulk flux contribution is illustrated in Fig 9.Fig. 9 shows that the bulk flux contribution of phenolis approximately 30% of the total mass flux. Thescatter of data in Fig. 9 is presumably due to an errorcaused by translating the sorption data from the phenolsorption graph in Ref. [17]. The water sorption level ofpolyether-block-polyamide membrane is approxi-mately 0.02 gH20/g at phenol feed concentration of31-994 ppm. Thus, referring to Eq. (17) or Eq. (21) itis clear that the bulk flux contribution for water isminimal As a result bulk mass flux has benefitted theseparation process by enhancing the mass flux ofphenol to a much larger degree than water, thus

A

8=

00= .=r,

0.0400.0350.0300.0250.0200.0150.0100.0050.000 -

0

' I ' ' ' ~ ' ' 'polyether-block-polyam ides46 Ilm, 50"C

D" , I , , , I , , , i , , , i

200 400 600 800Phenol F e e d C o n c e n t r a t io n (ppm)

1000

Fig. 8. Phenol sorption level as a function of feed concentration.

0 . 58

o. 4Co0 0 . 3;-r~ 0 . 2o~ 0 .1,,.r

0

' I , ' I , I ' ' ' i , ,

p o l y e t h e r - b l o c k - p o l y a m i d e s46 l t tm, 50 "C

___., . . . . . . . ". . . . . . . . . . . _" . . . . . . . . . . . . . . . . .

I i i i I , , , I , , , I , , ,

2 0 0 4 0 0 6 0 0 8 0 0 10 0 0Phenol F eed Concentration (ppm)

Fig. 9. Bulk flux contribution of phenol as a function of phenolfeed concentration.

improving the removal of phenol from the feed solu-tion.

6 . C o n c l u s i o n

The objective o f this paper is to show that the frameof reference term cannot always be ignored even if thesorption level is minimal. In the case of membrane-based separation of multicomponent mixtures, theassumption of negligible bulk flux based on lowsorption levels is erroneous when the relative massflux of the other component is high. In the twoexamples discussed above, the sorption level of thecomponents is relatively small; however, significantbulk flux contribution is observed The above discus-sion shows that keeping the frame of reference term inthe transport equation can make a difference in thefundamental interpretation of experimental data.

In the case of binary systems in which the mass fluxof the component to be separated is much higher thanthe other component (high 'r') and the slower corn-/ a v g \ponent has a moderate-to-high sorption level (w8 ),the separation process can be inefficient even thoughthe relative mobility and solubility favor the desiredcomponent to be separated (component A) tremen-dously. The first example showed that even at lowsorption levels of CH4, the separation processbecomes less efficient due to the enhanced bulk fluxof CH4. If the sorption level of mobile CH4 is doubled,the result would have been very detrimental to the


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H.D. Kamaruddin, W.J. Koros/Journal of Membrane Science 135 (1997) 147-159 1 5 7se pa r a t i on p r oc e s s . I t is t he r e f o r e impe r a t i ve t o r e duc e kDt he bu lk f l ux i n o r de r t o a c h i e ve be t t e r s e pa r a t ion . T h i s Mp o i n t s h o u l d b e k e p t i n m i n d w h e n s e l e ct i n g a m e m - nbr a ne f o r a pa r t i c u l a r s e pa r a t i on p r oc e s s . S inc e bu lk n bulkm a s s f lu x o f th e s l o w e r c o m p o n e n t ( c o m p o n e n t B ) is n a i f fpr opo r t i ona l t o t he so r p t i on l e ve l a nd ( 1 + r ) , t he r e - Nf or e , min im iz ing t he so r p t i on l e ve l o f t he s l ow e r pc o m p o n e n t w i l l i m p r o v e t h e s e p a ra t io n . R e d u c i n g P' r ' i s no t de s i ra b l e be c a use t he i de a l m e m br a n e se l e c -t i v i t y i s a l so r e duc e d . I f t he so r p t i on l e ve l o f t he s l ow e r P .c o m p o n e n t i s r e d u c e d w i t h o u t t h e e x p e n s e o f t h emo bi l i t y a nd so lub i l i ty s e l e c t iv i t i e s t he n t he m e m - rb r a n e s e p a r at i o n w i l l b e i m p r o v e d . T h e i m p r o v e m e n t Sto t he se pa r a t i on i s a f unc t ion o f t he r e duc t ion o f t h e ~"is o r p ti o n l e v e l o f t h e s l o w e r c o m p o n e n t , a

I n t he de r iva t i on o f t he e qua t ions , t he d i f f us ion / ~ p e r v a p( m o b i l i t y ) c o e f fi c i e n t o f t h e c o m p o n e n t s i s a s s u m e d / ~ e v a pt o be c ons t a n t . T h i s a s sum pt ion sh ou ld be qu i t e a c c u - f l m e mr a t e i n t he c a se o f t he tw o e xa m ple s i l lu s t r a t e d i n t h i s ~vp a p e r b e c a u s e o f t h e l o w l e v e l o f s o r p ti o n . H o w e v e r , pth i s a s sum pt ion c a n be e r r one o us i n t he c a se o f a 1 -[bulkh igh ly sw e l l i ng sys t e m , t ha t is , h igh so r p t i on l e ve l o ft h e c o m p o n e n t s w h e r e t h e d i f fu s i o n c o e f f i ci e n t o f t h e 7,1. Subscriptspe r m e a n t s c a n be a s t r ong f unc t ion o f t he loc a lp e r m e a n t c o n c e n t r a ti o n . T h e c o n c e n t r a t io n p r o f il e s i c o m p o n e n t io f t h e c o m p o n e n t s i n t h e m e m b r a n e c a n d e v i a t e f r o m 1 f e e d s id ethe p r e d i c t e d va lue us ing c ons t a n t d i f f us ion c oe f f i - 2 pe r m e a te s i dec i e n t , D im. T o ge t a n a c c ur a t e r e p r e se n t a t i on o f t hepe r m e a n t t r a nspor t , a r e l a t ionsh ip o f d i f fus ion c oe f f i -c i e n t a s a f unc t ion o f t he pe ne t r a n t c onc e n t r a t i on ha sto be ob t a ine d a nd u t i l i z e d i n the t r a nspor t e qua t ions .A n o t h e r w a y w o u l d b e t o e m p l o y a f i n i t e e l e m e n tm e t h o d .

7. Notations

d u a l - m o d e H e n r y ' s c o n s t a n t ( a t m - 1 )m o l e c u l a r w e ig h t ( g / m o l )m a s s f l u x ( g / c m 2 s )b u l k m a s s f l u x ( g / c m 2 s )d i f fu s i o n m a s s f l u x ( g / c m 2 s )g a s m o l a r f l u x ( c m 3 ( S T P ) / c m z s)p r e s s u r e ( c m H g )o b s e r v e d m e m b r a n e p e r m e a b i l i t y ( c m 3( S TP ) c m / c m 2 s cm H g )d i f f u s i o n - b a s e d m e m b r a n e p e r m e a b i l i t y( cm 3 ( S T P ) c m / c m 2 s c m H g )r a t i o o f ma ss f l ux , nA/n8s o l u b il i ty c o e f f i c ie n t ( c m 3 ( S T P ) / c m 3 c m H g )s o rb e d m o l a r v o l u m e ( c m 3 / m o l )ide a l me mbr a ne se l e c t i v i t yove r a l l pe r va por a t i on se pa r a t i on f a c to re va por a t i on se pa r a t i on f a c to rm e m b r a n e s e p a r a t i o n f a c t o rc o n c e n t r a t io n ( g / g )d e n s it y ( g / c m 3 )f r a c t i on o f bu lk f l ux c on t r i bu t ion

AcknowledgementsT h e a u thor s w ou ld l i ke to e xpr e s s t he i r g r a t it ude t o

t h e D e p a r t m e n t o f E n e r g y ( D O E ) a n d S e p a r a t i o nR e s e a r c h P r o g r a m ( S R P ) a t t h e U n i v e r s i t y o f T e x a sa t A us t i n f o r sponsor ing t h i s p r o j e c t .

bCCn '

DimD ~D of ~F~

d u a l - m o d e a f f in i t y c o n s t a n t ( a tm - 1 )c o n c e n t ra t io n ( c m 3 ( S T P ) / c m 3)d u a l - m o d e c a p a c i t y c o n s t a n t ( c m 3 ( S T P ) /c m 3 )d i f f u s io n c o e f f i c ie n t ( c m 2 / s )d i f fu s i o n c o e f f i c i e n t o f L a n g m u i r ' s p o p u l a -t i o n ( c m 2 / s )d i f f u s i o n c o e f f i c i e n t o f H e n r y ' s p o p u l a t i o n( c m 2 / s )f u g a c i t y ( c m H g )DHi/DDi

Appendix AA v e r a g e c o n c e n t r a t i o n s f o r e a c h c o m p o n e n t ,a v gWa g a nd t v~ , c a n b e ob t a ine d by f i rs t de t e r m in-

ing t he c onc e n t r a t i on p r o f i l e s i ns ide t he me mbr a ne ,(.z3A ( X ) and ~v8 (x) , wh ich i s obta in ed b y integ ra t -i ng ( 1 ) a nd ( 2 ) w i th t he f o l l ow ing bounda r y c ond i -t ions :X = 0 ; 0 .)A = OJA1 ~ 03B = OJB1

= X; ~ = ~ (X ) , ~B = ~B(X)


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158 H.D. K amaruddin , W.J. Koros/Journal o f M embrane Science 135 (1997) 147-159

nA = (hA Jr- nB Jr np)OJA -- pDAm- dWA-~b d w an B = ( nA + n B + n p) W a - - p B m - ~

1 - [ 1 - W At(1 + 1 /r ) ] e xp [ ~ ]W A ( X )= (1 + 1 / r )a n d

1 - [1 - W Sl(1 + r)]e xp [nB(~r)x]L pusm .I ( A . 4 )~B(X) = (1 + r)

T h e c o n c e n t r a ti o n o f A a n d B i n si d e t h e m e m b r a n e o ft h i ck n es s f can b e av e rag ed as fo l l o ws :

1avg f ~ & A ( X ) d X ( ( 1 - I - I / r ) )~.z3A - - ~ - -

( 1 - - ( . ( 1 - ~ at( 1-k -n ~l1/r)p[)Am)~~/r)/I[ " f n a ( ( 1 - t - 1 / r ) " ~ 1 ) } k e x p k p b- ~- m ) -

1 WA1 ( A . 9 )(A .1) waVg -- (1 + l / r ) l n [1 / (1 - - WAt (1 + i / r ) ) ]( A . 2 ) E q s . ( A . 1 0 ) a n d ( A . 1 1 ) a r e o b t a i n e d b y e v a l u a t i n g

Eq . (A . 4 ) w i t h t h e fo l l o wi n g b o u n d ary co n d i t i o n s :X = O; W e(X ) ~--- a)B1

1 - [ 1 - w i n ( 1 + r ) ] (A . 1 0 )(A.3) ~BI = (1 + r )x = ~ ; W B ( X )= 0 ( d o w n s t r e a m p r e s s u r e i sa p p r o a c h i n g z e r o )

[nB((lp_~B_~r).]0 = 1 - [ 1 - wB I (1 + r ) ] ex p [ (A . 1 1 )A s i m p l e r ex p res s i o n o f w~ g i s o b t a i n ed b y r ea r r an -g i n g Eq . (A . 4 ) an d s u b s t i tu t i n g Eq s . (A . 1 0 ) an d(A . 1 1 )~dBvg __ 1 0)B1( l + r ) I n [ l / ( 1 - a :m ( 1 + r ) ) ] ( A . 1 2 )

(A.5)a n dw~vg f ~ w B ( x ) d x ( 1 )

-

x ( 1 - ( . ( 1 - ~Bl(lnB((l+--r)P[)~m)"~+) ]f [nB((1 + - 1 )k e x p [ P D B m r ) J } _ ( A . 6 )

Eq s . (A . 7 ) an d (A . 8 ) a re o b t a i n ed b y ev a l u a t i n gE q . ( A . 3 ) w i t h t h e f o l lo w i n g b o u n d a r y c o n d i t io n s :X = 0 ; O J A ( X) = ~ A 1

I - [ 1 - W A I (1 + l / r ) ] ( A . 7 )03A1 : ( 1 + l / r )x = f ; W A ( X ) = 0 ( d o w n s t r e a m p r e s s u r e i s a p p r o a -c h i n g z e r o )0 = 1 - - [ 1 - - W A l ( l + ! ) ] e x p [ n A f ( ~ l p ~ A 1 / r ) ]

( A . 8 )the expres s ion for Wa g can b e s impl i f ied by subs t i tu t -i n g Eq s . ( A . 7 ) a n d ( A . 8 ) t o Eq . ( A . 5 ) .

R e f e r e n c e s[1] R.B . Bird , W .E. Stew art and E .N . Lightfoot , TransportPhenomena, John Wiley and Sons, New York, 1960.[2] D .R. Paul and O.M . Ebra-Lima, Pressure-induced diffusion of

organic l iquids through highly sw ollen polymer membranes,J. A ppl. Polym. Sci., 14 (1970) 2201.[3] D .R . Paul and D .H. Carranza, Pressure-induced diffusion inswollen butyl rubber membranes, J. Polym. Sci., 41 (1973)69 .[4] D .R . Paul, Relation between hydraulic permeabili ty anddiffusion in hom ogeneous swollen membranes, J. P olym. Sci.,11 (1973 ) 289.[5] O.M . Ebra-Lima and D .R . Paul, Hydraulic permeation ofliquids through swollen polymeric networks. I. Poly(vinylalcohol)-water, J. Appl. Polym. Sci., 19 (1975) 19.[6] D .R . Paul, J.D . Paciotti and O .M . Ebra-Lima, Hydraulicpermeation of l iquids through sw ollen polymeric networks. II.Liquid mixtures, J. Appl. Polym. Sci., 19 (1975) 1837.[7] D .R . Paul and O .M . Ebra-Lima, H ydraulic Permeationo f l i q u id s t h ro u g h swo l l e n p o ly m e r i c n e two rk s , l l IA generalized correlation, J. Appl. Polym. Sci., 19 (1975)2759.[8] J. C omyn, in J. C omyn (Ed.), Polymer Permeabili ty, ElsevierAppl. Sc. Ltd, London, 1985, Ch. 1.[9] J.L . Duda, J.S . Vrentas, S.T . Ju and H.T. Liu, Prediction ofdiffusion coefficients for polymer-solven t sy stems, AICh E J.,28 (1982) 279.[10] J .S. Vrentas and J.L. Duda, M olecular diffusion in polymersolutions, AIChE J., 25(1) (1979) 1.


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o f po lypyr ro lones fo r ga s s epa ra t i on m em branes , TheUnivers i ty of Texas a t Aust in , 1994.

[12] W.J. Koros, Ph.D. The sis, Sorp tion and transpo rt in glassypolymers , The U nivers i ty of Texas a t Aust in , 1977.

[13] W.R. Vieth, J.M. How ell an d J.H. Hsieh , Dual-so rptiontheory, J. Mem b. Sci. , 1 (1976) 177.

[14] H .B . Hopfen berg and V . Stannet , in R .N. Haw ard (Ed.) , ThePhy sics of Gla ssy State, Appl. Sci. Publ. Ltd. , London , 1973,Ch. 9 .

[15] W.J . Koros , Model for sorpt ion of mixed gasses in glassypolymers, J. Polym. Sci. Polym. Phys. Ed. , 18 (1980) 981.

[16] W .J. Koros, R.T. Che rn, V. Stannett and H.B. H opfenberg,A m ode l f o r pe rm ea t i on o f m ixed gases and vapor s i nglassy polymers, J. Polym. Sci. Polym. Phys. Ed., 19 (1981)1513.

[17] K .W . Bodd eker and G . B engston, Pervaporat ion of lowvola til i ty arom atics from water, J. Memb. Sci. , 53 (1990 ) 143.

[18] J .G . W ijmans and R .W. Baker , A s imple predic t ive t rea tmentof the perme ation process in pervaporation, J. Memb. Sci. , 79(1993) 101.

[19] J.M. Prausnitz, R.N. L ichte ntha ler and E.G. de Azevedo,Molecu lar Thermodyn amics of Fluid-phase Equil ibr ia , 2ndedn., Prentice-Hall, NJ, 1986.

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