Prediction of mass-transfer coefficient with suction

8/9/2019 Prediction of mass-transfer coefficient with suction

1/13

E L S E V I E R

J o u r n a l o f M e m b r a n e S c i e n c e 1 2 8 1 9 9 7 ) 1 1 9 - 1 3 1

rnal

f

B R A N E

S C I E N C E

P r e d i c t i o n o f m a s s t r a n s f e r c o e f f i c ie n t w i t h s u c t i o n i n th e

a p p l i c a t i o n s o f r ev e r s e o s m o s i s a n d u lt ra f il tr a ti o n

S . D e , P . K . B h a t t a c h a r y a *

Depar tment o f Chemica l Eng ineering Indian Ins t i tu te o f Technology a t Kanpur K an pu r- 208016 Ind ia

R ec e ived 22 Apr i l 1996; r ece ived in r ev i s ed fo rm 5 Aug us t 1996 ; accep ted 11 Octo ber 1996

A bst ract

Sherwood-number re la t ions for predic t ion of the mass- t ransfer coeff ic ient for developing concent ra t ion boundary- layer

have been o bta ined for lam inar f low-regime from f i rst pr inc iples. The c om mo n f low-modules, namely, rec tangular channel,

tubu lar and radial cross-flow are considered. T he relationships deve loped include the effect of suction thro ugh the me mb rane.

Relevant relations for estimation o f ma ss-tran sfer coefficient for cross-flow reverse o sm osis and ultrafi ltration are formulated.

The Sherw ood -num ber re la t ions developed are com pared w i th the standard corre lat ions to quant i fy the effec t of the suc tion.

The pro posed S herw ood re la t ions are used in conjunct ion wi th the osmo t ic-pressure model to predic t the permeate f lux in

reverse osm osis an d o smo t ic-pressure g overne d ul tra fi lt rat ion.

Keywords Mass-transfer coefficient; Suction; Laminar flow; Osmotic pressure: Cross flow; Reverse osmosis; Ultrafi l tration

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

T h e d e s ig n o f p r es s u re d r i v en m e m b r a n e - s e p a r a t i o n

p r o c e s s e s , l i k e re v e r s e o s m o s i s ( R O ) a n d u l t r a f il tr a t io n

( U F ) , a r e g e n e r a l l y b a s e d o n t h e m a s s - t r a n s f e r c o e f f i -

c i e n t ( k ) f o r th e r e l e v a n t f l o w - c o n f i g u r a t i o n a n d f l o w -

r e g i m e . T h e m a s s - t r a n s f e r c o e f f i c i e n t s u s e d f o r s u c h

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

o b t a i n e d f r o m h e a t - m a s s - t r a n s f e r a n a l o g i e s . T h e m a j o r

d r a w b a c k s o f th e u s e o f s u c h S h e r w o o d - n u m b e r c o r -

r e l a t i o n s w i t h r e g a r d t o R O / U F a r e : ( a ) t h e y a r e d e -

r i v e d f o r f l o w t h r o u g h a n o n - p o r o u s c o n d u i t ; h e n c e ,

t h e e f f e c t o f s u c t i o n c a n n o t b e c o n s i d e r e d ; ( b ) c h a n g e s

i n p r o p e r ti e s l i k e v i s c o s i t y a n d d e n s i t y d u e t o c o n c e n -

t r a t i o n p o l a r i z a t i o n n e a r t h e m e m b r a n e s u r f a c e c a n n o t

* C o r r e s p o n d i n g a u t h o r ,

0 3 7 6 - 7 3 8 8 / 9 7 / 1 7 . 0 0

1997 E l s ev ie r Sc ience B . V. Al l r i gh t s r es e rved .

P I I S 0 3 7 6 - 7 3 8 8 9 6 ) 0 0 3 1 3 - 4

b e t a k e n i n t o c o n s i d e r a t i o n ; ( c ) i t i s t a c i t l y a s s u m e d

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

o p e d o v e r m o s t o f t he c h a n n e l l e n g th , w h i c h m a y n o t

b e t h e c a s e f o r R O / U F ; ( d ) th e o s m o t i c p r e s s u r e , b u i lt

u p n e a r t h e m e m b r a n e s u r f a c e , c a n n o t b e c o n s i d e r e d

i n s u c h a p p r o a c h e s ; a n d , f i n a l l y , ( e ) t h e m a s s - t r a n s f e r

c o e f f i c i e n t i s a s s u m e d t o b e i n d e p e n d e n t o f p r e s su r e ,

w h i c h m a y n o t b e v a li d f o r R O / U F o p e r at io n s .

T h e r e f o r e , t h e u s e o f s t a n d a r d c o r r e l a t i o n s l e a d t o a n

i n a c c u r a t e e s t i m a t i o n o f t h e m a s s - t r a n s f e r c o e f f i c ie n t

a n d h e n c e , a n i n c o r r e c t p r e d i c ti o n o f th e p e r m e a t e

f lu x . O n e w a y t o a v o i d t h i s i s t o p e r f o r m a d e t a i l e d

s i m u l a t io n , s o l v i n g r e le v a n t m o m e n t u m a n d s o l u t e

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

t i o n s [ 1 - 4 ] . B u t s u c h m e t h o d s m a y n o t b e v e r y a t t r a c -

t iv e f r o m a d e s i g n e r s p o i n t o f v i e w o w i n g t o t h e i r

e x t e n s i v e c o m p u t a t i o n a l e f f o r t a n d c o m p l i c a t i o n s .


2/13

120 S. De. PK. Bhattacharya/Journal of Membrane Science 128 1997) 119-131

T h e m a s s - t r a n s f e r c o r r e l a t i o n s f o r m e m b r a n e -

s e p a r a t i o n p r o c e s s e s w e r e r e v i e w e d i n d e t a i l [ 5 , 6 ] .

I t w a s c o n c l u d e d , i n b o t h o f t h e r e v i e w s , t h a t t h e

p r e s e n t c o r r e l a t i o n s n e e d t o b e mo d i f i e d i n l i g h t o f

the i r l imi ta t ions d i scussed ear l i e r . In fac t , i t was

s u g g e s t e d t h a t ma s s - t r a n s f e r c o r r e l a t io n s s h o u l d b e

d e v e l o p e d , b a s e d o n e x p e r i m e n t a l t e c h n iq u e s , n a m e l y ,

t h e v e l o c i t y - v a r i a t i o n t e c h n i q u e o r o s mo t i c - p r e s s u r e

mo d e l [ 5 - 7 ] . E a c h t e c h n i q u e h a s i t s o w n d i s a d v a n -

tages which were d i scussed in de ta i l [5 ,6 ] .

A n o t h e r a l t e r n a t i v e me t h o d , w h i c h w a s q u i t e s u c -

c e s s f u ll y e m p l o y e d i n o u r e a r li e r w o r k [ 8 - 1 1 ] o n U E

i n c l u d e s t h e d e v e l o p me n t o f a s e c o n d c o r r e l a t i o n f o r

c o n c e n t r a t i o n p o l a r i z a t i o n in t e r ms o f p o l a r i z e d l a y e r -

r e s i s t a n c e a l o n g w i t h a s t a n d a r d ma s s - t r a n s f e r c o r re -

l a t io n l i k e t h e L e v e q u e c o r r e l a t i o n f o r l a m i n a r f l o w i n

a chann el o r Co l to n s co r re l a t ion in a s t i r red ce l l [3 ].

Bu t s u c h a p p r o a c h e s a r e s o l u te a n d s y s t e m s p e c i f ic . I n

add i t ion , i t i s d i f f i cu l t t o work wi th two cor re la t ions

s i mu l t a n e o u s l y .

T h e r o l e o f s u c t io n i n ma s s t r a n s f e r t h r o u g h p o r o u s

me mb r a n e s i s v e r y i mp o r t a n t . I t h a s b e e n i d e n t i f i e d

e a r l i e r [ 5 ,6 ] t h a t t h e e f f e c t o f s u c t i o n o n m a s s - t r a n s f e r

c o e f f i c i e n t i s t w o - f o l d . F i r s t , i t e n h a n c e s t h e ma s s

t r a n s f e r f r o m t h e s u r f a c e t o t h e b u l k ; a n d , s e c o n d , i t

s t a b i li z e s t h e l a mi n a r - f l o w c o n d i t i o n i n t h e c o n d u i t b y

delay ing the l aminar - to - tu rbu len t t rans i t ion ( typ ica l ly ,

c r i t ic a l Re y n o l d s n u m b e r i s s h if t e d f r o m 2 1 0 0 t o 4 0 0 0

in the p resence o f suc t ion [6 ] ) .

T h e r e f o r e , i t s e e me d p o s s i b l e t h a t a g e n e r a l i z e d

ma s s - t r a n s f e r r e l a t i o n ma y b e o b t a i n e d t h e o r e t i c a l l y

f o r l a mi n a r f l o w f r o m f i r s t p r i n c i p l e s . T h e p r e s e n t

w o r k a i m s t o d e v e l o p a g e n e r a l i z e d m a s s - t r a n s f e r

re l a t ion , inc lud ing the e f fec t s o f suc t ion over a deve l -

o p i n g c o n c e n t r a t i o n b o u n d a r y - l a y e r . Su c h r e l a t i o n s

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

p r e d i c t p e r me a t e f l u x f o r o s mo t i c - p r e s s u r e g o v e r n e d

U F a n d a l s o f o r RO . Fu r t h e r , t h e t h e o r e t i c a l w o r k i s

e x t e n d e d t o i n c l u d e a l l t h e f l o w mo d u l e s , u s u a l l y

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

n a me l y , r e c t a n g u l a r c h a n n e l , t u b u l a r a n d r a d i a l

c r o s s - f l o w c o n f i g u r a t io n s .

t h e b a s i c a p p r o a c h o f s o l v i n g s i mu l t a n e o u s l y , t h e

g o v e r n i n g s o l u te - m a s s a n d m o m e n t u m - b a l a n c e e q u a -

t ions a long wi th the boundary cond i t ions .

T h e f l o w c o n f i g u r a t io n i n a c l o s e d c o n d u i t i s s h o w n

in Fig . l a . F ig . lb dep ic t s the f low geomet ry o f a

rad ia l c ross - f low ce l l . The f lu id i s a l lowed to f low

t a n g e n t i a ll y o v e r th e me mb r a n e s u r f a c e . T h e u p p e r

b o u n d a r y o f th e c h a n n e l i s i mp e r v i o u s f o r t h e r e c t a n -

gu lar and rad ia l ce l l s . The permeate f lux i s a func t ion

o f t h e c h a n n e l l e n g t h f o r t h e r e c t a n g u l a r a n d t u b u l a r

mo dules ; and fo r a rad ia l mod u le , i t is a func t ion o f the

r a d i u s o f th e c e l l. T h e c o n c e n t r a t i o n b o u n d a r y - l a y e r

d e v e l o p s o v e r t h e e f f e c t i v e l e n g t h o f t h e me mb r a n e .

A s s u m p t i o n s m a d e i n t hi s mo d e l a r e : ( a ) t h e f l o w i s

s t e a d y ; ( b ) t h e d i f f u s i o n a l o n g t h e me mb r a n e i s n e g -

l i g ib l e , c o m p a r e d t o th e c o n v e c t i o n i n t h e s a me d i r e c -

t ion ; (c ) the f low i s l aminar and fu l ly deve loped ; (d )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x=0 , y=0 8

V w x)

N o n p e r m e a t i n g s e c ti o n L

F e e d

T o p D i s k

o t t o m D i s k

P e r m e a t e

y = 2 h

2 h

M e m b r a n e

2 Theory

I n t h is s e c t i o n , a n a t t e mp t h a s b e e n ma d e t o d e v e l o p

a g e n e r a l i z e d ma s s - t r a n s f e r - c o e f f i c i e n t r e l a t i o n f r o m

F i g . 1 . S c h e m a t i c d i a g r a m o f a ) t h e f lo w c o n f i g u r a t i o n i n a

r e c t a n g u l a r c o n d u i t , a n d b ) t h e f lo w c o n f i g u r a t i o n i n a r a d i al

c ros s - f low ce l l .


3/13

S. De. P.K. Bhattacharya/Journal of Membrane Science 128 1997) 119-131 121

t h e p e r me a t e v e l o c i t y i s s ma l l e n o u g h , c o mp a r e d t o

t h e f e e d v e l o c i t y , k e e p i n g t h e p a r a b o l i c v e l o c i t y p r o -

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

t u b u l a r mo d u l e s ; e ) t h e c o n c e n t r a t i o n a t t h e m e m -

b r a n e s u r f a c e is c o n st a n t ; a n d f ) t h e p h y s i c a l p r o p e r -

t i es o f the so lu t ion a re cons tan t .

W i t h t h e a b o v e - me n t i o n e d a s s u mp t i o n s , t h e s o l u t e

ma s s - b a l a n c e e q u a t i o n o v e r a d i f f e r e n t i a l e l e me n t i n

t h e c o n d u i t g i v e s

V V c

= V D V c ) 1 )

wh ere the ve lo c i ty vec to r , V , i s the resu l t an t o f the

a x i a l v e l o c i ty , u , a n d th e t r a n s v e r s e - v e l o c i t y c o m p o -

n e n t , v . A s s u m i n g n o p e r m a n e n t i r r e v e r s ib l e ) a d s o r p -

t i o n o f t h e s o l u te o n t h e me m b r a n e s u r f a c e , w h i c h ma y

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

n o n - a d s o r b i n g m e m b r a n e ,

v ~ -Vw 2)

w i t h i n t h e t h i n c o n c e n t r a t i o n - b o u n d a r y - l a y e r .

T h e b o u n d a r y c o n d i t i o n s o f E q . 1 ) a r e a s f o l lo w s :

f o r r e c t a n g u l a r a n d t u b u l a r mo d u l e s ,

c = c 0 a t x = 0 3)

and , fo r the rad ia l c ross - f low ce l l ,

c = c 0 at r = 0 4)

T h e a s s u m p t i o n o f c o n s t a n t s o l u te c o n c e n t r a t i o n a t t h e

me mb r a n e s u r f a c e r e s u l t s i n t h e f o l l o w i n g b o u n d a r y

cond i t ion :

C = C m a t y = 0 5)

A t t h e me mb r a n e s u r f a c e , t h e c o n c e n t r a t i o n i s g r e a t e r

t h a n t h e b u l k ; b a c k d i f f u s i o n o c c u r s f r o m t h e s u r f a c e

to the bu lk so lu t ion due to the concen t ra t ion g rad ien t .

T h i s i s o p p o s i t e t o t h e c o n v e c t i v e mo v e me n t o f t h e

s o l u t e p a r t i c l e s t o w a r d s t h e me mb r a n e . A t s t e a d y

s ta t e , t he ne t resu l t o f t hese two oppos ing f luxes i s

e q u a l t o t h e c o n v e c t i v e f l u x o f th e p e r me a t i n g s o l ut i o n.

T h e r e f o r e , o n e h a s t h e f o l l o w i n g b o u n d a r y c o n d i t i o n

a t t h e me mb r a n e s u r f a c e :

O c

V wCm + D -~ y = V wCp

at y = 0 6)

or ,

O c

V w Cm R r + D ~ y = 0

at y = 0 7)

whe re R r i s t he in t r ins i c re j ec t ion o f the so lu te by the

m e m b r a n e , w h i c h m a y b e a s s u m e d t o b e c o n s ta n t f o r a

m e m b r a n e - s o l u t e s y s t e m [ 2 , 1 2 ] .

F ina l ly , i n the bu lk , t he so lu te concen t ra t ion i s

c o n s t a n t b e y o n d t h e c o n c e n t r a t i o n b o u n d a r y - l a y e r ,

t h e t h i c k n e s s o f w h i c h i s n e g l i g i b l e c o mp a r e d t o t h e

h a l f - c h a n n e l h e i g h t . H e n c e , a c o mmo n w a y t o r e p r e -

sen t the boundary cond i t ion fo r the bu lk so lu t ion i s

[131

c = c 0 a t y = ~c 8 )

F r o m t h i s p o i n t o n w a r d s , e s t i ma t i o n o f t h e ma s s -

t r a n s f e r c o e f f i c i e n t f o r d i ff e r e n t f l o w m o d u l e s i s p r e -

sen ted separa t e ly .

2 . 1 . F l o w t h r o u g h a r e c t a n g u l a r c e l l

T h e s o l u te m a s s - b a l a n c e e q u a t i o n , f o r fl o w th r o u g h

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

e a r l i e r , ma y b e w r i t t e n i n t h e f o l l o w i n g f o r m .

O c O c 0 2 c

u Ox - V w ~ y -= D Or-. 9)

A s imi l a r i ty so lu t ion fo r Eq . 9 ) is ob ta in ed by def in -

i n g a d i me n s i o n l e s s v a r i a b l e l u mp e d p a r a m e t e r ) ,

r / = y 1 0 )

a n d d i me n s i o n l e s s c o n c e n t r a t i o n c a n b e e x p r e s s e d b y

c* r l) = c / co

11)

I n t e r ms o f c ~ a n d ~ , E q . 9 ) b e c o m e s a n o r d i n a r y

d i f fe ren t i a l equa t ion o f second o rder , as fo l lows :

12)

T h e a x i a l v e l o c i t y p r o f i l e ma y b e t a k e n a s

u = ~ u 0 1 - 13 )

W i t h i n t h e th i n c o n c e n t r a t i o n - b o u n d a r y - l a y e r , w h e n y

i s s ma l l c o m p a r e d to th e h a l f- c h a n n e l h e i g h t , E q . 1 3 )

c a n b e a p p r o x i ma t e d as b y n e g l e c t i n g y 2 / h : )

3 u o y

u - - 14)

h

Fo r c o n s t a n t p h y s i c a l p r o p e r t i e s a n d c h a n n e l g e o me -

V w X = c o n s t a n t . T h u s , a n o n - d im e n s i o n a l f o r m o f

ry, 1/3


4/13

122

S. De. PK. Bhattacharya/Journal of Membrane Science 128 1997) 119-131

t h i s c a n b e d e f i n e d a s :

V w = A 1

1 5 )

T h e r e f o r e , E q . 1 2 ) b e c o m e s

d 2 c *

d c *

d~72 - r?2 + A 1 ) d r / 1 6 )

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

c * = 1 a t ~ / = ~ 1 7 )

a n d

d c *

d---~+A1Rrc* = 0 a t ~7 = 0 18 )

T h e s o l u t i o n o f E q . 1 6 ) , a lo n g w i t h t h e b o u n d a r y

c o n d i t i o n s E q s . 1 7 ) a n d 1 8 ) ), c a n b e w r i t t e n a s

c * 0 7 ) = K l f o ~ l e x p ( - ~ - A 9 7 ) d ~ + K 2 1 9 )

w h e r e

A 1 R r

K 1 - -

2 0 )

1 - - A 1 R r I 1

1

/ 2 - - 2 1 )

1 A 1 R r I 1

a n d ,

f 0 T ]

1 = e x p - ~ - - A l r / )d r / 2 2 )

T h e a v e r a g e f lu x o v e r t h e m e m b r a n e l e n g t h c a n b e

o b t a i n e d f r o m E q . 1 5 ) a s :

-1 o L f uo D 2)

l / 3 a

Vw : V w X ) dx : 1 .5 ~ , ~ - - ) , 2 3 )

N o w , f o r a r e c t a n g u l a r c h a n n e l , t h e e q u i v a l e n t h y d r a u -

l i c d i a m e t e r c a n b e d e f i n e d b y :

d e = 4 h 2 4 )

T h e r e f o r e , A 1 c a n b e e x p r e s s e d f r o m E q s . 2 3 ) a n d

2 4 ) , i n t e r m s o f P e w = V w d e / D , a s

A 1 = 0 . 4 2 A 1 2 5 )

w h e r e A l = P e w / R e S c de~L)1 / 3 , Re = puod f f a n d

S c = / p D . H e n c e , t h e i n t e g r a l, I 1 , g i v e n b y E q . 2 2 )

c a n b e r e w r i t t e n a s

I i = f o ~ e X p [ - ~ - O .4 2A lr lld ~ 2 6 )

N o w , t h e m a s s - t r a n s f e r c o e f f i c ie n t , k , i s d e f i n e d a n d

o b t a i n e d f r o m a s o lu t e m a s s - b a l a n c e a t th e m e m b r a n e

s u r f a c e , a s

k(cm - CO) = -D(O ~yy) 2 7 )

y = 0

I n t e rm s o f n o n - d i m e n s i o n a l f o r m s o f c a n d y, t h e

a b o v e e q u a t io n c a n b e r e p r e s e n t e d b y :

U0 ) 1 / 3 d c * ~ 2 8 )

k ( c m - 1 ) = - D h x D \ d ~ / n = 0

S u b s t i t u t i n g v a l u e s o f c m a n d d c * / d ~ a t 7 7= 0 f r o m

E q . 1 9 ) , w e h a v e

_ / u 0 , ~1 /3

k ( K2 - l ) = - t g ~ , ~ ) K 1 2 9 )

o r

: 3 0 )

E x p r e s s i n g t h e m a s s - t r a n s f e r c o e f f ic i e n t , k, i n t e r m s o f

t he S h e r w o o d n u m b e r

(kdf fD)

a s a f u n c t i o n o f d i m e n -

s i o n l e s s c h a n n e l l e n g t h ,

x*(x /L) ,

w e c a n w r i t e

4 1 / 3

S h x * ) = ~ R e S c de / L ) U 3( x * ) - 1 / 3 3 1 )

a n d t h e a v e r a g e S h e r w o o d n u m b e r o v e r l e ng t h , L , c a n

b e e x p r e s s e d a s

/ 0

-h = S h x * ) d x * = 2 . 3 8 1

R e S c d e / L ) l / 3

I i 3 2 )

w h e r e I1 i s g i v e n b y E q . 2 6 ) .

2 .2 . F l ow t hrough a t ubu l ar m odu l e

F o r s t e a d y f l o w t h r o u g h a t u b u l a r m o d u l e u n d e r t h e

p r e v i o u s l y m e n t i o n e d a s s u m p t i o n s , t h e s o lu t e m a s s -

b a l a n c e e q u a t i o n c a n b e w r i t te n a s :

Oc o c O rO C

UO x - V w o r -- r O r \ O r / 3 3 )

C o n s i d e r i n g a th i n c o n c e n t r a t io n - b o u n d a r y - l a y e r

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


5/13

S. De. P.K. Bhattacharya/Journal of Membrane Science 128 1997) 119-131

123

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

the wal l w ere f l a t . I f the d i s t ance f rom the wal l i s

d e n o t e d a s

y = R - r ,

t h e f l ui d m a y b e r e g a r d e d a s b e i n g

c o n f i n e d b e t w e e n a f l a t ma s s - t r a n s f e r s u r f a c e e x t e n d -

i n g f r o m y = 0 t o y= cx ~ . T h e r e f o r e , t h e s o l u te m a s s -

b a l a n c e e q u a t i o n ( E q . ( 3 3 )) c a n a g a i n b e e x p r e s s e d b y

Eq. (9) .

T h e f u l l y d e v e l o p e d v e l o c i t y p ro f i l e i n a t u b e c a n b e

d e s c r i b e d b y

u = 2 u o 1 - (3 4)

I n t h e c o n c e n t r a t i o n b o u n d a r y - l a y e r n e a r t he w a l l , th e

a b o v e e x p r e s s i o n f o r t h e v e l o c i t y p r o f i l e ma y b e

e x p r e s s e d a s :

[ ) 2 1

= 2 u 0 1 - (3 5)

N e g l e c t i n g h i g h e r o r d e r t e r m s ( y Z / R2 ) f o r t h e t h i n

c o n c e n t r a t i o n - b o u n d a r y - l a y e r , t h e a x i a l - v e l o c i t y p r o -

f i l e ma y b e a p p r o x i ma t e d a s :

V

u = 4u0 R (36)

T h e s i mi l a r i t y p a r a me t e r c h o s e n i n t h i s c a s e i s :

( U 0 ,~1 /3

4 ' = Y x ~ / ( 3 7 )

I n t e r ms o f c * a n d 4 ' , t h e s o l u t e ma s s - b a l a n c e e q u a t i o n

( E q . ( 3 3 ) ) c a n b e e x p r e s s e d i n t h e f o l l o w i n g f o r m:

d 2c * ( ~ ) d r * ( 38 )

d 0 2 _ _ 4 2 + A 2 4 dr/

w h e r e

( x D ~ 1 / 3

A2 = Vw \uod---Sj (39)

T h e a v e r a g e f l u x o v e r th e l e n g t h o f t he m o d u l e c a n b e

d e f i n e d b y E q . ( 2 3 ) , E x p r e s s i n g t h e a v e r a g e f l u x i n

t e r m s o f P e w ( V w d / D ) , A 2 c a n b e w r i t t e n a s :

9

A2 = ~,~2 (40)

w h e r e A 2 = Pe w / ( Re Sc d/L ) ~/3. T h e i n i t i a l a n d b o u n d -

a r y c o n d i t i o n s i n t e r m s o f c * a n d 4 ' r e m a i n t h e s a me , a s

Eqs . (3 ) , (5 ) , (7 ) and (8 ). The so lu t ion o f Eq . (38) wi th

t h e b o u n d a r y c o n d i t i o n s p r o v i d e s t h e c o n c e n t r a t i o n

prof i l e in the tubu lar module , as fo l lows :

fo ~ ( 8 4 3 A ~ O ) d 6 + K 4

(41)

* (6 ) = K3 exp 9 '

w h e r e

K 3 -

K 4

A2Rr

42)

1 - A2R r l2

1 - A2Rr l2 (43)

and

2 = exp 9 ~A ?0 dO (44)

P r o c e e d i n g e x a c t l y a s f o r r e c t a n g u l a r c h a n n e l , t h e

Sh e r w o o d - n u mb e r p r o f i l e a s a f u n c t i o n o f t h e mo d u l e

l e n g t h c a n b e o b t a i n e d f r o m

Sh(x*) = 1 (Re

c

d/L) 1/3 x*) -1/3 (45)

a n d t h e a v e r a g e S h e r w o o d n u m b e r o v e r t h e m o d u l e

l e n g t h c a n b e e x p r e s s e d a s

= __1 5 R e S c d / L ) l / 3 (46)

2 . 3 . F l o w t h r o u g h a r a d i a l c r o s s - f l o w c e l l

T h e g e o me t r y o f a r a d i a l c r o s s - f l o w c e l l i s b e s t

r e p r e s e n t e d b y a n a x i s y mme t r i c c y l i n d r i c a l - c o o r d i -

na te sys tem, where r i s the rad ia l and y i s the t rans -

v e r s e d i r e c t i o n ( F i g . l b ) . H o w e v e r , f o r c o n v e n i e n c e ,

w e a s s u me a t w o - d i me n s i o n a l c a r t e s i a n - c o o r d i n a t e

sys tem, where the rad ia l d i rec t ion , r , i s no t a rad ia l

coord ina te bu t a car t es i an ax i s . Such s impl i f i ca t ions

ma k e t h e mo d e l e q u a t i o n s s i mp l e r a n d d o n o t a l t e r t h e

resu l t s to any s ign i f i can t ex ten t . The refo re , the s t eady-

s t a te s o l u t e ma s s - b a l a n c e e q u a t i o n w i t h in t h e c o n c e n -

t r a ti o n b o u n d a r y - l a y e r c a n b e w r i t te n i n t h e f o l l o w i n g

f o r m:

OC OC 02C

U ~ r - V w ~y = D - - ( 47 )

Oy2

T h e r a d i a l - v e l o c i t y p r o f i l e w i t h i n t h e b o u n d a r y l a y e r

can be expressed [4 ] as :

3 Q y (48)

u(r , y ) - 47rrh2


6/13

124 S . D e . P . K . B h a t t a c h a r y a / J o u r n a l o f M e m b r a n e S c i en c e 1 2 8 1 9 9 7 ) 1 1 9 -1 3 1

w here Q i s the ave rage vo lum et r i c f low - ra te , g iven by

Q = 4 7 r R h u o ,

and Uo is the av erage v eloci ty in the

condui t .

The s imi la r i ty pa ramete r in th i s case m ay be chos en

as~

Q ) 1/3

= y ~ (49)

In terms of c* and ~ , Eq. (47) can be wr i t ten in the

fo l low ing fo rm:

d 2 c * ( ~ ) d c *

d ~ 2 - - - + A 3 ~ d ~ (50 )

w here

V w 7 h ; D ) 1 /3

= - - r 2 /3

A3 ~ - (51)

The ave rage f lux over the membrane r ad ius can be

def ined as

f 0

w = - ~ V w ( r ) r d r (52)

Expres s ing the ave rage f lux in t e rms o f

P e w ( V w h / D ) ,

A 3 c a n be wri t ten as :

A3 = 0.42~3 (53)

w here ~ 3 = Pew/(Re

S c

h [ R ) 1 / 3 , R e = p u o h / # a n d S c = /

( p D ) .

The boundary cond i t ions in t e rms o f c* and

remain the same as Eqs . (4) , (5) , (7) and (8) . The

solu t ion of Eq. (50) resul ts in the fo l lowing concen-

t ra t ion prof i le :

~3

- - A 3 ~ d ~ + g 6 5 4 )

* ~ ) = K , f ~ e x p - ~

w here

A 3 R r

K5 -- (55)

1 - A3Rr13

K 6 - - ( 5 6 )

1 - A3R r/ '3

a n d

e c

~ 3 _ 0 ' 4 2 X 3 ~ d ~

/ 3 = f 0 e xp ( - ~-

5 7 )

P roceed ing exac t ly a s in the cas e o f r ec tangu la r

channe l , the S herw ood-number p ro f i l e a s a func t ion

o f channe l r ad ius can be ob ta ined as :

41/3

Sh(r*) = ~ (Re Sc h / R ) l / 3 ( r * ) - 2 / 3 (58)

and the ave rage S herw ood number ove r the channe l

r ad ius can be exp res s ed by

Sh = __2'381 (R e Sc h / R ) 1 /3 (59)

/3

In the next sect ion , the in tegrals ,

11-13, are

e x a m i n e d

for d if ferent domains of suct ion (Pew) .

2 . 3 . 1 . C a s e 1. N o S u c t i o n ; P e w = O

For the rectangular cross - f low cel l , the in tegral 11

takes the fo l low ing fo rm w hen the re i s no s uc tion :

/ '1 = J O

exp(- - r /3 /3) dr /

= 3 - 2 / 3 F ( 1 / 3 ) (60)

= 1.2879

The corresponding express ion for Sh is :

g ~ = 1 . 8 5 R e S c

a e / L ) 1 /3

(61)

which is ident ical to the Leveque solu t ion [5 ,6] for

heat t ransfer in a non-porous channel . S imilar ly , for a

tubu la r modu le , the ave rage S herw ood nu mbe r may be

obtained f rom Eq. (46) for Pew=0, as fo l lows:

S---h = 1. 62 (R e Sc

d / L ) 1 / 3

(62)

Th is exp res s ion fo r ave rage S herw ood number i s

aga in iden t i ca l to the Levequ e s o lu t ion [14 ] fo r hea t

t ransfer in a non-porous tube.

For a radia l cross - f low cel l , the average Sherwood

num ber m ay be ob ta ined f rom Eq . (59 ) a s :

S---h = 1. 47 (R e Sc h / R ) 1 / 3 (63)

2 . 3 .2 . C a s e 2 : R O / U F s y s te m

F or a typ ica l R O /U F s ys tem, ~ 1 .2 ,3 va ry f rom ve ry

low values up to 10 . Th e beh aviou r of the in tegrals 1 /

I 1 ,2 ,3 in th is range of -~1,z,3 d ic ta tes the depe nden ce of

the ave rage S herw ood number on the s uc t ion ( i . e .

Pew) . To v isual ize the var ia t ion of 1/11,2,3 (g iven by

Eqs. (26), (44) an d (57)) for this range, 1/ / 1 ,2 , 3 is

evaluate d for ,~1 ,2 ,3 varyin g in the range 0-1 0 by

numerical in tegrat ion and is presented in F ig . 2 . In

Fig . 2 , the symbols represent the numer ical ly in te-

grated v alues and the l ines represent the bes t- f i t data .


7/13

S . D e , P . K . B h a n a c h a ~ a / J o u r n a l o f M e m b r a n e S c i e n c e 1 2 8 ( 1 9 9 7 ) I 1 9 - 1 3 1 1 2 5

8

~ 4

0

0

I

l g

/ /

i3

/

t

i 2 /

13 223

[ 3 f

~ r T w n ~ n ~ ; ~ I l r

2 4 6 8 10

/ ~ 1 , 2 , 3

Fig. 2. Variation of

1 H i 2 3

with A1.2,3. Solid line is for rectangular

cell Eq. 64)), dotted line is for tubular mod ule Eq. 65)) and long

dashed l ine is for radial cross-flow cell Eq. 66)). Symbols

represent the nu merically integrated values, obtained from Eq. 22)

circle), Eq. 44) box ) and Eq. 57) triangle).

F r o m t h is f i g u r e, i t m a y b e o b s e r v e d t h a t th e v a r i a t i o n

o f 1 / I 1 2 3 w i t h A I . 2 3 i s no t l i nea r . In a l l t he ca se s , t he

c o r r e l a t i o n c o e f f i c i e n t s a r e g r e a t e r th a n 0 . 9 9 9 9 . T h e r e -

f o r e , t h e a v e r a g e S h e r w o o d n u m b e r s f o r d if f e r e n t

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

e q u a t i o n s :

F o r a r e c t a n g u l a r c r o s s - f l o w c e l l :

S h : 1 . 8 5 ( R e S c d e / L ) /3

[ 1 . 0 + 0 . 3 2 A , + 0 . 0 2 A ~

- - 8 . 0 5 1 0 - 4 A ~ ] 6 4 )

F o r a t u b u l a r m o d u l e :

S h = 1 . 6 2 ( R e S c d / L ) 1/3

[1 .0 + 0 .37A 2 + 0 .03A~

- 1 . 0 5 > 1 0 -3 A ~ ] 6 5 )

F o r a r a d i a l c r o s s - f l o w c e l l :

S h = 1 . 4 6 7 R e S c

h / R ) 1 / 3

[1 .0 q - 0 .41A3 + 0 .03A 2

- 1 . 2 5 1 0 3 A ~] 6 6 )

2 .4 . A p p l i c a t i o n o f S h e r w o o d - n u m b e r r e l a ti o n s m

R O a n d U F f o r p r e d i c t io n o f f l u x

I n a n o s m o t i c - p r e s s u r e c o n t r o l l e d m e m b r a n e -

s e p a r a t i o n p r o c e s s , p e r m e a t e f l u x c a n s i m p l y b e

e x p r e s s ed b y t h e p h e n o m e n o l o g i c a l e q u a ti o n ,

V w = L p A p - A T r) 6 7 )

w h e r e

z _ ~ 7 r : 7 1 -m - - 7 1- p ( 6 8 )

T h e o s m o t i c p r e s s u r e o f th e s o l u t i o n c a n b e e x p r e s s e d

a s a f u n c t i o n o f s o l u t e c o n c e n t r a t i o n , a s :

-)

7r = C~zc + ~2c- + ~3 c 3 69)

I n t e r m s o f Pe ~,,, E q . 6 7 ) m a y b e w r i t t e n a s

Pew = Bi 1 .0 -

A T r / ~ P )

7 0 )

w h e r e B ~ = L p A P d e / D ; i t m a y b e n o t e d t h a t f o r a

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

o f B j s h o u l d b e r e p l a c e d b y d a n d h , r es p e c t iv e l y .

N o w , t h e a v e r a g e s o l v e n t - f l u x t h r o u g h t h e m e m -

b r a n e i s g i v e n a s :

_o c)

Vw cm -

C p) : \ ~ J , : 0 ( 7 1 )

I n t e rm s o f a v e r a g e S h e r w o o d n u m b e r a n d n o n - d i m e n -

s i o n a l f l u x P e w ) f r o m E q s . 2 7 ) a n d 7 1 ) , o n e c a n

o b t a i n

Pe ,~ . S~ h 1

= Rr - co /cm )

7 2 )

T h e e x p r e s s i o n s o f S h f o r d i ff e r e n t f lo w c o n f i g u r a t i o n s

a r e p r e s e n t e d i n E q s . 6 4 ) - 6 6 ) . T h e r e f o r e , a s im u l t a -

n e o u s s o l u t io n o f E q s . 6 8 ) , 7 0 ) a n d 7 2 ) p r o v i d e s th e

p r e d i c t e d v a l u e o f P ew a n d h e n c e , p e r m e a t e f lu x .

H o w e v e r , i t m a y b e n o t e d h e r e t h a t a p a r t f r o m t h e

o p e r a t i n g c o n d i t i o n s A p , U o, C o ) , t h e c h a r a c t e r i s t i c

r e t e n t io n f o r s o l u t e - m e m b r a n e s y s t e m R r) i s r e q u i r e d

t o p r e d i c t t h e p e r m e a t e f lu x . G e n e r a l l y , t h i s c h a r a c t e r -

i s t i c r e t e n t i o n p a r a m e t e r i s o b t a i n e d f r o m a s e p a r a t e

s e t o f e x p e r i m e n t s .

3 R e s u l t s a n d d i s c u s s i o n

I n th i s s e c t i o n , s e v e r a l i m p l i c a t i o n s o f t h e S h e r -

w o o d - n u m b e r r e la t io n s , d e v e l o p e d h e r e , a r e e x a m -

i n e d . T h e t y p i c a l v a l u e o f th e g r o u p R e S c

d f fL)

f o r th e r e c t a n g u l a r c e ll o r R e S c d/L, f o r t h e t u b u l a r

a n d R e S c h/R, f o r r a d i a l c r o s s - f l o w c e l l ) v a r i e s i n t h e

r a n g e 1 0 3 - 1 0 6 , in m e m b r a n e s e p a r a t io n p r o c e s s e s .

C o r r e s p o n d i n g p e r m e a t e f lu x e s , i n te r m s o f t h e s u c t i o n


8/13

126 s . De . P .K . Bha t tac harya /Jo urna l o f Mem brane Sc ience 128 1997) 119-131

p a r a m e t e r P e w ), c a n b e in t h e r a n g e 1 - 5 0 0 . S h e r -

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

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

s h o w n i n F i g . 3 a . S i m i l a r p r o f il e s f o r t u b u l a r a n d

r a d i a l m o d u l e s a r e d e p i c t e d i n F ig . 3 b a n d F i g . 3 c ,

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

E q s . 3 1 ) , 4 5 ) a n d 5 8 ) , w h e r e t h e i n t e g r a l s I i , / 2

a n d 1 3 w e r e e v a l u a t e d n u m e r i c a l l y . I n t h e s e f i g u r e s , th e

s u c t i o n p a r a m e t e r , P ew , v a r i e s i n t h e r a n g e 0 - 3 0 0 ; t h e

s o l i d l i n e s a r e f o r R e S c d e / L o r R e S c d / L o r R e S c h~

R ) = - 1 0 3 a n d t h e d a s h e d l i n e s a r e f o r R e S c

d e / L

o r

R e S c

d / L

o r R e S c

h / R )

= 1 0 5 . I t c a n b e o b s e r v e d f r o m

6 0 0 ~ 1 a

4 0 0 ~ I ~ ~ \

2 0 O

2222 . . . . . a

r/?

0

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2

x / L

5 0 0

4 0 0

3

2 0 O

1 0 0

b

t

i \

2

0 ' l l l l ' ' ' ' l l l ' J ' ' t l r l ' l l ' T ' ' l l l ' ' l l ' '' l ' ' ' l i ' i ~ ' ' l ' ' '' ~ l l l l

0 . 0 0 . 2 0 . 4 O ~ 0 . 8 1 . 0 1 . 2

X

6 0 0

4 0 0

2 0 0

Ij

0 . 0

I P i l l I I I ~ [ I , , ,, 1 1 I [ 1 1 J I I I I l ~ [ , l l la I I I I I ' I I t I I I t t [ I I I I 1 ~ 1 e l

0 . 2 0 . 4 0 , 6 0 . 8 1 . 0 1 . 2

r

Fig. 3. a) Variation of local Sherwoo d numb er along the channel length for different values of suction, for a rectangular cross-flow cell. 1

Pew=0; 2: Pe ,,=50; 3: Pew=100; 4: Pew=200; and 5: Pew =300. Solid l ines are for Re Sc dJL =l O 3 and dashed l ines are for Re Sc de/L=-lO5.

b) Variation of local Sherw ood num ber along the m odule length for different values of suction, for a tubular module. 1: Pew=0; 2: Pew =50; 3:

Pew=lO0; 4: Pew=200; and 5: Pew=300. Solid l ines are for Re Sc

d / L = l O 3

and dashed lines are for Re

Scd /L= lO 5 .

c) Variation of local

Sherw ood num ber along the channel radius for different values of suction, for a radial cross-flow cell. 1: Pew=0; 2: Pew =50; 3: Pew = 100; 4:

Pew=200; a nd 5: Pew=300. Solid l ines are for Re Sc h / R - l O 3 and dashed l ines are for Re Sc h/R= lO 5 .


9/13

S. De. P.K. Bhattacharya /Jou rna l of Mem brane Science 128 1997) 119-131 127

the figures that the Sherwood number decreases shar-

ply near the entrance and then gradually for the rest of

the conduit. Local Sherwood number increases with

an increase in the suction (as Pew increases). This

leads to an increased mass transfer from the surface to

the bulk, in agree ment with the qualitative description

of the effects of suction on mass-transfer coefficient by

Gekas and Hallstr om [6]. At higher Re Sc d J L , the

Sherwood number is larger for the same suction. For

the same solute and system geometry, an increase in

Re increases forced convection and, consequently, the

growth of the concentration boundary-layer is mini-

mized so that the Shelwood number increases.

Now, for the description of a realistic mass-transfer

operation in a conduit, it is convenient to work with an

average mass-transfer co efficient and an average Sher-

wood-number relationship. The effects of suction on

the average Sherwood number, as estimated from

a 6 b 7

5

q 4 }

2

1 2 3 4 5 1 2 3 4

Pe~ Pew

C 9

7

3

3

i

5

i 2 3 4 5

P e w

Fig. 4. (a) Variation of ~/S hno suction with Pe,~ for different Re Sc

de~L,

in a rectangular cell. 1: Re c de/L=103; 2: Re Sc d e / L 1 0 4 ; 3:

Re Sc

dJL=105;

and 4: Re Sc

d J L = l O 6 .

(b) Variation of Sh/Sh,o suction with Pew for different Re Sc

d J L

in a tubular module. 1: Re Sc

d~

L=103: 2: Re Sc

d/L=104;

3: Re Sc

dlL=105;

and 4: Re Sc

d /L=lO 6 .

(c) Variation of Sh/Shno suction with Pe,~, for different Re Sc

d J L ,

in a

radial cross-flow cell. 1: Re Sc h/R=103; 2: Re Sc hIR=104; 3: Re Sc h/R=105; and 4: Re Sc h /R=lO 6 .


10/13

128 S De. P.K. Bhattacharya/Journal of Membrane Science 128 1997) 119-131

E q s . ( 6 4 ) - ( 6 6 ) , c a n b e c o m p a r e d w i t h th e a v e r a g e

Sh e r w o o d n u mb e r w i t h n o s u c t i o n (t h e st a n d a r d L e v e -

q u e e q u a t i o n ( E q s . ( 6 1 ) - ( 6 3 ) ) f o r t h e d i f f e r e n t o p e r a t -

i n g c o n d i t i o n s a n d f l o w g e o me t r i e s . T h e r a t i o o f Sh

wi th su c t ion to tha t wi thou t suc t ion versus Pew , fo r

d i f f e r e n t v a l u e s o f Re Sc

de /L

fo r a rec tang u lar ce l l , i s

p l o t t e d i n F ig . 4 a . Fo r a r e a li s ti c U F s y s t e m, Re Sc

de/

L is o f the o rder o f 105-106. W hen Re Sc de /L i s 105, a

Pe w o f t h e o r d e r o f 5 0 0 r e s u l t s i n a b o u t a 6 - f o l d

i n c r e a s e i n t h e a v e r a g e S h e r w o o d n u m b e r c o m p a r e d

t o th a t g i v e n b y L e v e q u e e q u a t i o n , w h e r e t h e e n h a n c e -

m e n t f a c t o r is ~ 3 f o r R e S c de /L = lO 6 . T h e e f f e c t s o f

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

m odu le i s p rese n ted in F ig . 4b . In th i s case , the e f fec t s

o f s u c t i o n a r e e v e n g r e a t e r. W h e n R e Sc

d / L =

105, the

e n h a n c e m e n t f a c t o r i s ~ 7 , w h i l e fo r R e S c

d / L =

106, it

i s ,- ~3 .5 , f o r Pe w = 5 0 0 . T h e r a t i o o f Sh e r w o o d n u m b e r

wi th and wi thou t suc t ion fo r a rad ia l c ross - f low ce l l i s

p rese n ted in F ig . 4c . Fr om the f igure , i t can b e

o b s e r v e d t h a t t h e e n h a n c e me n t r a t i o i s 7 . 7 f o r

R e S c h / R = l O 5 a n d 3 .8 f o r Re Sc h/R~ - lO 6 , f o r

Pe w = 5 0 0 . H e n c e , t h e e f f e c t o f s u c ti o n o n Sh e r w o o d

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

mo d u l e s . T h e r e f o r e , i t c a n b e c o n c l u d e d t h a t t h e r e

e x i s t s a s i g n i f i c a n t e f f e c t o f s u c t i o n o n Sh e r w o o d

n u mb e r a n d , h e n c e , o n ma s s t r a n s f e r .

T h e p r o p o s e d S h e r w o o d - n u m b e r r e l at i on s c a n a l s o

b e u s e d t o p r e d i c t t he p e r m e a t e f l u x i n RO a n d U F . Fo r

RO i n a r e c t a n g u l a r c h a n n e l , E q . ( 6 4 ) ma s s - t r a n s f e r

c o e f f i c i e n t a l o n g w i t h E q . ( 7 2 ) f o r t h e o s mo t i c p r e s -

su re and Eq . (70) were so lved i t e ra t ive ly us ing the

N e w t o n - R a p h s o n t e ch n iq u e . T h e e x p e r i m e n t a l d a t a o f

Mer ten e t a l . [15] were used fo r th i s purpose and the

s i mu l a t i o n s w e r e c a r r i e d o u t f o r d i f f e r e n t c h a n n e l

d i m e n s i o n s L/de). T h e p r e d i c t e d c h a n g e i n Pe w w i t h

Re i s shown in F ig . 5 . Th e f igure ind ica tes a very c lose

a g r e e m e n t b e t w e e n t h e p r e d i c t e d a n d e x p e r i m e n t a l

Pe w ( e x p e r i me n t a l d a t a c o r r e s p o n d t o

L /de =16 . 56 ) .

I n t e r e s t i n g l y , t h e a g r e e me n t i s e x c e l l e n t u p t o

R e = 2 6 2 0 . I t w a s e x p e c t e d t h at l a m i n a r - to - t u rb u l e n c e

t r a n s i t i o n w o u l d o c c u r i n t h e r a n g e 2 0 0 0 - 2 2 0 0 . Bu t

suc t ion has s t ab i l i zed the boundary l ayer , l ead ing to a

de lay in the onse t o f tu rbu len t f low [6 ] .

T h e p r e d i c t i o n o f f l u x i n U F w a s c a r r i e d o u t f o r

d e x t r a n a n d PE G s o l u t i o n s i n t h e r e c t a n g u l a r c r o s s -

f l o w c e l l . T h e ma s s - t r a n s f e r c o e f f i c i e n t i n t h i s c a s e

w a s e v a l u a t e d u s i n g E q . ( 6 4 ) . T h i s e q u a t i o n , a l o n g

w i t h t h e o s mo t i c - p r e s s u r e m o d e l ( E q s . ( 7 0 ) a n d ( 7 2 ))

15

09

10

t i i l , l i l ~ , l i , i i , J , l , , , l l l i , ,

I

2 0 0 0 3 0 0 0

e

F i g . 5 . V a r i a t io n o f d i m e n s i o n l e s s f l u x P e w ) w i t h R e f o r R O

s y s t e m . S o l i d l i n e s a r e p r e d ic t e d f l u x a n d s y m b o l s a r e t h e

exp er im enta l da t a o f Me r t en e t a l. [ 15] . l : L /de =5 . 0 ; 2 : L /

d e = 1 6 . 5 6 ; 3 :

L/de=30.O;

4: L /de=60 . 0 ; 5 : L /de=150 . 0 ; and 6 : L /

d e = 3 0 0 . 0 .

w a s s o l v e d , a s d e s c r ib e d e a r li e r, t o o b t a i n t h e p e r me a t e

f lu x . T h e e x p e r i m e n t a l d a t a o f o u r e a r l i e r w o r k [ 1 6]

a r e c o n s i d e r e d f o r c o mp a r i s o n w i t h t h e p r e d i c t e d

values . In t r ins i c re t en t ion (Rr) fo r dex t ran was t aken

as 1 .0 and tha t fo r PE G, 0 .9 [16]. T he exp er im en ta l

a n d p r e d i c t e d v a l u e s o f t he p e r m e a t e f lu x f o r d e x tr a n ,

fo r a l l t he opera t ing cond i t ions , a re shown in F ig . 6 .

T h e f i g u r e i n d i c a t e s a n e x c e l l e n t ma t c h f o r t h e t w o .

Fo r PE G , a t a ll o p e r a t i n g c o n d i t io n s , t h e p r e d i c t e d a n d

exper imen ta l f lux va lues a re p lo t t ed in F ig . 7 . The

f i g u r e r e v e a l s a c l o s e a g r e e me n t b e t w e e n t h e e x p e r i -

me n t a l a n d c a l c u l a t e d f l u x v a l u e s . T h e ma x i mu m

d e v i a t i o n b e t w e e n th e v a l u e s i s 1 0 .

T h e c o m p a r i s o n o f p r e d ic t e d a n d e x p e r i m e n t a l

p e r me a t e f l u x e s f o r u l t r a f i l t r a t i o n o f PE G 6 K i n a

rad ia l c ross - f low ce l l was a l so car r i ed ou t . In th i s case ,

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

f r o m E q . ( 6 6) . A s m e n t i o n e d e a r li e r , th e o s m o t i c -

p ressu re m ode l Eq . (70) , a long wi th Eqs . (66) and

( 7 2 ), w a s s o l v e d i t e r at i ve l y . T h e e x p e r i me n t a l d a t a f o r

U F o f P E G a r e o b ta i n e d f r o m G a n g u l y [ 17 ], f o r

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

Rr w a s t a k e n a s 0 . 8 8 [ 1 7 ] . T h e c o mp a r i s o n b e t w e e n

e x p e r i me n t a l a n d p r e d i c t e d p e r me a t e f l u x e s i s p r e -

sen ted in the non-d im ens io na l ( i . e . in t e rm s o f Pew)


11/13

S . D e . P .K . B h a n a c h a r y a / J o u r n a l o M e m b r a n e S c i e nc e 1 2 8 ~1 9 9 7 ) 1 1 9 - 1 3 1 129

~ 9. . . W a t e r / ~ F l u x

-~ / ~ . . AS, ,> s - a - 5 . .........:::fl::::::

~ , ~ :'~ . ::::::{ %% ...A . . . . . .

.

. ' . : ~ l

. . . . . . .

i

3

:~ : . . .

/ . S : j c b

. , ' , - : : - - '

, , * , , q ? . - '

0

150

300 450 600

A P ( k P a )

Fig . 6 . Var ia tion of permeate f lux w i th pressure for UF o f dext ran

( T- 20 ). l : c o - 1 0 k g m 3; 2 : e o - 3 0 k g m 3 ; a nd 3 : C o - - 5 0 k g m 3

S y m b o l s re p r e s en t : o p e n c i r cl e , a o = 0 . 4 4 m s 1 b o x ,

Uo =0.38 m s i ; and t r iangle , uo =O .30m s - l . Curves are the

predicted values of the permeate flux.

18

/

~ +1 0~ / / /

v / /~ / / /10 5g

C

N /

4

U , / /

0 I I I I I I t l t l l r l l l I ' l l l l l l t I I 1 1 1

0 4 e~ 12

Experimental Flux x lO m3/m2.s)

Fig. 7. Fit t ing between predicted and experimental permeate flux

for UF o f PEG 6K in a rec tangular channel . Dashed l ines a re for

10% deviations.

35

5

q)

o

is_,

/ /

/ / /

0 / ' / O / /

/ / / / / ( ~

+ 1 0 ~ / / ~ ~

I i i i , I i i [ i , i i i p i J i i i I i i i i i i i

1 5 2 5 3 5

Exper imental Pe,~

Fig. 8. Fit t ing between predicted and experimental permeate flux

for UF o f PEG 6K in a radia l c ross- llow ce l l . Dashed l ines a re for

10% devia t ions .

include the effects of suction and can be used for an

accurate prediction of permeate flux in both cross-flow

RO and UF.

4 C o n c l u s i o n s

General Sherwood-number relations for cross-flow

RO and UF, including the effects of suction for

different flow geometries, were obtained from first

principles. Suction through the porous membrane had

a significant effect on the mass-transfer coefficient.

The proposed Sherwood-number relations were used

to predict mass-transfer coefficient and, in turn, the

permeate flux for both RO and UE The simple rela-

tions developed in this work to quantify the effects of

suction on mass-transfer coefficient should be of

immense help to the process and design engineers.

5 L i s t o f s y m b o l s

form, in Fig. 8, for all the experimental conditions.

Most of the predicted values lie within 4-10% of the

experimental data.

Therefore, the Sherwood-number relationships

developed in this work for different flow geometries

A1,2.3

B1

c

C *

Constants defined in Eqs. (15), (39) and

(51 ), respectively

Term defined in Eq. (70)

Solute concentration, kg m ~

Dimensionless solute concentration

C/Co)


12/13

130


D

de

d

h

11.2.3

k

K1, 2 , 3 , 4 , 5 , 6

L

L p

P E G

P e w

Q

F

/-

R

R e

R O

R r

S c

S h

S h

U

U o

V

w

)w

V

x

x

Y

U F

D i f f u s i v i t y , m 2 s - 1

E q u i v a l e n t d i a m e t e r , m

D i a m e t e r o f t h e tu b e , m

H a l f - c h a n n e l h e i g h t , m

I n t e g r a l d e f i n e d b y E q s . ( 2 2 ) , ( 4 4 ) a n d

( 5 7 ) , r e s p e c t i v e l y

M a s s - t r a n s f e r c o e f f i c ie n t , m s - 1

C o n s t a n t s d e f i n e d b y E q s . ( 2 0 ) , (2 1 ) ,

( 4 2 ) , ( 4 3 ) , ( 5 5 ) a n d ( 5 6 ) , r e s p e c t i v e l y

C h a n n e l l e n g t h , m

M e m b r a n e p e r m e a b il it y , m 3 N - 1

S 1

P o l y e t h y l e n e g l y c o l

W a l l p e c l e t n u m b e r

V o l u m e t r i c f l o w - r a t e , m 3 s - t

R a d i a l c o o r d i n a t e , m

D i m e n s i o n l e s s r a d i a l d i s t a n c e ,

r/R)

C e l l r a d i u s , m

R e y n o l d s n u m b e r

R e v e r s e o s m o s i s

R e a l r e j e c t i o n , ( 1 - Cp/Cm)

S c h i m d t n u m b e r

L o c a l S h e r w o o d n u m b e r

A v e r a g e S h e r w o o d n u m b e r

A x i a l v e l o c i t y , m s - 1

A v e r a g e b u l k v e l o c i t y , m s - 1

V e l o c i t y , m s - 1

A v e r a g e p e r m e a t e f lu x , m 3 m - 2 s - 1

L o c a l p e r m e a t e f l u x , m 3 m 2 s - 1

T r a n s v e r s e v e l o c i t y , m s - 1

A x i a l d i s t a n c e , m

D i m e n s i o n l e s s a x i a l d i s t a n c e , x/L)

N o r m a l d i s t a n c e , m

U l t r a f i l t r a t i o n

G r e e k s y m b o l s

A P

A n

71

P

A1,2,3

P r e s s u r e d i f f e r e n t i a l , P a

O s m o t i c p r e s s u r e d i f f e r e n t i a l , P a

O s m o t i c p r e s s u r e , P a

P a r a m e t e r d e f i n e d b y E q . ( 1 0 )


D e n s i t y , k g m - 3

V i s c o s i t y , P a s


P e w / ( R e S c de[L)1/3, P e w / ( R e S c d/ L) 1/3

a n d P e w / ( R e S c h /R ) 1 /3 , r e s p e c t i v e l y

S u b s c r i p t s

m M e m b r a n e s u r fa c e c o n d i ti o n

o B u l k c o n d i t i o n

p P e r m e a t e c o n d i t i o n

A p p e n d i x

6 . P h y s i c a l p r o p e r t i e s o f t h e s o l u t es

T h e d i f f u s i o n c o e f f i c i e n t o f d e x t r a n ( T - 2 0 ) w a s

t a k e n a s 6 . 7 5 x 1 0 - l t m e s - [ 1 8 ]. T h e d i f f u s i o n c o e f -

f i ci e n t o f P E G ( i n m 2 s 1) w a s o b t a i n e d f r o m t h e

e m p i r i c a l e q u a t i o n , f o r a p o l y m e r i c s o l u t i o n [ 1 6 ] :

D p E G = 1 . 5 0 7 8 8 1 0 - 1 ( A 1 )

O s m o t i c p r e s s u r e f o r d e x t r a n i s o b t a i n e d f r o m t h e

c o r r e l a t i o n d e v e l o p e d b y W i j m a n s e t al . [ 1 9 ], w h i c h i s

g i v e n a s :

7r - - ( 0 . 3 7 5 c + 7 . 5 2 c 2 + 7 6 . 4 c 3 ) 1 05 ( A 2 )

w he re 7 r i s i n Pa a nd c i s i n g m l i .

T h e o s m o t i c p r e ss u r e f o r P E G w a s c a l c u la t e d f r o m

F l o r y ' s e q u a t i o n [ 4 , 2 0 ] .

R e f e r e n c e s

[ 1] C . K l e i n s t r e u e r a n d M . S . P a l le r , L a m i n a r d i l u te s u s p e n s i o n

f l o w s i n p l a t e a n d f r a m e u l t r a f i l t r a t i o n u n i t s , A I C h E J . , 2 9

(1983) 529 .

[2 ] C . R . B ou cha rd , P .J . C ar r eau , T . M at s u uara and S . Sour i r a j an ,

M o d e l i n g o f u l t r a fi l tr a t i o n : p r e d i c t i o n s o f c o n c e n t r a t i o n

p o l a r i z a t i o n e f f e c ts , J. M e m b r a n e S c i . , 9 7 ( 1 9 9 4 ) 2 1 5 .

[ 3] S . B h a t t a c h a r je e , A . S h a r m a a n d P . K . B h a t t a c h a r y a , S u r f a c e

i n t e r a c t io n s i n o s m o t i c p r e s s u r e c o n t r o l l e d f l u x d e c l i n e d u r i n g

u l t r a f i l t r a t ion , Langmui r , 10 (1994) 4710 .

[ 4] S . G a n g u l y a n d P .K . B h a t t a c h a r y a , D e v e l o p m e n t o f c o n c e n -

t r a t ion p ro f i l e and p r ed ic t ion o f f lux fo r u l t r a f i l tr a t ion in a

r ad ia l c ros s f low ce l l , J . Membrane Sc i . , 97 (1994) 287 .

[ 5 ] G . B . v a n d e n B e r g , I . G . R a c z a n d C . A . S m o l d e r s , M a s s

t r a n s f e r c o e f f i c ie n t s i n c r o s s f l o w u l t r af i l tr a t io n , J . M e m b r a n e

Sci . , 47 (1989) 25 .

[ 6 ] V . G e k a s a n d B . H a l l s t r o m , M a s s t r a n s f e r i n t h e m e m b r a n e

c o n c e n t r a t i o n p o l a r i z a t i o n l a y e r u n d e r t u r b u l e n t c r o s s f lo w . I .

C r i t ic a l l i t e r a tu r e r e v i e w a n d a d a p t a t i o n o f e x i s t i n g S h e r w o o d

c o r r e l a ti o n s t o m e m b r a n e o p e r a t io n s , J. M e m b r a n e S c i ., 8 0

(1987) 153.


13/13


131

[ 7] S . D e , S . B h a t t a c h a r y a a n d E K , B h a t t a c h a r y a, D e v e l o p m e n t

o f co r re la t ions fo r mass t r an s fe r coe f f ic ien t in u l t r a f i l t r a t ion

sys tem s , Dev . Che m. M in . Proce sse s , 3 (1995 ) 187 .

[8 ] T .K. Podda r , R .P . S ingh and EK . Bha t tacha rya , Ul t ra f i l t r a t ion

f l u x a n d r e j e c t i o n c h a r a c t e r i s t i c s o f b l a c k l i q u o r a n d

p o l y e t h y l e n e g l y c o l, C h e m . E n g . C o m m u n . , 7 5 ( 1 9 8 9 ) 3 9 .

[ 9] S . D a s g u p t a a n d E K . B h a t t a ch a r y a , C o m p a r a t i v e l im i t i n g f lu x

a n a l y s i s o f b l a c k l i q u o r a n d p o l y e t h y l e n e g l y c o l i n

u l t r a f i l t r a t ion , Chem. Eng . Commun . , 93 (1990 ) 193 .

[10] S . Sr idha r and P .K. Bha t tacha rya , L imi t in g f lux phe nom ena in

u l t r a f i l t r a t ion o f k ra f t b lack l iquo r , J . M embrane Sc i . , 57

(1991) 187.

[11] C . Bha t tacha r jee and EK. Bha t tacha rya , F lux dec l ine ana ly s i s

in u l t r a f i l t r a t ion o f k ra f t b lack l iquo r , J . M embrane Sc i . , 82

(1993) 1 .

[12 ] W.S. Opong and A.L. Zydney , Di f fu s ive and convec t ive

p r o t e i n t ra n s p o r t th r o u g h a s y m m e t r i c m e m b r a n e s , A I C h E J . ,

37 (1991) 1497.

[ 13 ] J . S S h e n a n d R . E P r o b s t e i n , O n t h e p r e d i c t i o n s o f l i m i t in g

f lux in l amina r u l t r a f i l t r a t ion o f mac romolecu la r so lu te s , Ind .

Eng . Chem. Fundam. , 16 (1977 ) 459 .

[14 ] EB. Bungay , H .K. Lonsda le and M .N. de Pinho (Eds ) ,

S y n t h e t i c M e m b r a n e s : S c i e n c e , E n g i n e e r i n g a n d A p p l i c a -

t ions , D . Re ide l Pub l i sh ing Co . , Dodrech t , 1983 .

[15 ] U . M er ten , H .K. Lonsda le and R.L . Ri ley , Bounda ry laye r

e f fec t s in r eve rse o smos is , Ind . Eng . Chem. Fundam. , 3

(1964) 210.

[16] S . De and EK. Bh a t tacha rya , F lux p red ic t ion o f b lack l iquo r

in c ro ss f lov , u l t r a f i l t r a t ion u s ing low and h igh re jec t ing

mem branes , J . M em bran e Sc i . , 109 (1996) 1 (19.

[17 ] S . Gangu ly , Deve lopmen t o f concen t ra t ion p ro f i l e in c ro ss

f low u l t r a f i l t r a t ion u s ing k ra f t b lack l iquo r in compar i son to

po lye th y lene g lyco l , M . Tech . Thes i s , I IT , Kanpu r , 1991.

[18 ] L . Leb run and G. Jun te r , D i f fu s ion o f dex t ran th rough

mic ropo rous membrane f i l t e r s , J . M embrane Sc i . , 88 (1994 )

253.

[19] J .G . W i jman s . S . Nakao , J .W.A. van den Be rg , ER. Troe l s t r a

a n d C . A . S m o l d e r s , H y d r o d y n a m i c r e s i s t a n c e o f c o n c e n t r a -

t i o n p o l a r i z a t i o n b o u n d a r y l a y e r s i n u l t r a f i l t r a t i o n , J .

M embrane Sc i . , 22 (1985 ) 117 .

[ 20 ] E J , F l o r y , P r i n c i p l e s o f P o l y m e r C h e m i s t r y . C o m e l l U n i -

versi ty Press . I thaca , NY, 1953.

Date post:	01-Jun-2018
Category:	Documents
Upload:	mahesh8760
View:	219 times
Download:	0 times

Prediction of mass-transfer coefficient with suction

Documents