21 Theoretical Descriptions of Membrane Filtration of Colloids and Fine Particles-An Assessment and...

8/19/2019 21 Theoretical Descriptions of Membrane Filtration of Colloids and Fine Particles-An Assessment and Review

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Adva nces in Colloid and Interfac e Science, 56 (1995) 141-200 141

Elsev ier Science B.V.

00232 A

T H E O R E T I C A L D E S C R I P T I O N S O F M E M B R A N E

F I L T R A T I O N O F C O L L O I D S A N D F I N E P A R T I C L E S :

A N A S S E S S M E N T A N D R E V I E W

W .

R I C H A R D B O W E N * , F R A N K

J E N N E R

Biochemical Engineering Group, Department of Chemical Engineering,

University College of Swansea, University of Wales, Swansea, SA2 8PP, UK

C O N T E N T S

A b s t r a c t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

1.1 T h e o r y of U l t r a f i l t r a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

1.2 Co n c e n t r a t i o n Po l a r i z a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . 144

2. Ge l -Po la r i z a t i o n Mode l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

2.1 L i m i t a t i o n s of t h e Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

2 .2 Mode l Deve l opmen t s wi th Var i ab l e Phys i c a l P roper t i e s . . . . . . . . . 148

3. O s m o t i c P r e s s u r e Mode l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4. Re s i s t a n c e Mod e l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.1 G e l - Po l a r i z a t i o n - R e s i s t a n c e Model . . . . . . . . . . . . . . . . . . . . . 160

4.2 Bo u n d a r y L a y e r R e s i s t a n c e Model . . . . . . . . . . . . . . . . . . . . . 162

5. A l t e r n a t i v e Mo de l s for Colloidal F i l t r a t i o n . . . . . . . . . . . . . . . . . . . 164

5 .1 Ine r t i a l Migra t i on Mode l s

tubular-pinch effect) . . . . . . . . . . . . . .

165

5.2 S h e a r - I n d u c e d Hy d r o d y n a m i c Co n v e c t i o n Mode l . . . . . . . . . . . . . 170

5.3 S he a r - I n d u c e d Hy d r o d y n a m i c Di f fu s ion Mode l . . . . . . . . . . . . . . 172

5.4 E r o s i o n Mo de l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5.4.1 Sc ou r Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5.4.2 T u r b u l e n t Bu r s t Model . . . . . . . . . . . . . . . . . . . . . . . . 179

5.5 Fr i c t i o n Force Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

5.6 Pa r t i c l e A d h e s i o n Mode l . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

5.7 Pore B loc k in g Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5.8 Su r f a c e R e n e w a l Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5.9 Pa r t i c l e - P a r t i c l e I n t e r a c t i o n s Model . . . . . . . . . . . . . . . . . . . . 186

6. Co n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

7. A c k n o w l e d g e m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

8. N o m e n c l a t u r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

9. Re f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

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

0001-8686/95/$29.00 © 1995 -- Elsev ier Science B.V. All righ ts reserved.

SSD10001-8686 94)00232-0


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ABSTRACT

M em brane separa t i on t echno logy i s a novel and h igh ly innovat ive p rocess eng ineer ing

opera t ion . M em brane p rocesses ex i s t fo r m os t o f t he f l u id separa t i ons encoun tered i n

indus t ry . T he m os t wide ly used a re m em brane u l t r a f il t r a ti on and m icro f i lt r a ti on , p res -

su re d r iven p rocesses wh ich a re capab le o f separa t i ng par t i c l es i n t he app rox imate s ize

rang es of 1 to 100 nm and 0 .1 to 10 pm, respect ively .

The des ign o f m em brane separa t i on p rocesses , l ike a l l o ther p rocesses , r equ i res

quan t i t a t i ve express ions r e l a t i ng mater i a l p roper t i es t o separa t i on per fo rmance . The

fac to r s con t ro ll ing t h e per fo rmance o f u l t r a - and micro f i lt r a ti on a re ex t ens ive ly r ev i ewed .

The re ha ve been a n um ber o f semina l ap proaches i n t h i s fi eld. M os t have bee n based on

the r a t e l imi ti ng e f fec ts o f t he concen t ra ti on po l ar isa t ion o f t he sepa ra t ed co lloids a t t h e

membrane su r face . Var ious r i go rous , empi r i ca l and in tu i t i ve model s ex i s t , wh ich have

been c r it ica l ly assesse d i n t e rm s o f t he i r p red i c ti ve capab i l i t y and app l icab i li ty . T he

dec i s ion as t o wh ich o f t he m em brane f i lt r a ti on model s i s the m os t co r rec t i n p red i c t ing

perm eat ion r a t es i s a mat t e r o f d i ff i cu lty and appea r s t o depend on the na tu re o f t he

d i sper s ion t o separa t ed . Recom men dat ions a re mad e as t o wh ich o f t he ex i s ti ng model s

can b e m os t ap propr i a t e ly app l i ed t o d i f f e ren t t ypes o f d i spers ions .

1. I N T R O D U C T I O N

M e m b r a n e s e p a r a t i o n t e c h n o l o g y i s a n o v e l a n d h i g h l y i n n o v a t i v e

p r o c e s s e n g i n e e r i n g o p e r a t i o n . M e m b r a n e f i l t r a t i o n p r o c e s s e s a re n o w a -

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

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

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

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

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

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

i n e x i s t i n g p r o d u c t i o n p r o c e s s e s , a s w e l l a s r e l a t i v e l y l o w c a p i t a l a n d

r u n n i n g c o s ts . T h e f o r m e r a d v a n t a g e s m a k e m e m b r a n e p r o c e s s e s e v e n

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

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

1 . D i s p e r s i o n s o f c o l l o id s a n d f i n e p a r t ic l e s , e s p e c i a l l y t h o s e w h i c h a r e

c o m p r e s s i b l e , h a v e a d e n s i t y c l o s e t o t h a t o f t h e l i q u id p h a s e , h a v e h i g h

v i s c o s i t y , o r a r e g e l a t i n o u s .

2 . B i o l o g i c a l m a t e r i a l s , w h i c h o f t e n fa l l i n t h e c o l lo i d a l s iz e r a n g e a n d

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

3 . L o w m o l e c u l a r w e i g h t , n o n - v o la t i l e o r g a n ic s o r p h a r m a c e u t i c a l s a n d

d i s s o l v e d s a l t s .

T h e v a r i o u s m e m b r a n e s e p a r a t i o n m e t h o d s c a n b e d i v id e d a c c o r d in g


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t o t h e i r s e p a r a t i o n c h a r a c t e r i s t i c s w h i c h m a y b e c la s s i fi e d b y t h e s iz e -

r a n g e o f m a t e r i a l s s e p a r a t e d a n d t h e a p p l ie d d r i v i n g fo rc e. U l t r a f i l t r a -

t i o n a n d m i c r o f i l t r a t io n s e p a r a t e s o l u t e s a n d c o ll o id a l p a r t i c l e s o f t h e s i ze

o f a p p r o x i m a t e l y 1 t o 1 0 0 n m a n d 0 .1 t o 1 0 ~tm r e s p e c t i v e l y , u s i n g a

p r e s s u r e d i f f e re n c e o f u s u a l l y 1 00 to 5 0 0 k P a a s t h e d r i v i n g f orc e.

T o d a y u l t r a f i l t r a t i o n a n d m i c r o f il t ra t io n m e m b r a n e s a r e m o s t c om -

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

c e l l u lo s e - a c e t a t e , p o l y c a r b o n a t e a n d a n u m b e r o f o t h e r a d v a n c e d p o ly -

m e r s . H o w e v e r , r e c e n t d e v e l o p m e n t s o f i n o r g a n i c m e m b r a n e s c o m p o se d

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

v a n t a g e o u s p r o p e r t i e s c o m p a r e d t o p o ly m e r i c t y p e s , l i k e h i g h e r t e m p e r a -

t u r e s t a b i l i t y , i n c r e a s e d r e s i s t a n c e t o f o u l in g a n d n a r r o w e r p o r e si z e

d i s t r i b u t i o n , d e s p i t e t h e i r h i g h c a p i t a l co s ts .

V a r i o u s c o n f i g u r a t i o n s e x i s t t o s u p p o r t o r c o n t a i n t h e m e m b r a n e s .

W h i c h c o n f i g u r a t io n , f o r i n s t a n c e t u b u l a r m o d u l e s , h o ll o w f i b re m o d u l e s ,

p l a t e - a n d - f r a m e m o d u l e s o r s p i ra l - w o u n d m o d u l e s i s to b e u s e d d e p e n d s

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

t i o n s .

I n g e n e r a l , t h r e e f i l tr a t i o n m o d e s c a n be d i s ti n g u i s h e d : u n s t i r r e d a n d

s t i r r e d d e a d - e n d f i l t r a t i o n a n d c r o s s - f l o w f i l t r a t i o n . I n t h e u n s t i r r e d

d e a d - e n d f i l tr a t i o n th e s o l u t io n is p u t u n d e r p r e s s u r e w i t h o u t a n y a g i t a -

t i o n in t h e l i q u id . I n t h e s t i r r e d d e a d - e n d m o d e a g i t a t i o n i s p r o v i d e d w i t h

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

t a n g e n t i a l l y o v e r t h e m e m b r a n e s u rf a ce .

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

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

e n c o u n t e r e d i n in d u s t r y . T h e d e v e l o p m e n t o f q u a n t i t a t i v e p r e d i ct iv e

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

m e m b r a n e s e p a r a t i o n p r o ce s se s in th e p r o ce s s i n d u s t r i e s . T h e d e s i g n a n d

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

r e q u i r e q u a n t i t a t i v e e x p r e s s io n s r e l a t i n g m a t e r i a l p r o p e r ti e s to s e p a r a -

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

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

e n a a t t h e m e m b r a n e s u r f a c e .

T h i s a r t i c l e p r e s e n t s a r e v ie w a n d a s s e s s m e n t o f t h e e x i s t i n g f i l tr a t i o n

m o d e l s f o r c o ll o id a l a n d f i n e p a r t i c l e d i s p e r s i o n s w i t h e m p h a s i s o n t h e i r

q u a l i t a t i v e a n d q u a n t i t a t i v e p r e d i c t iv e c a p a b i l i ty , t h e i r s c i e n t if i c b a s i s ,

a n d t h e i r l i m i t a t i o n s .


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1 .1 T h e o r y o f U l t r a f il t r a t io n

Ultrafiltration is a pressure-driven membrane process by which macro-

molecu lar solutes and/or colloidal particles are sep ara ted from a solvent,

usual ly water. The relationship between the applied ultrafilt ration pres-

sure and the rate of permeation (flux) for a pure solvent feed flowing

under laminar conditions in tortuous membrane channels may be de-

scribed by the C arm an-Ko zen y equation (Carman (1938))

J-IAPl (1)

R m

where J is the flux (volumetric rate per unit area), Ap the t ra ns mem brane

pre ssu re difference, p the solvent viscosity and R m the membrane resis-

tance. The general approach to describe ultrafiltr ation in the presence of

a solute is given by

J = l A P ] - IA~I (2)

~ R m + R s)

where An is the difference in osmotic pres sure across the mem bra ne, and

R s rep res ent s reversible (concentrated layer, filter cake) and sometimes

irreversib le (foulants) deposition of solute (or solids) onto the membra ne

surface.

1 . 2 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

The separation of solute and solvent takes place at the membrane

surface where the solvent passes throug h the memb rane and the ret ained

solute causes th e local concent ration to increase, an effect whic h is known

as 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 . Thereby a concentration profle is estab-

lished within a boundary film generat ed by the h ydrodynamic conditions

(see Fig. 1).

With the higher concentration at the me mbr ane surface, there will be

a tendency of solute to diffuse back into the bulk solution according to

Fick's law of diffusion. A solute mass balance above the membrane

surface at steady state condition and with the assumption that


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Cm It

I

dy

J c

Membrane

Cp C~

< <

Cm

Cb

Y

Fig . 1 . Concen t ra t ion po la r i za t ion a t a membrane su r face .

1. solvent and solute densit ies are similar,

2. the diffusion coefficient is cons tan t and

3. concentration gradients parallel to the membra ne are negligible

compared with the concentration gradients orthogonal to the

membrane,

gives the rate of convective transpor t of solute towards the membr ane

surface equal to the rate of solute leakage through the mem bra ne plus

the rate of solute due to back-diffusion,

d e

J c = D-d--~y + J C (3)

where c and Cp are solute concentrations in the boundary layer and in the

permeate, respectively and D is the diffusion coefficient of the solute in

the solvent. An integration of Eq. (3) over the boundary layer thickness

5 with the bound ary conditions

c y = 5 ) = C b c y = O ) = Cm

gives the well-known film model relationship:


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C m - - C p

J = k s In [ cb _ cp

(4)

where k s = 19 /5 , the overall mass transfer coefficient of the solute in the

boundary layer, and c m is the concentration at the mem brane surface.

The overall mass tra nsf er coefficient is usually obtained from correla-

tions of the form

Sh - ~ - K Re a S c b (5)

where the constants K, a, b , c vary with the flow regime (see Blat t et al.

(1970), Por te r (1972a), Gekas and Halls trSm (1987)). The above boundary

layer theory applies to mass transfer controlled systems where the

permeate flux is ind ependen t of pressure (no pressure term in the model).

2. GEL-POLARIZATION MODEL

Several investigators (Blatt et al. (1970), Porter (1972a,b), Henry

(1972), Kozinski and Lightfoot (1972), F ane et al. (1981)) found from plots

of flux versus applied ultrafiltration pressure of most macromolecular

solution and colloidal dispersion experiments t hat the steady sta te flux

reaches asymptotically a limiting value where fu rther increase in applied

pressu re r esults in minimal increase in pe rmeate flux. The existence of

this flux plat eau cannot be explained from the basic principle of the film

model of concentration polarisation.

Michaels (1968) and Blatt et al. (1970) forwarded a hypothesi s that as

the concentration at the memb ran e surface increases due to polarization

the macrosolute reaches its solubility limit and precipitates on the

membra ne surface to form solid or thioxotropic gels. For colloidal disper-

sions the g e l l a y e r is expected to resemble a layer of close-packed spheres

(Porter (1972a)). As a consequence, a constant gel layer concent ration is

rapidly reached, which is expected to be virtually independent of bulk

solution concentration, applied pressure, fluid flow conditions or mem-

brane characteristics. According to Michaels' model, an increase in ap-

plied press ure produces a t emporary increase in flux, which brings more

solute to the gel layer and increases its thickness (increase in hydrauli c

resi stance to solvent flow), the reby reducing the flux to the original level.


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F o r 1 0 0 % s o l u t e r e j e c t i o n ( % = 0 ) E q . ( 4 ) c a n t h u s b e w r i t t e n i n t h e

f o r m

J l i m =

k s

i n

I - ~ | ( 6 )

~ ) C b

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

J l i m

i s t h e l i m i t i n g f lu x .

B l a t t e t a l . ( 1 9 7 0 ) , P o r t e r ( 1 9 7 2 a , b ) , G o l d s m i t h ( 1 9 7 1 ) , H e n r y ( 1 9 7 2 )

a n d M a d s e n (1 97 7 ), a m o n g m a n y o t h er s , w h o h a v e s t u d i e d m a c r o s o l u t e

a n d c o ll oi da l u l t r a f i l t r a t i o n , h a v e o b t a i n e d e x p e r i m e n t a l r e s u l t s f or th e

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

n a m e l y ,

J lim i s i n d e p e n d e n t o f a p p l i e d p r e s s u r e ,

J lim i s s e m i - l o g a r i t h m i c a l l y r e l a t e d t o Cb,

J lim a p p r o a c h e s z e ro a t a l i m i t i n g b u l k c o n c e n t r a t i o n

Cb,lim

w h i c h i s

e q u a l t h e g e l c o n c e n t r a t i o n Cg ( or t h e c a k e - c o n c e n t r a t i o n i n t h e

c a s e o f c o l l o id a l d i s p e r s i o n s ) ,

J lim m a y b e m o d i f e d b y f a c to r s w h i c h a l t e r t h e o ve r a ll m a s s t r a n s f e r

c o e f f i c i e n t

k s

P o r t e r ( 1 9 7 2 a ) r e p o r t e d t h a t t h e a g r e e m e n t b e t w e e n t h e o r e t i c a l a n d

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

s a i d t o b e w i t h i n 1 5 t o 3 0 % u s i n g t h e L ~ v S q u e ( l a m i n a r f l o w ) a n d

D i t t u s - B o e l t e r ( t u r b u l e n t f lo w ) m a s s t r a n s f e r r e l a t i o n s h i p s f o r f l ow in

n o n - p o r o us c h a n n e l a n d t u b u l a r s y s t e m s . T h e s u c c es s o f t h e r e l a t i o n s h i p s

i n i n d i c a t i n g t h e v a r i a t i o n ( po w e r d e p e n d e nc e ) o f u l t r a f i l t r a t i o n f l u x w i t h

c h a n n e l g e o m e t r y a n d f l u i d v e l o c i ty i s g r a t if y i n g . H o w e v e r , f o r c o l lo i d a l

d i s p e r s i o n s , e x p e r i m e n t a l f l u x v a l u e s a r e o f te n o n e t o t w o o r d e r s o f

m a g n i t u d e h i g h e r t h a n t h os e i n di c a te d b y t h e L ~ v~ qu e a n d D i t t u s - B o e l t e r

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

n o t e d b y B l a t t e t a l . ( 19 7 0 ).

2.1 L imi ta t ions o f the Model

T h e g e l - p o l a r i z a t i o n m o d e l h a s p r o v e d t o b e u s e f u l i n t h e i n t e r p r e t a -

t i o n o f u l t r a f i l t r a t i o n p e r f o r m a n c e ( F a n e ( 1 98 6)) . H o w e v e r , i t b e a r s s o m e

i m p l i c a t i o n s w h i c h p u t i t s f o u n d a t i o n i n d o u b t.

P o r t e r ( 19 7 2 a ) n o te d t h a t f or t h e u l t r a f l l t r a t i o n o f h u m a n a l b u m i n i n

s p i r a l f l o w c h a n n e l p l a t e s w i t h 1 0 m m a n d 3 0 m m c h a n n e l - h e i g h t f o r


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laminar and turbulent flow, respectively, the plots of flux versus bulk

concent ration gave surprisingly different gel concentrations. A fact which

cannot be accounted for by the gel-polarization model.

Nakao et al. (1979), who have measured the concent ration of the gel

layer at steady state conditions for macromolecular solutions, found th at

the gel concentration is not constant but rather a function of bulk

concentration and cross-flow velocity, which confirmed Porter's experi-

men tal observations. Furthermore, they found from experiments using a

bulk concent ration equal to the calculated gel concentration (obtained by

extrapola tion of flux versus In

cb

plots), that the permeate flux was not

zero as predicted by the gel-polarization model.

Fane et al. (1981) observed experimenta lly that the gel-polarized

behav iour with identical solutions and hydrodynamics produced different

limiting flux values when membranes of differing permeability were

used. This is in contradiction to the gel-polarization model, since no

dependence of membrane properties is implied.

Blatt et al. (1970) and Porter (1972a) ascertained that colloidal ul-

trafll tration flux is several times higher than predicted by the gel-polari-

zation model and the conventional mass transfer correlations. This has

been termed, the

flux paradox

for colloidal dispersions. Porter related

this finding to the tubular-pinch effect (Segr~ and Silberberg (1962)),

which is sometimes known as the radial migration phenomenon. The

tubular-pinch effect as well as further explanations are presented in

Section 5.

Thus the gel-polarization model appears to have some physical limi-

tations although in some cases it still remains the most convenien t model

from a pract ical point of view (Fane (1986)).

2.2 Model Developments with Variable Physical Properties

Kozinski and Lightfoot (1972) developed a theoretical model for pre-

dicting permeate flux through a rotating disk ultrafiltration memb ran e

tak ing into consideration concentration dependent diffusivity and viscosity.

The model is based on the concentration-diffus ion equation (in a form

equivale nt to Eq. (7)) and the equations of motion (in a form equiva lent

to Eq. (8), (9)), which have to be solved simultaneously to give a numerical

solution. Exper imental studies were performed with bovine serum albumin

(BSA) solutions and the results compared quite well with t he numer ical


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predictions for situations in which the protein has not denatured. However,

the model could not account for the experimental finding that the filtra-

tion rate increased by 50% when ionic str eng th was lowered by a factor

of 5. They suggested that these effects might result from the instability

of bovine serum albumin at low ionic strength and in particular its

tende ncy to undergo polymerization.

Their numerica l solution was found to be very sensitive to the exact

behaviour of the diffusivity, osmotic pressure and viscosity at high

concentrations. The lack of sufficient diffusion data (see Fig. 2) may be

the greatest source of uncertainty in the prediction of ultrafiltration

per formance (Kozinski and Lightfoot (1972), Shen and Probs tein (1977),

Probstein (1978), Trettin and Doshi (1980a,b), Vilker et al. (1981)).

Kozinski and Lightfoot extended their model to parallel plate systems

9 W

E

¢D

~ 6

5

3

O

= 3

i t 3

2

1

0

z~

X 0 0.1M buffer, pH 4.7 Keller e t a / 1971))

X r l 0 .15M NaCI , pH 7 .4

Shen and Pr obste i n

1977))

0 .15M N aCI, pH 7 .1 - 7 .7 Ph l l ies e t a L 1976))

Z~ Ibid. PilUes

et a l

1976))

V 0 .1M NaCI (Doherty and enedek (1974))

onic

strength and pH l iterature data

, I , I ~ I , / , I

1 2 3 6

Albumin Concentration /(g L-l)

Fig. 2. Diffusion coefficient versus concentra tion of bovine serum alb umi n (BSA) solutions

(Shen a nd Pro bst ein (1977)).


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150

and considered the limitations of Blatt's et al. (1970) thin channel

ultraf iltra tion da ta for bovine seru m albumin. The predictions were low

but well within the accuracy of Blatt's data.

Shen and Probstein (1977) proposed that the physical model of

Michaels (1968) is essential ly correct in its qualit ative behav iour and th at

the quanti tat ive disa greement is principally a consequence of not consid-

ering the variable transport properties of the macromolecular solution,

in particular, the variability of the diffusion coefficient. They obtained

out of the concentration-diffusion equation for steady, Newtonian, fully

developed lam inar flow in a parallel plate system

o c o c o / o

u ~xx + v ~y - 8y (c) (7)

with a concentration dependent diffusion coefficient D c ) and u , v the

velocity components longitudinal and normal to the membrane surface,

respectively, that the limiting flux is proportional to the dimensionless

quan ti ty (D(cg) /D(c b))2/3. This result suggests simply the replacing of the

diffusion coefficient evaluated at c b by one evaluated at Cg in the mass

transfer correlation for laminar channel flow of the gel-polarization

model. This finding is consistent with Kozinski's and Lightfoot's (1972)

observation that the critical region for mass transfer is adjacent to the

membrane surface.

Shen and Probstein (1977) compared the calculated results for the

limit ing flux of the ir modified film model with that of an exact numerical

solution developed out of the boundary layer concentration-diffusion

equation with variable diffusivity (Eq. (7)), coupled to the channel mo-

me ntu m equation

~u

= ~(c) 8y (8)

with variable viscosity and the continuity equation

~u ~v

+ = = o ( 9 )

oy

The agreement proved to be quite good for a bovine serum albumin

solution of known properties despite the ra the r complex behav iour of the


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151

diffusion coefficient through the boundary layer.

Many meas urem en ts of bovine serum albumin diffusivity (Fig. 2) at

dilute concentrations (usually below 1 wt.%) have been reported in the

lite ratu re, where solution properties like buffer type, pH-value (electrical

charge on the micro-ions), and ionic stren gth have negligible effects on

the diffusion coefficient. However, ther e is a lack of data at h igh concen-

trat ions which is especially limiting since the above stated factors cannot

be ignored as the interaction among the charged macro-ions becomes

more pronounced. Furthermore, Shen and Probstein (1977) and Trettin

and Doshi (1980b) stated that even existing data scattered appreciably

for the same solution conditions.

Shen and Probste in (1977) tested the ir numerical model for the sensi-

tivity of the dependen ce of the limiting flux on viscosity. Their predicted

flux values for a cons tant bulk viscosity were at most 50% high er th an

the calculations whe n the dimensionless viscosity was tak en to vary from

1 to infinity in the concent ration boundary layer as th e bulk concentra-

tion increases to the gel concentration. Therefore, they concluded that

the limiting flux will be much more depen dent on the variation of the

diffusion coefficient than the viscosity coefficient. This finding fu rther

confirmed the assu mption of a constant viscosity which was adopted in

deriving the modified film model equation.

Shen and Probstein (1977) compared also Blatt's et al. (1970) experi-

men tal data of bovine serum a lbumin in 0.9% saline-water solution with

the ir calculated data using the modified film-model. The predictions were

consistently higher by a small amount, due to uncerta inties by the

extrapolation of diffusion coefficient data at high gel concen trations and

unkno wn pH-values ofBla tt's et al. (1970) experiments. However, Tre ttin

and Doshi (1980a) noted that Shen and Probstein (1977) as well as

Kozinski and Lightfoot (1972) misinterpreted the system design of Blatt's

et al. (1970) experiments, which led to them using a cross-flow velocity

four times h igh er th an reality. Use of the correct velocity in the predic-

tions of the modified model of Shen and Probstein would resul t in 24 to

34% lower flux values than mea sured by Blatt et al. (1970).

Probstein et al. (1978) developed an integral method based on the

par tia l diffe rent ial equat ions of cont inui ty (Eq. (9)) and diffusion (Eq. (7))

with a concentration dep ende nt diffusivity governing the behaviour of a

developing concent ration boundary layer of constant density fluid flowing

under laminar conditions in a plane channel. The result is valid when

the gelling concentration is large compared to the bulk concentration and


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152

justifies analytically the result previously obtained by Shen and Probstein

(1977), namely the replacing of the bulk diffusivity by the diffusivity at

the gel concentration in the widely used formula of Michaels (1968).

Comparison of the analytic result for the limiting flux with both exact

numerica l solution (Shen and Probstein (1977)) and with experiments for

bovine serum albumin in 0.15 M saline water at pH 7.4 showed excellent

agreement. Possible limitations of the experimental data of Probstein et

al. (1978) are that flux measurements were taken only over a narrow

rang e of low solute concentrations.

Trett in and Doshi (1980a) have developed an integral me thod solution

to the solute mass balance equation (Eq. (7)) for a parallel plate system

using a variable non-l inear concentration profile (Doshi et al. (1971)) in

the boundary layer. Excellent agreement was found, for the case of a

constant diffusion coefficient, between the calculated limiting flux values

of the integral method solution and an exact numerical solution, where

Eq. (7) was trans formed to an ordinary differential equation by a method

described by Shen and Probste in (1977). For a concentra tion dep end en t

diffusion coefficient the integral method was satisfactory. By a compari-

son of their integral method solution with the film theory model of

Michaels (1968) for constant fluid properties ag reeme nt was found for

Fg

< 4, where Fg is the ratio of the gel concentration to the feed concentration.

For higher values

o fFg

the integral solution deviates from the logarithmical

behaviour of the film model and predicts higher permea te flux rates. This

would mean tha t extrapolation of experimental values in flux versus In

cb

plots may not be valid and may underpredict the gel concentration.

Despite existing predictive differences, the same upsloping flux behav-

iour was observed when they compared Probstein's et al. (1978) integral

method relationship, which is a special case of Trettin and Doshi's

integral method solution, with the film model. However, experimental

evidence for their theoretical investiga tion was not presented.

Trett in and Doshi (1980b) carried on in the development of a constant

property integral method solution for ultrafilt ration in an u nsti rred batch

cell which agreed well with an exact solution but not with the film theory

model. The deviat ion was more tha n 25% for all values

of Fg

greater than

4, which is similar to the results obtained by Trettin and Doshi (1980a)

for the parallel plate system. Agreement was found to be excellent

between experimental data for bovine serum albumin over a wide con-

centration and pressure range in the unstirred batch cell and the int egral

model predictions, with the average error less than 3%, while the film


13/60


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154

ship via an osmotic pressure mechan ism ( lap I - I Anl ---- 0), which has

been confirmed experimentally. Their observation may not apply to other

macromolecular solutions since they migh t display much lower osmotic

pressures in concentrated solutions.

Aimar and Sanchez (1986) developed a semi-empirical model based on

the film theory model and osmotic pressure model (see Section 3) consid-

ering the dependenc e of the mass trans fer coefficient on permeat e flux,

which has already been proposed by Jonsson (1984) as a qualitative

explanation of the existence of a limiting flux. Concentra tion depe nde nt

viscosity is used in the model, since the variation in diffusivity is demon-

stra ted to be of less influence compared to the viscosity varia tion on th e

mass transfer coefficient for bovine serum albumin solutions (data used

from Kozinski and Lightfoot (1972) and Phil lies et al. (1976)). They

compared t he values of the limiting flux of thei r model predictions with

experimental data from several authors for different solutes, module

geometries and me mbra nes after determining a few param eters depend-

ing on hydrodynamic conditions and solute properties first. Excellent

agreement was found for all kinds of ultrafiltration experiments. Fur-

thermore, a complete model is deduced which calculates the permeate

flux from zero to the limiting value. Again, the experimental data was

well represen ted by the model over the whole range of applied pressures.

Therefore, the assumptions made in the model seem to be valid, especially

the decrease of the mass trans fer coefficient with increasing applied

pressure (increasing membr ane wall concentration) for the l imiting flux

case. They concluded tha t the limiting membr ane wall concentration (for

limiting flux) is a function of the bulk concentration, the hydrodynamic

conditions and the physicochemical solution properties.

Gill et al. (1988) model led the effect of viscosity on the concentration

polarization in the ent rance region of a t hin rec tangular channel module

until gel-conditions are reached on the membrane surface. They men-

tioned that Shen and Probstein (1977) neglected the entranc e length in

their model by assuming that the gel concentration is present at the

channel inlet. However, Gill's et al. model could only predict tha t the gel

concentration is reached after a shorter distance from the channel en-

trance w hen a concentration dependent viscosity is used rathe r tha n a

constant viscosity. Quantitative predictions were limited because of

several assumptions made in the model.


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155

TABLE 1

Osmotic pressures of macromolecular solutions. (a) Vilker et al. (1981), (b) Jonsson (1984),

(c) Wijmans et al. (1985)

Conc. Osmotic pressure

(g/L) (kPa)

BSAa Wheyb Dextran

pH 5.4 proteins

T10b T70 c T500 c

100 7 63 72 21 13

200 25 126 216 97 84

300 60 298 595 284 267

400 134 685 1300 - -

3. OSMOTIC PRESSURE MODEL

At typical ul trafi l t rat ion feed concentrat ions, macrosolutes have a

negligible osmotic pre ssu re and cons eque ntly osmotic effects are fre-

quently ignored (Blatt et al. (1970), Porter (1972a)). However, with the

effect of concentrat ion polar izat ion at the mem bra ne surface the surface

conce ntrat i on could be of one to two orders of mag nit ude higher, wh ere

osmotic pre ssu re s m ay be impo rt an t (see Table 1), as long as the bound -

ary lay er rema ins New toni an and gelat ion or precipi tat ion does not occur

(Fane (1986)).

Kedem and Katchalsky (1958) derived the osmotic pressure model ,

which describes the permeate flux in relat ion to the osmotic pressure

difference An crea ted by the concentrat ion difference betw een the two

sides of the membr ane:

j : I A P l - I A n l

~t R m

i i )

where

R m

is the me mb ra ne resistance, ~ the solvent viscosi ty and An =

n c m) - n Cp) with the concentrat ions c m and Cp a t the memb rane surface

and in the permeate, respectively. The osmotic pressure n is often repre-

sente d in te rms of a polynomial


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156

= a l c + a 2 c 2 + a 3 c 3 (12)

where a 1 is the coefficient in van' t Hoffs law for infinitely dilute solutions

and a2, a 3 rep resent the non-ideality of the solution.

Goldsmith (1971) who first applied the osmotic pressure model has

shown that under circumstances where no gel layer is expected, that is

for a low molecular weight po lyethylene glycol (mean Mwt = 15500

Daltons) as the solute with a limiting concentration less th an 10 wt.%,

an almost limiting flux can be obtained for laminar and turbulent flow

conditions, whereby the J versus In

c b

plot is linear for a given pressure

drop Ap.

Kozinski and Lightfoot (1972) examined low polarization ultrafiltra-

tion in a rotating disk geometry. They numerically integrated the con-

cent ration-dif fusion equation (Eq. (7)) and the equations of motion (Eq.

(8), (9)) subject to the osmotic pressure boundary condition and found

tha t concen tration polarization could account for experimenta lly ob-

served reductions in permeate flux when ultrafiltering bovine serum

albumin solutions.

Leung and Probstein (1979) developed an integral solution method

based on the two-dimensional concentration-diffus ion equation (Eq. (7))

and the osmotic pressure model with concentration depende nt diffusivity

and osmotic pressure for the ultra filtration of bovine serum albumin in

steady, plane laminar channel flow under low polarization conditions.

The integral solution was checked with a finite difference solution for the

case of a linea r osmotic pressure--concentration relation and a constant

diffusivity, where the agreement was excellent. Experimental ultrafil-

trat e fluxes were found to agree very well with the theoretica lly calcu-

lated values of the integra l solution method using Vilker's et al. (1981)

non-linear osmotic pressure data and a diffusivity relation ob tained by

linearly interpolati ng the diffusivity values between the gel and dilute

concentra tion limits.

Vilker et al. (1981) concluded from osmotic pressure measurements

and ultrafiltration experiments of bovine serum albumin in an unstir red

cell, tha t the permea te flux is limited by the osmotic pressure, since also

no gel formation was observed under t he applied operation conditions.

Wijmans et al. (1984) provided an inte res ting analysis of the osmotic

pressure model. Assuming complete rejection

Cp

= 0) the osmotic pres-

sure difference in equation (11) is approximated by

(13)

T z = ~ c m ) = a c m


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157

with n > 1. c m itsel f is de pen dent on perm eat e flux according to the film

model (Eq. (4)), which gives

la p[ - a c~ exp

n J / k s )

J = (14)

g R m

Wijmans et al. (1984) demonstrated how the derivatives of Eq. (14)

provide insight into the ultrafiltrat ion process. The flux-pressure derivat ive

[ 1 1

J n

- ~ R m + , - : - ( i @ i - J g R m )

~IApl

i1_1

= g R ~ + - - I A n

k s

(15)

gives the asymptotes

~J

~ l a p J

~. gR in )

-1 for lap [ ~ O, or [ An[ ~ 0

and

~J

2 1 @ 1

--> 0 for ]Apl

--~ oo, o r

[An[

> > J g R m

Thus the flux-pressure profile commences at low Ap with a slope similar

to pure solvent flow and as Ap increases the slope declines and approaches

zero at high pressure, which is similar to the gel-polarization model.

The fl ux-concentr ation relationship could be examine d by rearrang-

ing Eq. (14), taking the logarithm and differentia ting to give

In c b l a p I j

n gRm

gR,nks ; 1

= - k s l+niAnl)

( 1 6 )


18/60

1 5 8

which shows that when polarization is significant, tha t is,

nlA l

IAPl >>

J g R m

or -- >> 1, then:

g R m k s

~J

__> k s

In

c b

The same prediction for the limit ing slope in a J versus In c b plot is given

in the gel-polarization model.

The limiting concentration Cb,l i m for J ~ 0 is obtained from equation

(14) wh en

Ap =a c n

b , l im = 7~(Cb , l im)

(17)

which gives an osmotic pressure equal to the applied pressure. This also

implies tha t Cb,l i m -- f(~9), an impor tant difference from the gel-polariza-

tion model which predicts tha t

C b , l i m - -- -

Cg :/: ~ p ) .

Wijmans et al. (1984) concluded from their theoretical ana lysis that,

(1) lower values for the membrane resistance

R m

lead to a more

pronounced osmotic pressure effect, that is flux limitation at lower

applied pressures,

(2) the mem bran e resistance has a diminishing importance as the

osmotic pressure increases, that is as the bulk concentration increases,

and,

(3) the osmotic pressure limitation will be expected in ultr afilt ration

of med ium macrosolutes (104-105 Mwt).

Jonsson (1984) has analysed the ultrafiltration of dextran solutions

(macrosolutes of med ium molecular weight) and demon strat ed tha t the

osmotic pressure model could produce a semi-logarithmic plot of Jve rs us

in Cb, which gives a linear relation at bulk concentra tions above 7 wt.%,

intersec ting the In cb-axis at a concentration of about 37 wt.%. However,

Jonss on concluded tha t this is not the gel concentration, but the concen-

tration at which the solution has an osmotic pressure equal to the applied

pressure. Fur the r evidence for the absence of a gel layer was obtained by

direct observation of the hydrodynamic flow of the polarization layer.

Cli~on et al. (1984) solved the two-dimensionalconcentration-diffusion

equation with constant diffusivity, coupled to the m omem tu m equat ion

with variab le viscosity for a hollow-fibre membr ane geometry in order to


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159

d e s c ri b e t h e g r o w t h o f t h e c o n c e n t r a ti o n p o l a r i z a ti o n l a y e r a l o n g t h e

m e m b r a n e . T h e o b t a i n e d i n t e g r a l s o lu t io n w a s c o m b i n e d w i t h t h e o s m o t i c

p r e s s u r e m o d e l t o y i e l d a n u m e r i c a l s o l u ti o n , w h e r e t h e p e r m e a t e f lu x ,

t a k e n s e p a r a t e l y i n s e v e r a l s e c ti o n s a l o n g t h e m e m b r a n e b u n d l e , w a s

a s s u m e d t o b e c o m p l e t e l y l i m i t e d b y t h e o s m o t ic p r e s s u r e e ff ec t. V e r y

g oo d a g r e e m e n t b e t w e e n o b s e rv e d a n d c a l c u la t e d v a l u e s w a s o f te n f ou n d ,

b u t g e n e r a l l y b e t t e r f or d e x t r a n T 7 0 t h a n P V P ( p o ly v i n y lp y r ro l id o n e )

s o l ut io n s . T h e y p r o p o se d t h a t t h i s w a s d u e t o a b e t t e r r e p r e s e n t a t i o n o f

t h e v i s c o s i ty v a r i a t i o n s b y t h e i r c o n c e n t r a ti o n d e p e n d e n t v i s c o s it y r e la -

t i o n s h i p f o r t h e d e x t r a n s o lu t io n s . F u r t h e r m o r e , f or h i g h b u l k c o n c e n t ra -

t i o n s o f P V P a d i s t i n c t d i v er g e n c e b e t w e e n m e a s u r e d a n d c a l c u l a te d f l u x

v a l u e s w a s o b s e rv e d b e y o n d a c e r t a i n d i s t a n c e a l o n g t h e m e m b r a n e .

C l i f t o n e t a l. (1 9 8 4) a l s o s u g g e s t e d t h a t t h e p o l a r i z a t i o n l a y e r b r e a k s

d o w n d u e t o i n s t a b i l i t i e s i n th e b o u n d a r y l a y e r a f t e r a c e rt a i n p o i n t a l o n g

t h e m e m b r a n e , b e c om i n g t h i n n e r a n d t h u s a l lo w i n g a m o re r a p i d m a s s

t r a n s f e r t o t a k e p l ac e r e s u l t i n g i n t h e o b s e rv e d h i g h e r f lu x v a l u e s. S u c h

h y d r o d y n a m i c d i s t u r b a n c e s h a v e b e e n o b se rv e d d i r e c tl y b y M a d s e n

( 1 97 7 ) a n d J o n s s o n ( 19 8 4) u s i n g a c o lo u r ed m a c r o m o l e c u l a r s o l u te .

C h o e e t a l . ( 1 9 8 6 ) c o n f i r m e d J o n s s o n ' s ( 1 9 8 4 ) o b s e r v a t i o n s t h a t i n

u l t r a f i l t r a t i o n w i t h d e x t r a n s o lu t io n s , th e o s m o ti c p r e s s u r e i s t h e o n l y

r e s i s t a n c e w h i c h n e e d s t o b e t a k e n i n to a c c o u n t to e x p l a in t h e p e r m e a t e

f l u x d e c l i n e .

4 . RE S IS TANC E M O DE LS

U l t r a f i l t r a t i o n p e r f o rm a n c e c a n a l so b e i n t e r p r e t e d b y a r e si s t a n c e -i n -

s e r i e s r e l a t i o n s h i p , w h i c h c a n b e o b t a i n e d f r o m e q u a t i o n (2) b y n e g l e c t i n g

t h e o s m o t i c p r e s s u r e t e r m ,

J ] SP l (18 )

~ t R m + R s )

A c c o r d i n g to t h e f i l t r a t i o n t h e o r y t h e r e s i s t a n c e o f p o l a r iz e d s o li d s c a n b e

w r i t t e n a s

R~ = o~ m p (19)

A m


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160

where m p is the mass of deposited particles, A m the membran e area and

the specific resistance of the deposit, which can be approximately

related to the properties for spherical particles by the C arma n-Ko zeny

rela tionsh ip (Carman (1938))

180(1 - ~)

= (20)

ppdp2a3

where ~ is the void volume of the cake, pp the densi ty of the particles and

d p

the mean diameter of the particles. A general observation would be

tha t the smaller the particles are then the greate r the specific resistance

will be. Equation (18), (19) and (20) apply equally well to particulate

filtra tion (microfiltration) and colloidal ultrafi ltra tion (Fane (1984)).

For dead-end (unstirred) filtration under constant pressure condi-

tions, without any particle back-transport, R s increases with time, be-

cause

m p = V c b (21)

where Vis the total volume filtered and

c b

the bulk concentration, so that

the combination of Eqs. (18), (19) and (21) with

J = A m 1 d V / d t

gives by

integration the well-known constant pressure filtration equation (in a

form first sugges ted by Underwood (1926)).

t ~ R m ~t c¢ c b

+ V (22)

V - A m l A P I 2 A 2 ] A p l

Equation (22) yields a stra ight line on plotting experimental data of t/V

versus V which allows determinat ion of the specific resistance a and the

membr ane resistance

R m.

4 .1 G e l - P o l a r i z a t i o n R e s i s t a n c e M o d e l

Fane (1986) expressed the gel-polarization model in te rms of resistances,

J = lAP] (23)

~ t R m + R b l + R g )


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where Rg is due to the gel layer at limiting concentration Cg, and Rbl

repre sents the resistance of the viscous, but non-gelled boundary layer.

If the concentration polarization at t he mem bran e surface increases from

zero to the gel concentration due to an increase in pressure, Rg is zero

and the flux is pressure dependent. After the gel concentration is reached

Rg is established and any furthe r increase in lap I simply increases the

thickness of the gel layer, and increases

Rg,

which predicts the pressure

ind epe ndent flux behaviour.

Chudacek and Fane (1984) used the filtration model to describe the

dynamics of polarization in stirred or cross-flow ultrafiltration. The

appropr iate form of the resis tance model (Eq. (18)) is

J= IAPl (24)

Rm + Rsd + Rsr)

where Rsd is the resistance which would be caused by deposition of all

convectively transpo rted solute (equivalent to R s in Eq. (19)), and Rsr is

the resi stance removed by stirring or cross-flow. They assumed th at the

remova l of solute (back-transport) is constant and equal to the convective

solute trans port at steady state = Jss Cb)so that Eq. (21) is not any more

valid. Equation (24) becomes

j :

~tIRm + g/nm-dsst)° ~Cb I

(25)

The quantity Jss can e ither be expressed by the film model relationship

(Eqn. (4)) or by exper imen tal ly det ermin ed values

Of Jss.

Chudacek and Fane's (1984) numerical solutions of the model using

experimentally determined a-values and measured Jss-values, tended

slightly to overpredict initial experimental flux values for their three

solutes (bovine serum albumin, dextran T2000, Syton X30 silica) used.

They explained the discrepancy as due to membrane-solute interactions

where the incoming solute may obstruct the entrance pores of the mem-

brane resulting in an additional resistance for which the model takes no

account.


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4 .2 B o u n d a r y L a y e r R e si s ta n c e M o d e l

Wijmans et al. (1985) proposed the boundary layer resistance model

for cross-flow ult rafil trat ion of dex tran solutions wher e no gel-formation

will occur. The basic principle of the model is the correspondence of the

permeability of a concentrated solute layer for solvent flow and the

permeability of a solute in a stagnant solution, as occurring during a

sedimentation experiment. This relationship (Mijnlieffand Jaspers (1971))

can be described by

~s

p = (26)c(1 -

v l / v o )

wh erep is the permeability of a concentrate d solute layer of concentrati on

c, v0 and v I are the par tial specific volumes of the solvent and the solute

respectively and s is the sedimentation coefficient at concentration c

usual ly described by

1 1

s s 0

-- - - (1 +

KlC + K2 c2 +

K3c3)

(27)

wh ere So, K1, K 2 and K 3 are constants .

The permeability depends on the concentration and since there is a

concentration profile in the boundary layer, the permeability will be a

function of the coordinate y perpend icular to the membrane . Since, the

hydrodynamic resistance of the boundary layer Rbl is generall y defined

a s

f

5

Rbz = rbldY

(28)

0

where

rbl

is the specific resistance of the boundary l ayer

rbl

= app(1 -e ))

and equal to the reciprocal of the permeability. Thus, th e boundary layer

resistance is

f

5 1

R~l = dy

o P(Y)

(29)

Substituting Eq. (26), (27) with c y) = c b e x p J y / D ) into Eq. (28) and

inte grat ing over the bo unda ry layer thickness 5, results in


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D 1 - v l l [ c m - - ~ - c 2 - c 2 )+- - - ~ c 3 c~) C m -C 4) 1

bl ~tSo Vo)L Cb + K1 K2 K3

_ _ . _ _ ~ _ 4

(3O)

w h e r e c m i s t h e s o l u t e c o n c e n t r a t io n a t t h e m e m b r a n e s u r fa c e , w h i c h

c o u ld b e d e t e r m i n e d v i a t h e f i lm m o d e l r e l a t i o n s h i p ( Eq . (4 )) i f

k s

i s

k n o w n .

C o m b i n i n g E q . (1 8) w i t h

R s = Rbl ,

(30) an d (4) fo r

Cp =

0 , t h e b o u n d a r y

l a y e r r e s i s t a n c e m o d e l is o b t a in e d i n w h i c h t h e p e r m e a t e f lu x is t h e s i n g l e

u n k n o w n p a r a m e t e r .

j =

I Pl

Vl l[c - + @ + c4-

m+ ~ 1 - VoJL 2 3 4

(31)

w i t h

C m = C b exp J/ k s)

T h u s , t h e f l u x c a n b e c a l c u l a t e d i f t h e p r o c es s c o n d i t i o n s ( A p, Rm, Cb, k s)

a n d t h e p h y s i c a l p r o p e r ti e s o f t h e s o l u t e - s o l v e n t s y s t e m (s, D ) a r e k n o w n .

W i j m a n s e t a l. (1 9 85 ), w h o c a l c u l a t e d t h e m a s s t r a n s f e r c o e f fi c ie n t b y

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

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

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

t a l f l u x v a l u e s .

N a k a o e t a l. ( 19 8 6) w h o e x a m i n e d t h 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 e f f e ct

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

a d a p t e d t o t h e c a k e f i l t r a ti o n th e o r y t o a n a l y s e t h e e x p e r i m e n t a l f l u x

b e h a v i o u r o f d e x t r a n a n d p o l y e t h y l e n e g l yc o l s o l u ti o n s . T h e r e f o r e a s t e p

c o n c e n t r a t i o n p r o fi le a n d a t i m e - i n d e p e n d e n t c o n c e n t r a t io n i n t h e b o u n d -

a r y l a y e r w i t h o u t a n y s o lu t e b a c k - t r a n s p o r t w e r e a s su m e d . T h e s i m p l e

m o d e l w o r k e d w e l l i n p r e d i c ti n g t h e e x p e r i m e n t a l f l u x b e h a v i o u r b u t o n l y

w i t h t h e n e e d o f s e v e r a l o th e r e x p e r i m e n t s t o o b t a i n t h e n e c e s s a r y

p a r a m e t e r s .

V a n d e n B e r g a n d S m o l d e rs ( 1 98 9) a n a l y s e d t h e c o n c e n t r a t i o n p o l ar i-

z a ti o n p h e n o m e n a o f b o v in e s e r u m a l b u m i n d u r i n g u n s t i r r e d d e a d -e n d

u l t r a f i l t r a t i o n b y a d a p t i n g t h e b o u n d a r y l a y e r re s i s t a n c e m o d el in


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164

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

d i f f u s i o n e q u a t i o n ( E q . (1 0)). T h e i r n u m e r i c a l a p p r o a c h c o m p a r e d t o

N a k a o ' s e t a l . ( 1 9 8 6 ) m o d e l r e q u i r e d n o a s s u m p t i o n s c o n c e r n i n g 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 , t h e c o n c e n t r a t i o n p r o f il e o r t h e s p e c if ic

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

w i t h t h e e x p e r i m e n t a l d a t a f or a ll f ee d co n c e n t ra t io n s , m e m b r a n e r e s is -

t a n c e s ( r e t e n t i o n s ) a n d a p p l i e d p r e s s u r e s i n v e s t i g a t e d .

5 . ALTERNATIVE MOD ELS F OR CO LLOIDAL FILTRATION

C o l l o i d a l d i s p e r s i o n s h a v e b e e n s u b j e c t e d t o u n s t i r r e d , s t i r r e d a n d

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

m a n y i n v e s t i g a to r s . F l u x p r e d i c ti o n s a r e f r e q u e n t l y d e s cr ib e d b y t h e

w i d e l y a c c e p t e d g e l - p o l a r i z a t i o n m o d e l o f M i c h a e l s ( 1 96 8 ),

J -- ~- In

= k s

In (32)

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

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

k T

D o

(33)

3 ~ ~i d p

a l t h o u g h t h i s s t r i c t l y a p p li e s o n l y t o d i lu t e s o l u t io n s , w h e r e k is t h e

B o l t z m a n n c o n s t a n t , T t h e a b s o l u t e t e m p e r a t u r e , ~t t h e v i s c o s i t y o f t h e

s o l v e n t , a n d

d p

t h e p a r t i c l e d i a m e t e r . T h e i n f e r e n c e f r o m E q . ( 3 2 ) a n d

( 33 ) i s t h a t a s p a r t i c l e s i z e i n c r e a s e s s o f l u x d e c r e a s e s . H o w e v e r , a s

p a r t i c l e s i z e i n c r e a s e s f r o m c o l lo i da l u p w a r d s , b a c k - t r a n s p o r t b e c o m e s

n o n - d i f f u s i v e a n d E q . (3 2) n o l o n g e r s t r i c t l y a p p l ie s .

B l a t t e t a l . ( 1 9 7 0 ) w h o h a v e c a r r i e d o u t u l t r a f i l t r a t i o n e x p e r i m e n t s

w i t h s k i m m e d m i l k , p o l y m e r la t ex , a n d c l a y d i s p e r si o n s f o u n d t h a t t h e

f i lm t h e o r y m o d e l e x p r e s s e d b y E q . (3 2) a n d ( 33 ) d r a s t i c a l l y u n d e r p r e -

d i ct s e x p e r i m e n t a l p e r m e a t e r a te s . T h e s e o b s e r v a t io n s l e d t h e m t o t h e

c o n c l u s i o n t h a t e i t h e r : (1 ) t h e b a c k - d i f f u s i o n o f p a r t i c l e s f r o m t h e p o l a r -

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

B r o w n i a n m o t io n o r, (2) t h e t r a n s m e m b r a n e f lu x is n o t l i m i t e d b y t h e

h y d r a u l i c r e s i s t a n c e o f t h e p o l ar iz e d l a y e r o v e r a n y r e a s o n a b l e r a n g e o f


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165

l a y e r th i c k n e s s . B l a t t e t a l. f a v o u r e d t h e s e c o n d e x p l a n a t i o n to b e th e

m o r e r e a s o n a b l e , s i n c e a c a k e f o r m e d b y m i c r o n s i z e p a r t i c l e s h a s a

r e l a t i v e l y lo w s p e ci fi c r e s i s t a n c e c o m p a r e d w i t h a m a c r o m o l e c u l a r c ak e .

P o r t e r ( 19 7 2 a ,b ) w h o o b s e r ve d e s s e n t i a l l y t h e s a m e f lu x a u g m e n t a t i o n

c o m p a r e d w i t h t h e f i lm m o d e l p r e d i c t io n s i n m o r e t h a n 4 0 c o ll o id a l

d i s p e r s i o n s h e s t u d i e d , d i s p u t e d B l a t t ' s e t a l . ( 1 9 7 0 ) h y p o t h e s i s a n d

a r g u e d t h a t , i f t h e p o l a r iz e d l a y e r is n o t t h e l i m i t i n g fa c to r , th e f l u x

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

a p p l i e d p r e s s u r e . H o w e v e r , h e o b s e r v e d t h e o p p o s i t e t o b e t r u e , d e c r e a s-

i n g f lu x w i t h i n c r e a s i n g b u l k c o n c e n t r a t io n . T h e r e fo r e , P o r t e r h y p o t h e -

s iz e d t h e a u g m e n t e d b a c k - t r a n s p o r t o f t h e l a t e r a l ( r a d ia l ) m i g r a t i o n o f

p a r t ic l e s k n o w n a s t h e

tubu lar-pinch e ffec t

( S eg r ~ a n d S i l b e r b e r g ( 19 6 2 ))

t o b e r e l e v a n t f o r t h e e n h a n c e d f l ux . B u t n o q u a n t i t a t i v e p r e d i ct io n s o f

t h e e n h a n c e d m a s s t r a n s f e r o f p a r t i c le s a w a y f r om t h e m e m b r a n e s u r fa c e

c o u l d b e m a d e a t t h a t s t ag e .

H e n r y ( 1 9 7 2 ) p o i n t e d o u t t h e l o n g t r a n s i e n t f lu x d e c li n e i n t h e u l t r a f il -

t r a t i o n o f c o l lo i d a l d i s p e r si o n s . H e f o u n d i n g e n e r a l t h a t t h e d e p e n d e n c e

o f t h e p e r m e a t i o n r a t e o n fl u i d r a t e ( or s h e a r r a t e ) i s g r e a t e r i n c o ll o id a l

u l t r a f i l t r a t i o n t h a n t h a t p r e d ic t ed b y t h e f i l m t h e o r y m o d e l. T h i s f i n d in g

l e d H e n r y t o t h e c o n c lu s io n , l i k e P o r t e r ( 19 7 2 a ,b ), t h a t a n a d d i t i o n a l

p a r t ic l e b a c k - t r a n s p o r t m e c h a n i s m i s p r e s e n t i n s u c h s y s t e m s .

5.1 Ine r t ial M igra t ion M odel Tu bu lar-Pinch Ef fec t)

S e g r ~ a n d S i l b e r b e r g ( 19 6 2 ), w o r k i n g w i t h d i l u t e d i s p e r s i o n s o f r i g id ,

s p h e r ic a l , n e u t r a l l y b u o y a n t p a r t ic l e s ( m e a n d i a m e t e r s f r o m 0 . 32 t o

1 .7 l m m ) t r a n s p o r t e d a l o n g i n P o i s e u i ll e f lo w t h r o u g h a n o n - p o r o u s t u b e ,

w e r e t h e f i r s t t o p u b l i s h t h e i r o b s e r v a t i o n s o f t h e

tubular-pinch e f fec t ,

w h e r e b y t h e p a r t ic l e s a r e s u b je c t to r a d i a l f or ce s w h i c h m i g r a t e t h e m

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

e q u i l i b r i u m p o s i t io n a t a b o u t 0 . 6 tu b e r a d i i f r o m t h e a x i s , i r r e s p e c t iv e o f

t h e t u b e e n t r a n c e r a d i a l p o s i t io n o f t h e s p h e r e s . T h e o r i g i n o f t h e s e e f fe c ts

w e r e f o u n d t o l i e i n t h e i n e r t i a o f t h e f lu i d. S e g r~ a n d S i l b e r b e r g d e v e lo p e d

a n e m p i r i c a l e q u a t i o n t o c o r r e l a t e t h e i r d a t a f o r t h e r a d i a l m i g r a t i o n

v e l o c i t y v r,

2.84

( r )

= r 1 - ~-; (34 )

vr

0 17uRet[d(d-~t

d t / 2


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w h e r e u i s t h e a v e r a g e f l u i d a x i a l v e l o c it y , R e t = u 2 R / v t h e R e y n o l d s

n u m b e r ,

dp

t h e m e a n p a r ti c le d i a m e t e r ,

d t

t h e t u b e d i a m e t e r , r t h e r a d i a l

c o o r d i n a te , a n d r * t h e e q u i l i b r i u m r a d i a l p o s i ti o n o f t h e p a r t i c l e w h i c h

d e c r e a s e s a s dp/d t i n c re a s e s. T h e i r o b se r v a ti o n s h a v e s p a w n e d a n u m b e r

o f t h e o r e t ic a l a n d e x p e r i m e n t a l s t u d i e s e x p l a i n i n g a n d q u a n t i f y i n g t h i s

e f f e c t (Bre n n e r (1 966) ) .

C o x a n d B r e n n e r ( 1 9 68 ) w e r e t h e f i r s t t o a t t e m p t a f u ll t h e o r e t i c a l

t r e a t m e n t o f t h e p r o b le m o f r i g id s p h e r e s i n l a m i n a r f lo w f re e l y r o t a t i n g

a n d t r a n s l a t i n g p a r a l l e l t o t h e t u b e a x is w h i c h g a v e a s a t i s fa c t o r y

f u n d a m e n t a l e x p l a n a t i o n o f t h e r a d i a l m i g r a t i o n p h e n o m e n o n i n a t u b e

o f f i n i te r a d i u s . T h e y o b t a i n e d t h e f i r s t -o r d e r s o l u t i o n o f t h e N a v i e r -

S t o k e s e q u a t i o n i n c l u d i n g th e i n e r t i a l t e r m s a n d c o n v e r t e d t h e l a t e r a l

f or ce , r e q u i r e d t o m a i n t a i n t h e s p h e r e s a t a f ix e d r , i n t o a n e q u i v a l e n t

r a d i a l m i g r a t i o n v e lo c it y b y a p p l y i n g S t o ke s ' la w . F o r t h e n e u t r a l l y

b u o y a n t c a se t h e y f o u n d

v r = O . 5 u R e t C l p l 3 f - ~ t )

w h e r e f r / d t )

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

E q u a t i o n ( 35 ) i s o f t h e s a m e f o r m a s t h e e m p i r i c a l o n e ( E q . ( 34 )) fr o m

S e g r ~ a n d S i l b e r b e r g ( 1 9 62 ).

V a s s e u r a n d C o x (1 97 6 ) h a v e a n a l y s e d t h e l a t e r a l m i g r a ti o n ,

Vl,

of a

s o li d s p h e r i c a l p a r t i c l e in l a m i n a r c h a n n e l f lo w u s i n g t h e m e t h o d d e v el -

o p e d b y C o x a n d B r e n n e r ( 19 68 ). F o r t h e c a s e o f a n e u t r a l l y b u o y a n t

s p h e r e t h e y o b t a in e d

v l = 4 u

Rech f (13) (3 6)

w h e r e R e ch = u 2 h /v , h i s t h e h a l f c h a n n e l h e i g h t a n d f ([3) d e p e n d s o n t h e

p a r t i c l e p o s i t i o n i n t h e c h a n n e l .

I n g e n e r a l , v i r t u a l l y a l l s t u d i e s o f t h e m i g r a t i o n p h e n o m e n a , w h e t h e r

t h e o r e t i c a l o r e x p e r i m e n t a l ( B r e n n e r ( 1 96 6) ) a r r i v e a t t h e f o l lo w i n g f o r m

f o r t h e m i g r a t i o n v e l o c it y e x p r e s s i o n ( G r e e n a n d B e l f o r t (1 9 80 ))

Vl = K u Re ~dp ln f r)

(37)


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w h e r e

d t

i s th e t u b e d i a m e t e r o r e q u a l t o 2 h t h e c h a n n e l h e i g h t , R e = R e t

o r R ech t h e v a l u e o f n l i e s b e t w e e n 2 .8 4 a n d 4 , a n d f ( r ) i s a f u n c t i o n o f

t h e e q u i l i b r i u m p o s i t io n o f t h e p a r t i c l e i n t h e t u b e o r c h a n n e l . A c c o r d in g

t o E q . (3 7 ), t h e l a t e r a l ( r a d i a l ) m i g r a t i o n v e l o c i ty d e p e n d s s t r o n g l y o n t h e

a v e r a g e a x i a l f l u i d v e l o c i t y a n d t h e p a r t i c l e s i ze .

M a d s e n ( 19 7 7 ) s e m i - e m p i r ic a l l y a n a l y s e d t h e p r o b le m o f p a r t ic l e

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

B r e n n e r ( 19 68 ) a n d t h e f i lm t h e o r y m o de l. H e f o u n d t h a t e x p e r i m e n t a l

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

a f a c t o r o f 5 0 b y t h e p a r t i c l e m i g r a t i o n m o d e l . H i s p r i m a r y c o n c l u s io n

w a s t h a t p a r t ic l e m i g r a t i o n c a n o n l y a c c o u n t fo r a p a r t o f t h e h i g h

p e r m e a t e f l u x o b s e rv e d i n u l t r a f i l t r a t i o n o f c o ll o id a l d i s p e r s i o n s .

G r e e n a n d B e l f o r t ( 19 8 0) d e v e l o p ed a m o d e l f o r c o ll o id a l u l t r a f i l t r a t i o n

i n c o r p o r a t i n g t h e l a t e r a l m i g r a t i o n e ff ec t i n t o t h e s t a n d a r d f l t r a t i o n

t h e o r y b y u s i n g S e g r ~ a n d S i l b e r b e r g ' s (1 9 62 ) e m p i r i c a l r e l a t i o n f o r t h e

c a s e o f f lo w in a n o n - p o ro u s d u c t. I t w a s a s s u m e d t h a t t h e p e r m e a t i o n d r a g

f or ce a n d t h e m i g r a t i o n l i ft d r a g f or ce c o ul d b e v e c t o r ia l l y a d d e d . F u r t h e r ,

i t w a s a s s u m e d t h a t t h e t h i c k n e s s o f t h e ( i m m o b il e) c a k e l a y e r a d j u s t s

i t s e l f t o t h e p o i n t w h e r e t h e c o n v e c ti v e f lo w of p a r t i c l e s t o w a r d s t h e

m e m b r a n e i s b a l a n c e d b y t h e l if t v e l o ci ty a w a y fr o m t h e m e m b r a n e . I n

o r d e r t o o b t a i n t h i s c o n d i t io n A l t e n a e t a l . ( 19 8 3) f o u n d a v a l u e f o r t h e c a k e

l a y e r t h i c k n e s s o f 0 .7 t i m e s t h e c h a n n e l h e ig h t . T h e y c o m m e n t e d t h a t

w h e t h e r s u c h a l a r g e v a l u e f o r t h e c a k e l a y e r i s r e a l is t i c, is q u e s ti o n a b l e .

H o w e v e r , G r e e n a n d B e l f o r t' s ( 19 8 0) m o d e l p r e d i c t io n s f o r t h e s t e a d y

s t a t e f l u x w e r e w i t h i n t h e r i g h t o r d e r -o f - m a g n i tu d e c o m p a r e d w i t h

P o r t e r ' s ( 19 7 2) t h i n c h a n n e l u l t r a f i l t r a t i o n d a t a o f s t y r e n e - b u t a d i e n e

p o l y m e r l a t e x. T h e y m e n t i o n e d t h a t M a d s e n (1 97 7 ) a p p l ie d t h e l i ft

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

m i g r a t i o n v e l o c it y a t t h e o r i g i n a l t u b e r a d i u s d ] 2 a n d n o t, a s i t s h o u l d

b e f or a s t e a d y s t a t e f l u x , a t d ] 2 - 5c, w h e r e 5c i s th e c a k e l a y e r t h i c k n e s s .

T h e y d e m o n s t r a t e d t h a t t h i s s l i g h t d if fe r en c e in e f fe c ti ve t u b e r a d i u s ,

n e g l e c t ed b y M a d s e n , p r o v id e d t h e m a r g i n b e t w e e n c o r r ec t a n d g r o s s ly

i n c o r r e c t p r e d i c t i o n s o f s t e a d y s t a t e f lu x .

A l t e n a e t a l. (1 9 83 ) a n d A l t e n a a n d B e l f o rt ( 19 8 4) s t u d i e d t h e o r e t i c a l l y

t h e e ff ec t of l a t e r a l m i g r a t i o n o f s p h e r i c a l r ig i d n e u t r a l l y b u o y a n t p a r t i -

c l es m o v i n g i n a l a m i n a r f lo w f i el d i n a p o r o u s c h a n n e l . T h e y e x t e n d e d

C o x a n d B r e n n e r ' s ( 1 9 68 ) a n a l y s i s fo r p a r t i c l e m o t i o n i n a n o n - p o r o u s

d u c t t o i n c l u d e t h e e f fe c t o f t h e w a l l p o r o s i ty . A l t e n a a n d B e l f o r t ( 19 8 4)

s h o w e d t h a t t h e a s s u m p t i o n o f a n o -s l ip c o n d it i o n f or t h e t a n g e n t i a l f lo w


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at the mem bran e wall and the negligibility of the disturbance velocity

within the porous wall are good approximations for typical conditions

found in hyperfiltration and ultrafiltration flow channels. They have

analysed t ha t the inerti ally induced velocity (tubular-pinch effect) and

the permeat ion drag velocity due to convection into the porous wall can

be vectorially added when ~ _= Rep •2, where ~. is the ratio of wall

permeat ion velocity to the mean axial fluid velocity, Rep is the particle

Reynolds number based on the mean axial velocity and the particle radius

and K is the ratio of particle radius to channel height. If ~ > Rep ~2, the equil ibrium posit ion moves

closer to the porous wall and finally coincides with it re sulting in particle

capture. Further, ~


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TABLE 2

Calcula tions o f the ratio of permeation and lift velocity as a function of particle size for a

given channel gemoetry and axial centre line velocity (J = 4x105 m s -1) (Alterna et al.

(1983))

h or d~/2 u m d p / 2 J / v l

(mm) (m s 1) (~tm)

Hollow-fibre 0.1 0.1 0.1 4x 104

1.0 40

10.0 0.04

Flat-p late 1.0 0.5

Tubular 10.0 1.5

0.1 2×105

1.0 200

10.0 0.2

0.1 2×108

1.0 2×103

10.0 2

~

1 0

0 8

0 6

0 4

0 2

0 . 0

0 01

• 2.5 vol.%

• l o v o l .

I I I I I I I II I I I I I I I II , i i i i i i iI I

0 1 0 1 0 0 10 00

Partic le s ize d / (~ m )

Fig. 3. Influence of particle size on permeate flux in aqueous dispersions of 2.5 vol% and

10 vol% particle concentration (Fane (1984)). (Lines are for illustration purposes only).


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particle size increases from 25 nm to 20 pm, the flux passes through a

minimu m at about 0.1 ~m particle size due to the fact that the polariza-

tion control changes from diffusive (decreasing with particle size) to

non-diffusive (increasing with particle size), such as lateral migration

and/or scour effects (Fane et al. (1982)).

5 . 2 S h e a r - I n d u c e d H y d r o d y n a m i c C o n v e c t i o n M o d e l

Madsen (1977), Green and Belfort (1980), Altena et al. (1983) and

Altena and Belfort (1984) have found from their hydrodynamic calculations

that the inerti al lift velocity is often less than the permeat ion velocity in

typical cross-flow filtration systems. As a result, a concentrated layer of

deposited particles is formed on the membra ne surface. If this cake layer

would built up indefinitely, it would plug the tube or channel, but this is

not observed in practice. Instead, Blatt et al. (1970) have hypothes ized

that the cake layer accumulates only until the hydrodynamic shear

exerted by flow of dispersion causes the cake to flow tang enti ally along

the membrane surface at a rate which balances the deposition of particles.

Although Porter (1972a,b), Henry (1972), and others have reported an

increase in the permeation flux with increasing tangenti al shear, indi-

cating that a high shear rate is effective in reducing the cake layer

thickness, this process is not well understood (Davis and Leighton (1987)).

Leonard and Vassilieff (1984), using the method of characteristics,

solved the unsteady, two-dimensional convective equat ion by neglecting

the diffusion ter m (no diffusive particle transport) for the axial mov eme nt

of deposited particles along a membrane surface. They described the

simultaneous convective deposition of particles onto the membrane and

the sweeping of this cake layer in response to the fluid shear along the

mem brane surface where it is discharged with t he fluid at t he exit of the

filtration device. Leonard and Vassilieff were able to obtain analytical

solutions for the developing cake layer thickness over time until steady

state is reached by assuming that the velocity profile in the vicinity of

the cake layer is linear, the permeate flux is invariant with time and

position along the membra ne and the cake layer is trea ted as a Newton-

ian fluid with the same effective viscosity as the bulk dispersion. Thei r

simple convection model has been used to predict the steady state

performance of plasmapheresis devices, yielding good correspondence

with exper imental observations.


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D a v i s a n d B i r d s e ll (1 98 7 ) p r e s e n t e d a s t e a d y s t r a t i f i e d l a m i n a r f lo w

m o d e l w h i c h r e l a x e s s o m e o f t h e s i m p l i fi c a ti o n s i m p o s e d b y L e o n a r d a n d

V a s s i l i e f f ( 19 8 4 ). I n p a r t i c u l a r , f u l l y d e v e l o p e d p a r a b o l i c v e l o c i t y p r o f i l e s

w e r e d e t e r m i n e d f or b o t h t h e d i s p e r s io n a n d t h e c a k e l ay e r , t h e c h a n n e l

p r e s s u r e a n d t h e p e r m e a t e f lu x w e r e a ll ow e d to v a r y a l o n g t h e c h a n n e l ,

a n d t h e c a k e l a y e r w a s a s s i g n e d a c o n c e n t r a t i o n - d e p e n d e n t e ff ec ti ve

v i s c o s i ty h i g h e r t h a n t h a t o f t h e b u l k d i s p er s io n s . T h e y d e v e lo p e d o u t o f

t h e n o n - d i m e n s i o n a l N a v i e r - S t o k e s e q u a t io n , b y n e g l e c t in g t h e t r a n -

s i e n t a n d c o n v e c ti v e i n e r t i a t e r m s , a n d t h e c o n t i n u i t y e q u a t i o n t h e

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

y i e l d s p a r a b o l i c f lo w p r o fi le s i n b o t h t h e d i s p e r s i o n s a n d t h e c a k e l a y e r .

A l t h o u g h t h e g o v e r n i n g e q u a t i o n s w e r e d e v el o pe d f o r t h e f lo w o f t h e

d i s p e r s i o n s , t h e y w e r e a l so v a l i d fo r t h e c a k e l a y e r , p r o v i d e d t h a t t h e

v i s c o u s t e r m s a r e m u l t i p l i e d b y t h e v i s c o s i t y r a t i o ~ = ~c / ~b. A d d i t i o n a l l y ,

t h e s t e a d y s t a t e d i f f e re n t i a l m a s s b a l a n c e s f o r t h e b u l k m a t e r i a l ( l i q u id

p l u s s o li ds ) a n d f or t h e s o li d m a t e r i a l f lo w i n g t h r o u g h t h e c h a n n e l s e r v e d

a s a u x i l i a r y e q u a t i o n s i n t h e d e t e r m i n a t i o n o f t h e c a k e l a y e r t h i c k n e s s ,

t h e p r e s s u r e d ro p , a n d t h e c h a n g e o f t h e p r e s s u r e d r op o v e r t h e c h a n n e l

d o w n s t r e a m d i s ta n c e , a l l a s a f u n c t i o n of t h e l o n g i t u d i n a l d i s ta n c e f r o m

t h e c h a n n e l e n t r a n c e , w h i c h r e s u l t e d f u r t h e r i n t h e p e r m e a t i o n f lu x . F o r

d i l u t e d i s p e r s i o n s , t h a t i s 5c / h


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3175 ~tm and 1054 ~m mean diameter, were used, to ensure better

observation of the marked particle and to make sure tha t Brownian and

electrokinetic forces were negligible. Eckstein et al. found that their

calcula ted values of the self-diffusion coefficient were not of high accuracy

owing to the nature and size of the experimental errors, but correct within

a factor of two. However, the self-diffusion coefficients showed the tr end

of increasing linearly in the range of 0 to 20% particle volume fraction,

and to be constant in the concent ration range of 20 to 50%.

Leigh ton and Acrivos (1987a) carried out similar experiment s but only

mea sure d the t ransi t time of a marked particle immersed in the disper-

sions to complete a circuit of a Couette device, since they developed an

utilization method, based on the unste ady two-dimensional density dif-

fusion equation, which obviated the need of measuring the radial particle

position in the Couette gap. This greatly simplified the experimental

apparatus compared to Eckstein's et al. (1977) and as a consequence

reduced the expe rimenta l errors. Leighton and Acrivos (1987 a) compared

the ir measured self-diffusion coefficients (for particle volume fractions of

5 to 40%) in dispersions of 645 ~tm and 389 ~m acrylic spheres with tha t

observed by Eckstein et al. (1977). They found that the self-diffusion

coefficient is proportional to the shear r ate and t he square of the particle

radius and to be an increasing function of particle volume fraction ~,

approximately equal to 0.5~ 2 at low concentrations, which is in contras t

to the linear dependence for low concentrations reported by Eckste in et

al. (1977). However, for partic le volume fractions above 20% the self-dif-

fusion coefficients devia ted rapidly from the constant values of Ecks tein

et al. (at ¢ = 40 vol.% approximately 5 times). Leighton and Acrivos

(1987a) demonstrated that Eckstein's et al. (1977) experiments at high

concentrations were limited by the presence of the Couette walls, since

the ir particle diameter to gap-width ratio was only 1/8 compared to 1/32

of Leighton and Acrivos.

Leighton and Acrivos (1987b) demonstrated in the course of their

exper iment al stud y of the behaviour of concentrated dispersions (30-50

vol.%) of neu tra lly buoyant polys tyrene particles (46 pm and 87 pm me an

diamete r) in New tonian fluids using the Couette device, tha t the short-

term viscosity-increase and the long-term viscosity-decrease phenome-

non observed in the Couette gap is due to particle migration across the

gap-width and out of the sheared gap respectively, forced by the shear-

induced diffusion mechanism. Their experiments enabled t hem to infer

the values of the effective diffusivity, which consists of the sum of the


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r a n d o m s e lf - d if f us i o n ( E c k s t e i n e t a l . ( 19 7 7 ) a n d L e i g h t o n a n d A c r iv o s

( 1 98 7 a )) a n d t h e n o n - r a n d o m d r i f t p r o c e ss ( m i g r a t i o n o f p a r t i c l e s a l o n g

c o n c e n t r a t i o n g r a d i e n t s ) o c c u r r i n g in c o n c e n t r a t e d d i s p e r s i o n s o f n o n -

u n i f o r m c o m p o s i t i o n . L e i g h t o n a n d A

21 Theoretical Descriptions of Membrane Filtration of Colloids and Fine Particles-An Assessment and...

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