Analysis of dialysis coupled with ultrafiltration in cross-flow membrane modules

(

"

'"

t',

SCIENCE ELSEVIER

Journal of Membrane Science 134 (1997) 151-162

Analysis of dialysis coupled with ultrafiltration in cross-flow membrane modules H.M. Yeh a'*, T.W. Cheng a, Y.J.

Chen b

aDepartment of Chemical Engineering, Tamkang University, Tamsui 251, Taiwan, ROC bDepartment of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan, ROC Received 23 August 1996; received in revised form 11 April 1997; accepted 23 April 1997

Abstract Analytical solution of solute concentration for the systems of dialysis coupled with ultrafiltration in cross-flow membrane modules was obtained by the method of perturbation, and thus separation efficiencies could be calculated for various operating and design conditions. The overall mass transfer coefficient was assumed to be constant and the concentration polarization and pressure drop on both compartments were neglected in solving this problem. Moreover, the mean values of permeation flux and sieving coefficient were taken and were treated as constant. It was found that increasing the flow rate in retentate phase is more beneficial to the mass transfer than increasing in dialysate phase. The performance of dialysis can be improved significantly by the effect of ultrafiltration, especially for the systems with low mass transfer coefficient. The present analysis could be applied to all kinds of configurations, such as parallel-flow, radial-flow and spiral-wound types, but with the limitations of laminar flow and low viscosity and high diffusivity of the fluids.

Keywords: Dialysis; Ultrafiltration; Cross-flow; Spiral-wound membrane module

1.

Introduction

The driving forces for the transport of molecules through the membrane can be the gradients in concentration, pressure, electrical potential, or temperature [1]. Dialysis may be considered as a concentration driven process. In fact, the driving force for dialysis is the difference in chemical potential between the feed and dialysate. If the feed and dialysate are composed of the same solvent differing only slightly in concentration, it may be assumed that the solute concentrations are proportional to their activities. Thus *Corresponding author. Fax: 886-2620-9887. 0376-7388/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved.

PII S 0 3 7 6 - 7 3 8 8 ( 9 7 ) 0 0 1 2 7 - 0

concentration gradients may be used to satisfactorily describe the driving force. The famous application of dialysis is hemodialysis for removing the metabolic waste from blood, but the rate of solute removal decreases sharply with the increasing of solute molecular weight. Ultrafiltration is the process driven by transmembrane pressure for the concentration of macromolecular solution or the recovery of valuable constituents [2-4]. Grimsurd and Babb [5] and Cooney et al. [6] analyzed the effect of dialysis in plate-type and tubular-type membranes, respectively, with the considerations that the flow is fully developed and that the fluid is incompressible. They also assumed that the

152

H.M. Yeh et al./Journal of Membrane Science 134 (1997) 151-162

concentration of solute in the dialysate is constant. Popovich et al. [7] considered both the effects of dialysis and ultrafiltration in plate-type membranes. Jagannathan and Shettigar [8] discussed both the effects in hollow-fiber membrane modules, their results showed that the clearance of the solute was affected significantly by the ultrafiltration flux, the solute permeability in membrane and the concentration of dialysate. Abbas and Tyagi [9] corrected the axial velocity in Jagannathan's paper and assumed the ultrafiltration flux is constant along the length of a module. An analytical solution for the concentration profile in the duct was thus obtained. The design of the above hollow-fiber membrane module is analogous to the shell-and-tube heat exchanger in parallel-flow type, that is a bundle of hollow-fiber potted in a cylindrical holder, for the larger specific area per unit volume. However, the flow maldistribution exits in the parallel-flow type module, and the mass transfer areas were not large as expected [10,11]. The fraction of effective transfer areas was only about 20% in the countercurrent flow arrangement, as shown in the experimental study by Tanigaki et al. [12]. Moreover, Tharakan and Chau [13] thought that the parallel-flow type module has never a uniform distribution of ultrafiltration flux, and proposed the using of cross-flow systems in which the flow directions of both streams besides a membrane are orthogonal, therefore, the transmembrane pressure is almost uniform along the length of the fiber. The facts that at an equivalent Reynolds number, the mass transfer rate in cross-flow module is higher than in parallel-flow module, were proved by Yang and Cussler [14,15] and Wickramasinghe et al. [16]. For the operations of absorption or extraction in the membrane contactors, the mass transfer coefficient in cross-flow module is at least two times more than that in parallel-flow module [15]. It was also shown by Knops et al. [17] that in microfiltration and ultrafiltration at an equivalent Reynolds number, feed flowing outside and crossing the fibers (permeate flowing inside the fibers) gives higher permeate fluxes than conventional tangential filtration. The other type of cross-flow systems is the spiralwound membrane module. In this module, membranes are supported by the spacers, the feed and the permeate move across the membrane in orthogonal flow. The flows are disturbed by the spacers, and the mass

transfer coefficient is enhanced by this disturbance. Spiral-wound module is usually used in pressuredriven membrane processes, some mathematical models were available for the design of this module [18-20], but the analysis of coupled dialysis and ultrafiltration in a spiral-wound module is still lack in the literature. It is the purpose of present paper to develop an explicit analytical procedure for the design of crossflow/spiral-wound modules with the effects of coupled dialysis and ultrafiltration. This analytical method uses a perturbation technique to solve the mass conservation equations of the process, then the explicit analytical equations for solute concentration were derived.

2.

Theory

The cross-flow membrane modules may be classified into four types based on the membrane form and its arrangement: fiat-plate, hollow-fiber, radial hollowfiber and spiral-wound types. The cross-flow hollow-fiber type is shown in Fig. 1, in which there are spacers between the hollow-fibers for increasing disturbance and, therefore, the fluid is Q~

T !

lil!l li

Q~

s,ao°r/ hollow-fiber

ii i! fPl T

Q~ Fig. 1. Cross-flow hollow-fiber membrane module.

H.M. Yeh et al./Journal of Membrane Science 134 (1997) 151-162

QD~

"4

OBo

ODi

I 1

I

Q) 0

LJ.

!

i

I I I J/

I I ~) I I I O i.-~ .4- hI I I

"4 Ld--

• I I O s---t -r I I O i I -r-v

l-'t"

153

"4

r"l-~

"4"

L.L

~ I...t d. I I

"4

4-4-

~ l-4

"4" I-1-,

~ l'-r

porous tube

s

e Q

I

P I

I I I

Qao

I

:1

I

I

-4I I I I

I

T

I!

~

QBo

I

P

Qoo

I

,41.

I

I

I

I

~ I . . I . -II!

I

I

I

I I

I I

k F

Fig. 3. Spiral-woundmembranemodule.

I

iI

I

I

spacer~ Qoo hollow_fibe~/~ poroustube

l*

Qs,

spacer

flow passages I

Fig. 2. Radial cross-flow hollow-fiber membrane module.

Fig. 4. A simplified diagram for the theoretical analysis of crossflow membrane module.

more uniform between the fibers. Fig. 2 is the radial cross-flow hollow-fiber type, the feed is pressured into the porous tube and flows in the radial direction across the hollow-fibers. In spiral-wound module as shown in Fig. 3, which is made from the flat membranes, the feed flows through the module in the direction parallel to the central porous tube, while the dialysate flows around the central porous tube and reaches there. Since the thicknesses of the spacers and the membranes are very thin, their curvatures can be neglected. Therefore, this module may be treated as a cross-flow plate module, as shown in Fig. 4.

With the driving forces of concentration gradient (dialysis) or transmembrane pressure (ultrafiltration), the solute is transported from the feed side through the membrane to the dialysate side. The concentration of the solute is analyzed in following sections.

2.1. The governing equations 2.1.1. Cross-flow hollow-fiber module and spiral-wound module Consider the mass transfer in a controlled element of a cross-flow module as shown in Fig. 5. The spacers

H.M. Yeh et al./Journal of Membrane Science 134 (1997) 151-162

154 ~x

Integrating Eq. (4) with the use of the inlet conditions: FB(or QB/W) = FBi(or QBi/W) at x=0 and Fo(or Qo/L) = F19i(or Q19i/L) at y=0, one obtains

///////////// .

/11/11111111/~

//////////////2 IIIIIIIIII/~Z ,/I I I.......

~v o,

ay

F,

FB = (QBi/W) -

Zx

Vw dx

(5)

Vw dy

(6)

.(., / / / /

/

/

/ /

/ / X/

IIIIIIIIIIIAA ..... ~ ..... Fo

/~

\ ~'membrane

F19 = (Q19i/L) +

II

Fo

~0 y

spacers

Fig. 5. A controlled element for the mass transfer analysis of crossflow system.

with negligible mass transfer resistance are inserted between the membranes as supporters. The analysis is based on the following assumptions: (i) the overall mass transfer coefficient of solute is constant; (ii) two streams in retentate and permeate phases flow laminarly and orthogonally with each other; (iii) the fluids are Newtonian and their physical properties are fixed; and (iv) the concentration polarization and pressure drop on both compartments are neglected. The mass conservation equation for solute is

FBCBAyIx - FBCBAy]x+~ = K(CB - C19)A3cAy + VwOCBAxA y = FDC19Axly+exy- F19CoAXly

(1)

where FB and F19 are the flow rates per unit length in retentate phase and dialysate phase, respectively. K is the overall mass transfer coefficient of the solute, Vwis the ultrafiltration flux, and 0 is the membrane sieving coefficient for solute. Where 0=1 for not rejected by the membrane, 0<1 for partially rejected and 0=0 for completely rejected. The mass conservation equation for solution is

FBAylx - FBAylx+~x = V w A x A y (2)

= F19z~kXly+Ay -- F19z~-kXly

With the combination of Eqs. (1) and (2) and taking the limit as both Ax and Ay approach to zero, two partial differential equations for concentrations and flow rates are obtained

O(FBCB) _ K(CB - Co) + VwOCB - 0(F19C19)

Ox

Oy (3)

-

OFa Ox -

=

Vw

--

OFD Oy

(4)

Rearrangement of Eq. (3) with the use of Eqs. (4)-(6) results in two dimensionless equations for solute concentrations 04e a -1 a f~ CdE [(1 + ¢(0 - 1))(B - (19] 0¢

0(19 b Or/ - 1 + b f J ~bd~7[(1 + ~0)(B - (1 + ~)(D]

(7) (8)

which should be subjected to the following inlet conditions:

(9)

~8(~ = 0, '7) = 1

(10) where the dimensionless terms are defined as

CB Co LWK. ~ = L ; ~7=W; ~B=-C~Bi; ~°=~iBi; a = QBi , LWK Vw CDi b = QDi ' ~ = --g-' e CBi (11) 2.1.2. Radial cross-flow hollow-fiber module Similarly, the concentration equations for radial cross-flow hollow-fiber modules, as shown in Fig. 6, can be obtained by following the same procedure performed in Section 2.1.1 The results are exactly the same as those in Eqs. (7),(8),(9) and (11), except that ~ and ~7 in Eq. (11) are replaced by (r ~ - ~2R0~).

= (R02_ t~2R2),

z

~7= ~,

(12)

2.2. Concentration profiles of the solute The solutions of Eqs. (7) and (8) represent the solute distributions in the cross-flow type modules, in which the dimensionless terms a, b, and e are the operating parameters of the system, and the terms ¢ and 0 are effected by the rejection characteristics of solute and vary along the membrane module. The variation of 0 is very complicated. For simplicity,

155

H.M. Yeh et al./Journal of Membrane Science 134 (1997) 151-162

o~o

' 0000~

0 0 cO , ~ .

'00 ."6 0 .... ••

r..Ro

o% g

0 •

o

~8(~', r/') ---- (1 - a'~,~)~+0-1 + a(1 - a ~ ) ~ +O-1 spacer

porous tube

flow passages

Fig. 6. The cross-section of the hollow-fiber module with radial cross-flow pattern. therefore, a mean value 0 will be taken for it as a constant in this analysis [21]. The value of ~b is the function of operating conditions, such as applied pressure, solute concentration, flow rate, and membrane properties. However, a mean value ~ will be also represented for it as a constant in this work.

2.2.1. Case I: 0 = 1

( 1 - a¢( - , - )- - ¢0 (D((,~7)d( I , , fo ~ I -

(18)

Similarly, the solution of Eq. (17) associated with the boundary condition, Eq. (10), is 1 _

¢~((,~)=~(1+b6,7)

1

r/(

×

fo

_

_1_

1

* +b(1 +~0)(1 +b¢,7) +

1 l + b~')~(B(~', ~')d~'

(19)

Substitution of Eq. (19) into Eq. (18) yields

~B=I-- [ 1-(l-a~)~+0-1]

']

[

1--~(11 --~-b~/)~) 1~)(0- 1) ~ 1

As the solute is not rejected by the membrane, the sieving coefficient is equal to unity, and Eqs. (7) and (8) can be solved by the method of Laplace transformation with the use of boundary conditions, Eqs. (9) and (10). The solution is [22] ~B=l-(l-e)

The concentrations (B and ~o are functions of ~ and ~7. In Eq. (16), if~7 is treated as a constant ff and ~o as an unknown function of ~, then Eq. (16) is reduced to an ordinary differential equation associated with the boundary condition, Eq. (9), its solution for ~ is

x

~

(17)

o

o

"~

hollow-fiber/

b Or/ = 1 + b~rl [(1 + ~0)~B - (1 + ~)(D]

--- O ° ° o ° -

.'

o o

~---R

f0 e x p ( - ( ~ + ~ ) ) J 0 ( / 4 ~ ) c 1 ~ (13)

where

+ ab(l + ~0)(1 -

a~)~ ÷°-1(1 + b~vl-~ -1

+

x

,

(1 - a~() ~ (1 + b~}v')7~Bd(dr[ (2O)

Since Eq. (20) is rather complicated in calculation even with numerical method, it will be further solved by the method of perturbation. Introducing a new parameter A into the equation, one obtains

~B=l_[l_(l_a~b~)~+O_ll

= In(1 - a~b~)/~b

l

e(l+b~b~/) _ 1'~_' 1]

(14)

and

_

= (1 + ~) ln(1 + b(brl)/~b

(15)

_1_

1

+ Aab(1 + ~0-)(1 - a~b~)¢+°-l(1 + be'q) ¢ x for/fo~(1 - a~P)-~-°(1 ~ (21) 1 - + b~P)~Bd~'d~ 1

2.2.2. Case H: 0 # 1 After the substitution of the average values ~ and 0, Eqs. (7) and (8) can be rewritten as 0~n O~ -- 1 -

a a¢~

Expanding (~ in power series with respect to A, we have

CB = if(B0)(~, 7]) "1- )~(B1)(~, ~) + ~k2~(B2)(~,T]) +''' [(1 + ~ ( 0 -

1))(n - (D]

(16)

(22)

156

H.M. Yeh et al./Journal of Membrane Science 134 (1997) 151-162

Substituting Eq. (22) into Eq. (21) and comparing the coefficients of the same degree in A, results in a set of recurrence relation

((B°)=l-[1-(1-a~)'~ i

~B: E ~(n) -

n=0

+0-1] -

× 1-

or

e +1+@(0-1)

l 1

1+~(0-1)

~

n-1

tyn 1 -- E(li=0 ~- ~)i/3i 1 +~0 1+~(0-1)

x

(23)

J

m

{1 + ~ 0 ~ n

--n=0 ~ k - i - ' ~ )

×/3. 1 - ~ ( 1 + ~ ( o - 1//%

~

_

_l_

((1) = ab(1 + ~0)(1 - a~¢)¢+°-1(1 + b~b~)

i=0 E oo 1 + ~(o - 1) ~_~o(1 + ~0)%/3.

1

× fo~fo ~(1 - a~b~') ' _ _ l _0 o(1 + b~r/')~f(°)d('dr] 1

By introducing the properties of an and/3.:

(24) 1

~(0-1) (1 + (po)n a = (1 - a ~ ) l+f

=o \ Y ~ - )

~(") = ab(1 + ~0)(1 -- a~)~+0-1(1 + b~/) -~-1 × fo fo (1-a~b~') ¢ (l+bga/')¢((~-l)d~'dTf (25) Successive substitutions of above equations would find the general form of ~(~"). With the definition of following terms

(30)

(31)

"

.=o 1 + ~ ( 0 - 1)

/3n ----(1 + b~r]) l+~(o-1)

(32)

Eq. (30) may be reduced to

_~0-0) @:(1-a~)

o~ .-1

'+¢--EET,

n=0 m=O

man/3,n

(33)

where

(-1)" a. - n!

in(1 - a ~ )

(1 - a ~ ) ~ +°-1

'I+ ln(1 + b~/) 1.(1 + b~q) ll¢ /3,, = n.l

(26)

(1 + ~0)n(1 +

~3) m-n

- e(1 + ~ ( 0 - 1)) n-m-1 (1 + ~/~)m

(34)

(27)

2.3. Cup-mixing concentration and separation efficiency

((B") is expressed as

The cup-mixing concentration for the solute in retentated stream at outlet is defined as

" 1 - :_~o(i + ~)%

= \-i-V~./

~{nm =

e

(

-t 1 + ~ ( 0 - 1 )

in-1

1+~0

1+~(0-1)

x/3n 1 - E ( l + ~ ( 0 - 1 ) ) ' a i

i=0

)"

(Oo = f l F ~ ( ~ = 1,~)@(~ = 1,~)d~ fo1 F. (~ = 1, r/)dr/

.]

~(1 + ¢o)" ~./3~ 1 + ~}(0- 1)

(35)

Since the permeate flux is assumed to be uniform, FB (~ = 1, ~) = FBi -- I/wL, then (28)

(Bo =

As A=I, Eq. (21) is reduced to Eq. (20), therefore, the solution of Eq. (20) is

Let

fib -~--ffi0)(~, 7]) -~- (B(1)(~, ~]) q- ffi2) (~, ~) + ' ' "

6m =

(29)

/o 14B(~ =

f0

/3mdr/

1, ~7)d~

(36)

(37)

157

H.M. Yeh et al./Journal o f Membrane Science 134 (1997) 151-162

Inserting Eq. (27) into above equation yields 6m ~-

6m-l

b~l.l

--

ln(1 + b~)

(1 + b~b)-7

rate is zero, then (38)

and the initial term is 60 =

l[1-

+ b~)--~]

(1

(39)

Finally, the integral equation, Eq. (36), can be rewritten as ~(1 o)

_

n=0

(41)

QBi(CBi--Coi)(~+~)

There are two forms of separation efficiency defined by the following equations M

1 - (Bo 1 -- e

-

-

t

a~b(-Bo - 1- e

(42)

and w

=

M LWK(CBi

-

Col)

-

(46)

=

and the solute concentration in the retentate phase, Eq. (33), is reduced to oo n-1 a n b m n

~

~ (

r/~ exp(-(a~+br/)) (47)

M -- QBiCBi -- QBoCBo

QBi(CBi -- C o i )

1- e

'~nml~=0

m=0

The amount of mass transfer in the module is

X :

(45)

Z T , m6mC~n(~= 1) (40)

=

(44)

fl,~l~=0 = ~.Vbmr/~ exp(-b~7)

CBl~0 =-1 - ( 1 - e) ~

o¢ n - 1

~Bo = ( 1 - - a ¢ ) --i77- - Z

~.an~ n exp(--a~)

O~nl~3=0 =

1 - (Bo ~-~o -a(1 - e) t- 1 - e

This equation is similar to the temperature profile in the cross-flow heat exchanger analyzed by Nusselt [23]. 3.2. Exit cup-mixing concentration

The exit cup-mixing concentrations of solute in retentate phase are calculated by Eq. (40) and the results are plotted in Fig. 7 for various values of a and b. For the case of 0 = 1, the solute is not rejected by the membrane, and the exit concentration decreases when the retentate flow rate decreases (or a increases), as well as when the dialysate flow rate increases (or b

(43) 1.5 / b=oo

3. Results and discussion The parameters a and b are analogous to the NTU (number of transfer unit) in the heat exchanger. When the operating conditions are known, the terms c~,, "/,m, and 6m Can be calculated from the Eqs. (26), (34) and (38), and then, cup-mixing concentration and separation efficiency, X or w, can be determined, respectively, from Eqs. (40) and (42), or Eq. (43).

~

/

1.0

0.5

.

.

.

.

.

0=0.6

3.1. Analogous to heat transfer equation

When the systems possess only diffusive transport with the absence of convective transport through the membrane, such as artificial lung (gas/liquid contactor), artificial kidney (dialyzer), etc., the ultrafiltration

oo oo

,

I 0.2

,

l 04

,

I 06

, 08

a Fig. 7. Solute concentration in the outlet of retentate phase for ~ = 1 . 0 and e=0.

158

H.M. Yeh et al./Journal of Membrane Science 134 (1997) 151-162

decreases). On the other hand, when the flow rate of retentate is very large (or the value of a approaches zero), the residence time of solute in the module is too short for solute to transfer through the membrane, and the exit concentration is almost equal to the inlet concentration. For the case of 0 < 1, the solute is partially rejected by the membrane, while the solvent is removed freely by the effect of ultrafiltration, and the exit solute concentration in retantate phase is higher than that in the case of 0 = 1. Further, the exit concentration is greater than the inlet concentration when flow rate of dialysate is small (or the value of b is large).

(a)

1.0

~=O2|b/~=O =~.o

-

0.8

....

0=0.6

0.6

Z 0.4

0.2

-t

3.3. Separation efficiency

0.0 0.5

,

I

,

I

0.7

,

I

0.9

,

I

1.1

,

1.3

] 1.5

(0 Fig. 8(a) and (b) are plotted for X vs. w under various values of b/a and sieving coefficient for a ~ = 0 . 2 and 0.5, respectively. The term b/a is the ratio of the flow rate in retentate phase to that in dialysate phase (QBi/QDi), and the term a ~ is the ratio of volumetric ultrafiltration rate to the inlet flow rate in retantate phase (Qw/QBi). Fig. 8(a) and (b) are the illustrations showing the relations between two different definitions of separation efficiency. If more parameters are concerned, these figures may be used for design purpose. For instance, consider the case: a ~ = 0 . 2 , bla=l.O, X=0.5 and 0 - 1 . 0 , the value of read in Fig. 8(a) is 0.88, and then the required area for mass transfer can be calculated from Eq. (43). The separation efficiency, a;, vs. number of transfer units, a, is plotted in Fig. 9 with b as parameter. The mass transfer rate of solute increases as the inlet flow rate in dialysate phase (Qoi) increases (or b decreases) under the fixed mass transfer areas. Moreover, the effect of the inlet flow rate in retentate phase (QBi) o n mass transfer rate is more noticeable. For the case of = 1, as well as most of the case of 0 < 1 with larger Qoi (or smaller b), the mass transfer rate increases as QBiincreases (or a decreases). This is because that the increase in QBi enhances the effect of dialysis. This enhancement in dialysis is due to the nearly indeclinable concentration of solute in retentate phase maintained by the high flow rate. For the particular case of < 1 with smaller Qoi (or larger b), the value of ~ is small but increases to approach unity as a increases (or QBidecreases). In this case, the low mass transfer rate of solute is due to the partial rejection of solute by the

(b)

08r,j 0.6 - -

Z

i

/

....

.....

b/a = =

__

-

6 . . . . . . . . . . . . . . . . . . . .

0.4

a ~ =t25 0.2

00

-

, 0.5

-

0=I.0

.....

0=0.6

I 0.7

,

I

,

0.9

I 1.1

,

I 1.3

, 1.5

O3 Fig. 8. T h e s e p a r a t i o n efficiency, X, as a f u n c t i o n o f w for o p e r a t i n g conditions: (a) a ~ = 0 . 2 a n d e = 0 ; (b) a ~ = 0 . 5 and e = 0 .

effect of membrane ultrafiltration, and then the increment is due to the enhancement of the effect of dialysis because of increasing concentration gradient across the membrane, which is created by partial rejection and accumulation of solute in retentate phase of low flow rate, as shown in Fig. 7. The influence of membrane area on mass transfer rate is shown in Fig. 10, which was plotted for X vs. a under the conditions of constant flow rate. It is seen that the mass transfer rate increases as the membrane area increases.

H.M. Yeh et al./Journal of Membrane Science 134 (1997) 151-162 2.5

159

1.0

b/a= l.O

L

--

0=1.0

. . . . .

, = 0 . 6

~f~,;

0.8 2.0

-

/

• # •

0.6

0)

Z

1.5

0.4

1.0

. . . . . . . .

~- -

-~,-

. . . . . . . . . .

-

0.2

o.s

,

0.0

I

,

0.2

I

,

0.4

a

I

,

0.6

I

,

0.8

1.0

Fig. 9. The separation efficiency, w, as a function of the number of transfer units, a, for operating conditions: ~ = 1 . 0 and e = 0 .

1.0

oo

0.0

,

I

0.2

a=0.2 =0.1 I

,

,

0.4

a~

I

,

0.6

I

,

0.8

1.0

Fig. 11. The separation efficiency, X, at different ultrafiltration rates for operating concditions: bla=l.O and e=0.

tion effect a ~ = 0.2, the value of X read in Fig. 11 is 0.274, this value is about three times the value without the effect of ultrafiltration (X¢=_o = 0.091). It is evident that for higher value of a ~ , the improvement in separation efficiency X is more significant.

o= 1.o

S.;6//,

0.8

l - ~ ~

0.6

%

600 0.4

b/a = L O 500

0 = 1.0

~=0.6

0.2

j

400

0.0 0.0

,

I 02

,

I 04

,

a

I 06

,

I 08

,

10

Fig. 10. The separation efficiency, X, as a function of the number of transfer units, a, for operating conditions: ~ = 1 . 0 and e = 0 .

E (%)30o

~

j

s

200

3.4. Enhancementof separation efficiency by ultrafiltration 0.0

The effect o f ultrafiltration rates on separation efficiency X is shown in Fig. 11. The mass transfer rate increases as the ultrafiltration rate increases. For the case, e ---- 0, 0 ---- 1.0, and the degree o f ultrafiltra-

0.1

0.2

0.3

0.4

0.5

aC~ Fig. 12. The enhancement in separation efficiency, E, by the effect of ultrafiltration for operation condition: b/a= 1.0 and e=O.

H.M. Yeh et al./Journal of Membrane Science 134 (1997) 151-162

160

The enhancement of separation efficiency with ultrafiltration is best illustrated by calculating the percentage increase in separation based on that without ultrafiltration E -

X X -r : 0

Xr:0

_

w -

~vr= 0 _

M

~vr:0

-

Mr:

0

(48)

M~:0

This enhancement E is plotted vs. ultrafiltration rate as shown in Fig. 12. The improvement in separation efficiency by the effect of ultrafiltration is significant, especially for the systems with low mass transfer coefficient.

Table 1 Calculated results of numericalexample

Qni QDi O.w o2 (ml/min) (ml/min) (ml/min) (-)

X (-)

For urea, 0=1.0, K=4.342× 10 -6 m]s 200 500 0 0.4154 200 500 20 0.4332 200 500 40 0.4552 200 500 80 0.4958 200 500 160 0.5549 200 1000 0 0.4200 200 1000 20 0.4590 200 1000 40 0.4780 200 1000 80 0.5121 200 1000 160 0.5582 400 500 0 0.5412 400 500 40 0.6041 400 500 80 0.6732 400 500 160 0.8095 400 500 320 1.0550

0.7358 0.7674 0.8064 0.8783 0.9830 0.7441 0.8130 0.8468 0.9073 0.9888 0.4793 0.5351 0.5962 0.7170 0.9345

For inulin, 0-0.61, K=6.050x 10 -7 m/s 200 500 0 0.8605 200 500 20 1.0777 200 500 40 1.3162 200 500 80 1.8274 200 500 160 3.0721 200 1000 0 0.8288 200 1000 20 1.0993 200 1000 40 1.3405 200 1000 80 1.8553 200 1000 160 3.0971 400 500 0 0.9110 400 500 40 1.3718 400 500 80 1.8673 400 500 160 2.9393 400 500 320 5.6332

0.2124 0.2660 0.3249 0.4511 0.7583 0.2046 0.2714 0.3309 0.4580 0.7645 0.1124 0.1693 0.2305 0.3628 0.6952

E (%)

4.29 9.59 19.37 33.60 9.26 13.80 21.93 32.89 11.64 24.39 49.59 94.97

25.24 52.97 112.38 257.02 32.65 61.73 123.85 273.66 50.62 105.07 222.78 518.51

4. Numerical example For the purpose of illustration, let us employ the experimental system of Kunitomo et al. [24] and Sakai and Mineshima [25]. In the experiment, they removed urea and inulin from blood by the filtryzer B-1-M. This hollow-fiber module has an effective membrane area of 1.36 m 2 and sieving coefficients of 1.0 and 0.61 for urea and inulin, respectively. The overall mass transfer coefficients for urea and inulin in this module system are 4 . 3 4 2 x 1 0 -6 and 6 . 0 5 0 x 1 0 - 7 m / s , respectively. As the membrane module is designed in a cross-flow type for removal of urea and inulin from blood, the enhancement E in separation efficiency with the effect of ultra filtration will be estimated by present theory. For the operating conditions: Coi=O (or e=0), Q B i : 2 0 0 - 4 0 0 ml min 1, and Q o i : 5 0 0 1000 ml min -1, the computed results, under various mean volumetric ultrafiltration rates (Qw) or LW~'w are listed in Table 1. It is seen that under these operating conditions the enhancement increases as the ultrafiltration rate increases, especially for low sieving coefficient. The enhancement increases also when QBi or aDi increases.

5. Conclusions The separation efficiency of dialysis process coupled with the effect of ultrafiltration in cross-flow membrane modules was analyzed thoroughly. The partial differential equations for solute concentration distributions in retentate and dialysate phases were derived based on mass balances, and then the concentration distributions were solved by the method of perturbation with the consideration of uniform permeate flux. After obtaining the outlet concentrations, two separation efficiencies were defined and calculated under various values of parameters, such as sieving coefficient (0), ultrafiltration flux (~'w) and volumetric flow rates (Qni and Qoi). The enhancement in separation efficiency of a dialysis process in cross-flow membrane modules can be substantially achieved by the effect of ultrafiltration. It was found that the enhancement is significantly increased with increasing ultrafiltration rate, especially for the systems with low mass transfer coefficient. Furthermore, increasing the flow rate in retentate phase is more

H.M. Yeh et al./Journal of Membrane Science 134 (1997) 151-162

beneficial to the mass transfer than increasing in dialysate phase. It is believed that the present analysis may be also applied to all kinds of configurations, such as parallel-flow, radial-flow and spiral-wound types. The ultrafiltration flux which is actually the function of operating conditions, such as solute concentration, fluid velocity and transmembrane pressure, and thus varies every place in the system of interest, was treated as a mean value in present study. The analysis with nonuniform ultrafiltration flux and the optimum design of practical cross-flow modules will be the further works.

A X

161

ratio of radius of inner porous tube to that of outer porous tube in radial crossflow pattern coefficient in perturbation method dimensionless coordinate, x/L, or z/L separation efficiency defined by Eq. (42) dimensionless ultrafiltration flux, Vw/K separation efficiency defined by Eq. (43)

6.2. Subscrip~ B D i o

retentate phase dialysate phase inlet outlet

6. List of symbols

C

FB Fo K L M Q

aw r

Ro

Vw W x,y,z

numbers of transfer unit in retentate phase, LWK/QBi numbers of transfer unit in dialysate phase, LWK/Qoi solute concentration (kg/m 3) flow rate in retentate phase per unit width of membrane (m3/s m) flow rate in dialysate phase per unit length of membrane (m3/s m) overall mass transfer coefficient (m/s) membrane length (m) mass transfer rate (kg/s) volumetric flow rate (m3/s) ultrafiltration rate, LWVw (m3/s) radial coordinate (m) radius of outer porous tube in radial cross-flow pattern (m) ultrafiltration flux (m/s) membrane width (m) coordinate (m)

6.1. Greek symbols OLn,~m,'ff nm C

expansion array for solute concentration in retentate phase ratio of inlet solute concentration in dialysate phase to that in retentate phase, CDi/CBi dimensionless solute concentration, CICBi dimensionless coordinate, y/W sieving coefficient

6.3. Superscripts -

mean

Acknowledgements The authors wish to express their thanks to the National Science Council for financial aid.

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