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Papers

Osmotic-pressure model with permeability analysis for ultrafiltration in hollow-fiber membrane modules

Ho-Miig Yeh

Department of Chemical Engineering, Tamkang University, Tamsui, Taiwan, ROC

Tung-Wen Cheng

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan, ROC

The effects of operating conditions on the permeate flux for ultrafiltration of aqueous solutions of Dextran and Polyvinylpyrrolidone in hollow-fiber membrane modules have been investigated experimen- tally based on the osmotic-pressure model with the permeability analysis. The correlation equations for pure-water permeability after solute adsorption and fouling on the membrane have been obtained as functions of the transmembrane pressure and solution concentration. The theoretic prediction of permeate fluxes agrees well in tendency with the experimental data.

Keywords: ultrafiltration; hollow fiber; osmotic-pressure model; permeability

Introduction

Ultrafiltration of macromolecular solutions has become an increasingly important separation process. On an industrial scale, the following applications have proved to be economically attractive and useful’: electrocoat paint recovery, latex recovery, polyvinyl alcohol re- covery, oil-water separations, and miscellaneous phar- maceutical and biologic separations.

The rapid development of this process was made possible by the advent of anisotropic, high-flux mem- brane capable of distinguishing among molecular and colloidal species in 10-A to IO-pm size range. One of the common ultrafiltration designs is the hollow-fiber membrane module in which the membrane is formed on the inside of tiny polymer cylinders that are then bundled and potted into a tube-and-shell arrangement.

Because this process is a pressure-driven membrane separation process, the pressure applied to the working fluid provided the driving potential to force the solvent to flow through the membrane. Typical driving pres- sure for ultrafiltration systems are in the range of 10 to 100 psi. However, beyond a certain value of applied

Address reprint requests to Dr. H.-M. Yeh at the Department of Chemical Engineering, Tamkang University, Tamsui, Taiwan, ROC. Received 6 July 1992; accepted 15 September 1992

pressure, the membrane permeation rate is limited by the presence of a concentration polarization layer, which increases the effective membrane thickness and so reduces its hydraulic permeability. Furthermore, concentration solutions of macromolecules have quite appreciable osmotic pressure.’ At the high concentra- tions found in ultrafiltration polarization layers, the osmotic pressure even can be the same order of magni- tude as the applied pressures generally used in ultrafil- tration.3 The permeate flux of ultrafiltration of macro- molecular solution is usually analyzed by the gel polarization mode1,4-i’ osmotic-pressure mode1,3J2-19 or resistance-in-series mode1.19~20

The influence of resistance caused by solute adsorp- tion at the membrane on permeate flux during ultrafil- tration of macromolecular solution has been investi- gated experimentally by Nabetani et al., with the use of a batch-type ultrafiltration celLi In their work, mem- branes were first soaked in solutions of various concen- trations. After reaching solute adsorption equilibrium, permeate fluxes for pure water were then measured. They found that soaking in more concentrated solution resulted in lower values of permeate flux and then adopted these batch-type adsorption data for the cross- flow plate-and-frame-type ultraliltration system. So, they neglected the effect of applied pressure on the

0 1993 Butterworth-Heinemann Sep. Technol., 1993, vol. 3, April 91

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Osmotic-pressure model: H.-M. Yeh and T.-W. Cheng

solute adsorption or the permeability of the mem- solution, there is no osmotic pressure present, and branes. Equation 1 reduces to

In practical operations, the resistance caused by sol- ute adsorption and fouling is not only dependent of the solution concentration, but is also dependent on the operating condition such as applied pressure. There- fore, in this study, we correlated the pure-water perme- ability with parameters of solution concentration and applied pressure in a tube-and-shell-type hollow-fiber ultrafiltration system. Furthermore, we investigated the influence of resistances caused by membrane itself, and by solute adsorption and fouling at the membrane on permeate flux during ultrafiltration of macromolecu- lar solution based on the osmotic-pressure model.

J, = L,AP, or L, = J,,,/AP (4)

in which J,,, is the permeate flux of pure-water ultrafil- tration. Actually, pure-water permeability of a fresh membrane before solute adsorption and fouling is re- lated with the intrinsic resistance of a membrane R, as

L, =& (5)

Nabetali et al.19 pointed out that during ultrafiltra- tion operation, the resistance caused by solute adsorp- tion and fouling on the membrane Rfmust be taken into consideration, and Equation 1 should be modified as

Osmotic-pressure model

Kedem and Katchalsky” gave the following expression for estimating permeate flux of ultrafiltration J,.

J, = L,(AP - ANT) (1)

in which AP and AT are transmembrane pressure and osmotic pressure difference across the membrane, and are determined by

Ap=p+p*_p 2 3 (2)

and

A?r = rr(C,) - m(C,) s r(C,,J (3)

L, is the solvent permeability of a fresh membrane. According to Equation 1, the effective pressure is re- duced as the osmotic pressure of the retentate in- creases, and will lead to declined permeate flux. When ultrafiltration is carried out for pure water instead of

J, = L;[AP - r(C,)] = & [AP - dC,)l (6) m f

In Equation 6, LL denotes solvent permeability of a used membrane after solute adsorption and fouling. When ultrafiltration is carried out for pure water with the use of a membrane that was employed in prior for ultrafiltration of solutions, Equation 6 reduces to

J,‘,, = LLAP, or LI, = JI.IAP (7)

Equation 7 can also be rewritten from Equation 4 with J,,. and Lp replaced by J,:. and Li, respectively. Both R,,, and Rf as well as L, and Li can be determined by experimental results.

The relation between osmotic pressure n(C,,) and the concentration of a macromolecular solution on the membrane surface C, can be represented as*

r(C,,) = A,C,, + A&f, + A,C; (8)

Notation

A,, A,, A3 constants defined in Equation 8 a, b, c constants defined in Equation 12 C solute concentration, wt% c solute concentration, g - mL_’

bulk solute concentration, wt% Cb

G! solute concentration on membrane surface, wt%

CP permeate solute concentration, wt% D diffusion coefficient, m* * s-’

Jv volume permeate flux for solution ultrafiltration, m3 * m-* . s-’ volume permeate flux for pure-water JW ultrafiltration before solute adsorption and fouling, m3 - me2 * s-’

JI, volume permeate flux for pure-water ultrafiltration after solute adsorption and fouling, m3 * m-* - SC’

k mass-transfer coefficient, m * s-’ L length of hollow fiber, m

4 pure-water permeability before solute adsorption and fouling, m3 * Pa-’ . m-2 . s-1

LI, pure-water permeability after solute adsorption and fouling, m3 . Pa-’ * m-2 . s-I

PI 7 p2 inlet, outlet pressure of the tubeside, Pa P3 permeate pressure of the shellside, Pa AP transmembrane pressure defined by

Equation 2, Pa

Rf resistance owing to solute adsorption and fouling, Pa . m* * s - mm3

R, intrinsic resistance of membrane, Pa * m2 . s . m-3

rrn radius of hollow-fiber, m u feed flow velocity, m * se1

Greek letters

7T osmotic pressure, Pa

92 Sep. Technol., 1993, vol. 3, April

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Osmotic-pressure model: H.-M. Yeh and T.-W. Cheng

Where A,, AZ, and A, are constants and can be deter- mined by experimental data.

In membrane ultrafiltration processes, solutes re- jected by the membrane accumulate on the membrane surface and form a concentration polarization layer there. At steady state, the quantity of solutes conveyed by the solvent to the membrane is equal to those that diffuse back. Accordingly, a material balance for sol- utes within the concentration boundary layer results in4S

C J A=expL! Cb k

where the mass-transfer coefficient k is given by the Leveque equation

In Equation 10, u denotes the flow velocity of feed solution, D is the diffusivity of solute in solvent, and T,,, and L are the radius and length of hollow fibers, respectively. For theoretic estimation, J, and C,,, may be solved from Equations 6 and 9 simultaneously.

Experiment

Apparatus and materials

The flow diagram of ultrafiltration apparatus is shown in Figure 1. Amicon model HlP30-20 hollow-fiber car- tridges (Amicon Co., Danvers, MA) were employed for membrane ultrafiltration. The hollow fibers (I.D. 5 x 10m4 m; effective length 0.153 m) are made of polysulfone, and the total effective membrane area in a cartridge is 0.06 m2.

The aqueous solutions of Dextran T500 (Pharmacia, Uppsala, Sweden, A4,, = 170,300) and Polyvinylpyrroli- done (PVP)-360 (Sigma Co., M, = 360,000) were used as tested solutions for ultrafiltration. Dextran T500 and PVP-360 were more than 99% retained by the mem- brane used. The solvent was ion-exchange pure water.

The feed solution was circulated by high-pressure pump with variable speed motor (L-07553-20, Cole- Parmer Co., Chicago, IL) and feed flow was measured with flow meter (L-03217-34, Cole-Parmeter Co.). The measurement of pressure was detected by pressure transmitter (model 891.14.525, Wika).

Experimental conditions and procedure

The concentrations of feed solution were 0.1, 0.2, 0.5, 1.0 and 2.0 wt% for both Dextran T500 and PVP-360 systems. The feed-flow velocities were 0.051, 0.102, 0 204, and 0.306 m/s, and the feed inlet pressures were 30, 50, 70, 100, and 140 kPa. In all experiments, feed solution temperature was kept at 25°C by a thermostat. During an experimental run, both permeate and reten- tate were recycled back to the feed tank to keep feed concentration constant.

At the beginning of experiment, a fresh hollow-fiber cartridge was employed for determination of the intrin-

1 . feed tank 6 . flow meter 2. pump 7 . permeate

3 . pressure gauge 8 . cqllector

4 . hollow fiber module 9, stirrer

5 . pressure control valve 10. thermostat

Figure 1 Flow diagram of experimental apparatus.

sic resistance of membrane R, as well as L,. Thus, permeate fluxes of ultrafiltration for feed water J, were measured under various transmembrane pressure AP and feed flow velocity u. The experimental data were presented in Tables la and 2a.

Then, the feed water was replaced with the tested solution. Until steady state had been reached, perme- ate fluxes of ultrafiltration for feed solution J, were measured under each operating condition. Values of permeate flux reached steady state within 30 to 120 minutes. These experimental data are plotted in Fig- ures 2-5 for Dextran T500 solutions and in Figures 6-9 for PVP-360 solutions.

After each feed solution run, which was generally over 2 days, the solution was replaced back with pure water, and the membrane cartridges used were em- ployed for pure-water ultrafiltration to determine pure- water permeabilities LI, through the membrane after solute adsorption and fouling. Permeate fluxes JL , thus obtained, were presented in Tables lb and 2b.

Results and discussion

Dextran T500 system

It was found from the experimental data that a straight line of l/J, vs. IlAP can be constructed by the least- square method. The value of Lp (or l/R,) for the mem- branes employed in ultrafiltration of Dextran T500 sys- tem was determined in Figure 10 based on Equation 4, with the use of experimental data listed in Table I. The result is

Lp = 4.14 x lO-‘O m3 * Pa-’ * me2 * se1 (11)

Sep. Technol., 1993, vol. 3, April 93

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Osmotic-pressure model: H.-M. Yeh and T.-W. Cheng

Table 1 Experimental data of pure water permeate flux for pure-water ultrafiltration, Dextran T500 system. (a) Before solute adsorption and fouling; (b) After solute adsorption and fouling

A

u m . s-l

AP x 1O-5 Pa

J, x IO6 m3. m-2 . s-1

0.102 0.095 3.83 0.102 0.248 i 0.08 0.102 0.455 18.36 0.102 0.655 25.97 0.102 0.958 37.45 0.102 1.355 51.67 0.051 0.664 25.57 0.204 0.643 25.10 0.306 0.626 24.72

B

Cb = 0.05 wt% Cb = 0.1 w-t% Cb = 0.5 w-t% Cb = 1.0 wt%

u m . s-l

BP x 1O-5 J;, x IO6 AP x 1O-5 J:, x lo6 AP x 1O-5 m3. m-2. s-1 Pa m3. m-2. s-l

J:, x lo6 Pa Pa m3 . m-2 . s-1

AP x 1O-5 Pa

J&x lo6 m3 . m-2 . s-1

0.102 0.253 a.37 0.249 7.80 0.250 7.28 0.250 5.87 0.102 0.452 13.80 0.449 i 2.87 0.452 11.37 0.449 a.97 0.102 0.654 19.17 0.650 16.84 0.652 14.83 0.652 ii.58 0.102 0.955 26.41 0.951 21.78 0.954 18.62 0.953 15.30 0.102 i .35a 35.91 i .348 28.42 1.351 23.83 1.351 17.46 0.051 0.660 19.45 0.662 16.35 0.660 18.46 0.655 lo.88 0.204 0.641 21.04 0.635 16.91 0.636 17.57 0.632 11.60 0.306 0.623 18.60 0.613 16.66 0.611 17.00 0.609 11.26

Table 2 Experimental data of pure-water permeate flux for pure-water ultrafiltration, PVP-360 system. (a) Before solute adsorption and fouling; (b) After solute adsorption and fouling

A

u m . s-l

AP x 1O-5 Pa

J, x lo6 m3 . m-2 . s-1

0.102 0.106 14.51 0.102 0.257 35.19 0.102 0.527 69.64 0.102 0.723 92.73 0.102 1.019 126.37 0.102 1.414 167.94

B

Cb = 0.1 wt% Cb = 0.2 wt% Cb = 0.5 wt% Cb = 1.0 wt%

u m - s-l

AP x 1O-5 J:, x lo6 AP x 1O-5 m3 . m-2 . s-1

J:, x lo6 AP x 1O-5 m3 . m-2 . s-1

J:, x IO6

Pa Pa Pa m3. m-2 . s-1 AP x 1O-5 J:, x lo6

Pa m3 . m-2 . s-1

0.102 0.102 0.102 0.102 0.102 0.051 0.204 0.306

0.264 13.66 0.464 23.38 0.664 32.75 0.964 46.55

0.665 35.01 0.658 34.88 0.649 34.67

0.260 ii.89 0.266 11.59 0.263 9.44 0.459 19.47 0.466 18.65 0.466 13.10 0.660 26.16 0.667 24.63 0.666 15.60 0.957 35.65 0.966 32.47 0.968 18.46 1.356 47.21 1.364 42.35 1.367 21.50 0.662 25.73 0.664 23.44 0.669 14.16 0.649 26.82 0.651 24.89 0.658 17.71 0.644 26.91 0.643 25.43 0.646 19.05

94 Sep. Technol., 1993, vol. 3, April

Page 5: Osmotic-pressure model with permeability analysis for ultrafiltration in hollow-fiber membrane modules

A

‘: E u-l

2 V

‘T 0

X

3

i

0.6

0.4

0.2

OL ’ 1 I I I I 0 1 2 3 4 5

l/AP x105 (Pa-‘)

Figure2 Experimental and theoretic values of permeatefluxfor ultrafiltration of Dextran T500 system: u = 0.051 m . s-l.

cb(Wt %)

0 0.05 a 0.1 . 0.5 A 1.0

~

average slope = b = 0.55

I I I

10.0 10.5 11.0 11.5 12.0

It7 AP

Figure3 \ Experimental and theoretic values of permeatefluxfor ultrafiltration of Dextran T500 system: u = 0.102 m . s-l.

The values of LL , determined by Equation 7 with the use of data in Table 1, are shown in Table 3. It is seen from Table 3 that Li varies with the change of trans- membrane pressure as well as the bulk concentration of the solution, which was ultraflltrated before in the hollow-fiber cartridge used but is nearly independent of the flow velocity. Therefore, we assume the correlate equation of LI, as

LL = Li exp[-a(AP)bC;] (12)

Osmotic-pressure model: H.-M. Yeh and T.-W. Cheng

Based on Equation 12, the values of a, b, and c could be determined by the graphic method. The procedures were shown in Figures II and 12 with the use of the data given in Table 3 for u = 0.102 m/s, and the result is

L; = 4.14 x lo-” exp[- 1.66 x 10-3(AP)oW~31]

(13)

The equation for estimating the osmotic pressure of aqueous solution of Dextran TSOO at 25°C was given by Vink** as

-6.0 I-

n : d

a -6.5

s

a

2 -7.0

-I

Y V I -7.5

\ c -

-8.0 -4

intercept = Ln a =-6.40

a =1.66~10-~

I I I I

-3 -2 -1 0 1

In Cb

Figure 4 Experimental and theoretic values of permeate flux for ultrafiltration of Dextran T500 system: u = 0.204 m . s-l.

10

u=O.O51 ms-’

Cb(wt%) Experimental Theoretical a- 0.1

0.2 a0 -- h

P:Z 0 --- 7-

rn A - --__ 7 2.0 n _________

E 6- 0 z 0 0

“0 4- 0 - X

-: / n, c -_ _z _, _‘I -?- -1

2- 0 c-- I._:_ _&__________-c---- __&____---

_ /- I-_-b____-,‘---- . I n

.

o- .D

I I I 0.4 0.8 1.2

aPx10m5(pa)

1.6

Figure 5 Experimental and theoretic values of permeate flux for ultrafiltration of Dextran T500 system: u = 0.306 m . s-J.

Sep. Technol., 1993, vol. 3, April 95

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Osmotic-pressure model: H.-M. Yeh and T.-W. Cheng

1 u=O.l02ms-’

, o _ C&t%) Experimental Theoretical

0.1 0 0.2

$ __ - -

7 ?.S

-__

ul 6- A -____

Y 2.0 n ________

E - 0

“E s- 0

V 0 A

a--

_I _x _r- _~,~~_-~ *_________*-------

m n

o! I I I

-I

‘0 0.4 0.8 1.2 1.6

AP ~10-~( Pa)

Figure 6 Experimental and theoretic values of permeate flux for ultrafiltration of PVP-360 system: u = 0.051 m . s-l.

u=0,204ms-’

C&wt%) Experimental Theoretical

0.1 0 0.2 A -- 0.5 : --- li.: ----_

m _.____--_

0 0

0 h

_F_ / C- _____*-_--- ________*-_ ,‘4 ,__.t”----li--‘- . .a-- . 8

n

‘..

- _-

-em-

0.4 0.8 1.2 1.6

APx10-5(Pa)

Figure 7 Experimental and theoretic values of permeate flux for ultrafiltration of PVP-360 system: u = 0.102 m . s-l.

n- x 1O-5 = 0.0655~,,, + 10.38c, + 75.3c;3, (14)

the diffusion data of Dextran T500 was suggested by Wijmans et a1.23 as

D x 10” = 1.204 + 26.14c, - 41.67c; + 21.32c;

(15)

The unit of concentrations c in Equations 14 and 15 is g - mL-‘.

The relation between c and C for Dextran TSOO

solution at 25°C has been obtained experimentally in the present work. The relation is nearly linear as

c = O.OllC (16)

Substitution of Equation 16 into Equations 14 and 15 results in

7~ x 1O-5 = 7.205 x 10-4C, + 1.256

x 10-Y; + 1.002 x 10-v; (17)

1

1

- 7-

ul N

‘E "E

V

"0

X -:

5-

2-

Q-

6-

3-

o- 0

U=0.306ms-’

Cb(wt”lo) Experimental Theoretical

-_ ___ -- ___

2.0 n .-______ 0

0

0.4 0.8 1.2 1.6

wxi0-5(pa)

Figure 6 Experimental and theoretic values of permeate flux for ultrafiltration of PVP-360 system: u = 0.204 m . s-l.

u ~0.051 ms-’

C,( wt%) Experimental Theoretical

- 0.1

::52

0

2 -- --- 1.0 A -_-- 2.0 l -____-

0 u

v-- A-

_ /“, HA- h - “-I,___ / l / c 9 - -,‘_,_L.3__________-~-..-

- , y--+---. n n

..= / . * I I I

0.4 0.8 1.2 1

APx10-5(Pa)

5

Figure 9 Experimental and theoretic values of permeate flux for ultrafiltration of PVP-360 system: u = 0.306 m . s-‘.

96 Sep. Technol., 1993, vol. 3, April

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Osmotic-pressure model: H.-M. Yeh and T.-W. Cheng

12

10

u &IO2 mr’

_ cb(wt%) Experimental Theoretical

0.1 0

0.2 0.5 $ -- ---

A - --- n _____ _

--

l ’ /

0 I I 1

0 0.4 0.8 1.2 1

APx10-5(Pa)

.6

Figure 10 Relation between l/J, and IIAP for Dextran T500 system.

/

and

D x 10” = 1.204 + 2.875 x IO-‘Ch - 5.042

x 10-3C; + 2.838 x lO-%Z; (18)

Finally, the permeate fluxes of ultrafiltration J, for Dextran T500 solutions in the present study may be predicted from Equation 6 with the use of Equations 9, 10, 13, 17, and 18. The theoretic values of permeate flux were also plotted in Figures 2-5 for comparison.

PVP-360 system

By following the same procedure as that performed in the preceding section for Dextran TSOO system, the permeability equation for PVP-360 system that was obtained from Table 2 and the physical properties of PVP-360 in aqueous solution22.24 are

Cb(wt%) Experimental Theoretical

16- 0.1 0.2

0

; --

0.5 --- 1.0 A _---

12- 2.0 n _______

0 O 0

-1 OAA A A

l -

G& k- _-- _____~ .-me--

. / /..--

_.I--

/ --

. . _____, ----a

O,L--------- 0.4 0.8 1.2 1

A PX 1 o-5(pa)

I.1

Figure 11 Relation between LLIL, and AP for Dextran T500

2ot

“=0.204ms’

L; = 13.94 x 10-t’ exp[-0.2837(AP)0W~20] (19)

r x lo-’ = 2.343 x 10-3C, + 1.967

x 10-3C2 + 2 209 x 10-4C; (20) m ’

D x 10” = 4.25 + 4.96 tanh(O.l6C, - 0.507) (21)

Similarly, the permeate fluxes of ultrafiltration J, for PVP-360 solutions in present study may be predicted from Equation 6 with the use of Equations 9, 10, 19, 20, and 2 1. The theoretic values of permeate flux were also plotted in Figures 6-9 for comparison.

Conclusion

The effects of transmembrane pressure, flow velocity, and feed concentration on the permeate flux for ultra- filtration of aqueous solutions of Dextran TSOO and

Table 3 Pure-water permeability for Dextran T500 system

Cb = 0.05 wt% Cb = 0.1 W/a Cb = 0.5 wt% Cb = l.Owt%

” AP x lO-5

m's-'

L; x 10" APx 1O-5 APx 1O-5 AP x 1O-5 Pa m3.pa-' .m-2.s-1

L; x 10"

Pa &.pa-'. m-2.s-l L; x 10" L; x 10"

Pa &. pa-1 .m-2.s-' Pa m3.pa-'.m-2.s-'

0.102 0.253 33.16 0.249 31.42 0.250 29.21 0.250 23.54

0.102 0.452 30.56 0.449 26.69 0.452 25.15 0.449 20.00

0.102 0.654 29.32 0.650 25.94 0.652 22.76 0.652 17.77

0.102 0.955 27.65 0.951 22.90 0.954 19.53 0.953 16.07

0.102 1.356 26.46 1.346 21.09 1.351 17.64 1.351 12.93

0.051 0.660 29.50 0.662 24.72 0.660 27.99 0.655 16.63

0.204 0.641 32.83 0.635 26.66 0.636 27.63 0.632 16.36

0.306 0.623 29.69 0.613 27.22 0.611 27.62 0.603 16.50

L~.aftersolute adsorption and fouling. Lp, before solute adsorption and fouling.41.40 x lo-" rn3. Pa-' me'. s-‘.

Sep. Technol., 1993, vol. 3, April 97

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Osmotic-pressure model: H.-M. Yeh and T.-W. Cheng

h

‘rn ‘;” _E

t

u=0.306ms-’

25 Cb(wt%) Experimental Theoretical

-- --- --_- ___-___

I I

E 15- 0 0

0 0 (D 0

;; lo- A A A

0 A -:

L OO

I 1 , I 0.4 0.8 1.2 1.6

A Px10-5(Pa)

Figure 12 Relation between -ln(L;lL,)lAP”~55and &for Dextran T500 system.

PVP-360 in hollow-fiber membrane modules have been investigated experimentally based on the osmotic-pres- sure model with the correlated permeability equations.

It is found from the experimental results that the pure-water permeabilities after solute adsorption and fouling LL , vary with the changes of transmembrane pressure and the solution concentration, but are nearly independent of the flow velocity. The correlation equa- tions for pure-water permeabilities after solute adsorp- tion and fouling on the membrane have been obtained as functions of the transmembrane pressure and solu- tion concentration.

The permeate fluxes calculated from the theoretic equations agree well in tendency with the experimental data. The deviation of theoretic prediction from experi- mental results may be due to the inaccurate estimation of the solute concentration on the membrane surface by Equation 9. The use of another estimated method2s,26 for the membrane surface concentration to improve the predicted value of permeate flux will be our further works.

Acknowledgment

The authors thank the Chinese National Science Coun- cil for financial aid.

References 1. Porter, M.C. Membrane filtration. In Handbook ofSeparation

Techniques for Chemical Engineers. P.A. Schweizter, ed. New York: McGraw-Hill, 1979, sec. 2.1

2. Flory, P.J. Principles of Polymer Chemistry. Ithaca: Cornell University Press, 1953, pp. 279-282

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