Membrane ultrafiltration in hollow-fiber module with the consideration of pressure declination along the fibers

Separation and Purification Technology 13 (1998) 171–180

Membrane ultrafiltration in hollow-fiber module with the consideration of pressure declination along the fibers H.M. Yeh a,*, T.W. Cheng a, H.H. Wu a a Department of Chemical Engineering, Tamkang University, Tamsui, Taipei 251, Taiwan Received 20 August 1997; received in revised form 4 December 1997; accepted 9 December 1997

Abstract The effects of operating conditions on the permeate flux for ultrafiltration of aqueous solution of PVP-360 in hollow-fiber membrane modules, have been investigated based on the resistance-in-series model coupled with the gelpolarization model under the consideration of the declination of transmembrane pressure along the axial direction in the hollow fibers. It is seen that correlation prediction may be improved if the more precise, but rather complicated, integral equation derived can be solved and if more experimental data are available. © 1998 Elsevier Science B.V. Keywords: Gel polarization; Hollow fiber; Membrane; Resistance in series; Ultrafiltration

Nomenclature A C b C i C g C m D D i F J J9 J lim k k i L m N

proportional constant defined by Eqs. (2) and (4) (m3 m−2 s−1 Pa−1) bulk solute concentration (wt%) concentration of feed solution (wt%) solute concentration in the gel layer (wt%) solute concentration on membrane surface (wt%) diffusion coefficient (m2 s−1) diffusion coefficient at the inlet (m2 s−1) modified factor defined by Eq. (24) volume permeate flux of solution (m3 m−2 s−1) average value of J in a hollow-fiber module (m3 m−2 s−1) limiting permeate flux (m3 m−2 s−1) mass-transfer coefficient (m s−1) mass-transfer coefficient at the inlet (m s−1) length of hollow fiber (m) constant defined by Eq. (15) (Pa s m−3) number of hollow fiber in a membrane module

* Corresponding author. 1383-5866/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S1 3 8 3 -5 8 6 6 ( 9 8 ) 0 0 04 1 - 0

n P P i P p DP DP i Q i Q L Q 9 R r m u i u: z

constant defined by Eq. (16) pressure distribution of the tube side (Pa) pressure at the inlet of a hollow fiber (Pa) permeate pressure of the shell side (Pa) transmembrane pressure, P−P (Pa) P P −P (Pa) i p volume flow rate at the inlet of a hollow-fiber module, Nu pr2 (m3 s−1) i m volume flow rate at the outlet of a hollow-fiber module (m3 s−1) average volume flow rate in a hollow-fiber module (m3 s−1) proportional constant defined by Eq. (4) (Pa m2 s m−3) inside radius of hollow fiber (m) mean axial velocity at fiber inlet (m s−1) average value of axial velocity through a hollow-fiber tube (m s−1) axial coordinate (m)

Greek letters m j

viscosity of solution (Pa s) z/L

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1. Introduction Recently, membrane ultrafiltration has been applied in a wide variety of fields, from the chemical industry (such as electrocoat paint recovery, latex processing, textile size recovery and recovery of lubricating oil ) to medical applications (such as kidney dialysis operations) and even to biotechnology applications (such as concentration of milk, egg white, juice, pectin and sugar, and recovery of protein from cheese whey, animal blood, gelatin and glue). Ultrafiltration is a pressure-driven membrane process used for the separation of macrosolutes from a solvent, usually water. The rapid development of this process was made possible by the advent of an isotropic, high-flux membrane capable of distinguishing between molecular and colloidal species in the 0.001–10 mm size range. One of the common ultrafiltration designs is the hollowfiber membrane module in which the membrane is formed on the inside of tiny polymer cylinders that are then bundled and potted into a tube-andshell arrangement. The advantages of this arrangement are the low cost of investment and operation, easy flow control and cleaning, and high specific surface area per unit volume. A number of mathematical models are available in the literature that attempt to describe the mechanism of transport through a membrane. Permeate flux of ultrafiltration is mainly analyzed by use of one of following models: the gel-polarization model [1–10], the osmotic-pressure model [11– 20], or the resistance-in-series model [19,21,22]. The gel-polarization model applies to the case that the transmembrane pressure is sufficiently large enough to form a gel layer on the membrane surface and the membrane permeation rate is limited without the influence of further increasing transmembrane pressure. In the osmotic-pressure model, permeate flux reduction results from the decrease in effective transmembrane pressure that occurs as the osmotic pressure of the retentate increases. In the resistance-in-series model, permeate flux decreases due to the resistance caused by fouling or solute adsorption and concentration polarization. In this study, with the consideration of pressure declination along the tubes, the com-

bined models of resistance in series and gel polarization for analyzing the permeate flux of hollowfiber ultrafiltration will be introduced. The effects of various parameters on permeate flux will also be discussed.

2. Theory 2.1. Permeate flux The model was developed to simulate forcedconvection ultrafiltration in a horizontal hollowflber membrane module. Since ultrafiltration is a pressure-driven membrane process, the pressure applied to the working fluid provides the driving potential to force the solvent to flow through the membrane. Therefore, when there is in absence of transmembrane pressure DP, no permeation occurs, i.e. as DP=0,

(1)

J=0.

Typical driving pressures for ultrafiltration systems are in the range of 10–100 psi. For small applied pressure the solvent flux J through the membrane is observed to be proportional to the applied pressure. Therefore, the second condition is as DP is small, J=(constant)DP=ADP.

(2)

However, as the pressure is increased further, the flux begins to drop below that which would result from a linear flux-pressure behavior. Eventually, a limiting flux J is reached where lim any further pressure increase no longer results in any increase in flux, i.e. as DP2, J=J

. (3) lim Accordingly, we may define the following relation DP

=

DP

, (4) ) R+(DP/J ) lim lim which satisfies three conditions in Eqs. (1)–(3). In the above equations, A is a proportional constant to be determined experimentally, while the transJ=

(1/A)+(DP/J

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membrane pressure is defined as DP=P(z)−P , (5) p where P(z) is the pressure distribution of the tube side along the axial direction, z, of a hollow fiber and P is the permeate pressure of the shell side. p It is noted that Eq. (4) is actually another form of expression for the resistance-in-series model with wDP replaced by DP/J , which repressents lim the resistance due to the concentration polarization–gel layer, while R denotes the intrinsic resistance of the membrane coupled with the resistances due to other fouling phenomena such as solute adsorption.

%Q , therefore, the arithmetic mean of volume i flow rate will be taken, i.e. Q +Q L =Q −(pr LN)J9 , Q 9= i i m 2

(11)

and the pressure distribution is obtained by substituting Eq. (11) into Eq. (8). The result is P(z)=P − i

CA B A B DA B 8mL

Q− i

Npr4 m

8mL2 r3 m

z

J9

L

. (12)

Finally, the transmembrane pressure is DP(z)=P(z)−P =DP −(mQ −nJ9 ) (z/L), p i i where

(13)

2.2. The pressure distribution Since the membrane permeation rate is small compared to the volume flow rate in a hollowfiber module, it can be assumed that the local declination in hydraulic pressure within the fiber of radius r is simply given by the Hagen–Poiseuille equation:

DP =P −P , i i p 8mL 8u mL m= = i , Npr4 Q r2 m i m and

(14)

8mQ 9

8mu: , (6) dz Npr4 r2 m m where the average volume rate Q 9 in a module of N hollow fibers, is related to the average velocity u: as

dP

=−

=−

Q (7) 9 =pr2 Nu: . m Integrating Eq. (6) with the use of boundary condition P=P , at z=0, we have i 8mQ 9 P(z)=P − z. (8) i Npr4 m The volumetric flow rates at the inlet and outlet of a hollow-fiber module of effective length L are related to each other by

A B

Q =Q −2pr LNJ9 =pr2 Nu −2pr LNJ9 , (9) L i m m i m in which the average permeate flux is defined as J9 =

1

P

L

J(z) dz. (10) 0 Since we may assume that (Q −Q )=2pr LNJ9 i L m L

(15)

8mL2

. (16) r3 m If the effect of permeation on volumetric flow rate is considered, from Eq. (9) n=

Q(z)=Q −2pr N i m

P

z

J(z) dz. (17) 0 Replacing Q 9 by Q in Eq. (6) and integrating with the inlet condition yields

DP(z)=DP −mQ (z/L)+2(n/L2) i i

P P z

z

0

0

×J(z) dz dz.

(18)

2.3. Average permeate flux Substitution of Eq. (13) into Eq. (4) yields the expression for local permeate flux of a hollowfiber module J(z) J lim

=

DP −(mQ −nJ9 ) (z/L) i i . RJ +[DP −(mQ −nJ9 ) (z/L)] lim i i

(19)

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Thus, the average permeate flux of a hollowfiber membrane module can be obtained by substituting Eq. (19) into Eq. (10). The result is J9 J lim

=1+

C A

RJ mQ −nJ9 lim i ln 1− (mQ −nJ9 ) RJ +DP i lim i

BD

. (20)

On the other hand, substitution of Eqs. (14) and (18) into Eq. (10) yields a more precise but rather complicated expression for average permeate flux J9 J lim

=

P

1

{1+RJ [DP −mQ j+2n lim i i

0 ×J(j) dj dj]−1}−1 dj,

P P j

j

0

0 (21)

3.2. Gel-polarization model for J

where (22)

j=z/L. 2.4. Determination of J

was made of polysulfone and the total fiber number and total effective membrane area were 250 and 600 cm2, respectively. The test solute was polyvinylpyrrolidone (PVP-360, Sigma Co., M =360 000). The solvent was deionized water. n The concentration of feed solutions were 0.1, 0.5, 1.0 and 2.0 wt% PVP-360, the feed flow velocities were 0.0723, 0.1209, 0.1684 and 0.2195 m s−1, and the feed inlet transmembrane pressures were 19, 38, 57, 77, 96 and 115 kPa. In all experiments, the feed solution temperature was controlled at 25°C by a thermostat. The experimental data of solution permeate flux J9 obtained under various operating conditions are given in Figs. 1–4, while the values of J and R thus determined by Eq. (23), are lim listed in Table 1.

lim

and R

Eq. (4) can be rewritten with the use of average experimental values, J9 and (DP) as exp exp 1 R 1 = + . (23) (J9 ) J (DP) exp lim exp Therefore, from a straight line plot of 1/(J9 ) versus 1/(DP) at a certain velocity, u , exp exp i and feed concentration, c , the value of 1/J (the i lim intersection at ordinate) and R (the slope), as well as J and A, may be determined experimentally lim as a function of u and c . i i

3. Comparison of correlation predictions with experimental results 3.1. Illustration For the purpose of illustration, consider the experimental data of Yeh and Wu’s work [23,24] as follows. In their experimental work, an Amicon model H1P 30-20 hollow-fiber cartridge (Amicon Corp., Danvers, MA) was used. The fiber (r =2.5×10−4 m, effective length L=0.153 m) m

lim

In hollow-fiber membrane ultrafiltration processes, solutes that are rejected 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. Since the rejection of ultrafiltration for macromolecules is generally very high, the solute concentration in the permeate may be neglected. Accordingly, a material balance for the solute results in the so-called concentration polarization model C (24) J=k ln m , C b where C and C are solute concentrations at m b membrane surface and in the bulk fluid, respectively, and k is the average mass-transfer coefficient. The Graetz solutions [25] for convective heat transfer in laminar flow channels, suitably modified for mass transfer, may be used to evalute the masstransfer coefficient in a hollow fiber:

A B

u D2 1/3 2r b (25) , 100
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175

Fig. 1. Comparison of correlation predictions with experimental results for u =0.0723 m s−1. i

length of a hollow fiber, respectively, and Re and Sc are the Reynolds and Schmidt numbers, respectively. Under high-pressure operation the concentration at the membrane surface can even rise to a point of incipient gel precipitation. When the membrane surface concentration is very high and a gel layer is formed, any further pressure increase no longer results in any increase in flux. In this case, gel layer concentration C is employed instead g of C and Eq. (24) becomes the gel polarization m model C J =k ln g . (26) lim C b It is shown by Eq. (26) that the limiting flux J becomes zero as the solute concentration in lim the bulk fluid, C, approaches the gel concentration, C . In this conventional gel polarization g model, the concentration of the gel layer may be

considered to be constant and dependent only on the kinds of solute and membrane used. According to conventional treatment of the gel polarization model, limiting permeate flux J lim versus ln C plots are straight lines as explained b by Eq. (26), and these lines merge at one point on a concentration axis when permeate flux is zero, which gives the value of C . Many investigators g pointed out that the values of C thus obtained g are not the real concentrations in the gel layers because these values differ among membranes; some of the values are not realistic and can be larger than 100 wt% [6,19]. Nevertheless, here we will merely consider the value of C as a parameter g in the modified gel polarization model, and its value will be determined by an alternative way as follows [10]. Since the permeate flux J is low compared with the flow velocity u, we assume that the bulk concentration and velocity are approximately the

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Fig. 2. Comparison of correlation predictions with experimental results for u =0.1209 m s−1. i

same as those of inlet values, i.e. C #C , and b i u #u . Further, the diffusivity coefficient in b i Eq. (25) is hard to estimate precisely because the concentration within the boundary layer is still uncertain. For convenience here, we evaluate the mass-transfer coefficient with the inlet value of the diffusion coefficient, D , and Eq. (26) is corrected i by a modified factor F. Eqs. (25) and (26) become

A B

u D2 1/3 k =1.62 i i , i 2r L m C J =k F ln g , lim i C i or

(27)

(28)

J /k =F ln C −F ln C . (29) lim i g i According to Eq. (29), if a straight line of J /k versus ln C can be constructed from the lim i i experimental data by the method of least squares,

C and F can be determined because ln C is the g g intersection at the concentration axis, which gives the value of C , while −F is the slope of this g straight line. The diffusion coefficient for PVP-360 solution of concentration C at 25°C can be estimated by i the following correlation equation [26 ]: D ×1011=4.25+4.96 tanh(0.16C i i −0.507) m2 s−1.

(30)

The experimental values of J /k were callim i culated from Table 1 and Eqs. (27) and (30) with the given values r =2.5×10−4 m and m L=0.153 m. The results are also presented in Table 1. The values of C and F for ultrafiltration g of a PVP-360 solution in an Amicon model H1P30-20 hollow-fiber cartridge were determined as shown in Fig. 5. The results are C =4.915 wt.% and F=1.975. g

H.M. Yeh et al. / Separation and Purification Technology 13 (1998) 171–180

177

Fig. 3. Comparison of correlation predictions with experimental results for u =0.1684 m s−1. i

Substitution of C , F and Eq. (27) into Eq. (28) g yields a correlation equation of J for ultrafiltralim tion of PVP-360 aqueous solution by Amicon model H1P30-20 hollow-fiber cartridge

A B A B

u D2 1/3 4.915 m s−1. J =3.2 i i ln lim 2r L C m i

(31)

3.3. Correlation equations of R and m Since R is a function of u and C as shown in i i Table 1, Yeh and Wu [23,24] obtained the correlation equation of R from the experimental data as R=(0.5+1.5u−0.036 e0.726 sinh Ci ) i ×109 Pa m2 s m−3.

(32)

The viscosity of PVP-360 aqueous solution at

25°C was estimated as m=0.89×10−3 e0.875Ci Pa s.

(33)

3.4. Comparison Correlation predictions of average permeate fluxes J9 under various u C , and DP were calcui i i lated from Eqs. (20) and (31) by trial-and-error method. The results were compared with the experimental data as shown in Figs. 1–4. The theoretical predictions agree well in tendency with the experimental results for higher feed concentrations (1.0 wt%
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Fig. 4. Comparison of correlation predictions with experimental results for u =0.2195 m s−1. i

4. Conclusion A correlation equation for predicting the permeate flux of hollow-fiber membrane ultrafiltration has been derived from three essential conditions with the consideration of the declination of transmembrane pressure along the axial direction in the hollow fibers. It was found that this correlation equation is actually the one based on the resistance-in-series model. The equation for the limiting flux based on gel-polarization model was modified from Eq. (26) and a modified factor F was introduced as shown in Eq. (28). The gel-layer concentration C and the modified factor F were g determined by an unconventional, but rather convenient, method as shown in Fig. 5. An illustration for ultrafiltration of PVP-360 aqueous solution by an Amicon model H1P30-20 hollow-fiber cartridge made of polysulfone was given. The average per-

meate fluxes with the consideration of permeation on the change of transmembrane pressure were thus calculated, and the results are presented in Figs. 1–4 for comparison with experimental data. Both correlation predictions and experimental results show that the average permeate flux increases as the transmembrane pressure or feed velocity increases, but it decreases when the solution concentration increases. It is also seen from Figs. 1–4 that correlation predictions with the consideration of permeation on the change of transmembrane pressure qualitatively agree with experimental data. It is believed that correlation predictions may be improved if the more precise but rather complicated expression, Eq. (21), for calculation of average permeate flux can be employed and if more experimental data are available for constructing the correlation equations for J and R. lim

H.M. Yeh et al. / Separation and Purification Technology 13 (1998) 171–180

179

Fig. 5. Relation between limiting flux and feed concentration.

References Table 1 Fitting parameters for various operating conditions C i (wt.%)

u i (m s−1)

R×10−9 (Pa m2 s m−3)

J ×105 lim (m s−1)

J /k lim i (–)

0.1 – – – – 0.5 – – – – 1.0 – – – – 2.0 – – –

0.0723 0.1209 0.1684 0.2195 – 0.0723 0.1209 0.1684 0.2195 – 0.0723 0.1209 0.1684 0.2195 – 0.0723 0.1209 0.1684 0.2195

2.445 2.424 2.458 2.359 – 2.934 2.951 3.001 2.866 – 3.847 3.456 3.613 3.518 – 24.805 24.871 24.294 –

0.731 1.025 1.324 1.526 – 0.395 0.593 0.785 0.951 – 0.316 0.503 0.659 0.820 – 0.285 0.420 – –

6.251 7.390 8.548 9.015 – 3.116 3.939 4.670 5.183 – 2.268 3.041 3.569 4.066 – 1.733 2.149 – –

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