Modeling of the Permeate Flux during Microfiltration of BSA-Adsorbed Microspheres in a Stirred Cell

Journal of Colloid and Interface Science 228, 270–278 (2000) doi:10.1006/jcis.2000.6940, available online at http://www.idealibrary.com on

Modeling of the Permeate Flux during Microfiltration of BSA-Adsorbed Microspheres in a Stirred Cell Sung-Wook Choi,∗ Jeong-Yeol Yoon,† Seungjoo Haam,∗ Joon-Ki Jung,‡ Jung-Hyun Kim,∗,1 and Woo-Sik Kim∗ ∗ Nanosphere Process and Technology Laboratory, Department of Chemical Engineering, Yonsei University, 134 Shinchon-dong, Sudaemoon-ku, Seoul 120-749, Korea; †Biomedical Engineering IDP, University of California, Los Angeles, California 90095; and ‡Seoul 120-749 Korea Biopilot Plant Division, Korea Research Institute of Bioscience and Biotechnology, Taejon 305-333, Korea Received October 8, 1999; accepted May 1, 2000

A study on the variation of the permeate flux was performed in a stirred cell charged with microspheres, to investigate the effects of the stirrer speeds (300, 400, and 600 rpm) and the BSA concentration (0.1, 0.2, 0.4, and 0.8 g/L) under constant pressure. The permeate flux increased over time before the saturation point, but it began to decrease after that point. An increase of the BSA concentration and the stirrer speed resulted in the rapid increase of the permeate flux. This is contrary to the observation of the conventional filtration experiments using a stirred cell. A resistance-in-series model was employed for the modeling of the permeate flux. The cake resistance (Rc , induced by the concentration polarization of microspheres) and the fouling resistance (Rf , induced by the adsorption of BSA inside the membrane pore) must be considered simultaneously for the modeling. These modeling results were in good agreement with the experimental data. These can be applied to the special system considering both Rc and Rf as well as the general filtration systems using a stirred cell. °C 2000 Academic Press Key Words: microspheres; stirred cell; flux modeling; resistancein-series model.

INTRODUCTION

Membrane processes are attractive as alternatives to the separation and concentration of colloid suspensions, which include clay particles, protein, and other macromolecules. However, the major obstacle to membrane filtration is the flux decline due to concentration polarization and membrane fouling. Membrane fouling is the result of the accumulation of rejected particles on the top surface of the membrane (external fouling), or the deposition and adsorption of small particles at the pore entrances or within the internal pore of the membrane (internal fouling). Many researchers have tried to overcome this obstacle. Pouliot et al. observed that pH increase from 5 to 9 raised the permeate flux and decreased the tendency of fouling when using a charged UF/MF membrane (1). Guosh and Cui (2) showed that the transmission of lysozyme through the poly-

1

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sulfone ultrafiltration membrane was enhanced by pretreating the surface of the membrane (adsorption of another protein, myoglobin). In our previous paper (3–7), the relationship between microspheres and proteins was elucidated and batch separation of a mixed protein solution was carried out successfully. The new protein separation system, using a stirred cell charged with microspheres, was also suggested. Microspheres had sufficient surface area provided by the submicron size and could be directly used without ligand coupling. To make a commercially valuable process, the modeling of the permeate flux is essential. Other studies using stirred cells mostly involved a single component (8–10), while this study involved two components (microspheres and BSA). Gekas and Hallstrom (11) studied the crossflow microfiltration of a mixture of polymerized silica particles and BSA. According to their report, the flux decline was greater when the silica particles were present than in the case of BSA being alone in the solution, which meant that the flux decline was due mostly to irreversible mechanisms, whereas in the presence of silica the opposite was true. Bhattacharjee et al. (12) have proposed a predictive model for the flux decline in unstirred and stirred batches by unifying the osmotic pressure and gel layer models. These analyses were mostly suitable for the filtration of an aqueous solution of low MW solutes (osmotic pressure controlled) and high MW solutes (gel layer controlled). The colloidal fouling of membrane in crossflow filtration was decribed by a model of coagulation (13). Kelly et al. (14) have shown that the rate of the flux decline during the microfiltration of BSA is a function of the detailed physicochemical characteristics of the particular BSA solution. The mechanisms proposed by Tracey and Davis (15), to explain the flux decline due to protein fouling of microfiltration membranes, consisted of two consecutive steps. The first step involved protein adsorption or deposition, mainly on the pore walls and mouths. During the second step, a cake layer built up on the membrane surface due to the deposition and growth of protein aggregates. A systematic study on the effects of electrostatic interaction on permeate flux decline and deposit cake formation in crossflow membrane filtration of a colloidal suspension was reported (16). However, our work focused on flux

270

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behavior during filtration and flux modeling using a resistancein-series model. The ultimate goal of this study is the development of a protein separation system. As the basic research, filtration experiments using a stirred cell charged with microspheres were carried out for BSA as model protein at different concentrations of BSA and stirrer speeds. The present results will be helpful in analyzing filtration system using a stirred cell. THEORY

Many different models have been proposed to predict the permeate flux during UF/MF. The oldest model is the resistance model based on the cake filtration theory. For a constant pressure filtration, the permeate flux (J ) can be expressed by the resistance-in-series model, J = 1P/µRt Rt = Rm + Rc + Rf ,

[1]

(1 − ε)m p mp (1 − ε)2 = 180 2 3 , dp2 ε3 (1 − ε)Am ρp d p ε A m ρp

[6]

(1 − ε) , dp2 ε3

[7]

rc0 is defined as rc0 = 180

where rc0 is the cake resistance per unit solid volume. In most of the filtration systems, membrane fouling is common and also causes flux to decrease. Fouling resistance can be defined as (18) R f = α Md .

[8]

Md is the amount of deposit (time-dependent parameter) and can be expressed as Eq. [9], d Md /dt = kf (Md∗ − Md ).

[9]

Integration of Eq. [9] gives Md = Md∗ (1 − exp(−kf t)).

[10]

From Eq. [8] and Eq. [10], fouling resistance can be expressed as Rf = α Md∗ (1 − exp(−kf t)),

[11]

where α is a proportional constant, Md∗ is the maximum deposition value, and kf is the deposition rate. Deposition rate kf increased significantly with protein concentration, but the maximum deposition value, Md∗ , was notably insensitive to concentration (18). Therefore, Eq. [1] can be expressed as J=

h

1P

i . [12]

µ Rm + 180 d 2 ε3 Am ρpp + α Md∗ (1 − exp(−kf t)) (1−ε)m p

Zδc rc dδc ,

[3]

0

where rc is the specific cake resistance and δc is the cake layer thickness. If the cake layer is assumed to be homogeneous, the above equation becomes Rc = rc δc .

[4]

For homogeneous cake, the cake thickness is δc =

Rc = 180

[2]

where J is the permeate flux, 1P is the transmembrane pressure, µ is the viscosity of the permeate, Rt is the total resistance, Rm is the membrane resistance, Rc is the cake resistance, and Rf is the fouling resistance. The cake filtration theory has been successful in describing flux during MF/UF. Rm is the resistance of clean membrane, which can be obtained from the data of pure water flux. Rc is defined to include concentration polarization and deposition of colloids on the membrane surface and Rf is due to internal fouling in the membrane pores. Small material can penetrate the membrane and become absorbed inside the pore of the membrane; flux decline due to colloids is caused by the formation of a cake layer on the membrane surface (17). The specific cake resistance is defined as the resistance per unit thickness of a cake layer,

Rc =

combining with the Kozeny–Carmen equation gives Rc as

mp = volume of cake/area of membrane, [5] ρp (1 − ε)Am

where m p is the total dried mass of a cake layer, ρp is the density of a particle, ε is the porosity of a cake layer, and Am is the effective membrane area. Substituting Eq. [5] into Eq. [4] and

As shown in Eq. [12], the membrane resistance, Rm , cake resistance, Rc , and fouling resistance, Rf can be considered simultaneously. EXPERIMENT

Materials Styrene (S) and methacrylic acid (MAA), purchased from Junsei Chemical Co. (Japan), were purified by Aldrich’s inhibitor remover column (catalog number 306312). Potassium persulfate (KPS) as initiator was purchased from Samchun Pure Chemical Ind., Ltd. (Korea). The resulting microsphere was purified by mixed-bed ion exchange resins (Dowex MR-3, Sigma Chemical Co., USA, catalog number I-9005). Bovine serum albumin (BSA, Sigma Chemical Co., USA, catalog number A-7906) is a globular ellipsoid, 14 nm × 3.8 nm × 3.8 nm, and used as model protein. Its molecular weight is 67,000 and

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its isoelectric point (pI ) is known as 4.7. An acetic buffer of pH 4.5, and a mixture of sodium acetate and acetic acid (both from Yakuri Pure Chemicals Co., Ltd., Japan), were used. The selected membrane for the experiments was a PES planar membrane (Orange Scientific Co., Belgium) with pore size of 0.2 µm. All water used in the experiments was distilled and deionized, produced by Elgastat’s UHQ system (UK). Preparation and Characterization of Microspheres Styrene (S) was used as the main monomer and methacrylic acid (MAA) was used as comonomer to give the carboxyl groups on the surface of microsphere. Polymerization was preformed in a 1.5-L internally stirred jacketed reactor at 70◦ C, and nitrogen gas was injected in small doses to provide a nitrogen atmosphere. Table 1 gives the recipe for the synthesis of PS/PMAA microspheres. These microspheres were purified by mixed-bed ionexchange resins and further purified by serum replacement method. The serum replacement procedure was performed in a 400-mL stirred cell (Amicon, model 8400, USA) equipped with 0.2-µm membrane filters (19). The average particle diameter of the micropheres was measured by the particle size analyzer (Capillary Hydrodynamic Fractionation, CHDF-1100, Matec Applied Sciences, USA), and the number density of surface functional groups was determined by conductometric titration (HI 8633 conductivity meter, Portugal) using NaOH solution as the titrant. Procedure The experimental set-up is shown schematically in Fig. 1. A stirred cell (50 mL, Amicon, model 8050, USA) equipped with a 0.2-µm membrane filter (Orange Scientific Co., Belgium) was charged with 30 mL of latex, the dispersion of microspheres. The specific surface area was adjusted to 0.19 m2 /mL and the ionic strength was calibrated to 0.01, which was the same condition as in our previous studies (3). A 500-mL BSA solution was prepared with acetic buffer at different concentrations. All latexes were sonicated before use. The BSA concentrations were chosen at 0.1, 0.2, 0.4, and 0.8 g/L, and the stirrer speed was varied as 300, 400, and 600 rpm. The permeate flux was measured continuously with an electronic balance (precision plus, Ohaus Co., USA) by a data acquisition system. The electronic balance was connected to a PC through an RS 232 C interface. Eluted fractions were collected by using fraction collector and analyzed by UV spectrophotometry (UV160A, Shimadzu, Japan) to obtain breakthrough curves. TABLE 1 Recipe for the PS/PMAA Microspheres Styrene (g) MAA (g) KPS (g) DDI (g)

60 0.6 0.24 600

FIG. 1. Experimental apparatus.

I. Batch adsorption experiment. The typical batch-type experiment was carried out in a 25◦ C water bath to obtain the effective diameter of agglutinated microspheres and the adsorption isotherm. Prior to the adsorption experiments, BSA was dissolved in acetic buffer of pH 4.5, followed by the addition of microspheres. The surface area of the microspheres was 0.19 m2 /mL of a protein–microsphere mixture. The final ionic strength of the buffer was always 0.01. Sample solutions were rotated gently (to avoid dimerization of the BSA molecules) for 3 h, and the effective particle diameter of samples was determined by a zeta potential analyzer (Dynamic Laser Light Scattering, ZetaPlus, Brookhaven Instruments Co., USA). To determine the adsorption isotherm, the sample solution, rotated gently for 3 h, was centrifuged and filtered through a cellulose nitrate filter membrane (Whatman or Advantec; pore diameter 0.1–0.2 µm) to remove particles completely. The residual protein concentration was determined by UV (UV-160A, Shimadzu, Japan) adsorbance at 280 nm. Because a small amount of protein was absorbed on the filter membrane, residual protein concentrations have to be calibrated (3). The experimental results of effective particle size are the average of five measurements. II. Latex filtration experiment. The latex filtration test was performed without stirring to determine the cake resistance per unit solid volume before the main experiments. A 0.01% (wt/wt) latex solution was prepared with the buffered DDI water. The stirred cell and reservoir were filled with 30 mL and 200 mL of prepared latex solution, respectively, and then the filtration test started. III. Protein filtration experiment. The main filtration experiments consisted of the following three steps. The permeate flux was measured in all steps. —Pure water step. This is the measurement of the pure water flux through the membrane. This experiment gives the value of

FLUX MODELING DURING MICROFILTRATION

273

the membrane resistance, Rm . The experimental results for Rm are the average of five measurements. —First buffer step. At a constant stirrer speed, acetic buffer of pH 4.5 without protein was introduced into the stirred cell until steady state flux was attained. The permeate flux must be constant to investigate the flux variation of second protein solution step. In this step, microspheres within the stirred cell were purified again. —Second protein solution step. After the first buffer step, the prepared BSA solution was introduced into the stirred cell with step injection. RESULTS AND DISCUSSION

Characterization Results of PS/PMAA Microspheres PS/PMAA, the low carboxylated monodisperse microspheres, had the advantage of high selectivity (5, 7). Because of the low carboxylated microspheres, hydrophobic interaction was the most important aspect of protein adsorption (4). Zeta potential of PS/PMAA microspheres is reported in our previous paper (3). Electrostatic interaction can be considered as constant in the experiment with carboxylated PS/PMAA microspheres, because the carboxyl group will not dissociate in our experimental range of pH 4.5, and only sulfate groups, originated from initiators, contribute to the electrostatic interaction. The electrophoretic mobility and colloidal stability of the PS/PMAA microspheres declined with the amount of BSA adsorbed onto microspheres (20). The protein adsorption isotherm is showed in Fig. 6 with the effective particle size of microspheres. Particle growth realted to extent of BSA adsorption could be explained by Fig. 6. The curve of effective particle size and absorbed amount of BSA protein is similar to a Langmuir–Freundlich isotherm. Therefore, it is concluded that BSA adsorption and particle agglutination are closely related. Other characterization results are shown in Table 2.

FIG. 2. Permeate flux with time during the first buffer step at different stirrer speeds.

then reached steady state. The permeate flux becomes large as the stirrer speed is fast because of the high shear stress at the surface of a cake layer. Breakthrough Curve The breakthrough curves were obtained from the second protein solution step with respect to different BSA concentrations at the stirrer speed of 400 rpm. As shown in Fig. 3, the breakthrough point was inversely proportional to the BSA concentration. When BSA saturated the surface of the microspheres, they permeated through the membrane. After the breakthrough point, the shape of the breakthrough curve was shown similarly.

The Behavior of the Permeate Flux in the First Buffer Step Figure 2 shows the permeate flux with time before the second protein solution step. During latex filtration, microspheres within the stirred cell were convectively driven to the membrane surface where they accumulated and tended to form a cake. This particle build-up near the membrane surface is known as concentration polarization, and results in increasing hydraulic resistance to permeate flow. In brief, the permeate flux decreased with time due to cake formation during filtration and TABLE 2 Characterization Results of PS/PMAA Microspheres Nc Dn U ρp

0.45 nm−2 500 nm 1.004 1.05 kg/m3

FIG. 3. Breakthrough curve of BSA solution.

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The Modeling of the Permeate Flux Conventional membrane filtration has the problem of concentration polarization and membrane fouling. The permeate flux decreased with time for those reasons. However, the mechanism of the newly suggested protein separation system involves the selective adsorption of protein onto microspheres. Therefore, Eq. [12] must be modified for the modeling of the permeate flux. The value of Rf was zero until the microspheres were saturated with BSA and rapidly increased after saturation. Therefore, we can employ the unit step function for the modification of Rf and α Md∗ , defined as rf . Modified Rf was expressed as Rf = rf (1 − exp(−kf tU (Vt − Vm ))), U (Vt − Vm ) =

0

(Vt < Vm )

1

(Vt > Vm ),

[13] [14]

where rf is the plateau fouling resistance, U (Vt − Vm ) is the unit step function, Vt is the permeate volume, and Vm is the accumulated elution volume until saturation. The values of Vm could be obtained from Fig. 3 at different BSA concentrations, respectively. The modified equation for the newly suggested protein separation system was established as follows: J= FIG. 4. Schematic diagram illustrating flux variation with time.

h

µ Rm +

1P mp 180 (1−n) dp2 n 3 Am ρp

+ rf (1 − exp(−kf tU (Vt − Vm )))

i.

[15]

The Behavior of the Permeate Flux in the Second Protein Solution Step Figure 4 gives an explanation of the behavior of the permeate flux as a schematic diagram. The permeate flux was decreased at region I (first buffer step). After step injection of BSA solution, the permeate flux increased with time until the breakthrough point (region II), but it began to decrease after that point (region III). This is in contrast to the conventional filtration experiments. This phenomenon could be explained as follows: BSA-adsorbed microspheres agglutinated each other and this resulted in the increase of effective diameter of agglutinated microspheres. It is well known that the large-sized particle causes flux to increase (21–24). After saturating the surface of themicrospheres, BSA permeated through the membrane and some of them absorbed inside the pore of membrane. This phenomenon resulted in flux decline as observed in the general filtration. The permeate flux was increased during the adsorption of the BSA onto the surface of microspheres. However, the permeate flux decreased after the microspheres were fully saturated with BSA. Experimental data are shown in Figs. 9, 10, and 11. As the BSA concentration increased, the permeate flux rapidly increased as expected, and there was little effect of the BSA concentration on the maximum value of the permeate flux. However, the stirrer speed affected the maximum value of the permeate flux.

The modeling of the permeate flux at region I in Fig. 4 has been done sufficiently by other researchers (25). Lee and Clark (26) reported the effects on flux decline of particle size, feed concentration, stirrer speed, and transmembrane pressure during filtration of colloidal suspensions. However, this is beyond our focus. Our emphasis is on the filtration models which are applied to predict the permeate flux at regions II and III in Fig. 4. The modeling procedure is shown in Fig. 5. The overall equation, Eq. [15], including Rm , Rc , and Rf , was established. The experiment with pure water gave the value of the membrane resistance, Rm , and the porosity of a cake layer, ε, could be obtained by the latex filtration experiment. It was assumed that the porosity of a cake layer was constant during filtration. The total mass of a cake layer at each stirrer speed could be determined using Eq. [15] ignoring Rf . The effective particle size with bulk concentration of BSA was determined by the batch adsorption experiment. The values of rf and kf were determined by fitting to the experimental data. Porosity of a cake layer. The value of cake resistance per unit solid volume, rc0 , obtained by the latex filtration experiment under unstirred conditions, was 6.75 × 1015 m−2 . The porosity of a cake layer, ε, could be also determined from rc0 . Under unstirred conditions, microspheres were carried convectively to the membrane surface where they accumulated and provided an increasing barrier to solvent flow. The value of ε obtained experimentally was 4.0. Details were presented elsewhere (27).

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FIG. 7. 1/J 2 versus time under unstirred conditions. FIG. 5. Flow diagram of the modeling procedure.

Experimental data under unstirred conditions are shown as Fig. 7 (including the regression result). We could determine the porosity of a cake layer in the consequence of calculations from experimental data at regions II and III using Eq. [15], in which Rf is ignored, under stirred conditions. It was observed that the porosity of a cake layer decreased as the effective particle diameter increased. The porosity of a cake layer was relatively constant between 0.3 and 0.4, and hence ε could be fixed to its average value of 3.5 to facilitate flux modeling. Total Particle mass of a cake layer. The total particle mass of a cake layer, m p , dependent on the stirrer speed could be assumed to be constant during the second protein solution step and was determined as follows. The permeate concentration of microspheres was nearly zero because the membrane pore size was small enough to retain all the microspheres. The concentrations were indirectly measured by UV spectrophotometry (UV-160A,

Shimadzu, Japan). There was a linear relationship between concentration of microspheres and absorbance. When the permeate flux was in the steady state, the absorbance of retentate was measured. Therefore, the total particle mass of a cake layer could be found at each stirrer speed: 300, 400, and 600 rpm. The total particle mass of a cake layer, m p , was decreased with stirrer speed, because the increased stirrer speed caused the thickness of a cake layer to decrease. Effective particle diameter. The effective particle diameter of agglutinated microspheres, determined by a laser light scattering method after a batch adsorption experiment, is a function of the BSA concentration. These experimental results are shown in Fig. 6. It was assumed that if primary particles aggregated into doublets, triplets, and larger particles, these aggregates could effectively behave as larger primary particles. Similar concepts were used by Clark and Flora in a study of floc restructuring (28). Therefore, the effective particle diameter of BSA-adsorbed microspheres could be expressed by dp = f (Cs ),

[16]

where dp is the particle diameter and Cs is the amount of adsorbed BSA per unit surface area. The curve of effective particle size versus the amount of adsorbed BSA is similar to a Langmuir–Freundlich isotherm as shown in Fig. 6. The typical Langmuir–Freundlich isotherm is expressed as 1/n

Cs = Cm

FIG. 6. Effective particle diameter and protein adsorption isotherm.

K Cb

1/n

1 + K Cb

,

[17]

where K is the adsorption constant, n is the exponential factor, Cb is the bulk concentration of BSA, and Cm is the amount

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adsorbed at equilibrium. In our previous paper (3), the value of n is 0.5, which means that one of two active sites is valid for adsorption (29). Therefore, we propose the equation of effective particle diameter with Cs similar to a Langmuir–Freundlich isotherm, 1/n

dp = Dm

K Cb

1/n

1 + K Cb

+ Dn ,

[18]

where Dn is the diameter of particles unadsorbed by BSA (=5.0 × 10−7 m). The values of n and K obtained by a nonlinear regression method from experimental data were 0.5 and 7.5, respectively. The physical meaning of K is related to the adsorption kinetics and the large value of K indicates the faster rate of enlargement of effective particle size. Dm + Dn (=1.3 × 10−6 m) is the maximum effective particle diameter when microspheres are fully saturated with BSA. Fouling resistance. The values of rf and kf of Eq. [15] could be obtained empirically by fitting to the experimental data. The values of rf and kf are shown in Fig. 8. It was found that rf and kf values decreased with an increase in the stirrer speed and were almost insensitive to the BSA concentration. When the permeate flux was high, there was not enough time to adsorb inside pore

FIG. 8. Values of rf and kf .

FIG. 9. Comparison of modeling results with experimental data at a stirrer speed of 300 rpm.

of the membrane. Therefore, the permeate flux was affected by the stirrer speed more than the BSA concentration. The other necessary parameters were determined as follows: Rm = 2.6 × 109 m−1 (membrane resistance), Am = 0.00134 m2 (effective membrane area), µ = 0.001 Pa · s (dynamic viscosity of pure water), and ρp = 1.05 kg/m3 (density of PS/PMAA microspheres). The modeling of the permeate flux was carried out using parameters determined in the above section. The experimental filtration data for various BSA concentrations and stirrer speeds are compared with the modeling results in Figs. 9, 10, and 11.

FIG. 10. Comparison of modeling results with experimental data at a stirrer speed of 400 rpm.

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Dm + D n Dn J kf K mp Md Md∗ n

FIG. 11. Comparison of modeling results with experimental data at a stirrer speed of 600 rpm.

Although the model tends to predict a slightly lower flux than the experimental results at region III, the modeling results of the permeate flux are quite consistent with the experimental data.

Nc rc rc0 rf Rc Rm Rf t U α δc ε µ ρp 1P

maximum agglutinated particle diameter (m) average particle diameter (m) permeate flux (m/s) empirical deposition rate (s−1 ) adsorption constant for L–F isotherm (dimensionless) total particle mass of a cake layer (kg) amount of deposit (µg/m2 ) empirical maximum deposition value (µg/m2 ) exponential factor of L–F isotherm (dimensionless) number density of surface carboxyl group (nm−2 ) specific cake resistance (m/kg) resistance per unit solid volume (m−2 ) plateau fouling resistance (m−1 ) cake layer resistance (m−1 ) membrane resistance (m−1 ) fouling resistance (m−1 ) filtration time (s) uniformity (dimensionless) proportional constant (m/µg) thickness of a cake layer (m) porosity of a cake layer (dimensionless) dynamic viscosity (Pa · s) density of particle (kg/m3 ) transmembrane pressure (Pa) ACKNOWLEDGMENT

CONCLUSION

From the study of the permeate flux in the filtration using a stirred cell charged with microspheres, the following conclusions can be drawn. First, the permeate flux was increased during the adsorption of the BSA onto the surface of microspheres, but after the microspheres were fully saturated with BSA, the permeate flux decreased. Second, with increasing BSA concentration and stirrer speed, the permeate flux was rapidly increased. Third, the modeling results obtained using a modified equation were in good agreement with the experimental results. The numerical model of microfiltration developed here successfully explains the fundamental mechanism for the newly suggested protein separation system. This model provides a useful tool to understand the effect of protein concentration and stirrer speed on the system using a stirred cell. Also, we can extend these results to other filtration experiments using stirred cells. NOMENCLATURE

Am Cs Cm C/C0 Cb dp

membrane filtration area (m2 ) amount of adsorbed BSA per unit surface area (mg/m2 ) adsorbed amount at equilibrium (mg/m2 ) eluted fraction of BSA (dimensionless) bulk concentration of BSA (mg/mL) particle diameter (m)

The authors acknowledge the financial support of Bioproducts Research Center, Yonsei University, Grant 94U4-1005-00-01-6 and the Korea Institute of S&T Evaluation and Planning (KISTEP) made in the program year of 1999.

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