Semiempirical Modeling of Cross-Flow Microfiltration with Periodic

Figure 2 Typical microfiltration results of a yeast suspension through a 0.2 μm polypropylene membrane. (a) Forward filtration: ΔPf = 10 psi, φ = 0...
0 downloads 0 Views 423KB Size
Download PDF
2920

Ind. Eng. Chem. Res. 1996, 35, 2920-2928

Semiempirical Modeling of Cross-Flow Microfiltration with Periodic Reverse Filtration Hanuman Mallubhotla and Georges Belfort* Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180

High-frequency reverse filtration is an effective method of improving membrane performance. Models based on blocking filtration laws have been successfully used for flux prediction during discontinuous semibatch processes. In this paper, we present two semiempirical models to predict the performance of a continuous cross-flow filtration system with back-pulsing. Parameters evaluated from batch cell experiments are shown to be useful in scale-up and design of such systems. Both models are applied to results obtained for this work and to published results and are shown to be effective tools for prediction of process variables. Methods of process optimization are also discussed. Introduction Back-pulsing, or reversing the flow direction through a membrane, is a common method to clean membrane modules for reuse. [Various authors have used different terms to describe the process “back-pulsing”. For example, Matsumoto et al. (1987, 1988) and Xu et al. (1995) have used the term backwashing, Rodgers and Sparks (1991, 1992, 1993) and Rodgers and Miller (1993) have used the term backpulsing, while Jonsson and Wenten (1994) used the term back-shocking. Redkar and Davis (1995) coined the term “periodic reverse filtration”.] Matsumoto et al. (1987) were able to maintain a constant permeation flux much higher than the pseudo-steady-state flux for the concentration of yeast by cross-flow filtration with back-pulsing. By comparing different methods of back-pulsing, Matsumoto et al. (1988) have shown that back-flushing with the filtrate is more effective than back-pulsing with air. With back-pulsing, Nipkow et al. (1989) reported initial improvement of 42% in the mean permeate flux of a microfiltration cell recycle pilot-scale system for continuous cultivation of Clostridium thermosulfurogenes. Improved solute flux by as much as 2 orders of magnitude for various cross-flow rates was reported for a binary protein mixture ultrafiltration with backpulsing by Rodgers and Sparks (1991). It was determined that transmembrane pressure pulsing altered the concentration polarization boundary layer by translation of body forces through the membrane and minute but significant membrane motion. This increased the solvent flux in ultrafiltration (Rodgers and Sparks, 1992). Rodgers and Sparks (1993) also determined that changes in feed concentration caused the most significant difference in flux enhancement due to back-pulsing. This was primarily attributed to a limited available flux range during the redevelopment phase of the polarization layer. Rodgers and Miller (1993) report increased apparent pore size and initial sieving coefficients for bovine serum albumin ultrafiltration with back-pulsing. Nikolov et al. (1993) report that synchronized pulsations in the feed and the permeate lead to a better performance of the membrane than independent pulsations. Jonsson and Wenten (1994) were able to maintain a high constant flux during microfiltration of beer * To whom correspondence should be addressed. Telephone: (518) 276-6948. Facsimile: (518) 276-4030. E-mail: [email protected].

S0888-5885(95)00719-6 CCC: $12.00

for 3 days with back-pulsing. They have used a reversed membrane; i.e., the support layer faced the upstream (feed) side. The rate of flux decline was lower for such a configuration. They have also achieved almost 100% protein transmission through the membrane. Using cake filtration theory and assuming that the back-pulsing period depends on the geometry of the filter module, Xu et al. (1995) optimized a two-stage discontinuous microfiltration process. Imperfect flux restoration due to fouling and chemical cleaning were also discussed. Redkar and Davis (1995) have obtained excellent experimental results for various combinations of transmembrane pressures, back-pulsing, and forward filtration times. Although the theory they have developed is qualitatively useful, quantitative predictions are far from acceptable. In the next section, we propose two models for predicting optimum flux during cross-flow filtration with high-frequency reverse filtration. Both models are modifications of the theory proposed by Redkar and Davis (1995). We have used experimental data from their reference extensively for purposes of comparison. Although the principles are the same, the second model assumes a different flux decline equation. Batch cell experiments were performed to evaluate the applicability of these models. Optimization of the process with respect to transmembrane pressure ratio, back-pulsing time and other parameters is discussed. Theory Consider one time cycle of a cross-flow with reverse filtration process. The flux declines from t ) 0 to t ) t1. Back-pulsing is done from t ) t1 till t ) t2. Here,

t1 ) tf, t2 - t1 ) tb, and t2 ) tf + tb

(1)

where, tf is the forward filtration time and tb is the time for reverse filtration. Equation 1 defines the limits of integration in the following two models. Model I. Following the theoretical model developed by Redkar and Davis (1995), we use the following equation for forward flux prediction:

J+ ) J0{1 + (t/τ1)}-0.5

(2)

where J+ is the forward flux, J0 is the initial flux, t is the time, and τ1 is the time constant for cake growth which can be determined experimentally (Redkar and © 1996 American Chemical Society

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2921

Figure 1. Normalized flux decline curve and a family of exponential decay curves. J0 is the initial or pure water flux.

Davis, 1993). For reverse flux, however, we assume that the cake is not instantly lifted off the membrane. In fact, the cake removal is delayed similar to forward filtration. Consequently, for reverse filtration, the flux is given by

J- ) RJ0[1 - {1 + (t/τ2)}-0.5]

(3)

where R ) ∆Pb/∆Pf is the ratio of the magnitudes of the reverse and forward transmembrane pressures. The average global flux is then given by

〈J〉 ) J0

∫0t

1

{

1+

}

t τ1

∫tt

-0.5

dt - RJ0

2

1

[ {

} ]

t τ2

1- 1+

-0.5

dt

t2 (4) As we show later, the assumption that the permeability of the clean membrane is recovered after each cycle due to back-pulsing is not valid either (Redkar and Davis, 1995). Integrating the numerator in eq 4, gives

[ [{ } ] } { } ]] [{

tf 0.5 - 1 - RJ0tb + τ1 tf + tb 0.5 tf 0.5 1+ - 1+ /(tf + tb) (5) τ2 τ2

〈J〉 ) 2J0τ1 1 + 2RJ0

Writing 〈J〉 as u/v, where u and v are the numerator and denominator on the right hand side of eq 5,

d〈J〉 v(du/dtf) - u(dv/dtf) v(du/dtf) - u ) ) dtf v2 v2

(6)

since dv/dtf ) 1. The maximum global flux is given by 〈J〉max ) du/dtf, or

〈J〉

max

[{

) J0

}

tmax f 1+ τ1

-0.5

({

}

+ tb tmax f +R 1+ τ2

{

-

-0.5

(7)

The corresponding optimum forward filtration time, tmax can be obtained by solving (by trial and error) f

v(du/dtf) ) u

decline curve intersects these curves at points corresponding to different time constants. In general then, the flux equation can be written as

{ }

t J ) exp J0 f(t)

(9)

where f(t) defines the time constants. Assuming a simple linear expression for f(t), we write

-0.5

} )]

tmax f 1+ τ2

Figure 2. Typical microfiltration results of a yeast suspension through a 0.2 µm polypropylene membrane. (a) Forward filtration: ∆Pf ) 10 psi, φ ) 0.00126, τ1 ) 89.82 s (for model I); A ) 312.0 s and B ) 0.45 (for model II). (b) Reverse filtration: ∆Pb ) 10 psi, τ2 ) 4.98 s for model I and 363.6 s for model II. J0 ) 0.1463 cm/s.

(8)

Model II. Consider a family of exponential-decay curves as shown in Figure 1. The normalized flux

{

}

t J ) exp J0 A + Bt

(10)

where the constants A and B can be determined experimentally. Note that this model predicts a steadystate flux at large times (J ) J0 exp(-1/B) as t f ∞). For reverse filtration, we assume that the rate of flux increase is proportional to (J∞ - J-) where J∞ is the final flux (Nagata et al., 1989). Thus,

dJ-/dt ) k(J∞ - J-)

(11)

2922 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

maximum global flux is given by

[ {

〈J〉max ) J0 exp -

tmax f A + Btmax f

}

-

]

/τ2) (14) R{1 - exp(-tb/τ2)} exp(-tmax f The corresponding optimal forward filtration time can be estimated numerically. Notice that an equation written for reverse flux similar to model I, based on eq 10, would also reduce to eq 12, since at large times Jf J∞; consequently, B f 0. Thus, the model shows that reverse filtration flux (without irreversible fouling) should necessarily be an exponential function of time. Also, similar to the gel-polarization model, at given transmembrane pressure and shear rate values, for all feed concentrations, a fixed wall concentration and gel layer thickness are established during fouling. Since this layer offers the greatest resistance to reverse filtration, it follows that the time constant τ2 should be independent of feed concentration during fouling. Without fouling, the flux is given by

〈J〉 )

J0 (t - Rtb) tf + tb f

(15)

For t e Rtb, 〈J〉 e 0. The experimental data is therefore plotted as (tf - Rtb) versus flux. The models given above are valid for tf > Rtb. Note that the time is measured from t ) 0, in both models. Experimental Section

Figure 3. Typical microfiltration results of a yeast suspension through a 0.2 µm polyethersulfone membrane. (a) Forward filtration: ∆Pf ) 5 psi, φ ) 0.0036, τ1 ) 0.9 s for skin and 4.24 s for support filtration (model I); A ) 25.92 s, B ) 0.23 for skin and A ) 119.22 s, B ) 0.031 for support filtration (model II). (b) Reverse filtration: ∆Pb ) 5 psi, τ2 ) 217.2 s for skin and 247.2 s for support filtration for model I and 492.0 s for skin and 552.0 s for support filtration for model II. J0 ) 0.1444 cm/s.

with the boundary condition, at t ) 0, J- ) 0. Integrating eq 11,

J-/J∞ ) 1 - e-t/τ2

(12)

where τ2 ) 1/k. J∞ can be replaced by RJ0 without significant error, provided there is no irreversible fouling (see Figure 2b). The global flux is then given by

〈J〉 )

[∫ exp{- A +t Bt} dt - Rt +

J0 tf + tb

tf

0

b

]

Rτ2{1 - exp(-tb/τ2)} exp(-tf/τ2) (13) Analytical integration of the expression exp[-t/(A + Bt)] is possible and involves exponential integrals. It is more convenient, however, to evaluate it numerically. The

Batch cell microfiltration experiments were conducted with yeast suspensions to test the two models proposed. Amicon batch filtration cells (Model 8050, Amicon, Inc., Beverly, MA) with a membrane superficial area of 13.4 cm2; symmetric polypropylene and asymmetric polyethersulfone membranes (Akzo Nobel, Accurel PP, Type 2E HF, Micro PES, Type 2F, Poretics Corporation, Livermore, CA), both of 0.2 µm nominal pore size were used. A new piece of polypropylene membrane was used for each experimental run. The asymmetric polyethersulfone membranes were cleaned by dipping in 0.1 N HNO3 and NaOH solutions successively, for 30 min each and washed with deionized water. This restored the water flux to within 1% of its original value and enabled us to conduct experiments with the skin facing the upstream side (called the skin filtration) and the support layer facing the upstream side (called the support filtration) with the same membrane. The skin and support water filtration fluxes were within 1% of each other. The medium for growing yeast was prepared by adding 5 g of KH2PO4, 2 g of (NH4)2SO4, 0.4 g of MgSO4 (crystal), 3 g of yeast extract, and 100 g of glucose to 1 L of deionized water. One gram of Fleischmann’s active dry yeast was then added to the medium and allowed to grow for 36-48 h. The concentration was monitored with a Hach turbidimeter (Model 2100 A, Hach Instruments, Ames, IA). After achieving constant turbidity (indicating that the cells are killed), the suspensions are stored at 10 °C. Suspensions of different concentrations are made by diluting the mother broth with deionized water. The pH of the suspensions was adjusted to 4.6 with an acetic acid-sodium acetate buffer.

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2923

Figure 4. Variation of model parameters with volume fraction of yeast. (a) A, (b) B, (c) τ1, and (d) τ2.

Figure 5. Global flux versus net forward filtration time. R ) 1.0, tb ) 2.0 s, φ ) 0.03. τ1 ) 0.945 s and τ2 ) 14.92 s for model I. A ) 4.5 s, B ) 0.4, and τ2 ) 20.0 s for model II. Data taken from Redkar and Davis (1995).

All the experiments were conducted at 27 ( 1 °C. A peristaltic pump was used to pump the suspension (∼1 L volume) through the membrane cell (and maintain the transmembrane pressure), and no appreciable temperature changes were observed. Experiments with polypropylene membranes were conducted at a trans-

membrane pressure of 10 ( 1 psi, while those with polyethersulfone membranes were conducted at 5 ( 1 psi, because the polyethersulfone membranes had higher water permeability. Each experimental run included the following steps: (i) evaluation of the pure water flux, (ii) dead-ended filtration of the yeast suspension with the permeate returned to the feed, (iii) cleaning the flow system with deionized water, and (iv) once-through (with permeate not returned to the feed) reverse filtration of deionized water with the fouled membrane. The membranes were cleaned as required. With polyethersulfone membranes, skin filtration was followed by cleaning and support filtration in every case. This was done so as to eliminate the effect of any irreversible pore plugging that might have occurred during support layer filtration. Results and Discussion The proposed models are evaluated for their applicability using the results of batch cell experiments. The models are then employed to predict the results of cross-flow filtration with high-frequency reverse filtra-

2924 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

Figure 6. Global flux versus net forward filtration time for various values of R. The arrow shows the direction of R increase for (a) model I and (b) model II. tb ) 2.0 s and φ ) 0.03. Data taken from Redkar and Davis (1995).

tion published by Redkar and Davis (1995). Finally, optimization of the model parameters is discussed. Evaluation. Figure 2 shows the typical results of microfiltration of a yeast suspension with a polypropylene membrane. Both models fit reasonably well for forward filtration, while for reverse filtration, model II is far better than model I. The results were similar for polyethersulfone membranes, both for skin and support layer filtration (Figure 3). These results validate the proposed theory that reverse filtration flux should be an exponential function of time and that the cake removal is gradual. Note that with support filtration the values of τ1 and A are increased significantly, while the value of τ2 is almost unchanged. Fitted values of A, B, τ1, and τ2 are plotted as functions of the volume fraction of yeast, φ, in Figure 4. The plots are made on a log-log scale only to improve the visibility of the data. Reported values of τ1 and optimized values of A and B for the data of Redkar and Davis (1995) are shown for comparison. In general, it is seen that A, B and τ1 decrease with increasing feed concentration (Figure 4 (a), (b) and (c)). As reported by Jonsson and Wenten (1994), although reversed membrane filtration reduced the rate of the flux decline, the ultimate flux was almost zero, because of pore plugging. This is reflected in the drastic reduction of the value of B for support filtration and increase in the values of A and τ1. The values of τ2 remained independent of concentration as was suggested in the theoretical section (Figure 4d). Larger differences in the values of τ2 than were obtained for skin and support filtration are expected. Similar time constants were observed probably because the batch cell experiments were not conducted for time periods long enough to cause significant pore plugging. Another possible explanation is that it is more difficult for cells/suspended solids to move into a porous

Figure 7. Global flux versus net forward filtration time for different reverse filtration times. R ) 1.0 and φ ) 0.03. The arrow shows the direction of tb increase for (a) model I and (b) model II. Data taken from Redkar and Davis (1995).

Figure 8. Global flux versus net forward filtration time for different bulk volume fractions. R ) 1.0 and tb ) 2.0 s. τ1 ) 8.9, 0.945, and 0.42 s for model I, A ) 24.0, 4.5, and 1.5 s, B ) 0.5, 0.4, and 0.4 for model II, for φ ) 0.003, 0.03, and 0.06, respectively. τ2 ) 14.92 s for model I and 20.0 s for model II for all φ. Data taken from Redkar and Davis (1995).

structure that progressively becomes more dense, than to move out of a porous structure that becomes more and more open. In the former case, the suspended solids fill the porous structure slowly causing a low rate of flux decline. The eventual thickness of the mass transport boundary layer, however, is increased by the magnitude of the thickness of the support layer itself. This results in a very low final flux value. In the latter case, the highest resistance is experienced near the skin, and is similar to the resistance offered by a gel layer. Thus, the values of the time constant, τ2, are similar for both skin and support layer filtration. Note that for all concentrations tested, the values of τ2 are slightly higher for support layer filtration. Application. Figure 5 shows both models applied to experimental data given by Redkar and Davis (1995). In this case, R ) 1.0 and tb ) 2.0 s. The fits are optimized with respect to the parameters τ1 and τ2 in the case of model I and A, B, and τ2 in the case of model

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2925

Figure 9. Optimum flux and net forward filtration time as functions of back-pulsing time. φ ) 0.03 and R ) 1.0. (a) τ1 ) 0.945 s and τ2 ) 14.92 s for model I. (b) A ) 4.5 s, B ) 0.4, and τ2 ) 20.0 s for model II.

II. The optimization is achieved by minimizing the value of ∑((Jexpt - Jmodel)/Jexpt)2. The optimum values are τ1 ) 0.945 s and τ2 ) 14.92 s for model I and A ) 4.5 s, B ) 0.4, and τ2 ) 20.0 s for model II. These values are used to evaluate the performance of these models with other data. Figure 6 shows the global average flux as a function of the net forward filtration time. Both models predict the trend of increasing global flux with decreasing pressure ratio, R. The experimental data for R ) 0.5 and R ) 1.0 seem to follow this. At R ) 0, however, there is no actual reverse filtration and the membrane channel behaves as a channel with solid walls for the duration of back-pulsing. This causes some cleaning (because of absence of hydrodynamic forces dragging the cells/suspended solids to the membrane surface) and an improvement in performance, but not as high as with reverse filtration. With increasing R, loss of permeate becomes significant and hence the reduction in global flux. With highly fouling materials like proteins, a threshold value of R will be required to obtain a global flux higher than that obtained with normal filtration. In such cases, the optimum value of R will be sharply

Figure 10. Optimum flux and net forward filtration time for different pressure ratios for model I. τ1 ) 0.945 s and τ2 ) 14.92 s. φ ) 0.03. (a) tb ) 2.0 s, and (b) tb ) 0.01 s.

defined. A fouling factor similar to that discussed by Xu et al. (1995) can be introduced into the present models, but will have to be determined experimentally. For highly fouling solutions/suspensions, a hybrid process of pressure and electroosmotic back-washing may be more effective (Bowen and Sabuni, 1995). Figure 7 shows the global average flux plotted against the net forward filtration time for various back-pulsing times. Both models predict the trend of higher global flux with decreasing back-pulsing times, correctly, although model II does a slightly better job. Figure 8 shows the flux as a function of the net forward filtration time for three different concentrations. Notice that the value of τ2, the time constant for reverse filtration, remains unaffected for both models. The optimized values of τ1 (for model I) are 8.9, 0.945, and 0.42 s for φ ) 0.003, 0.03 and 0.06, respectively. These are comparable to the values reported by Redkar and Davis (1995) (τ1 ) 6.9, 0.67, and 0.32 s for φ ) 0.003, 0.03, and 0.06, respectively). The discrepancy arises, because of the ill-fitting nature of model I for reverse filtration (see Figure 2b). The optimized values for model II are A ) 24.0, 4.5, and 1.5 s, and B ) 0.5, 0.4, and 0.4, for φ ) 0.003, 0.03, and 0.06, respectively. τ2 remained unchanged. While τ1 and A decrease with

2926 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

Figure 11. Optimum flux and net forward filtration time for different pressure ratios for model II. A ) 4.5 s, B ) 0.4, and τ2 ) 20.0 s. φ ) 0.03. (a) tb ) 2.0 s, and (b) tb ) 0.01 s.

increasing concentration, B seems to be a weak function of concentration (see Figure 4b). The values of B calculated from the steady-state fluxes are 0.261, 0.242, and 0.224, for φ ) 0.003, 0.03, and 0.06, respectively. Table 1 shows the summary of the predicted results and compares them with measured values. While both models predict values of 〈J〉max and tmax well, the f steady-state-flux values, Js, predicted by model II are off by an order of magnitude. This is because the differences in the values of B are amplified by the exponential expression.

Figure 12. Effect of model I parameters on optimum flux and net forward filtration time. R ) 1.0, tb ) 2.0 s, and φ ) 0.03. (a) Effect of τ2; τ1 ) 0.945 s. (b) Effect of τ1; τ2 ) 14.92 s.

Optimization. It is generally observed that decreasing the back-pulsing time, tb (Redkar and Davis, 1995; Jonsson and Wenten, 1994; Xu et al., 1995), and increasing τ1 by reversing the membrane (Jonsson and Wenten, 1994) improve the performance in terms of flux achieved. The proposed models can be used to optimize these parameters for better performance. Both models predict improved fluxes with decreasing back-pulsing times (Figure 9). Notice that an increased flux is weighed against a smaller forward filtration time, tf, and

Table 1. Theoretical and Measured Maximum Fluxes and Corresponding Optimum Forward Filtration Times model I

measuredb

model II

modelb

conditionsa

〈J〉max (cm/s)

tmax (s) f

〈J〉max (cm/s)

tmax (s) f

Js (cm/s)

〈J〉max (cm/s)

tmax (s) f

Js (cm/s)

〈J〉max (cm/s)

tmax (s) f

R ) 1.0 R ) 0.5 R ) 0.0 tb ) 0.5 tb ) 1.0 tb ) 4.0 φ ) 0.003 φ ) 0.06

0.062 0.067 0.07 0.10 0.082 0.042 0.110 0.042

7 7 7 2.5 4 11 10 6

0.07 0.075 0.082 0.112 0.093 0.042 0.110 0.045

7 7 7 2.5 5 11 10 5

0.082 0.082 0.082 0.082 0.082 0.082 0.1353 0.082

0.074 ( 0.009 0.077 ( 0.008 0.045 ( 0.003 0.119 0.102 ( 0.006 0.045 ( 0.006 0.116 ( 0.006 0.044 ( 0.004

7 7 8 3 6 8 9 8

0.0028 0.0028 0.0028 0.0028 0.0028 0.0028 0.0038 0.0020

0.033 0.044 0.064 0.066 0.048 0.021 0.091 0.020

19 10 4 4 8 48 19 25

a Unless otherwise specified, the operating conditions are R ) 1.0, t ) 2.0 s, and φ ) 0.03. b Measured data and model predictions from b Redkar and Davis (1995).

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2927

Figure 10 shows predictions by model I for 〈J〉max and tmax as functions of the pressure ratio, R for two backf pulsing times. Generally, using the smallest possible pressure ratio is advisable (provided there is no irreversible fouling), but at small back-pulsing times, both 〈J〉max and tmax become insensitive to R. Similar results f are obtained from model II (Figure 11). Except at low values of τ2, 〈J〉max and tmax are relaf tively independent of τ2 for given τ1 (Figure 12a). On the other hand, increasing τ1 has a significant and beneficial effect on performance (Figure 12b). A larger filtration cycle can be used, while obtaining a larger flux with a membrane that has a smaller tendency to foul (i.e., has a larger τ1). Model II shows a similar relationship between A and 〈J〉max and tmax (Figure 13a). As shown in Figure 13b, f increasing the value of B also results in better performance. Again, both 〈J〉max and tmax are relatively f independent of τ2 at large values of τ2. Clearly, membrane systems with large values of τ1 or A and B and low values of τ2 need to be developed. Higher values of the parameter τ1 and A represent a decrease in the rate of fouling, while a higher value of B represents a decrease in the amount of fouling. Thick, porous membranes seem to offer better performance capabilities, as is suggested by the reversed membrane filtration results. However, this would reduce the value of the parameter B, which changes with the orientation and properties of the membrane. An optimal arrangement can be arrived at by analyzing the process parameters as shown above. Conclusions Two semiempirical models have been proposed to predict results of continuous cross-flow filtration with periodic reverse filtration. Results obtained from batch cell were used to validate the models. Both models have also been applied to results published by Redkar and Davis (1995) with a high degree of accuracy. It is shown that continuous cross-flow filtration with back-shocking can be optimized in terms of the transmembrane pressure ratio, R, and back-pulsing time, tb. Directions of further developments are indicated. Acknowledgment We are especially honored to be part of the Eli Ruckenstein Special Issue of Industrial & Engineering Chemistry Research. Professor Ruckenstein’s work has had an important influence on our research. We thank him for this. We also thank Roland Bu¨chele, Akzo Nobel Faser AG, Wuppertal, Germany, for supplying the polypropylene and polyethersulfone microfiltration membranes. Nomenclature

Figure 13. Effect of model II parameters on optimum flux and net forward filtration time. R ) 1.0, tb ) 2.0 s, and φ ) 0.03. (a) Effect of A; B ) 0.4 and τ2 ) 20.0 s. (b) Effect of B; A ) 4.5 s and τ2 ) 20.0 s. (c) Effect of τ2; A ) 4.5 s and B ) 0.4.

a smaller filtration cycle. Also, decreasing the backpulsing time below 0.01 s will not improve the performance significantly for the given process parameters.

A ) time constant for forward filtration in model II, s B ) constant for forward filtration in model II 〈J〉 ) average permeate flux per cycle, cm/s 〈J〉max ) maximum average permeate flux cycle, cm/s J0 ) clean membrane flux, cm/s Js ) steady-state permeate flux, cm/s ∆P ) transmembrane pressure, psi or kPa ∆Pb ) magnitude of reverse filtration transmembrane pressure, psi or kPa ∆Pf ) magnitude of forward filtration transmembrane pressure, psi or kPa

2928 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 t ) time elapsed since the start of experiment, s tb ) reverse filtration time, s tf ) forward filtration time, s ) forward filtration time at which maximum average tmax f flux is observed, s Greek Letters R ) ratio of reverse and forward-filtration pressures, ∆Pb/ ∆Pf τ1 ) time constant for forward filtration in model I, s τ2 ) time constant for reverse filtration in models I and II, s φ ) bulk volume fraction of cells

Literature Cited Bowen, W. R.; Sabuni, H. A. M. Electroosmotic Membrane Backwashing. Ind. Eng. Chem. Res. 1994, 33, 1245. Jonsson, G.; Wenten, I. G. Control of Concentration Polarization, Fouling and Protein Transmission of Microfiltration Processes within the Agro Based Industry. Presented at the Workshop on Membrane Technology in Agro Based Industry, Kualalumpur, 1994. Matsumoto, K.; Katsuyama, M.; Ohya, H. Separation of Yeast by Cross-Flow Filtration With Back-Washing. J. Ferment. Technol. 1987, 65 (1), 77. Matsumoto, K.; Kawahara, M.; Ohya, H. Cross-Flow Filtration of Yeast by Microporous Ceramic Membrane with Backflushing. J. Ferment. Technol. 1988, 62 (2), 199. Nagata, N.; Herouvis, K. J.; Dziewulski, D. M.; Belfort, G. CrossFlow Membrane Microfiltration of a Bacterial Fermentation Broth. Biotechnol. Bioeng. 1989, 34, 447.

Nikolov, N. D.; Mavrov, V.; Nikolova J. D. Ultrafiltration in a Tubular Membrane under Simultaneous Action of Pulsating Pressures in Permeate and Feed Solution. J. Membr. Sci. 1993, 83, 167. Nipkow, A.; Zeikus, J. G.; Gerhardt, P. Microfiltration Cell-Recycle Pilot System for Continuous Thermoanaerobic Production of exo-β-Amylase. Biotechnol. Bioeng. 1989, 34, 1075. Redkar, S. G.; Davis, R. H. Crossflow Microfiltration of Yeast Suspensions in Tubular Filters. Biotechnol. Prog. 1993, 9, 625. Redkar, S. G.; Davis, R. H. Crossflow Microfiltration with High Frequency Reverse Filtration. AIChE J. 1995, 41, 501. Rodgers, V. G. J.; Sparks, R. E. Reduction of Membrane Fouling in Protein Ultrafiltration. AIChE J. 1991, 37, 1517. Rodgers, V. G. J.; Sparks, R. E. Effect of Transmembrane Pressure Pulsing on Concentration Polarization. J. Membr. Sci. 1992, 68, 149. Rodgers, V. G. J.; Miller, K. D. Analysis of Steric Hindrance Reduction in Pulsed Protein Ultrafiltration. J. Membr. Sci. 1993, 85, 39. Rodgers, V. G. J.; Sparks, R. E. Effects of Solution Properties on Polarization Redevelopment and Flux in Pressure Pulsed Ultrafiltration. J. Membr. Sci. 1993, 78, 163. Xu, Y.; Dodds, J.; Leclerc, D. Optimization of a Discontinuous Microfiltration-Backwash Process. Chem. Eng. J. 1995, 57, 247.

Received for review December 1, 1995 Revised manuscript received May 6, 1996 Accepted May 7, 1996X IE950719T X Abstract published in Advance ACS Abstracts, August 15, 1996.