Protein Fouling of Asymmetric and Composite Microfiltration

Initial fouling occurs by pore blockage, with a cake layer then forming over those regions covered by foulant. Model calculations are in excellent agr...
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Ind. Eng. Chem. Res. 2001, 40, 1412-1421

Protein Fouling of Asymmetric and Composite Microfiltration Membranes Chia-Chi Ho and Andrew L. Zydney* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716

Although many microfiltration membranes have asymmetric or composite structures, there is little understanding of the effects of this multilayer structure on fouling. A new model was developed that explicitly accounts for fluid flow through two layers: an upper layer with noninterconnected pores and a substructure with highly interconnected pores. Initial fouling occurs by pore blockage, with a cake layer then forming over those regions covered by foulant. Model calculations are in excellent agreement with experimental data for the filtrate flux and resistance during constant-pressure filtration of bovine serum albumin. The highly interconnected pores within the support structure reduce the rate of flux decline by shunting more fluid through the open pores. The extra resistance provided by the support also reduces the relative importance of the protein deposit. The results provide important insights into the effects of pore morphology on membrane fouling. Introduction Membrane microfiltration is currently used in a variety of applications, including cell harvesting, sterile filtration, and particle removal for the production of ultrapure water. Although most microfiltration processes use homogeneous (or symmetric) membranes, there is growing interest in the use of both asymmetric and composite membranes. This is particularly true for particle removal applications in the semiconductor industry1 and virus clearance in biopharmaceutical applications.2 Asymmetric membranes have a very thin skin layer that determines the membrane selectivity and a relatively thick porous substructure that provides the necessary mechanical stability. Composite membranes can be produced by dip coating, interfacial polymerization, or plasma polymerization.3 The different layers in these membranes are chosen to provide the desired combination of selectivity, throughput, and mechanical and chemical stability. Ho and Zydney4 have recently shown that the detailed pore morphology and structure can also have a strong effect on membrane fouling. Isotropic membranes with a highly interconnected pore structure foul more slowly than membranes with straight-through pores because the filtrate is able to flow under and around any pore blockage on the membrane surface as it percolates through the interconnected pore structure. In contrast, the straight-through pores in track-etch membranes are readily blocked by protein aggregates. Model composite membranes formed by a sandwich arrangement of a track-etch membrane in series with an isotropic membrane appear to have an intermediate rate of fouling because of the percolating flow through the interconnected pores in the isotropic membrane. Ho and Zydney4 developed a semiquantitative model for the effect of the membrane morphology on the initial rate of protein fouling, but no attempt was made to quantify the fouling behavior over the full course of filtration. * All correspondence should be addressed to: Andrew Zydney, Department of Chemical Engineering, University of Delaware, Newark, DE 19716. Phone: 302-831-2399. Fax: 302-831-1048. E-mail: [email protected].

Several studies of protein fouling during microfiltration5,6 have shown that the initial flux decline is due to the deposition of large protein aggregates on the membrane surface. These aggregates block the membrane pores, with continued filtration leading to the growth of a protein cake or deposit on the membrane surface. Ho and Zydney7 recently developed a mathematical model for the flux decline through track-etched microfiltration membranes with straight-through pores that accounts for both the pore blockage and the cake growth phenomena. The objective of this study was to extend this model to include the effects of the complex pore morphology and internal flow distribution in asymmetric and composite (multilayer) membranes on the fouling behavior. The model calculations were shown to be in excellent agreement with experimental data for bovine serum albumin (BSA) filtration through a variety of membranes with different internal pore structures. Model Development To simplify the mathematical development, the membrane was assumed to be composed of two distinct structural layers, each having a given resistance to fluid flow (R1 and R2). Fluid flow in the top layer (R1) is assumed to occur through a structure with noninterconnected pores, whereas the bottom layer (R2) is assumed to have a highly interconnected pore structure, as is characteristic of the substructure in most asymmetric membranes. The assumption of a noninterconnected pore structure in the top layer will clearly be valid for the composite membranes studied by Ho and Zydney4 in which the upper layer was formed using a track-etch polycarbonate membrane. Most asymmetric membranes can also be described as having a noninterconnected pore structure in the skin layer as the skin is so thin that there is insufficient time (or distance) for the fluid to flow around and under any pore blockage on the membrane surface. Any transverse or lateral flow in these asymmetric membranes will occur within the relatively thick substructure. Figure 1 shows a schematic representation of the membrane structure. The initial fouling during protein microfiltration occurs by the deposition of large protein aggregates on

10.1021/ie000810j CCC: $20.00 © 2001 American Chemical Society Published on Web 02/10/2001

Ind. Eng. Chem. Res., Vol. 40, No. 5, 2001 1413

Figure 1. Schematic diagram showing the underlying structure of the asymmetric and composite membranes.

the membrane surface, as discussed by Kelly et al.8 We assume that these “blocked” pores allow some fluid flow through the blockage, as discussed by Ho and Zydney.7 The resistance of this protein blockage (Rp) increases with time as additional protein adds to the growing deposit on the membrane surface. The total fluid flow rate through the upper layer of the membrane is thus given by the sum of the flow rates through the open and blocked pores

Q ) JopenAopen + JblockedAblocked

Jblocked )

∆P1 µR1 ∆P1

µ(R1 + Rp)

(2) (3)

∆P2 µR2

(4)

where ∆P2 is the pressure drop across the bottom layer. Fluid flow through the bottom layer of the membrane is assumed to occur throughout the total membrane area

A ) Aopen + Ablocked

(5)

because of the highly interconnected pore structure in

(6)

where X is the fraction of open pores at any time (X ) Aopen/A) and Q0 is the volumetric filtrate flow rate through the unfouled membrane

Q0 )

∆P A µ(R1 + R2)

(7)

The filtrate fluxes through the open (Jopen) and blocked (Jblocked) pores can also be evaluated in terms of the fraction of open pores using eqs 1-6 as

R12 + R1R2 + R1Rp + R2Rp Jopen ) 2 J0 R +R R +R R +R R X 1

where µ is the solution viscosity and ∆P1 is the pressure drop across just the top layer of the membrane including the protein deposit. Note that, in writing eq 1, we have implicitly assumed that the flow through every blocked pore is the same, i.e., that the resistance of the protein deposit is uniform over the membrane surface even though different pores are blocked at different times. Ho and Zydney7 examined the spatial variation of the protein resistance for flow through membranes with noninterconnected pores and showed that the inhomogeneity of the protein deposit has a negligible effect on the total flow rate because of the inherent self-leveling aspect of the deposition process. In this case, the pores that are covered first have a slower rate of cake growth because of the smaller filtrate flux through these regions with greater resistance. This spatial variation in the cake growth rate leads to a relatively uniform thickness for the overall deposit. The total filtrate flow rate through the bottom layer of the membrane must be equal to that through the upper layer

Q)

R12 + R1R2 + R1RpX + R2RpX Q ) Q0 R12 + R1R2 + R1Rp + R2RpX

(1)

where Aopen and Ablocked are the areas of the open and covered regions on the membrane surface, respectively. The fluxes through the open and blocked pores are both given by Darcy’s law

Jopen )

this layer. Thus, the effects of any pore blockage on the upper layer are “lost” in the bottom layer because the lateral fluid flow makes the entire substructure equally accessible to the filtrate. Note that a small pressure gradient in the transverse direction exists in the upper region of the bottom layer because of the lateral flow. However, this region is sufficiently thin relative to the overall thickness of the substructure that this effect can be neglected. This is discussed in more detail subsequently. Equations 1-5 can be combined to evaluate the total filtrate flow rate through the fouled membrane in terms of the total pressure drop across the two-layer structure (∆P), yielding

Jblocked ) J0 R

1

2

1

p

2

R12 + R1R2 2 1

(8)

p

+ R1R2 + R1Rp + R2RpX

(9)

Note that Jopen actually becomes greater than J0 as the pores become blocked, i.e., as X decreases, because the fluid flow is shunted away from the blocked pores and through the open pores. This effect is not seen in a homogeneous (single-layer) membrane with noninterconnected pores (i.e., when R2 ) 0), because there is no opportunity for the fluid to flow laterally as it moves through such a pore structure. To evaluate the filtrate flow rate as a function of time, we need to develop expressions for the fraction of open pores (X) and the protein layer resistance (Rp) in terms of the filtration time. The rate of pore blockage is assumed to be proportional to the convective flux of protein aggregates toward the open (unblocked) surface of the membrane7

dX ) -RJopenXCb dt

(10)

where Cb is the bulk protein concentration and R is a pore blockage parameter that is equal to the membrane area blocked per unit mass of protein convected to the membrane surface.7 R will be proportional to the fraction of protein present as large protein aggregates, with each aggregate that is convected to the membrane assumed to deposit on the surface. Equation 10 also provides the basis for the development of the classical pore blockage model2,9 with Jopen ) Q/A since flow only occurs through the open pores. Note that eq 10 neglects any back transport or aggregate removal from the membrane surface, both of which should be negligible in dead-end

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or normal flow filtration processes. The growth of the protein deposit resistance (Rp) is assumed to be proportional to the convective protein transport through the existing protein layer

dRp dmp ) R′ ) f ′R′JblockedCb dt dt

(11)

where R′ is the specific protein layer resistance and mp is the mass of the protein deposit on the membrane per unit area of the deposit. f ′ is the fraction of the bulk protein that contributes to the growth of the deposit. It will be equal to the fraction of protein present in the form of large aggregates under conditions where only the aggregates are able to add to the growing deposit. Equation 11 is also used to develop the classical cake filtration model,2,9 although in that case, Jblocked is set equal to Q/A as the cake layer is assumed to be uniform over the entire membrane surface. Equations 8-11 can be integrated numerically using an appropriate numerical scheme. Alternatively, eqs 10 and 11 can be combined to give

f ′R′Jblocked dRp )dX RJopenX

(12)

x

1-

[(

) (

)]

Rp0 2 f ′R′ ln(X) - Rp0 1 + R1 R 2R1

- R1 (13)

where we have used the condition that Rp ) Rp0 when X ) 1. Rp0 is thus the additional resistance provided by the protein aggregate when it first lands on the membrane surface. According to eq 13, the protein layer resistance Rp is a unique function of the fraction of open pores (X), irrespective of the value of R2. This unusual result occurs because the ratio of Jblocked to Jopen depends only on R1, Rp, and X. The resistance of the substructure (R2) alters the flux through the open and blocked pores in exactly the same way, making Jopen/Jblocked independent of R2. Equation 13 can be used to evaluate Jopen as a function of X, converting eq 10 into a separable ordinary differential equation. Equation 10 can then be solved numerically for X(t) using a simple Euler integration, or it can be integrated analytically with the final result expressed in terms of the imaginary error function. Once X is known as a function of time, the resistance of the protein deposit can be evaluated directly from eq 13 with the filtrate flow rate given by eq 6. Approximate Solution. An approximate analytical solution for the flux can be developed when the rate of cake growth is relatively slow compared to the initial rate of aggregate deposition, i.e., when

| |

1 dX 1 dRp , Rp0 dt X dt

Using eqs 8-11, this corresponds to

R1 (R1 + Rp0)Rp0

(f R′R′) , 1

ln X +

R2Rp0(X - 1 - ln X) 2

R1 + R1R2 + R1Rp0 + R2Rp0

) -RJ0Cbt (15)

The total flow rate is again given by eq 6, with X evaluated as function of time by iterative solution of eq 15 and Rp given by eq 13. Note that, when R2 ) 0, i.e., for a homogeneous membrane, eq 15 predicts an exponential decay in the fraction of open pores with time, consistent with the analysis presented previously by Ho and Zydney.7 The presence of the membrane substructure, described by the second term on the left-hand side of eq 15, increases the rate of pore blockage, and thus decreases the value of X at any time t, by shunting more of the flow through the open pores. Materials and Methods

Equation 12 is a first-order ordinary differential equation that can be directly integrated using eqs 8 and 9 to evaluate Jopen and Jblocked to give

Rp ) R1

which ends up being valid for the system examined in this study. Under these conditions, the rate of pore blockage can be approximated using eq 10 with Jopen evaluated from eq 8 but with the resistance of the protein deposit assumed to be constant at its initial value, i.e., Rp ) Rp0. This reduces eq 10 to a simple differential equation for X, which can be integrated directly to give

(14)

Bovine serum albumin (BSA) solutions were prepared by dissolving crystalline BSA powder (Fraction V heatshock-precipitated BSA, catalog number A7906, Sigma) in PBS consisting of 0.03 M KH2PO4, 0.03 M Na2HPO4‚ 7H2O, and 0.03 M NaOH. The BSA solutions were freshly prepared before each experiment and used within 8 h of preparation. Data were obtained using 0.16-µm pore size polyethersulfone Omega membranes (Pall Filtron, Northborough, MA) and 0.2-µm pore size polycarbonate tracketched (PCTE) membranes (Osmonics, Minnetonka, MN). Composite membrane structures were formed using an 0.2-µm PCTE membrane placed in series with one or more 0.2-µm polyvinylidene fluoride (PVDF) membranes (Millipore, Bedford, MA) in a sandwich configuration. All filtration experiments were conducted using a 25mm-diameter stirred ultrafiltration cell (model 8010, Amicon Corp., Danvers, MA) connected to an airpressurized acrylic solution reservoir. The stirred cell and solution reservoir were initially filled with PBS, with the saline flux measured until steady state was attained (usually within 30 min). The stirred cell was then quickly emptied, refilled with a BSA solution, and attached to a fresh reservoir containing additional BSA solution. The system was repressurized (within 1 min), and the filtrate flow rate was measured by timed collection using a digital balance (model 1580, Sartorious, Edgewood, NY). At the end of the filtration, the stirred cell was rinsed with PBS, and the steady-state PBS flux was reevaluated. All experiments were performed at room temperature (22 ( 2 °C). The amount of protein deposited on the membrane was evaluated by direct weighing. A clean (dry) membrane was initially weighed using a Sartorious balance with an accuracy of 0.1 mg. The membrane was then used to filter a preset volume of 2 g/L BSA solution at 14 kPa. The membrane was then air-dried for 12 h and reweighed to estimate the mass of the deposited protein. Additional details on the experimental procedures are provided by Ho and Zydney.7

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Figure 2. Filtrate flux (top panel) and rate of flux decline (bottom panel) for filtration of 2 g/L BSA solutions through the PES, PCTE, PCTE/PVDF, and PCTE/4PVDF membranes at an initial flux of (1.7 ( 0.03) × 10-4 m s-1. Solid and dashed curves are model calculations using the full and approximate solutions, respectively, with the best-fit values of the model parameters given in Table 1.

Results and Analysis Model Validation and Parameter Estimation. Figure 2 shows typical experimental data for the normalized filtrate flow rate (top panel) and the rate of flux decline (bottom panel) as a function of time during the constant-pressure filtration of 2 g/L BSA solutions through the 0.16-µm asymmetric PES membrane, 0.2µm polycarbonate track-etch (PCTE) membrane, and two composite membranes formed by placing a PCTE membrane directly on top of either one or four layers of 0.2-µm PVDF membranes. In each case, the transmembrane pressure was adjusted to achieve an initial flux of (1.75 ( 0.03) × 10-4 m s-1, with the pressure values given in Figure 2. The initial flux measured with the BSA solution was within 10% of the steady-state saline flux evaluated immediately prior to the protein filtration. The rate of flux decline (K) was evaluated directly from the filtrate flow rate data as

K)-

1 dQ Q dt

( )

(16)

All derivatives were evaluated numerically using appropriate finite difference representations accurate to O(∆t2). The solid and dashed curves in Figure 2 are model calculations that are described subsequently. The flow rate declined most rapidly for the PES membrane, with the flow rate decreasing to less than

20% of its initial value after only 10 min of filtration. The flux decline for the composite membrane was less pronounced than that for the PCTE membrane alone, although the long-time behavior was similar. The effect of the underlying PVDF membranes on the flux through the PCTE composite membranes was quite pronounced. For example, the flow rate through the PCTE/4PVDF composite membrane is 65% greater than that through the PCTE membrane alone at t ) 18 min. Hydraulic permeability measurements and scanning electron microscopy of the individual layers in the composite membrane structure showed that all fouling occurred in the top PCTE layer; there was no observable fouling in any of the underlying PVDF membranes. The composite membranes also showed a distinct maximum in the instantaneous rate of flux decline, whereas the rate of flux decline for the PCTE membrane alone decreased monotonically with time. In addition, the initial rate of flux decline for the PCTE/PVDF and PCTE/4PVDF membranes was much lower than that for the PCTE membrane. The asymmetric PES membrane also showed a weak maximum in the rate of flux decline, with this maximum value occurring at very short filtration times. The solid curves in the top and bottom panels of Figure 2 are the results using the full model presented earlier in this manuscript. The dashed curves are the results for the approximate analytical solution (eq 15). The values of R1 and R2 for the composite membranes were determined from the resistance of the isolated PCTE [R1 ) (3.3 ( 0.3) × 1010 m-1] and PVDF membrane [R2 ) (4.5 ( 0.5) × 1010 m-1 per PVDF layer]. The resistance of the substructure for the asymmetric PES membrane [R2 ) (2.6 ( 0.2) × 1010 m-1] was evaluated from saline flux data obtained with a PES membrane in which the skin layer had first been removed by carefully scraping off the upper skin. The resistance of the skin layer [R1 ) (7.2 ( 0.3) × 1010 m-1] was then evaluated from the total membrane resistance by simply subtracting off the resistance of the substructure. The best-fit values of R, Rp0, and f ′R′ for all four membranes were determined by minimizing the sum of the squared residuals between the data and the model calculations using the method of steepest descent, and the results are summarized in Table 1. The R and Rp0 values for the PCTE membrane and for the composite membranes with the PCTE membrane as the top layer are very similar, with variations of less than 7% for R and less than 30% for Rp0. In contrast, the best-fit values of f ′R′ increase with increasing substructure resistance (R2). This behavior will be discussed in more detail subsequently. The larger value of R for the PES membrane is likely to be due to the smaller pore size, and thus the greater efficiency of aggregate capture, for the PES membrane. The slightly larger value of Rp0 for the PES membrane might be due to the difference in average size of the aggregates captured on the different pore size membranes or it could be due to differences in surface porosity or morphology between the PCTE and PES membranes. The additional resistance provided by the protein aggregates (Rpo) is nearly 10 times the resistance of the upper layer (R1) for both the PES and composite membranes, which leads to the fairly rapid initial flux decline seen in these experiments. The model calculations are in very good agreement with the experimental data for both the flow rate and the rate of flux decline, accurately capturing the maximum in the rate of flux decline (K) seen for the

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Table 1. Resistance Data and Best-Fit Parameter Values for Experiments in Figure 2 membrane

R1 (m-1 × 10-10)

R2 (m-1 × 10-10)

R (m2 kg-1)

Rp0 (m-1 × 10-11)

f ′R′ (m kg-1 × 10-12)

PES PCTE PCTE/PVDF PCTE/4PVDF

7.2 ( 0.3 3.3 ( 0.3 3.3 ( 0.3 3.3 ( 0.3

2.6 ( 0.2 4.5 ( 0.5 16.0 ( 2.0

10.6 ( 0.2 2.8 ( 0.1 2.9 ( 0.1 2.7 ( 0.1

5.5 ( 0.2 4.1 ( 0.2 3.9 ( 0.2 3.0 ( 0.5

3.5 ( 0.2 0.74 ( 0.03 2.2 ( 0.1 5.7 ( 0.2

asymmetric PES and composite PCTE/PVDF membranes. The calculations using the approximate model (dashed curves) are indistinguishable from those of the full model for the PES, PCTE, and PCTE/PVDF membranes, with only a small (less than 1.3%) difference seen for the PCTE/4PVDF membrane. This is consistent with the inequality in eq 14 as {R1/[(R1 + Rp0)Rp0]}(f ′R′/ R) ) 0.069 for the PES, 0.048 for the PCTE, and 0.7 for the PCTE/4PVDF membrane. The larger value for the PCTE/4PVDF membrane explains the observable deviation between the model and data in Figure 2. Similar agreement was obtained at other bulk protein concentrations using the same values of R, Rp0, and f ′R′. The resistance of the substructure for the asymmetric PES membrane is only about one-third as large as the resistance of the skin, whereas the resistance of the supporting structure in the PCTE/PVDF composite membranes is actually greater than that for the skin. The greater values of R2/R1 for the composite membranes lead to the more pronounced maximum in the rate of flux decline seen in the lower panel of Figure 2. Although the model parameters (R, Rp0, and f ′R′) were all evaluated directly from the flux decline data, the best-fit values of these parameters are in good agreement with independent estimates developed by Ho and Zydney.7 In particular, the pore blockage parameter R can be estimated directly from the size and concentration of the BSA aggregates as measured by light scattering4 as R ) 2.9 ( 2.6 m2/kg. Similarly, Rp0 can be estimated from the size of the aggregates using the Kozeny-Carman equation7 as Rp0 ) 4 × 1011 m-1. The specific resistance of the protein deposit (R′) can be evaluated from measurements of the saline flux through the fouled membrane. This is discussed in more detail in the next section. One of the implicit assumptions in the flux decline model for the asymmetric and composite membranes is that the flow in the membrane substructure is completely uniform even though the fluid only enters the lower region through distinct open pores in the upper layer. In reality, there must be some lateral flow, and thus a lateral pressure gradient, in the membrane substructure beneath the nonporous (or blocked) regions of the upper layer. The magnitude of this lateral flow was evaluated numerically using the theoretical model for flow in a porous membrane developed by Ho and Zydney.10 The fluid velocities in both the normal and transverse (parallel to the membrane surface) directions were assumed to be given by Darcy’s law expressions, yielding a modified form of Laplace’s equation for the local pressure. The ratio of the normal (kz) and transverse (kx) permeabilities for the PVDF membrane was evaluated experimentally by measuring the fluid flow rate through a membrane in which the upper and lower surfaces were blocked in an overlapping fashion.11 The data yielded kx/kz ) 1.3 ( 0.1, which is consistent with an isotropic pore structure in these homogeneous membranes. The pores in the PCTE membrane were assumed to be arranged in a hexagonal array, allowing a Krogh cylinder-type model to be used to describe the

Figure 3. Model simulations for the pressure distribution (top panel) and fluid streamlines (bottom panel) in the PCTE/PVDF composite membrane.

flow through a pore and the surrounding region of the membrane.10 Model simulations for the pressure distribution and fluid streamlines in the PCTE/PVDF composite membrane are shown in the top and bottom panels in Figure 3. Results are shown only for the bottom 0.5 µm of the 10-µm PCTE membrane and the top 0.5 µm of the 120-µm PVDF membrane to highlight the flow distribution at the interface between the top and bottom layers. The high permeability of the PVDF membrane allows the fluid to flow laterally under the nonporous regions of the PCTE membrane over a vertical distance of only about 0.2 µm, which is less than 0.2% of the total thickness of the PVDF layer. The maximum variation in the local hydrostatic pressure between the region directly under the polycarbonate and that directly under the pore was only 12 Pa, which is less than 0.15% of the pressure variation in the normal flow direction. Additional simulations performed with highly fouled membranes with greater spacing between the open pores showed that the lateral pressure variations within the substructure are negligible under all conditions. Thus, the assumption of uniform flow/ pressure throughout the lower layer provides a highly accurate description of the actual flow profiles in this region of the composite membrane. Similar results were obtained for the asymmetric PES membranes, with the flow becoming uniform within a very small distance into the substructure. Flux/Fouling Analysis. The much smaller values for the initial rate of flux decline seen for the composite membranes in Figure 2 can be understood more clearly by examining the variation in flux through the open and blocked pores during the course of the filtration. Model calculations are shown in Figure 4 for the PCTE, PCTE/ PVDF, and PCTE/4PVDF membranes using the parameter values in Table 1. The normalized flux through the

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Jblocked R1 ) J0 R1 + Rp0

Figure 4. Predicted values of the normalized flux though the open (top panel) and blocked pores (bottom panel) for the PCTE composite membranes.

open pores in the PCTE membrane (i.e., with R2 ) 0) is constant because the pressure drop (∆P) and membrane resistance (R1) both remain constant during the filtration. In contrast, the flux through the open pores in the composite membranes increases significantly as pores become blocked because more of the flow is shunted through the unblocked pores. In this case, the decline in the total flow rate through the membrane reduces the pressure drop across the bottom layer, which causes a concomitant increase in the pressure drop across the upper layer to maintain ∆P ) ∆P1 + ∆P2 ) constant. The net result is that the normalized flux through the open pores for the PCTE/4PVDF membrane becomes more than 5 times as large as that for the PCTE membrane alone for t > 30 min. At very long times, the total flow rate becomes very small, causing ∆P2 to approach zero, at which point

R2 Jopen )1+ J0 R1

(17)

from eqs 2 and 7 with ∆P1 ) ∆P. Thus, the long-time asymptote for the PCTE/PVDF membrane is Jopen/J0 ) 2.4, whereas that for the PCTE/4PVDF membrane is 5.5. The normalized flux through the blocked pores shortly after the start of the filtration, when only a single protein aggregate has been deposited on the pores, can be evaluated from eq 9 with X ) 1 as

(18)

which is completely independent of R2. The lack of any initial dependence on R2 arises because essentially all of the fluid flow occurs through the unblocked pores, so ∆P2 is simply equal to its initial value. The flux through the blocked pores in the PCTE membrane decreases slowly with time because of the growth in the thickness and resistance of the protein deposit. In contrast, the flux through the blocked pores in the composite membranes initially increases with filtration time because of the increase in ∆P1 arising from the reduction in ∆P2 that occurs as the flux declines. Jblocked then goes through a maximum before decreasing at long times because of the increase in resistance of the protein deposit. However, Jblocked/J0 for the composite membranes remains larger than that for the PCTE membrane alone because the percentage increase in the total resistance to flow (R1 + R2 + Rp)/(R1 + R2) is reduced as R2 increases. Thus, the effect of the protein deposit is less pronounced on a membrane with greater initial resistance to flow. The larger values of Jopen/J0 and Jblocked/J0 for the composite membranes are what cause the greater overall flux through these membranes, as seen in Figure 2. The maximum in Jblocked/J0, in combination with the initial increase in Jopen/J0, causes the initial increase and then the maximum in the rate of flux decline (K). Although the presence of the support structure increases the flux through both the open and blocked pores, it also increases the rate at which the pores become blocked. This is shown in Figure 5 for the PCTE composite membranes, with the model calculations again developed using the parameter values in Table 1 for both the full (solid curves) and approximate (dashed curves) solutions. The results for the PCTE membrane alone show a linear dependence of ln(X) with time. The fraction of open pores at any time t decreases with increasing resistance of the bottom layer (R2) because of the greater convective flux of aggregates toward the open pores, as seen in Figure 4. The calculations using the approximate solution are identical to those for the full model when R2 ) 0 as Jopen remains constant under these conditions (the key approximation in developing eq 15). There is some deviation between the two models for the composite membranes, especially at long times. The approximate solution overpredicts the fraction of open pores because the actual value of Jopen will be greater than that predicted by assuming Rp ) Rp0. The filtrate flow rate data in Figure 2 have been replotted in terms of the total resistance

Rtotal )

∆P A µQ

(19)

in Figure 6. The solid and dashed lines are the results from the full model and approximate solutions, respectively, using the parameter values in Table 1. The PCTE membrane has the smallest initial resistance, whereas the PCTE/4PVDF composite has the greatest initial resistance. The total resistance for the PES membrane increases sharply at the start of the filtration, becoming greater than that for the PCTE/4PVDF composite for times between 5 and 30 min. This rapid initial increase is due to the greater rate of pore blockage for the smaller-pore-size PES membrane as characterized by

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Figure 5. Model predictions for the fraction of open pores as a function of filtration time for the PCTE composite membranes. The solid curves are the full model, and the dashed curves are the approximate analytical solution.

Figure 7. Flux decline analysis for BSA filtration through the PES, PCTE, and PCTE/4PVDF membranes. Solid curves are calculations using the full model with parameter values given in Table 1.

volume. The exponent n is characteristic of the fouling model, such that n ) 0 for cake filtration, n ) 1 for intermediate blocking, n ) 3/2 for pore constriction (also called standard blocking), and n ) 2 for complete pore blocking,9 with these values determined for homogeneous membranes with completely noninterconnected pore structures. The filtrate flow rate data in Figure 2 have been replotted in the form suggested by eq 20 with the required derivatives evaluated in terms of the flow rate as

Figure 6. Total resistance as a function of time for BSA filtration through PES, PCTE, PCTE/PVDF, and PCTE/4PVDF membranes. Solid and dashed curves are model calculations using the full and approximate solutions, respectively, with the best-fit values of the model parameters given in Table 1.

the higher R value. The total resistance for the PCTE composite membranes increases quite slowly at first, but Rtotal then begins to increase more rapidly before leveling out to some extent at very long times. The net result is that the total resistance plot is concave up during the early stage of the filtration and becomes concave down at long times. Tracey and Davis5 have shown that this behavior is consistent with a transition from a poreblockage to a cake-formation fouling mechanism for flow through a homogeneous membrane with straightthrough noninterconnected pores. The transition in concavity, given by the inflection point in the Rtotal curves, is shifted to a shorter time as R2 increases because of the faster rate of pore blockage for the composite membrane structures (Figure 5). One of the classical approaches to analyzing membrane fouling is to plot the flux decline data in the form6,7,9

d2t dt n )k 2 dV dV

( )

(20)

where t is the filtration time and V is the total filtered

dt 1 ) dV Q

(21)

1 dQ d2 t )- 3 2 dV Q dt

(22)

dQ/dt was determined by numerically differentiating the flow rate versus time data using the IMSL routine DCSDER to take the derivative of a series of piecewise cubic polynomials that were fit to the raw data. The solid curves in Figure 7 are the full model calculations using the parameter values in Table 1. At low dt/dV (high J), the data for the PCTE membrane yield a linear relationship with slope approximately equal to 2.0, consistent with a pore-blockage mechanism. The model calculations yield a slope of 1.9, with the slight reduction from 2 arising from the fluid flow though the blocked pores. This is discussed in more detail by Ho and Zydney.7 The data for the PES and PCTE/4PVDF membranes have a slope much greater than 2 for very small dt/dV, i.e., at the start of the filtration. This behavior cannot be explained by any of the classical fouling models, but it is very accurately described by the model accounting for the effect of the substructure on membrane fouling. In this case, the resistance of the substructure causes the fluid flow to be shunted through the open pores, increasing the rate of pore blockage (Figure 5) and thus the slope on the d2t/dV2 plot in Figure 7. At long times, i.e., large dt/dV, the model and data give a region with zero slope, corresponding to a cake-formation fouling mechanism. The transition from predominantly pore blockage to predominantly cake filtration occurs near the maximum in d2t/dV2. This maximum value arises because of the large reduction in the rate of flux decline that occurs during the

Ind. Eng. Chem. Res., Vol. 40, No. 5, 2001 1419 Table 2. Initial Flow Rate and Best-Fit Parameter Values for Experiments in Figure 8 membrane

Q0 (m3 s-1 × 108)

a (m2 kg-1)

Rp0 (m-1 × 10-11)

f ′R′ (m kg-1 × 10-12)

PCTE PCTE PCTE PCTE/4PVDF PCTE/4PVDF PCTE/4PVDF

3.2 7.0 10.2 3.2 7.0 10.2

2.8 ( 0.1 2.8 ( 0.1 2.6 ( 0.1 2.8 ( 0.1 2.7 ( 0.1 2.4 ( 0.1

4.0 ( 0.2 4.1 ( 0.2 4.0 ( 0.2 3.0 ( 0.5 3.0 ( 0.5 3.7 ( 0.5

0.35 ( 0.02 0.74 ( 0.03 1.1 ( 0.1 1.6 ( 0.2 5.7 ( 0.2 8.4 ( 0.4

transition between the pore-blockage and cake-filtration mechanisms. The location of the maximum occurs at a higher dt/dV value (lower J) for the PCTE membrane compared to the PCTE/4PVDF membrane because of the smaller flux through the open and blocked pores for the PCTE membrane alone (Figure 4). Pressure Effects. As shown in Table 1, the best-fit values of f ′R′ for the PCTE/PVDF and PCTE/4PVDF membranes were significantly greater than the value determined for the PCTE membrane alone. Previous studies have demonstrated that the specific resistance (R′) of a BSA deposit increases significantly with increasing pressure7 because of the compressibility of the protein deposit. The higher values of f ′R′ for the PCTE composite membranes are thus likely to be due to the greater transmembrane pressure used in these experiments to achieve the same initial flow rate for these higher-resistance membranes. To explore this effect in more detail, a series of filtration experiments was performed at different pressures, with the results for the PCTE and PCTE/4PVDF membranes shown in the top and bottom panels in Figure 8, respectively. The transmembrane pressures applied to the PCTE/4PVDF membranes were chosen so that the initial flow rates for the three PCTE/4PVDF runs were identical to those for the three PCTE runs in the upper panel of Figure 8 (Q0 ) 3.2 × 10-8 m3s-1, 7.0 × 10-8 m3s-1, and 10.2 × 10-8 m3s-1). The normalized flow rate declines more rapidly for the runs with a higher pressure because of the greater convective flow rate through these membranes. The solid curves are the model calculations given by the full model using the parameter values listed in Table 2. The values of R and Rp0 are very similar to the values determined previously (Table 1), with the values for f ′R′ increasing with increasing pressure because of the compressibility of the BSA deposit. The model calculations are in excellent agreement with the data for the flux decline at different transmembrane pressures for both the PCTE and composite membranes. The specific resistances of the BSA deposits were evaluated from the best-fit values of f ′R′ in Table 2 using f ′ ) 0.0003, with the results shown as the solid symbols in Figure 9. This value of f ′ is equal to the fractional concentration of BSA aggregates in solution determined by Ho and Zydney4 from light scattering measurements. The results are plotted as a function of the pressure drop across the upper layer of the membrane. This was simply the applied transmembrane pressure for the PCTE membrane, but for the PCTE/ 4PVDF membrane, this pressure drop was calculated from the measured flow rate (Q) at the end of a 100min filtration as

∆P1 ) ∆P -

QµR2 A

(23)

Although the pressure drop across the upper layer

Figure 8. Filtrate flux for 2 g/L BSA solutions through the PCTE (top panel) and PCTE/4PVDF (bottom panel) membranes at different transmembrane pressures. Solid curves are calculations using the full model with best-fit values of the model parameters given in Table 2.

varies during the filtration, the best-fit values of f ′R′ are determined primarily by the behavior during the latter stage of the filtration. Also shown for comparison are the specific resistances evaluated independently by measuring the steady-state saline flux through a heavily fouled membrane at different transmembrane pressures for the PCTE (open circles) and PCTE/4PVDF (open triangles) membranes. In this case, both data sets were obtained with a single fouled PCTE membrane, first for the membrane alone and then for the same membrane placed on top of four clean PVDF membranes. The resistance of the protein layer was evaluated from eq 6 with X ) 0 using the previously determined values of R1 and R2 (Table 1). The specific resistance (R′) was then calculated as R′ ) Rp/mp, where the mass of the protein deposit (mpA) was evaluated as 0.2 mg by direct weighing of the clean and fouled PCTE membranes. The values of the specific resistance of the protein deposit on the PCTE and PCTE/4PVDF membranes, evaluated from both the flux decline data and the saline flux

1420

Ind. Eng. Chem. Res., Vol. 40, No. 5, 2001

Figure 9. Specific protein layer resistance as a function of the pressure drop across the upper layer of the membrane (∆P1). Solid symbols represent the values obtained by fitting the flux data in Figures 2 and 8. Open circles and triangles are data obtained from steady-state saline flux measurements through heavily fouled membranes. Solid line is simple linear regression fit.

measurements, are in excellent agreement when plotted as a function of the pressure drop across the upper layer. The specific resistance values determined from the flux decline data for the PES and PCTE/PVDF membranes (values from Table 1) are also in good agreement with these results. The slightly larger value of R′ seen for the PES membrane is likely to be due to the use of f ′ ) 0.0003 even though the smaller pores on the PES membrane are able to retain a larger fraction of the protein aggregates. These results clearly demonstrate that the differences in f ′R′ seen in Tables 1 and 2 are a direct result of the compressibility of the protein deposit. The specific resistance data in Figure 9 are highly linear when plotted on this log-log graph. This behavior is consistent with the power-law relationship used previously by Porter12 and Belter et al.13 to describe the compressibility of different filter cakes

R′ ) k(∆P1)s

(24)

where k is a constant related to the size and shape of the particles forming the deposit and s is the cake compressibility, which varies between 0 for an incompressible layer and 1 for a highly compressible layer. Simple linear regression of the data in Figure 9 yield s ) 0.97 and k ) 5.9 × 1011 m kg-1 (m2 N-1)0.97. Model calculations using these values of s and k, in combination with the best-fit values of R and Rp0 given in Table 1, were in very good agreement with the experimental data for all of the PCTE composite membranes over the full range of pressures (2.3-53 kPa), with deviations between the experimental data and model predictions for the flow rate of typically less than 5%. Conclusions The theoretical model developed in this study provides the first quantitative analysis of the fouling behavior of asymmetric or composite membranes that explicitly accounts for the fluid flow through the individual layers of the membrane. The presence of an underlying membrane substructure has several effects on the filtrate flux and fouling characteristics. First, the substructure

causes more of the fluid flow to be shunted through the open pores as the membrane becomes fouled. This increase in flux through the open pores is directly related to the highly interconnected pore morphology within the substructure, which allows the fluid to travel laterally as it percolates down through this layer. Thus, the fluid flux in the substructure remains spatially uniform even as some of the pores on the upstream surface of the asymmetric or composite membrane become blocked. The net result is that the rate of flux decline decreases as the resistance of the substructure increases. This effect is very pronounced for the composite membranes in which R2 is much larger than R1. This effect is fairly small for the asymmetric PES membrane, although it does result in an initial increase in the rate of flux decline, as seen in Figure 2. It should be noted that larger-pore-size asymmetric membranes14 tend to have much larger values of R2/R1 and are thus expected to show flux decline behavior similar to that seen with the composite PCTE/PVDF membranes examined in this study. The membrane substructure also increases the flux through the blocked pores by reducing the relative increase in the overall resistance to flow caused by the protein deposit. This phenomenon dominates at long times when the membrane surface is almost entirely covered by a protein layer and the flux occurs primarily through the blocked pores. In contrast to the shunting of the fluid flow to the open pores, this latter effect is completely independent of the interconnectivity of the pores within the membrane structure. It is simply due to the increase in the resistance of the unfouled membrane and the corresponding reduction in the relative importance of the resistance of the protein deposit or cake layer. Finally, the presence of the substructure increases the pressure drop across the protein layer over the course of the filtration, which causes a significant increase in the specific resistance of the deposit for highly compressible protein layers such as those formed with BSA. This variation of R′ with pressure could be of even greater importance in systems operated at constant filtrate flux in which the applied transmembrane pressure increases throughout the filtration as the membrane fouls. The theoretical model developed in this study should provide an appropriate framework for the analysis of fouling phenomena using asymmetric or composite membranes in these filtration systems. Acknowledgment This work was supported in part by a grant from the National Science Foundation. Literature Cited (1) Fisher, W. G. Particle Interaction with Integrated Circuits, Particle Control for Semiconductor Manufacturing; Marcel Dekker: New York, 1990. (2) Zeman, L. J.; Zydney, A. L. Microfiltration and Ultrafiltration: Principles and Applications; Marcel Dekker: New York, 1996. (3) Mulder, M. Basic Principles of Membrane Technology; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1996. (4) Ho, C. C.; Zydney, A. L. Effect of Membrane Morphology on the Initial Rate of Protein Fouling during Microfiltration. J. Membr. Sci. 1999, 155, 261. (5) Tracey, E. M.; Davis, R. H. BSA Fouling of Track-Etched Polycarbonate Microfiltration Membranes. J. Colloid Interface Sci. 1994, 167, 104.

Ind. Eng. Chem. Res., Vol. 40, No. 5, 2001 1421 (6) Bowen, W. R.; Calvo, J. I.; Herna´ndez, A. Steps of Membrane Blocking in Flux Decline during Protein Microfiltration. J. Membr. Sci. 1995, 101, 153. (7) Ho, C. C.; Zydney, A. L. A Combined Pore Blockage and Cake Filtration Model for Protein Fouling during Microfiltration. J. Colloid Interface Sci. 2000, 232, 389. (8) Kelly, S. T.; Opong, W. S.; Zydney, A. L. The Influence of Protein Aggregates on the Fouling of Microfiltration Membranes during Stirred Cell Filtration. J. Membr. Sci. 1993, 80, 175. (9) Hermia, J. Constant Pressure Blocking Filtration Laws: Application to Power Law Non-Newtonian Fluids. Trans. Inst. Chem. Eng. 1982, 60, 183. (10) Ho, C. C.; Zydney, A. L. Theoretical Analysis of the Effect of Membrane Morphology on Fouling during Microfiltration. Sep. Sci. Technol. 1999, 34, 2461. (11) Ho, C. C.; Zydney, A. L. Measurement of Membrane Pore Interconnectivity. J. Membr. Sci. 1999, 155, 261.

(12) Porter, M. C. What, When, and Why of MembranessMF, UF, and RO. In What the Filter Man Needs to Know About Filtration; AIChE Symposium Series No. 171; American Institute of Chemical Engineers: New York, 1977. (13) Belter, P. A.; Cusseler, E. L.; Hu, W. S. Bioseparationss DownStream Processing for Biotechnology; John Wiley & Sons: New York, 1988. (14) The Filter Book; Pall Corporation: East Hills, NY, 1999.

Received for review September 5, 2000 Revised manuscript received January 5, 2001 Accepted January 10, 2001 IE000810J