Combined models of membrane fouling: Development and application to microfiltration and ultrafiltration of biological fluids
Introduction
Sterilizing grade membranes are used throughout biotech purification processes to remove bacteria from fermentation media, buffers and process streams. Additionally there is an increasing use of filters for the removal of viruses from these streams. The membrane area required by these process steps can be determined by limitations in the membrane permeability, membrane capacity, or a combination of the two. For operations run at a fixed trans-membrane pressure, capacity is defined as the amount of fluid per membrane area that can be processed until the flux declines to a set fraction of the initial flow. For operations run at a fixed flow rate, capacity is determined when the trans-membrane pressure increases to some set multiple of the initial value.
Fouling of a membrane can occur by deposition of particles inside or on top of the membrane [1], [2]. There are four mechanistic models that are typically used to describe fouling. Complete blocking assumes that particles seal off pore entrances and prevent flow. Intermediate blocking is similar to complete blocking but assumes that a portion of the particles seal off pores and the rest accumulate on top of other deposited particles. Cake filtration occurs when particles accumulate on the surface of a membrane in a permeable cake of increasing thickness that adds resistance to flow. Standard blocking assumes that particles accumulate inside membranes on the walls of straight cylindrical pores. As particles are deposited, the pores become constricted and the permeability of the membrane is reduced. Each of these mechanisms have been used individually or in combinations to explain experimental observations. Bowen and Gan [3] observed that BSA fouled microporous aluminum oxide, PVDF, and polycarbonate membranes internally because stirring of the system had no effect on flux decline and because the standard blocking mechanism provided good curve fits. Marshall et al. concluded that proteins can foul microporous membranes both by deposition within the membrane pores and deposition on the membrane surface [4]. Hlavacek and Bouchet [5] fit the intermediate, complete and standard models to data for the fouling of microporous track-etched polycarbonate, cellulose, and PVDF membranes by BSA and found that the intermediate model provided the best fits. Tracey and Davis [6] found that data for the fouling of microporous track-etched membranes by BSA could be fit initially by either the standard or complete model and subsequently by the cake model. Bowen et al. [7] observed that the flux decline during the fouling of microporous track-etched membranes by BSA did not follow any of the individual fouling models and likely occurred through a combination of complete blocking, standard blocking, and cake formation. The results of the different studies indicate microporous membranes can foul by both surface and pore deposition and the mechanisms vary with system conditions.
A number of models of the effects of deposition of different particles sizes on the permeability of porous media have been developed. Particles can deposit due to inertial impaction, interception, sedimentation, electrostatic forces, diffusion and straining. A detailed description of these mechanisms and the models that have been developed to describe them is provided by Tien [8]. The particles that typically foul microporous membranes are too small to foul by impaction or sedimentation. Typically particles are strained, resulting in pore blockage or caking, or particles are adsorbed due to the effects of diffusion, interception, or electrostatic forces.
Recently a model was developed that used a two-stage mechanism to describe membrane fouling [9]. Fouling occurred initially through complete pore blocking. The deposited aggregates were assumed to be permeable. Flow through the blocked areas resulted in the deposition of a cake, reducing flux further. This was the first model that accounted for the combined effects of two fouling mechanisms. The model used three fitted parameters and an approximate solution was provided that allowed calculation of flux as a function of time without integrating. Plots of volume versus time during constant pressure operation or pressure versus time during constant flow operation were obtained by numerical integration of the relevant equations. The values of the fitted parameters were similar to the values obtained by independent experiments. The authors found this model provided good data fits and made accurate membrane sizing estimates for the fouling of microporous track-etched membranes by five proteins.
In this study the previous modeling work was expanded by using a new method to combine the four individual fouling mechanisms. The goal was to develop models that described combined-mechanism fouling with two fitted parameters and with explicit equations for constant flow and constant pressure operation. Five new fouling models that accounted for the combined effects of the different individual fouling mechanisms were derived from Darcy's law. Explicit equations were derived that related pressure to time during constant flow operation, and volume to time during constant pressure operation. The models all used two fitted parameters and reduced to the individual models in the absence of the second fouling mechanisms. The combined models were assessed through testing under constant pressure and constant flow modes with solutions of bovine serum albumin and human IgG. Bovine serum albumin was filtered through virus retention membranes and human IgG was filtered through sterilizing grade microporous membranes. The combined caking and complete blockage model was the most useful, as it was able to provide good fits of both data sets, and provide good fits of each of the other individual model predictions. The combined cake-standard and cake-intermediate models also provided good data fits and may be applicable to systems where these models are consistent with the experimentally observed fouling mechanisms.
Section snippets
Modeling
The flow rate Q can be calculated as a function of resistance R and membrane frontal area A using Darcy's lawwhere P is the trans-membrane pressure and μ is the solution viscosity.
Proteins
Bovine serum albumin (BSA, Sigma, St. Louis, MO) was prepared at 2.5 mg/mL concentration in pH 7.2 PBS buffer (DF2314-15-0, Fisher Scientific, Hampton, NH). Human plasma IgG (Seracare, Oceanside, CA) was prepared in pH 7.2 PBS at a concentration of 10.0 mg/mL.
Membrane testing equipment: constant pressure
The fouling solution was contained in a pressure vessel (XX1100000, Millipore, Bedford, MA). Durapore PVDF based membrane with a nominal pore size rating of 0.22 μm was run at 10 psi during constant pressure tests. Viresolve 180, a PVDF
Constant pressure: sterile filtration of human plasma IgG
The combined models were applied to the constant pressure sterile filtration of the human plasma IgG solution to determine if they would provide better fits of the experimental data than other models. Three experiments were performed using the 13 mm membrane holders. The permeate volume was measured as a function of time until the flux had declined 95% from its initial value of 1.13 × 10−3 m/s.
The permeate volume versus time data was fit using the combined models and all other typical fouling
Conclusions
To account for the combined effects of different individual fouling mechanisms, five new fouling models were generated. Explicit equations were derived from Darcy's law that related pressure to time during constant flow operation and volume to time during constant pressure operation. The models used two parameters and reduced to the equations for the individual models in the absence of the second fouling mechanism.
Generally there is a balance between physical detail and numerically complexity
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