J. CHEM. SOC. FARADAY TRANS., 1995, 91(19), 3389-3398 Theory of Concentration Polarization in Crossflow Filtration Lianfa Song and Menachem Eiimelech*? School of Engineering and Applied Science, University of California, Los Angeles, CA 90095-1593, USA A novel theory is developed for concentration polarization of non-interacting particles in crossflow-filtration systems. This theory reveals that the extent of concentration polarization, a s well as the behaviour of the per- meate flux, are characterized by an important dimensionless filtration number (N, = 4m:Ai3/3k~). There is a critical value of the filtration number for a given suspension and operational conditions. When the filtration number is smaller than the critical value, a polarization layer exists directly over the membrane surface and the wall particle concentration is determined by the pressure and temperature. At higher filtration numbers, a cake layer of retained particles forms between the polarization layer and the membrane surface. Mathematical models are constructed for both cases and analytical solutions for the permeate flux are derived. An increase in permeate flux with increasing pressure is predicted for all operational conditions. 1. Introduction Crossflow filtration refers to a pressure-driven separation process in which the permeate flow is perpendicular to the feed flow. Separation in crossflow filtration is usually achieved by a membrane displaying different permeability to the solvent and solute (or particles). Crossflow filtration pro- cesses are conventionally classified as reverse osmosis (hyperfiltration), ultrafiltration, and microfiltration, depend- ing on membrane pore size and operation conditions. Cross- flow filtration has marked advantages over dead-end filtration and has been widely used to separate solutes from solutions or colloidal and particulate materials from suspen- sions. Concentration polarization is a phenomenon in which the solute or particle concentration in the vicinity of the mem- brane surface is higher than that in the bulk. This phenome- non, inherent to all crossflow filtration processes, occurs as long as the membrane shows different permeability for the various components of the solution or suspension. The resulting concentrated layer at the membrane surface increases the filter resistance and consequently reduces the permeate flux through the membrane. Concentration polar- ization is of considerable interest since a high permeate rate is most desirable in filtration processes. As a result, much research effort has been expended in this area. During the past two decades, numerous theoretical attempts have been made in an effort to understand concen- tration polarization and to predict permeate rate in crossflow-filtration systems. A representative achievement of these studies was development of the gel-layer model.',2 This model has been used as a tool for directly interpreting experi- mental data in ultrafiltration and as a basis for further theo- retical development in microfiltration. In the latter case, shear-induced-diffusion and inertial lift are incorporated into the theory in order to obtain a better fit with experimental data.3-5 Another popular theory in this field is the osmotic pressure model. This model, however, has been shown to be fundamentally equivalent to the gel-layer model6 The gel-layer theory is derived from a single mass balance equation'-3 with two unknowns: the permeate velocity and the particle concentration distribution over the membrane. Therefore, it is impossible to evaluate the permeate velocity from this theory without additional assumptions. The theory becomes a semi-empirical model as it assumes a fixed surface particle concentration and adapts a mass transfer coefficient from theories of convective heat transfer to impermeable sur- faces. Neither has been theoretically proven valid for cross- flow filtration. Besides fundamental shortcomings, practical applications of this model are indeed limited. Although the gel-layer model has been developed to predict the so-called 'limiting flux', the theory cannot specify explicitly the criteria necessary for achievement of limiting flux. Furthermore, the model cannot be used for conditions where the permeate velocity is pressure dependent (i.e. before limiting flux is attained). The osmotic-pressure model has similar conceptual prob- lems. In this model, the wall particle concentration must be estimated from certain empirical relationships or simplified assumptions. This is one of the reasons why empirical expres- sions are often used to predict permeate flux in reverse osmosis systems.'*' Evidently, this model cannot be applied to microfiltration and most ultrafiltration systems since osmotic pressure is negligible in these cases. In this paper, a novel theory for concentration polarization is developed, based on the hydrodynamics and thermodyna- mics of particle suspensions. It is shown that another funda- mental relationship (force or energy balance) is required, in addition to mass balance, to describe completely the concen- tration polarization in crossflow filtration. Similar to the gel- layer model, uniform non-interacting ('hard') spherical particles are assumed in the mathematical derivation of the theory. It is expected that this new theory will provide a solid foundation for understanding and predicting permeate flux in crossflow-filtration systems. 2. Physics in Crossflow Filtration Crossflow-filtration systems can have different configurations (e.g. channel or tube). In the following sections, without loss of generality, a rectangular channel will be used in the theo- retical analysis and a complete rejection of the solute or par- ticles by the membrane is assumed. Several fundamental physical principles are involved in the separation process by crossflow filtration. For instance, the flow field and drag force in crossflow filtration are described by basic theories of hydrodynamics and, more importantly, many bulk properties of particle suspensions in crossflow filtration are governed by thermodynamic principles. In this section, the important fea- tures of the flow field and driving force in crossflow filtration,