Ultrafiltration of colloidal dispersions - A theoretical model of the concentration polarization phenomena

A general thermodynamic model of the concentration polarization phenomena of colloidal particles at a membrane surface is presented. The model is based on the balance between a thermodynamic force, due to the osmotic pressure gradient, and a frictional force, due to the fluid flow around each particle. A cell model description is used to model the concentration dependence of the thermodynamic force as well as the how properties in the concentrated colloidal solution. Equilibrium thermodynamics of the colloidal system can be used in the cell calculations since local equilibrium is assumed in the neighborhood of each colloidal particle (i.e., in each cell). This means that the concentration dependence of the osmotic pressure can be obtained, either from an experimental determination or from a theoretical model of the bulk properties of the colloidal system. To exemplify the usefulness of the model when establishing the influence of different operating parameters, such as the transmembrane pressure, the fluid shear, or different solution properties, such as concentration, particle size, pH, and ionic strength, a model system of charged spherical colloidal particles is used. The interaction between the particles is in the presented examples assumed to be a combination of electrostatic interactions, calculated from the Poisson-Boltzmann equation, dispersion forces, calculated as additive 1/r(6) interactions, and a hard sphere interaction calculated from the Carnahan-Starling equation. (C) 1996 Academic Press, Inc.