Modelling the effects of particle polydispersity in crossflow filtration

Abstract

A mathematical model of crossflow filtration which accounts for particle polydispersity is developed. The model combines classical filtration theory with the concept of the cut-off particle diameter (i.e., the diameter below which particles will deposit on the membrane), to yield a single differential equation for the filter cake thickness as a function of time. Using an analysis of the hydrodynamic forces acting on a depositing particle, the cut-off diameter is expressed as a function of cake thickness. The differential equation is then solved numerically by considering the cake to be incompressible and to be composed of discrete layers of different average particle size, the average particle size being greatest in the layer adjacent to the membrane, and lowest in the outermost layer. Steady state is shown to be attained when the fraction of suspension particles which are below the cut-off diameter, approaches zero. It is demonstrated that the steady state filtrate flux increases with transmembrane pressure and crossflow velocity, and decreases with increasing membrane resistance. The average specific cake resistance is found to increase with membrane resistance and crossflow velocity (due to deposition of smaller particles), but to decrease with increasing transmembrane pressure (due to deposition of larger particles). Thus, the particle polydispersity effects modelled here cannot explain experimental observations of fluxes which decrease with increasing crossflow velocity, because the decrease in cake-height more than compensates for the increase in specific cake resistance.