A new model for calculating specific resistance of aggregated colloidal cake layers in membrane filtration processes

A simple model to evaluate hydrodynamic cake resistance due to filtered aggregates is developed in this study. An aggregate is treated as a hydrodynamically as well as geometrically equivalent solid core with a porous shell. Creeping flow past a swarm of the composite spheres is solved using Stokes' equation and Brinkman's extension of Darcy's law. The dimensionless drag force (Ω) exerted on the composite sphere is analytically determined by four parameters: radius of the solid core, thickness of the porous shell, permeability of the aggregate, and occupancy fraction as defined in this paper. In certain limiting cases, Ω converges to pre-existing analytical solutions for (i) an isolated impermeable sphere, (ii) an isolated uniformly porous sphere, (iii) an isolated composite sphere, (iv) a swarm of impermeable spheres, and (v) a swarm of uniformly porous spheres. This expression is then used to predict the specific resistance of aggregate cake formed on membrane surfaces.