A SIMILARITY SOLUTION FOR STEADY-STATE CROSS-FLOW MICROFILTRATION

A similarity solution is presented for the convective-diffusion equation governing the steady-state concentration-polarization boundary layer in crossflow microfiltration of the particles, under conditions where a thin stagnant layer of particles deposited on the microporous membrane surface provides the controlling resistance to filtration. The analysis employs concentration-dependent shear viscosities and shear-induced hydrodynamic diffusivities based on empirical correlations for suspensions of rigid spheres. The resulting permeate flux is vw(x) = τw(a4/3x) 1 3 ν t-w/μ0, where x is filter entrance, τw is the wall shear stress exerted on the boundary layer by the tangential flow of bulk suspension through the filter channel, a is the particle radius, and μ0 is the characteristic viscosity. The dimensionless permeate flux, ν -w (φb), depends only on the particle volume fraction in the bulk suspension, φb, and is given by ν -w = 0.0581φb- 1 3 when the suspension is dilute (φb < 0.10). The results for the permeate flux and for the concentration and velocity profiles show that the approximate solution of Davis and Leighton [Chem. Engng Sci. 42, 275-281 (1987)] and Romero and Davis [J. Membrane Sci. 39, 157-185 (1988)], which neglects axial convection in the differential particle mass balance but retains it when integrating across the entire boundary layer, is exact in the dilute limit and accurate to within a few percent for nondilute suspensions. The solution may easily be extended to other suspensions having different dependencies of viscosity and diffusivity on concentration. © 1990.