Dimensional analysis of steady state flux for microfiltration and ultrafiltration membranes

Dimensional analysis of the mass, length and time shows that the steady state flux observed for microfiltration or ultrafiltration through inorganic composite membrane can be expressed using two dimensionless numbers. The shear stress number N_S compares the shear stress against the membrane wall to the driving pressure, while the resistance number N_f compares the convective cross-flow transport to the drived transport through a layer, whose resistance is the sum of all the resistances induced by the different processes which limit the mass transport. Experimental data obtained in ultrafiltration of hydrocarbon emulsions and microfiltration of methanogenic bacteria suspensions and secondary treated wastewater were recalculated in terms of these dimensionless groups. Straight lines were plotted whose slope depends solely on the suspension and the membrane and not on the solute concentration. A negative slope and a positive intersection with the N_S axis means that a cake layer or a polarization layer can be completely eliminated at a critical cross-flow velocity; this was the case for an inorganic particles suspension and for the methanogenic suspension. A straight line of negative slope followed by a plateau means that an irreversible fouling is superimposed to the reversible phenomenon; this was observed for a secondary treated wastewater. A positive slope means that fouling predominates; this was observed with hydrocarbon emulsions.