Simple Theoretical Results on Reversible Fouling in Cross-Flow Membrane Filtration

In cross-flow membrane filtration, fouling results from material deposit which clogs the membrane inner surface. This hinders filtration, which experiences the so-called limiting flux. Among the models proposed by the literature, we retain a simple one: a steady-state reversible fouling is modelled with the use of a single additional parameter, i.e., N d , the ratio of the critical concentration for deposition to the feed concentration at inlet. To focus on fouling, viscous pressure drop and osmotic (counter-)pressure have been chosen low. It results in a minimal model of fouling. Solved thoroughly with the numerical means appropriate to enforce the nonlinear coupling between permeation and concentration polarization, the model delivers novel information. It first shows that permeation is utterly governed by solute transfer, the relevant non-dimensional quantities being hence limited to N d and P e i n , the transverse Péclet number. Furthermore, when the role played by N d and moderate P e i n (say P e i n < 40 ) is investigated, all results can be interpreted with the use of a single non-dimensional parameter, F l , the so-called fouling number, which simply reads F l ≡ P e i n N d - 1 . Now rendered possible, the overall fit of the numerical data allows us to put forward analytical final expressions, which involve all the physical parameters and allow us to retrieve the experimental trends.