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

• Main Authors: Pierre Haldenwang, Braulio Bernales, Pierrette Guichardon, Nelson Ibaseta
• Format: Article
• Language: English
• Published: MDPI AG 2019-04-01
• Series: Membranes
• Subjects: membrane separation; cross-flow filtration; polarization of concentration; limiting flux; reversible fouling; Starling–Darcy boundary conditions
• Online Access: https://www.mdpi.com/2077-0375/9/4/48

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., <inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>d</mi> </msub> </semantics> </math> </inline-formula>, 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>d</mi> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics> </math> </inline-formula>, the transverse P&#233;clet number. Furthermore, when the role played by <inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>d</mi> </msub> </semantics> </math> </inline-formula> and moderate <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics> </math> </inline-formula> (say <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>&lt;</mo> <mn>40</mn> </mrow> </semantics> </math> </inline-formula>) is investigated, all results can be interpreted with the use of a single non-dimensional parameter, <inline-formula> <math display="inline"> <semantics> <msub> <mi>F</mi> <mi>l</mi> </msub> </semantics> </math> </inline-formula>, the so-called fouling number, which simply reads <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>F</mi> <mi>l</mi> </msub> <mo>&#8801;</mo> <mi>P</mi> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <msubsup> <mi>N</mi> <mi>d</mi> <mrow> <mo>&#8722;</mo> <mn>1</mn> </mrow> </msubsup> </mrow> </semantics> </math> </inline-formula>. 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.