Flow and fouling in elastic membrane filters with hierarchical branching pore morphology

Abstract

Filtration is widely used in industry; therefore, prediction of filtration efficacy and analysis of filter performance are essential. Real membranes have complex internal geometry: pores inside the membrane branch and interconnect with each other, which must be taken into account in mathematical models of filtration. Membrane fouling, as an unavoidable consequence of removing particles, occurs in the course of filtration and deteriorates the membrane permeability. In addition, for membranes made of elastic materials, the pressure within the membrane results in expansion of the pore radii. The pore expansion competes with particle deposition to delay fouling and, thus, influences filtration performance. In this paper, we develop a mathematical model of flow and fouling of such elastic membrane filters with multi-layer bifurcating (hierarchical) interior morphology. Two filtration forcing mechanisms through the membrane are considered: (i) constant pressure drop and (ii) constant flux. We investigate how filtration behaves under these two forcing mechanisms and mathematically describe the morphology change due to fouling coupled to elastic pore expansion. In particular, we obtain an analytical solution for the deformation of the elastic pore walls, which is easily incorporated into the filtration model. Our model provides a quantitative mathematical framework to predict the impact of hierarchical pore morphology and the elasticity of pore walls on filtration performance.