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.
Membrane filtration fouling is a very complex process and is determined by many properties such as the membrane internal morphology, membrane pore structure, flow rate and contaminant properties. In a very slow filtration process or during the late stage of filtration, when the flow rate is naturally low and Péclet number is small, particle diffusion is essential and cannot be neglected, while in typical filtration models, especially in moderate and fast filtration process, the main contribution stems from the particle advection. The objectives of this study is to formulate mathematical models that can (i) investigate how filtration process varies under possible effects of particles diffusion; and (ii) describe how membrane morphology evolves and investigate the filtration performance during the filtration process. We also compare the results with the case that diffusion is less important and make a prediction about what kind of membrane filter pore structure should be employed to achieve a particular optimum filtration performance. According to our results, the filtrate and efficiency of particle separation are found to be under the trade-off relationship, and the selection of the membrane properties depends on the requirement of the filtration.
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