The purpose of this thesis is to formulate and investigate new mathematical models for membrane filtration. The work presented is divided into six chapters. In the first chapter the problem is introduced and motivated. In the second chapter, a new mathematical model for flow and fouling in a pleated membrane filter is presented. Pleated membrane filters are widely used in many applications, and offer significantly better surface area to volume ratios than equal area unpleated membrane filters. However, their filtration characteristics are markedly inferior to those of equivalent unpleated membrane filters in dead-end filtration. While several hypotheses have been advanced for this, one possibility is that the flow field induced by the pleating leads to spatially nonuniform fouling of the filter, which in turn degrades performance. This hypothesis is investigated by developing a simplified model for the flow and fouling within a pleated membrane filter. The model accounts for the pleated membrane geometry (which affects the flow), for porous support layers surrounding the membrane, and for two membrane fouling mechanisms: (i) adsorption of very small particles within membrane pores; and (ii) blocking of entire pores by large particles. Asymptotic techniques are used based on the small pleat aspect ratio to solve the model, and solutions are compared to those for the closest-equivalent unpleated filter.
In the third and fourth chapters, mathematical models are proposed to describe the effects of filter membrane morphology on filtration efficiency. A reasonable question that membrane filter manufacturers may ask is: what is the optimal configuration of filter membranes, in terms of internal morphology (pore size and shape), to achieve the most efficient filtration? In order to answer this question, a robust measure of filtration performance must be first proposed. Filter membrane performance can be measured in a number of different ways. As filtration occurs, the membrane becomes blocked, or fouled, by the impurities in the feed solution, and any performance measure must take account of this. For example, one performance measure might be the total throughput - the amount of filtered feed solution - at the end of filtration process, when the membrane is so badly blocked that it is deemed no longer functional. A simplified mathematical model is proposed, which (i) characterizes membrane internal pore structure via pore or permeability profiles in the depth of the membrane; (ii) accounts for various membrane fouling mechanisms (adsorption and blocking in Chapter 3, and cake formation in Chapter 4); and (iii) defines a measure of filter performance; and (iv) predicts the optimum pore or permeability profile for the chosen performance measure.
In the fifth chapter, a model for more complex pore morphology is described. Many models have been proposed to describe particle capture by membrane filters and the associated fluid dynamics, but most such models are based on a very simple structure in which the pores of the membrane are assumed to be simple circularly-cylindrical tubes spanning the depth of the membrane. Real membranes used in applications usually have much more complex geometry, with interconnected pores which may branch and bifurcate. Pores are also typically larger on the upstream side of the membrane than on the downstream side. An idealized mathematical model is presented, in which a membrane consists of a series of bifurcating pores, which decrease in size as the membrane is traversed. The membrane’s permeability decreases as the filtration progresses, ultimately falling to zero. The dependence of filtration efficiency on the characteristics of the branching structure is discussed.
The sixth chapter concludes the thesis with a discussion of future work.
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