| Samenvatting: | Filtration is the removal of particles from a fluid suspension by forcing it, either using a constant flow rate or pressure drop, through a porous material, which causes particles to become captured in the pores. Filtration using membranes is a well-studied industrial process with multiple important applications. Experimental observations of particle–pore interactions to gain insight into the filtration process are difficult and sometimes destructive. Some existing mathematical approaches either ignore important microscale information, such as the particle- and pore-size distribution or the connectivity of the pores, or are too computationally expensive for filter scale experiments. In this thesis we motivate, derive, solve, and discuss three novel models for particle filtration: (I) A size-structured model. In this work, the particle and pore sizes are treated as independent variables, which gives rise to a system of partial integro–differential equations within which the filter is treated as a macroscale continuum. We demonstrate that consideration of size structure allows for greater understanding of the filtration process; (II) A network model. Here, the filter is treated as a network composed of pores, which are modelled as edges, and junctions between pores, which are modelled as nodes. The connectivity of the network changes as particles deposit in pores. This gives rise to a system of ordinary algebraic–differential equations for variables defined at each pore and junction of the network. We show that this consideration of the structure of the microscale leads to physically intuitive behaviour of macroscale quantities, such as the flow rate and pressure; (III) A multiscale model. In this work, we extend the method of network homogenisation to apply to networks whose connectivity varies in time. We use this to derive a system of partial differential equations defined on the continuous macroscale geometry that is coupled, via its parameters, with variables that are defined on the discrete network that defines the microscale. We show that solutions of this multiscale model match those of the network model from (II) in a particular asymptotic limit, even though they are obtained at considerably reduced computational cost. These three models increase our understanding of the filtration process. The computational tractability of the models means that they are suitable candidates for filtration process optimisation tasks that involve repeated solution in future work.
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