# Using mathematics to optimise the performance of filtration membranes

By Victoria Pereira

Membrane filtration is a process used for the clarification, purification, and separation of fluid mixtures. In a typical filtration system, a mixture of fluid and contaminant particles is passed through a porous membrane; the fluid passes through while the particles are retained, either on the surface of the membrane or within the membrane structure. Filtration has many important applications from filtering of blood to purifying water and air. Smart Separations Ltd. have developed a filtration device comprising angled membranes in a concertinaed structure (see figure a). This structure optimises the available surface area of the filtration membrane.

To fully realize the potential of the new filtration device, a comprehensive analysis of the performance and efficiency of the membrane is vital. The modelling challenge is to provide quantification on how the membrane performs under operation. In Oxford, we have been developing a mathematical model of the flow through the filtration device to study how the geometry and system parameters relate to the fluid flow rate. Specifically, the mathematical model is able to predict the optimal configuration, that is the angle and position, of a membrane of given thickness and permeance that maximises steady-state flux through the filter.

We find that, for a membrane of fixed angle and physical properties, there can exist multiple membrane positions that aximise the flux for an applied pressure difference, thus uncovering the existence of bifurcating optima in a non-trivial coupled lubrication flow problem. More generally, we show that, while the maximal flux achievable depends on the membrane thickness and permeance, the optimal membrane configuration is always in one of two setups: centred
and diagonal across the full domain; or angled and in the corner of the domain (see figure b). Practitioners may use the model presented herein to derive the optimal configuration for a membrane of specified thickness and permeance corresponding to the maximum possible flux through the device.