A multiscale mathematical model for particle filtration*

Particle filtration is a well-studied industrial process with many applications such as food production and sewage treatment. As filtration occurs, particles deposit in pores, which causes the filter to clog. This results in membrane-downtime for cleaning or replacement, which is expensive.

Experimental observations of particle–pore interactions that lead to clogging are difficult and destructive, while mathematical approaches are often too computationally expensive for filter-scale experiments, since membranes can consist of billions pores.

In this talk, we develop a model that couples a dynamical network system on the microscale with a macroscale PDE system. We construct a discrete version of the usual method of multiple scales, and use this to homogenise the flow and deposition of particles through an arbitrarily-connected periodic network of channels, which models the pores.

The result is an effective system consisting of Darcy’s equation and an advection-reaction equation. Microscopic information enters via the permeability and adhesivity, which are parameters of this system. These are given in terms of the channel conductances by solving a low-dimensional, linear, algebraic problem, whose simplicity leads to decreased computational complexity compared to standard approaches.