The contamination crisis and a filtration fix: Two modelling frameworks for particle filtration

By Arkady Wey

Filtration
Filtration is the separation of particles from fluid, usually by passing it through a membrane (Figure 1). It is used extensively for wastewater decontamination, and for harmful-particulate removal from exhaust gases in combustion plants. There are few larger problems facing humanity: The WHO has predicted that 10% of all deaths worldwide are linked to polluted air, and a further 6% have been attributed to contaminated water supplies.

Figure 1: Diagram of filtration. A particle-laden fluid is driven through a porous membrane. Some particles deposit inside the filter so the concentration of particles exiting the filter is less than that entering it. Contaminants in air and water contribute to around 10% and 6% of deaths worldwide. Improving filtration technology may decrease these numbers.

Filtration is also essential for the creation of new products and cleaning used ones, and is therefore one of the most prevalent physical processes, with a huge and diverse set of socially and economically important applications. In households, filters can be found in vacuum cleaners and water purifiers. In the pharmaceutical industry, filtration is used for the separation of plasma from blood. In biotechnology, it plays an important role in cell harvesting. 

Filter membranes and clogging 
For many of these applications, one commonly used membrane material is the expanded polytetrafluoroethylene (ePTFE) manufactured by U.S. materials manufacturer W.L. Gore & Associates, Inc (Gore). On the macroscale, ePTFE is a thin semi-permeable sheet. On the microscale, it consists of tiny fibres that cross over each other to form a complex network of pore space (Figure 2). 

Figure 2: Photographs of ePTFE. On the macroscale, it is a smooth sheet.  On the microscale, it consists of thin fibres crossing over each other randomly.

Our mathematical models
At the University of Oxford’s Mathematical Institute, mathematicians Arkady Wey, Chris Breward, Jon Chapman and Ian Griffiths have been collaborating with Uwe Beuscher and Vasudevan Venkateshwaran at Gore, to develop new mathematical frameworks for filtration. Together, we have derived two models: A size-structured model and a network–continuum multiscale model.

When fluid passes through ePTFE, particles deposit within this pore network. Deposition eventually causes the membrane to clog, which results in decreased filter performance and downtime for replacement. Filter improvement for clogging reduction thus has significant socio-economic benefits. Manufacturers such as Gore are broadly interested in extending the lifetime (time until the membrane must be replaced) while obtaining a specified particle retention efficiency (proportion of particles removed from the fluid) by tweaking membrane microscale properties (pore size-distribution and structure).

In the size-structured framework, ePTFE is modelled as a solid that is permeated by tunnels that represent pores (Figure 3). A cross-section through any depth of the membrane contains a distribution of pores of different sizes. When a fluid, containing particles with another distribution of sizes, is passed through these, particles stick to pore walls and decrease pore radii. Our model tracks changes in the size distributions of particles and pores as this process occurs, via a coupled system of PDEs.

Figure 3: Diagram of the size-structured model. (Left) ePTFE modelled as a solid permeated by tunnels of pore-space. (Right-Top) Top-view of a cross-section of the model geometry. (Right-Bottom) Side-view of a cross-section of the model. Particles advect through pore space and deposit on pore walls. 

In the multiscale framework, the microscale is modelled as a periodic network composed of a repeating cell containing N nodes (Figure 4). Pore space consists of conductive channels (edges) that are connected to each other at junctions (nodes). Edges shrink as particles deposit within them. We have developed an asymptotic method to average the flow and particle deposition through this network. The result is an ‘effective’ model (EM) that consists of Darcy’s equation for porous-media flow and an advection–reaction equation for the particle concentration. These PDEs are coupled to the underlying NM via the parameters of the system, given in terms of the fluid-conductances of edges, which are found by solution of a linear-algebraic problem of size N.  Unlike in usual PDEMs, information about the dynamic state of the underlying microscopic network enters the macroscopic system, meaning that more empirical data about the physical situation can be employed.

Figure 4: Diagram of the multiscale model. (Left) ePTFE modelled as a network of edges and nodes. (Right) A particle depositing on a pore wall and decreasing the pore radius. Each pore is modelled as an edge.

Achievements
Using the size-structured model, we are able to track changes to the pore distribution inside the filter, allowing us to identify which pores clog first, and where this happens. We find explicit solutions in several industrially-relevant parameter regimes, which means simulation is possible at a fraction of the computational cost of many other PDEMs.

In the multiscale model, we find that the cell-size, N, need only be small before the cell behaves similarly to the microscale network as a whole. This has two important implications. On one hand, the EM retains the accuracy of the associated NM because the repeating cell is representative of the network. On the other hand, the EM is almost as cheap as the uncoupled PDEM, because the linear-algebraic system that causes coupling is small (size N). 

Figure 5: Illustrative graphs of results. (Left) Concentration as a function of filter depth, from which the particle retention efficiency can be calculated. (Right) Fluid velocity as a function of time, from which filter lifetime can be calculated.

Both models are useful tools for predicting a filter’s particle retention and lifetime (Figure 5). The computational feasibility of both models also opens them up for the potential to aid filter optimisation. Future work may involve answering design questions like: “Given a desired particle retention efficiency and filter lifetime, what is the optimal size distribution and structure of the membrane pore space?”. This may lead to increased filter performance, and ultimately to the reduction of harmful particles and excess waste. 

Acknowledgements
This work is funded by the EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling (InFoMM). 

The work was recognised by The Parliamentary & Scientific Committee via the Gold Mathematics STEM for Britain 2023 award (Figure 6), and SIAM 3-Minute Thesis prizes.

Figure 6: Photograph of Arkady Wey with The Parliamentary & Scientific Committee members at the STEM for Britain 2023 awards ceremony.
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