GOALI: Network Models for Membrane Filtration

Membrane filters are used in in an extraordinarily wide range of everyday applications ranging from industrial waste treatment to water purification to air conditioning to food production and, of particular relevance today, medical masks and vaccine manufacture. Since the goal of effective filtration is the removal by the filter of particles or impurities carried by the feed solution, fouling of the membrane is unavoidable, leading ultimately to filter failure. Improved understanding of the fouling mechanisms, critical in improving filter designs, is therefore the target of significant research efforts. In addition to considerations of how particles interact with and adhere to the membrane material, pore structure, size and shape are critical factors in determining filter functionality and efficiency. There is considerable industrial interest in designing and manufacturing efficient filters that allow for fine control of particle removal while maintaining a reasonable lifetime. Such filters typically have pore size that decreases in the membrane depth; and may incorporate varying degrees of connectivity between pores. Our work will use theoretical approaches (combined with experimental calibration) to study general networks of pores within filters and identify optimal pore network structures that maximize filter lifetime while meeting a prescribed impurity removal threshold. This project will also provide training and research and industrial experiences for undergraduate and graduate students.This project addresses issues of flow and fouling in porous membrane filters, where the pore structure may be modeled as a network of connected (sufficiently slender) tubes. Working with industrial collaborators at W.L. Gore & Associates and with a team of students, the PIs will derive new predictive mathematical models to describe situations of practical importance. Our models will be formulated on arbitrary pore networks (where both the length and radius of pores can vary) and will describe fouling both by adsorption of small particulate contaminants at the pore walls, and by sieving of larger particles (with an arbitrary size distribution). We will study networks generated computationally (using a variant of the Random Geometric Graph construction) as well as real pore networks in non-proprietary manufactured membranes. We will also use methods from persistent homology (PH) to study connectivity features of such pore networks and how these evolve in time, to elucidate scaling relationships between geometric/topological features and filtration performance metrics. The work is of significant interest to our industrial collaborators, who will provide us with data and advise on industrial relevance via regular team meetings. This interaction will ensure that the project remains focused and will guarantee that we identify and address questions of real importance to applications. The experimental data provided will also allow us to identify appropriate ranges for unknown parameters in our models, and to test uncertain modeling assumptions. Additionally, we will formulate new modules for our Capstone Course in Applied Mathematics that will involve undergraduate students in the research by means of Monte-Carlo type simulations and persistent homology methods using available software libraries. The research will combine novel mathematical modeling with analytical approaches including graph theory, optimization, homogenization techniques, multiple scales analysis, asymptotic methods, stochastic analysis and probability theory, and efficient numerical techniques.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.