GOALI: Network Models for Membrane Filtration

膜过滤器在非常广泛的日常应用中使用，从工业废物处理到水净化，从空调到食品生产，以及今天特别相关的医用口罩和疫苗制造。由于有效过滤的目标是通过过滤器去除进料溶液中携带的颗粒或杂质，因此膜的污染是不可避免的，最终导致过滤器故障。因此，提高对污垢机理的理解是重大研究工作的目标，这对改进过滤器设计至关重要。除了考虑颗粒如何与膜材料相互作用并附着在膜材料上之外，孔结构、大小和形状是决定过滤器功能和效率的关键因素。工业上对设计和制造高效过滤器有相当大的兴趣，这些过滤器允许精细控制颗粒去除，同时保持合理的寿命。这种过滤器通常具有膜深度减小的孔大小；并且可以结合不同程度的孔之间的连通性。我们的工作将使用理论方法(结合实验校准)来研究过滤器内的一般孔隙网络，并确定最佳的孔隙网络结构，使过滤器寿命最大化，同时满足指定的杂质去除阈值。这个项目还将为本科生和研究生提供培训、研究和工业经验。这个项目解决了多孔膜过滤器中的流动和污染问题，其中的孔结构可以被模拟为连接的(足够细的)管的网络。与W.L.Gore&Amp；Associates的行业合作者和一个学生团队合作，PI将推导出新的预测性数学模型，以描述具有实际重要性的情况。我们的模型将建立在任意孔网络(其中孔的长度和半径都可以变化)上，并将通过在孔壁上吸附小颗粒污染物和筛选大颗粒(具有任意尺寸分布)来描述污垢。我们将研究通过计算生成的网络(使用随机几何图形构造的变体)以及非专利制造膜中的实际孔网络。我们还将使用持久同源(PH)的方法来研究这种孔网络的连通性特征以及这些连通性是如何随时间演变的，以阐明几何/拓扑特征与过滤性能指标之间的比例关系。我们的行业合作伙伴对这项工作非常感兴趣，他们将通过定期的团队会议为我们提供数据并就行业相关性提供建议。这种互动将确保项目保持重点，并保证我们确定和解决对应用程序真正重要的问题。提供的实验数据还将使我们能够为模型中的未知参数确定合适的范围，并测试不确定的建模假设。此外，我们将为我们的应用数学Capstone课程制定新的模块，利用现有的软件库，通过蒙特卡罗类型模拟和持久同源方法，让本科生参与研究。这项研究将把新颖的数学建模与分析方法相结合，包括图论、最优化、齐化技术、多尺度分析、渐近方法、随机分析和概率理论，以及有效的数值技术。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.