GOALI: Network Models for Membrane Filtration

膜过滤器用于非常广泛的日常应用，从工业废物处理到水净化到空气调节到食品生产，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携带的颗粒或杂质，因此膜的结垢是不可避免的，最终导致过滤器失效。因此，对结垢机制的更好理解是重要研究工作的目标，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和粘附之外，孔结构、尺寸和形状是决定过滤器功能和效率的关键因素。在设计和制造允许精细控制颗粒去除同时保持合理寿命的高效过滤器方面存在相当大的工业兴趣。这种过滤器通常具有在膜深度上减小的孔径;并且可以在孔之间结合不同程度的连通性。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，同时满足规定的杂质去除阈值。该项目还将为本科生和研究生提供培训、研究和工业经验。该项目解决多孔膜过滤器中的流动和污染问题，其中孔结构可以模拟为连接的（足够细的）管的网络。与W.L.的工业合作伙伴合作。戈尔协会和一个学生团队，PI将推导出新的预测数学模型来描述具有实际意义的情况。我们的模型将在任意孔隙网络上制定（其中孔隙的长度和半径都可以变化），并将通过在孔壁吸附小颗粒污染物和筛分较大颗粒（具有任意尺寸分布）来描述结垢。我们将研究计算生成的网络（使用随机几何图构造的变体）以及非专有制造膜中的真实的孔隙网络。我们还将使用持久同源性（PH）的方法来研究这种孔隙网络的连通性特征以及这些特征如何随时间演变，以阐明几何/拓扑特征与过滤性能指标之间的比例关系。我们的工业合作者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工业相关性提供建议。这种互动将确保项目保持重点，并保证我们识别和解决对应用程序具有真实的重要性的问题。所提供的实验数据还将使我们能够确定模型中未知参数的适当范围，并测试不确定的建模假设。此外，我们将制定新的模块，我们的顶点课程应用数学，将涉及本科生的研究通过蒙特-卡罗型模拟和持久的同源性方法，使用可用的软件库。该研究将结合联合收割机新的数学建模与分析方法，包括图论，优化，均匀化技术，多尺度分析，渐近方法，随机分析和概率论，以及有效的数值技术。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.