Membrane filters - essentially, thin sheets of porous material, which act to remove certain particles suspended in a fluid that passes through the medium - are in widespread industrial use, and represent a multi-billion dollar industry in the US alone. Major multinational companies manufacture a huge number of membrane-based filtration products, and maintain a keen interest in improving and optimizing their filters. Membrane filtration is used in applications as diverse as water purification; treatment of radioactive sludge; various purification processes in the biotech industry; and the cleaning of air and other gases. While the underlying applications and the details of the filtration may vary dramatically (gas vs liquid filtration; small vs large particle removal; slow vs fast throughput; rigid vs deformable particles), the broad engineering challenge of efficient filtration is the same: to achieve finely-controlled separation at low power consumption.
The desired separation control is to remove only those particles in a certain size range from the input flow (often referred to as "feed" or "challenge solution"); and the obvious resolution to the engineering challenge would appear to be: use the largest pore size and void fraction consistent with the separation requirement. However, these membrane characteristics (and hence the filter's behavior and performance) are far from constant over its lifetime: the particles removed from the feed are deposited within and on the membrane filter, fouling it and degrading the performance over time. The processes by which this fouling occurs are complex, and depend strongly on several factors, including: the internal structure of the membrane; the flow dynamics of the feed solution; and the type of particles in the feed (the shape, size, and chemistry affects how they are removed by the membrane).
Our work is concerned with the development, analysis and computational simulation of new models governing membrane filtration, in two situations of widespread practical interest: (i) Flow and fouling within pleated filter cartridges; and (ii) Fouling models for internally heterogeneous membranes. In both scenarios we are building models that account for an arbitrary particle size distribution within the feed solution, and account also for a distribution of membrane pore sizes. First-principles theoretical studies of these scenarios should be of interest to those carrying out fundamental experimental research on such systems, as well as to those seeking to extend the scope of current applications and improve on manufacturing processes. Our recent publications in this area, listed below, give an indication of projects completed to date. This work is supported by the National Science Foundation under grants DMS-1261596 and DMS-1615719.
Pleated membrane filters, which offer larger surface area to volume ratios than unpleated membrane filters, are used in a wide variety of applications. However, the performance of the pleated filter, as characterized by a flux-throughput plot, indicates that the equivalent unpleated filter provides better performance under the same pressure drop. Earlier work [Sanaei, Richardson, Witelski, and Cummings, J. Fluid Mech. 795, 36 (2016)] used a highly simplified membrane model to investigate how the pleating effect and membrane geometry affect this performance differential. In this work, we extend this line of investigation and use asymptotic methods to couple an outer problem for the flow within the pleated structure to an inner problem that accounts for the pore structure within the membrane. We use our model to formulate and address questions of optimal membrane design for a given filtration application.
Manufacturers of membrane filters have an interest in optimizing the internal pore structure of the membrane to achieve the most efficient filtration. As filtration occurs, the membrane becomes fouled by impurities in the feed solution, and any effective model of filter performance must account for this. In this paper, we present a simplified mathematical model, which (i) characterizes membrane internal pore structure via permeability or resistance gradients in the depth of the membrane; (ii) accounts for multiple membrane fouling mechanisms (adsorption, blocking, and cake formation); (iii) defines a measure of filter performance; and (iv) for given operating conditions, is able to predict the optimum permeability or resistance profile for the chosen performance measure.
Membrane filters are in widespread industrial use, and mathematical models to predict their efficacy are potentially very useful, as such models can suggest design modifications to improve filter performance and lifetime. Many models have been proposed to describe particle capture by membrane filters and the associated fluid dynamics, but most such models are based on a very simple structure in which the pores of the membrane are assumed to be simple circularly cylindrical tubes spanning the depth of the membrane. Real membranes used in applications usually have much more complex geometry, with interconnected pores that may branch and bifurcate. Pores are also typically larger on the upstream side of the membrane than on the downstream side. We present an idealized mathematical model, in which a membrane consists of a series of bifurcating pores, which decrease in size as the membrane is traversed. Feed solution is forced through the membrane by applied pressure and particles are removed from the feed by adsorption within pores (which shrinks them). Thus, the membrane?s permeability decreases as the filtration progresses. We discuss how filtration efficiency depends on the characteristics of the idealized branching structure.
Membrane filters are used extensively in microfiltration applications. The type of membrane used can vary widely depending on the particular application, but broadly speaking the requirements are to achieve fine control of separation, with low power consumption. The solution to this challenge might seem obvious: select the membrane with the largest pore size and void fraction consistent with the separation requirements. However, membrane fouling (an inevitable consequence of successful filtration) is a complicated process, which depends on many parameters other than membrane-pore size and void fraction; and which itself greatly affects the filtration process and membrane functionality. In this work we formulate mathematical models that can (i) account for the membrane internal morphology (internal structure, pore size and shape, etc.); (ii) describe fouling of membranes with specific morphology; and (iii) make some predictions as to what type of membrane morphology might offer optimum filtration performance.
Pleated membrane filters are widely used in many applications, and offer significantly better surface area to volume ratios than equal-area unpleated membrane filters. However, their filtration characteristics are markedly inferior to those of equivalent unpleated membrane filters in dead-end filtration. While several hypotheses have been advanced for this, one possibility is that the flow field induced by the pleating leads to spatially non-uniform fouling of the filter, which in turn degrades performance. In this paper we investigate this hypothesis by developing a simplified model for the flow and fouling within a pleated membrane filter. Our model accounts for the pleated membrane geometry (which affects the flow), for porous support layers surrounding the membrane, and for two membrane fouling mechanisms: (i) adsorption of very small particles within membrane pores; and (ii) blocking of entire pores by large particles. We use asymptotic techniques based on the small pleat aspect ratio to solve the model, and we compare solutions to those for the closest-equivalent unpleated filter.