Filtration

Pleated membrane filters are ubiquitous in many industrial filtration systems due to their high surface-area-to-volume ratio. However, their performance often falls short compared to flat non-pleated membrane filters of the same membrane surface area. This raises the question: What is the optimal initial internal pore structure of the membrane to achieve the most efficient filtration? To address this question, we first present a mathematical model describing the feed flow and particle transport within the complex geometry of a pleated filter based on our

previous work [Fong and Sanaei, “Flow and transport in a pleated filter,” Phys. Fluids 34, 097102 (2022)]. We then analyze the governing equations using asymptotic analysis by exploiting the small aspect ratios of the pleated membrane and filter cartridge. In the second part of the paper, we formulate a computationally efficient optimization problem aimed at determining the optimal initial pore shape to improve filtration performance. Depending on the initial average porosity, substantial differences in the computed optimal pore profile are observed. Furthermore, by varying a geometric parameter in our model, we investigate the influence of the pleat packing density on the optimal initial

pore shape.

A pleated membrane filter consists of a porous membrane layer, which is surrounded by two supporting layers, and the whole structure is pleated and placed into a cylindrical cartridge. Pleated membrane filters are used in a variety of industrial applications, since they offer more surface area to volume ratio that is not found in equivalent flat filters. In this work, we introduce a novel three dimensional model of a pleated membrane filter that consists of an empty region, a pleated region, and a hollow region. The advection diffusion equation is used to model contaminant concentration in the membrane pores along with Darcy’s law to model the flow within the membrane and support layers, while the Stokes equation is used for the flow in the empty region and the hollow region. We further use the key assumptions of our model based on small aspect ratios of the filter cartridge and the pleated membrane to simplify the governing equations, which can be easily solved by numerical methods. By performing these steps, we seek to discover an optimal pleat packing density to find the optimum filter performance, while not exceeding a threshold for the particle concentration at the filter outlet.

Filtration is widely used in industry, therefore prediction of filtration efficacy and analysis of filter performance are essential. Real membranes have complex internal geometry: pores inside the membrane branch and interconnect with each other, which must be taken into account in mathematical models of filtration. Membrane fouling, as an unavoidable consequence of removing particles, occurs in the course of filtration and deteriorates the membrane permeability. In addition, for membranes made of elastic materials, the pressure within the membrane results in expansion of the pore radii. The pore expansion competes with particle deposition to delay fouling and, thus, influences filtration performance. In this paper, we develop a mathematical model of flow and fouling of such elastic membrane filters with multi-layer bifurcating (hierarchical) interior morphology. Two filtration forcing mechanisms through the membrane are considered: (i) constant pressure drop; and (ii) constant flux. We investigate how filtration behaves under these two forcing mechanisms and mathematically describe the morphology change due to fouling coupled to elastic pore expansion. In particular, we obtain an analytical solution for the deformation of the elastic pore walls, which is easily incorporated into the filtration model. Our model provides a quantitative mathematical framework with which to predict the impact of hierarchical pore morphology and the elasticity of pore walls on filtration performance.

Pleated membrane filters are widely used to remove undesired impurities from a fluid in many applications. A filter membrane is sandwiched between porous support layers and then pleated and packed into an annular cylindrical cartridge with a central hollow duct for outflow. Although this arrangement offers a high surface filtration area to volume ratio, the filter performance is not as efficient as those of equivalent flat filters. In this paper, we use asymptotic methods to

simplify the flow throughout the cartridge to systematically investigate how the number of pleats or pleat packing density affects the performance of the pleated membrane filters. The model is used to determine an optimal number of pleats in order to achieve a particular optimum filtration performance. Our findings show that only the “just right”—neither too few nor too many—number of pleats gives optimum performance in a pleated filter cartridge.

Multilayered membrane filters, which consist of a stack of thin porous membranes with different properties (such as pore size and void fraction), are widely used in industrial applications to remove contaminants and undesired impurities (particles) from a solvent. It has been observed experimentally that the performance of well-designed multilayer structured membranes is markedly better than that of equivalent homogeneous membranes. Mathematical characterization and modeling of multilayered membranes can help our understanding of how the properties of each layer affect the performance of the overall membrane stack. In this work, a simplified mathematical model is presented to describe how the pore-scale properties of a multilayered membrane affect the overall filter performance.

Membrane science is a known area of study, which motivates the development and improvement of filtration technology in various industries. Membrane filters structures are equipped with specific properties, such as pore size and void fraction, which both vary depending on different applications. The requirement of filtration is to achieve the separation of particles and fluid, while minimizing the energy consumption at the same time. However, membrane fouling is inevitable during the filtration process and affects membrane functionality in any filtration stage. Membrane fouling is a very complex process and is determined by many properties such as the membrane internal morphology, internal membrane pore structure, flow rate and contaminant properties. In a very slow filtration process or during the late stage of filtration, when the flow rate is naturally low and P'eclet number is small, particle diffusion is essential and can not be neglected, while in typical filtration models, especially in moderate and fast filtration process, it is completely ignored. The objective of this study is to investigate how filtration process changes under possible effects brought by particle diffusion. We discuss how membrane morphology evolves and investigate the filtration performance during the filtration process. We also compare the results with the situation where diffusion is less important (in a fast filtration process) and make a prediction about what kind of membrane filter should be employed to achieve a particular optimum filtration performance in a slow process.

We study the influence of a membrane filter's internal pore structure on its flow and adsorptive fouling behaviour. Membrane performance is measured via (i) comparison of flux-throughput characteristics during filtration; and (ii) control of concentration of foulants at membrane pore outlets. Taking both measures into account, we address the merits and drawbacks of selected membrane pore structures. We first model layered planar membrane structures with intra-layer pore connections, and present comparisons between non-connected and connected structures. The model predicts that membrane filters with connected pore structures lead to higher total throughput than those with non-connected structures, over the filter lifetime. We also provide a sufficient criterion for the concentration of particles escaping the filter to achieve a maximum in time (indicative of a membrane filter whose particle retention capability can deteriorate). Additionally, we find that spatial inhomogeneity in the pore-size distribution influences performance of filters differently when their pores have different connectivity properties.

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 model of filter performance must account for this. In this work, a simplified mathematical model is presented, which (i) characterizes membrane internal pore structure via permeability or resistance gradients in the depth of the membrane; (ii) accounts for multiple simultaneous 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. Here, an idealized mathematical model is presented, 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, mathematical models are formulated, which 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 work this hypothesis is investigated by developing a simplified model for the flow and fouling within a pleated membrane filter. This 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. Asymptotic techniques, based on the small pleat aspect ratio, are used to solve the model, and solutions are compared to those for the closest-equivalent unpleated filter.