New Jersey Institute of Technology Digital Commons @ NJIT Dissertations Electronic Theses and Dissertations Spring 2017 Mathematical modeling of membrane filtration Pejman Sanaei New Jersey Institute of Technology Follow this and additional works at: https://digitalcommons.njit.edu/dissertations Part of the Mathematics Commons Recommended Citation Sanaei, Pejman, "Mathematical modeling of membrane filtration" (2017). Dissertations. 24. https://digitalcommons.njit.edu/dissertations/24 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at Digital Commons @ NJIT. It has been accepted for inclusion in Dissertations by an authorized administrator of Digital Commons @ NJIT. For more information, please contact [email protected]. Copyright Warning & Restrictions The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. 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Please Note: The author retains the copyright while the New Jersey Institute of Technology reserves the right to distribute this thesis or dissertation Printing note: If you do not wish to print this page, then select “Pages from: first page # to: last page #” on the print dialog screen The Van Houten library has removed some of the personal information and all signatures from the approval page and biographical sketches of theses and dissertations in order to protect the identity of NJIT graduates and faculty. ABSTRACT MATHEMATICAL MODELING OF MEMBRANE FILTRATION by Pejman Sanaei The purpose of this thesis is to formulate and investigate new mathematical models for membrane filtration. The work presented is divided into six chapters. In the first chapter the problem is introduced and motivated. In the second chapter, a new mathematical model for flow and fouling in a pleated membrane filter is presented. 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 nonuniform fouling of the filter, which in turn degrades performance. This hypothesis is investigated by developing a simplified model for the flow and fouling within a pleated membrane filter. The 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 are used based on the small pleat aspect ratio to solve the model, and solutions are compared to those for the closest-equivalent unpleated filter. In the third and fourth chapters, mathematical models are proposed to describe the effects of filter membrane morphology on filtration efficiency. A reasonable question that membrane filter manufacturers may ask is: what is the optimal configuration of filter membranes, in terms of internal morphology (pore size and shape), to achieve the most efficient filtration? In order to answer this question, a robust measure of filtration performance must be first proposed. Filter membrane performance can be measured in a number of different ways. As filtration occurs, the membrane becomes blocked, or fouled, by the impurities in the feed solution, and any performance measure must take account of this. For example, one performance measure might be the total throughput { the amount of filtered feed solution { at the end of filtration process, when the membrane is so badly blocked that it is deemed no longer functional. A simplified mathematical model is proposed, which (i) characterizes membrane internal pore structure via pore or permeability profiles in the depth of the membrane; (ii) accounts for various membrane fouling mechanisms (adsorption and blocking in Chapter 3, and cake formation in Chapter 4); and (iii) defines a measure of filter performance; and (iv) predicts the optimum pore or permeability profile for the chosen performance measure. In the fifth chapter, a model for more complex pore morphology is described. 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 which may branch and bifurcate. Pores are also typically larger on the upstream side of the membrane than on the downstream side. 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. The membrane's permeability decreases as the filtration progresses, ultimately falling to zero. The dependence of filtration efficiency on the characteristics of the branching structure is discussed. The sixth chapter concludes the thesis with a discussion of future work. MATHEMATICAL MODELING OF MEMBRANE FILTRATION by Pejman Sanaei A Dissertation Submitted to the Faculty of New Jersey Institute of Technology and Rutgers, The State University of New Jersey { Newark in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematical Sciences Department of Mathematical Sciences, NJIT Department of Mathematics and Computer Science, Rutgers-Newark August 2017 Copyright c 2017 by Pejman Sanaei ALL RIGHTS RESERVED APPROVAL PAGE MATHEMATICAL MODELING OF MEMBRANE FILTRATION Pejman Sanaei Linda J. Cummings, Dissertation Advisor Date Professor of Mathematics, New Jersey Institute of Technology Lou Kondic, Committee Member Date Professor of Mathematics, New Jersey Institute of Technology Michael S. Siegel, Committee Member Date Professor of Mathematics, New Jersey Institute of Technology Ian M. Griffiths, Committee Member Date Royal Society University Research Fellow, University of Oxford Emilie Dressaire, Committee Member Date Professor of Mechanical and Aerospace Engineering, New York University Tandon School of Engineering Anil Kumar, Committee Member Date Principal Process Engineer at Sanofi Genzyme BIOGRAPHICAL SKETCH Author: Pejman Sanaei Degree: Doctor of Philosophy Date: August 2017 Undergraduate and Graduate Education: • Doctor of Philosophy in Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 2017 • Master of Science in Applied Statistics, New Jersey Institute of Technology, Newark, NJ, 2016 • Master of Science in Pure Mathematics, Shiraz University, Iran, 2009 • Bachelor of Science in Mechanical Engineering, Shiraz University, Iran, 2006 Major: Mathematical Sciences Presentations and Publications: P. Sanaei, G.W. Richardson, T.P. Witelski and L.J. Cummings, “Flow and fouling in a pleated membrane filter,” Journal of Fluid Mechanics, 795:36-59, 2016. P. Sanaei and L.J. Cummings, “Flow and fouling in membrane filters: Effects of membrane morphology,” Journal of Fluid Mechanics, 818:744-771, 2017. P. Sanaei and L.J. Cummings, “Optimum permeability profile and fouling in membrane filters,” Preprint. P. Sanaei and L.J. Cummings, “Modeling flow and fouling in membrane filters with complex pore morphology,” Preprint. P. Sanaei, I.M. Griffiths, L.J. Cummings and S.L. Waters, “Curvature and stress driven tissue growth in a tissue engineering scaffold pore,” Preprint. P. Sanaei, L.J. Cummings and I.M. Griffiths, “Comparing a continuum and a network mathematical model for membrane filters,” In preparation. iv P. Sanaei and L.J. Cummings, \Fouling of a filter membrane with complex microstructure: A simplified mathematical model," Conference Presentation, Annual Meeting of Society for Industrial and Applied Mathematics, Pittsburgh, 2017. P. Sanaei and L.J. Cummings, \Modeling of membrane filters, Optimizing internal structure and morphology," Conference Presentation, Annual Meeting of Society for Industrial and Applied Mathematics, Student Days Talks, Pittsburgh, 2017. P. Sanaei and L.J. Cummings, \Mathematical modeling of pleated membrane filters,” Conference Presentation, Society for Industrial and Applied Mathematics{ Computational Science and Engineering, Atlanta, 2017. L.J. Cummings and P. Sanaei, \Optimizing internal structure of membrane filters,” Conference Presentation, 69th Bulletin of the American Physical Society Division of Fluid Dynamics, Portland, 2016. P. Sanaei and L.J. Cummings, \Modeling branching pore structures in membrane filters,” Conference Presentation, 69th Bulletin of the American Physical Society Division of Fluid Dynamics, Portland, 2016. P. Sanaei and L.J. Cummings, \Optimum permeability