Computational modeling of coupled free and porous media flow for membrane-based filtration systems: a review

We review different mathematical models proposed in literature to describe fluid-dynamic aspects in membrane-based water filtration systems. Firstly, we discuss the societal impact of water filtration, especially in the context of developing countries under emergency situations, and then review the basic concepts of membrane science that are necessary for a mathematical description of a filtration system. Secondly, we categorize the mathematical models available in the literature as (a) microscopic, if the pore-scale geometry of the membrane is accounted for; (b) reduced, if the membrane is treated as a geometrically lower-dimensional entity due to its small thickness compared to the free flow domain; (c) mesoscopic, if the characteristic geometrical dimension of the free flow domain and the porous domain is the same, and a multi-physics problem involving both incompressible fluid flow and porous media flow is considered. Implementation aspects of mesoscopic models in CFD software are also discussed with the help of relevant examples

Numerical models and optimisation techniques for a circular cross-flow filtration device

The main goal of this thesis is the hydrodynamical analysis of a membrane-based water purification system for potable use that exploits a circular cross-flow design. To that end, a description of the device is introduced and multiphysics problems involving both incompressible fluid flows and porous media flows are solved. The results from two modelling approaches are compared, one based on a coupled system formed by the Navier–Stokes equations and by Darcy’s law with ad-hoc coupling conditions, the other involving Brinkman’s equation. The computations are conducted using COMSOL, a commercial Finite Element solver, and the results are validated using experimental data. Computational modelling permits to identify and quantify flow instabilities (Dean vortices) close to the membrane’s surface, that may improve filtration performance as also evidenced by experimental work. The computation of Dean numbers indicate that the vortices become more pronounced for higher inlet pressure and enlarged aspect ratio of the free flow channel. Even if Brinkman’s model is easier to implement, the coupled Navier–Stokes–Darcy model permits to better represent the slip velocity on the interface. Both models require a large computational time so that they can become prohibitive to study several configurations of the device for optimisation purposes. For this reason, the Proper Generalized Decomposition (PGD) is introduced. It is able to obtain solutions of boundary value problems in closed form with explicit dependence on parameters that can be varied to achieve the optimal configuration of the system one wants to study. The focus shifts on the membrane domain only, where the use of the PGD to obtain a parametric solution depending both on material properties (the permeability of the membrane) and on operational ones (the inflow flux) is studied. The convergence speed of the PGD is found to deteriorate as the number of parameters increase. To overcome this, the linearity of the problem is exploited to define an improved algorithm. Finally, possible techniques to combine PGD and domain decomposition methods are explored in order to obtain a parametric solution of a multiphysics problem in a complex domain. More specifically, both the classical Multiplicative Schwarz method accelerated by PGD and the overlapping Arlequin method are considered. A variant of the latter is proposed that provides similar results to the original method but has the advantage of not requiring to compute quantities on the overlapping region. This could possibly result in a simpler implementation and reduced computational cost, especially in 3D cases. The methods presented and developed in this thesis can be used as the foundation towards the parametric optimisation of the filtration device.</div

Mathematical and numerical modelling of a circular cross-flow filtration module

We present a computational modelling framework to assess the fluid dynamic behaviour of a circular cross-flow filtration module for water purification. We study two modelling approaches, namely, the Navier-Stokes-Darcy and the one-domain models, that provide a different characterization of the flow in the interfacial region between the feed domain and the membrane surface. Extensive comparison of the numerical results obtained by the two approaches highlights significant differences in the predicted fluid tangential velocity on the membrane surface. Numerical modelling permits to gain a deeper understanding of the flow behaviour than the sole experimental work, e.g., by identifying Dean vortices inside the feed domain and by relating them to geometrical and flow characteristics. This study lays the basis for the optimization of the circular cross-flow filtration module

Mathematical and numerical modelling of hybrid filtration systems

We present a computational modelling framework to support the design and optimization of membrane-based water purification systems. Two modelling approaches are defined which differ in the way they describe the flow in the interfacial region between the feed domain and the membrane surface. Extensive comparison of the results obtained by the two methods highlights non-negligible differences in the predicted flow pattern, especially in the neighbourhood of the membrane. Numerical modelling and computer simulations permit to gain a deeper understanding of the flow behaviour than the sole experimental work, e.g., by identifying Dean vortices inside the feed domain and by relating them to geometrical and flow characteristics