A 2-D streamline upwind Petrov/Galerkin finite element model for concentration polarization in spiral wound reverse osmosis modules
Introduction
As a phenomenon inherently associated with membrane separations, concentration polarization has long been identified as a major problem that deteriorates the performance of reverse osmosis (RO) systems by reducing permeate flux and solute rejection [1], [2], [3], [4]. However, the phenomenon is still not well understood especially in practical systems employing spiral wound modules, in which spacers are widely used to support the channel and to promote mixing in the feed channel. Although previous studies have reported on the effects of filaments on hydrodynamic parameters such as velocity, wall stress and pressure drop [5], [6], [7], [8], [9], [10], [11], [12], a quantitative link between the geometric configurations of the spacers/filaments and concentration polarization is still unavailable.
In concentration polarization, solute transport that is dependent on momentum transfer also affects momentum transfer by prescribing velocity boundary conditions (permeate) on the membrane surface and concentration-dependent viscosity in the feed channel [2], [13], [14]. This interaction makes concentration polarization substantially different from the typical mass transfer problem in channels with impermeable walls or with sinks/sources in the wall. To accurately and reliably describe such interactions, a coupled model of solute transport and momentum transfer in the feed channel is needed.
Concentration polarization in RO channels has been simulated since 1960s. The models used fall into either the category of analytical models [3], [4], [15], [16], [17], [18] or numerical models [2], [13], [14], [19], [20], [21], [22], [23], [24]. The analytical models, which are based on some assumptions and simplifications, are usually inapplicable to the practical spiral wound modules due to complicated geometry of the feed channel with spacers. This problem usually can be solved in numerical models because most numerical solvers can deal with different geometrical configurations of computational domain. Srinivasan et al. [13] developed a coupled model for concentration polarization in RO systems. However, this model can only be applied to empty channels (without spacers) because the governing equations were solved in the boundary layer rather than in the whole channel. There are a few uncoupled numerical models that solved the solute transport equations in the whole channel [21], [23], [24] employing some simplifications in the velocity field [25] and/or empirical velocity profiles [5], [7]. In these models, simplified velocity fields were used, so the interaction of mass and momentum transfer, which features membrane separation processes, was not adequately described. To solve this problem, Wiley and Fletcher [14] developed a coupled concentration polarization model using the commercial computational fluid dynamic (CFD) software (CFX). The model was, however, only applied to empty channels and not tested in spacer-filled channels.
The governing equations of concentration polarization, Navier–Stokes equations and convection–diffusion equation, have been studied extensively in fluid mechanics and mass transport problems. However, there are difficulties in obtaining numerical solutions of these equations when they are applied to study concentration polarization in the RO feed channel. Firstly, the geometry of the domain of interest is extremely narrow and long in typical spiral wound RO modules. The height of the feed channel is usually less than 1 mm and the length in the order of meters. Moreover, the thickness of concentration polarization layer is usually far less than the channel height. Therefore the required number of elements or cells or grids is enormous in order to satisfy the requirement of aspect ratio of the elements and to capture the sharp gradient of salt concentration near the membrane surface. Secondly, for concentration polarization, Navier–Stokes equations and convection–diffusion equation are coupled not only in the domain but also on the boundary (membrane surface); most general-purpose CFD software packages are not designed for this type of problems. Therefore, it is necessary to develop a specially designed numerical model for concentration polarization to reliably simulate this phenomenon in spiral wound modules.
In convection dominated problems, streamline upwind Petrov/Galerkin (SUPG) finite element formulation can produce wiggle-free solutions without refinement. In the last 20 years, SUPG finite element has been applied to many kinds of mass, heat and momentum transfer problems and now is one of the preferred methods for this type of problems. Brooks and Hughes [26] developed and introduced this method systematically for incompressible Navier–Stokes equations. Later, this method was successfully used in many solute transport problems [27].
In this paper, a finite element model specially designed for the study of concentration polarization in spiral wound RO modules was developed. To effectively achieve accurate and reliable solutions for the convection dominated momentum and mass transfer problem in the feed channel, the streamline upwind Petrov/Galerkin (SUPG) method was employed in the model to solve the coupled governing equations (Navier–Stokes equations and convection–diffusion equation) numerically. With the numerical model developed, the effect of element number on solution accuracy was studied. The simulated velocity and concentration profiles demonstrated that concentration polarization in spiral wound membrane models could be adequately described with this numerical model.
Section snippets
Governing equations
In practical spiral wound RO modules, because the channel is too narrow and the permeate velocity is usually at least 3 or 4 orders lower than the mean crossflow velocity which is usually less than 0.5 m/s, turbulence is unlikely to be fully developed and laminar flow can be assumed [9], [28]. Gauwbergen and Baeyens [9] and Schock and Miquel [29] have shown that the bent envelope in spiral wound modules can be reasonably approximated by an unwound flat membrane channel. The 2-D governing
Results and discussion
Numerical simulations were carried out on supercomputers (SGI Origin 2000 with 16 MIPS R10,000 CPUs and 6 Gbytes physical memory and Compaq GS320 with 22 EV67 Alpha21264 CPUs and 16 Gbytes physical memory) at the Supercomputing & Visualization Unit (SVU) of the National University of Singapore.
To validate the numerical model developed, the numerical simulation results from the model were compared with published experimental data [37]. The effects of meshing schemes on the accuracy of
Conclusions
Numerical simulation is one of the most efficient methods to study concentration polarization in real systems and to optimization of channel/module design. A streamline upwind Petrov/Galerkin finite element model for concentration polarization in spiral wound RO modules was developed by solving the coupled convection–diffusion and Navier–Stokes equations so that the interactions between momentum transfer and solute transport was adequately represented.
Meshing quality is critical for a stable
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