A review of computational fluid dynamics applications in pressure-driven membrane filtration

Abstract

Ongoing advances in computational performance and numerics have led to computational fluid dynamics (CFD) becoming a ubiquitous modelling tool. However, CFD methods have only been adopted to simulate pressure-driven membrane filtration systems relatively recently. This paper reviews various approaches to describing the behaviour of these systems using CFD, beginning with the hydrodynamics of membrane channels, including discussion of laminar, turbulent, and transition flow regimes, with reference to the effects of osmotic pressure, concentration polarisation, and cake formation. The use of CFD in describing mass transfer through the membrane itself is then discussed, followed by some concluding comments on commercial membrane simulation packages and future research directions in membrane CFD.

1 Introduction

Computational fluid dynamics (CFD) refers to the use of numerical methods with digital computers to solve fluid flow problems. The basic premise of CFD is replacing the partial differential equations which govern viscous fluid flow with discretised algebraic equations that approximate the partial differential equations (Munson et al. 2002). These equations are then solved numerically to obtain solutions for problems where analytical solutions to the governing equations are not available.

Computational fluid dynamics methods have relatively recently become a viable tool for analysis of membrane filtration systems, due to increases in computing performance and developments in numerical and mathematical methods. This paper gives a brief overview of the basic principles of pressure-driven membrane filtration and CFD, then discusses in detail various approaches to describing the behaviour of pressure-driven membrane filtration systems using CFD.

1.1 Basic principles of pressure-driven membrane filtration

The pressure-driven membrane filtration processes pass a feed stream through a semi-permeable membrane (generally in a crossflow configuration), and use hydraulic pressure to facilitate mass transfer through the membrane to achieve separation into permeate (‘clean’) and concentrate (‘dirty’) streams. These processes include (in order of decreasing pollutant size removed): microfiltration (MF), ultrafiltration (UF), nanofiltration (NF), and reverse osmosis (RO).

Multiple physical phenomena, including membrane fouling, cake formation, and concentration polarisation (where rejected solutes accumulate next to the membrane surface, forming a boundary layer, shown in Fig. 1), make modelling of these processes not a trivial task. Considerable research effort has been expended in describing membrane flux (usually denoted J) in terms of these mass-transfer phenomena: for example, the classical film theory (Kim and Hoek 2005), gel polarisation (Sablani et al. 2001), osmotic pressure (Mulder 1991) and boundary layer resistance models (Wijmans et al. 1985) of concentration polarisation; the resistance in series and pore flow models for fouled membranes (Schäfer 2001); and many more. At the smaller scales of NF and RO, solute charge effects play a large role, represented in models such as the extended Nernst–Planck equation (Mohammad et al. 2002), and variations thereof, including the Donnan steric partitioning model (DSPM) (Bowen and Mukhtar 1996), and extensions of the DSPM with dielectric effects (DSPM-DE) (Bandini and Vezzani 2003; Szymczyk and Fievet 2005); though the details of these and other mass-transfer models are beyond the scope of this review.

Fig. 1

Typical crossflow membrane filtration showing concentration polarisation layer formation

While the mass transfer through a membrane is presumably linked to the hydrodynamic conditions of the membrane channel, the extent and significance of this linkage has been debated, as the flux of fluid through the membrane is typically orders of magnitude smaller than the crossflow velocity in the membrane channel. The use of CFD methods offers opportunities to investigate membrane channel hydrodynamics, as well as their effect on membrane flux, and is becoming increasingly popular in membrane research. We begin with a brief discussion of the basics of CFD.

1.2 Basic principles of CFD

Viscous fluid flow is governed by the basic principles of conservation of mass, and conservation of linear momentum, which can be described by the following equations, respectively:

$$\frac{\partial \rho }{\partial t} + \nabla \cdot (\rho v) = 0$$

(1)

$$\frac{\partial (\rho v)}{\partial t} + \nabla \cdot (\rho vu) = - \frac{\partial P}{\partial y} + \nabla \cdot (\mu \nabla v) + SM_{y}$$

(2)

where ρ is the density of the fluid (kg m⁻³), v is the component of the fluid velocity in the x direction (m s⁻¹), μ is the fluid viscosity (Pa s), P is the pressure (Pa), t is time (s), and SM _y is a source term in the y direction (kg m⁻² s⁻²) (Ghidossi et al. 2006).

To solve the partial differential equations governing fluid flow, appropriate numerical discretisation schemes must be employed. The three main classes of methods are finite difference methods (FDM), finite element methods (FEM) and finite volume methods (FVM). A detailed discussion of these methods can be found elsewhere (Abbott and Basco 1989; Chung 2002; Ferziger and Peric 1996). The main characteristics of each method are listed in Table 1.

Table 1 Summary of characteristics of discretisation methods (compiled from Chung 2002)

Theoretically, it is possible to solve the governing equations of viscous flow via computational methods for both laminar and turbulent flow regimes. However, in turbulent flows, there is a much larger range of length and time scales than in laminar flows, making the equations for turbulent flow more difficult and computationally expensive to solve (Ferziger and Peric 1996). The problem of turbulence can be approached in three main ways (Chung 2002):

• direct numerical simulation (DNS), where refined computational meshes are used to resolve all scales of turbulence, large and small. This is generally not feasible for problems of practical interest;

• Reynolds averaged Navier–Stokes (RANS) methods, where turbulent motion is described in terms of time-averaged quantities, so that time-dependent flow quantities are given in terms of a time-averaged mean component and a time-varying fluctuating component. All scales of turbulence are modelled (as opposed to directly computed) using these methods;

• large eddy simulation (LES), where large scale turbulence is directly computed and small-scale turbulence is modelled using time-averaged methods. This method is becoming increasingly popular, as it offers additional resolution of large-scale turbulence without the heavy computational cost of DNS methods.

A detailed treatment of individual turbulence models is beyond the scope of this paper. Some of the further discussion will refer to certain turbulence models; the reader is referred elsewhere for details (e.g. Chung 2002; Ferziger and Peric 1996).

2 Modelling of membrane processes under laminar conditions

2.1 Flow through porous membranes

Investigations into the hydrodynamics of membrane channels largely began with Berman’s (1953) study of laminar flow in a rectangular channel with porous walls, in which expressions were derived for fluid velocity and pressure as functions of the channel dimensions, fluid properties and position within the channel. The expressions were derived from the Navier–Stokes equations with the following assumptions: (1) a steady state prevails; (2) the fluid is incompressible; (3) no external forces act on the fluid; (4) the flow is laminar and (5) the velocity of the fluid leaving the walls of the channel is independent of position. The primary results derived were that pressure drop in a porous channel is significantly less than a non-porous channel, and that the velocity profile observed within the channel was flatter near the centre and more curved near the walls than the classical parabolic profile of Poiseuille flow.

Later studies were conducted by Friedman and Gillis (1967) for a porous tube with a viscous fluid, which were further refined by Mizushina et al. (1971) for the case of variable radial mass flux, and by Galowin and De Santis (1974) for the case of variable wall suction (all cited in Ghidossi et al. 2006). Berman’s approach was also extended by Karode (2001), who assumed that the velocity of fluid exiting the porous wall was not constant, but was proportional to the local transmembrane pressure (TMP). This revised analytical solution showed good agreement with Berman’s results for the case of constant wall velocity, as shown in Fig. 2.

Fig. 2

Comparison of dimensionless axial pressure drop for CFD model with analytical solutions for hydrodynamic verification of model (Keir and Jegatheesan 2012)

Recent numerical treatments of laminar flow through a porous membrane have followed a similar approach, by using the Navier–Stokes equations to describe the motion of the fluid through the main ‘channel’ of the membrane module, and the Darcy equation to describe the fluid flow through the porous walls of the membrane module. For the case of flow through a cylindrical tube, the continuity and Navier–Stokes equations can be written in cylindrical coordinates as:

$$\frac{{\partial v_{r} }}{\partial r} + \frac{{v_{r} }}{r} + \frac{{\partial v_{z} }}{\partial z} = 0$$

(3)

$$\rho v_{r} \frac{{\partial v_{r} }}{\partial r} + \rho v_{z} \frac{{\partial v_{z} }}{\partial z} + \frac{\partial P}{\partial r} - \frac{\partial }{\partial r}\left( {2\mu \frac{{\partial v_{r} }}{\partial r}} \right) - 2\mu \frac{{\partial v_{r} }}{\partial r} + 2\mu \frac{{v_{r} }}{{r^{2} }} - \frac{\partial }{\partial z}\left[ {\mu \left( {\frac{{\partial v_{z} }}{\partial r} + \frac{{\partial v_{r} }}{\partial z}} \right)} \right] - \rho g_{r} = 0$$

(4)

$$\rho v_{r} \frac{{\partial v_{z} }}{\partial r} + \rho v_{z} \frac{{\partial v_{z} }}{\partial z} + \frac{\partial P}{\partial z} - \frac{\partial }{\partial z}\left( {2\mu \frac{{\partial v_{z} }}{\partial z}} \right) - \frac{\mu }{r}\left( {\frac{{\partial v_{z} }}{\partial r} + \frac{{\partial v_{r} }}{\partial z}} \right) - \frac{\partial }{\partial r}\left[ {\mu \left( {\frac{{\partial v_{z} }}{\partial r} + \frac{{\partial v_{r} }}{\partial z}} \right)} \right] - \rho g_{z} = 0$$

(5)

where r and z represent the radial and axial coordinates (m), respectively, and g is the acceleration due to gravity (m s⁻²).

Similarly, the Darcy equation can be written as:

$$\frac{{\mu_{f} }}{{L_{r} }}v_{r} + \frac{\partial P}{\partial r} = 0$$

(6)

$$\frac{{\mu_{f} }}{{L_{z} }}v_{z} + \frac{\partial P}{\partial z} = 0$$

(7)

where μ _f is the viscosity of the filtrate (Pa s), and L is the permeability of the porous wall (m²).

An example of this approach is the work by Nassehi (1998), who presented a method for coupling the two sets of equations using FEM. The lack of second-order derivatives in the Darcy equation (which are present in the Navier–Stokes equations) presents difficulties in coupling the two sets of equations. Hence, the Brinkmann equation which does contain second-order terms (shown below) has sometimes been used in place of the Darcy equation, though this is not ideal, as the Brinkmann equation is limited to high porosities generally not found in membrane modules. The Brinkmann equation is given by:

$$\nabla P - \mu^{*} \nabla^{2} v = - \frac{\mu }{L}v$$

(8)

where $\mu^{*}$ is the effective viscosity of the fluid flowing through the porous wall (Pa s). Nassehi resolved this issue by developing a Galerkin finite element scheme based on the use of C ⁰ Lagrange elements, in which the layer of elements joining the porous wall to the rest of the solution domain are cast in such a form to represent a combined flow field. That is, elements in the stiffness matrices corresponding to the nodes located on the porous wall are replaced by the discretised form of the Darcy equation, while the remainder of the matrices remain as per the Navier–Stokes equations. Results demonstrated the scheme was flexible in application and accurately preserved flow continuity.

Another method of coupling the Navier–Stokes and Darcy equations was demonstrated by Damak et al. (2004) by assuming the permeability of the wall L _v (m²) to be the same in the axial and radial direction, and assuming the viscosity of the inflow and filtrate to be identical (denoted by μ). This allows the flow through the porous wall to be taken simply as a boundary condition for the Navier–Stokes equations governing the free flow through the tube. The boundary condition can be written as:

$$v_{r} = \frac{{L_{v} }}{\mu } + \frac{\nabla P}{e},v_{z} = 0\;{\text{at}}\;r = R,\;{\text{and}}\;0 \le z \le L$$

(9)

where e is the thickness of the porous wall (m), and R and L are the radius and length of the tube, respectively (m). The justification for setting the axial velocity at the wall to zero is given by work carried out by Schmitz and Prat (1995; cited in Damak et al. 2004), which demonstrated that the slip velocity at the porous wall surface was negligible. The Navier–Stokes equations can then be solved by an implicit finite difference method. The model was verified by comparison against velocity profiles experimentally measured by Gouverneur (1991; cited in Damak et al. 2004), and showed good agreement. The results also demonstrated that the deviation for the parabolic Poiseuille profile previously predicted by Berman (1953) was directly dependent on the wall suction (i.e. the local TMP).

2.2 Concentration-polarisation effects

The inclusion of concentration polarisation effects into the treatment of porous walls introduces additional difficulties. The boundary layer formed by concentration polarisation can be viewed as a kind of second porous wall with a lower permeability than the membrane, as shown in Fig. 1. The primary difficulty in modelling this situation is determining appropriate concentration boundary conditions at the wall, as concentrations will be continually changing at the wall, and the wall geometry itself may change over time due to particle deposition. Several approaches have been developed to account for this, and are discussed below.

A relatively simple approach to this problem was used by Huang and Morrissey (1999). They used FEM to simulate the development of the concentration polarisation boundary layer in a rectangular crossflow ultrafiltration membrane by solving the Navier–Stokes equations to determine fluid velocities in a manner similar to Berman (1953). These were then used to solve the convective-diffusive mass transfer equation using FEM, which describes the formation of the boundary layer:

$$u\frac{\partial C}{\partial x} + \frac{v}{H}\frac{\partial C}{\partial \lambda } = \frac{D}{{H^{2} }}\frac{{\partial^{2} C}}{{\partial \lambda^{2} }}$$

(10)

with the boundary conditions

$$C = C_{0} \;{\text{at}}\;x = 0$$

(11)

$$\frac{\partial C}{\partial \lambda } = 0\;{\text{at}}\;\lambda = 0$$

(12)

$$\frac{\partial C}{\partial \lambda } = \frac{{J_{W} C}}{D}\;{\text{at}}\;\lambda = 1$$

(13)

where C is the solute concentration (g L⁻¹), H is the channel height (m), D is the solute diffusion coefficient (m² s⁻¹), λ is the dimensionless channel height, and J _W is the permeate flux on the membrane wall (m s⁻¹). The finite element package Pdease2D^® was used, with a fine mesh near the membrane surface to accommodate the large concentration gradient in the vicinity of the boundary layer. Results from the simulations were compared to experimental results using bovine serum albumin (BSA) solutions. The results demonstrated a linear relationship between the diffusion coefficient and the thickness of the concentration polarisation boundary layer, as predicted by the classical film theory and gel polarisation models. The model also corresponded well with experimental predictions for the mass transfer coefficient and concentration on the membrane surface.

Richardson and Nassehi (2003) used a more sophisticated approach by assuming that the solution does not remain evenly mixed in domains with porous walls. At each time step, the physical properties of the fluid (viscosity and density) are updated using relations for the concentration calculated at each time step. In their model, the fluid flow in the bulk of the channel is described by the Navier–Stokes equations, while the mass transfer through the porous wall is described by the two-dimensional convective dispersion equation, which is written as:

$$\frac{\partial C}{\partial t} + v_{x} \frac{\partial C}{\partial x} + v_{y} \frac{\partial C}{\partial y} = \left[ {\frac{\partial }{\partial x}D(C)} \right]\frac{\partial C}{\partial x} + \left[ {\frac{\partial }{\partial y}D(C)} \right]\frac{\partial C}{\partial y}$$

(14)

The solution of the flow and convective dispersion equations is then carried out in a decoupled fashion (i.e. in separate steps, with the Navier–Stokes equations solved to provide velocity and pressure fields, with the velocity field then used to solve the convective dispersion equation). The density and viscosity are then updated using current values of velocity and concentration, and the process repeated until the solution converges. The density is calculated in terms of a solids volume fraction Φ which can be determined from the concentration, and the densities of the solid and fluid (ρ _s and ρ _f , respectively, kg m⁻³), as written below:

$$\rho = \rho_{s} \varPhi + \rho_{f} (1 - \varPhi )$$

(15)

The viscosity is calculated by a two-step process, where the viscosity is first updated using the following power law model:

$$\mu = \mu_{o} \dot{\gamma }^{n - 1}$$

(16)

where n is the power law index, μ _o is the consistency coefficient (Pa sⁿ), and $\dot{\gamma }$ is the shear rate (s⁻¹). The viscosity is then updated based on the solids volume fraction via a relationship such as the following:

$$\frac{{\mu_{f} }}{\mu } = 1 + \varLambda \varPhi$$

(17)

where μ _f is the viscosity of the carrying fluid (Pa s), and Λ is a dimensionless empirical factor found experimentally. The equations were solved by use of a streamline upwind Petrov–Galerkin FEM technique (Richardson and Nassehi 2003). Results indicated the model was capable of predicting both flow and mass transport patterns for a variety of domains including flat and curved porous walls.

2.3 Cake formation effects

Geissler and Werner (1995) simulated crossflow filtration in a flat-sheet membrane system considering the formation of a cake layer upon the membrane. The growth of the cake layer was modelled by considering both the convective transport of particles towards the membrane and the diffusive transport in the other direction due to concentration polarisation. The following assumptions were made in developing the model:

• the flow is two-dimensional, incompressible, laminar, and Newtonian, characterised by a local suspension viscosity η _sus ;

• the particles within the fluid are neutrally buoyant;

• the permeate contains no particles; and

• the cake layer formed is immobile.

The equations thus derived to describe the particle transport within the system are as follows:

$$\frac{\partial }{\partial x}\left( {uC - D^{*} \frac{\partial C}{\partial x}} \right) + \frac{\partial }{\partial y}\left( {vC - D^{*} \frac{\partial C}{\partial y}} \right) = 0$$

(18)

$$D^{*} = \hat{D}_{{\phi_{v} }} r_{p}^{2} \frac{\partial u}{\partial y}$$

(19)

$$v_{F} (x) = \frac{\varDelta P(x)}{{\mu [R_{M} + R_{k} (x)]}}$$

(20)

$$R_{k} (x) = \frac{{\delta_{k} (x)}}{{K_{k} }}$$

(21)

$$d\delta_{k} (x) = \frac{{\left\lfloor {d\dot{m}_{k} (x) - d\dot{m}_{dif} (x)} \right\rfloor dt}}{{\rho_{p} (1 - \varepsilon_{k} )dA(x)}}$$

(22)

where D* is the shear-induced diffusion coefficient (m² s⁻¹), $\hat{D}_{{\phi_{v} }}$ is the dimensionless diffusion coefficient, r _p is the particle radius (m), u and v are the velocities in the axial and normal dimensions with respect to the membrane (m s⁻¹), R _k is the hydraulic resistance of the cake (m⁻¹), v _F is the permeate velocity (m s⁻¹), K _k is the cake permeability (m), δ _k is the local cake height (m), ε _k is the cake porosity (dimensionless), $\dot{m}$ is the mass flux (kg s⁻¹) and the subscripts _k and _dif represent the cake and diffusion, respectively.

The model was verified against experimental observations of permeate flux and cake layer height, and showed good agreement (within 15 % of predicted values). However, the model requires empirical parameters (particularly the hydraulic resistance of the cake layer) to be determined by analysis of experimental data in order to achieve useful results.

Lee and Clark (1998) developed a more advanced model of cake formation for a rectangular crossflow ultrafiltration system. Again, mass transfer is expressed by the two-dimensional convective dispersion equation, and the steady-state concentration field can be obtained by solving the following equations:

$$u\frac{\partial C}{\partial x} + v\frac{\partial C}{\partial y} = D_{y} \frac{{\partial^{2} C}}{{\partial y^{2} }}$$

(23)

with the boundary conditions:

$$C(x = 0,y) = C_{B}$$

(24)

$$\left( {\frac{\partial C}{\partial y}} \right)_{y = 0} = 0$$

(25)

$$\left( {vC - D_{y} \frac{\partial C}{\partial y}} \right)_{{y = H_{0} }} = 0$$

(26)

where the half height of the channel is given by H ₀ (m), and D _y is the diffusion coefficient in the transverse direction (m² s⁻¹). Equation (26) can be interpreted as meaning there is no particle accumulation on the membrane surface (i.e. a steady-state has been reached), which is valid for a membrane with complete rejection. The steady-state model is then adapted to provide a step-wise pseudo-steady-state model of the flux decline due to cake formation. This is done by assuming the particle concentration at the membrane surface C _W does not exceed a limiting value C _max = ρ _p (1 − ε _k ) (both g L⁻¹), and that the excess of particles will combine to form a cake layer at the surface.

The solution algorithm proposed is as follows; first, the concentration profile is obtained by solving the steady-state model using the velocity field at the previous time step. Second, it is determined whether flux decline occurs by comparing the concentration profile with the calculated C _max—if greater than C _max, a cake layer will be formed at that location. The cake growth is calculated by a mass balance between the mass convection rate and the mass accumulation rate at the membrane surface. The permeate flux decline is then calculated using a resistance-in-series model so the local permeate flux is known at each axial location. Third, a new velocity field is calculated using the updated permeate flux as a boundary condition, and concentration profiles are again calculated using the steady-state model. The second and third steps are then repeated until C _W  ≤ C _max at every axial location.

The model results presented by Lee and Clark showed good agreement with experimental results; however the model still requires the input of the specific resistance of the cake deposit, which can only be determined by dead-end filtration experiments.

2.4 The influence of osmotic pressure

Osmotic pressure plays an important role in RO and NF membranes. Several authors have attempted to include the effects of osmotic pressure within CFD models, by incorporating boundary conditions for the porous membrane wall which can be expressed as a function of the osmotic pressure.

Geraldes et al. (2001) developed a numerical model using FVM to simulate NF processes in both spiral-wound and plate-and-frame systems, which can be modelled as a slit-type channel configuration (i.e. the height of the channel is very small compared to the length and width of the channel). These geometries mean that the effect of the concentration polarisation boundary layer is a significant operating issue. The model developed by Geraldes et al. (2001) uses CFD for the modelling of the fluid phase, with boundary conditions that take into account the solute transport occurring within the membrane, including osmotic pressure effects. The transport equations used for modelling of the fluid were as follows:

$$\frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} = 0$$

(27)

$$\frac{\partial (\rho uu)}{\partial x} + \frac{\partial (\rho vu)}{\partial y} = - \frac{\partial p}{\partial x} + 2\frac{\partial }{\partial x}\left( {\mu \frac{\partial u}{\partial x}} \right) + \frac{\partial }{\partial y}\left[ {\mu \left( {\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}} \right)} \right] - \frac{2}{3}\frac{\partial }{\partial x}\left[ {\mu \left( {\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}} \right)} \right]$$

(28)

$$\frac{\partial (\rho uv)}{\partial x} + \frac{\partial (\rho vv)}{\partial y} = - \frac{\partial p}{\partial y} + 2\frac{\partial }{\partial y}\left( {\mu \frac{\partial v}{\partial y}} \right) + \frac{\partial }{\partial x}\left[ {\mu \left( {\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}} \right)} \right] - \frac{2}{3}\frac{\partial }{\partial y}\left[ {\mu \left( {\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}} \right)} \right]$$

(29)

$$\frac{{\partial (\rho u\omega_{A} )}}{\partial x} + \frac{{\partial (\rho v\omega_{A} )}}{\partial y} = \frac{\partial }{\partial x}\left[ {\rho D_{AB} \left( {\frac{{\partial \omega_{A} }}{\partial x}} \right)} \right] + \frac{\partial }{\partial y}\left[ {\rho D_{AB} \left( {\frac{{\partial \omega_{A} }}{\partial y}} \right)} \right]$$

(30)

where ω _A is the mass fraction of species A within the solution (dimensionless), and D _AB is the binary mass-diffusion coefficient (m² s⁻¹). Considering the channel to have length l and half height h (both m), the boundary conditions are given by:

$$x = 0,\forall y,u = u_{0} ,v = 0,\omega_{A} = \omega_{A0}$$

(31)

$$x = l,\forall y,\frac{\partial u}{\partial x} = 0,\frac{\partial v}{\partial x} = 0,\frac{{\partial \omega_{A} }}{\partial x} = 0$$

(32)

$$y = 0,\forall x,u = 0,\rho v = \rho_{p} v_{p} ,\rho D_{AB} \frac{{\partial \omega_{A} }}{\partial y} = \rho_{p} v_{p} \omega_{A} f^{{\prime }}$$

(33)

$$y = 2h,\forall x,u = 0,v = 0,\frac{{\partial \omega_{A} }}{\partial y} = 0$$

(34)

where u ₀ is the uniform inlet velocity (m s⁻¹), ω _A0 is the uniform mass fraction at the inlet (dimensionless), f′ is the intrinsic rejection coefficient (dimensionless) and the p subscript denotes properties relating to the permeate. The intrinsic rejection coefficient f′ can be described by the Deen hindered transport model, which for the case of Pe ≪ 1 can be simplified as:

$$f^{{\prime }} = \left[ {1 + \left( {\frac{8u}{{r_{0}^{2} }}} \right)\left( {\frac{{HD_{AB}^{0} }}{\varDelta P}} \right)} \right]^{ - 1}$$

(35)

where r ₀ is the average pore radius of the membrane (m), H is a hindrance factor for solute diffusive transport and $D_{AB}^{0}$ is the binary mass-diffusion coefficient at infinite dilution (m² s⁻¹). The permeate velocity v _p (m s⁻¹) is dependent upon the TMP and the osmotic pressure. However, studies by other researchers (e.g. Niwa et al. 1988; Tanimura et al. 1991; cited in Geraldes et al. 2001) have established that permeate velocity declines with increased solute concentration at the membrane surface (i.e. in the presence of a concentration polarisation layer). The authors thus introduced a concentration dependent correction factor Θ to account for this, in a manner similar to Rosén and Trägårdh (1993; cited in Geraldes et al. 2001). The permeate velocity is then given by:

$$v_{p} = - \varTheta \omega_{Am} A_{h} (\varDelta P - \varDelta \varPi )$$

(36)

$$\varDelta \varPi = \varPi (\omega_{Am} - \omega_{Ap} )$$

(37)

where A _h is the hydraulic permeability of the membrane, (m² s kg⁻¹) and the subscripts m and p refer to the membrane surface and the permeate, respectively. The above equations were discretised using a finite volume formulation. Comparison of the computational results showed good agreement with experimental results, as shown in Fig. 3. Essentially identical approaches to describe the permeate velocity, albeit without the use of a correction factor, were also employed by Fletcher and Wiley (2004) and Miranda and Campos (2001).

Fig. 3

Comparison of predicted permeate fluxes for CFD model of Keir and Jegatheesan (2012) incorporating osmotic pressure correction factor with experimental and modelled fluxes from Geraldes et al. (2001) using experimentally determined rejection factors. Collector numbers represent specified physical locations in membrane channel, with both sucrose and PEG1000 solutions considered for two channel Reynolds numbers, N _Re for varying transmembrane pressures

3 Modelling of membrane processes under turbulent conditions

The studies previously mentioned in this paper have all concerned laminar flows within the membrane module. However, it has been widely suggested that turbulent conditions do occur in practice in UF systems (Belfort 1989; Pellerin et al. 1995). Further, the mixing afforded by turbulent flow may provide significant benefits in reducing concentration polarisation effects. Investigation of turbulent membrane processes is an obvious application of CFD, though it poses significantly more theoretical difficulties than the laminar case.

One of the obvious questions relating to the use of CFD to simulate membrane processes is the question of when turbulent models need to be employed. Typically, flow in narrow channels remains laminar up to Reynolds numbers of approximately 2,000. However, studies have indicated that this is not the case for some types of membrane modules; for example, Schwinge et al.(2002a) examined the transition to turbulence (i.e. the appearance of unsteady flow structures) in spiral-wound membrane modules, concluding that transition may occur in spacer-filled channels at Reynolds numbers of 300–400. Ranade and Kumar (2006b) performed direct numerical simulations (DNS) on spacer-filled channels, which indicated a transition to turbulence at a Reynolds number of approximately 350. Work by the present authors (Keir 2012) using DNS of the spacer-filled channel geometries presented by Wardeh and Morvan (2008) indicated unsteady flow structures were only apparent at Reynolds numbers greater than 400, as shown in Fig. 4.

Fig. 4

Predicted fluid streamlines for spacer-filled channel (Keir 2012) showing steady flow for channel Reynolds numbers up to 400 (upper), and unsteady flow for channel Reynolds number of 800 (lower). Unsteady flow is visualised over three successive timesteps, with individual vortices numbered to track their movement through spacer-filled membrane channel

Belfort and Nagata (1985) also showed the transition to turbulence in tubular channels is delayed to a Reynolds number of approximately 4,000. There is limited information in the literature for other types of membrane modules.

However, the appearance of unsteady flow structures and the transition to turbulence does not necessarily mean that employing conventional turbulence models is the best approach. Turbulence models are suitable only for high Reynolds numbers (generally greater than 30,000), at which point turbulence can be assumed to be fully developed and isotropic (Fimbres-Weihs and Wiley 2010). Pellerin et al. (1995) provide some examples of UF and MF flow with Reynolds numbers of 20,000 and 30,000, respectively, based on the guidelines of Belfort and Nagata (1985). However, the majority of membrane flows are below this level; for example, it is reported that for spiral-wound membrane modules, most flows lie in the range N _Re = 1,000–3,000 (e.g. Belfort 1989; Koutsou et al. 2004). Given this fact, other methods to describe unsteady flows in these flow regimes may be more suitable, such as DNS, which we discuss in Sect. 3.2. We first begin however with a review of studies using conventional turbulence models.

3.1 Studies using Reynolds averaged Navier–Stokes methods

Pellerin et al.(1995) proposed a general hydrodynamic model with the ability to incorporate turbulent effects, as well as complex geometries and pressure-related boundary conditions (to describe the mass transfer through the membrane surface). The transport equation used can be written in general terms as:

$$\frac{\partial (\rho \varphi )}{\partial t} + \nabla (\rho u\varphi - \varGamma_{\varphi } \nabla_{\varphi } ) = S_{\varphi }$$

(38)

where Γ _φ is the exchange coefficient, S _φ is the volumetric source rate, and φ is a general field variable. Turbulent effects in the field variable are represented using a RANS approach via a combination of a time-averaged component $\left\langle \varphi \right\rangle$ and a time-varying component $\varphi^{{\prime }}$:

$$\varphi = \left\langle \varphi \right\rangle + \varphi^{{\prime }}$$

(39)

The two equation k-ε turbulence model was used to resolve the turbulent motion, where k is the turbulent kinetic energy and ε is the turbulent energy dissipation rate. Further discussion on the k-ε model can be found in a multitude of sources (e.g. Ferziger and Peric 1996; Pellerin et al. 1995). Boundary conditions at the membrane wall are given by the transmembrane velocity v _w using the Darcy equation:

$$v_{w} = A(\varDelta P - \sigma \varDelta \varPi )$$

(40)

where A is a membrane-specific coefficient (m² s kg⁻¹). The unitless reflection coefficient σ is a measure of the rejection of the membrane, and was set to one (very high rejection). Finally, the concentration field of the solute can be written in the generalised transport equation format as:

$$\frac{{\partial \left[ {\rho \left( {\frac{{\rho_{k} }}{\rho }} \right)} \right]}}{\partial t} + \nabla \left[ {\rho u\left( {\frac{{\rho_{k} }}{\rho }} \right) - D\rho \nabla \left( {\frac{{\rho_{k} }}{\rho }} \right)} \right] = 0$$

(41)

where ρ _k is the dimensionless density fraction of species k. The above equations were discretised in finite volume form using the computer code TURCOM, and simulations were run to represent RO, UF and MF conditions. The results showed good agreement with classical analytical solutions (Berman 1953; Gupta and Levy 1976; Pellerin et al. 1995) for low Reynolds numbers. However, significant deviation from the velocity profiles in the analytical solutions was observed for high Reynolds number cases (as high as 50,000).

Ranade and Kumar (2006a) also carried out simulations using the k-ε turbulence model to investigate spacer effects in spiral-wound membranes. The turbulence model was only employed for Reynolds numbers greater than 300, based on recommendations from the literature and the authors’ own direct numerical simulations. The commercial package FLUENT was used for all simulations. The authors successfully managed to model a variety of spacer geometries, with adequate agreement with experimental results reported by Da Costa et al. (1991, 1994; cited in Ranade and Kumar 2006a). They also noted that full CFD simulations of an entire membrane module under turbulent conditions remain very computationally expensive, but good results could be obtained by considering a representative unit ‘cell’ of the module.

Cao et al. (2001) also employed a RANS approach in simulating turbulence effects caused by net-type spacers in a narrow channel. The authors made use of the RNG k-ε turbulence model, which differs slightly in formulation from the classic k-ε model used by Pellerin et al. (1995). The RNG model aims to describe the effects of small eddies in terms of larger motions and a modified viscosity term in order to eliminate these small-scale eddies from the governing equations. The RNG model is particularly effective when small-scale flows are significantly anisotropic, as such anisotropic flows tend to either become decoupled from the large-scale flow or give their energy to large-scale eddies, rather than interacting directly with the large-scale flow by increasing the dissipation of large eddies, as was demonstrated by Sivashinsky and Yakhot (1985). The region adjacent to the membrane surface is typically highly anisotropic, making the RNG approach suitable to this application. Simulations were carried out with the FLUENT package for a variety of geometric spacer configurations, details of which are not necessary to be presented here.

3.2 Studies using direct numerical simulation methods

DNS methods allow small-scale and large-scale turbulent effects to be calculated directly, without resorting to the approximation of turbulence models. This is considered to be a more useful approach for most membrane filtration conditions, where Reynolds numbers are sufficiently low (generally less than 30,000) that turbulence is not fully developed, and conventional turbulence models are not strictly applicable. However, this approach is extremely computationally expensive for most problems of practical interest. Nonetheless, several studies have been undertaken using these methods.

Miyake et al. (1995) made use of a DNS approach, where all scales of turbulence are directly computed, to simulate turbulent flow with a periodic pressure gradient in a rectangular channel with one porous wall. Using spectral methods developed by Kim et al. (1987; cited in Miyake et al. 1995) to solve the three-dimensional Navier–Stokes equations, a flow was simulated with periodic pressure boundary conditions on the porous wall. The study indicated that for this flow, good results (compared to the direct numerical solution) could be obtained by large-eddy simulation (LES) models, as the periodic pressure gradient has less influence on small-scale eddies compared to large eddies. This result is interesting, given the relative scarcity of LES models in the membrane modelling literature. However, the applicability of this result to more realistic membrane operating conditions is uncertain.

More recently, several investigations have been made using commercial CFD codes to perform transient simulations with very small timesteps and fine computational meshes to resolve unsteady flows in membrane channels. Some examples of this approach can be seen in studies by Schwinge et al. (2002a, b, c) and Fimbres-Weihs et al. (2006; Fimbres-Weihs and Wiley 2008) using the ANSYS CFX code, and Koutsou et al. (2004) using FLUENT. These models were used to study the effect of spacers in rectangular membrane channels, with all cases considered in these studies simplified by the assumption of impermeable walls or the use of a ‘dissolving-wall’ boundary condition, where the wall concentration is set to a high fixed value to simulate the effects of concentration polarisation. Koutsou et al. (2004) found that instabilities in the vicinity of the cylinder began to occur at a Reynolds number of 60, and further instabilities in the form of wall eddies began to occur at a Reynolds number of 78. Schwinge et al.(2002a) also identified transition to unsteady flow occurring at Reynolds numbers between 300 and 600, dependent on the geometrical configuration of the spacers. These models provide useful insight into the unsteady flow behaviour of these systems, but the simplified wall boundary conditions employed are not entirely physically representative of a real membrane system.

Newer studies have incorporated more realistic descriptions of membrane channel permeation, such as that of Wardeh and Morvan (2008). This study simulated unsteady flow in 2D spacer-filled channels for conditions similar to that of Schwinge et al. (2002a), but incorporated a permeable wall condition using a Darcy’s law type expression. It appears reasonable that further progress in this area should inevitably include the effects of membrane permeation in this manner to develop a more fully representative hydrodynamic model.

Finally, the simulation of flow in spacer-filled channels for these conditions has also led to the use of 3D models becoming somewhat common. While 3D models provide a more accurate representation of real flow conditions, they also incur greatly increased computational expense. Consequently, initial studies used low spatial resolutions (Karode and Kumar 2001) or neglected the membrane mass transport processes by assuming impermeable walls (e.g. Karode and Kumar 2001; Koutsou et al. 2007; Ranade and Kumar 2006b). The most important and common method of reducing computational expense for these 3D simulations however is through use of a unit cell approach, using spatially periodic or ‘wrapped’ boundary conditions. An example of a 3D unit cell geometry for a spacer filled channel is shown in Fig. 5.

Fig. 5

3D quarter unit-cell geometry (Keir 2012) corresponding to 2D model dimensions used in Wardeh and Morvan (2008), with inset detail of coarse 3D mesh (9.7 million elements)

The unit cell approach exploits the fact that spatially periodic (but not necessarily temporally periodic) flow structures tend to occur in spacer-filled channels between filaments, and hence simulation of only a single unit cell between repeating filament structures is required. There are multiple examples of this approach using the dissolving-wall boundary condition to ease computational requirements (Fimbres-Weihs and Wiley 2007; Koutsou et al. 2009; Li et al. 2002, 2005; Santos et al. 2007), with one study by Lau et al. (2009) employing a true permeable wall boundary condition. These studies have also almost universally used the approach of direct numerical simulation of unsteady flows by use of laminar models with very fine spatial and temporal discretisation. However, due to the expense of 3D simulations, this approach is still limited to the unit-cell approach, where spatially periodic flow structures occur. In this respect, direct numerical simulation in this fashion remains more feasible in a general case by use of 2D models.

3.3 Summary of turbulent membrane models

It is clear that nearly the entirety of research in the literature related to turbulence modelling in membrane modules focuses on the modelling and optimisation of spacers and turbulence promoters in spiral-wound membranes. In addition, most applications of turbulence models have employed the standard k-ε model, or the RNG k-ε model, which are suited only to high Reynolds numbers which may not be common to a wide range of membrane applications. Little work has been done in applying turbulent CFD techniques to other membrane configurations, and correspondingly little work has been done using LES techniques, even though these are becoming increasingly popular within the discipline of CFD at large. However, the use of transient laminar flow simulations with very fine meshes and timestepping has also been shown to well describe unsteady flow behaviour for lower Reynolds numbers, which is where the majority of membrane applications are likely to occur.

4 Prediction of membrane mass transfer

The majority of the CFD models discussed so far in this paper have generally attempted to model only the hydrodynamic conditions within the membrane channel or module in question. Most neglect what is the most important property of the membrane, which is the ability to selectively remove constituents from a feed stream; that is, the transfer of mass through the membrane itself. A variety of models have been developed to describe this mass transfer process, as briefly touched upon in Sect. 1.1. By coupling a predictive mass transfer model to a CFD model, a reasonably general model of the membrane filtration process would be potentially realisable.

However, due to the different removal mechanisms in the different types of pressure-driven membrane filtration, it is difficult for a single model to describe membrane mass transfer generally. For the lower-pressure processes, where particulate constituents in the feed stream are common and the membrane pores are relatively large, one might surmise that it might be possible to explicitly model the flow through the porous membrane microstructure within the CFD model itself. That is, if a geometric model of the membrane pore structures could be developed, standard Lagrangian particle transport models included in commercial CFD packages could be used, in conjunction with some method for modelling particle attachment and detachment within and on the membrane microstructure, to predict the performance of these systems. In these particle transport models, particles are essentially injected into the CFD flow domain, and then tracked through the domain by solving differential equations describing the particulate position, velocity, temperature and mass (ANSYS Inc. 2010). However, some serious difficulties hinder this approach:

• The geometry of membrane microstructures is very complex, and can vary significantly between different membranes, as well as spatially over a single membrane, so it is difficult to construct a geometry for general use;

• Implementing a model to describe both particle–particle and particle–membrane interactions is very difficult; a range of models have been developed for other process engineering applications (ANSYS Inc. 2010) but it is unknown how applicable these are to membrane applications;

• Finally, and most significantly, the computational expense incurred by such simulations is prohibitively large at present. This is especially so when considering the combination of the complexity of particle–particle interactions, particle–membrane interactions, and the membrane microstructure geometry.

For now, at least, this method appears mostly infeasible. Considering the higher-pressure processes (NF and RO), on the other hand, it is more conceivable that a general predictive model may be developed. In these processes, the feed constituents tend to be ions or molecules dissolved into solution, and the membrane pores are relatively small. As these pore sizes begin to approach the scale of individual water molecules, attempting to model the microstructure of the membrane using CFD is futile, as conventional descriptions of hydrodynamics do not apply at these scales. Instead, a more useful approach is to use a mathematical model to predict the rejection of the solutes within the feed solution, which is then applied as a boundary flux condition to a CFD model of the bulk of the membrane channel. In this manner, the CFD model can be used to resolve the flow structures and concentration polarisation effects within the membrane channel, even for complex geometries, while the rejection model can be used to describe the mass transfer through the membrane itself. A few attempts have been made using this coupled CFD approach to describe membrane filtration, which are summarised in the following.

Geraldes et al. (2001) used a Deen hindered transport model to predict the rejection of various charged and uncharged compounds in single-solute solutions by a NF membrane in conjunction with a CFD model, as previously discussed in Sect. 2.4. In this approach, the mass transfer (rejection) of the solute depends on the local pressure differential across the membrane, obtained from the CFD solution. However, the rejection calculation, as per Eq. (35) depends on the fitted parameter (8μ/r ²₀ )(HD ⁰ _AB /ΔP), which is constant for a given membrane-single component solution system. This makes the approach difficult to apply generally, and even more so when multiple-component solutions are considered.

An additional significant difficulty in modelling multiple-component solutions in CFD is describing the diffusion of each solute throughout the solution. The classical way to deal with this in CFD is to use a modified version of Fick’s law, where the rate of diffusion of each component is governed by a diffusivity coefficient, assuming that each component in the solution diffuses independently (Bhattacharyya et al. 1990; Fimbres-Weihs and Wiley 2010; Krishna and Wesselingh 1997). However, this method does not necessarily ensure electroneutrality of the solution in the case of charged solutes. To overcome this, a Fick’s law approach can be employed assuming that salts do not dissociate, and a mean diffusion term is used for each individual salt.

In some cases, however, the diffusion of one component may depend on other components. For this situation, a modification of Fick’s law was put forward by Onsager (Onsager 1945), using a matrix of diffusion coefficients, effectively describing the dependence of the diffusivity of each component on all other components. While this approach is general and theoretically sound, determination of these diffusion coefficient matrices is very difficult, and has only been performed for a handful of solutions.

An alternative approach is the use of the Maxwell–Stefan diffusion model (Fimbres-Weihs and Wiley 2010; Krishna and Wesselingh 1997), which considers diffusion to result from the superposition of different frictional forces between the component molecules. The formulation of the Maxwell–Stefan approach is complex and will not be treated in detail here, but has been used for some membrane applications with success. However, it still effectively requires use of a matrix of diffusion coefficients which are difficult to determine, which makes it difficult to use for a general case.

However, simplification of the Maxwell–Stefan diffusion model leads to an approach which is relatively common in membrane mass-transfer models, which is the Nernst–Planck model (see Sect. 1.1). As per Fick’s law, this requires only a single diffusion coefficient for each component, but also takes into account electrical effects (Fimbres-Weihs and Wiley 2010). Fimbres-Weihs and Wiley (2010) describe a method using this approach coupled with the commercial CFD package ANSYS CFX-10.0. This package (and most other commercial packages) assume that multicomponent species transport occurs via Fick’s law where each component diffuses independently. This makes it impossible to directly specify the mass fluxes for each component directly (via the extended Nernst–Planck equations). Instead, a mass source term must be included in the model to cancel out the Fickian diffusion term in the default transport equations, and then add the corresponding diffusive term from the extended Nernst–Planck equations (Fimbres-Weihs and Wiley 2010). In this manner, the components can be allowed to diffuse based on the diffusion rates of other components, while guaranteeing the electroneutrality of the solution is conserved. The Fickian diffusion value used in the inbuilt transport equations is thus a dummy value, which should not affect the simulation as the term is cancelled out by the additional source term (Fimbres-Weihs and Wiley 2010).

Fimbres-Weihs and Wiley (2010) then used this method to simulate the flow of a NaCl:KCl solution through a 2D rectangular membrane channel. No-slip boundary conditions were enforced along the channel walls, with either a fixed wall concentration boundary condition or a constant flux boundary condition imposed at the membrane surface. This method yielded converged solutions which showed different behaviour to that predicted by the classical Fick’s Law approach. However, the use of the more rigorous diffusion model presented some difficulties. In particular, the inclusion of the source term into the governing equations for the commercial code CFX for the entire flow domain caused the mass fraction transport equations to become numerically stiff, particularly near the membrane surface. In addition, it was found that the value of the dummy diffusion coefficient significantly affected the simulation results, even though the term should have simply cancelled out (Fimbres-Weihs and Wiley 2010). It is also probable that the inclusion of the source term and the associated expense of solving the Nernst–Planck equations would have significantly increased the overall computational expense. For these reasons, the authors recommended that future work in coupling a multi-component diffusion model to a CFD simulation over the entire flow domain should avoid the use of source terms, and focus on instead directly specifying the diffusive terms within the inbuilt transport equations.

Déon et al. (2011) also adopted a coupled model of sorts, using the DSPM-DE (a Donnan steric pore model including dielectric exclusion effects, as referenced in Sect. 1.1) to describe the mass transfer through the membrane, and a 2D model using the extended Nernst–Planck equations to describe the diffusion of multiple ions through the concentration polarisation boundary layer. This approach is similar in principle to that of Fimbres-Weihs and Wiley (2010), except that CFD is not used to determine the hydrodynamics of the membrane channel. Instead, classical hydrodynamic equations for turbulent pipe flow are used to determine the velocity profiles in the concentration polarisation layer. A similar approach was also used by Bhattacharjee et al. (2001), though using the conventional DSPM and a linear velocity profile in the concentration polarisation layer. This method was able to reproduce experimental rejection data well, and has the advantages of providing a rigorous description of the transport of ions in a multiple-ion solution both within the membrane itself and within the concentration polarisation boundary layer. This is an improvement over the Fick’s law approach, however, the formulation of Déon et al. (2011) using classical 2D velocity profiles within the concentration polarisation layers means the model is not completely general in terms of the hydrodynamics. That is, in complex membrane channel geometries, the concentration polarisation boundary layer hydrodynamics may not follow the simplifying assumptions imposed by the use of turbulent pipe flow velocity profiles. For this reason, the more general approach including the coupled CFD model (capable of describing the hydrodynamics for arbitrary geometries) used by Fimbres-Weihs and Wiley (2010) is potentially more widely applicable. However, this method still suffers from numerical difficulties as the extended Nernst–Planck equations become very stiff as discussed previously.

Finally, the current author (Keir 2012) presented a model which coupled CFD simulations of membrane channel using ANSYS CFX, and a DSPM mass-transfer model. This model was similar to that presented by Fimbres-Weihs and Wiley (2010), with the notable exception that a Fick’s law approach was used throughout the bulk hydrodynamic model to ensure solution electroneutrality. The DSPM was then used to calculate rejection of individual solution components (ions) within the concentration polarisation boundary layer, allowing for variable mass transfer of different ions through the concentration polarisation layer and membrane. Some illustrative results from the model simulations are shown in Figs. 6 and 7 for a multi-ionic solution in a spacer-filled membrane channel, showing the potential of the coupled CFD-mass transfer approach to predict the interactions between the channel hydrodynamics and membrane mass transfer.

Fig. 6

Predicted variation in mass transfer parameters within spacer-filled channel with spatially varying permeate flux and solute rejection (NaCl:Na₂SO₄ with molar ratio 0.2:0.8, overall ionic strength 34.1 eq m⁻³, Reynolds number 100) (Keir 2012)

Fig. 7

Predicted transverse variation in normalised solute concentration as a function of non-dimensionalised distance from lower membrane wall within spacer-filled channel with spatially varying permeate flux and solute rejection (NaCl:Na₂SO₄ with molar ratio 0.2:0.8, overall ionic strength 34.1 eq m⁻³, Reynolds number 100) (Keir 2012)

5 Comments on commercial membrane simulation packages

Several commercially available software packages for simulating membrane performance exist, generally produced ‘in-house’ by major membrane manufacturers. The most prominent examples of this are the ROPRO^® software developed by Koch Membrane Systems, the ROSA software developed by Dow Water & Process Solutions, the TorayDS software developed by Toray Membrane, and the IMSDesign software developed by Hydranautics (Kucera 2010).

As these packages are generally developed ‘in-house’ and are of significant commercial value, there is relatively little information describing the internal computational and numerical techniques implemented in these packages. However, these packages in general are intended to aid process engineers in designing appropriate membrane treatment trains, i.e. large-scale configurations of membrane modules and arrays using ‘off-the-shelf’ commercially available membrane modules (usually those of the manufacturer supplying the software). In general, these packages provide a design solution encompassing array sizing, operating pressures, scaling indices, and product and concentrate water qualities, with predictions based on nominal membrane performance under steady-state design conditions (Kucera 2010). They are not intended for more fundamental studies of membrane performance, either in terms of hydrodynamics or membrane mass transfer.

Computational fluid dynamics models, on the other hand, are still at this stage generally unsuitable for design purposes, due to their high computational cost and parameterisation requirements. However, there is potential for the use of CFD techniques in detailed models to predict the performance of specific systems, where accurately representing all physical processes involved, and the dynamic nature of the system, is important. CFD techniques are not a competing alternative to these commercial design tools, but rather offer the opportunity for the improvement of commercial tools, if not in routine design, in the areas of performance prediction, process control and optimisation. Perhaps in this area, the potential of CFD modelling of membrane processes may be more fully realised, and work towards this goal remains a challenging yet potentially rewarding endeavour.

6 Summary

It is apparent that CFD techniques are a useful and increasingly accessible method for describing the behaviour of membrane filtration systems. However, developing a general model for the pressure-driven membrane processes (MF, UF, NF, and RO) presents some difficulties. The differences in removal mechanisms between the lower-pressure processes (MF and UF) and the higher-pressure processes (NF and RO) means that a general model of pollutant removal is perhaps unattainable at present. In particular, the particulate nature of pollutants in MF and UF processes requires a separate and considerably more expensive approach to CFD modelling. In addition, the fouling behaviour of such systems is also highly difficult to predict in a general sense.

On the other hand, if non-particulate pollutants only are considered (i.e. chemical species which are mixed within the feed solution), then the possibility of a general model becomes greater. Such a model would in principle be capable of describing the high-pressure processes of NF and RO, as well as some UF processes. It is also evident that these non-particulate pollutants will be present in the low-pressure processes of MF and UF, though their removal will be significantly lesser and their effect on membrane performance will be small. The converse however is not true; high-pressure processes are generally not required to treat such large pollutants which are removed by a range of pre-treatment processes.

In addition, modelling of a wide range of membrane processes introduces the difficult topic of turbulence. While UF and MF processes may in some instances operate in flow regimes where turbulence is fully developed (i.e. Reynolds numbers greater than 30,000), the majority of processes lie in lower Reynolds number regimes, where flow is either laminar (steady), or unsteady without isotropic turbulence, and hence the use of conventional turbulence models cannot be justified. These unsteady flows can be fully resolved, however, if appropriately fine spatial and temporal discretisation is employed, which has been shown to be feasible for conditions encountered in NF and RO.

The use of 2D versus 3D flow models is also of interest. 3D models should be able to more fully describe the complex flow patterns in membrane modules, and provide the most general implementation of the model possible. However, the great computational expense incurred by 3D models has meant that they can only be feasibly be employed for small representative portions of the membrane channel where spatially periodic flow conditions exist, which is not the case in general. The use of 2D models allows computational costs to be kept to a reasonable level such that simulations can be performed for a general case, and also include mass-transfer processes as discussed above.