# Transfer of Monovalent Salts through Nanofiltration Membranes: A Model Combining Transport through Pores and the Polarization Layer

Sébastien Déon, Patrick Dutournié, Patrick Bourseau

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Sébastien Déon, Patrick Dutournié, Patrick Bourseau. Transfer of Monovalent Salts through Nanofiltration Membranes: A Model Combining Transport through Pores and the Polarization Layer. Industrial and engineering chemistry research, 2007, 46 (21), pp.6752-6761. 10.1021/ie061687k. hal-00399007

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# Transfer of Monovalent Salts through Nanofiltration Membranes: A Model Combining Transport through Pores and the Polarization Layer

Sébastien Déon, Patrick Dutournié, and Patrick Bourseau*

Laboratoire d'Etudes Thermiques Energétique et Environnement, Université de Bretagne Sud, rue St-Maudé, BP 92116, 56321 Lorient Cedex, France

Nanofiltration is a very useful separation process having many industrial and environmental applications. The prediction of separation performances requires the understanding of the various physical phenomena governing the solute transfer through the membrane. In this paper, a two-dimensional model called the pores and polarization transport model (PPTM) is presented, which combines the convective-diffusive-migration transport within the pores with the transport through the concentration polarization layer. This model was developed for predictive purpose to evaluate the efficiency of separation with a minimum of characterization experiments. Model parameters assessment is presented, and experimental results obtained for various monovalent salt solutions (NaCl and KCl) are compared with simulations to validate the model. This model can predict the influence of the various operating parameters on the polarization layer and the experimental rejections of single salt solutions.

## Introduction

Nanofiltration (NF) is a membrane process which has many industrial and environmental applications. Indeed, it presents many advantages compared to other processes such as distillation because of its easy process control, lower energy consumption, and environmental respect, i.e., it does not require chemical addition. Salt rejection of nanofiltration membranes is governed by steric and electric as well as electrostatic effects, which allow separation between small or weakly charged ions and bigger or more charged ions. For this reason, nanofiltration membranes were found to be suitable for a pretreatment prior to a desalination process by reverse osmosis, electrodialysis, or a nonmembrane process (e.g., distillation). For example, Al-Sofi¹ has reported that NF pretreatment prevents membrane fouling and scaling and lowers the pressure required to operate a reverse osmosis step. Nanofiltration is also a particularly well-suited process for charged solutes fractionation and a very competitive process for industrial and environmental purposes, including the partial desalination of brackish,² seawater,³ and industrial solutions,⁴ for example, for the production of drinking or isotonic water.⁵,⁶ It was also proved that nanofiltration can play a determinant role in removing chemical and biological contaminants in drinking water production.⁷

In the past few years, many authors have studied the mass transfer through nanofiltration membranes which is usually described by a one-dimensional (1-D) vision of transport within perfect porous media.⁸⁻¹¹ With this point of view, separation selectivity is mainly governed by transport within the pores and by the equilibrium partitioning at the membrane-solution interfaces. Nevertheless, it was observed experimentally that selectivity can also be controlled by transport through the polarization layer resulting from the solutes accumulation near the wall due to permeation. The polarization layer is frequently neglected in models, or estimated from empirical formula-tions.¹²,¹³ In this study, the polarization layer is modeled from a two-dimensional vision of mass transport as it was performed

within the laminar sublayer by some authors.¹⁴,¹⁵ Moreover, it is here assumed that the polarization layer can also be spread out in the other hydrodynamic layers. The model presented in this paper is a global two-dimensional model including interface equilibrium partitioning and transports within the pores and the polarization layer. It can be used either to assess some membrane characteristics or to estimate the fluxes and rejections for given experimental conditions. In addition, model predictions of salt rejections are compared to experimental results.

## Theoretical Background

Three main approaches have been proposed in the literature for the transport of ionic compounds in membrane pores relying on the following: (a) the extended Nernst-Planck equation expressing the flux of a solute as the sum of terms of transport by convection, diffusion, and electromigration,¹⁶,¹⁷ (b) the Stefan-Maxwell equation stating that the transport of a solute results from the balance between the driving force on the solute, namely its generalized electrochemical potential, and the friction forces exerted by all the other components on it (solvent and membrane included),¹⁸ (c) the thermodynamics of irreversible processes (TPI) which requires few parameters in its simplified form¹⁹,²⁰ but which are difficult to estimate in the case of ions due to their dependence on concentration. The approach which is most widely used is the extended Nernst-Planck one, probably because it requires less parameters than the Stefan-Maxwell approach and TPI. It also has a wider range of use than the simplified thermodynamic approach (e.g., ionic compounds and mixtures) which assumed no electrical effect.

The 1-D modeling of transfer through the membrane presented in this paper is based upon the Donnan steric pore model (DSPM) proposed by Bowen et al.¹⁰,²¹ for which separation is controlled by pore transport and equilibrium partitioning at the membrane interface. But, it is considered in this paper that the separation selectivity also depends on solute accumulation close to the membrane wall which is described here by a two-dimensional model based on a differential mass balance. The

<footnote>* To whom all correspondence should be addressed. Tel.: +33 2 97 87 45 31. Fax: +33 2 97 87 45 00. E-mail: Patrick.bourseau@univ-ubs.fr.</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1. Concentration profiles through the membrane and the polarization layer for a NaCl 0.1 M solution (ΔP = 20 bar and Uᵢ = 2 m s⁻¹).</figcaption></figure>

rejection of ionic solutes by nanofiltration membranes is thus mainly governed by the three following phenomena (see Figure 1):

(i) the transport of the solute through the polarization layer due to its accumulation at the membrane wall—this accumulation changes the solute concentration at the wall and influences the following step;

(ii) the solute equilibrium partitioning at the membrane—external solutions interfaces that results from electrochemical potential equality on both sides of the interface;

(iii) the solute transport through the membrane pores due to the electrochemical gradient along the pore.

The Nernst–Planck approach allows one to take account of the various physical phenomena influencing the rejection of ionic species such as equilibrium partitioning between the solutions inside/outside the membrane resulting from steric, electrostatic, and dielectric effects and the transport by convection, diffusion, and electromigration through the membrane. However, this interesting physical meaning should not cause one to overlook the limits of the approach. Indeed, the membrane is supposed to be a perfect porous medium respecting the classical laws of continuous media mechanics. Porosity is represented by a set of cylindrical pores of the same radius. Although it allows the correct description of the variation of permeation flux with pressure, the adequacy of this model with reality can be questioned by the fact the pore diameter (about 1 nm in this work) is hardly greater than the size of water molecules (about 0.3 nm). The physical meaning of the water and solution viscosities as well as the notion of viscous flows in such small pores could be undoubtedly discussed. On the other hand, recent works of Yaroshchuck,²² Szymczyk and Fievet,²³ Bandini and Vezzani,²⁴ Bowen et al.,²⁵ and other authors seem to prove that dielectric effects play a significant role in the rejection of ionic species by nanofiltration membranes besides steric and electrostatic effects. But, if these effects are not completely understood nor their interactions with electric effects, their intensity depends on the pore size in such a way that it is probably not realistic to consider a single (mean) pore radius and to ignore the pore size distribution.

**Transport through Membrane Pores.** Solute transport through the membrane is similar to transport of solid spheres within cylindrical single nanometer-sized pores uniformly dispersed. With this approach, the flux of a given ion i transmitted through the pores is given by the extended Nernst–Planck equation¹⁶,²⁶ as the sum of three components

respectively due to diffusion, convection, and electromigration. The influence of the pore wall on diffusion and convection is introduced in the model through hindrance factors Kᵢ,ₗ and Kᵢ,ₗ.²⁷,²⁸ The flux of the i-ion jᵢ is finally expressed as follows:²⁵

$$j_i(x) = -c_i D_{i,p} \frac{d[\ln \gamma_i]}{dx} - \frac{D_{i,p}}{RT} \overline{c_i V_i} \frac{dp}{dx} - D_{i,p} \frac{dc_i}{dx} - \frac{z_i c_i D_{i,p}}{RT} F \frac{d\psi}{dx} + K_{i,c} c_i V \quad (1a)$$

where cᵢ, γᵢ are respectively the concentration and the activity coefficient of an i-ion. Dᵢ,ₚ is the i-ion diffusivity within the pores equal to the product of the diffusivity at infinite dilution by the diffusion hindrance factor (Dᵢ,ₚ = Kᵢ,ₗDᵢ,ₐ). The diffusivity in the pore can be considered at infinite dilution because ion concentrations in the pore are much lower than in the bulk. The term dψ/dx is the electrical gradient along the pore, and the solvent velocity within the pore V is calculated by the Hagen–Poiseuille equation:

$$V = \frac{r_p^2 \Delta P_e}{8 \eta \Delta x} = \frac{r_p^2 [\Delta P - \Delta \pi]}{8 \eta \Delta x} \quad (2)$$

with η being the dynamic viscosity corrected as a function of the salt concentration from handbook data,²⁹ rₚ being the average pore radius, Δx being the membrane thickness, ΔP being the applied pressure, and Δπ being the osmotic pressure difference on both sides of the membrane estimated from the Van't Hoff relation

$$\Delta\pi = RT \sum (C_{i,m} - C_{i,p}) \quad (3)$$

The concentration variations along the pores are small enough to neglect the influence of the activity coefficient gradient on the electrochemical gradient. A sensitivity study has also shown that the pressure gradient term (second term in eq 1) is also negligible compared to the others so that only the three last terms are required to model the mass transport within the pores:

$$j_i(x) = -D_{i,p} \frac{dc_i}{dx} - \frac{z_i c_i D_{i,p}}{RT} F \frac{d\psi}{dx} + K_{i,c} c_i V \quad (1b)$$

**Equilibrium Partitioning.** Equilibrium partitioning is a phenomenon of great importance governing the separation of ionic solutes. It is described by assuming that thermodynamic equilibrium is reached, corresponding to the equality of the i-ion electrochemical potentials on both sides of the interface.³⁰ This partitioning also takes account of the change of activity coefficients γᵢ between the solutions on both sides of the membrane–solution interfaces, as well as the steric and dielectric exclusions at the pore entrance. The ratio of concentrations on both sides of the interface is finally expressed as follows:

$$\frac{c_i}{C_i} = \frac{\gamma_{i,s}}{\gamma_{i,m}} \phi_i \exp \left( \frac{-\Delta W_{i,born}}{k_B T} \right) \exp \left( \frac{-z_i F}{RT} \Delta \psi_D \right) = \phi_i' \exp \left( \frac{-z_i F}{RT} \Delta \psi_D \right) \quad (4)$$

where cᵢ and Cᵢ are the concentrations respectively within the pores and the solution. The term Δψₔ is the electrical potential between the both sides of the interface (also called the
Donnan potential), and the steric partitioning coefficient $\phi_i$ is defined by$^{31}$

$$ \phi_i = \left(1 - \frac{r_i}{r_p}\right)^2 = (1 - \lambda_i)^2 \quad (5) $$

The dielectric effects are described by the Born solvation energy barrier $\Delta W_{i,born}$ describing the influence of the difference between the dielectric constants of the free solution $\epsilon_s$ and the solution within the pores $\epsilon_p$. $^{32}$

$$ \Delta W_{i,born} = \frac{z_i^2 e^2}{8\pi\epsilon_0 r_i^3 \epsilon_p} \left(\frac{1}{\epsilon_p} - \frac{1}{\epsilon_s}\right) \quad (6) $$

It should be noted that some authors$^{22,23,33}$ introduce another dielectric effect caused by interactions between ions and the so-called “image charges” generated at interfaces having different dielectric properties.

Equilibrium partitioning is used in an operating form obtained by eliminating the Donnan potential $\Delta\psi_D$ between eq 4 written for two ions i and j, which allows one to link the concentrations of the two ions by

$$ \left(\frac{c_i}{\phi_i C_i}\right)^{(1/z_i)} = \left(\frac{c_j}{\phi_j C_j}\right)^{(1/z_j)} \quad (7) $$

**Concentration Polarization Layer.** The third phenomenon governing the separation selectivity occurs when permeation through the membrane causes solute accumulation close to the wall. This polarization thus introduces a wall concentration $C_{i,w}$ different from the bulk feed concentration $C_{i,r}$. It was shown in Figure 1 that solute accumulation close to the membrane wall is quite significant and cannot be neglected.

This polarization phenomenon leads to the definitions of two rejections, the observed ($R_{i,obs}$) and real ($R_{i,m}$) rejections, respectively defined by the following:

$$ R_{i,obs} = 1 - \frac{C_{i,p}}{C_{i,r}} \quad (8) $$

$$ R_{i,m} = 1 - \frac{C_{i,p}}{C_{i,w}} \quad (9) $$

The former corresponds to an experimental value which depends on solute accumulation at the membrane wall, and its prediction requires the modeling of the concentration polarization layer. The latter defines the intrinsic rejection of the membrane (for a given permeation flux) and is totally independent of the hydrodynamic conditions. Unfortunately, this intrinsic value cannot be determined experimentally and has to be estimated by an indirect method or modeled as presented in this paragraph.

**(A) Velocity Variation Method (VVM).** Usually, accumulation of solutes close to the membrane wall is neglected or assessed by an indirect method such as the velocity variation method (VVM).$^{34,35}$ This approach consists in extrapolating the experimental values of observed rejection $R_{i,obs}$ at various velocities $U_t$ to an infinite velocity. The obtained values of rejection are the real rejections $R_{i,m}$ corresponding to the absence of the concentration polarization layer. For given permeation fluxes, the extrapolation of observed rejections at various tangential velocities $U_t$ gives the real rejections according to the stagnant

film model:$^{36}$

$$ \ln\left(\frac{1 - R_{i,obs}}{R_{i,obs}}\right) = \ln\left(\frac{1 - R_{i,m}^{\text{VVM}}}{R_{i,m}^{\text{VVM}}}\right) + \frac{J_v}{k} \quad (10) $$

where the mass transfer coefficient $k$ is given by $k = KU_t^b$ based on an empirical correlation for the Sherwood number. In this study, the Deissler equation$^{37}$ was chosen: $Sh = aRe^bSc^c$ with $a = 0.023$, $b = 0.875$, and $c = 0.25$. $J_v$ is the volumetric permeation flux expressed as

$$ J_v = \frac{L_p}{\eta} (\Delta P - \Delta \pi) \quad (11) $$

where $L_p$ is the membrane permeability. It must be quoted that for the partial rejection of only one neutral compound, the osmotic pressure difference $\Delta\pi$ must be weighted by a reflection coefficient $\sigma$, defined from irreversible thermodynamics as the limiting value of the compound rejection when the permeation flux tends to infinity. But in the case of salts, this limiting value varies with concentration and then $\sigma$ also varies. Moreover, in the case of mixtures, there would be as many limiting values as there are molecules. For this reason, the expression giving $J_v$ is written as (11).$^{25}$ It can be noticed that the osmotic pressure term $\Delta\pi$ is low compared to pressure term $\Delta P$ for monovalent salts. For divalent salts, this term is more important but rejections are higher and $\sigma$ would be then close to unity.

It should be also noted that classical equations do not take into account the effect of suction and roughness on $k$, but some modified relations are available in the literature.$^{12,38}$

This method cannot be applied in a predictive way, and it has the drawback of using a global transfer coefficient which does not take account of the layer axial heterogeneity. For this reason, VVM is used in this study to assess the model parameters characterizing the membrane from experimental data with the 1-D membrane transport model.

**(b) Two-Dimensional Transport through the Polarization Layer.** To take account of polarization layer heterogeneity along the membrane, a 2-D model based on a steady-state differential mass balance has been developed:$^{39}$

$$ u(y) \frac{\partial C_i(y,z)}{\partial z} + v(y,z) \frac{\partial C_i(y,z)}{\partial y} = D_i(y) \frac{\partial^2 C_i(y,z)}{\partial y^2} \quad (12) $$

where $u$ and $v$ are respectively the radial ($y$) and axial ($z$) velocities and $D_i$ is the effective diffusivity defined from the pseudolaminar model as the sum of molecular and turbulent diffusivity. The polarization layer thickness (tens of micrometers) is sufficiently small compared to the membrane diameter (12.7 mm) to neglect the membrane curvature (i.e., the membrane is assumed to be flat and eq 12 is written in Cartesian coordinates).

It is usually assumed that the polarization layer is wholly contained in the laminar sublayer. In this study, the three main hydrodynamic sublayers were taken into account for velocity and diffusion profiles. Indeed, simulations showed that the polarization layer is spread out not only within the laminar sublayer$^{40}$ but also within the buffer layer. Velocity profiles are determined from classical hydrodynamic equations for turbulent flow in a tubular pipe.$^{41}$ The turbulent diffusivity $D_t$ introduced in the buffer layer diffusivity is estimated from the turbulent kinematic viscosity through Prandtl's mixing length.$^{42}$ At the membrane wall, the mass
<figure><img src="image_3.png" /><figcaption>Figure 2. Numerical procedure for PPTM solving.</figcaption></figure>

balance becomes the following:

$$J_v(C_{i,w} - C_{i,p}) = -D_{i,w} \left(\frac{\partial C_i}{\partial y}\right) \quad (13)$$

A sensibility study of the various model parameters has shown that experimental tendencies are predicted only by adjusting the diffusivity of the salt at the wall $D_{i,w}$ in boundary eq 13. This so-called “wall diffusivity” becomes an important variable of polarization evolution with conditions, and its values are assessed by fitting experimental data obtained with a specific experiment. So, the only parameter of the polarization modeling is an $\alpha$ coefficient which represents the influence of the membrane wall on the diffusivity at the interface. This increase in diffusivity is assumed to be due to hydrodynamic phenomena such as membrane roughness or suction and is thus linked to the tangential velocity $U_t$:

$$D_{i,w} = \alpha(U_t)D_{i,m} \quad (14)$$

A similar phenomenon was modeled for biofilms by Picioreanu et al.⁴³ who presented a model showing that mass transfer rates are greatly increased by the roughness of the biofilm and especially when the Reynolds number increases. It was also shown by several authors³⁸,⁴¹ that suction enhances the mass transfer coefficient, but this phenomenon seems to occur mainly for high fluxes and could probably be neglected in our case.

## Model Elaboration

The resolution of the pores and polarization transport model (PPTM) presented in Figure 2 consists in solving the 2-D polarization layer model that includes the 1-D modeling of the pore transport as a subroutine.

**1-D Pore Transport Subroutine.** The concentrations at the pore entry for the first ion $c_i(0)$ are calculated knowing all the wall concentrations $C_{i,w}$ by solving the electroneutrality equation with the equilibrium partitioning (eq 7):

$$z_1 c_1(0) + \sum_{j=2}^{n} \left[ z_j \phi'_j C_{j,w} \left( \frac{c_1(0)}{\phi'_1 C_{1,w}} \right)^{z_j/z_1} \right] + X_d = 0 \quad (15)$$

Knowing the concentration of the first ion $c_1(0)$, concentrations of the others ions $j$ at the pore inlet are calculated through eq 7.

The mass conservation along the pores $j_i(x) = j_i = VC_{i,p}$ is introduced into the extended Nernst-Planck equation (1) which becomes the following:

$$\frac{dc_i}{dx} = \frac{V}{D_{i,p}}(K_{i,c}c_i - C_{i,p}) - \frac{z_i c_i}{RT} F \frac{d\psi}{dx} \quad (16)$$

In this equation, the electric gradient along the pore $d\psi/dx$ is the same for all ions. It can be calculated by derivating the
<table><caption>**Table 1.** Constant Parameters</caption><thead><tr><th>parameters</th><th>values</th><th>references</th></tr></thead><tbody><tr><td>salt diffusivity at infinite dilution D<sub>∞</sub> (10<sup>-9</sup> m<sup>2</sup> s<sup>-1</sup>)</td><td>D<sub>NaCl</sub> = 1.607/D<sub>KCl</sub> = 1.993</td><td>Robinson & Stokes<sup>51</sup></td></tr><tr><td>solute Stokes radius r<sub>i</sub> (10<sup>-10</sup> m)</td><td>r<sub>Na<sup>+</sup></sub> = 1.84/r<sub>KCl</sub> = 1.21/r<sub>K<sup>+</sup></sub> = 1.095</td><td>Bowen & Welfoot<sup>25</sup></td></tr><tr><td>membrane radius and length</td><td>r<sub>M</sub> = 6.35 × 10<sup>-3</sup> m L<sup>-1</sup> = 1.2 m</td><td></td></tr><tr><td>temperature</td><td>T = 298 K</td><td></td></tr><tr><td>hindrance factor for convection</td><td>K<sub>i,c</sub> = (2 − φ)(1 + 0.054λ − 0.988λ<sup>2</sup> − 0.441λ<sup>3</sup>)</td><td>Bowen & Mukhtar<sup>21</sup></td></tr><tr><td>hindrance factor for diffusion</td><td>K<sub>i,d</sub> = (1 − 2.30λ + 1.154λ<sup>2</sup> + 0.224λ<sup>3</sup>)</td><td>Bowen & Mukhtar<sup>21</sup></td></tr><tr><td>dielectric constant of water at 25 °C</td><td>ε<sub>b</sub> = 79.4</td><td>Lide<sup>29</sup></td></tr></tbody></table>

electroneutrality equation and introduced in the resulting equation ($\sum_{i=1}^{n} z_i (dc_i/dx) = 0$), the summation over all ions of eq 16:

$$ \frac{d\psi}{dx} = \frac{\sum_{i=1}^{n} z_i V}{\frac{F}{RT} \sum_{i=1}^{n} z_i^2 c_i} \quad (17) $$

The concentration evolution of the various ions along the pores is calculated with eqs 16 and 17. Then, the permeate concentrations $C_{i,p}$ are determined via the equilibrium partitioning at the outlet interface (eq 7). These concentrations are used to calculate the osmotic pressure difference (eq 3) and the velocity $V$ (eq 2) of the following iteration. The 1-D model converges when $C_{i,p}$ becomes constant from one iteration to another.

**2-D Global Model (PPTM).** The 2-D model consists of solving eqs 12 and 13 with $C_{i,p}$ estimated with the 1-D model (cf. previous paragraph) and the following boundary conditions:

$$ z = 0, \forall y \ge 0 \quad C_i(0,y) = C_{i,r} \quad (18a) $$

$$ \forall z \ge 0, y \ge \delta(z) \quad C_i(z,y) = C_{i,r} \quad (18b) $$

with $y$ and $z$ respectively being the radial and axial directions and $\delta$ being the polarization layer thickness.

The hydrodynamic layers are discretized by the finite difference method. The set of discretized equations is expressed in a matrix form and solved directly by a Gaussian elimination method.

Once the concentration profile in the polarization layer has been determined all along the membrane (m steps from $z = 0$ to $L$), the average permeate concentrations $\bar{C}_{i,p}$ and permeation flux $\bar{J}_v$ are calculated by eqs 19 and 20. These quantities are interesting because they can be obtained experimentally. Convergence is obtained when $\bar{C}_{i,p}$ values do not vary from one iteration to another.

$$ \bar{C}_{i,p} = \frac{\sum_{1}^{m} J_v(z) C_{i,p}(z)}{\sum_{1}^{m} J_v(z)} \quad (19) $$

$$ \bar{J}_v = \frac{\sum_{1}^{m} J_v(z)}{m} \quad (20) $$

## Experimental Aspects

**Equipment.** The experimental setup is a semi-industrial pilot-plant equipped with a tubular AFC 40 membrane (4.8 ×

$10^{-2}$ m$^2$) from PCI Membrane Systems Ltd. The radius and the length of this membrane are given in Table 1. The concentration is kept constant by recycling the solution into the feeding tank, and the temperature of the solution is regulated manually at 25 °C by circulation in a countercurrent heat exchanger. The permeate is weighted and analyzed to determine permeation fluxes and permeate concentrations.

**Solutions.** Solutions of NaCl and KCl were made with mineral salts of high purity obtained from Acros Organics. Water flux measurements, cleaning, and salt solutions preparation were performed using demineralized water with a residual conductivity lower than 0.1 µS cm$^{-1}$.

**Analyses.** Ions concentrations in permeate and retentate were analyzed with a double electrode probe conductimeter. Concentrations of glucose solutions were determined with an enzymatic colorimetric method using a GOD-PAP kit coupled with a UV-visible spectrometer.

**Protocols.** Before each experiment, the pilot is cleaned and filled with demineralized water. Water flux is then measured for various applied pressures to check that the hydrodynamic membrane characteristics do not change between experiments. Membrane permeability $L_p$ is also calculated from water flux evolution with applied pressure. Then, the feed tank is emptied and filled with the salt solution which is homogenized by circulation through the retentate circuit with high flow velocity for 30 min. Then, the pilot is run during one night at $Q = 1$ m$^3$ h$^{-1}$ ($U_t = 2.2$ m s$^{-1}$) and $\Delta P = 10$ bar to reach ionic adsorption equilibrium into the membrane pores. After these preliminary steps, observed rejections are determined for different pressures and velocities. For each experimental assay, the permeate flux is calculated by measuring the permeation flow rate and both permeate and retentate are sampled for analyzes. Observed rejections are then calculated from eq 8 and plotted with respect to permeate flux for each velocity. From these experimental curves, real rejections are determined with the velocity variation method (see the Concentration Polarization Layer section).

## Results and Discussion

The various data required in this study are given in the Table 1.

**Determination of Model Parameters.** The model requires the determination of some parameters characterizing the membrane and the solutions. This section deals with the description of the experimental protocol of parameter determination and the presentation of the results obtained with this protocol.

<table><caption>**Table 2.** Fitted Parameters to Predict NaCl Observed Rejections</caption><thead><tr><th rowspan="2">L<sub>p</sub> × 10<sup>13</sup><br>(m<sup>3</sup> m<sup>-2</sup>)</th><th colspan="3">1-D membrane<br>transport model</th><th colspan="2">2-D polarization<br>model</th></tr><tr><th>r<sub>p</sub><br>(nm)</th><th>ε<sub>p</sub><br>(-)</th><th>K<sub>ad</sub></th><th>1/n</th><th>A</th><th>B</th></tr></thead><tbody><tr><td>1.39–1.46</td><td>0.54</td><td>59.8</td><td>6.69</td><td>0.65</td><td>0.57</td><td>1.52</td></tr></tbody></table>
<figure><img src="image_4.png" /><figcaption>Figure 3. Experimental (●) and simulated (−) evolution of NaCl rejection with permeation flux for various concentrations.</figcaption></figure>

The various model parameters required to use the PPTM for predictive purposes are given in Table 2.

**(A) Parameters of the 1-D Pores Transport.** The 1-D pores transport model requires the experimental determination of four physical parameters, i.e., the membrane permeability ($L_p$), mean pore radius ($r_p$), dielectric constant of the solution within the pores ($\epsilon_p$), and the membrane charge density ($X_d$), which is a function of the ion concentration within the pores. The membrane permeability $L_p$ is calculated before each experiment by measuring the evolution of the permeation flux of pure water with transmembrane pressure. The mean pore radius $r_p$ is assessed by fitting the experimental $R_{i,m}$ curves with the 1-D pore transport model for a solution of glucose for which only steric effects occur. The dielectric constant within the pores $\epsilon_p$ is estimated from an experiment carried out with a concentrated salt solution (0.4 M) for which membrane charges are supposed to be totally screened by the solution charges and for which the equilibrium partitioning is governed only by steric and dielectric effects.33,44 Indeed, Figure 3 shows that, above 0.4 M, the rejection does not depend anymore on the salt concentration. The previous parameters are estimated once for all from specific experiments and are supposed to be constant whatever the experimental conditions. The membrane charge density $X_d$ depends on the ion concentration of the solution with which the membrane is in contact, and it has, thus, to be determined for various salt concentrations to establish a relation linking $X_d$ to concentration. This determination can be done either experimentally45–47 (through streaming or membrane potentials) or through the model by fitting experimental data, as done here. The NaCl rejection evolution with concentration used to estimate the membrane charge densities is presented in Figure 3.

It can be concluded from this figure, that the real rejection evolution with permeation flux is well-fitted by the model. The values of $X_d$ assessed in this way for each NaCl concentration are drawn in Figure 4.

The curves obtained for the three smaller concentrations (0.01–0.1 M) are consistent with an adsorption law. Indeed, $X_d$ seems to follow a Freundlich isotherm as it has been reported by several authors:18,24,25,47–49

$$|X_d| = K_{ad}C_{i,w}^{1/n} \qquad (21)$$

The charge densities obtained for concentrations greater than 100 mM do not respect the same law which could be explained by a screening effect occurring at high concentration as was

<figure><img src="image_5.png" /><figcaption>Figure 4. Evolution of the membrane charge density with concentration.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5. Assessment of model parameters $\epsilon_p$ and $\alpha$ with a 0.4 M NaCl solution.</figcaption></figure>

reported by Bowen et al.44 The adsorption isotherm (in logarithmic form) can thus be estimated from experiments obtained for three diluted solutions as it is linear. For this reason, the model presented here is mainly suitable for concentrations lower than 0.1 M.

**(B) Parameter of the 2-D Polarization Layer Transport.** As indicated previously, the polarization modeling requires the introduction of an $\alpha$ parameter to describe the diffusivity increase at the membrane wall. Indeed, experimental results cannot be correctly reproduced by the model without increasing ion diffusivity at the membrane wall. This phenomenon can be attributed to hydrodynamic perturbations induced by roughness and suction which enhance the mass transfer. This enhancement can probably be linked to velocities in the axial ($U_i$) and radial ($J_v$) directions, but to simplify the parameter determination, the value of $\alpha$ is assumed to depend only on $U_i$. The observed and real rejections evolution with permeation flux is presented in Figure 5. The rejection evolution with permeation flux for each velocity is well-fitted by a single $\alpha$ value which seems to be consistent with the assumption that $\alpha$ is only a function of the tangential velocity (see Figure 5). The $\alpha$ value is assessed with the 2-D model from observed rejections $R_{i,obs}$ curves obtained for various tangential velocities (cf. Figure 5).

The variation of $\alpha$ with velocity $U_i$ is presented and discussed in the following paragraph. This $\alpha$ parameter seems to increase linearly with $U_i$ (cf. Figure 6) which justifies its physical interpretation and the assumption that the diffusivity increase close to the membrane wall is mainly imputable to axial velocity. The relation between $\alpha$ and the velocity are given by the
following relation:

$$ \alpha = AU_t + B \quad (22) $$

The two parameters included in the model (A and B) for the prediction of the so-called wall diffusivity are also given in Table 2.

Finally, the model requires seven parameters for a given salt that are determined in the following way:

- $L_p$, the membrane permeability, is determined through an independent experiment by measuring the water flux, without any fitting. As the flux permeation can slightly change from an experiment to another, $L_p$ is determined before each experiment in order to minimize experimental uncertainty on the permeation flux.

- $r_p$, the average pore radius, is fitted with experimental rejection values of a neutral solute (glucose).

- $\epsilon_p$, the dielectric constant within the pores, as well as $K_{ad}$ and $n$ for the evolution of $X_d$ with salt concentration are fitted with the rejection of salt solutions, respectively a highly concentrated solution for $\epsilon_p$, and three diluted ones for the $X_d$ evolution.

- A and B for the evolution of $\alpha$ with velocity are fitted with the observed rejections of salt solutions using the same experiments as for $\epsilon_p$, $K_{ad}$, and $n$.

The sets of experiments to fit $r_p$, $\epsilon_p$, $K_{ad}$, and $n$ consist in determining the evolution of the rejection of glucose ($r_p$) or salt with respect to the permeation flux (transmembrane pressure) for various tangential velocities in order to calculate the real rejections. The considered parameters are thus fitted from these real rejections with the 1-D model of solute transport through the membrane. On the opposite side, A and B values are fitted with the experimental observed rejections $R_{obs}$, obtained at various velocities, using the 2-D model including transport through the membrane and through the polarization layer.

**Model Validation.** The polarization layer model developed here takes account of the influence of turbulence on velocity and concentration profiles. Consequently, the 2-D model validation is mainly carried out on the concentration polarization prediction. Results obtained with the 2-D global model for a 0.05 M NaCl solution are presented in Figure 7. For this simulation, the membrane charge density $X_d$ is calculated for each membrane wall concentration $C_{i,w}$ estimated by the 2-D polarization model from the adsorption isotherm (eq 21). The values of $\alpha$ were estimated from the values of the coefficients A and B obtained from previous assays (see Table 2), so that there is no fitted parameter. Figure 7a depicts the comparison between experimental and simulated evolutions of the real and observed rejection ($R_{i,m}$ and $R_{i,obs}$) curves with the permeation flux at various tangential velocities. The simulated evolution of the permeate flux with applied pressure is also compared to an experimental one in Figure 7b.

A very good agreement was found between experimental results and simulations carried out with parameters previously assessed for both rejections and permeation fluxes. Indeed, for a given applied pressure and velocity, flux and rejections of NaCl solutions are well-predicted by the model.

These results seem to show that the model can be used for predictive purposes through only a few characterization experiments in the case of NaCl solutions. The final part of the PPTM validation consists in observing the influence of the salt. For this purpose, experiments with KCl were carried out with three concentrations (0.05, 0.2, and 0.4 M). The dielectric constant of the solution within the pores was reassessed from KCl concentrated solution, and the obtained value (56.8) is close to

<figure><img src="image_7.png" /><figcaption>Figure 6. $\alpha$ coefficients versus tangential velocity for various experiments.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7. Experimental (●) and simulated (−) results of rejections (a) and fluxes (b) obtained for a 0.05 M NaCl solution.</figcaption></figure>

that of NaCl solution (59.8) but slightly smaller. The membrane charge density was also determined, and the value is the same as that obtained with NaCl solution (−67 mM for NaCl and −68 for KCl) which is coherent with the fact that adsorption phenomena are mainly due to predominant adsorption of negative ions on hydrophobic sites of the membrane.⁵⁰ Finally, the linear relation between $\alpha$ and $U_t$ obtained with NaCl solutions is checked for KCl solutions, and results are presented in Figure 6.

This figure confirms that $\alpha$ and thus the so-called wall diffusivity increase quite linearly with the tangential velocity in the membrane whatever the concentration or the salt. This coefficient can thus be easily assessed by a linear law which can be obtained from an experiment with a given salt at a single concentration. So at first glance, this coefficient can be estimated
by the obtained relation $\alpha = (0.57 U_t + 1.52) \pm 0.17$ but it should be noticed that it is strongly dependent on the membrane surface and has to be assessed for a given membrane like the other model parameters.

Last, the PPTM requires only five simple sets of experiments to assess all the parameters, i.e., the filtration of a glucose solution for the average pore radius, a concentrated salt solution for the dielectric constant within the pores, and three diluted solutions for adsorption isotherm and $\alpha$ evolution with velocity. Each set of experiments consists of measuring permeation flux and permeate and retentate concentrations for various pressures and velocities. A water flux determination is performed before each set of experiments.

The model correctly describes the filtration of a single monovalent salt solution in the presence of a polarization layer, by taking explicitly into account the increase of the salt concentration at the membrane wall. Indeed, the influence of the main experimental parameters (feed concentration, applied pressure, and tangential velocity) on real and observed rejections and fluxes are well-predicted.

## Conclusion

In this study, a 2-D model predicting the transport of ionic solutes through nanofiltration membranes (PPTM) has been developed. It includes the main phenomena governing the separation of charged molecules, i.e., the transport through the polarization layer, the interface equilibrium partitioning, and the transport within the membrane pores. Transport through the polarization layer is based on a differential mass balance in the flow assuming that the polarization layer is spread out not only in the laminar sublayer. The turbulent velocity profiles are obtained from classical relations of fluid mechanics, and eddy transport is described through a turbulent diffusivity profile. The influence of the main operating parameters, i.e., transmembrane pressure, tangential velocity, and solute concentration, are correctly described by the model, provided that a diffusivity enhancement at the membrane wall is taken into consideration. The parameter $\alpha$ representing this diffusivity increase at the wall was identified for various conditions. This parameter clearly increases with the tangential velocity but not much with pressure, concentration, and salt nature. These observations are consistent with the physical meaning of this parameter which is imputed to a hydrodynamic phenomenon. Indeed, this diffusivity increase is supposed to be generated by the membrane surface such as roughness or porosity, leading to some back-transport.

Finally, this predictive model has seven parameters (the membrane permeability $L_p$ can be considered either as experimental data or as a parameter, but its values are assessed experimentally without any fitting procedure). Five of them concern the transfer through the membrane: the membrane permeability ($L_p$), the mean pore radius ($r_p$), the dielectric constant of the solution within the pores ($\epsilon_p$), and $K_{ad}$ and $1/n$ for the membrane charge density ($X_d$) evolution with concentration. The modeling of the polarization layer requires only the assessment of the $\alpha$ parameter evolution with velocity through A and B coefficients. The first four parameters can be estimated easily by the 1-D membrane transport model with respectively water, glucose, and concentrated salt solution experiments (by fitting real rejections). For diluted solutions ($C_{i,r} < 0.1$ M), it was shown that $X_d$ can be linked to the salt wall concentration $C_{i,w}$ through a Freundlich adsorption isotherm. The $\alpha$ parameter (A and B) can be assessed by fitting experimental observed rejection $R_{i,obs}$ curves with the 2-D model from one previous set of experiments.

Once all the model parameters have been identified, simulations for NaCl solutions correctly predict the velocity and pressure influences on permeation flux $J_v$ and observed rejection $R_{i,obs}$ in the presence of a polarization layer.

This model is perfectly suitable for monovalent salts, and this procedure can be easily generalized to multivalent salts. It can be expected that multivalent salts behave in the same way and the main challenge will be to extend this predictive 2-D model to mixed salt solutions and perhaps to complex solutions such as seawater.

## Nomenclature

$A, B$ = parameters of the $\alpha$ law

$a_i$ = activity of ion $i$ (mol m$^{-3}$)

$A_k$ = membrane porosity (dimensionless)

$c_i$ = concentration of ion $i$ within the pore (mol m$^{-3}$)

$C_i$ = concentration of ion $i$ within the polarization layer (mol m$^{-3}$)

$C_{i,p}$ = permeate concentration of ion $i$ (mol m$^{-3}$)

$C_{i,r}$ = bulk concentration of ion $i$ (mol m$^{-3}$)

$C_{i,w}$ = wall concentration of ion $i$ (mol m$^{-3}$)

$D_{i,so}$ = molecular diffusion coefficient of ion $i$ at infinite dilution (m$^2$ s$^{-1}$)

$D_{i,p}$ = pore diffusion coefficient of ion $i$ (m$^2$ s$^{-1}$)

$D_{i,w}$ = diffusion coefficient of ion $i$ at the wall (m$^2$ s$^{-1}$)

$e$ = electronic charge (1.602177 × 10$^{-19}$ C)

$F$ = Faraday constant (96487 C mol$^{-1}$)

$I$ = ionic strength (mol m$^{-3}$)

$j_i$ = ionic flux of ion $i$ (mol m$^{-2}$ s$^{-1}$)

$J_v$ = volumetric permeation flux (m$^3$ m$^{-2}$ s$^{-1}$)

$J_w$ = water permeation flux (m$^3$ m$^{-2}$ s$^{-1}$)

$k$ = mass transfer coefficient (m s$^{-1}$)

$k_B$ = Boltzmann constant (1.38066 × 10$^{-23}$ J K$^{-1}$)

$K_{i,c}$ = ionic hindrance factor for convection (dimensionless)

$K_{i,d}$ = ionic hindrance factor for diffusion (dimensionless)

$K_{ad}$, $1/n$ = constants in the adsorption law

$L$ = membrane length

$L_p$ = water permeability (m$^3$ m$^{-2}$)

$P$ = pressure (Pa)

$R$ = universal gas constant (8.314 J mol$^{-1}$ K$^{-1}$)

$r_i$ = Stokes radius of ion $i$ (m)

$r_M$ = membrane radius (m)

$R_{i,m}$ = real rejection of ion $i$ (dimensionless)

$R_{i,obs}$ = observed rejection of ion $i$ (dimensionless)

$r_p$ = average pore radius (m)

$T$ = temperature (K)

$u$ = axial velocity within the flow (m s$^{-1}$)

$U_t$ = average tangential velocity (m s$^{-1}$)

$v$ = radial velocity within the flow (m s$^{-1}$)

$V$ = solvent velocity (m s$^{-1}$)

$\overline{v}_i$ = partial molar volume of ion $i$ (m$^3$ mol$^{-1}$)

$x$ = axial position within the pore (m)

$X_d$ = membrane effective charge density (mol m$^{-3}$)

$y$ = radial wall distance (m)

$y_+$ = normalized wall distance (dimensionless)

$z$ = axial position along the membrane (m)

$z_i$ = valence of ion $i$ (dimensionless)

## Greek Letters

$\alpha$ = factor manifesting diffusion increase at the wall (dimensionless)

$\Delta P$ = applied pressure (Pa)

$\Delta x$ = membrane thickness (m)

$\Delta \psi_D$ = Donnan potential (V)
$\Delta\pi$ = osmotic pressure difference (Pa)

$\phi_i$ = steric partition coefficient (dimensionless)

$\phi'_i$ = partition coefficient including activity coefficient, steric, and dielectric effects

$\psi$ = electrical potential within the pore (V)

$\delta$ = polarization layer thickness (m)

$\epsilon_0$ = permittivity of free space (8.85419 × 10⁻¹⁹ C)

$\epsilon_b$ = bulk dielectric constant (dimensionless)

$\epsilon_p$ = pore dielectric constant (dimensionless)

$\gamma_{i,m}$ = activity coefficient of ion *i* in the pore side of the interface (dimensionless)

$\gamma_{i,s}$ = activity coefficient of ion *i* in the solution side of the interface (dimensionless)

$\eta$ = dynamic viscosity within pores (Pa s)

$\eta_0$ = bulk viscosity (Pa s)

$\nu$ = kinematic viscosity (m² s⁻¹)

$\lambda_i$ = ratio of ion stokes radius to pore radius (dimensionless)

## Literature Cited

(1) Al-Sofi, A.-K. M. Seawater desalination—SWCC experience and vision. *Desalination* **2001**, *135*, 121.

(2) Paugam, L.; Diawara, C. K.; Schlumpf, J. P.; Jaouen, P.; Quemeneur, F. Transfer of monovalent anions and nitrates especially through nanofiltration membranes in brackish water conditions. *Sep. Purif. Technol.* **2004**, *40*, 237.

(3) Pontie, M.; Lhassani, A.; Diawara, C. K.; Elana, A.; Innocent, C.; Aureau, D.; Rumeau, M.; Croue, J. P.; Buisson, H.; Hemery, P. Seawater nanofiltration for the elaboration of usable salty waters. *Desalination* **2004**, *167*, 347.

(4) Yu, S.; Gao, C.; Su, H.; Liu, M. Nanofiltration used for desalination and concentration in dye production. *Desalination* **2001**, *140*, 97.

(5) De Witte, J.-P. Surface water potabilisation by means of a novel nanofiltration element. *Desalination Annu. Meet. Eur. Desalination Soc. Desalination Environ.* **1997**, *108*, 153.

(6) Oumar Anne, C.; Trebouet, D.; Jaouen, P.; Quemeneur, F. Nanofiltration of seawater: fractionation of mono- and multi-valent cations. *Desalination* **2001**, *140*, 67.

(7) Van der Bruggen, B.; Vandecasteele, C. Removal of pollutants from surface water and groundwater by nanofiltration: overview of possible applications in the drinking water industry. *Environ. Pollut.* **2003**, *122*, 435.

(8) Combe, C.; Guizard, C.; Aimar, P.; Sanchez, V. Experimental determination of four characteristics used to predict the retention of a ceramic nanofiltration membrane. *J. Membr. Sci.* **1997**, *129*, 147.

(9) Palmeri, J.; Sandeaux, J.; Sandeaux, R.; Lefebvre, X.; David, P.; Guizard, C.; Amblard, P.; Diaz, J.-F.; Lamaze, B. Modeling of multi-electrolyte transport in charged ceramic and organic nanofilters using the computer simulation program NanoFlux. *Desalination* **2002**, *147*, 231.

(10) Bowen, W. R.; Mohammad, A. W.; Hilal, N. Characterisation of nanofiltration membranes for predictive purposes—use of salts, uncharged solutes and atomic force microscopy. *J. Membr. Sci.* **1997**, *126*, 91.

(11) Szymczyk, A.; Labbez, C.; Fievet, P.; Vidonne, A.; Foissy, A.; Pagetti, J. Contribution of convection, diffusion and migration to electrolyte transport through nanofiltration membranes. *Adv. Colloid Interface Sci.* **2003**, *103*, 77.

(12) Gekas, V.; Olund, K. Mass transfer in the membrane concentration polarization layer under turbulent cross flow: II. Application to the characterization of ultrafiltration membranes. *J. Membr. Sci.* **1988**, *37*, 145.

(13) Afonso, M. D.; De Pinho, M. N. Mass transfer modeling for salt transport in amphoteric nanofiltration membranes. *Ind. Eng. Chem. Res.* **1998**, *37*, 4118.

(14) Minnikanti, V. S.; DasGupta, S.; De, S. Prediction of mass transfer coefficient with suction for turbulent flow in cross flow ultrafiltration. *J. Membr. Sci.* **1999**, *157*, 227.

(15) Bhattacharjee, S.; Kim, A. S.; Elimelech, M. Concentration Polarization of Interacting Solute Particles in Cross-Flow Membrane Filtration. *J. Colloid Interface Sci.* **1999**, *212*, 81.

(16) Dresner, L. Some remarks on the integration of extended Nernst-Planck equation in the hyperfiltration of multicomponent solutions. *Desalination* **1972**, *10*, 26.

(17) Gross, R. J.; Osterle, J. F. Membrane transport characteristics of ultrafine capillaries. *J. Chem. Phys.* **1968**, *49*, 228.

(18) Straatsma, J.; Bargeman, G.; van der Horst, H. C.; Wesselingh, J. A. Can nanofiltration be fully predicted by a model? *J. Membr. Sci.* **2002**, *198*, 273.

(19) Kedem, O.; Katchalsky, A. A physical interpretation of phenomenological coefficients of membrane permeability. *J. Gen. Physiol.* **1961**, *45*, 143.

(20) Spiegler, K. S.; Kedem, O. Thermodynamics of hyperfiltration (Reverse Osmosis): Criteria for coefficient membranes. *Desalination* **1966**, *1*, 311.

(21) Bowen, W. R; Mukhtar, H. Characterisation and prediction of separation performance of nanofiltration membranes. *J. Membr. Sci.* **1996**, *112*, 263.

(22) Yaroshchuk, A. E. Non-steric mechanisms of nanofiltration: superposition of Donnan and dielectric exclusion. *Sep. Purif. Technol.* **2001**, *22*–23, 143.

(23) Szymczyk, A.; Fievet, P. Investigating transport properties of nanofiltration membranes by means of a steric, electric and dielectric exclusion model. *J. Membr. Sci.* **2005**, *252*, 77.

(24) Bandini, S.; Vezzani, D. Nanofiltration modeling: the role of dielectric exclusion in membrane characterization. *Chem. Eng. Sci.* **2003**, *58*, 3303.

(25) Bowen, W. R.; Welfoot, J. S. Modelling the performance of membrane nanofiltration-critical assessment and model development. *Chem. Eng. Sci.* **2002**, *57*, 1121.

(26) Schlögl, R. Membrane permeation in system far from equilibrium. *Ber. Bunsen-Ges. Phys. Chem.* **1966**, *70*, 400.

(27) Deen, W. M. Hindered transport of large molecules in liquid -filled pores. *AIChE. J.* **1987**, *33*, 1409.

(28) Anderson, J. L.; Quinn, J. A. Restricted transport in small pores: a model for steric exclusion and hindered particle motion. *Biophys. J.* **1974**, *14*, 130.

(29) Lide, L. R. *Handbook of chemistry and physics*, 79th ed.; CRC Press: Boca Raton, FL, 1999.

(30) Donnan, F. G. Theory of membrane equilibria and membrane potentials in the presence of non-dialysing electrolytes. A contribution to physical-chemical physiology. *J. Membr. Sci.* **1995**, *100*, 45.

(31) Ferry, J. D. Statistical evaluation of sieve constants in ultrafiltration. *J. Gen. Physiol.* **1935**, *20*, 95.

(32) Born, M. Volumen and hydratationswärme der ionen. *Z. Phys. Chem.* **1920**, *1*, 45.

(33) Yaroshchuk, A. E. Dielectric exclusion of ions from membranes. *Adv. Colloid Interface Sci.* **2000**, *85*, 193.

(34) Jonsson, G.; Boesen, C. E. Concentration polarization in a reverse osmosis test cell. *Desalination* **1977**, *21*, 1.

(35) Nakao, S.; Kimura, S. Analysis of solutes rejection in ultrafiltration. *J. Chem. Eng. Jpn.* **1981**, *14*, 32.

(36) Van den Berg, G. B.; Racz, I. G.; Smolders, C. A. Mass transfer coefficients in cross-flow ultrafiltration. *J. Membr. Sci.* **1989**, *47*, 25.

(37) Wijmans, J. G.; Nakao, S.; Van Den Berg, J. W. A.; Troelstra, F. R.; Smolders, C. A. Hydrodynamic resistance of concentration polarization boundary layers in ultrafiltration. *J. Membr. Sci.* **1985**, *22*, 117.

(38) Gekas, V.; Hallstrom, B. Mass transfer in the membrane concentration polarization layer under turbulent cross flow: I. Critical literature review and adaptation of existing sherwood correlations to membrane operations. *J. Membr. Sci.* **1987**, *30*, 153.

(39) Bhattacharjee, S.; Chen, J. C.; Elimelech, M. Coupled model of concentration polarization and pore transport in cross flow nanofiltration. *AIChE. J.* **2001**, *47*, 2733.

(40) Déon, S.; Dutournié, P.; Bourseau, P. Modeling nanofiltration with Nernst-Planck approach and polarization layer. *AIChE. J.*, accepted for publication.

(41) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. *Transport phenomena*; John Wiley & Sons Inc: New York, 2002.

(42) Padet, J. *Fluides en écoulements*; Masson, Paris, 1990.

(43) Picioreanu, C.; Van Loosdrecht, M. C. M.; Heijnen, J. J. A theoretical study on the effect of surface roughness on mass transport and transformation in biofilms. *Biotechnol. Bioeng.* **2000**, *68*, 355.

(44) Bowen, W. R.; Welfoot, J. S.; Williams, P. M. Linearized Transport Model for Nanofiltration: Development and Assessment. *AIChE. J.* **2002**, *48*, 760.

(45) Szymczyk, A.; Fievet, P.; Reggiani, J. C.; Pagetti, J. Characterisation of surface properties of ceramic membranes by streaming and membrane potentials. *J. Membr. Sci.* **1998**, *146*, 277.

(46) Fievet, P.; Sbai, M.; Szymczyk, A.; Magnenet, C.; Labbez, C.; Vidonne, A. A new tangential streaming potential setup for the electrokinetic characterisation of tubular membranes. *Sep. Sci. Technol.* **2004**, *39*, 2931.
(47) Hagmeyer, G.; Gimbel, R. Modelling the rejection of nanofiltration membranes using zeta potential measurements. *Sep. Purif. Technol.* **1999**, *15*, 19.

(48) Afonso, M. D. Surface charge on loose nanofiltration membranes. *Desalination* Int. Congr. Membr. Membr. Process. **2006**, *191*, 262.

(49) Peeters, J. M. M.; Mulder, M. H. V.; Strathmann, H. Streaming potential measurements as a characterization method for nanofiltration membranes. *Colloids Surf. A Physicochem. Eng. Aspects* **1999**, *150*, 247.

(50) Bandini, S. Modelling the mechanism of charge formation in NF membranes: Theory and application. *J. Membr. Sci.* **2005**, *264*, 75.

(51) Robinson, R. A.; Stokes, R. H. *Electrolytes solutions*; Butterworths: Markham, Canada, 1965.

<figure><img src="image_9.png" /></figure>