Modelling the separation by nanofiltration of a multi-ionic solution relevant to an industrial process

Journal of Membrane Science 322 (2008) 320–330

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Modelling the separation by nanofiltration of a multi-ionic solution relevant to an industrial process A.I. Cavaco Morão a , A. Szymczyk b , P. Fievet c , A.M. Brites Alves a,∗ a

Instituto Superior Técnico-Department of Chemical and Biological Engineering, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal Université Rennes 1/ENSCR, Chimie et Ingénierie des Procédés, UMR 6226 CNRS, 263 avenue du général Leclerc, 35042 Rennes Cedex, France c Institut UTINAM, UMR CNRS 6213, Université de Franche-Comté, 16 route de Gray, Besanc¸on Cedex 25030, France b

a r t i c l e

i n f o

Article history: Received 27 March 2008 Received in revised form 30 May 2008 Accepted 5 June 2008 Available online 11 June 2008 Keywords: Nanofiltration Multi-ionic solutions Transport model Membrane charge Concentration polarization

a b s t r a c t The current work focuses on the application of nanofiltration (NF) to the isolation of a pharmaceutical product, clavulanate (CA− ), from clarified fermentation broths, which show a complex composition with five main identified ions (K+ , Cl− , NH4 + , SO4 2− and CA− ). Our aim is to predict the rejection rates of these five ions, with the NF membrane Desal-DK, which may influence the separation of CA− and play a role in the whole downstream process. Concentration polarization of this multi-ionic solution was addressed both through the application of the extended Nernst–Planck equations in the film layer, assuming limiting values for the boundary layer thickness, and by applying a new methodology which dispenses this arbitrary assumption. The latter approach was chosen to calculate the intrinsic rejection rates. The transport through the membrane pores was modelled using the Steric, Electric, Dielectric and Exclusion (SEDE) model in order to predict the intrinsic rejections of the five ions in solution. The membrane volume charge density was assessed from tangential streaming potential measurements for the system membrane/five ions solution. With a single fitted parameter, εp (the dielectric constant of the solution inside pores) and considering that the ion selectivity is governed by steric, Donnan and Born dielectric exclusion mechanisms, the predicted intrinsic rejections rates of the five ions are in good agreement with the experimental values. © 2008 Elsevier B.V. All rights reserved.

1. Introduction In the recent years, nanofiltration (NF) is applied to more and more industrial processes [1]. Aiming the design of new NF processes and the optimisation of existing membrane applications, the development of good predictive tools is desired. A practical and useful model should predict quantitative process data from accessible physical properties data and be physically realistic. The Steric, Electric and Dielectric Exclusion (SEDE) model developed by Szymczyk and Fievet [2] for NF is based on the extended Nernst–Planck equation to describe the mass transfer across the membrane and accounts for the ions distribution at the pore inlet and outlet through equilibrium partitioning relations. This model has been shown to provide a relatively good description of the experimental data for both symmetric and asymmetric single salts [2–5]. The present challenge is to extend this model to the case of complex multi-component separations and to validate

∗ Corresponding author. Tel.: +351 218417216. E-mail address: [email protected] (A.M. Brites Alves). 0376-7388/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2008.06.003

this predictive method with industrially relevant NF applications. In order to predict the NF performance of either single salt or multi-ionic solutions, it is essential to consider the mass transfer both across the membrane and within the film layer at the vicinity of the feed-solution/membrane interface. Without an accurate determination of the concentration polarization, the intrinsic rejection rate of the ions is imprecise and the consistency of a transport model can hardly be checked. Nevertheless, the issue of concentration polarization has been largely overlooked in the few published works dealing with the prediction of rejection rates of multi-ionic solutions [6–9]. The few works addressing this phenomenon account for the coupling of ionic fluxes through the application of the extended Nernst–Planck equations in the film layer at the vicinity of the membrane surface. The estimation of the thickness of this layer is not straightforward and its choice is largely arbitrary [10,11]. Recently, Geraldes and Afonso [12] proposed a new simplified methodology which overcomes the need of assuming a film layer thickness. This approach will be applied in this work to estimate concentration polarization of the five ions solution.

A.I. Cavaco Morão et al. / Journal of Membrane Science 322 (2008) 320–330

The current work focuses on the application of NF to the isolation of a pharmaceutical product, clavulanic acid, produced industrially by fermentation of Streptomyces clavuligerus. The first step of the downstream process of this product is a solid–liquid separation (ultrafiltration/diafiltration) followed by NF to concentrate the clarified broth. Besides clavulanate, the clarified fermentation broths contain a complex mixture of ions (e.g. K+ , Cl− , H2 PO4 − , NH4 + , SO4 2− ) and uncharged solutes which influence the NF separation performance [13]. The main innovative aspect of the present work is the prediction of the ions rejection rates contained in a complex solution of five ions which mimics the industrial NF feed in the isolation process of clavulanate. With this purpose, concentration polarization was carefully analyzed using both the new aforementioned approach and the Nernst–Planck equations applied to the film layer. The application of the SEDE model was extended to a multi-component solution. An extensive work to characterize the membrane was carried out and three out of the four parameters of the SEDE model were evaluated from independent experimental techniques, namely the pore radius, rp , the membrane thickness to porosity x /Ak and the membrane volume charge density, X. Some models applied to multi-ionic solutions, described in the literature, calculate the membrane charge of a ternary ion mixture as a fitted parameter [6] or from the experimental results of membrane charge for the single salt solutions and applying a mixing rule according to the mole fractions of the ions [14]. Other authors predict the rejection of ternary and quaternary solutions estimating the volume charge of the membrane contacting these solutions using the effective charge density fitted by the Donnan-Steric partitioning pore model for NaCl at several concentrations. The charge of the membrane contacting the multi-ionic solution is then calculated using an interpolation based on the salt concentration [9] or a weighting scheme based on the ion strength of each salt in the mixture [7]. In contrast, in the current work, X was inferred from tangential streaming potential (TSP) measurements with the membrane contacting the multi-ionic solution. Furthermore, aiming the development of a correlation between the charge of the membrane contacting the five ions solution and each ion, TSP measurements were also carried out for single salt solutions.

2. Theory The modelling of mass transport in NF (v.d. Fig. 1) accounts for: (1) concentration polarization at the feed side; (2) equilibrium partitioning of species at the feed/membrane interface; (3) solute transport through the pores by a combination of convection, diffusion and electromigration (for charged solutes only); (4) equilibrium partitioning of species at the membrane/permeate interface. The SEDE model comprises steps 2–4. The SEDE model is a four-parameter model, provided that the dielectric constant of the active layer material, εm is known. The model parameters are the pore radius rp , the membrane thickness to porosity ratio x/Ak , the membrane volume charge density X and the dielectric constant of the solution-filled pores (εp ). The latter is the single fitting parameter whereas the remaining parameters are estimated experimentally. The inputs of the SEDE model are the volumetric permeate fluxes, Jv and the concentration at the membrane interface, Ci,m . The outputs of the SEDE model are the intrinsic rejection rates, Ri = 1 − Ci,p /Ci,m . Jv as well as Ci,f are determined experimentally. The value of Ci,m cannot be obtained directly from experiments because methodologies to measure the concentration profile in the thin boundary layer are not available for most of the systems or are not accurate

321

Fig. 1. Simplified representation of concentration gradients (single salt 1:1) adjacent to the membrane due to concentration polarization, partition at the pores entrance, concentration gradients inside the membrane pores and partition at the pores outlet. Subscripts: (−) anion and (+) cation.

enough and therefore indirect methods are required [15]. Thus, Ci,m is calculated through boundary layer modelling. 2.1. Concentration polarization of multi-ionic solutions Concentration polarization of uncharged solutes and binary ionic solutions can be estimated by the film theory using the appropriate mass transfer coefficient and effective diffusivities of the solutes. Some authors working with mixtures of ions assume that the phenomenon of concentration polarization is negligible [6,7] while others assume that the ionic fluxes are not coupled and use the film theory to calculate the concentration polarization for each ion [8,13]. For multi-ionic solutions, as the several ions move at distinct rates within the film layer a definition of the effective diffusivity of the ions mixture lacks. Thus, in order to estimate the concentration polarization of multi-ionic mixtures, the application of the extended Nernst–Planck equation (without steric hindrance factors) in the film layer requires that a single layer thickness is imposed [10–12]. Recently, an approximate methodology to predict concentration polarization, discarding the arbitrary assumption of the film thickness in the vicinity of the membrane surface, was suggested by Geraldes and Afonso [12]. The main assumption of this approach is the use of the mass transfer coefficient of each ion, as if they were independent of each other, i.e., the mass transfer coefficient of a solute does not depend on the diffusivities of the remaining solutes. Besides, the model was derived for dilute solutions whereas the friction interactions between the solutes are negligible and only the interaction between each solute and the solvent are considered. Assuming unhindered transport, the total flux of ion i at the feed solution–membrane interface is given by the sum of back diffusion flux, convection flux and the migration flux due to the electrical potential gradient, , as follows [12]: Jv (1 − Ri )Ci,m = Ji,m + Jv C i,m − zi Ci,m Di

F  RT

(1)

For dilute solutions the back diffusion flux of the ion i, Ji,m may be given by [16]: Ji,m = −ki• (Ci,m − Ci,f )

(2)

where ki• is the mass transfer coefficient of the ion i corrected for high mass transfer rates. ki• is related with the convectional mass

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transfer coefficient for vanishing mass transfer rates (ki ) by ki• = ki [16], where  is the mass transfer correction factor which depends only on the ratio i = Jv /ki . It was demonstrated [17,18] that for i ≤ 1 the correction factor derived from the film theory may be used: i = i +

i exp(i ) − 1

(3)

Substituting Eq. (2) in Eq. (1) yields: Jv Ci,m Ri = ki i (Ci,m − Ci,f ) + zi Ci.m Di

F  RT

(4)

The electrical potential gradient at the feed solution/membrane interface, , enforces the electroneutrality requirement: n 

zi Ci,m = 0

(5)

i=1

Therefore, provided that ki is known and the volumetric permeation flux Jv , the feed concentration Ci,f and the permeate concentration Ci,p are determined experimentally, Ci,m can be calculated. Considering a solution of n ions, the estimation of Ci,m relies on the resolution of a set of n + 1 algebraic equations: Eq. (4) applied to each ion (n equations) together with Eq. (5). In a previous work [18], a mass transfer coefficient correlation was obtained for plate-and-frame membrane module used in this work (64 < Re < 570, 450 < Sci < 8900): Shi =

ki h = 0.143Re0.46 Sci0.37 Di

(6)

Variations of the solutions properties, such as diffusivity, density and viscosity within the boundary layer were neglected, which is valid for dilute solutions. 2.2. SEDE model—feed/membrane interface, transport through the membrane and fed/permeate interface The SEDE model has been extensively described in the literature [2–5]. This homogeneous 1D model neglects the radial variations of electrical potential and ion concentration inside pores due to the strong overlap of the electrical double layers inside narrow pores of NF membranes. The separation of solutes results from the transport across the membrane, described by the extended Nernst–Planck (ENP) equations and interfacial phenomena including steric hindrance, Donnan exclusion and dielectric exclusion. The latter has two different contributions (1) Born dielectric effect, related to the lowering of the dielectric constant of the solution inside nanodimensional pores, εp < εb and (2) the interactions between the ions in solution with the polarization charges induced by ions in solution at the pore walls interface because of the different dielectric constants of the two media, i.e. pore walls and the pore filling solution (εm < εp ). The interactions of the ions with the polarized surface are frequently described in terms of the so-called “image forces”.

Within the scope of the SEDE model, the excess solvation energy resulting from the difference in the dielectric constant between the bulk solution and inside the membrane pores is accounted for by a modified Born equation that considers the radius of the cavity formed by the ion i in the solvent, ri,cav [19] instead of the ionic radius, as stated in the original Born model. The interaction between the ions and the induced polarization charges (by the ions) at the pore walls was extensively studied by Yaroshchuk [20]. This author derived approximate expressions for the interaction energy due to image forces for cylindrical and slitlike pores which have been applied to the SEDE model in previous works [2–5]. The magnitude of image forces is strongly dependent on the pore geometry (slit-like or cylindrical) which makes difficult the description of experimental data because the membrane pore geometry is not well defined [2]. The dielectric exclusion in cylindrical pores is stronger than in slit-like pores [20]. The best rejection rates predictions using the SEDE model were obtained assuming cylindrical pores for 1 kDa cut-off ceramic membranes manufactured by Tami-Industries (rp = 1.7 nm) [4]. On the other hand, in other works, with narrower membrane pores, such as AFC 40 (PCI) [3,5], AFC 30 (PCI) [2,5] and Desal DK (Osmonics GE) [21] membranes (rp = 0.54, 0.50 and 0.54 nm, respectively), the best description of experimental data was obtained assuming slit-like pores. Apparently, for membranes with tight pores the assumption of a slit geometry works out better. The solutes in liquid filled pores of molecular dimensions have reduced diffusivities, a phenomenon explained by a combination of particle–wall hydrodynamic interactions and steric restrictions. Simplified theoretical expressions [22] which account for these effects have been available for many years to calculate the hindrance factors for convection and diffusion of uncharged spheres in pores. These expressions have been applied in the former versions of SEDE model and depend on the solute to pore size ratio defined as i = ri,s /rp , where ri,s is the Stokes radius of the solute i. Nevertheless, all these correlations were derived assuming symmetrically positioned spheres (i.e., in the pore central axis), known as the centreline approximation. More recently the theory of hindered transport in pores was reviewed by Dechadilok and Deen [23] and new equations were derived (v.d. Table 1) for diffusive and convective hindrance factors of spheres which are properly averaged over the pore cross section. These new equations were applied in this work to calculate hindrance factors for convection and diffusion. Two distinct approaches have been applied to estimate the membrane volume charge density, X: (1) fitting the rejection rate of salts at various concentrations and correlating the fitted volume charge densities versus concentration by a Freundlich isotherm and (2) from tangential streaming potential measurements. The first approach can either rely just on the volume charge density fitted for NaCl [10,21,24] or on the fitted values for different salts [6,25,26]. The latter strategy defeats the purpose of having a predictive model. Both procedures often lead to unrealistic vol-

Table 1 Hindrance factors for diffusion and convection in cylindrical and slit-like pores, 0 ≤ i < 1 Cylindrical pores

Slit pores

˚i = (1 − i )2

˚i = (1 − i )

Diffusion Ki,d =

Diffusion

H(i ) ˚i

1.915213 i

with H(i ) = 1 + − 2.819034 i

ln i − 1.56034i + 0.5281552 +

+ 0.2707885 i

Convection  Ki,c =

9  8 i

1+3.867i −1.9072 −0.834 i

1+1.867i −0.7412 i

 3 i

i

+ 1.101156 i

− 0.4359337 i

H(i ) with H(i ) = ˚i 9  ln  i i − 1.19358i 16

Ki,d = 1+

+ 0.42853 − 0.31924 + 0.084285 i

i

i

Convection W (i ) with W (i ) = ˚i 1 − 3.022 + 5.7763 − 12.36754 i i i

Ki,c =

+ 18.97755 − 15.21856 + 4.85257 i

i

i

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ume charges masking the role of distinct mechanisms not fully accounted in such models, such as dielectric exclusion [2,5,21]. The experimental assessment of the surface charge of NF membranes active layer is not an easy task. The measurements of TSP have been pointed out as the most appropriate experimental technique for this purpose [3,7]. When this technique is applied to porous materials like NF membranes, a non-negligible share of the conduction current is likely to flow through the membrane body filled with the electrolyte solution and must be accounted for in the calculation of zeta potential,  [27,28]. Neither the classical Helmholtz–Smoluchowski equation nor the related relations to account for the surface conductivity can be applied to calculate the zeta potential based on the TSP obtained for such a system. This phenomenon can be taken into account for instance by carrying out a series of measurements at various channel heights as demonstrated theoretically by Yaroshchuk and Ribitsch [27] and first applied by Fievet et al. [28]. For rectangular slit channels the reciprocal streaming potential coefficient (P/V) is given by

 P  V

I=0

=

0 2 hm m + ε0 εb  εo εb 

1

(7)

h

Fig. 2. Equilibrium dissociation of clavulanic acid (HCA) into clavulanate (CA− ).

It will be assumed that the gradient of activity coefficients inside the membrane can be neglected, which implies that either the concentrations within pore are small or their variations are very small [1]. There are evidences of water structural changes in confined systems influencing its physical properties, such as viscosity [30]. However, as quantitative studies dealing with the increase of water viscosity in pores lack, in this study the bulk water viscosity is used.

3. Experimental

where I is the electric current, 0 is the solution conductivity, hm is the effective thickness in which the conduction current flows inside the porous body of the membrane, m is the electrical conductivity inside the porous body of the membrane and h is the height of the channel formed by the two membranes. The previous equation neglects the contribution of the surface conductivity in the channel which is likely to be a reasonable approximation since the ratio of the channel height to the Debye height is usually taken greater than 200 [28]. The electrokinetic surface charge density, ek , can be estimated from zeta potential results through the Gouy–Chapman equation:

3.1. Membranes and chemicals



3.2. Nanofiltration experiments



ek = −sign()

2ε0 εb RT



 z F  i

Ci,f exp −

i

RT



−1

or 2 ek (cylindrical pores) rp F

(10)

Given that these relations correspond to the assumption of constant charge density, at a given bulk concentration, on the membrane surface and inside pores, the volume charge density, X, should be less than XTSP . A simple correction can be introduced by considering the dielectric constant of the solution inside pores, εp , instead of εb in Eq. (8) [3] leading to:



X = XTSP

εp εb

The membrane used for this study was the thin film composite membrane Desal DK (GE Osmonics). All chemicals except potassium clavulanate were of analytical grade and solutions were prepared with milli-Q quality water. The potassium clavulanate (KCA, MW = 237 g/mol, Fig. 2) used for the NF and tangential streaming potential measurements was supplied by CIPAN S.A. (lot 315.45 B 008).

(8)

In order to apply the SEDE model, the volume charge density must be obtained, usually from the streaming potential measurements:

XTSP = − ek (slit-like pores) (9) rp F

XTSP = −

323

(11)

It should be stressed that this correction is based on the arbitrary assumption that the zeta potential inside the pores is the same as that of the external membrane surface. Nevertheless, it is known that when two charged surfaces drew together, the surface charge density decreases and tends to zero at contact, a phenomena, known as “charge regulation” [29]. As the introduction of the charge regulation theory in the SEDE model is out of the scope of this work, the values of X determined from the TSP measurements can be considered as rough estimation of the true volume charge density. Nevertheless, it must be kept in mind that this represents the upper limiting case of charge density.

All filtration experiments were carried out on a plate-and-frame LabStak M20 unit (Alfa Laval). The experiments were performed with a membrane area of 0.072 m2 (two pairs of membranes). The effective channel height of the feed channels is 0.46 mm [18]. The experimental set-up used in this study and the plate-and-frame module are described in [13,18]. The feed solution was pumped at constant velocity yielding an average cross flow velocity of 1.23 m/s. In order to avoid too high clavulanate degradation, the feed temperature was kept at 15 ± 2 ◦ C. Preceding first use, the membranes were cleaned in order to remove their preservatives. This cleaning procedure included three steps: (1) rinsing with deionised water; (2) recirculation of deionised water, for 45 min, at 45 ◦ C and pH 10.5, adjusted with NaOH; (3) rinsing with deionised water to neutral pH. After cleaning membranes were compacted for 2 h at 40 bar and the pure water flux (Jw ) of the membrane was measured at several transmembrane pressure differences. This cleaning protocol was performed, also after each experiment in order to restore the hydraulic permeability (Lp ). NF experiments were performed at variable permeation flux by varying the transmembrane pressure in the range 6–40 bar and at pH 5.5, the same pH of ultrafiltration (UF) permeates of industrial fermentation broths. The experiments were carried out in total recycling mode, i.e., the retentate and permeate were both recycled back to the feed tank in order to keep the feed concentration constant. Only a minimum volume of feed solution and permeate (ca. 8 ml), required to determine the ions concentrations (Ci,p and Ci,f ) and for the volumetric permeation flux determination, was taken at each permeation flux, after 10 min of permeation to ensure the system reached a steady-state. Concentrate and permeate flow rates were measured by weighing (±0.01 g).

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3.3. Characterization of the process streams

where Ki,c and Ki,d are the convective and diffusive hindrance factors (v.d. Table 1). Pe denotes the Peclet permeation number defined as

The model solutions were prepared considering the pH (5.5 ± 0.2) and concentrations of (NH4 )2 SO4 (0.5 g/l), KCl (0.21 g/l) and KCA (2.0 g/l) in the ultrafiltration permeates of industrial fermentation broths [13]. According to the characterization previously performed the broths also contain KH2 PO4 [13]. Nevertheless the concentration of this ion corresponds only to 1% of the total concentration of ions and therefore for simplification it will be neglected. The rejections of the uncharged solutes, glucose and glycerine, which were found to exist in the broth are not studied here as well. Even though these species might have some impact on the overall rejections, at this stage, the modelling will be limited to the five ions solution and the interactions with uncharged solutes will not be considered in the current work. Clavulanic acid is a weak organic acid and so, depending on the pH, there are undissociated and dissociated molecules in the solution. The pH of the feed and permeate of the five ions model solution remained constant during NF (pH 5.5 ± 0.2). According to the dissociation equilibrium (pKa = 2.5, vide Fig. 2) at this pH clavulanic acid is largely dissociated because [CA− ] > 99% for pH > 4.5 and the percentage of dissociated acid increases with pH. Therefore, it is assumed that clavulanic acid is completely dissociated and behaves like a monovalent ion. The diffusion coefficient and radius of clavulanate are not readily available in the literature. The diffusion coefficient at infinite dilution of clavulanate was experimentally determined from measurements of the electrical conductivity of different concentration solutions [31,32]. Even thought the shape the clavulanate is likely to differ from a sphere, its radius (ri,s ) was determined assuming that it behaves as an hard sphere and the Stokes–Einstein relationship was applied. Thus, the influence of the real shape of this molecule on rejection is not accounted for. The cavity radii of the ions (ri,cav ) were based on the covalent radii for cations and the ionic radii for anions, increasing these radii by 7% [19]. The values of the radii and diffusivities of the ions considered in these works are shown in Table 2 as well as their source and the approximations needed to calculate these values. 3.4. Characterization of the membrane 3.4.1. Determination of the membrane structural characteristics The effective pore size (rp ) as well as the membrane active layer thickness to porosity ratio (x/Ak ) were estimated through independent experiments, by fitting the rejection rates of uncharged solutes (glycerol rs = 0.258 nm, xylose rs = 0.302 nm and glucose rs = 0.355 nm, 0.5 g/l) to the following equation: Ri = 1 −

Ci,p Ci,m

=1−

˚i Ki,c 1 − [(1 − ˚i Ki,c )(exp(−Pe))]

(12)

Ki,c Jv x Ki,d Di,∞ Ak

(13)

The structural parameter x/Ak can be determined from the water volume flux, Jw , considering the Hagen–Poiseuille equation: Jw = Lp P =

Jw = Lp P =

rp2

P

(cylindrical pores)

(14)

P

(slit-like pores)

(15)

8 (x/Ak ) rp2 3 (x/Ak )

Since x/Ak are parameters of the active layer, the application of these equations imply that the pressure drop occurring through the macroporous sublayer(s) is negligible so that P can be attributed to the active layer, an assumption widely made in the literature. 3.4.2. Electrokinetic characterization of the membrane The zeta potential of the membrane was determined from TSP measurements for the various single salt solutions (KCl, (NH4 )2 SO4 , KCA) and the five ions solution. The solutions tested were prepared with the same composition and pH as those used in the NF experiments. TSP experiments were conducted with a ZETACAD zetameter (CAD Instrumentation). The TSP cell and the exact measuring procedure can be found elsewhere [28,34]. Measurements were carried out at several channel heights (at least four) by using Teflon spacers of several thicknesses. The length and the width of the channel were 75 and 25 mm, respectively. Measurements were performed at room temperature. The zeta potential was obtained from the intercept of the plot of P/V versus the reciprocal channel height according to Eq. (7). 3.5. Analytical methods For NF experiments aiming membrane characterization with neutral solutes the feed and permeate concentrations were determined by differential refractive index (LKB Bromma 2142, Pharmacia). In order to calculate the observed rejections of the ions retentate and permeate concentrations were quantified using distinct analytical methods. Clavulanate concentration was determined using a spectrophotometric technique described elsewhere [35]. Ion chromatography (IC) was used for chloride and sulphate determination. A Dionex ICS-2500 chromatograph was used with a column AS14A (250 mm × 4 mm). The eluent used was 16 mM Na2 CO3 1 mM NaHCO3 at a flow rate of 1 ml min−1 . Potassium was determined using a Jenway PFP7 flame photometer. Conductivity was measured using a conductimeter (Crison GLP31) and pH was measured by a pH meter (Crison Basic 20). 3.6. Comparison between the SEDE predictions and the rejections based on experimental results

Table 2 Diffusion coefficients of the solutes at 15 ◦ C, Stokes radius, cavity radius

K+ Cl− CA− NH4 + SO4 2−

Pe =

Di × 10−9 (m2 s−1 )

ri,s (nm)

ri,cav (nm)

1.557 [31] 1.621 [31] 0.625a 1.562 [31] 0.846 [31]

0.112 0.108 0.265 0.112 0.207

0.217 [19] 0.194 [19] 0.283b 0.213 [19] 0.246c

Throughout this study it was necessary to compare the intrinsic rejections rates based on the experimental data with the model predictions (Ri,model ). The quality of the fit was evaluated through the use of a least-squares objective function, Sy . If there are n solutes and j data points for each solute Sy is defined as follows:



a

Determined experimentally. The ionic radius was calculated based on the molar volume of clavulanic acid, assuming a spherical molecule. c Based on the crystallographic radius of sulfate [33]. b

Sy =

n j 1

1

(Ri − Ri,model )2 jn − 1

.

(16)

A.I. Cavaco Morão et al. / Journal of Membrane Science 322 (2008) 320–330

325

Table 3 Structural parameters of the membrane Desal DK

rp (nm) x /Ak (␮m)

Slit-like pores

Cylindrical pores

0.33 3.89

0.46 2.76

4. Results and discussion 4.1. Membrane characterization The average pore size and the membrane active layer thickness to porosity ratio were determined from the data of intrinsic rejection rate of uncharged solutes with different Stokes radii and from the hydraulic permeability of the membrane (Lp = 9.72 × 10−12 m s−1 Pa−1 , 15 ◦ C) by the simultaneous solution of Eq. (12) and the Hagen–Poiseuille equation (Eq. (14) or (15)). The iterative method to solve simultaneously these equations was: (1) initialization of rp and x/Ak through the Hagen–Poiseuille equation; (2) substitution of x/Ak into Eq. (12) and calculation of a new value for rp was calculated; (3) determination of a new value of x/Ak by means of the Hagen–Poiseuille relationship; (4) repeat steps (2) and (3) until the deviation between the rejections predicted by Eq. (12) and the intrinsic rejection rates obtained from the experiments is minimized. The values of the membrane structural parameters, obtained considering either cylindrical or slit-like pores are shown in Table 3. The membrane Desal DK is a commonly used NF membrane and its structural parameters are reported in the literature, assuming cylindrical pores, for distinct batches of membranes and using different characterization methods. The values of rp found in the literature are in the range 0.42–0.59 nm and x/Ak in the range 1.76–3.13 ␮m [11,21,30,36,37]. The values of rp and x/Ak obtained in this work, assuming cylindrical pores (Table 3), are within the range of variation of these previous works. Clearly, the rp obtained is narrower if the membrane is modelled as a bundle of identical slit-like pores of half-width rp . In many reported works in the literature [10,21,24,30] the membrane charge at constant pH has been related to the ionic concentration of solution contacting the membrane assuming a Freundlich isotherm and some authors have claimed that it is independent of the electrolyte type [10,24,30]. This assumption would be particularly advantageous in the case of multi-component electrolyte solutions since the membrane charge could be easily inferred from the characterization with a model solution, e.g. NaCl. Nevertheless there are experimental [3,5] and theoretical [38] evidences that this simple approach does not have physical consistency. Thus, in order to obtain reliable values of volume charge density to be used in the model this parameter was assessed through tangential streaming potential measurements for the membrane contacting KCl, (NH4 )2 SO4 , KCA binary solutions and the five ions solution. The measured results of P/V as function of the reciprocal channel height are shown in Fig. 3 for the five ions solution. The zeta potential was obtained from the intercept of this plot using the extrapolation method (Eq. (7)). The zeta potential of the membrane contacting the binary solutions was assessed in the same way as described for the five ions solution. The slope of the regressed line plotted in Fig. 3 is rather low indicating that the contribution of the porous body to the conduction phenomenon is, in this case, very low since the SP is almost independent of the channel height. The electrokinetic surface charge densities of the membrane contacting the distinct solutions were computed by Eq. (8), using the respective values of concentration and zeta potential. The volume charge densities calculated by Eqs. (9) and (10) are shown in

Fig. 3. Reciprocal streaming potential coefficient (P/V) vs. reciprocal channel height (1/h), for the five ions solution (Ck+ = 11.5 mol m−3 , CNH4 + = 8.7 mol m−3 , CSO 2− = 4.4 mol m−3 , CCl− = 3.1 mol m−3 , and CCA− = 8.4 mol m−3 ). 4

Fig. 4. There is not any self-evident relation between the volume charge density and the ion concentration. Even if only the single salt solutions were considered, a Freundlich-type isotherm could not be assumed because the KCA solution would not follow the same trend. Also the approach of characterizing the membrane charge with single salt solutions and use a mixing rule according to the molar concentration of the ions to estimate the charge of the membrane contacting the multi-ionic solution [14] is not evident. Some authors have studied the role of the electrolyte type (NaCl and CaCl2 ) on the charge formation based on the volume charges fitted by the DSPM&DE (Donnan steric pore model and dielectric exclusion) model [38]. According to these authors, for an electrolyte without specific affinity to adsorb on the membrane surface (e.g. NaCl) the membrane charge is due to the competitive adsorption of cations and anions: at lower concentrations the binding of anions to the hydrophobic groups at the active layer of the membrane prevails due to their lower hydration radii and the charge of the membrane gets more and more negative. On the other hand as concentration increases furthermore, the membrane binding sites become saturated and counter ions in solution contribute to the shielding of the membrane charges. Thus, at constant pH, such a charging mechanism can be mathematically described by a Freundlich type

Fig. 4. Volume charge density inferred by tangential streaming potential at pH 5.5.

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isotherm. Unlike for NaCl, Mazzoni et al. [38] have observed that for CaCl2 , the membrane charge does not show a monotone increasing trend with the salt concentration, which was attributed to the competition between the preferentially adsorbed Ca2+ and Cl− . Considering the different nature of the CA− ion, which is a weak organic acid, larger then the remaining salts considered, one could think either about a specific adsorption of CA− or, more likely, an unfavourable adsorption of counter ions at larger concentration, for instance due to the steric hindrance of the larger anion molecules. However, for the five ions mixture, which contains CA− , the charge density of the membrane contacting this solution follows the same trend as KCA and (NH4 )2 SO4 . Thus, the interpretation of the values depicted in Fig. 4 is not straightforward. The mechanisms of the membrane charge formation and the hardly predictable interactions among the complex mixture of ions and between the ions/membrane need to be further studied. The lack of studies on the membrane charge formation in the presence of co-ions, namely SO4 2− and mixtures with more than two ions which are typically found in industrial applications have been recognised in recent studies [24]. The experimental determination of the charge of membranes contacting multi-ionic solutions is not yet reported in the literature and the number of measurements of the present work seems to be rather reduced to draw conclusions. Moreover, these results clearly point out that the characterization of the membranes with NaCl and application of the fitted volume charge densities to predict rejection of distinct solutions may be erroneous. The volume charge densities based on the zeta potential values and corrected by Eq. (11) will be introduced in the SEDE model in order to predict the rejection rates of the ions bearing in mind the limitations of these values. 4.2. Modelling 4.2.1. Concentration polarization of multi-ionic solutions For binary solutions, the thickness of the boundary layer (ıi = Di /ki ) was determined using the effective diffusivity [16] and the mass transfer coefficient calculated by Eq. (6). For multi-ionic solutions, the concentration of ions on the outer membrane surface (Ci,m ) was calculated numerically using the extended Nernst–Planck equation without considering steric hindrance effects. In the case of the five ions solution, the choice of the boundary layer thickness is not straightforward, as aforementioned. Therefore, the computation of the concentration on the membrane surface, Ci,m , was performed for two limiting cases, i.e., assuming that the transport is controlled by the solute with the

Fig. 5. Variation of the concentration polarization index ( i ) with the permeation flux for the five ions solution at pH 5.5. Ci,m was calculated by solving the set of algebraic equations—Eqs. (4) and (5) (symbols) and the Nernst–Planck equations for the limiting thicknesses (lines). Calculations using = 999.2 kg m−3 and = 0.00114 Pa s.

highest (Cl− ) and lowest (CA− ) diffusivity, corresponding to the largest and the thinnest boundary layer thicknesses, respectively (Fig. 5). Ci,m was also evaluated applying Eq. (4) to each ion together with Eq. (5) and solving the six algebraic equations simultaneously. The results of the calculated concentration polarization indexes ( i = Ci,m /Ci,f − 1) are shown in Fig. 5. If the transport is controlled by the ion with the smallest diffusivity, corresponding to the thinnest boundary layer (ımin ), concentration polarization is much smaller than if the transport is controlled by the ion with the largest diffusivity (ımax ). Both assumptions appear, however, like rather simplistic approaches, not fully reflecting the coupling of the ionic fluxes. In some works reported in the literature, it is assumed, without further justification, that the transport is controlled by the ion with the lowest diffusivity [10] while other authors simply use the average concentration polarization layer thickness [11]. In the work of Lefebvre et al. [39] concentration polarization is neglected but for multi-ionic solutions these author’s assume that the transport through the membrane is controlled by the ion with the maximum diffusivity. Thus the choice of the boundary layer thickness is merely arbitrary even though it leads to significant differences in the concentration at the membrane surface, especially at high permeation fluxes as Fig. 5 clearly shows. Thus, an approach that overcomes the need of this assumption is necessary. Fig. 5 shows the values of i computed with Ci,m predicted by the simultaneous resolution of Eqs. (4) and (5). The predictions obtained with this model and the Nernst–Planck equations using ımin are rather close, suggesting that the transport in the film layer may indeed be controlled mainly by the slowest ion. On the other hand if ımax is considered, concentration polarization is, apparently, strongly overestimated. Fig. 5 also shows that in the case of the divalent ion, SO4 2− , when Ci,m was calculated by Nernst–Planck equations, i < 0 (Ci,m is smaller than Ci,f ) and decreases with the permeation flux. This behaviour was observed previously by Bowen and Mohammad [10], that refer, in their work, that the concentration at the membrane interface can be lower than in the bulk feed solution due to the coupling of the ion fluxes. The modelling of the mass transport in the boundary layer based on the work of Geraldes and Afonso [12] will be used in subsequent work to determine the concentration polarization and the intrinsic rejections, based on the experimental results of observed rejection (Ri,obs = 1 − Ci,p /Ci,f ). 4.2.2. Solute transport through NF membranes The rejection rates of the ions in the five ions mixture were predicted using the SEDE model assuming slit-like pores, with the structural parameters depicted in Table 3 and the volume charge density derived from TSP measurements (Fig. 4) corrected by Eq. (11). The dielectric constant of the membrane (εm ) was fixed at 3, which is the value extensively used for polyamide membranes [1,3,4,21]. The bulk solution dielectric constant was set to the value of dielectric constant of the water at 25 ◦ C, εb = 78.54. Thus, the single parameter fitted in the current application of SEDE model is the effective dielectric constant inside pores, εp . The SEDE model assumes that the charge density is constant in the radial direction. Wang et al. [40] showed that this approximation does not introduce significant errors for a ratio of Debye length (D ) to pore radius (D /rp ) larger than 3.3. These authors used a membrane with a surface charge density ∼10-fold larger than the value that we have determined experimentally for DesalDK in the 5 five ions solution. Therefore, for the five ions solution herein studied, the radial variation of charge inside the membrane Desal DK pores should not be significant because D /rp > 3.8. SEDE predictions of rejection rates are compared with the intrinsic rejection rates based on the experimental results of observed

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Fig. 6. Comparison between the SEDE predictions (lines) and intrinsic rejection rates based on the experimental data (symbols) of each ion in the five ions mixture. Pores were modelled as slits.

rejections in Fig. 6 (Sy = 0.16). The model predictions strongly overestimate the rejections of all ions except for clavulanate and sulphate, even neglecting the Born exclusion, εp = εb . Simulations were also performed assuming cylindrical pore geometry (results not shown) and it was found that the rejection rates are further overestimated which is consistent with the reports found in the literature [1,3,20]. The value fixed for εm is the polyamide typical dielectric constant, nevertheless it is reasonable to consider that the dielectric constant of the wet polymer might be higher than

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this value. In fact, if εm = 22 is assumed, then the model predicts the experimental rejections fairly well (Fig. 6b, Sy = 0.031). The value of 22 for εm seems, however, unrealistically high. Dielectric spectroscopy studies with polysulfone NF membranes showed that the dielectric constant of the wet polymer rises from 3–3.5 to 6.2–5.6 in LiCl solutions [43]. On the other hand, if only the Born dielectric effect is considered, the experimental data are pretty well predicted (Fig. 7) with εp = 62, which appears to be a physically realistic value. This fitted value is in good agreement with values obtained by other authors which have fitted εp in the range 42–63 [2–5] depending on the electrolyte type, concentration and pore radius. Considering these results it appears reasonable to consider the solvation energy barrier as the dominant dielectric exclusion mechanism, under the experimental conditions presented in this work. It has been claimed, however, that the “image forces” are a fundamental dielectric exclusion mechanism [41]. The “image forces” dielectric exclusion occurs due to the interaction of the ions in the pores filling solution with the induced charges on the pores inner surface. An ion located in the medium of a higher dielectric constant (solvent inside pores) induces polarization charges of the same sign as its own at the interfaces with the medium of a lower dielectric constant such as a polymeric membrane. As mentioned previously, the volume charge density inferred from TSP even corrected by Eq. (11), X, still remains an overestimated value of the charge density inside pores since it does not account for the charge regulation process. This raised the issue whether the overestimated intrinsic rejections obtained when considering image forces were due to the use of an overestimated membrane volume charge density. Nevertheless, accounting for the charge regulation phenomenon would further increase the complexity and the number of fitted parameters making the use of the SEDE model less appropriate for predictive purposes [3]. We have carried out a set of additional calculations by setting the effective volume charge density to zero, which represents the lower limiting case for which the fixed charge density would simply vanish, and intermediate cases with X/10 and X/4. Fig. 8 shows the model predictions obtained with reduced values of volume charge density and setting εm = 3. Only if X is set to zero (Fig. 8a), εm can be fixed to the “standard” value of 3 without overestimating the ions rejection rates and, in order to have a good description of the experimental data, the Born effect is also needed (εp is fitted). If the membrane volume charge density is set to a value equal or greater than X/10 then εm must be fitted to a value larger than three otherwise the rejection rates are overestimated (Fig. 8b and c). In these cases, the Born exclusion mechanism was neglected (εb = εp ) as its inclusion would only contribute to increase the rejection rates. According to these results, in order to have only one fitted parameter, one must use X = 0 or the value of X inferred from TSP (corrected by Eq. (11)). Otherwise, to have a good prediction of the experimental data, if the Born exclusion mechanism is included, either εm must different than 3, which inevitably leads to the increase of the number of fitted parameters to two (εm and εp ), or the image forces neglected. Table 4 summarizes the results obtained until this point. As mentioned previously, εm = 22 is not a physically realistic value

Table 4 Model fitted parameters which lead to the smallest deviations to the rejection rates based on the experimental data

Fig. 7. Comparison between the SEDE predictions (lines) and intrinsic rejection rates based on the experimental data (symbols) of each ion in the five ions mixture. Image forces were neglected and εp was fitted (Sy = 0.031). Pores were modelled as slits.

X (equiv. m−3 )

εp

εm

Sy

−112 −126 0

62 εb 58

no image forces 22 3

0.031 0.031 0.032

Fig. 7 Fig. 6b Fig. 8a

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Fig. 9. Comparison between the SEDE predictions (lines) and intrinsic rejection rates based on the experimental data (symbols) of each ion in the five ions mixture. Dielectric exclusion was neglected and pores were modelled as cylinders.

Fig. 8. Rejection rates predicted with reduced volume charge densities: X = 0 (a), X/10 (b) and X/4 (c). Pores were modelled as slits.

compared with the values of polymers dielectric constants reported in the literature. Also, the assumption of zero charge inside the membrane does not appear the most realistic approach. The equations derived by Yaroshshcuk [20] to account for the image forces are approximated and this exclusion mechanism is overestimated for very small pores. Another explanation might be related with the high ionic strength of the solution or the complex interactions between the charges of the different ions in solution and their interaction with the membrane fixed and induced charges. Considering these limitations, the best approach to predict the experimental rejections of the multi-ionic system considered here relies on the use of volume charge density X (corrected by Eq. (11)) inferred from TSP (since it is an experimental value), the adjustment of the value of εp to 62 and the neglect of the exclusion due to image forces. Nevertheless, without complementary experiments, for instance to determine the dielectric constant of the solution filling the pores, the accurate contribution of the dielectric mechanisms or the charge regulation mechanism cannot be evaluated. The lack of experimental studies about the behaviour of the dielectric constant of electrolyte solutions when confined inside charged nanopores makes difficult any rigorous quantitative discussion about εp fitted by the model. As a matter of fact, the physical

meaning of εp is rather doubtful for several reasons. First of all, as aforementioned, εp is a fitted parameter thus compensating the weaknesses of the other parameters introduced in the model and even any eventual limitations related with the exclusion mechanisms considered. On the other hand, it is very well known that the Born model strongly overestimates the solvation energy, especially for cations [19,41]. For this reason an improved version of the Born model was applied using the cavity radius instead of the ionic radius. Nevertheless, the values of the ionic radius or the covalent radius for the polyatomic ions are not always readily available in the literature and approximate values had to be used, as mentioned in Table 2. Other modified versions using different radius have been proposed [42] but unless the ionic radius of the ions are accurately known these efforts are not likely to improve the model. We also have performed further calculations neglecting dielectric exclusion. Both for slit-like and cylindrical pore geometry, the rejection rates of all ions are underestimated. Nevertheless, the assumption of cylindrical pore geometry significantly improves the quality of the predictions as we obtained the values of Sy = 0.24 and 0.088 for slit-like and cylindrical pores, respectively. The comparison between the experimental and fitted values for the cylindrical pore geometry is depicted in Fig. 9. Even for the design point of view these predictions are not satisfactory: for Cl− the deviations from the experimental results are about 30% and less than 10% for the other ions. Nevertheless, the ions rejection order is well predicted and the model is applied without any fitted parameter. Most likely, these results are obtained because there was not any divalent cation considered in the mixture. In fact, the crucial role of dielectric exclusion to predict the rejection of divalent counter-ions has been clearly demonstrated in the last years [2–6,20,21,24,30] and, it must be stressed, the inclusion of dielectric effects clearly improve the description of the experimental results. It is evident in Figs. 6–9 that the experimental rejection of K+ is higher than NH4 + whereas the predicted rejections are coincident and that the variation of the fitted parameters does not change this fact. This happens because both monovalent cations have very close diffusivity coefficients, Stokes and cavity radius (v.d. Table 2) and so the separation mechanisms included in the SEDE model do not distinguish these ions. Nevertheless, accordingly to the Hofmeister series, K+ is more kosmotropic than NH4 + [44] and so it is more strongly hydrated and might eventually have a larger hydrated radius leading to a higher rejection, as experimentally observed.

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5. Conclusions Surprisingly, despite all the limitations of the SEDE model, the predicted intrinsic rejections rates of a five ions solution are quite satisfactory considering steric, Donnan and Born dielectric exclusion mechanisms, with a single fitted parameter, εp . Concentration polarization is a critical factor determining the intrinsic rejections mainly at high permeation fluxes. Nevertheless, it was shown that for the complex system herein studied its prediction is not straightforward due to the coupling of the flux of the different charged and sized ions. Even using a different approach [12], mathematically less complex and avoiding the choice of a boundary layer thickness, the accuracy of the different methodologies to determine the concentration polarization cannot be evaluated since the “real” values of concentration at the membrane surface are not available. The major weakness of the present approach is the requirement to experimentally determine the effective membrane volume charge density for each electrolyte solution. For multi-component electrolyte solutions there is not any straightforward mean to obtain the adsorption isotherms describing ion adsorption and a large number of time consuming tangential streaming potential measurements are required. On the other hand, the results clearly point out that the characterization of the membranes with NaCl and application of the fitted volume charge densities to predict rejection of distinct solutions may be erroneous. The assumption that the charge on the membrane top surface of the active layer is the same as inside pores is surely a rough approximation, especially for a membrane with quite narrow pores, as Desal DK, that should lead to a strong charge regulation effect. Most likely, the introduction of a fitted parameter in the model, εp , implicitly compensates these shortcomings.

Acknowledgments The PhD grant SFRH/BD/18496/2004 from Fundac¸ão para a Ciência e Tecnologia (Portugal) is gratefully acknowledged by Cavaco Morão, A.I. We wish also to thank to CIPAN S.A. (Portugal) for supplying the potassium clavulanate and for their assistance in the realization of the NF experiments.

Nomenclature C Ci,f Ci,m Ci,p Di e F h hm I Ji,m Jv Jw ki Ki,c

molar concentration of the solution molar concentration of solute i in the feed solution molar concentration of solute i at the membrane surface molar concentration of solute i in the permeate diffusivity of solute i at infinite dilution electron charge Faraday constant, 96486.7 C equiv.−1 channel height effective thickness in which the conduction current flows inside the porous body of the membrane ionic strength back diffusion flux of the ion i permeation flux pure water permeation flux mass-transfer coefficient solute i hydrodynamic coefficient accounting for the effect of pore walls on convective transport

Ki,d Lp ri,cav ri,s rp P R Re Ri Ri,obs Sy Sci Shi T v0 V X XTSP x/Ak zi

hydrodynamic coefficient for hindered transport inside pores hydraulic permeability cavity radius of ion i stokes radius of ion i pore radius pressure difference gas constant, 8.314 J mol−1 K−1 Reynolds number, Re = v0 h/ . intrinsic rejection rate of solute i, Ri = 1 −Ci,p /Ci,m observed rejection rate of solute i, Ri,obs = 1 − Ci,p /Ci,f least squares objective function Schmidt number, Sci = / Di Sherwood number, Shi = ki h/Di absolute temperature. average cross-flow velocity at the membrane channel streaming potential membrane volume charge density membrane volume charge density derived from TSP membrane active layer thickness to porosity ratio charge number of ion i

Greek letters i Concentration polarization index of ion i, i = Ci,m /Ci,f − 1 ε0 vacuum permittivity, 8.8542 × 10−12 C2 J−1 m−1 εb dielectric constant of the bulk solution, 78.54 at 25 ◦ C dielectric constant of the membrane εm εp dielectric constant of the solution inside pores  zeta potential 0 solution conductivity Debye length, D = [␧b ␧0 RT/(2F2 I)]1/2 D i ratio of solute radius to pore radius, i = ri,s /rp m conductivity inside the porous body of the membrane dynamic viscosity  electrical potential gradient at the feedsolution/membrane interface  mass-transfer coefficient correction factor,  = • ki /ki

density

ek electrokinetic charge density ratio, i = Jv /ki i ˚i equilibrium partition coefficient for steric interactions (v.d. Table 1)

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