Kinetic modeling and selectivity of anion exchange in Donnan dialysis

Journal of Membrane Science 479 (2015) 132–140

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Kinetic modeling and selectivity of anion exchange in Donnan dialysis Adam Beck n, Mathias Ernst Institute for Water Resources and Water Supply, Hamburg University of Technology, DE-21073 Hamburg, Germany

art ic l e i nf o

a b s t r a c t

Article history: Received 13 August 2014 Received in revised form 26 November 2014 Accepted 23 December 2014 Available online 2 January 2015

The objective of this work was to compare selective sorption and transport behavior of a Selemion AMV membrane for different anions with a theoretically derived kinetic model describing the Donnan dialysis (DD) process. This analysis resulted in a suggested relation for the diffusivity of small ions through “nanochannels” of ion exchange membranes. Mass transfer through boundary layers and membrane diffusion were modeled on the basis of the Nernst–Planck equation by introducing constant diffusivity ratios between the exchanging counter ions. To indentify the kinetic and selectivity coefficients, DD batch experiments with sodium nitrate, sulfate or dihydrogen phosphate as feed electrolytes and sodium chloride as receiver electrolyte were conducted. The derived kinetic model simulated the measured concentration changes very precisely after fitting three concentration-independent parameters and the concentration-dependent permeability coefficient. The selectivity sequence was found to be nitrate> sulfate>dihydrogen phosphate>chloride, while this sequence is strongly connected to activity in solution and in the membrane. This influence was very significant for sulfate, which resulted in higher removal efficiency than expected. Regarding diffusivity the identified sequence was nitrate>sulfate>chloride> dihydrogen phosphate. These results led to a correlation that describes diffusivity of counter ions through nanochannels as a function of hydrated cross section area and valence. & 2014 Elsevier B.V. All rights reserved.

Keywords: Selectivity Diffusivity Mass transfer Activity Selemion AMV

1. Introduction Donnan dialysis (DD) utilizes the properties of ion exchange membranes (IEMs) to exchange counter ions (opposite charge like the IEM surface) between a receiver and feed solution. To maintain electroneutrality in both solutions, the depleted feed ions are replaced by driving ions from the receiver and vice versa. The feed ions are enriched in the receiver solution by a factor which depends on the salinity and volume ratio between receiver and feed. Theoretically, the equimolar electrolyte concentration of both solutions should remain constant but co-ion (same charge like the IEM surface) leakage can lead to a salination of the feed and osmotic water flow can dilute the receiver. DD is a promising process since it uses a spontaneous “reaction” which offers the opportunity to replace and enrich troublesome or valuable ions with very low energy demand since no external driving forces are required except for feed and receiver pumping. However, the costs for IEMs are high (US $300–500 per m² in 2010) [1] which generates the necessity for a process optimization and a precise prediction of required membrane area. Experimental investigations in the field of DD were presented in several studies. Especially those of Wisniewski et al. are relevant in the following. They conducted several studies aiming at the exchange

n

Corresponding author. Tel.: þ 49 40 428783920. E-mail address: [email protected] (A. Beck).

http://dx.doi.org/10.1016/j.memsci.2014.12.037 0376-7388/& 2014 Elsevier B.V. All rights reserved.

of troublesome ions like nitrate, sulfate and bicarbonate [2–4], with emphasis on removing these troublesome ions prior to an electrodialytic desalination process. They observed a good removal of ions in single and multi component feed solutions and significant differences in co-ion leakage and counter ion flux for different IEMs. The latter observation was referred to differences in water content and porosity of the analyzed membranes. Recent work by Długołęcki et al. revealed that the electrical resistance of IEMs depends on the electrolyte concentration in solution, especially at low solution concentrations below 100 eq/m³ [5]. The DD exchange process can be theoretically simulated by combining the Navier–Stokes, Nernst–Planck and Poisson equation [6]. Although the results seem to be in good accordance with real behavior the numerical effort is immense and empirical determination of values like surface potential and mean pore diameter –which are usually not provided by the IEM manufacturer– is still required. A semi-empirical model can reduce the complexity by combining experimentally-calculated kinetic and selectivity coefficients with an electro kinetic model. Usually, the Nernst–Planck equation (NPE) is used to derive the functional dependency between concentration and exchange rate, like it was done in the following examples. Ho et al. studied the exchange behavior of sodium and hydronium ions by deriving a model which considers an unsteady state at the beginning of batch experiments due to saturation of the IEM with counter ions [7,8]. To reduce the numerical effort, Ho et al. assumed a constant diffusivity ratio between exchanging counter ions, calculated by using

A. Beck, M. Ernst / Journal of Membrane Science 479 (2015) 132–140

the respective diffusion coefficients in dilute aqueous solution. A significant increase of diffusion and mass transfer rate with rising feed ion concentration was observed. This connection between concentration and diffusivity was also observed by Miyoshi et al. They found that the diffusion coefficient is strongly affected by the counter ion concentration but scarcely affected by the co-ion species [9]. Their study, focused on the behavior of feed and driving ions of different valence, led to the result that monovalent driving ions are generating higher fluxes [10]. Preceding studies with Hasson et al., on modeling phosphate removal by assuming equal diffusion coefficients for both counter ions, led to a simple and applicable model which combines mass transfer through boundary layers on the IEM and diffusion resistance through the IEM within one equation [11]. The objective of the present study was to derive a kinetic model that considers different diffusivities and selectivities of involved counter ions for modeling the transport through the membrane and through boundary layers on the same. This was done with constant and concentration-independent parameters fitted with DD experiments. Two different diffusivity ratios were used, one for the transport through the boundary layer on the feed side and the other for the diffusion through the IEM matrix. Former were calculated using respective diffusion coefficients in dilute aqueous solution while latter were fitted with experimental DD data. The selective sorption process onto the IEM matrix was theoretically analyzed and selectivity was identified to depend on the total electrolyte concentration and ion activity in solution and in the membrane. The influence of ion activity and selectivity was determined experimentally and combined within an “effective separation factor”. The presented numerical fitting procedure allows identifying the following four important parameters by conducting simple bi-ionic DD batch experiments: mass transfer coefficients, effective separation factors, diffusivity ratios and permeability coefficients of analyzed counter ions. The main novelties of the present study can be manifested in three points: 1) Derivation of a mass transfer equation which can model counter ion exchange through boundary layers on the membrane. 2) Consideration of ion activity differences between feed and receiver phase as a parameter which influences the removal efficiencies in DD. 3) Identification of diffusivity ratios between involved counter ions as a method to understand transport processes through IEMs.

2. Theory With the following theoretical approach it was intended to enable a clearer understanding of exchange processes through ion exchange membranes (IEM) by fitting four characteristic parameters on the basis of bi-ionic DD batch experiments. To model the DD process it is necessary to account for three different mechanisms. The exchange of counter ions is initiated in the respective bulk solution. The ions have to be transported from the bulk to the surface of the membrane which constitutes the first resistance; the mass transfer resistance. Prior to the transport through the membrane, the counter ions are adsorbed onto the IEM polymer structure and the resulting surface concentration depends on selectivities and concentrations in solution. Subsequent transport through the membrane constitutes the main – diffusive – transport resistance, before the ions are again desorbed and convectively transported to the opposite bulk solution. To simplify the modeling of DD, equimolar fractions (Eq. 1) were introduced, since electrokinetic transport and sorption processes in ion exchange materials rather depend on the regarded ion fraction than ion concentration [12]. They comprise the molar concentration of respective counter ion ci ½mol=m3  with the valence zi referred to the overall equimolar electrolyte concentration represented by the

133

constant equimolar co-ion concentration c þ ½eq=m3 . ci zi cþ

Xi ¼

ð1Þ

Electroneutrality conditions are summarized in Eq. (2). Electroneutrality appears in three different aspects. The first is the simple and well known additive variant (left), the second considers electroneutrality in the development of ionic fraction gradients (middle) and the third considers the fact that no net current flux can arise since no external electrical field is applied (right). 1¼

X i

Xi

0¼

X dX i dx i

0¼

X

zi  J i

ð2Þ

i

The mass balance equation (Eq. 3) refers to a batch system with a F=R concentration change solely attributed to the flux J i of the exchanging counter ions. The initial equilibration of the IEM with the surrounding solution is neglected since this saturation process is rather fast compared to the overall duration of the ion exchange between feed (F) and receiver (R) solution. The dependence between flux and concentration of respective counter ion is not linear and will be calculated numerically by combining the different transport processes described in the following. V F=R is the solution volume and Am the IEM area, while the superscript m denotes properties of the membrane. F=R

dðci

V F=R Þ F=R ¼ Am  J i ðcFi ; cRi Þ dt

ð3Þ

2.1. Membrane kinetics The derivation of membrane kinetics is similar to other approaches [7–10,13]. They all base on the simplified Nernst–Planck equation (NPE) (Eq. 4) with Dm i as membrane diffusion coefficient and cm A as counter ion concentration in the membrane. F is the Faraday constant, Rg the universal gas constant and T the absolute temperature. The simplifications of the general NPE are the negligence of pressure gradient, activity gradient and convective transport. Although these neglected terms may be present in DD (e.g. convective osmotic water flow and osmotic pressure difference), they are commonly disregarded because they are considered to be of minor importance in the calculation of the flux. Additionally, their negligence heavily simplifies the derivation of membrane kinetics and enables to model the kinetic behavior with an analytically derived equation. Hence, these simplifications will be also used in the following, while the negligence of activity gradients in the derived transport equation will be compensated by considering activity differences in the derivation of thermodynamic sorption equilibria (see Section 2.3).  m  dcA m zi F dφ Ji ¼  Dm ð4Þ c A i Rg T dx dx Rearranging and combining Eq. (4) for feed and driving ion with the same electrical potential gradient for both ions lead to following flux equation after introducing the electroneutrality conditions:  m m 2 m  m dcm A DA DCl ðzA cA þ cCl Þ JA ¼  ð5Þ m m m dx z2A Dm A cA þ DCl cCl As one can see in Eq. (5), the diffusion coefficient of respective counter ion is multiplied with its concentration in the membrane. This leads to a coupling effect of diffusivity and concentration in the membrane and represents the non linear part of Eq. (5), distinguishing this bi-ionic transport equation from the common Fick's law of diffusion. Nevertheless, it is shown in this and in other studies [5,7–11] that an additional dependence between

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A. Beck, M. Ernst / Journal of Membrane Science 479 (2015) 132–140

concentration and membrane diffusivity exists, that cannot be explained by the NPE. Integration of Eq. (5) is possible by applying the pseudo-steady state condition on the flux J A , which assumes that the flux is constant within a short increment of time. The diffusivity ratio for the counter ions within the membrane θm , is a constant parameter and will be regarded as concentration-independent. The fractions on F=R m both sides of the membrane X A will be calculated with the sorption equilibrium at the solution/membrane interface presented in Section 2.3. The advantage of the permeability coefficient P m A, generated by using fraction instead of concentrations, is that unknown parameters like fixed ion concentration referred to the water phase in the membrane cm þ and thickness of the membrane after swelling δm , are combined to one parameter which can be fitted with experimental DD data. "

JA ¼ Pm A

!#  m    θ 1 1 þ θm zA  1 X Fm ðzA 1Þ  Fm A  m  X A  X Rm þ   2 U ln A z A θ zA  1 1 þ θm zA  1 X Rm θ m zA  1 A

m Dm Dm P m A cþ Pm ; θm ¼ Am ¼ Am A ¼ δm DCl P Cl

JA ¼

w1 FI P FA 1  ð1  zA θA 3 ÞX A  F X A X FI A FI w zA 1  ð1  zA θA ÞX A

P FA ¼ kA cFþ

ð11Þ

2.3. Equilibria

A preceding study illustrated that the mass transfer resistance of the laminar boundary layers constitutes a major part to the overall resistance in the exchange process [11]. Especially the film resistance on the feed side was identified to be of great importance. Theoretical and experimental evidences were given to the fact that the resistance of regarded boundary layer increases with decreasing electrolyte concentration. Using these previous findings, only the film resistance on the feed side will be considered in the modeling of the exchange process. If ions are transported within an electrolyte solution the different species are interacting with each other and the resulting transport rate can be calculated by considering electrical potential φ and different diffusivities. This was done by applying the simplified NPE (see Eq. 4) on the region directly at the membrane surface ðFIÞ, where the no-slip condition can be adopted (Eq. 7). Using this approach, the diffusive part of the NPE can be replaced by the well-known mass transfer equation, taking into account convective transport processes with the mass transfer coefficient ki . The diffusion coefficients Di will be the respective values in infinite aqueous dilution [14].

 

dci z F dφ

F dφ

F FI FI  ci i J i ¼  Di ¼ k ðc  c Þ þ D z c U ð7Þ i i i i i i Rg T dx FI Rg T dx FI dx Rearranging and combining Eq. (7) (right) for the feed ion (A  z) and the driving ion chloride (Cl  ) with the same electrical potential gradient dφ=dx for both ions, lead to following flux equation after introducing the electroneutrality conditions: FI 2 kA DCl ðcFþ zA cFI A Þ þ zA kCl DA cA ðcFA  cFI AÞ 2 D cFI DCl ðcFþ zA cFI Þ þ z A A A A

The ratio θw can be used to correlate both mass transfer coefficient and diffusion coefficients. This enables to eliminate the diffusion coefficients of both counter ions and replace the mass transfer coefficient of chloride kCl in Eq. (8). By following this derivation, it is possible to obtain a mass transfer equation (Eq. 11) with only one fitting parameter, namely the mass transfer coefficient for the feed ion kA ½m=s. Like already shown in preceding study [11] the permeability of the boundary layer for counter ions is proportional to the electrolyte concentration (Eq. 11 down) in respective solution.

ð6Þ

2.2. Mass transfer characterization

JA ¼

θw (Eq. 10) will be calculated with the self-diffusion coefficients in water [14]. 

2=3  2=3 kA DA

¼ ¼ θw ð10Þ kCl DCl water

ð8Þ

To correlate mass transfer resistance for feed ions with the respective value for chloride, an empirical correlation for mass transfer in laminar flow is used. The Graetz–Leveque equation [15] (Eq. 9 left) can be transformed to a term showing the functional dependence between ki and Di (Eq. 9 right). The same dependence would be the result by using the Grober equation [15] for turbulent flow within the entrance region of channels, which might be present within the dialyzer channels used in this study.  1=3 Sc ¼ η ρDi kd d Sh ¼ i h ¼ 1:86 Re Sc h 3 ki ¼ Di 2=3 U const ð9Þ Di L This finding can be used to simplify the numerical fitting of the mass transfer coefficient kA , while the aqueous diffusivity ratios

There are two exchange equilibria which have to be considered in DD. Primarily, there is the interface equilibrium which establishes throughout the entire process since solution and membrane surface are in direct contact. The sorption reaction takes place between the fixed functional groups of the membrane and the dissolved counter ions. In contrast to the transport processes presented above, the activity ai (Eq. 12 left) difference of respective ions will be considered in the modeling of thermodynamic equilibria. Since the ionic strength of IEMs and receiver phase is high, the activity coefficient γ i cannot be assumed to be one and an activity difference between receiver and feed can significantly influence the exchange process. According to the Debye-Hückel equation the activity influence is even more significant for multivalent ions. The fact that this difference was neglected within the derivation of the transport equation (see Eq. 6) is compensated by considering it in the modeling of sorption equilibria. Following equation refers to the sorption equilibrium at the interface between solution and membrane surface for the feed ion [A  zA] and driving-ion [Cl  ]. The superscript m denotes values on the membrane surface.  z A  zA  z A a  am γ  γm c  cm A A A ¼  ClzA U  ClzA ð12Þ K ACl ¼  ClzA m m m aCl  aA γ Cl  γ A cCl  cA To overcome the necessity to calculate each single activity coefficient, they will be combined to a single activity parameter Γ for the respective interface and counter ion pair. Additionally, the introduction of equimolar fractions extracts a term showing the dependence on the ionic concentration difference between both phases (Eq. 13). For equal valence of feed and driving ions, this influence cancels out.  z A  1 cþ ðX Þz A  X m A  Γ   Cl ð13Þ K ACl ¼ m z A cþ Xm  XA Cl The separation factor r ACl is an ionic-strength and activity independent parameter showing the preference of membranes for certain ions (Eq. 14 left). But since the calculation of Γ is very complex, the effective separation factor r~ ACl is introduced (Eq. 14 right). rA ðX ÞzA  X m ðX Þz A  X m A A or Cl ¼ r~ ACl ¼  Cl r ACl ¼ Γ  Cl z A z A Γ Xm  X Xm  XA A Cl Cl

ð14Þ

After introducing the electroneutrality condition, Eq. (14) (right) will be solved to obtain the membrane feed ion fraction X m A as a function of the fraction in adjacent solution X A with the parameter r~ ACl .

A. Beck, M. Ernst / Journal of Membrane Science 479 (2015) 132–140

This requires the solution of a polynomial equation with the order zA . For the bivalent case the defining zeros have to be distinguished into two different cases; for r~ ACl 4 1 the negative version of the quadratic formula has to be used, while for r~ ACl o 1 the opposite. However, a remaining problem is to find the relation between the activity parameter Γ on the feed (superscript F) and on the receiver (superscript R) side. To obtain this relation, one has to analyze the second important equilibrium at the end of the DD process. Following the initial theory of Donnan [16], where equilibrium between receiver and feed solution can be predicted by the electrochemical potential, it was found in this current work that the ionic activity difference between the interfaces on the feed and receiver side can contribute to a significant increase of the removal efficiency. This shift is even more significant if the valences of feed and driving ion differs. Calculating the activity coefficients for mixtures is difficult and it is even more difficult for the complex IEM matrix. Van der Stegen et al. modeled the activities in IEMs with the use of a modified Pitzer equation and the results show accordance with experimentally derived coefficients [17]. Nevertheless, to model ion transport in DD it was found in present study that not the single activity values are needed but relations between activities on feed and receiver side of the IEM [see Γ F =Γ R in Eq. 16]. To obtain these ratios, equilibrium concentrations can be used to calculate the relation between effective separation factors on feed ðr~ F Þ and receiver ðr~ R Þ side. In equilibrium, the ion flux and the ionic gradients disappear which is illustrated in the following: F FI F for J A ¼ J Cl ¼ 0 - X Fm ¼ X Rm ¼ Xm i i i;eq 4 X i ¼ X i ¼ X i;eq

Debye–Hückel law (Eq. 17) to calculate the mean electrolyte activity [18], which in turn is independent from the electrolyte species. With γ 7 as mean activity coefficient, z as the valence of ion species and I as ionic strength of the solution, it is possible to calculate the mean activity coefficient in the feed solution at the beginning and at the end of the exchange process, respectively. pffiffi   log γ 7 ¼ 0:5091  z þ z  I ð17Þ Considering this and remembering the fact that the overall electrolyte concentration remains constant in DD, the exchange of monovalent feed ions with monovalent driving ions should not change the mean activity coefficients. However, if the feed ion is bivalent and the driving ion monovalent this leads to a change of the mean activity coefficient. For such a system with a constant equimolar feed concentration of 10 meq/l, the initial feed solution has a mean activity coefficient of γ 17 ¼ 0:75. After exchange of all bivalent ions by monovalent ions the resulting value becomes γ 27 ¼ 0:89. Using a constant value in-between these borders would lead to an error lower than 10 %, of course only regarding the activity coefficient in the feed solution. The resulting error for the effective separation factor, which combines selectivity and activity properties of the system (see Eqs. 12–14), is assumed to be smaller since the activity coefficients in the membrane — which can also be regarded as a high concentrated electrolyte — are considered to be less affected by the concentration changes (see explanation above) than the activity coefficients of the feed solution.

ð15Þ

This can be used to eliminate the membrane activities by dividing Eq. (14) for the feed interface by the same for the receiver interface which leads to:  z A  R zA X FCl;eq  X RA;eq γ  γ FA Γ R r~ AF Cl ¼  ¼  Cl zA ¼ ¼ Γ FR ð16Þ z A AR γ FCl  γ RA Γ F r~ Cl XR  XF Cl;eq

135

A;eq

After the experimentally determined equilibrium fractions are introduced into Eq. (16), the ratio Γ FR is obtained. This will be used for the numerical modeling by multiplying the fitted effective separation factor r~ F for the feed side with the ratio Γ FR to obtain the effective separation factor for the receiver side r~ R . For a ratio of Γ FR ¼ 1; Eq. (16) converges into the well-known Donnan equilibrium between two phases of different ionic strengths separated by an ideal permselective membrane, while a value bigger than unity leads to an improvement of the removal efficiency. The activity coefficients vary due to concentration changes within the solution, but the intention behind a constant activity ratio is to adjust the equilibrium to which the transient model is converging. Introducing a concentration dependency would mean to use different models (Debye–Hückel, Pitzer equation etc.) within the three different phases which are present in Donnan dialysis (i.e. feed, receiver and membrane) as well as compensate for the concentration gradients within these phases. To overcome these problems and at the same time get a very exact fit of experimental data, it was convenient to use the equilibrium concentrations to consider the activity relation between feed and receiver solution. Nevertheless, based on theoretical deliberations it is possible to evaluate the effect of a negligence of varying activity coefficients. In solutions where one electrolyte has a high concentration and another a comparably low concentration (i.e. feed ions in receiver and membrane phase) the activity of the low concentrated ion can be assumed to be constant, since its activity mainly depends on the overall ionic strength, which in turn in mainly controlled by the higher concentrated electrolyte [18]. This argument does not hold for the feed solution, since this solution commonly has a concentration below 10 mM. However, this allows us to use the

3. Experimental The membrane contactor system (Fig. 1) consisted of two solution reservoirs, associated pumps and a rectangular dialysis cell. The flow channels had the overall dimensions of 400  50  7.5 mm3, respectively. The receiver and feed solution in reservoirs plus residual volume in tubes and in the dialysis cell had a volume of 0.5 l, respectively. The two compartments in the cell were separated by a homogeneous Selemion AMV anion exchange membrane (see Table 1) with an effective area of 0.02 m². The recycle flow rate through each compartment was 1.8 l/min which resulted in a Reynolds number of 1175, calculated by assuming a flow between two parallel plates, a hydraulic diameter of dh ¼4A/U (cross section area A, wetted perimeter U) and the dynamic viscosity of water at 25 1C (η¼0.89 mPas). Chemical pure sodium chloride (NaCl), sodium nitrate (NaNO3), anhydrous sodium sulfate (Na2SO4) and monohydrated sodium dihydrogen phosphate (NaH2PO4  H2O) were used to prepare the solutions for feed and receiver. The receiver

Fig. 1. Membrane contactor system for Donnan dialysis recycled batch experiments.

Table 1 Key data of the Selemion AMV membrane used in the DD experiments [Manufacturer information]. IECn [meq/ Thickness gdry] [μm]

Water content [gH2 O =g dry ]

Area resistance [Ω cm²]

Functional group

1.9–2.0

0.19

1.5–3.0

Quaternary amine

n

110–120

ion exchange capacity.

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A. Beck, M. Ernst / Journal of Membrane Science 479 (2015) 132–140

Table 2 Initial feed and receiver concentrations for the experiments with single electrolyte solutions. Feed electrolytes

Receiver electrolyte

Run

NaNO3 [mM]

NaH2PO4 [mM]

Na2SO4 [mM]

NaCl [mM]

C1 C2 C3 C4

0.5 0.9 2.0 4.0

0.05 0.5 1.0 2.1

0.5 1.0 2.0 4.0

100 100 100 100

solution had a concentration of 100 mM sodium chloride for each experiment, while the feed concentration was varied for each anion species according to the values listed in Table 2. Prior to an experiment, the membrane was equilibrated in 2 l of 100 mM NaCl solution for at least 12 h. The conductivity measurements for all experiments and solutions were conducted with a CONSORT conductometer K610. The concentration measurements of nitrate were conducted with a SHIMADZU UV–vis-spectrophotometer UVmini-1240 according to the procedure described in the literature [19]. Since the measurement of nitrate is influenced by chloride, two calibration curves were prepared to find the absorbance coefficient defined by the Beer–Lambert law. The concentration measurements for sulfate and phosphate were conducted with a HACH LANGE VIS-spectrophotometer DR 2800 by adding HACH PERMACHEM reagents SulfaVer 4 or PhosVer 3 for 10 ml samples, respectively. The recommended concentration ranges, were 2–70 mg/l- SO4 2 for sulfate, 0.02–2.50 mg/l- PO4 3 for phosphate and 0.14–42 mg/l-N for nitrate. Since phosphate has different protonated species depending on the pH, this can influence its transport properties in DD. However, the pH was not adjusted during the experiments because this would simultaneously add additional ions into the system. The experiments with phosphate were conducted in a pH range between 6 and 7 in the feed, which results in a H2PO4 fraction of 61–94% calculated with Eq. (18) [11]. !1 ½H2 PO4  ½H þ  10  7:2 10  19:5 þ ¼ þ 1 þ ð18Þ C TP ½H þ  ½H þ  10  2:1

4. Numerical For the treatment of experimental data, a numerical simulation was designed to enable the fitting of concentration profiles by the variation of concentration-dependent membrane permeability coefficient P m A and the concentration-independent effective separation factors r~ ACl , diffusivity ratios θm and mass transfer coefficients kCl . Since several runs with varying initial feed concentrations were conducted for each feed ion species, the concentration independent factors were fitted by finding the set of parameters leading to a reduction of the cumulated fitting error for all runs with respective species. The concentrationdependent permeability coefficient was fitted for each single run. This was done by implementing the equations presented in Section 2 into an algorithm developed with MATLABs. Since an analytical integration of the two transport equations (Eqs. 6 and 11) in combination with the interface equilibrium equation (Eq. 14) and the mass balance (Eq. 3) is very complex, numerical integration was chosen. Therefore the different equations are solved for each time increment Δτ separately and the resulting flux is inserted into following mass balance equation, which is then numerically integrated over time to simulate the concentration profiles. F=R dðci

      V F=R Þ mi F=R t j ¼ mi F=R t j  1 þ  Δτ ¼ mi F=R t j  1 þ Am dt    J i F=R t j  1  Δτ

ð19Þ

The normalized fitting error E is calculated with the receiver data since the concentration profile in the feed is significantly influenced by a saturation processes taking place at the initial period of the experiments. The objective function representing the normalized error between experimental X RA;exp and simulated data points X RA;cal with the number of measurements nexp is:



R

R X X A;cal  X A;exp cR  Fþ ð20Þ E¼ nexp cþ The following fitting procedure was adjusted to the present problem to ensure a plausible set of parameters. The permeability coefficient was observed to have the biggest influence within the modeling equations and was therefore fitted separately with the golden section method [7] and Eq. (20) for each iteration step. The starting value of kCl ¼ 8:5  10  6 m=s was calculated with the Graetz–Leveque equation (Eq. 9). The initial effective separation factors r~ ACl and diffusivity ratios θm for the respective feed ion were set to be unity. Since the mass transfer coefficient is connected with the effective separation factor by the fact that a decrease of kCl can be compensated by an increase of r~ ACl , these two parameters were adjusted at first, while P m A was fitted for each variation step. Throughout this procedure, the fitting error was reduced by up to 70 %. The identified mass transfer coefficients were used in the subsequent procedure. The second procedure optimizes the separation factor in combination with the diffusivity ratios θm , while P m A was fitted for each variation step. The two parameters are coupled by the fact that a decrease of r~ ACl can be compensated by an increase of θm . It was found that the fitting error was only further reduced by approximately 1%. Furthermore, this reduction was scattered over a broad interval of combinations of r~ ACl and θm . An additional dependency was introduced to calculate a confined set of parameters. The diffusivity ratios and corresponding separation factors within the identified interval were chosen, which led to one permeability coefficient profile for chloride. This procedure enables to find the main parameters in ion exchange processes with simple bi-ionic DD batch experiments.

5. Results and discussion The fitting results calculated with the derived equations in Section 2 and the numerical procedure described in Section 4 are summarized in Table 3. The identified coefficients were found to be the optimal values for all initial feed concentrations of respective biionic system. To compare the mass transfer coefficients between used feed ions, they were normalized by multiplying them with the respective diffusivity ratio in aqueous solution θw to obtain the mass transfer coefficient of chloride kCl . These ratios were calculated with self-diffusion coefficients of respective ions in water [14]. 5.1. Mass transfer characterization The values of the mass transfer coefficient for chloride in Table 3 were used to quantify the influence of the mass transfer Table 3 Selectivity and kinetic parameters of the AMV membrane and feed mass transfer coefficients for chloride ðkCl Þ identified for each bi-ionic system. Diffusivity ratios used for the boundary layer on the feed side: w w θw SO4 =Cl ¼ 0:53; θNO3 =Cl ¼ 0:94; θH2 PO4 =Cl ¼ 0:42: Counter ion pairs A  z/Cl 

Mass transfer coeff.

Diffusivity ratio θm ½  

Effect. Sep. factor

kCl ½10  6 m=s

NO3 /Cl  H2PO4 /Cl  SO4 2/Cl 

25.0 71.0 20.0 71.0 27.0 70.5

1.50 7 0.05 0.69 7 0.03 1.45 7 0.05

5.5 7 0.20 3.7 7 0.10 5.3 7 0.15

(feed side) r~ ACl ½  

A. Beck, M. Ernst / Journal of Membrane Science 479 (2015) 132–140

resistance through the boundary layer on the feed side. This was necessary to ensure that this resistance does not affect the fitting of the other kinetic parameters. However, the values vary although they are referred to chloride and the flow conditions were kept constant in all runs. A possible explanation for this observation could be that the coefficients were fitted for each feed ion species separately and that slight differences in the fluid velocity in respective runs could have led to different values. However, the Graetz–Leveque equation (Eq. 9 left) shows that the presented difference between the mass transfer coefficient of phosphate and sulfate would only result if the fluid velocity in the experiments with sulfate was approx. 2.5 times (1.8 for the Grober equation) higher. Although slight variations in flow velocity cannot be avoided, a variation by a factor between 1.8–2.5, is very unlikely. The more probable reason would be that the used diffusivity ratios for the calculation of mass transfer in Eq. (9) – calculated with the respective self-diffusion coefficient in water – are different from their actual values. A reduction of θw H2 PO4 =Cl to 0.3 and a slight increase of θw SO4 =Cl to 0.6 would lead to the more or less same mass transfer coefficient of chloride. However, the fitting of the diffusivity ratio in the boundary layer was avoided. Instead the mass transfer coefficient was fitted, to ensure that the influence of mass transfer resistance does not affect the fitting value for the membrane diffusion resistance [see Section 5.2]. Studies focusing the influence of mass transfer would need experiments with different flow velocities, while the concentration influence was already described by previous studies [11]. Since the experimental procedure in present study was based on a constant flow velocity and varying feed concentrations, a detailed characterization of the diffusivity ratios in the boundary layers, would not be convincing.

137

The dependence between concentration and exchange rate is also visible in the concentration profiles for different initial concentrations. Fig. 3 shows this observation exemplarily for nitrate, while the phosphate and sulfate show similar profiles. The fact that equilibrium is reached in more or less the same time period shows that the transport rate increases proportionally with feed concentration. Note that the transport in DD is not comparable with a normal diffusion process, where the transport gradient increases with increasing feed concentration. Since the interface equilibrium is a function of fractions in the solution (see Eq. 14), the feed ion concentration in the membrane should not depend on its concentration but on its fraction at the interface. The second observation, which was also approved by the concentration profiles measured for the same initial equivalent feed concentration but different feed-ion species (Fig. 4), is that the permeability of analyzed feed ions follows the order nitrate4sulfate4phosphate. Looking at Fig. 4, it is quite obvious that the fitting procedure was applied on the receiver data and not on the feed concentrations. This was due to the fact that the feed profile was significantly influenced by a sorption process in the membrane at the beginning of each run. This is most significant for the experiments with nitrate. The fact that the permeability of nitrate and sulfate has such similar values is rather unexpected, since the respective diffusion coefficients in infinite aqueous dilution differ by a factor of two. A possible explanation may be that the diffusion in an electrical

5.2. Membrane kinetics Fig. 2 shows the fitted permeability coefficients for the Selemion AMV membrane. It was explained in Section 4 that the feed ion permeability coefficients were identified by finding the set of coefficients for all used feed ions resulting in only one continuous permeability coefficient profile for chloride. This represents a reasonable way to find permeability coefficients with ion exchange experiments using different feed ion species and a uniform driving ion. To improve the visibility of data the last data points for sulfate [8.04 eq/m³, 4.97  10  5 mol/(m s)] and chloride [8.04 eq⧸m³, 3.43  10  5 mol/(m2 s)] are excluded. Fig. 2 illustrates the mentioned fact that a functional dependence is missing within the transport equation derived on the basis of the NPE. This fact was already observed in previous studies [5,7–11] and indicates that the electrolyte concentration – especially at low concentrations less than 100 eq/m³ [5] – contributes significantly to the resistance of the IEM matrix.

Fig. 3. Measured and simulated concentration profiles of nitrate in the receiver solution for each initial feed concentration and a constant receiver NaCl-concentration of 100 eq/m³.

Permeability [10 mol/(m s)]

30 SO

NO

20

Cl

10 H PO

0 0

1

2

3

4

5

Feed Electrolyte Concentration [eq/m ] Fig. 2. Fitted permeability coefficients as a function of initial equivalent feed electrolyte concentration and constant receiver concentration of Ā eq/m³ for a Selemion AMV membrane. Excluded data points: P_SO4 [8.04 eq/m3, 49.7  10  6 mol/(m2 s)] and P_Cl [8.04 eq/m3, 34.3  10  6 mol/(m2 s)].

Fig. 4. Measured and simulated concentration profiles of each bi-ionic system for an initial equivalent feed concentration of approx. 2 eq/m³ and a constant receiver NaCl-concentration of 100 eq/m³.

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potential gradient is influenced by the ion valence which enhances the transport for multivalent ions since only one outer friction zone contains more charges. This may explain the better permeabilities referred to the equivalent concentration of sulfate. The fact that sulfate and nitrate are showing better diffusivities than chloride can be explained by looking at their hydrated radius r hyd (chloride i 0.3187 nm, nitrate 0.265ax  0.3451eq, dihydrogen phosphate 0.377 nm and sulfate 0.3815 nm) [20]. Nitrate has an asymmetric atomic structure and its radius can be split into an equatorial (eq) and an axial (ax) radius, while the axial value is significantly below the hydrated radius of chloride. Analyzing these observations leads to the outcome that diffusivity ratios through nanochannels of the IEM can be calculated using following correlation:  2 zA U r hyd Cl θm   2 zCl U r hyd A

ð21Þ

Inserting the hydrated radii and valences leads to more or less the same values (see below) like they were found by the fitting procedure of measured concentration profiles (see Table 3):

are present in IEMs, whereas the Stokes law is only applicable for an infinite size of the surrounding fluid body. While the ions are transported through these channels, they occupy a certain amount of the overall cross section, whereas the occupied fraction is proportional to their own cross section area. Hence, diffusion resistance of regarded ion is proportional to its cross section area because it is connected to the probability to collide with other ions or surrounding water molecules while the confined space in the channel increases the effort to displace these particles to obey the conservation of mass. Displacing these particles leads to an acceleration of surrounding molecules. Since the cross section area of IEM channels and small ions are in the same order of magnitude, the acceleration is even more significant the smaller the mean channel diameter is. However, it is questionable if this relation holds for hydronium and hydroxide ions. The so-called Grotthuss mechanism describes the transport of these ions to be controlled by quantum tunneling of protons which leads to an extra conductivity, not explainable by normal Brownian motion effects [18]. Nevertheless, for common cations and anions Eq. (21) might represent a useful relationship for the calculation of diffusion coefficients in nanochannels.

m m θm SO4 =Cl ¼ 1:4; θ NO3 =Cl ¼ 1:45; θ H2 PO4 =Cl ¼ 0:71

5.3. Equilibria Fig. 5 illustrates the dependence between feed ion concentration in solution and equivalent mass of feed ions adsorbed by the AMV membrane. This was identified by calculating the difference between initial feed ion mass in solution and the mass in receiver and feed solution after exchange equilibrium was reached. The abscissa represents the theoretical initial feed ion concentration excluding the amount adsorbed by the membrane. Looking at the values for the separation factors listed in Table 3, the selectivity sequence of the AMV membrane and involved ions is nitrate 4sulfate4dihydrogen phosphate4 chloride while the absorbed feed ion masses shown in Fig. 5 are indicating a selectivity sequence of sulfate4nitrate4dihydrogen phosphate. It was shown with Eq. (14), that the effective separation factor is the ratio of the real separation factor divided by the activity term Γ, which in turn is representing a relation of respective activity coefficients of involved ion in solution and in the membrane. Therefore it is not possible to conclude the real selectivity of the AMV membrane for respective ions with the values listed in Table 3. Extending the model with equations which calculate the activities in solution and in the membrane could overcome this problem. Van der Stegen et al. [17] managed to model the activity of ions in the IEM matrix with reasonable outcome by extending the Pitzer equation with a shielding term which accounts for the decreased scope of fixed charged groups in the membrane. Nevertheless, the concept of an effective separation factor is an easy way 0.5

Absorbed Mass [meq]

Eq. (21) describes the diffusivity of regarded ion to be proportional to the reciprocal of its hydrated cross section area multiplied by its comprised charge. The smaller axial radius of nitrate is used since it is a one-dimensional transport and it is more likely that the smaller cross section is important for the flow resistance in the trajectory. Whether this equation is valid for other anion species and applicable for cations, which show a much higher ratio between their hydrated and crystal radii [20], is a question of further investigation. However, it is out of doubt that this equation can only be applied if the mean pore diameter of the IEM is sufficiently large to pass the counter ions. According to the so-called two phase model, homogeneous ion exchange membranes (IEM) like Selemion AMV consist of a gel phase and an interstitial phase. The interstitial phase can be regarded as an electro neutral solution in the meso- and macropores of ion exchange membrane. However, studies have shown that the ion transport properties are mainly taking place in the gel phase [21]. Based on a theoretical model which describes the membrane microstructure as an array of cylindrical pores of identical radius and experimental results on equilibrium uptake of different counter-ion species, the mean pore radius of a Selemion AMV membrane was calculated. It was found that this model predicts an average pore radius of 2.4 nm [22]. To visualize the micro-pore structure of IEM, small-angle X-ray or neutron scattering (SAXS/SANS) is used, where the refraction of the membrane sample is detected. On the basis of certain models, this detection data are interpreted and parameters like porosity, pore-diameter and pore-structure can be obtained. This was done for a Nafion proton exchange membrane (PEM) and it was found that the scattering data supports the thesis that this membrane contains elongated parallel, cylindrical nanochannels with an average radius of 1.2 nm surrounded by functional groups [23]. Since the Selemion AMV membrane shows a higher conductivity than a PEM Nafion membrane, the two-fold higher radius seems to be reasonable. These findings support the result of the present study, namely that the nanochannels play a central role in the transport processes of counter ions through ion exchange membranes. Eq. (21) does not correlate with the Stokes law, where the diffusion coefficient is proportional to the reciprocal of the spherical perimeter, but it is questionable if the macro-viscosity of a fluid can be applied to the micro-viscosity relevant for the movement of small -and maybe none spherical- ions [18]. Furthermore, Eq. (21) may only be valid for confined channels like they

SO42-

0.4 0.3

NO 3-

0.2 0.1

H 2 PO4-

0 0

2

4

6

8

Feed Electrolyte Conc. [eq/m³] Fig. 5. Adsorbed feed ion masses for each DD experiment plotted against the theoretical initial feed concentration excluding the adsorbed feed ion masses.

Table 4 Experimentally and theoretically determined values regarding ion exchange equilibrium between receiver and feed solutions (with RA as removal efficiency and Γ FR as activity parameter ratio). Nitrate

Sulfate

Run no.

RA;theo [%]

RA;exp [%]

Γ FR ½  

RA;theo [%]

RA;exp [%]

Γ FR ½  

1 2 3 4

99.53 99.10 98.14 96.21

– 98.69 98.01 96.65

– 0.68 0.94 0.97

99.02 98.06 96.05 92.14

– – 99.42 99.43

– – 7.15 15.67

Leaked NaCl-Mass [mmol]

A. Beck, M. Ernst / Journal of Membrane Science 479 (2015) 132–140

139

0.12 0.1

NaCl

0.08 0.06 0.04 0.02 0 0

to overcome the necessity to calculate activity coefficients by measuring the equilibrium concentrations. Some of these theoretical –neglecting activity influence– and experimental removal efficiencies (Eq. 22) are listed in Table 4. The theoretical removal efficiencies were calculated with the use of Eq. (16), respective electroneutrality and mass balance conditions and by setting the activity parameter to unity. ! V R cRA;eq RA ¼ 1  F  100% ð22Þ V cFA;0 Furthermore, experimentally calculated activity parameter ratios Γ FR (see Eq. 16) are presented in Table 4, which were used to explain deviations between measured equilibrium concentrations and concentrations which would be expected if the activity influence would be neglected. A value of Γ FR 4 1 is related to a reduced activity of feed ions in the receiver solution leading to an improvement of the removal efficiency, since “inactive” feed ions do not contribute to electrochemical equilibrium. For experiments where the equilibrium concentrations are below the detection limits (see Section 3), experimental removal efficiencies are missing in Table 4. For experiments with phosphate, equilibrium could not be reached within duration of the experiments. Looking at the values of Γ FR for sulfate in Table 4, it is quite obvious that the activity of bivalent ions is significantly reduced in the receiver leading to much higher removal efficiencies. This effect is less significant for nitrate. This observation gives rise to the suggestion that it might be possible to intentionally reduce the activity of the desired ions in the receiver solution to influence the selective removal of these ions out of the feed solution. If the intention is to remove multivalent ions, this can be achieved by increasing the ionic strength of the receiver solution. However, such an increase of the receiver salinity is accompanied by an increase of the co-ion leakage. In present study, the co-ion is sodium and the leakage is quantified in the following.

5.4. Co-ion leakage Additionally to DD experiments, a run with DI-Water as feed solution and 100 mM-NaCl receiver solution was carried out to investigate osmotic water flow and co-ion leakage. Osmotic water flow could not be detected within the measureable accuracy (72.5 ml). Although osmotic water flow could not be measured, it is very unlikely that it does not occur. Primarily, the osmotic pressure difference between receiver and feed can lead to a dilution of the receiver solution. Additionally the exchange of hydrated counter ions will contribute to a water exchange between both solutions, whereas the net water flow of latter phenomena depends on the hydration numbers of respective ion. Fig. 6 illustrates leakage of sodium into the feed solution by plotting the change of molar amount of sodium chloride against time. The leaked co-ion mass was calculated by monitoring the

2

4 Time [h]

6

8

Fig. 6. Leaked NaCl mass from receiver to feed solution during the experiment evaluating osmosis and co-ion leakage.

rising conductivity of the feed. The temperature influence was taken into account [24]. In DD, a low diffusion resistance for IEMs is preferential since there is no external driving force compensating the transport resistance. In preceding studies, Fumasep FAB and Selemion AMV membrane were compared and a 30-fold higher resistance was measured for the FAB membrane [11]. For design, this can only be compensated by a 30-fold higher AEM area. Since IEMs are comparatively expensive, this would significantly contribute to investment costs. However, good diffusivity properties are usually connected with a high leakage of co-ions from receiver to feed solution and consequently by a salination of the feed solution. Looking at Fig. 6, it is evident that the salination of the feed is of major importance. The exclusion of co-ions is improved by lower receiver salinities, which represents a consequence of the Donnan potential and can be further improved by using bigger and multivalent co-ions in the receiver solution.

6. Conclusion The ion exchange process in Donnan dialysis with a Selemion AMV membrane was analyzed and the selectivity and transport properties for nitrate, dihydrogen phosphate, sulfate and chloride were modeled with a semi-empirical kinetic model. The mass transfer resistance through the boundary layer on the feed side represents a significant part of the overall resistance. To consider this influence, a novel equation (see Eq. 11) for bi-ionic ion exchange through boundary layers on the membrane was derived by using the Nernst–Planck equation (NPE) and the no-slip condition to introduce the common mass transfer coefficient. The activity influence on the separation factor was introduced and calculated with the equilibrium concentrations. The calculated activity parameters correlate the different activity influences between feed and receiver side which led to two different effective separation factors (see Eqs. 13–16). The activity influence was most significant for the bivalent anion sulfate and it was shown that the experimental removal efficiency is much greater than the expected removal efficiency calculated with the Donnan equilibrium equations and activity negligence. The permeability of counter ions increases with feed ion concentration while the diffusivity sequence was nitrate 4sulfate 4chloride 4dihydrogen phosphate. Since chloride was used in each run an additional dependency could be implemented, which postulates that the diffusion coefficient of chloride should not depend on the feed ion species and therefore should be the same for each bi-ionic system. The fitted AMV diffusivity ratios, between the diffusion coefficient of respective feed ion and the driving ion

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chloride, were found to correlate to a suggested relation (see Eq. 21). It describes ion diffusivity to depend on the reciprocal of their hydrated cross section area multiplied with their comprised charge. It was possible to recalculate each fitted diffusivity ratio and it is a question of further investigation if this correlation could be extended for different ions and membrane materials. This could lead to a new understanding of transport processes of ions through nanochannels of ion exchange membranes.

Acknowledgment The publication was partially supported by the German Federal Ministry of Education and Research (BMBF) through a scholarship for the first author within the Young Scientist Exchange Program (Grant KIT1101/YSEP70) as part of the German-Israeli Research Cooperation in Water Technology. The initiator of the scientific exchange and a constant support was Prof. em. Wolfgang Calmano. The help of Dipl.-Ing. Jan Benecke, in the final phrasing of the manuscript, is highly appreciated. The experimental part of this study was conducted at the Technion, supervised by Prof. em. David Hasson and Prof. Raphael Semiat from the Wolfson Department of Chemical Engineering. Prof. Raphael Semiat provided the Rabin Desalination Laboratory and Prof. em. David Hasson offered a comprehensive introduction into the field of Donnan dialysis and a review of the final manuscript.

Symbols Xi ci cþ x zi Ji Di ki F Rg T φ θ Pi δm K ACl r~ ACl γ Γ r hyd i V RA I Δτ

Equimolar ion-fraction [–] Molar ion conc. [mol/m³] Total equimolar electrolyte concentration [eq/m³] Distance [m] Valence [–] Ion flux [mol/(m² s)] Diffusion coefficient [m²/s] Mass transfer coefficient [m/s] Faraday constant [C/mol] Universal gas constant [J/(mol K)] Temperature [K] Potential [V] Diffusivity ratio between counter ions [–] Permeability [mol/(m² s)] Membrane thickness [m] Selectivity coefficient between counter ions [–] Effective separation factor between counter ions [–] Activity coefficient [–] Activity parameter [–] Hydrated ion radius [nm] Solution volume [m³] Feed ion removal efficiency [%] Ionic strength [mol/m³] Time increment [s]

Subscripts i A Cl

Ion species Feed ion Driving ion chloride

eq 0 cal exp

Equilibrium concentration Initial concentration Simulated data Experimental data

Superscripts F R m w I

Feed solution Receiver solution Membrane phase Bulk phase Interface

References [1] T.A. Davis, Donnan dialysis, Desalin. Water Resour– Membr. Process. II (2010) 230–241. [2] Jacek Wiśniewski, Agnieszka Różańska, Tomasz Winnicki, Removal of troublesome anions from water by means of Donnan dialysis, Desalination 182 (2005) 339–346. [3] Jacek Wiśniewski, Agnieszka Różańska, Donnan dialysis for hardness removal from water before electrodialytic desalination, Desalination 212 (2007) 251–260. [4] Jacek Wiśniewski, Agnieszka Różańska, Donnan dialysis with anion-exchange membranes as a pretreatment step before electrodialytic desalination, Desalination 191 (2006) 210–218. [5] Piotr Długołęcki, Benoît Anet, Sybrand J. Metz, Kitty Nijmeijer, Matthias Wessling, Transport limitations in ion exchange membranes at low salt concentrations, J. Membr. Sci. 346 (2013) 163–171. [6] Eleanor H. Cwirko, Ruben G. Carbonell, A theoretical analysis of Donnan dialysis across charged porous membranes, J. Membr. Sci. 48 (1990) 155–179. [7] Chien-Cheng Ho, Dong-Syau Jan, Fuan-Nan Tsai, Membrane diffusioncontrolled kinetics of ionic transport, J. Membr. Sci. 81 (1993) 287–294. [8] Dong-Syau Jan, Chien-Cheng Ho, Fuan-Nan Tsai, Combined film and membrane diffusion-controlled transport of ions through charged membrane, J. Membr. Sci. 90 (1994) 109–115. [9] Hirofumi Miyoshi, Diffusion coefficients of ions through ion-exchange membranes for Donnan dialysis using ions of the same valence, Chem. Eng. Sci. 52 (1996) 1087–1096. [10] Hirofumi Miyoshi, Diffusion coefficients of ions through ion exchange membrane in Donnan dialysis using ions of different valence, J. Membr. Sci. 141 (1998) 101–110. [11] David Hasson, Adam Beck, Fiana Fingermana, Chen Tachman, Hilla Shemer, Raphael Semiat, Simple model for characterizing a Donnan dialysis process, Ind. Eng. Chem. Res. 53 (2014) 6094–6102. [12] G. Boari, L. Liberti, C. Merli, R. Passino, Exchange equilibria on anion resins, Desalination 15 (1974) 145–166. [13] D. Nwal Amang, S. Alexandrova, P. Schaetzel, Mass transfer characterization of Donnan dialysis in a bi-ionic chloride–nitrate system, J. Chem. Eng. 99 (2004) 69–76. [14] Yuan-Hu Li, Sandra Gregory, Diffusion of ions in sea water and deep sea sediments, Geochim. et Cosmochim. Acta 88 (1974) 703–714. [15] G.B. Berg, I.G. Rácz, C.A. Smolders, Mass transfer coefficients in cross-flow ultrafiltration, J. Membr. Sci. 47 (1989) 25–51. [16] F.G. Donnan, The theory of membrane equilibria, Chem. Rev. 1 (1925) 73–90. [17] J.H.G. Van der Stegen, H. Weerdenburg, A.J. van der Veen, J.A. Hogendoorn, G. F. Versteeg, Application of the Pitzer model for the estimation of activity coefficients of electrolytes in ion selective membranes, Fluid Phase Equilibria 157 (1999) 181–196. [18] Carl H. Hamann, Wolf Vielstich, Elektrochemie, 4th ed., Wiley-VCH, Weinheim (2005) 31. [19] Vernon L. Snoeyink, David Jenkins, Water Chemistry, Wiley, New York, 1980. [20] Yizhak Marcus, Ionic radii in aqueous solutions, Chem. Rev. 88 (1988) 1475–1498. [21] Xuan Tuan Le, Permselectivity and microstructure of anion exchange membranes, J. Colloid Interface Sci. 325 (2008) 215–222. [22] Tim Malewitz, Peter N. Pintauro, David Rear, Multicomponent absorption of anions in commercial anion-exchange membranes, J. Membr. Sci. 301 (2007) 171–179. [23] Klaus Schmidt-Rohr, Qiang Chen, Parallel cylindrical water nanochannels in Nafion fuel-cell membranes, Nat. Mater. 7 (2008) 75–83. [24] Hilla Shemer, Raphael Semiat, Impact of halogen based disinfectants in seawater on polyamide RO membranes, Desalination 273 (2011) 179–183.