Modelling non-stationary ion transfer in neutralization dialysis

Journal of Membrane Science 540 (2017) 60–70

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Modelling non-stationary ion transfer in neutralization dialysis c

A. Kozmai , M. Chérif a b c

a,b

a

a

a

, L. Dammak , M. Bdiri , C. Larchet , V. Nikonenko

c,⁎

MARK

Institut de Chimie et des Matériaux Paris-Est (ICMPE), UMR 7182 CNRS – Université Paris-Est Créteil, 2 Rue Henri Dunant, 94320 Thiais, France Laboratoire de Sciences des Matériaux et de l′Environnement (MESLAB), Université de Sfax pour le Sud, Route de la Soukra km 4 – BP no. 802, 3038 Sfax, Tunisia Membrane Institute, Kuban State University, 149 Stavropolskaya Street, 350040 Krasnodar, Russia

A R T I C L E I N F O

A B S T R A C T

Keywords: Neutralization dialysis Desalination Ion-exchange membranes Nernst-Planck equation Modelling

A non-steady state theoretical model is developed using the Nernst-Planck equations in order to study ion transport kinetics through Ion-Exchange Membranes (IEMs) during water desalination by Neutralization Dialysis (ND) in batch mode. The ND cell under study involves three compartments (acid, saline, and alkali) separated by two membranes (a cation-exchange and an anion-exchange ones) assumed ideally permselective and homogeneous. The presence of Diffusion Boundary Layers (DBLs) at the membrane-solution interface is taken into account in the saline compartment. The results of numerical simulation are compared with known experimental data. A good agreement is obtained between experimental and theoretical values representing the pH and the conductivity of the saline circulating solution as functions of time. These experimental results are also compared with the calculations made using the quasi-steady state model developed by Denisov et al. [10]. It is shown that the quasi-steady state approach is not applicable at the beginning of the ND process, during a few tens of minutes, where the concentration profiles in the membranes are far from linear. Within this stage, a few pH fluctuations are possible, while only one or two pH fluctuations occurs in the quasi-steady stage. The mechanism of these fluctuations, which are determined by the periodical change of the “leadership” between the cationexchange and the anion-exchange membranes, delivering the H+ and the OH– ions into the saline solution, respectively, is discussed.

1. Introduction The Neutralization Dialysis (ND) process was first suggested by Igawa et al. [1] for water desalination. It is based on the simultaneous use of two Donnan dialysis operations: one using acidic solution and a cation-exchange membrane (CEM) and the other one using alkaline solution and an anion-exchange membrane (AEM). The H+ ions penetrate across the CEM and the OH– ions penetrate across the AEM into the desalination compartment in exchange for the mineral ions initially present there, thus allowing the demineralization of the feed solution. This method has certain advantages over existing methods such as Distillation [2], Nanofiltration [3], Reverse Osmosis [4] and Electrodialysis [5]. ND is not technically sophisticated, it demands low investment costs, it is characterized by low energy consumptions; its implantation is easy for isolated locations. Thus, ND can be considered as a promising membrane process convenient mainly to the developing countries, which suffer from the water lack but have access to ground and surface brackish waters [6]. Moreover, many studies [7–9] reveal the efficiency of ND process for desalination of solutions containing organic substances (mono- and

⁎

Corresponding author. E-mail addresses: [email protected] (L. Dammak), [email protected] (V. Nikonenko).

http://dx.doi.org/10.1016/j.memsci.2017.06.039 Received 26 March 2017; Received in revised form 13 June 2017; Accepted 15 June 2017 Available online 19 June 2017 0376-7388/ © 2017 Elsevier B.V. All rights reserved.

oligosaccharides, polysaccharides, proteins). Different applications were considered such as separation of weak acids and bases [8], transport of glycine [9] and desalination of aqueous solutions of carbohydrates and milk whey [7]. However, this process is poorly studied both experimentally and theoretically. It is particularly difficult to maintain a predetermined value of pH of the product, while the applications require a specified value of pH [10–13]. Namely, for drinking water, the pH must be close to 7; for organic solutions, the desired value of pH depends on the form of the resulting product to be obtained. Moreover, low or high pH reduces the ion-exchange rate of desalination [13]. Apart from ND there are a number of other diffusion based membrane processes, such as diffusion dialysis, Donnan dialysis or reverse electrodialysis, which attract more and more attention. Besides, in electrodialysis (ED) concentration when the concentration gradients are significant, ion diffusion transport through the membrane must be taken into account. In the pulsed electric field mode of electrodialysis, during the pauses when the current does not flow, diffusion remains the only transport mechanism. In the great part of the mentioned above processes, transient diffusion is important. However, most of the known

Journal of Membrane Science 540 (2017) 60–70

A. Kozmai et al.

12

theoretical models describing diffusion processes in membrane systems consider only steady or quasi-steady state diffusion. In particular, Denisov et al. [10] have proposed quasi-steady-state models for batch mode of ND; similar models for continuous and batch modes of ND were developed by Sato et al. [11]. The main quasi-steady state assumption is that the rates of the concentration profile formation in the solution and membranes are substantially greater than the rate of change in the inlet concentration. In the present study, we propose a more comprehensive non-steady state mathematical model. This allows describing desalination process in conditions where the quasi-steady state assumption is not valid, especially in the beginning of the process (the first 10 or 15 min). In particular, we examine the impact of the initial state of membranes on the exchange rate. We show that in the conditions of the experiment carried out earlier [13], this initial state was important, and the correct description of its effect allowed us to obtain a good agreement between the theoretical and experimental results.

(a)

10

pH / -

8

6

nonstat D=D(c)

nonstat D=const

4

stat D=const experimental values

2 0

0

20

40

60

80 100 t / min

120

2.5

/ mS cm-1

2.1. System under study

160

180 (b)

nonstat D=D(c) nonstat D=const

2.0

2. Theoretical

140

stat D=const experimental values

1.5 1.0 0.5

A system, consisted of three compartments filled with different solutions (HCl, NaOH and NaCl) separated by a cation- and an anionexchange membranes, is studied (Fig. 1). VA, VB and VD are the volumes of acid, alkali (base) and saline (under desalination) solutions, respectively, circulating through the corresponding compartments and intermediate tanks. Upper indexes A, B and D correspond to acid, base and desalination compartments, respectively. δ is the diffusion boundary layer (DBL) thickness assumed the same near the CEM and AEM. CEM (of thickness dc) and AEM (of thickness da) are supposed to be ideally selective (the co-ion transport is neglected), which is justified in relatively diluted solutions as used in this study. Indeed, the evaluation below shows that the co-ion flux under experimental conditions is about two orders of magnitude smaller than the counterion flux. The evaluation is carried out for the limiting case, where the membrane separates a 0.1 M NaCl solution and water. In this case, as it is known from literature [14,15], the diffusion permeability, P, of Neosepta AMX and CMX membranes is less than 10−8 cm2 s−1. The maximum value of the co-ion flux in the steady state through a CMX or AMX membrane in these conditions calculated using equation J = PΔc / d [14,16] is about 5 × 10−8 mmol cm−2 s−1 (here Δc is the difference in the electrolyte concentration on both sides of the membrane, d is the membrane thickness). Even if we assume that in the case of HCl (CMX) and NaOH (AMX) the diffusion permeability would be one order of magnitude higher than in NaCl solution, the co-ion flux (5 × 10−7 mmol cm−2 s−1) should be nearly 100 times lower than the counterion flux (which has the order of 10−5 mmol cm−2 s−1, Fig. 2c). Convective transport within the diffusion layers is neglected; it is taken into account implicitly by adjusting the DBL thickness.

0.0

0

20

40

60

4.5

4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

J×105 / mmol cm-2 s -1

4.0 3.5 3.0

2.5 2.0 1.5 1.0

Jc nonstat

0.5

Ja nonstat

0.0

80 100 t / min

0

20

40

60

120

140

160

180

(c)

0

80 100 t / min

5

120

10

140

15

20

160

180

Fig. 2. Calculated (using non-steady state (“nonstat”) and quasi-steady state (“stat”) models) and experimental time-dependences of the saline solution pH (a) and conductivity (b) in the desalination compartment as well as flux densities (c) across the CEM (Jc) and AEM (Ja). The values of parameters used in both models are the same and listed in Table A3 of Annex 3. The prehistory of the membranes (see Section 3) were taken into account.

Longitudinal concentration changes along the channel are not taken into account. It is assumed that the ion concentrations in the bulk of compartment D and in the corresponding intermediate tank are the same. In conditions of the experiment described by Cherif et al. [13], these assumptions are justified by a short length of the cell (10 cm) and the pipes (60 cm) used in the hydraulic circuit, as well as a high velocity of solution flowing. Indeed, the time of solution passage through the D compartment (about 5 s) as well as that through the overall circuit (about 40 s) are small compared with the time of one experimental run. During 40 s, the concentration of NaCl in the circuit can change only by 0.4%, which is small compared to the concentration measurement error. Hence, the material balance equations are applied to the circulating solution of volume VD as a whole. In the states close to the quasi-stationary one, ion fluxes across a membrane and two adjacent DBLs are close. As a consequence, concentration gradients as well as absolute variation of concentration in both DBLs adjoining the membrane should be close. In these conditions, the relative variation of concentration in the A or B compartments

Fig. 1. Scheme of the system under study. Concentration profiles are shown schematically for H+ (solid line), Na+ (dashed line), OH– (dash-dot line) and Cl– (points) ions.

61

Journal of Membrane Science 540 (2017) 60–70

A. Kozmai et al. D ∂CNa S (2) JNa = D V ∂t

where the concentration is essentially higher than that in the D compartment, is small. According to our calculations, the absolute variation in concentration in the diffusion layers in all the compartments of the system is no greater than 0.01 M that is 10% for acid and alkali compartments. Hence, it is possible to neglect the diffusion layers in the A and B compartments. At the membrane/solution interfaces, we assume the local thermodynamic equilibrium between exchangeable ions. Additionally, we assume local equilibrium of the water self-ionization reaction: H+ + OH– ⇌ H2O.

−

(1)

(2)

j

∂t

=−

−

A ∂CNa S (1) JNa = A V ∂t

(13)

C ja

C jc

(14)

(15)

At the diffusion layer/bulk boundary, the continuity of concentration is assumed:

(4)

Where Cj is the concentration, Jj is the flux density across the membrane, zj is the charge, Dj is the diffusion coefficient of ion j, X is the membrane exchange capacity, ω can take the values − 1, + 1 and 0 for an AEM, a CEM and a solution, respectively; t is the time, x is the coordinate normal to the membranes surfaces, R, T and F are the gas constant, temperature and Faraday constant, respectively. In the D compartment, all the ions are considered (j = H+, OH–, Na+, Cl–); in the CEM, j = H+, Na+; in the AEM, j = OH–, Cl–. The concentration changes in the A compartment of volume VA are (1) associated with the H+ (JH(1) ) and the Na+ ( JNa ) fluxes through the lefthand boundary of the CEM:

∂CHA S − A JH(1) = V ∂t

a ⋅CCls COH a s CCl⋅COH

(J jDBL1) x = dc = (J jc ) x = dc

∂Jj ∂x

Ka =

(12)

The flux continuity condition at the CEM interface with the saline solution ( x = dc ) leads to Eq. (15):

(3)

the equation of material balance

∂Cj

s CHc ⋅CNa c CNa⋅CHs

Kw = СH ⋅COH = 10−14

the condition of zero current flow

Σzj Jj = 0

Kc =

is the interfacial solution concentration of ion j, and are where the interfacial concentrations of these ions in the CEM and AEM, respectively; K c is the equilibrium coefficient for H+/Na+ exchange at the CEM, and K a is the similar parameter for OH−/Cl− exchange at the AEM interface. The equilibrium in solution between the H+ and OH− ions and the water molecules reads as:

the electroneutrality condition j

(11)

Cjs

⎟

Σzj Cj = ωX

(10)

The Nikolskii equations [17] describing local equilibrium at the membrane/solution boundaries are written as:

The following set of equations describes one-dimensional ion electrodiffusion transport in the membranes and in the diffusion layers occurring in the D compartment: the Nernst-Planck (N-P) equations ⎜

∂CClD S (3) JCl = D V ∂t

S (2) ∂ D (3) D (JH + JOH )= (CH − COH ) VD ∂t

2.2. Problem formulation

∂Cj F ∂φ ⎞ + zj Cj Jj = −Dj ⎛ RT ∂x ⎠ ⎝ ∂x

(9)

(Cj ) x = dc + δ = CjD

(16)

CjD

denotes the concentration of ion j in the saline solution bulk. where Similar boundary conditions are used for the AEM side. The electric potential can be deduced from the Nernst-Planck equations. Eqs. (1)–(3) give the following equations for the CEM ∂C c

c − DHc ) ∂xH (DNa ∂φc RT = ⋅ c c c ∂x F (DH − DNa ) CHc + DNa Xc

(17)

and for the AEM ∂C a

(5)

a a − DCl (DOH ) ∂OH ∂φa RT x = ⋅ a a a a a ∂x F (DOH − DCl ) COH + DCl X

(6)

∂φ DBL1, DBL2

(18)

Similarly, for the DBLs in the D compartment, we have:

∂x

where S is an active membrane surface area. The concentration changes in the B compartment of volume VB are (4) (4) associated with the OH– (JOH ) and the Cl– ( JCl ) fluxes through the righthand boundary of the AEM: B ∂COH S (4) JOH = B V ∂t

(7)

∂CClB S (4) J = V B Cl ∂t

(8)

⋅

=

s DOH

RT F DBL1, DBL2 ∂COH

∂x s DBL1, DBL2 DOH COH

+ DCls +

DBL1, DBL2 ∂CCl

∂x DCls CClDBL1, DBL2

− DHs +

DBL1, DBL2 ∂CH

∂x DHs CHDBL1, DBL2

s − DNa

DBL1, DBL2 ∂CNa

∂x

s DBL1, DBL2 + DNa CNa

(19) ∂φ DBL1, DBL2 ∂x

is then expressed through the concentrations of two sorts of ions by applying Eqs. (2) and (14). For the diffusion layer near the CEM, we obtain:

∂φ DBL1 RT = ∂x F

Note that a flux is considered as positive, if its direction coincides with the positive direction of axis x. The concentration changes in the saline solution of volume VD circulating through the D compartment (Fig. 1) and an intermediate tank are determined by the fluxes through the right-hand CEM (denoted by upper index 2) and the left-hand AEM (denoted by upper index 3) boundaries as described by Denisov et al. [10]:

⋅

⎡DCls − DHs + ⎢ ⎣ s (DOH

−

Kw DBL1) (CH

Kw DCls ) C DBL 1 H

2

s (DCls − DOH )⎤ ⎥ ⎦

+

(DHs

+

DBL1 ∂CH

∂x

DCls ) CHDBL1

s ) + (DCls − DNa

+

s (DNa

+

DBL1 ∂CNa

∂x

DBL1 DCls ) CNa

(20) and near the AEM we obtain: 62

Journal of Membrane Science 540 (2017) 60–70

A. Kozmai et al.

∂φ DBL2 RT = ∂x F

The initial concentration of NaCl solution circulating through the D compartment and an intermediate tank is 0.02 mol L−1, while the concentration of HCl and NaOH solutions are both equal to 0.1 mol L−1 as used by Cherif et al. [12]. The volume of saline solution is 0.5 L, and that of acid and alkali solutions is 1.5 L. The temperature during experiments was kept at 25 °C. The experiment was performed as follows. Initially the membranes were pretreated in compliance with French standard NF X 45-200, which ended with soaking the membranes in a 0.1 M NaCl solution. Further the membranes were taken out from the solution and installed in the cell. For one hour, the cell was fed with working solutions: a 0.1 M HCl and a 0.1 M NaOH solutions were circulated through the A and B compartments, respectively; a 0.02 M NaCl solution, through the D compartment. Next, the solutions were replaced by fresh ones and the registration of kinetic dependencies of pH and conductivity started.

DBL2

DBL2

Kw s s s s ⎤ ∂COH s ∂CCl ⎡DOH − DNa + DBL ) ∂x + (DCls − DNa 2 (DH − DNa ) ∂x (COH 2) ⎢ ⎥ ⎦ ⋅⎣ s K w s s s DBL2 s (DH − DNa ) C DBL + (DNa + DCls ) CClDBL2 2 + (DOH + DNa ) COH OH

(21) Then Eqs. (17)–(21) are substituted in Eq. (1) written for the CEM, AEM and the diffusion layers, respectively. The system of the partial differential equations (presented in Annex 1) obtained for fluxes, together with Eq. (4) are then solved numerically. The boundary-value problem above requests initial conditions related to the beginning of the operation. At the initial moment of time (t = 0), the concentration distribution in the diffusion layers is uniform and equal to the initial feed solution concentration. In the membranes, the concentration distribution depends on their prehistory. Generally, the initial conditions can be written as:

Сj (x )t = 0

0 c c ⎧C j , d ≤ x ≤ d + ⎪ 0 ⎪C j , 0 ≤ x ≤ δ for = ⎨C jc (x ), 0 ≤ x ≤ dc ⎪ a ⎪C j (x ), δ ≤ x ≤ da ⎩

4. Results and discussions

δ for the DBL near the CEM

4.1. Evaluation of the model parameters

the DBL near the AEM The exchange capacity of the CEM and AEM in the swollen state, Xwet , was found from the data, given by Chérif et al. [13] for the membranes in dry state, Xdry , as

for the CEM + δ for the AEM

(22)

where C j0 is the initial concentration of ion j in feed solution, functions C jc (x ) and C ja (x ) depend on the membrane state at t ≤ 0. These functions will be specified in Section 4.3.

Xwet = Xdry ρwet (1 − W )

where W is the water content, ρwet is the membrane density in swollen state. The values of the exchange capacity and other parameters used in Eq. (24) are gathered in Table 1. The individual ion diffusion coefficients in the membranes, Djk , are

2.3. Numerical solution of the non-steady state problem Both membranes, as well as the diffusion boundary layers, were discretized by elementary segments, when the uniform mesh was used. For each segment of the system we applied the algorithm first used by Larchet et al. [18] and described in details by Mareev et al. [19]. This algorithm implies the use of explicit Euler finite-difference approximation of derivatives. The time step, Δt, was determined in a way to satisfy the Courant–Friedrichs–Lewy condition (Δt has the value less than or equal to 10−5 s):

Δt ≤

(min(Δx ))2 max(Dj )

κ kRT

determined by using the relationship Djk = k 2 [20,21], where index k X F denotes the type of membrane (k = a, c); the membrane exchange c capacity X k and the conductivity κ k are known (Table 1): DNa = a a 3.6×10−7, DHc = 2.7 × 10−6, DCl = 3.0 × 10−7, DOH = 9.6 × 10−7 (in cm2 s−1). These values are coherent with those found by Denisov et al. [10] for the same type of ion-exchange membranes. The DBL thickness is mainly affected by the hydrodynamic conditions and the spacer type. Its estimation is based on the fact that in relatively long channels (8 cm in our case) without spacer, according to convective-diffusion model [22], δ is about 1/3 of the intermembrane spacing (equal to 450 µm), hence, about 150 µm. Secondly, we took into account the role of spacer: it is known [23] that the limiting current density in an electrodialysis desalination channel with a spacer (similar to that used in the current study) is about 1.8 times higher than the limiting current density in a channel without spacer at the same flow rate. Thus, the value of δ should be close to 150/1.8 ≈ 80 µm. The values of the Nikolskii constants are taken equal to unity. In fact, K a, c depends on the nature of ions. An ion-exchange membrane absorbs preferably counterions with smaller hydration radius, i.e. Na+ is absorbed better than H+, and Cl– better than OH– by cation-exchange and anion-exchange membranes, respectively [24,25]. However, for the sake of simplicity we use the assumption of an ideal ion-exchange membrane (no co-ions, no selectivity in ion exchange, constant diffusion coefficients). As for the solution, we consider two cases: when the ion diffusion coefficients, Djs , are independent of the concentration, and when Djs are functions of the local ionic strength. In the last case, we use the NernstHartley relation [26]

(23)

where Δx is the spatial step, Dj is the diffusion coefficient in membrane or in solution. 3. Experimental The ND process was carried out with the use of cation exchange (Neosepta® CMX, Astom, Japan) and anion exchange (Neosepta® AMX, Astom, Japan) membranes. Their main characteristics are given in Table 1. The membrane active area is 8 × 8 cm2. The intermembrane space is 0.045 cm (equal to the thickness of spacer). The liquid volumetric flow rate is 35 mL min−1; the corresponding linear flow velocity is 1.45 cm s− 1. Table 1 Physicochemical characteristics of membranes under study (experimental data). Membranes

CMX

AMX

Thickness, cm [13] Water content (wt%) [13] Membrane density (g cm−3 wet membrane) (our data) Ion-exchange capacity (meq g−1 dry membrane) [13] Ion-exchange capacity (meq cm−3 wet membrane) (see Section 4.1) Conductivity (mS cm−1) in 0.05 M NaCl [13] Conductivity (mS cm−1): CMX in 0.05 M HCl, AMX in 0.05 M NaOH [13]

0.017 25 1.18 1.6 1.43

0.014 26 1.05 1.3 1.28

2.1 16.0

1.2 4.7

(24)

Djs = Dj0 g (Cj )

(25)

where Dj0 is the diffusion coefficient of ion j at infinite dilution; g is the activity factor [26]. The activity factor, g, can be found from the experimental data on average activity coefficients and molar concentrations of NaCl and HCl solutions. For the dilute solutions up to 0.02 mol L–1, g may be 63

Journal of Membrane Science 540 (2017) 60–70

A. Kozmai et al.

a, 3 = X a . In this case, the logamembranes, hence, CHc, 2 = X c and COH rithmic terms in Eqs. (27) and (28) are zero and, according to these c a = Jstat = 0 . Thus, no exchange occurs in this case. equations, Jstat Eqs. (27) and (28) can be applied after a certain time passes from c, 2 = X c, the beginning of operation. If we use conditions CHc,2 < < CNa a,3 a, 3 a COH < < CCl = X , which relates to the case where initially the concentrations of H+ and OH– ions are negligible in the saline solution, and substitute the numerical values of Djc, a , dc, a , X c, a from Table A3 (see c = 7.04 Annex 3) in Eqs. (27) and (28), we find in the case of CMX Jstat a × 10−5 mmol cm−2 s−1, and in the case of AMX, Jstat = 4.63 × 10−5 mmol cm−2 s−1. The obtained values are overestimated, since the initial conditions are artificial: really the boundary concentrations c,2 change very quickly and very soon CHc,2 becomes comparable with CNa , a,3 a,3 and COH with CCl . Numerical calculation of the fluxes at t = 1 min (which relates to the end of the essentially unsteady state) in the conditions described in the legend of Fig. 2 gives J c = 3.9 × 10−5 mmol cm−2 s−1 and J a = 1.5 × 10−5 mmol cm−2 s−1 (Fig. 2с). The calculations using Eqs. (27) and (28) and the boundary conc,2 centrations found numerically for t = 1 min (CHc,2 = 0.784, CNa = c a,3 a,3 = 0.675, CCl = 0.605, all in mmol cm−3), give Jstat = 1.74 0.647, COH a × 10−5 mmol cm−2 s−1, Jstat = 1.57 × 10−5 mmol cm−2 s−1. These values are already closer to those found numerically. The estimations above as well as the results shown in Fig. 2c show that in the beginning of ND process, the exchange rate of H+ and Na+ ions across the CEM is higher than that of OH– and Cl– ions across the AEM, in the used conditions. This difference in fluxes shifts the acidbase balance in the D compartment: pH decreases there at the beginning of operation (Fig. 2a). Simultaneously, the concentration of Na+ ions in the saline compartment decreases faster than that of Cl– ions (Fig. 3). Low concentration of Na+ ions and high concentration of H+ ions in the D compartment (and consequently the corresponding concentrations in the membrane at the interface with the D compartment, CHc,2 c,2 and CNa ) result in a decrease of the cation exchange rate across the CEM, in accordance to Eq. (27). On the contrary, relatively high concentration of Cl– ions and low concentration of OH– ions in the D compartment leads to an increase of the anion exchange rate across the AEM, Eq. (26). As a result, at about t = 20 min the fluxes J c and J a become equal (Fig. 2c), and then J a is greater than J c . Consequently, the tendency in the variation of pH changes: its value starts to increase (Fig. 2a). When the pH value approaches the neutral one (approx. at 95 min, Fig. 2a), the rate of pH variation sharply increases. However, a higher exchange rate of OH– and Cl– ions causes a more rapid decrease in concentration of Cl– ions in comparison with Na+ ions and the situation changes again: a greater mole fraction of Na+ ions in the solution determines a higher contribution of H+ and Na+ exchange

considered as independent on the electrolyte nature and approximated (using the approach suggested by Larchet et al. [16]) as

g = 0.7396I − 0.5184I 1/2 + 0.9977

(26)

0.5 ∑ Cj z j2

where I = is the ionic strength, Cj is the local molar concentration of ion j (H+, Na+, OH–, Cl–). 4.2. Model verification. Comparison to quasi-steady state model The experimental and computed time dependencies of electric conductivity and pH of the saline solution circulating through the D compartment (Fig. 1) is shown in Fig. 2a–c. Both quasi-steady and nonsteady models are applied. When applying the non-steady state model, the prehistory of the membranes described in Section 3 is taken into account (the details are given below). Similarly to Denisov et al. [10], we obtain equations for the steadystate fluxes through the membranes and adjacent diffusion layers. The fluxes are expressed through the ion concentrations in the solution bulk and at the membrane surface. For this purpose, we use Eqs. (1)–(3), (12), (13). The resulting equations are presented in Annex 2. Fig. 2c shows the time dependencies of ion flux densities through the CEM (J c ) and the AEM ( J a ) at the membrane/saline solution interfaces calculated numerically using Eqs. (A1.1)–(A1.4). J c and J a are the quantitative characteristic of ion exchange rate across the corresponding membrane. It can be seen from Fig. 2с that in the beginning of the process, the ion fluxes vary rapidly until the system attains a quasi-steady state. The stage of essentially unsteady state lasts about 10 min in the conditions described above. Within this stage, the concentration profiles in the membrane and solution approach a nearly linear form. The form of concentration profile may not be perfectly linear even in steady state because the (mutual) diffusion coefficients in solution and the membranes may be dependent on the local concentration. In order to better understand the behavior of the membrane system in neutralization dialysis, we begin our analysis by considering the system evolution starting from the time when the stage of essentially unsteady state is finished. In conditions used in calculations of the curves presented in Fig. 2, this time may be evaluated as t ≈ 10 min. c Under the assumption of quasi-steady state, the values of Jstat and a Jstat can be expressed as follows [10,13,24]: c Jstat =

c DHc DNa Xc c c c d (DH − DNa )

a Jstat =

a a a DOH DCl X Da Xa ⎞ ln ⎛⎜ a a, OH a a a a, 3 ⎟ da (DOH ) ⎝ DOH COH 3 + DCl CCl ⎠ − DCl

CHc, 2 +

c, 2 (CNa )

ln ⎜⎛ c c, ⎝ DH CH

a, 3 and COH – –

DHc X c ⎞ c 2 c, 2 ⎟ + DNa CNa ⎠

(27)

(28)

0.02

a, 3 (CCl )

C / mmol cm-3

where are the boundary concentrations of H+ (Na ) and OH (Cl ) ions in the membrane at the interface with the c, 2 a, 3 a, 3 D compartment, respectively; CHc, 2 , CNa , COH , and CCl depend on the boundary concentrations in the saline solution through the Nikolskii equation, (Eqs. (12) and (13)). According to Eqs. (27) and (28), the rate of ion-exchange across a membrane depends on two factors. The first one, which is before the logarithm, involves the diffusion coefficients, the ion-exchange capacity and the membrane thickness and does not depend on the saline solution composition. As it can be seen from Eqs. (27) and (28), the greater exchange capacity, X, the greater the counterion flux, J, and, on the contrary, the bigger the membrane thickness, d, the lower J. The increase in J can be achieved if the values of both counterion diffusion coefficients in membrane are high and close. The second one is a function of the saline solution composition. This logarithmic term increases with increasing the ratio of salt to H+/OH– ions concentration in the saline feed solution. If the salt concentration D in the feed solution is zero, CNa , CClD = 0, the salt ions are absent in both

0.002

Cl-

0.0195

0.0015

Na+

0.001

0.019 H+

0.0185 0.018

0

1

2

3

0.0005

4

5

0

t / min Fig. 3. Calculated (using non-steady state model) time-dependences of ions concentrations in the D compartment for the concentration distribution in membranes obtained by simulation of the processes prior the beginning of measurements. The moment of time corresponding to the maximum in conductivity in Fig. 2b is shown by the vertical dashed line.

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(Fig. 2c). As a result, in the case considered in Fig. 2a, the pH value in the time interval from 120 min to 180 min remains nearly constant showing that in the given conditions the exchange rates across the CEM and AEM are nearly the same. According to Eqs. (27) and (28), the final pH in the D compartment should be 7. This state relates to the situation where the concentration of Na+ in the CEM and that of Cl– in the AEM tend to zero. Then, as c a explained above, the exchange rates Jstat and Jstat are both equal to zero. As Fig. 2 shows, there is a good agreement between simulations and the experiment in the case where the non-steady state model is used and the concentration dependence of diffusion coefficients, D = D (C ) , is taken into account. However, when applying other assumptions, the agreement is not so good. As it can be seen from Fig. 2b, the nonlinear behavior of electrical conductivity at the beginning of the process cannot be described using a steady-state model, in particular, it is impossible to describe the maximum. This maximum is explained by the fact that in the initial period of time, the exchange rate of Н+/Na+ across the CEM is greater than that of ОН–/Cl– across the AEM (Fig. 2c). This leads to a rapid decrease in the saline solution pH (Fig. 2a). The Na+ ions are replaced by the essentially more mobile H+ ions, while the decrease in concentration of Cl– ions is insignificant (Fig. 3). As a result, the electrical conductivity of saline solution increases during the first 3 min. When applying the non-steady state model, the prehistory of the membranes was taken into account through the appropriate initial (t = 0) distribution of concentrations. According to the measurement procedure (Section 3), initially the CEM and the AEM installed in the cell were transferred in the Na+ and in the Cl– forms, respectively. Then the desalting process was carried out for one hour without registration of pH and conductivity. Only after these stages the registration was started. The distribution of concentrations in the membranes at the beginning of registrations, t = 0 was, therefore, a consequence of the membranes’ previous life. This distribution of concentrations was found by computation using the non-steady state model in the time interval from t = − 60 min to t = 0 starting from the initial state of the membranes in the Na+ and in the Cl– forms. The evolution of concentration profiles in membranes from t = − 60 to t = 0 min is shown in Fig. 4, and in the time interval from t = − 60 to t = 120 min in Fig. A4, Annex 4. In the case of quasi-steady state model, the distribution of concentrations in the membranes is determined by the condition of independence of flux densities from the coordinate within a membrane and the adjacent DBLs. Since this assumption does not hold at the beginning of data recording, the prehistory of membranes cannot be taken

Fig. 5. Time-dependences of ion fluxes through the CEM ( J c ) and the AEM ( J a ) computed using the non-steady state (“nonstat”) and the quasi-steady state (“stat”) models under the initial conditions described above in the text.

into account when applying the quasi-steady state model. In the proposed non-steady state model, we take into account the concentration dependence of electrolyte diffusion coefficients, D(C). In this case there is a good agreement between the computations and experimental data. If D(C) is not taken into account, in the initial period of time just after beginning of data recording, the values of diffusion coefficients used in calculations relate to the condition of infinite dilution. These values are too high as compared to the real ones. Consequently, the calculated rate of exchange process is faster than that observed in the experiment. As a result, the rate of acidification of saline solution and its conductivity calculated under the assumption of constant D are too high as compared to the experimental data (Fig. 2a and b). However, the quasi-steady state model is adequate at time scales substantially greater than the duration of concentration profiles formation in membranes. Fig. 5 shows a comparison of kinetic dependencies of fluxes across the CEM and the AEM calculated with the help of the steady state and the non-steady state models. Calculations are performed assuming that the initial state of the system in both cases is a quasi-steady state where the ion concentrations in the solution bulk are taken the same for both models as well as the same concentrations at the DBL/membrane interfaces. 4.3. Effect of membrane initial state One of the initial membrane states, namely, that corresponding to the experiment procedure, is discussed earlier (Fig. 2). In this section we will examine the time dependences of saline solution pH and conductivity, depending on various initial distributions of counterion concentration in the membranes. Four possible initial concentration distributions in the membranes are analysed below, namely, when the CMX and the AMX membranes are either in Na+ and Cl–, H+ and OH–, H+ and Cl– or Na+ and OH– forms, respectively. For the sake of simplicity, further calculations are performed under the assumption that the ion diffusion coefficients are independent of concentration. As described above, in the steady state model, the exchange rate across the membrane is characterized by J k (k = c, a), expressed by Eqs. (27) and (28). At the beginning of the process, this value is of the order of 10−5 mmol cm−2 s−1, and for the CEM it is greater than that for the AEM. 4.3.1. Influence of initial membrane state on solution pH in D compartment At the beginning of ND process, when it is substantially non-stationary, Eqs. (27) and (28) are not fulfilled. The pH change in the D compartment is due to the difference between the quantities J c ( x = dc ) and J a ( x = δ ) which determine the amount of H+ and OH– ions

Fig. 4. Computed (using the non-steady state model) concentration profiles of H+ (dashed line) and Na+ (solid line) ions in the CEM, and OH– (dashed line) and Cl– (solid line) ions in the AEM for the time prior to registration, the moments of time are indicated at the corresponding curves (in minutes). Here Cj/Xk is the dimensionless concentration, where for the CMX k = c and j = H+, Na+; for the AMX k = a and j = OH–, Cl–.

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this initial period. After reaching the quasi-steady state (which requires up to 20 min, as in the case of Na+ and OH– forms), the initial form of membranes has a negligible effect, and the process develops as described in details in Section 4.2. The important fact here is that the dominance of the exchange through one of the membranes in the first instants generates strong pH variations, which accelerates the ion exchange through the other membrane. When the “leadership” passes to the other membrane, pH varies in the opposite direction, and that results in stimulation of the exchange through the membrane, which was the leader initially. Thus, the “leadership” may pass form one membrane to the other several times; each time there is a change in the direction of pH variation (Fig. 6a and b). The resulting pattern is very similar to those observed in the biological evolution of systems, consisting of predators and preys [27]. In the case when membranes initially are in H+ and OH– forms in the very beginning of the process fluxes through the CEM and AEM take very high and close absolute values (of the order Jtk→ 0 ~ 10−3 mmol cm−2 s−1) which is two orders of magnitude greater than in the case of Na+ (CEM) and Cl– (AEM) forms. This leads to more frequent oscillations in the pH compared to other cases. These frequent oscillations are explained by the fact that at the beginning of ND process not the whole membrane takes part in the ion exchange, but only its surface layer bathing the solution in the D compartment. The thickness of this layer increases with time, and consequently, the period of pH oscillation increases rapidly from a few seconds up to tens of minutes when the quasi-steady state occurs (Fig. 6b). Kinetic dependencies of the electrical conductivity of the solution for the four different initial states of membranes described above are shown in Fig. 6c. It follows from our simulation, that the pH of saline solution may oscillate with time. In particular, as it can be seen in Fig. 6b, in the case where the membranes are initially in the H+ and OH– forms, the saline solution pH takes the value 7 at least four times in course of an initially non-steady state ND process. However, if the process is assumed quasisteady state from the beginning, the saline solution pH can take the value 7 no more than two times, as it was theoretically shown by Denisov et al. [10]. The reason is that in non-steady state, in the membrane, only a thin near-surface layer bathing the solution in the D compartment determines the ion-exchange rate. In the steady state, the ion exchange is governed by the concentration profile over the whole membrane. The concentration profile within a near-surface layer can change quickly, which explains the possibility of rapid variation of the flux through the membrane interface, hence, a rapid variation of saline solution pH.

Fig. 6. Calculated (using the non-steady state model) time-dependences of the saline solution рН (a), pH and fluxes through the CEM ( J c ) and the АEМ ( J a ) (b), and conductivity of solution (c) in the D compartment for various initial states of the CEM and the AEM.

5. Conclusion In this paper, a non-steady state model of ion transport occurring in the course of neutralization dialysis with two ion-exchange membranes is developed. The values of the parameters used for simulation are found from independent measurements of membrane properties (exchange capacity and membrane conductivity) or evaluated from literature data (diffusion boundary layers thickness). A good agreement is found between the simulated and experimental curves representing the pH and electric conductivity vs. time in the saline solution. We have compared this model with the quasi-steady state model developed by Denisov et al. We have found that the difference between these two models occurs at the beginning of ND process where the effect of initial membrane state is important. Four different cases of initial concentration distributions in the membranes are considered. It is found that in the case where the CEM is in the H+ form and the AEM is in the OH– form, high exchange rates across the surface layers of these membranes bathing the saline solution result in a few рН fluctuations of the saline solution. These fluctuations cannot be described by the quasisteady state model. However, the two models describe well the pH fluctuations occurring in the saline solution after the concentration

incoming to the D compartment, respectively. In the case, where initially the CEM is in the Na+ (or Н+) form and the AEM is in the Cl– form, at relatively short times of the ND process (< 10 min), J c > J a . This relation is also performed when the quasic a steady state is established: Jstat > Jstat , Eqs. (27) and (28). This causes a decrease in pH at the beginning of the process (Fig. 6a). When the CEM is in the Н+ form, at the beginning of the process the gradient of H+ ions at the CEM/saline solution boundary is very high and J c > > J a , hence, the pH of the saline solution decreases much faster than in the case where the CEM in the Na+ form. In the case of Na+ and OH– forms, at the beginning of the process J c < < J a and the pH of saline solution rapidly becomes alkaline. This happens since the H+ ions require much time to pass from the acid compartment through the CEM into the D compartment, while the OH– ions do it immediately. In all three cases considered above, the exchange across one of the membranes is essentially dominant approximately for the first 10 min. This results in a persistent trend of pH change in the D compartment: the pH change depends on which flux (H+ or OH–) is predominating in 66

Journal of Membrane Science 540 (2017) 60–70

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systems, at least in the case of ND. One can assume that these equations and parameters will be also important when describing other membrane based diffusion processes, such as diffusion dialysis, Donnan dialysis or reverse electrodialysis. In particular, it was shown that a special attention must be paid to the concentration dependence of ion diffusion coefficients in solution when there are rapid changes in solution pH. Thus, the theory developed in this paper can be also applied in this novel technique of clean energy generation.

profiles in the membrane approach a linear form (about 20 min after starting the operation). The oscillations in the pH saline solution are linked with the delay in variations of the concentration profiles in the membranes after changing saline solution composition. When ion exchange across one of the membranes is dominant, this generates strong pH variations in the solution, which accelerates the ion exchange through the other membrane. However, the passage of the “leadership” from one membrane to the other requires some time during which the pH continues to vary. Finely, the increasing rate of ion exchange across the other membrane results in the change in the direction of pH variation. Note that the electrodiffusion processes governing the neutralization dialysis are similar to those occurring in reverse electrodialysis [28,29]: in both cases, the driving force is the concentration gradient across a membrane. The results of our study allow elucidating what transport equations and what membrane parameters govern diffusion kinetics in membrane

Acknowledgements The study has been made within the French–Russian International Associated Laboratory “Ion-Exchange Membranes and Related Processes”. The authors are grateful to CNRS, France, and to RFBR, Russia (Grant #15-08-04522-a), for their financial support.

Annex 1 Resulting system of partial differential equations for fluxes in the case of non-steady state model Obtained differential equations for the fluxes in a CEM (upper index “c”) and in the saline DBL adjacent to this membrane (upper index “DBL1”), reads as: ∂C c

c ⎛ ∂C jc (DNa − DHc ) ∂xH ⎞ J jc = −Djc ⎜ + zj C jc c c c (DH − DNa) CHc + DNa X c ⎟⎟ ⎜ ∂x ⎝ ⎠

⎛ J jDBL1

=

⎜ −Djs

∂CjDBL1

⎜ ⎜ ⎝

∂x

+

⎡DCls − DHs + ⎢

zj CjDBL1 ⎣

(A1.1)

Kw DBL1) (CH

2

s (DCls − DOH )⎤ ⎥ ⎦

DBL1 ∂CH

∂x

s ) + (DCls − DNa

DBL1 ∂CNa

∂x

K

w s s s DBL1 s DBL1 (DOH − DCls ) C DBL + (DNa + DCls ) CNa 1 + (DH + DCl ) CH H

+

⎞ ⎟ ⎟ ⎟ ⎠

(A1.2)

+

where j = H , Na . Similarly, in the case of AEM (upper index “a”) and the adjacent DBL (upper index “DBL2”), we find:

J ja

=

⎛ ∂C ja

−Dja ⎜

⎜ ∂x ⎝

+

zj C ja

a ∂COH

a a (DOH − DCl ) a (DOH

−

∂x

a a DCl ) COH

+

⎞

a a⎟ DCl X ⎟

(A1.3)

⎠ DBL2

DBL2

J jDBL2

Kw s s s s ⎤ ∂COH s ∂CCl ⎛ ⎡DOH − DNa + DBL ) ∂x + (DCls − DNa 2 (DH − DNa ) ∂x (COH 2) ∂CjDBL2 ⎢ ⎥ ⎦ s⎜ DBL2 ⎣ + z j Cj = − Dj Kw s s s DBL2 s ⎜ ∂x (DHs − DNa ) C DBL + (DNa + DCls ) CClDBL2 2 + (DOH + DNa ) COH OH ⎜ ⎝ –

⎞ ⎟ ⎟ ⎟ ⎠

(A1.4)

–

j = OH , Cl . Annex 2 Solution to the steady-state problem For the CEM and adjacent diffusion layer from the desalination compartment side, we find:

Jc =

c 2DHc DNa Xc c d

⎛1 + ⎝

A CH

c ⎡⎛ (DHc − DNa ) 1+ ⎢⎝ ⎣

D s, D CNa = CHD + CNa −

CHs, D =

A CNa

−1

⎞ ⎠

A CNa A CH

− ⎛1 + ⎝

−1

⎞ ⎠

s, D CNa

−1

⎞

s, D CH ⎠

+ ⎛1 + ⎝

s, D CNa

−1

c ⎞ ⎤ + 2DNa ⎥ ⎦

s, D CH ⎠

(A2.1)

J cδ − CHs, D DHs , Na

(A2.2)

D 2DHs , Na (CHD + CNa )(DHs CHD + J c δ ) − (J c δ )2 s s D 2DH (DH , Na (CHD + CNa ) − J cδ )

(A2.3)

where

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Journal of Membrane Science 540 (2017) 60–70

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DHs , Na =

s 2DHs DNa s DHs − DNa

For the AEM and adjacent diffusion layer from the desalination compartment side, we find:

Ja =

a a a 2DOH DCl X a d

⎛1 + ⎝

J aδ s DOH , Cl

−1

− ⎛1 + ⎝

⎞

B COH ⎠

a a ⎡⎛ − DCl (DOH ) 1+ ⎢⎝ ⎣

D CCls, D = COH + CClD −

s, D COH =

B CCl

B CCl B COH

−1

⎞ ⎠

s, D CCl

−1

⎞

s, D COH ⎠

+ ⎛1 + ⎝

s, D CCl s, D COH

−1

a ⎞ ⎤ + 2DCl ⎠ ⎥ ⎦

(A2.4)

s, D − COH

s D D s D 2DOH , Cl (COH + CCl )(DOH COH + s s D D 2DOH (DOH , Cl (COH + CCl )

(A2.5)

J aδ ) − (J aδ )2 − J aδ )

(A2.6)

where s DOH , Cl =

s 2DOH DCls s DOH − DCls

To find J a, c , systems of Eqs. (A2.1)–(A2.6) are solved numerically using the Newton-Raphson method. J a, c are then substituted into Eqs. (5)–(11) of ND kinetics, and the latter are solved numerically with use of an explicit finite difference scheme. As can be seen from Eqs. (A2.1)–(A2.6), fluxes through membranes are determined by the ion concentrations in the bulks of compartments as well as by the DBL thickness near the membrane surface. Thus, when the bulk concentrations and the DBL thickness are given, one obtains a single possible distribution of concentrations in membranes, namely, concentrations in the interfaces from the both sides of membranes. Annex 3 See Table A3

Table A3 Model parameters. Parameter

Value

Description −6

2 −1

DHc

2.7 × 10

c DNa a DOH a DCl

3.6 × 10−7 cm2 s−1 9.6 × 10−7 cm2 s−1

δ

80 µm

da and dc

140 and 170 µm

DH0

9.3·10−5 cm2 s−1

Diffusion coefficients of H+ and Na+ ions in the CEM

cm s

Diffusion coefficients of OH– and Cl– ions in the AEM

3.0 × 10−7 cm2 s−1

0 DOH 0 DNa 0 DCl A CH B COH

5.3·10−5 cm2 s−1

D CNa D CCl CHD D COH Xc

0.02 mmol cm−3

Thickness of the DBL adjacent to the CEM in the desalination compartment assumed the same as that adjacent to the AEM Thickness of the AEM and CEM, respectively Diffusion coefficients of H+, OH–, Na+ and Cl– ions in solution at infinite dilution

1.33·10−5 cm2 s−1 2.03·10−5 cm2 s−1 0.1 mmol cm−3 0.1 mmol cm

Initial concentrations of H+ and OH– ions in the acid and alkaline compartments, respectively Initial concentrations of Na+, Cl–, H+ and OH– ions in the desalination compartment

−3

0.02 mmol cm−3 10−6 mmol cm−3 10−8 mmol cm−3

VA

1.43 mmol cm−3 1.28 mmol cm−3 1500 cm3

VB

1500 cm3

VD

500 cm3

Exchange capacity of the CEM and AEM Solutions volumes circulating through the acid (A), alkaline (B) and desalination (D) compartments

S F R T

64 cm2 96.485 C mmol−1 8.314·10−3 J mmol−1 K−1 298 K

Membrane working surface area Faraday constant Gas constant Absolute temperature

Xa

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Annex 4 Concentration profiles of ions H+, Na+ in the CEM and in the DBL1, as well as OH–, Cl– in the AEM and in the DBL2 for different moments of time (numbers on the plots, in minutes) computed using non-steady state model are shown on Figs. A4.1 and A4.2. Membranes was initially in Na+ (for the CEM) and Cl– (for the AEM) forms. Fig. A4.1. Calculated profiles for H+ and OH– ions concentration in the membranes (a) and the DBLs (b). The numbers indicate the time (in minutes) passed from the beginning of operation.

Fig. A4.2. Calculated profiles for Na+ and Cl– ions concentration in the membranes (a) and the DBLs (b). The numbers indicate the time (in minutes) passed from the beginning of operation.

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1260. [15] Е А Shutkina, Е Е Nevakshenova, N.D. Pismenskaya, S.A. Mareev, V.V. Nikonenko, Diffusion permeability of the anion-exchange membranes in sodium dihydrogen phosphate solution, Condens. Matter Inter. 17 (2015) 566. [16] C. Larchet, B. Auclair, V. Nikonenko, Approximate evaluation of water transport number in ion-exchange membranes, Electrochim. Acta 49 (2004) 1711. [17] B.P. Nikolskii, Theory of the glass electrode. I. Theoretical, J. Phys. Chem. 10 (1937) 495. [18] C. Larchet, S. Nouri, B. Auclair, L. Dammak, V. Nikonenko, Application of chronopotentiometry to determine the thickness of diffusion, Adv. Colloid Interface 139 (2008) 45. [19] S.A. Mareev, V.V. Nikonenko, A numerical experiment approach to modeling impedance: application to study a Warburg-type spectrum in a membrane system with diffusion coefficients depending on concentration, Electrochim. Acta 81 (2012) 268. [20] V.I. Zabolotsky, V.V. Nikonenko, Effect of structural membrane inhomogeneity on transport properties, J. Membr. Sci. 79 (1993) 181. [21] L. Dammak, R. Lteif, G. Bulvestre, G. Pourcelly, B. Auclair, Determination of the diffusion coefficients of ions in cation-exchange membranes, supposed to be homogeneous, from the electrical membrane conductivity and the equilibrium quantity of absorbed electrolyte, Electrochim. Acta 47 (2001) 451. [22] N.P. Gnusin, V.I. Zabolotskii, V.V. Nikonenko, M.K. Urtenov, Convective-diffusion model of electrodialytic desalination, limiting current and diffusion layer, Sov. Electrochem + 23 (1986) 298. [23] V.V. Nikonenko, N.D. Pismenskaya, V.I. Zabolotskii, Mass-transfer in narrow slit channel with spacer, Rus. J. Electrochem + 28 (1992) 1982. [24] F.G. Helfferich, Ion Exchange, McGraw-Hill, New York, 1962. [25] C. Bessière, L. Dammak, C. Larchet, B. Auclair, Détermination du coefficient d′affinité d′une membrane échangeuse de cations, Eur. Polym. J. 35 (1999) 899. [26] R.A. Robinson, R.H. Stokes, Electrolyte Solution, Butterworths, London, 1970. [27] A.J. Lotka, Elements of Physical Biology, Williams & Wilkins Company, 1925. [28] B.E. Logan, M. Elimelech, Membrane-based processes for sustainable power generation using water, Nature 488 (2012) 313. [29] D.A. Vermaas, M. Saakes, K. Nijmeijer, Power generation using profiled membranes in reverse electrodialysis, J. Membr. Sci. 385–386 (2011) 234.

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