Nernst-Planck transport theory for (reverse) electrodialysis: I. Effect of co-ion transport through the membranes

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1 Nernst-Planck transport theory for (reverse) electrodialysis: I. Effect of co-ion transport through the membranes M. Tedesco, H.V.M. Hamelers, and P.M. Biesheuvel Wetsus, European Centre of Excellence for Sustainable Water Technology, Oostergoweg 9, 8911 MA Leeuwarden, The Netherlands Abstract Electrodialysis (ED) and Reverse Electrodialysis (RED) are related technologies for water desalination and energy conversion, both based on the selective transport of ions through ion exchange membranes. Fundamental understanding of these processes requires the description of ion transfer phenomena both along and through the membranes. We develop a simple 2D model valid for ED and RED, extending the approach by Sonin and Probstein (Desalination 5, 1968, 293) by using the Nernst-Planck equation not only in the flow channels but also in the membranes. This model requires as only input parameters the geometrical features of the system, the membrane charge density, and the diffusion coefficients of ions in the channel and in the membrane. The effect of non-ideal behavior of the membrane due to the co-ion transport is discussed, evaluating the performance of the processes in terms of current efficiency and energy consumption (for ED), and salt flux efficiency and power density (for RED). Membrane properties such as permselectivity are calculated as outputs of the model, and depend on position in the channel. Keywords: Nernst-Planck equation, Donnan potential, ion exchange membranes, co-ion exclusion, concentration polarization. 1. Introduction Electrodialysis (ED) is a technology widely applied for desalination of brackish water sources since the 1950s. Recent developments in related processes, such as reverse electrodialysis (RED) [1], membrane capacitive deionization (MCDI) [2], and shock electrodialysis [3], further increase the number of applications and the scientific interest in electro-membrane technologies. The principle of both ED and RED is based on the selective transport of cations and anions through ion exchange membranes (IEMs), under either the influence of an applied voltage (ED) or a concentration difference (RED). A (R)ED system consists of an alternating series of cation and anion exchange membranes (CEMs and AEMs), which are separated by spacer channels that allow for water flow along the membranes (Fig. 1). Ideally, each IEM allows only the passage of counterions, i.e., Please cite this article as: M. Tedesco, H. V. M. Hamelers, P. M. Biesheuvel, Nernst-Planck transport theory for (reverse) electrodialysis: I. Effect of co-ion transport through the membranes, Journal of Membrane Science 510 (2016) address: michele.tedesco@wetsus.nl (M. Tedesco)

2 ions with a sign opposite to that of the fixed charge in the membranes (i.e., cations for CEMs and anions for AEMs), while co-ions are rejected. In ED, an external voltage is applied between two end-electrodes to generate an ionic current through the membrane stack: therefore, the feed stream is desalinated and concentrated in alternating channels, and the concentrate and diluate are collected (Fig. 1). Conversely, in the RED process, the system is fed with concentrated and dilute streams, exploiting the salt concentration difference as driving force to generate electric power. Figure 1: Principle of electrodialysis (ED) and reverse electrodialysis (RED). Note that the direction of the current is opposite in ED and RED (see arrows). The symbols + and denote the electrode at higher potential (+) relative to the other ( ). The same device can be operated at least in principle both in ED and RED mode, although it is commonly accepted that the cell design should be optimized for each process. As an example, the spacers used for commercial ED units have typically a thickness between mm [4], while for RED the use of a thinner channel (in the range of mm, or even smaller [5]) is generally preferred to reduce the electrical resistance and enhance the power output. Transport phenomena in ED and RED involve both the convective flux through spacer channels along ion exchange membranes (IEMs), and the electro-migrative flux of ions towards and through IEMs. Ideally, only counterions can pass through the membranes. However, because membranes do not behave ideally, also co-ions and water can cross the membranes, thus reducing the process performance. Therefore, modeling ion transport phenomena in ED and RED is a complex task, which requires a reliable description of all transport phenomena within channels and membranes, and at the membrane-solution interface. Up to now, different modeling approaches have been proposed in the literature to describe ion transport in ED and RED, based on (i) the Nernst-Planck equation [6 13], (ii) Stefan-Maxwell theory [14, 15], (iii) the irreversible thermodynamics formalism [16 18], and (iv) semi-empirical models that use experimentally established overall membrane properties [19 23]. When water transport is neglected, the most common approach is the Nernst-Planck (NP) equation, which requires knowledge 2

3 of diffusion coefficients of ions in solutions and within the membranes, as well as information of membrane parameters such as fixed charge density and porosity. A simple yet fundamental modeling approach for a full ED cell was proposed in 1968 by Sonin and Probstein [7], where in the channels the Nernst-Planck equation is combined with a simplified description of hydrodynamics. Despite some strong simplifying assumptions, especially the use of ideally permselective membranes, the model describes several of the main aspects involved in the ED process in a fundamental manner. In particular, the Sonin-Probstein approach is based on the exact steady-state solution of the NP-equation and local electroneutrality in the entire domain of the spacer channel, thus without the assumption of a prescribed thickness of diffusion boundary (or stagnant) layers (DBLs). In the present work, we aim to describe the (R)ED process taking into account the non-ideal behavior of membranes, and extend the model of Sonin and Probstein [7], allowing for both the counterion and co-ion to move through the membranes. The model can be used in a wide range of operating conditions (flow rates, concentrations, etc.) without the need to consider permselectivities or transport numbers as input parameters. 2. Model development 2.1. Definition of model geometry The repeating unit in an ED (or RED) stack consists of a concentrate channel, a cation exchange membrane (CEM), a diluate channel, and an anion exchange membrane (AEM). In the present work, like in ref. [7], to simplify the calculation, both AEMs and CEMs are assumed to have the same relevant properties, such as thickness, magnitude of the fixed charge density, ion diffusion coefficients and porosity. The feed streams are considered as 1:1 electrolyte solutions with completely dissociated ions, assuming diffusion coefficients for cation and anion equal to one another, both in the channel and in the membrane. Because of these assumptions, transport phenomena through AEMs and CEMs are the same in the present modeling framework. According to the resulting symmetry midplane in each channel, the computation can be restricted to one half of a concentrate channel and one half of a diluate channel, as well as one membrane (either a CEM or AEM, as shown in Fig. 2). A co-current flow arrangement is considered, as normally adopted in industrial ED units to avoid local pressure differences and minimize internal leakages [4]. As a consequence, both feed streams flow in the positive y-direction in Fig. 2, while the electric field is directed along the x-coordinate. For the sake of simplicity, water transport through the membranes is neglected in the present work. This is a strong assumption for ED, though water transport is smaller in RED, due to the counteracting effects of osmosis and electro-osmosis [24]. A constant temperature is assumed within the system for both ED and RED. This assumption is generally valid in real operation, provided that the power supply is not excessively high (for ED) and both feed streams have the same inlet temperature (for RED). 3

4 mid-plane mid-plane L outflow concentrate channel IEM diluate channel fluid flow profile fluid flow profile y 0 x h m inflow Figure 2: Simplified scheme of Sonin-Probstein model [7]. The computational domain consists of half of a concentrate channel, an ion exchange membrane (IEM) and half of a diluate channel. Perfect symmetry and a parabolic fluid flow profile are assumed for both streams in the channel. The x-coordinate starts at the mid-plane (x = 0) and goes toward the membrane, where x = h = δ sp/2. Both solutions flow along the membrane, in the y-direction, from y = 0 to y = L Modeling the spacer channel In the present model, we assume empty spacer channels, i.e., no volume reduction is considered due to the spacer. In each channel, the ion mass balance is given by c i t + J i = 0 (1) where c i is the ion concentration and J i is the ion flux, which is described by the extended Nernst- Planck (NP) equation J i = c i v D i ( c i + z i c i φ) (2) where v, D i, and z i are the fluid flow velocity, the ion diffusion coefficient and the ion valence, respectively, while φ is the dimensionless electric potential, to be multiplied by RT/F = kt/e to obtain the dimensional voltage. Ion concentrations are considered instead of activities in Eq. 2, i.e., we assume ideal thermodynamics in this work. The extended NP equation is valid both in solutions and in the membranes, though the three transport terms may have a different relevance in these regions. Considering ion transport in the channels, it is worth noting that no assumption is required in the present approach about the thickness of a diffusion boundary layer (DBL, or Nernst layer, or stagnant diffusion layer) next to the membrane. Substituting Eq. 2 into Eq. 1 leads to c i t = D i ( c i + z i c i φ) (c i v). (3) 4

5 Eq. 3 can be written for the salt concentration in solution, c, taking into account that the concentrations of cations, c +, and anions, c, are related by the local electroneutrality condition z i c i = 0. (4) i Therefore, c + = c = c in the channels for a solution with a single 1:1-valent electrolyte. Eq. 4 is valid at any position in the spacer channel, except for the nanoscale region at the membrane/solution interface where electrical double layers (EDLs) are formed. This concept will be discussed in Section 2.4. Combining Eqs. 3 and 4, the salt mass balance in the spacer channel becomes ( ) ( ) c t = D+ + D 2 D+ D c + (c φ) (c v). (5) 2 2 Eq. 5 can be notably simplified under the following assumptions: (1) the process is steady state; (2) ions in solutions have the same diffusion coefficient (D + = D = D); (3) diffusion and migration are neglected in the axial direction (because we assume that the gradients in y-direction, i.e., along the membrane, see Fig. 2, are very small with respect to the corresponding gradients in the x-direction); (4) the fluid velocity profile is already fully developed at the entrance of the channel. The final expression for the salt mass balance (as derived in [7]) now becomes D 2 c (x, y) c (x, y) x 2 = v (x). (6) y According to Eq. 6, a convective flux is considered only in the y-direction, with diffusion and migration only in the x-direction. Interestingly, such an assumption is well respected in reality, because the channel length, L, is generally orders of magnitude greater than the channel thickness (typical values for a laboratory-scale unit are L 10 cm and δ sp < 1 mm). Regarding the fluid velocity, the flow regime is laminar both in ED and in RED, with Reynolds number usually Re < 200 for ED [25] and Re < 100 for RED [26, 27]. For ED, the fluid velocity is typically in the range of 6-12 cm/s [28], which significantly reduces the effect of concentration polarization. In RED, fluid velocities are typically below 2-3 cm/s, because higher fluid velocities lead to unacceptable hydraulic losses [5, 29, 30]. A full description of fluid dynamics within spacer-filled channels [27, 30] is beyond the scope of the present work. For ED, a numerical solution of the Navier-Stokes equations fully coupled with the Nernst-Planck-Poisson equations has been reported for overlimiting current conditions [31]. The fluid flow profile is affected by the presence of spacers filaments in the channel. A parabolic flow profile is assumed in the present model, although other profiles can be considered. Therefore, the fluid velocity is described by the Hagen-Poiseuille equation ( ) x 2 v (x) = v 0 (1 h) where h is the half-thickness of the channel (h = δ sp /2) and v 0 is the maximum fluid velocity (at the mid-plane), which is 3/2 of the average velocity, v avg, for a slit-like structure with Poiseuille flow. The average fluid velocity is calculated as the ratio between the channel length and the residence time (v avg = L/τ). The reliability of this assumption is discussed in Section 3.7, where the results obtained with a parabolic flow profile are compared with calculations based on plug flow. In general, in the channel, the ionic current density in the x-direction, J ch, is given by the difference 5 (7)

6 of the fluxes of cation and anion, and using Eq. 2 leads to φ (x, y) J ch (y) = J + J = 2 D c (8) x where J ch can be multiplied by F to have the dimension of A/m 2. Note that J ch will change in the y-direction, i.e., through the channel, but is not dependent on the x-position ( J ch / x = 0), and thus is constant across the solution/membrane interfaces and through the membrane, having the same value of both sides of the membrane. The effective (or mixed-cup average [32]) concentration at each y position in the channel is evaluated as h vc (x, y) dx 0 c eff (y) = (9) v avg h which, for y = L, is equal to the salt concentration of the effluent streams Modeling the membrane Within the membrane domain, the electroneutrality condition should take into account also the fixed charge of the IEM z i c i + ωx = 0 (10) i where ω is the sign of fixed charge (ω = +1 for an AEM and ω = 1 for a CEM), and X is the molar concentration of membrane charge, defined per unit solution phase in the membrane pores [33]. In the case of an ideal membrane, co-ions are completely rejected and therefore the concentration of counterions within the IEM is equal to the concentration of fixed charge, X. The current density in the membrane, J ch, can be evaluated (based on Eqs. 2 and 10) as φ (x, y) J ch (y) = D m c T,m (x, y) (11) x where c T,m is the total ion concentration in the IEM (i.e., the sum of counterions and co-ions), and D m is the ion diffusion coefficient in the membrane, assumed to be equal for both ions; subscript m refers to variables calculated within the membrane, where the spatial coordinate runs from 0 to δ m (IEM thickness). Membrane porosity also affects the ion transport through the membrane. However, in a steady-state calculation, this parameter can be included in the ion diffusion coefficient in the membrane, D m, so that only one transport parameter, i.e., D m, is required as input for the model. The case of unequal diffusion coefficients in the membrane is discussed in Section 3.6. Based on Eqs. 2 and 10, the total ion flux in the membrane, J ions,m = J +,m + J,m, is given by ( ct,m (x, y) J ions,m (y) = D m ωx φ ) m (x, y) (12) x x which is independent of the x-position in the membrane (i.e., across the membrane thickness), because of the steady-state nature of the calculation Overall and boundary conditions The following boundary conditions are considered for the mass balance and the current density. For the mass balance equation (Eq. 6), c/ x = 0 is considered at the mid-plane, according to the assumption of perfect symmetry in the system. For the current density (in the x-direction), the boundary condition is given by equating Eqs. 8 and 11. 6

7 For the total ion flux in the membrane, J ions,m, the continuity of ion flux is considered as boundary condition at each interface, and related to the salt concentration gradient in solution as c (x, y) J ions,m (y) = 2 D. (13) x At the membrane/solution interface, an EDL is formed, and an electric potential difference arises due to the Donnan equilibrium. In case of ideal membranes, the sum of the two Donnan potentials (i.e., one on each side of the membrane) is given by φ c+d = ln(c c/c d ) [34]. However, in reality, each membrane is permeable for both cations and anions. As a consequence, the Donnan potential on each side of the membrane is also dependent of membrane charge density, X. An expression for the Donnan potential as a function of X can be derived from the electroneutrality conditions in solution and in the membrane (Eqs. 4 and 10), together with the Boltzmann relation [35 37] φ D = sinh 1 ωx 2 c. (14) Therefore, the total ions concentration in the membrane at each solution/membrane interface, c T,m, can be evaluated by c T,m = 2 c cosh ( φ D ) = X 2 + (2 c ) 2. (15) Note that c in Eq. 15 refers to the salt concentration at the membrane/solution interface on the solution-side, and is different for the concentrate/iem and diluate/iem interfaces. Finally, the cell pair voltage, V CP, is given by V CP = 2 V T ( φ c + φ D,c + φ IEM φ D,d + φ d ) (16) where φ c, φ IEM and φ d are the potential drops over the concentrate half-channel, the membrane, and the diluate half-channel, respectively. In Eq. 16, φ D,c and φ D,d refer to the two Donnan potentials arising at the two membrane/solution interfaces, while the factor 2 is because a cell pair is twice our computational domain of Fig. 2. This set of algebraic and ordinary differential equations (Eqs. 6 13) results in a self-consistent model, which only requires the definition of process conditions (flow rate, inlet concentrations), ion diffusion coefficients and membrane properties (thickness and charge density) as input to be solved. Interestingly, performance parameters such as permselectivity and transport number are not an input of the model, but can be calculated afterwards. After discretization of the differential equations (both along the x- and y-coordinate), the resulting model can be solved numerically by any AE solver. In the x-direction, the equations are discretized using the central difference approximation, while in the y-direction the implicit Euler method is adopted. For numerical stability, it is advantageous to rewrite the expressions for the current density (Eq. 11) and for the ion flux (Eq. 12) before discretization as J ch x = 0, J ions,m = 0. (17) x The integration expressed in Eq. 9 can be done efficiently using a generic analytical method (e.g., Simpson s rule). 7

8 3. Results and discussion 3.1. Overall settings for model calculations In order to simulate the performance of a stack operating under ED or RED mode, the same input parameters have been used for the membranes and stack geometry. The membranes have a thickness of δ m = 110 µm, a fixed charge density of X = 2 M and an ion diffusion coefficient 10 times lower than in solution for both ions (D m /D = 0.1, where D = m 2 /s in solution, calculated as the average of the diffusion coefficients of Na + and Cl in solution). An open spacer channel of δ sp = 250 µm thickness was considered, with a residence time τ = 30 s in both channels. For ED and RED, different operating conditions were modeled in terms of feed salt concentrations and cell pair voltage. Both processes are modeled considering a single (unsegmented) electrode, which establishes the same voltage V cp at each position along the membrane. For ED, the first stage of a seawater desalination process was simulated, considering a feed concentration of c 0 = 500 mm and an operating cell pair voltage of V cp = 100 mv (i.e., corresponding to an average current density of < J ch >= 60 ma/cm 2 in the investigated scenario). For RED, a power generation process between seawater (c c,0 = 500 mm) and fresh water (c d,0 = 17 mm) was modeled, with a cell pair voltage of V cp = 50 mv, i.e., roughly corresponding to the working condition of maximum power output Concentration profiles in ED and RED channels The selective transport of ions through the IEMs leads to concentration polarization, i.e., the formation of concentration differences between the bulk and the membrane/solution interface. Concentration polarization has in fact different effects on ED and RED, due to the directions of ion fluxes in the system (Fig. 3). In particular, in ED the net ion flux is directed from diluate to concentrate channels, due to migration. As a result, the membrane/solution interfaces are depleted in the diluate channels, and enriched in the concentrate channels and this might ultimately lead to limiting current conditions when the ion concentration at the diluate/iem interface becomes zero [38]. Conversely, in RED, the net ion flux has the opposite direction (i.e., from concentrate to diluate channels), driven by the salt concentration difference. Therefore, in RED salt depletion at the membrane occurs in the concentrate channel, and thus limiting current conditions cannot occur in RED. Fig. 4 shows the concentration profiles arising in diluate and concentrate channels for both processes, as function of the y-coordinate. As a result of the different transport properties in solution and in the membrane, the model predicts a gradual concentration change from bulk to interface. Note that our model does not assume a certain thickness of the diffusion boundary layer (DBL). In models based on such a DBL-approach, a stagnant layer is considered close to the membrane/solution interface, thus convection along the membrane is neglected in this region. Typically, in ED/RED systems the thickness of such a DBL, δ DBL, is in the order of δ DBL µm [39, 40], i.e., a significant fraction of the entire spacer channel thickness. Instead, in the present model, at each position in the spacer channel there is convective flow along the membrane, in line with expectation. This shows how the theoretical concept of the DBL is less appropriate for modeling of narrow channels, i.e., when the spacer thickness is lower than δ sp 1 mm. 8

9 Figure 3: Schematic representation of concentration profiles and ion fluxes (horizontal arrows) in ED and RED. Large and small arrows crossing the IEMs represent the flux of counterions and co-ions. Both for ED and RED the electrode at positive potential relative to the other, is drawn on the left-hand side. However, note that the ionic current is oppositely directed, when comparing ED and RED Effect of co-ion transport on (R)ED process efficiency In order to quantify the efficiency of the ion transport through the membranes in ED and RED, we define two related efficiencies, namely the current efficiency, λ (for ED), and the salt flux efficiency, ϑ (for RED). Current efficiency, λ, is defined in ED as the ratio between the salt flux, J salt, and the ionic current, λ = J salt /J ch [2]. In the symmetric model discussed in the present work, this measurable flux, J salt, is equal to the ions flux through each of the IEMs, i.e., J ions,m. For RED, it is better to define the efficiency as the opposite ratio, ϑ = J ch /J salt, which represents a salt flux efficiency of the process. As the inverse of the current efficiency used in ED, the salt flux efficiency, ϑ, quantifies the efficiency of the IEMs in RED. The calculated efficiencies are reported in Fig. 5.A-B as a function of the y-position in the channel. The integral efficiencies, < λ > and < ϑ >, can be calculated by separately integrating both the ionic current and the salt flux along the y-coordinate, and taking the ratio as Jsalt dy < λ >= Jch dy, < ϑ >= Jch dy Jsalt dy. (18) For ideal membranes, in which there is only a counterion flux, the efficiency of the (R)ED process is equal to λ, ϑ = 1. However, in the case of non-ideal membranes, the co-ion transport causes a significant decrease of the efficiencies, and λ and ϑ depend on the position along the membrane (Fig. 5). In general, λ and ϑ will depend both on operating conditions (inlet concentration, residence time, cell pair voltage), and on membrane properties (fixed charge density, ion diffusion coefficients). The fixed charge density of the membrane, X, significantly influences the efficiency of both processes, 9

10 salt concentration (mm) salt concentration (mm) ED concentrate channel outlet (y=1) 0.4 IEM RED IEM inlet (y=0) concentrate channel inlet (y=0) (seawater) outlet (y=1) midplane outlet (y=1) diluate channel x/h IEM-solution interface outlet (y=1) inlet (y=0) midplane diluate channel (fresh water) x/h IEM-solution interface Figure 4: Calculated concentration profiles arising in ED and RED. affecting the concentration of co-ions inside the membrane (as described by Eq. 10). This effect has been investigated for membrane charge densities, X, in the range of X = 1 5 M. As shown in Fig. 5, a fixed charge density X 2 M leads to current (and salt flux) efficiency λ, ϑ < 0.75 for the investigated cases. Therefore, a decrease in X leads to an increase of the co-ion transport and a reduction of the counterion transport through the membrane, both in ED and RED. This effect is clearly shown in Fig. 5.C-D, where both counterion and co-ion fluxes are plotted as a function of the (axial) y-coordinate. Note that in ED counterion and co-ion fluxes have different directions, because the driving force is the external applied voltage. In contrast, they have the same direction in RED, where the main driving force is the salt concentration difference between concentrate and diluate Calculation of membrane resistance and permselectivity The membrane electrical resistance, R IEM, and permselectivity, α, are the main properties usually adopted in the membrane manufacturing industry to characterize ion exchange membranes. In the present work, these properties are not required as input parameters to describe the (R)ED process performance, but can be calculated by the model afterwards. The membrane electrical resistance can be evaluated as the ratio of the (ohmic) voltage drop across the membrane over the ionic current, thus 10

11 J counter, -J co (mmol/(m 2 s)) J counter, J co (mmol/(m 2 s)) Current efficiency Salt flux efficiency ϑ ED A RED B X (M) X (M) ED C RED J counter D X (M) 2.0 J counter X (M) -J co J co y/l y/l Figure 5: Calculated efficiency and ion fluxes as a function of the position along the membrane, y/l, and of membrane charge density, X. A) Current efficiency, λ, in ED. B) Salt flux efficiency, ϑ, in RED. C) Counter- and co-ions fluxes in ED. D) Counter- and co-ions fluxes in RED. R IEM = V IEM I ch (19) where V IEM is the potential drop across one membrane excluding the Donnan potentials, and I ch is the ionic current density expressed as A/m 2 (I ch = F J ch ). Permselectivity is usually determined by measuring the potential difference across a membrane separating two solutions of different concentration (e.g., 0.5 M 1 M KCl, or 0.5 M M NaCl [4, 41]), although other experimental methods (e.g., through estimation of transport numbers [42]) can also be adopted. In the present work, α has been calculated as α = V real = φ D,c + φ IEM φ D,d (20) V ideal ln (c eff,c /c eff,d ) where V real is the voltage drop across one membrane, including the Donnan potentials at each membrane/solution interface, and V ideal is the ideal potential for 100% permselective membranes, in the absence of concentration polarization. Eq. 20 gives the actual value of membrane permselectivity, α, as a spatially dependent parameter along the membrane. Results are shown in Fig. 6, where both the membrane resistance (for ED and RED processes), and permselectivity (RED conditions), are 11

12 R IEM (Ω cm 2 ) Permselectivity reported. 2.0 RED A 1.0 B 1.5 ED X (M) X (M) y/l Figure 6: Calculated membrane resistance, R IEM, and permselectivity, α. A) Influence of membrane charge density, X, on IEM resistance in ED and RED. B) Local permselectivity, α, calculated for RED conditions as function of X. Fig. 6 shows that both membrane properties, R IEM and α, are significantly influenced by the membrane charge density, X. In particular, the calculated electrical resistance is in the range of R IEM = Ω cm 2 in ED, and up to R IEM = 2 Ω cm 2 in RED (Fig. 6.A). These values are in agreement with the typical resistance of commercial ion exchange membranes, usually in the range of R IEM = Ω cm 2 [43]. The lower resistance in ED with respect to RED conditions is due to the higher average salt concentration in bulk solution for ED (and therefore in the membrane, see Fig. 4 and Eqs. 11 and 15). At a high membrane charge density, X, we can assume that c T,m X, and the gradient of electric potential over the membrane is φ/ x φ IEM /δ m. Therefore, a limiting value of R IEM independent of conditions in solution can be calculated from Eqs. 11 and 16 as R IEM,lim = V IEM R T δ m F J ch F 2 (21) D m X which for instance gives a value of R IEM = 0.35 Ω cm 2 for X = 5 M, in close agreement with Fig. 6.A. In all the calculations, R IEM given by Eq. 19 was almost independent of the y-coordinate. This is because the total ion concentration in the membrane, c T,m, does not change appreciably in the y-direction (see Eqs. 11 and 15). The calculated values of permselectivity are in the range of α = (Fig. 6.B), with a strong dependence of α on the axial position and on the membrane charge density, X [44]. The benefit of increasing X on permselectivity can be attributed to the higher co-ion exclusion effect, according to the electroneutrality in the membrane (Eq. 10). Moreover, the dependance of the permselectivity on the y-position along the membrane is due to the significant change of local salt concentration, due to salt transport. The permselectivities shown in Fig. 6.B are much lower than the values typically reported for commercial IEMs in 0.01 M 0.5 M NaCl solutions of α = [43]. However, our predicted values refer to the actual permselectivity of the membrane during the RED process, i.e., 12

13 when the operating conditions within the system are very different from the open circuit conditions, in which the nominal permselectivities are normally evaluated [43] Calculation of energy consumption (in ED) and power density (in RED) Using the present model, we can estimate the overall performance of both processes, in order to assess the detrimental effect of co-ion transport on system behavior. For ED, we calculate the energy consumption, EC, as the ratio between the electric power and the volumetric flow rate of diluate product, Q d,out [19, 45] EC = V cp < I ch > S Q d,out (22) where < I ch > is the average current density, and S the membrane area. The energy consumption for ED was calculated for a water recovery ratio of 50%, until reaching a target concentration for the diluate of c d,out = 20 mm. For RED, the gross power density was calculated, i.e., the electric power generated by the system per total membrane area, thus P D = 1 2 V cp < I ch >. (23) Results are shown in Fig. 7, where the performance of the (R)ED process with non-ideal IEMs is compared with the ideal case, i.e., neglecting the co-ion transport (λ = ϑ = 1). For the ED process, the required current and the resulting energy consumption are shown as a function of the residence time, τ = τ d = τ c (Fig. 7.A,C). For the ideal membranes, where the salt flux is equal to the ionic current (λ = 1), < I ch > is inversely proportional to τ, as a result of the mass balance. On the other hand, the non-ideal membranes deviate from this behavior, especially at high residence time (Fig. 7.A). The energy consumption in case of ideal membranes decreases with increasing τ, until reaching a constant value of EC = 2.6 kwh/m 3 (Fig. 7.C). This value represents the minimum energy consumption to desalinate a salt solution from 500 mm to 20 mm with 50% water recovery. This theoretical limit, ECmin th, can be analytically derived by taking into account the theoretical (minimum) voltage in the cell pair Vcp,min th = 2 R T F ln c c,out (24) c d,out where c c,out, c d,out are the outlet salt concentrations. For ideal membranes, the mass balance can be rewritten as < J ch > S = Q d,out c (25) where c = c 0 c d,out, and c 0 is the inlet salt concentration. Substituting Eqs. 24 and 25 into Eq. 22, and for τ d = τ d the theoretical limit of energy consumption, ECmin th, is given by ECmin th = 2 R T c ln c 0 + c c 0 c. (26) Using non-ideal membranes, the minimum energy consumption is EC = 8.1 kwh/m 3 at < I ch >= 45 ma/cm 2, i.e., an increase by a factor of three when compared with the theoretical minimum value, EC th min (Fig. 7.C). This can be compared with the value of EC = 5.3 kwh/m3 suggested by Turek as the energy consumption for seawater desalination by ED [45]. For RED, the cell pair voltage and power density are shown as a function of the current in Fig. 7.B,D. A maximum power density of P D = 1.7 W/m 2 is predicted in the case of ideal membranes 13

14 EC (kwh/m 3 ) PD (W/m 2 ) <I ch > (ma/cm 2 ) V cp (V) ED A RED 0.15 B non-ideal ideal ED 14 C RED ideal D 12 non-ideal 1.5 non-ideal ideal EC th min = 2.6 kwh/m t (s) <I ch > (ma/cm 2 ) Figure 7: Effect of non-ideal behavior of membranes on the performance of ED and RED. A) Average current density, < I ch > in ED as a function of residence time, τ = τ d = τ c, (c d,out = 20 mm). B) Cell pair voltage, V cp, in RED as a function of average current density, < I ch >. C) Energy consumption, EC, in ED. D) Power density, P D, in RED. Dashed lines: ideal IEMs (λ = 1, ϑ = 1). Solid lines: non-ideal IEMs. (Fig. 7.D), while P D = 1.4 W/m 2 for non-ideal membranes. Therefore, co-ion transport causes a 20% decrease in power density, in accordance with the calculated permselectivities (Fig. 6.B). These values are reasonably in agreement with the experimental values of power output reported for RED systems under similar conditions [5, 46] Effect of different diffusion coefficients in the membrane Until now, we have assumed equal diffusion coefficients in the membrane for counterions, D m,ct, and co-ions, D m,co. However, these ion diffusion coefficients can be very different. This phenomenon has been numerically investigated, by removing the simplifying assumption of equal diffusion coefficients in the membrane, and considering different values both for D m,ct and for the ratio D m,co /D m,ct. Note that, in this case, Eqs. 11 and 12 are no longer valid, and must be replaced by their general expressions. Results of the model calculations are reported in Fig. 8, where the ratio D m,co /D m,ct varies from 0.01 to 10. Interestingly, a faster diffusion in the membrane for co-ions than for counterions, i.e., a 14

15 Current efficiency Salt flux efficiency ϑ ratio D m,co /D m,ct > 1, leads to a dramatic decrease of efficiency in both ED and RED (Fig. 8). This might be possible, taking into account that the electrostatic repulsion of the fixed charges might cause a faster transport for co-ions through the membrane pores, and likewise a slower transport for counterions. ED 1.0 A RED 1.0 B D m,ct /D ct D m,ct /D ct D m,co /D m,ct D m,co /D m,ct Figure 8: Calculated efficiency as a function of the ratio of diffusion coefficients of co-ions and counterions in the membrane, D m,co/d m,ct. A) Current efficiency, λ, in ED. B) Salt flux efficiency, ϑ, in RED. The diffusion coefficient of counterions is shown next to the curves Effect of fluid flow profiles in thin channels on process efficiency All the model results reported so far were based on the assumption of a parabolic fluid flow profile in the channels. Clearly, the fluid flow inside a spacer-filled channel is in reality more complex, because of the tortuous path of the fluid around the spacer filaments. A detailed description of fluid flow is helpful to calculate the pressure drops in the stack, and to investigate the effect of spacer geometry on the mass transport in the channel. This is especially relevant for systems, like in ED, where the channels are relatively wide (δ m 1 mm). However, in the case of thin spacers (δ m 200 µm), the impact of fluid flow might have a lower impact on the overall process performance. In this regard, the present model was used to simulate the effect on process efficiency, taking into account two simplified fluid flow profiles, i.e., parabolic and plug flow. Results are shown in Fig. 9, where the salt flux efficiency, ϑ, as well as counter- and co-ion fluxes, have been calculated for both conditions in RED. Results for ED are not shown in Fig. 9, as both cases are totally indistinguishable. For the investigated cases of δ m = 250 µm, the model calculations for plug flow and parabolic profile lead to negligible differences. In particular, going from parabolic profile to plug flow leads to only a 5% increase in the counterion fluxes in RED. This in turn leads to slightly higher values of permselectivity (not shown in Fig. 9), though the salt flux efficiency remains the same, because of the concomitant increase in the average current density. As a result, the assumption of a parabolic flow profile is valid for the calculation of the overall process performance in ED and RED, and can be 15

16 Salt flux efficiency ϑ J counter, J co (mmol/(m 2 s)) RED 1.0 A RED 1.0 plug flow B plug flow parabolic parabolic J counter J co y/l y/l Figure 9: Effect of fluid flow profiles on salt flux efficiency and ion fluxes in RED, as a function of the position along the membrane, y/l. A) Salt flux efficiency, ϑ. B) Counter- and co-ions fluxes. Membrane charge density is X = 2 M. replaced by the use of a plug flow profile, which is mathematically easier to implement. This analysis suggests that the optimization of spacer geometry should be mainly focused on reducing of pressure drops in the channels Outlook The model presented in this work can simulate, within the same fundamental modeling framework, the performance of both ED and RED processes. Moreover, the model is applicable to any other process in which ion transport phenomena through IEMs play an important role such as MCDI, and might be especially suitable to predict the behavior of novel applications involving hybrid ED-RED processes, e.g., in the field of energy storage systems [47]. Overall, the model shows a promising agreement with literature data. However, our calculation results are primarily meant to be illustrative of the present modeling framework, and future modifications are needed to describe data more quantitatively. Moreover, note that the model developed in this work is not valid in the overlimiting current regime, where other reactions normally occur, such as water splitting. In that regime, the ph variations in solutions must be taken into account to describe the transport phenomena involved. For a complete description, future modeling will include: (i) water transport [36], (ii) ion pairing, taking into account non-dissociated ions and ph-dependent reactions [48], and (iii) the effect of multi-valent ions [33]. 4. Conclusions The aim of this work has been to present a simple 2D model to describe the transport phenomena involved in electrodialysis (ED) and reverse electrodialysis (RED) using non-ideal membranes. For the first time, a single modeling framework has been adopted for simulating both processes, including the effect of co-ion transport. The model gives a fundamental description based on the Nernst-Planck 16

17 theory, predicting process performance in terms of current efficiency for ED, and salt flux efficiency for RED. The most relevant properties of ion exchange membranes (permselectivity and electrical resistance) are predicted as a function of the fixed charge density of the membranes, and the position in the channel. Moreover, the effect of co-ion transport has been assessed, showing typically in our calculation a 20% reduction of power density in RED, and compared to the theoretical minimum in ED an increase in energy consumption by a factor of 3. In order to fully describe all the main transport phenomena involved in ED and RED, future modeling efforts will include water transport through IEMs, unequal ion diffusion coefficients, and on the effects of mixtures of ions and ph fluctuations. Acknowledgments This work was performed in the cooperation framework of Wetsus, European Centre of Excellence for Sustainable Water Technology ( Wetsus is co funded by the Dutch Ministry of Economic Affairs and Ministry of Infrastructure and Environment, the Province of Fryslân, and the Northern Netherlands Provinces. The authors like to thank the participants of the research theme Blue Energy for fruitful discussions and financial support. Nomenclature c Salt concentration at the membrane/solution interface (mol m 3 ) c c,out, c d,out Outlet salt concentration of concentrate and dilute (mol m 3 ) c eff,c, c eff,d Effective salt concentration at each y position in the channel (mol m 3 ) c i Ion concentration in solution (mol m 3 ) C T,m Total ion concentration in the membrane (mol m 3 ) c +, c Concentration of cations and anions in solution (mol m 3 ) D +, D Diffusion coefficients of cations and anions in solution (m 2 s) D ct, D co Diffusion coefficient of counterions and co-ions in solution (m 2 s) D i Ion diffusion coefficient in solution (m 2 s) D m Ion diffusion coefficient in the membrane (m 2 s) D m,ct, D m,co Diffusion coefficient of counterions and co-ions in the membrane (m 2 s) EC Energy consumption in ED (kwh m 3 ) EC th min Theoretical limit of energy consumption in ED (kwh m 3 ) F Faraday constant (96485 C mol 1 ) h Half-thickness of the spacer channel (m) I ch Ionic current density (A m 2 ) J +, J Fluxes of cations and anions (mol m 2 s 1 ) J ch Ionic current density (mol m 2 s 1 ) J i Ion flux (mol m 2 s 1 ) J ions,m Total ion flux in the membrane (mol m 2 s 1 ) 17

18 J salt Salt flux (mol m 2 s 1 ) L P D Channel length (m) Power density in RED (W m 2 of total membrane area) Q d,out Volumetric flowrate of diluate product in ED (m 3 s 1 ) R Universal gas constant (8.314 J mol 1 K 1 ) R IEM Electrical resistance of the membrane (Ω m 2 ) S Membrane area (m 2 ) t Time (s) T Absolute temperature (298 K) v Fluid flow velocity (m s 1 ) v 0 Maximum fluid flow velocity (m s 1 ) v avg Average fluid flow velocity (m s 1 ) V cp V th cp,min V ideal V real V T x Cell pair voltage (V) Theoretical (minimum) cell pair voltage in ED (V) Ideal potential for 100% permselective membrane (V) Voltage drop across the membrane (V) Dimensional voltage, V T = R T/F (V) Spatial coordinate along the channel width (m) X Concentration of fixed charges in the membrane (mol m 3 ) y z i Ion valence (-) Spatial coordinate along the membrane (m) Greek letters α Permselectivity (-) δ m δ sp Membrane thickness (m) Channel thickness (m) c Inlet-outlet salt concentration difference in ED (mol m 3 ) φ c, φ d Dimensionless voltage drop over the concentrate and diluate half-channel (-) φ D,c, φ D,d Dimensionless Donnan potentials across concentrate/iem and diluate/iem interfaces (-) φ IEM Dimensionless voltage drop over the membrane (-) V IEM Voltage drop over the membrane (V) ϑ Salt flux efficiency in RED (-) λ Current efficiency in ED (-) τ Residence time of solutions in ED (s) φ Dimensionless electric potential in solution (-) φ m Dimensionless electric potential in the membrane (-) ω sign of fixed charges in the membrane (-) 18

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