4.1. Conductivity Measurement Results
The results of measuring the electrical conductivity of ion-exchange membranes of two batches in a wide range of sodium chloride concentrations are shown in
Figure 5. The results for each batch are available in
Supplementary File S1. The data for each membrane are used for the calculation of the electroneutral solution fraction. It is notable that while the standard deviation is less than 5% at maximum for each batch, for each membrane between two batches, it increases drastically.
Within the microheterogeneous model, the conductivity of a heterogeneous ion-exchange membrane is the geometric mean of the conductivities of the gel phase and the phase of an electrically neutral solution. Given the above, the conductivity of the membrane is expressed as follows:
where
,
, and
are the electrical conductivity of the membrane, gel, and electroneutral solution solution, S/cm;
A is the characteristic parameter that describes the spatial distribution of conducting phases in the membrane.
A = +1 for parallel and
A = –1 for series connected phases; in real samples, the
A parameter takes values in range 0.1–0.3.
In dilute solutions near the point of isoelectric conductivity (such a value of electrical conductivity when ), Equation (13) is simplified, and its linearization in bilogarithmic coordinates makes it possible to determine the value of the parameter f2.
The found values of the parameter
f2 and the coordinates of the isoelectric conductivity point for the membranes under study are given in
Table 3.
It can be seen that for heterogeneous Ralex membranes, the value of the f2 parameter is comparable to the value of this parameter for homogeneous membranes. Such values of this parameter were obtained in other works [
34,
35,
36]. In [
34], the authors suggested that such a value of the fraction of the intergel solution for these membranes is a consequence of the fact that the particles of the ion exchanger in these membranes are small enough and, at the same time, there are no macroscopic cavities inside the membrane that electrically neutral solutions can occupy. These structural features of Ralex membranes were demonstrated by Akberova et al. [
37] and Slouka et al. [
38].
From the perspective of the RED system, a large value of the
f2 parameter (MK-40, MA-41 membranes) can provide high electrical conductivity. This is especially significant for the MK-40 membrane, which has a mean electrical conductivity almost twice as high as the Ralex CM membrane (
Figure 5). At the same time, the electrically neutral solution in the membrane’s pores causes a decrease in permselectivity.
Another factor that attracts attention is the large scatter of electrical conductivity values among the samples under study (
Figure 5). While for Ralex membranes, the standard deviation values are lower, which suggests better repeatability of the production process, the values of the
f2 parameter for both Ralex CM and Ralex AMH membranes differ almost two-fold. The reason for this can be that different production routes are used for the preparation of different batches. For example, more prolonged milling of the ion-exchange resin results in smaller particles and a higher fraction of macropores in the Ralex CM membrane, as shown in [
37].
Regarding MK-40 and MA-41 membranes, a large scatter of their conductivity is also found in the literature, and Veerman gives a good collection of data in [
39]. As in the case of Ralex membranes, different production conditions seem to be the leading cause of such a wide range of electrical conductivity. According to Veerman, commercial membranes are not chemical compounds with unchanging properties; different lot numbers, years of production, and storage conditions can significantly influence their properties [
39].
4.2. Diffusion Permeability Results
As already mentioned in the introduction, the membrane potential is significantly influenced by the permselectivity of the membranes. Diffusion permeability is a value that characterizes the non-selective flux of electrolyte through an ion-exchange membrane.
The results of measuring the diffusion permeability of the studied ion-exchange membranes in 0.1–1 mol/L NaCl solutions are shown in
Figure 6. The results for each batch are available in
Supplementary File S1. The data for each membrane are used for the calculation of the
βj parameter.
The results obtained when measuring the diffusion permeability correlate well with the results obtained when studying the electrical conductivity. For Ralex membranes, low values of the f2 parameter are characteristic, which is reflected in the low dependence of the integral coefficient of diffusion permeability on the concentration of the external solution. At the same time, for the second batch, despite the same low value of the f2 parameter, the diffusion permeability is comparable to the diffusion permeability of MK-40 and MA-41 membranes. At the same time, no such dependence was revealed for MK-40 and MA-41 membranes. Thus, despite the different values of the integral coefficients of diffusion permeability obtained for different samples, their relationship does not change significantly.
The differential (also sometimes called “local” [
36]) diffusion permeability coefficient is used to calculate the transport numbers according to Equation (6). In contrast to the experimentally determined integral coefficient of diffusion permeability, which is the average value over the entire thickness of the ion-exchange membrane, the differential coefficient corresponds to the diffusion permeability of a thin ion-exchange film in equilibrium with a “virtual solution” with a particular concentration
c at a point in space
x. The following transformation is used:
where
is the parameter that characterizes the concentration profile in the ion-exchange membrane [
23] (linear at
, convex at
, or concave at
).
The parameter values found based on experimental data are shown in
Table 4.
4.3. Transport Numbers
According to the microheterogeneous model, the transport numbers are determined by the combined action of two factors: the electrical conductivity of the membrane (which is mainly determined by the counterion transport) and its diffusion permeability (which is determined by the co-ion transport). Moreover, both of these parameters depend on each other [
23]. Thus, membranes with high diffusion permeability are characterized by high electrical conductivity in concentrated solutions. The counterion transfer number for membranes with higher diffusion permeability will be lower, since high diffusion permeability means more co-ions are present in the membrane phase. High electrical conductivity, especially in dilute solutions, is an essential characteristic for the reverse electrodialyzer process, since it reduces the internal resistance of the electromembrane stack.
Counterion transport numbers were calculated using Equation (5). The results are presented in
Table 5.
It can be seen that for different samples of Ralex membranes that despite the differences in electrical conductivity and diffusion permeability, the transport numbers differ insignificantly, both for cation-exchange and anion-exchange membranes. For example, the maximum difference in concentrated (1 mol/L) solutions is ±0.01 for Ralex AMH.
For MK-40 and MA-41 membranes, the situation is different. For MK-40 membrane, the difference in transport numbers reaches ±0.05. On the other hand, the largest scatter was obtained for the MA-41 membrane, where, depending on the batch, the difference is ±0.1.
4.4. OCV and Power Density of the RED Stack
The raw data of the dependence of the power density on the current density in the external circuit are shown in
Figure 7.
It is known that the maximum power can be obtained when the resistance of the external load is equal to the internal resistance of the reverse electrodialysis module [
40]. The power density and internal resistance obtained on a laboratory reverse electrodialysis module are shown in
Figure 8.
The total inner resistance of the reverse electrodialyzer membrane stack (
Ri) can be divided into several components:
where
,
, and
are the ohmic and non-ohmic components of the resistance of the membrane stack and the resistance of the solution in the electrode chambers, Ohm.
The resistance of the solution in the electrode chambers can be measured directly.
The ohmic component of the resistance includes the resistances of ion-exchange membranes and high and low salinity solutions. Thus, for the entire membrane stack, the resistance can be found as:
where
h is the thickness of the gasket, m; superscript «
s» marks the surface resistance in Ohm·m
2. The conductivity of the solutions in the low and high concentration chambers must take into account the porosity factor of the spacer.
Non-ohmic components of resistance include the change in resistance along the channel length associated with a change in concentration and the contribution of the resistance of diffusion boundary layers at the membrane/solution interface:
The first term in parentheses is responsible for the change in resistance along the length of the channel with high and low salinity solutions. The second term represents the change in resistance in the enriched (from the side of the channel with a low salinity solution) and depleted (from the side of the channel with high salinity solution) diffusion layers.
In the general case, the non-ohmic components of the resistance of the membrane stack of an electrodialyzer are neglected based on the following considerations:
The change in the concentration of the solution with a not very long channel length is insignificant. Considering the values of the integral diffusion permeability coefficient given in
Section 4.2 and the linear velocity of the solution, and the geometric parameters of the electrodialyzer, the calculated concentration decrease in the high salinity solution is 0.002 mol/L. Such changes will not have a significant impact on resistance.
The resistance of diffusion layers is also generally not considered. In the case of an enriched diffusion layer (
), the concentration at the membrane surface is higher than in the bulk of the solution, and its resistance will be lower. In the case of a depleted diffusion layer (
), it is assumed that a separator is sufficiently effective, so the thickness of this layer is sufficiently small. The decrease of concentration in the depleted diffusion layer against the bulk of a high salinity solution is also assumed to be insignificant. The dependence of power density on the linear velocity of the solution in the high salinity channel verifies the later statement (see
Supplementary).
For comparison of electrodialysis modules of different designs, it is convenient to use specific values. The surface resistance per unit cell can be calculated as:
When considering the dependence of the resistance of the RED module on the concentration of a low salinity solution (
Figure 8), two sections can be distinguished: the first corresponds to a sharp drop in resistance as the concentration of the solution entering the low concentration compartment increases (conditional river water). A sharp drop in resistance is observed up to a solution concentration of 0.8 g/L. Further increase in the low salinity solution concentration does not significantly decrease the RED module resistance.
This nature of the dependence is associated with a change in the “limiting phase”, the material with the maximum resistance in the membrane stack. In solutions with a low concentration, ion-exchange membranes have higher conductivity, while the electrical conductivity is linearly dependent on its concentration. As the concentration of the solution approaches the concentration of
ciso, the contributions to the total conductivity of the ion-exchange membranes and the low salinity solution become comparable (
Table 6). Obviously, with a further increase in concentration and when
ciso is exceeded, the resistance of the membrane stack will remain practically constant.
As can be seen from the table, the main contribution to the resistance of the membrane stack in dilute solutions is made by the solution in the low salinity solution compartment. When using solutions with a higher concentration, the total contribution of ion-exchange membranes becomes more significant than the solution’s contribution. In this case, the resistance of the solution in the path with high salinity solution and the solution of the electrode chambers does not significantly affect the total ohmic resistance of the apparatus.
The maximum power density (
) obtained in the process of reverse electrodialysis depends on two parameters [
41]: the open circuit potential and the internal resistance of the apparatus.
As already indicated, the OCV value can be established knowing the characteristics of the membranes using Equations (2) and (5). The calculation results, as well as experimental data, are shown in
Figure 9. Due to the large scatter of the experimental data (see
Supplementary) and to minimize uncertainty in the characteristics, we used the average values between the test results of two batches for calculation of the OCV and power characteristics of the RED module.
From the results shown in
Figure 9, it can be seen that the calculated OCV value for the Ralex membrane pair is close to the ideally selective membrane pair, for which α = 1. Low selectivity of the MK-40 and MA-41 membranes negatively affects the open circuit potential (OCV). The experimentally observed OCV values for the Ralex membrane pair are lower than the theoretical ones and are close to the values calculated for the MK-40/MA-41 membrane pair. The deviation of the experimental values from the theoretical ones can be explained by the difference between the membrane samples, the presence of assembly defects in the electrodialysis module, and leakage currents.
Based on the obtained experimental data, the integral permselectivity of the Ralex membrane pair was calculated. The calculation was carried out similarly to the calculation of the apparent transfer numbers obtained by measuring the membrane potential, as the ratio of the experimental potential to the theoretical one:
where
is the experimental open circuit potential, V;
is the theoretical open circuit potential, calculated using Equation (1) with
α = 1, V.
The resulting integral permselectivity is essentially an average value that considers all those changes in the solution concentration along the length of the electrodialyzer channel, the dispersion of the membrane characteristics over the sample area, and the number of membranes of each type. The mean value found for the Ralex membrane pair in the investigated concentration range of the low salinity solution was 0.94.
The dependences of the electrical conductivity of ion-exchange membranes on the concentration of the external solution obtained in this work, together with the theoretically calculated values of the OCV potential, can be used to calculate the theoretical value of the power density (Equation (19)). The calculation results and their comparison with experimental data are presented in
Figure 10.
The calculated values of the power density are higher than the experimental results. This can be explained by the fact that when calculating according to Equation (20), the internal resistance of the membrane package does not consider the non-ohmic components of the resistance. Another possible reason can be the spacer shadow effect [
42]. The significant difference between the theoretical and experimental values observed at concentrations of a diluted solution above 0.5 g/L can be associated with both the underestimated value of the internal resistance and the underestimated values of the permselectivity of ion-exchange membranes in these solutions (
Figure 9).
Another possible explanation for the decrease in power density is the appearance of parasitic currents or leakage currents. This phenomenon is well known for the electrodialysis process, especially in the field of concentrated solutions [
43,
44] or bipolar electrodialysis [
45]. Some of the current generated by the reverse electrodialyzer tends to flow through a common collector with brine solution. This is because the electrical resistance of such a collector is significantly lower than the resistance of the rest of the components of the system and because it provides an almost direct path of current flow from the cathode to the anode. Mathematical modeling and experimental studies carried out in [
40,
46] showed that the share of power losses from the occurrence of parasitic currents increases with an increase in the number of pair cells and can reach 30–40%.
Let us consider the possible share of power losses associated with parasitic currents in this work. The investigated reverse electrodialyzer has a sufficiently large number of pair cells (
N = 16). Based on the literature data, this is approximately the lower limit at which the effects of parasitic currents begin to build. However, the share of power losses, in this case, does not exceed 5% (calculated value from [
40]) and is even less in the experiment. In the above-mentioned studies, the concentration of the brine solution was 5 M, which is much higher than the solution of 20 g/L (0.34 M) used in this work.
In addition, the investigated RED module has an elongated channel shape (the length-to-width ratio of the channel is 4:1), which makes it possible to use only one feed and drain channel, which should create additional resistance that prevents the parasitic flow of current. However, this channel geometry can be disadvantageous at low fluid flow rates, since dead zones can appear at the corners of the channel. The flow of the solution in stagnant zones is low, significantly reducing the concentration gradient between high and low concentrations chambers. Indirectly, the small influence of such zones is indicated by the absence of dependence of the power density on the flow through the module. However, to confirm this fact, additional experimental studies are needed.