## 1. Introduction

The energy economy is facing its most challenging decade, as it must transcend into a more climate-friendly one, as half of the emitted CO

${}_{2}$ due to energy generation and consumption has been targeted for reduction. To achieve this, the technologies used must be changed from those depending on the burning of fossil fuels into electricity and heat, towards technologies which provide electricity and store it in the form of chemical energy. Striving for renewable energy generation, energy storage systems, and renewable hydrogen production, reverse electrodialysis is one of the few technologies that could address all three of these needs [

1,

2,

3].

Salinity gradient energy (SGE)—particularly RED, which harvests energy produced by mixing two aqueous solutions with different salinities,—has received great interest in the literature [

2,

3,

4,

5,

6,

7,

8,

9,

10,

11] since its first use, which was reported by Pattle in 1954 [

12]. Concentration batteries have also been recently proposed and discussed, which couple salinity gradient energy (SGE) technologies for energy generation to their corresponding desalination technologies [

2,

4,

13]. Jalili et al. developed mathematical models to compare three types of energy storage systems: electrodialytic, osmotic, and capacitive batteries [

2]. Influential parameters, such as temperature and energy consumption of the pump, on the performance of different concentration batteries were also discussed in their work [

2] applying a mathematical model. They reported that the peak power densities of the energy storage systems increase at elevated temperature [

2].

A schematic of a simple RED stack is shown in

Figure 1. In general, a unit cell consists of a dilute solution compartment, a concentrated solution compartment, a cation exchange membrane, and an anion exchange membrane. By repeating unit cells and connecting the end points of the stack to an anode and a cathode compartment (where the electrode rinse solutions are present), a RED stack can be completed for converting an ionic flux into an electrical one [

2].

The electrical potential of a RED unit cell is always lower than the open-circuit potential, due to the ohmic resistance, concentration changes in the boundary layer, and concentration changes in the bulk solutions. The last two sources can be interpreted as non-ohmic resistances [

5,

14]. Non-ohmic resistance is mainly controlled by concentration polarization [

15], which has been investigated and discussed by several researchers in the literature [

15,

16,

17,

18,

19,

20,

21].

Although it has been agreed, by some researchers that increasing the flow velocity and the introduction of flow promoters (i.e., spacers) can mitigate the concentration polarization and enhance the mass transfer by disturbing the diffusive boundary layer [

16,

20,

22,

23], Vermaas et al. [

24] through an experimental work showed that at low Re numbers (less than 100), which are typically used for RED, introducing non-conductive sub-corrugation is not that beneficial to reduce the ohmic losses and increase the power density [

24]. They also showed that although the non-ohmic resistance (concentration boundary layer effects) decreases significantly when increasing the Reynolds number; the ohmic resistances are almost independent of the Re number at high Re numbers and dominates the power loss [

24]. Pawlowski et al. performed an extensive literature review of the development and application of corrugated membranes in electro-membrane-based processes [

25]. They reported the effect of corrugated membranes in the performance of reverse electrodialysis (RED), showing that electrodialysis (ED) is significantly influenced by the shape of the corrugation, Reynolds number, and ion concentrations. For high Reynolds numbers, corrugation creates eddies which lead to enhanced mass transfer, reduced deposition of foulants, and increased diffuse boundary layer thickness. In particular, they highlighted the role of conductive spacers in lowering the resistance of the RED stack, by eliminating spacer shadow effects [

25]. They foresaw the rapid progress of the design and manufacturing of corrugated membranes due to advances in CFD simulations and 3D printing technology [

25]. Gurreri et al. [

26] used CFD modeling to study fluid flow behavior in a reverse electrodialysis stack, aiming to address the effect of the spacer material on the pressure losses along the channel, evaluating the choice of a fiber-structure porous medium, instead of the commonly adopted net spacers, and investigated the influences of the distributor and channel configurations on fluid dynamics in a RED system [

26]. They documented that the total pressure loss in a RED stack is the sum of the pressure drop relevant to the feed distributor, the pressure drop inside the channel, and the pressure drop in the discharging collector [

26]. Simulations revealed that the spacer geometry may not necessarily be the main factor controlling the overall pressure drop. In addition, the pressure drop induced by a porous medium made of small fibers is larger than that for a typical net spacer; therefore, they might not be suitable for RED [

26]. Pawlowski et al. [

27] showed, by CFD modeling, that chevron-corrugated membranes have the highest net produced power density among several investigated profiled membranes, due to increased membrane area, reduction of the concentration polarization, and the proper trade-off between momentum and mass transfer [

27]. These results were validated also through experimental comparison [

28]. Cerva et al. [

29] presented a coupled study of one-dimensional CFD modeling with three-dimensional finite volume modeling for a flat channel, profiled membranes, and different spacer-filled corrugations in a RED stack. Then, they validated the overall model by comparison with experimental data measured in a laboratory [

29]. Their results showed that the boundary layer potential drop is significantly lower than the ohmic losses. In addition, woven spacers had the smallest boundary layer potential loss, followed by Overlapped Crossed Filaments (OCF) profiled membranes and then the flat channel, thus indicating that woven spacers provide the most efficient and effective mixing among the considered systems [

29]. The highest gross power density and the highest short-circuit current density were reported for OCF profiles, followed by the woven spacers and then the flat channel. However, the highest net power density per cell pair was provided by the flat channel, followed by OCF profiled membranes and then by the woven spacers [

29]. Mehdizadeh et al. [

30] experimentally studied several non-conductive spacers with different geometries and properties (e.g., different diameters, angles, distances, area fractions, and volume fractions) to understand the spacer shadow effect on the membrane and solution compartment resistances in RED. They reported a correlation between the spacer shadow effect on the membrane resistance and a combined parameter of spacer area fraction and spacer diameter [

30]. The spacer shadow effect on the solution compartment resistance was also correlated with the spacer area and volume fraction. They observed that the spacer area fraction had a dominant effect only for less porous spacers [

30]. Jalili et al. [

31,

32] used CFD modeling to examine the influence of flow velocities and spacer topology with respect to the transport of mass and momentum, as well as the flow channel resistivity of a RED unit cell. They reported that the resistivity of the dilute solution channel dominates over the resistivity of the concentrated solution channel and membranes in a RED unit cell [

32]. Similar observations have also been reported by Ortiz-Martinez et al. [

33]. The electrical potential of a RED unit cell was enhanced by reducing the flow velocity and introducing flow promoters in a dilute solution channel, due to reduced solution resistance [

32]. Introducing spacers in a concentrated solution channel or increasing the flow velocity in a dilute solution channel increases the resistivity and has adverse effects on the electrical potential [

32]. They also demonstrated that the mass transfer is higher for active membrane-integrated spacers, compared to inactive spacers, under similar flow velocity and spacer topology, due to increased active membrane area [

31]. They also concluded that cylindrical membrane-integrated corrugation is an optimum spacer geometry at low flow velocities, while triangular membrane-integrated corrugation is a better geometry at high flow velocities [

31]. Recently Dong et al. [

34] performed a CFD study of mass and momentum transfer for several types of profiled membrane channels in RED. Their work showed that conductive wavy sub-corrugations improved the mass transfer and reduced the concentration polarization (i.e., non-ohmic losses) [

34]. Furthermore, they showed that single-sided wave-profiled membranes had better performance, compared to single-sided pillar-profiled membranes; while single-sided profiled membranes had a smaller impact on the performance, compared to double-sided chevron-profiled membrane and woven spacer-filled channels [

34].

Long et al. reported a numerical study matched with experimental data for optimizing channel geometry and flow rate of the concentrated and diluted solutions with non-conductive spacers, to obtain maximum net power output by RED. They reported that the optimal channel thickness and flow rate in the concentrated solution compartment in a RED stack are, respectively, much less than those of the dilute solution compartment [

35]. In another work, they revealed that the optimal flow rates in the dilute and concentrated solution channels in an RED stack with varying flow rates along the flow direction to achieve maximum energy efficiency were lower than the optimal flow rates to obtain the maximum net power density. Therefore, an optimization study based on the Non-dominated Sorting Genetic Algorithm II (NSGA-II) was performed, in order to analyze the compromise between the net peak power density and the energy efficiency [

36]. Their work showed that the net power density at maximum energy efficiency was less than the peak power density [

36].

Several researchers have highlighted the potential use of waste heat in RED systems. Luo et al. [

37] reported that by using ammonium bicarbonate as a working fluid in a thermally driven electrochemical generator, waste heat could be converted to electricity [

37]. A maximum power density was obtained at an overall energy efficiency of 0.33 W m

${}^{-2}$, by operating a RED system with a dilute concentration of 0.02 M [

37]. Micari et al. [

38] reported the conversion of waste heat into electricity by coupling RED with membrane distillation (MD), resulting in considerable system energy efficiency improvement. The construction and operation of the first lab-scale prototype unit of a thermolytic reverse electrodialysis heat engine (t-RED HE) for converting low-temperature waste heat into electricity have been reported by Giacalone et al. [

39]. Ortiz-Imedio et al. [

33] documented the strong dependence of the performance of RED on temperature. They reported that the membrane resistance increased when reducing the temperature, and that the perm-selectivity reduced when increasing the temperature [

33]. Jalili et al. [

31] showed that increasing the temperature enhanced the mass transfer of dilute and concentrated solutions, due to higher diffusivity and lower viscosity at increased temperature. In another work, they reported that the open-circuit potential increased with increasing temperature [

2]. Contrary to the most of the literature, which has investigated salinity gradient energy at isothermal conditions, Long et al. [

40] addressed the asymmetric temperature influence in dilute and concentrated solution channels on the performance of nanofluidic power systems, using numerical simulation by coupling the Poisson–Nernst–Planck equation and the Navier–Stokes equation, as well as the energy-conservation equation. They observed that when the temperature of the concentrated solution channel is lower than the temperature of the dilute solution channel, the ion-concentration polarization is suppressed, ion diffusion along the osmotic direction enhances, and perm-selectivity increases; thus, the membrane potential improves [

40]. However when the temperature in the concentrated solution channel is higher than that of the dilute solution channel, the membrane potential reduces; although the diffusion current increases, due to the lower resistance [

40]. In another work [

41], they reported the influences of heat transfer and the membrane thermal conductivity in the performance of nanofluidic energy conversion systems. They reported that when the temperature of the concentrated solution channel is lower than the temperature of the dilute solution channel, a larger membrane thermal conductivity results, with reduced electrical power improvement; on the other hand, when the temperature of the concentrated solution channel is higher than the temperature of the dilute solution channel, the increased membrane thermal conductivity leads to enhanced power density [

41].

Although several studies have reported the application of CFD modeling for investigating momentum and mass transfer in order to determine the trade-off between the pressure loss and mass transfer in an RED channel [

16,

18,

19,

22,

23,

27], there have been limited CFD studies of electrical potential in an RED channel [

42,

43]. To the best of our knowledge, there have been no parametric studies which assessed the relative effect of relevant parameters on the net power density for a RED cell. In particular, addressing the influence of temperature, as proposed by Jalili et al. [

31], was not compared to the other parameters. The current work is an extension of the previously published works [

31,

32] by the current authors. We demonstrate that the electrical potential changes linearly with the height of the channel for a constant concentration profile, and that it follows a logarithmic trend with length of the channel height when the concentration profile varies linearly with the channel height [

32]. Other interesting observations of this work [

32] can be summarized as follows: First, the concentration gradient near the walls of the channel increase, due to reduced boundary layer thickness, with higher Re number. In fact, the concentration at the center of the channel is at its maximum for the concentrated solution channel and is at its minimum for the diluted solution channel [

32]. Second, the pressure drop for the dilute solution channel is lower than that in the concentrated solution channel, given similar Re number and channel geometry [

32]. This observation was also reported by Zhu et al. [

21], when conducting several experiments. Third, the resistance of the dilute solution is more dominant, compared to the resistance of the concentrated solution channel, which can be seen as a limiting factor for the power density of a RED stack. Reducing the Re number (i.e., reducing the velocity at a constant temperature) or introducing corrugation in a dilute solution channel reduces the resistivity of the dilute solution channel by increasing the thickness of the boundary layer, which provides a thicker and more conductive region in the flow channel and results in improved mixing by the developing wakes downstream from the spacers [

32]. An opposite trend was observed for the resistivity of the concentrated solution channel [

32]. This observation was also supported by Long et al. [

35].

This present work describes a numerical framework for simulation of the Navier–Stokes (NS) and Nernst–Planck (NP) system, based on the open source CFD platform OpenFOAM [

44], with the aim of predicting the influence of flow velocity, temperature, and geometry on concentration, pressure drop, electrical potential drop, and net power density. Factorial design [

45] is applied to address the relative effects of the parameters on the peak power density.

## 2. Theory and Governing Equations

The flow in the channel is considered to be two-dimensional, incompressible, steady-state, isothermal, and laminar. Physical properties such as density and viscosity are assumed to be constant. There is charge neutrality in the whole system, where only monovalent ions exist. The Navier–Stokes and Nernst–Plank equations [

42,

46,

47] are presented by Equation (

1) and Equation (

2), respectively.

for species

i, where

${C}_{i}$ is the concentration ([mol/m

${}^{3}$]),

${\mathcal{D}}_{i}$ is the diffusivity ([m

${}^{2}$/s]),

$\overrightarrow{u}$ is the fluid velocity ([m/s]), and

is the electrophoretic mobility ([m

${}^{2}$/Vs]), where

${z}_{i}$ is the valency,

$F=\mathrm{96,485.3}$ C/mol is the Faraday constant,

$R=8.314$ J/K·mol is the universal gas constant, and

T is the temperature (in Kelvin), while

$\varphi $ is the electrostatic potential ([V]).

Assuming two monovalent ionic species, denoted + and -, and using charge neutrality (i.e.,

${C}_{+}={C}_{-}=C$), Equation (

2) can be written as [

31,

32]:

where

$\mathcal{D}$ is the effective diffusivity for the salt and

C is the concentration. The effective diffusivity is assumed to be a function of temperature, using the published data by Bastug and Kuyucak [

48].

The electrical potential can be calculated from the conservation of electrical current density

$\overrightarrow{j}$ [

32],

The electrical current density is obtained by a weighted sum of the charged species, resulting in

where the advective flux cancels out, due to monovalent ions and charge neutrality. Combining Equations (

5) and (

6), we obtain the following relation [

31,

32]:

from which the electrostatic potential can be calculated, given a known concentration field in Equation (

4). The proposed framework essentially consists of four one-way coupled equations—namely the incompressible Navier–Stokes Equation (

1) which, together with continuity, determine the pressure and velocity fields; the concentration Equation (

4), which essentially is an advection–diffusion equation with a known velocity; and, finally, the equation for the electrostatic potential (

7), which is essentially reduced to a Poisson equation with a known source term. Given the domain and boundary conditions described in the following sections, the incompressible Navier–Stokes equations are solved by means of the simpleFoam solver in OpenFOAM, modified to account for concentration and potential following the steps described, for instance, in the openfoamwiki [

49].

The trade-off between maximum produced electrical potential and the current density provides the peak power density. The peak power density,

${P}_{RED}^{peak}$ (W/m

${}^{2}$), of a RED unit cell, the principal parameter of interest in the current work, can be expressed as follows: [

5,

11,

14]:

where

${r}_{unit\phantom{\rule{4pt}{0ex}}cell}$ and

${E}_{OCP}$ represent the area resistance of the unit cell and the open-circuit potential of the unit cell, correspondingly. The area resistance of the unit cell can be calculated by Equation (

9) [

5,

14]:

where

${r}_{AEM}$ and

${r}_{CEM}$ are the area resistances of the AEM and CEM, respectively, and

${r}_{c}$ and

${r}_{d}$ are the total area resistances for concentrated and dilute solution channels, respectively. The open-circuit potential depends upon the concentrations of dilute and concentrated channels as well as temperature, each of which are assumed fixed for a given setup in the current work. Assuming constant membrane properties, the only remaining variables are the area resistances of the channels. The total area resistance of the channels is calculated by dividing area-weighted average of electrical potential difference across the channel by the current density at the peak power density of RED unit cell, as shown by Equation (

10) [

14,

32]:

where

${r}_{j}$ is the total area resistance (ohmic and non-ohmic) of the concentrated or dilute channels,

j is the current density, and

is the difference in area-weighted average of electrical potential

$\mathsf{\Phi}$, calculated on the active membrane. The electrostatic potential across each channel, and thereby also the resistance, can be calculated based on the coupled Nernst–Planck and Navier–Stokes framework, presented in the theory and governing equations section. The formulation used in the current work accounts for both local values and gradients in concentration, and thus accounts for both ohmic and non-ohmic contributions. It should be noted that when dividing the potential drop by the imposed current, as in the above equation, non-ohmic contributions appear as an ohmic potential drop, although they are not of an ohmic nature [

14].

When operating a RED system, the diluted and concentrated solutions are pumped through the compartments between the membranes, which inevitably leads to an energy loss. The required pump power density for each channel can be estimated by Equation (

12) [

14]:

where

A is the membrane area,

Q is the volumetric flow rate through the channel,

H is the height of the channel,

L is the length of the channel,

u is the average velocity in the channel, and

$\mathsf{\Delta}p$ is the pressure drop across the channel length which will be estimated through CFD modeling. To reduce ohmic energy losses in RED systems, the channel height should be as thin as possible; however, as this leads to increased pumping losses, there is a need to find an optimum value though. There are several factors affecting the optimal thickness of the inter-membrane distance, dictated by flow velocity, salinity and hydrodynamic pressure drops, but generally 50–300

$\mathsf{\mu}$m is considered an optimum. This is for sterile particle free systems, but also fouling and other effects in nature can affects this further [

2,

50].

Given the energy consumption in the pump, the net peak power density can be calculated as:

In summary, the net peak power density can be calculated as follows:

- 1.
Coupled flow, concentration and potential fields are calculated through Equations (

1)–(

7).

- 2.
The potential difference across each channel is computed, allowing for determination the corresponding area resistances, as of Equations (

11) and (

10).

- 3.
Unit cell resistances and the peak power densities are calculated based on Equations (

8) and (

9).

- 4.
Pumping power is estimated using Equation (

12), considering the flow velocities, and pressure drop from Equation (

1).

- 5.
The net peak power density is finally computed as of Equation (

13).