Na-Ion Desalination (NID) Enabled by Na-Blocking Membranes and Symmetric Na-Intercalation: Porous-Electrode Modeling

Figure 1 shows the two-dimensional device simulated presently and its dimensions. The NID device is a symmetric Na-ion cell, meaning that it contains the same type of Na intercalant in both electrodes (rather than dissimilar intercalants used in conventional NIBs). These electrodes are chosen with common thickness w and length L (Fig. 1b). The electrodes are porous composites, which enables them to conduct electrons and ions along the x-direction while saline solution flows through them along the y-direction (Fig. 1b). The two electrodes can be isolated (Fig. 1a) by either a polymeric separator (commonly used in NIBs) or an anion-selective membrane that blocks Na ions.

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Next, we describe the operating concept for the present cell. At the beginning of the charge process electroactive material in the cathode acts as a source of Na, while electroactive material in the anode acts as a sink. Both electrodes contain aqueous NaCl with a certain initial concentration defined by the water source of choice. When charging starts, electrons are released into conductive carbon from the electroactive material in the cathode, which de-intercalates Na ions from the cathode electroactive material into the saline water that flows through the cathode (Fig. 1a, left inlay). This electrochemical reaction induces solution-phase charge-imbalance that drives Cl ions from the anode across the separator to the cathode and concentrates cathode solution with Na and Cl ions. A diluted solution forms in the anode (Fig. 1a, right inlay) in which electrons enter the anode electroactive material, causing Na ions to intercalate into electroactive material and forcing Cl ions to leave the anode. This process depletes anode solution with Na and Cl ions. Due to the cathode solution being concentrated and the anode solution being diluted simultaneously, effluent streams of two disparate salt concentrations are generated (Fig. 1a, top). The situation is reversed during the discharge cycle, where cathode solution depletes while anode solution concentrates in Na and Cl ions. Since the accumulating and depleting streams are switched during the discharge cycle, the concentrated effluent and desalinated effluent tanks would be switched when the current direction is switched in a practical, experimental device.

The simultaneous accumulation and depletion of electrolyte-phase salt ions in opposing electrodes is known as the salt depletion effect, which was first observed in rocking-chair Li-ion batteries.¹⁸ The proposed desalination method resembles both conventional CDI (due to porous electrodes) and electrodialysis (due to anion-exchange membrane) processes, however it differs from both because it uses Na-ion intercalants to store charge. In addition, the change in cell polarity between the charge and discharge cycles is similar to electrodialysis reversal,¹² which should decrease membrane fouling since there is no buildup of Cl ions at the anion perm-selective membrane.¹¹ Finally, the anion perm-selective membrane, using the Gibbs-Donnan effect,³⁶ can sustain a large concentration gradient between the electrodes, allowing significant desalination in the depleted electrode.

A porous-electrode model³⁷ is used here to model Na-ion intercalation, ion transport in flowing NaCl solution, and electron transport inside flow-through electrodes. Overpotential η drives Na-ion intercalation in solid electroactive particles and is defined as η = ϕ_s − ϕ_e − ϕ_eq, where ϕ_s, ϕ_e, and ϕ_eq are the solid-phase potential of the electronic conductor, solution-phase potential of Na⁺, and the equilibrium potential of Na-ion intercalation, respectively. The solid- and solution-phase potentials are coupled through their respective current-conservation equations (described later). The equilibrium potentials of NTP and NMO vary with the fraction of intercalated Na, x_Na (defined as the intercalated Na concentration divided by the terminal value). The equilibrium potentials used here, shown in Fig. 2, were estimated from the absolute electrode potentials (i.e., versus a Ag/AgCl reference) measured during charge and discharge of an experimental NMO/NTP cell (Ref. 38) at 0.6 C rate. The terminal concentration of intercalated Na (c_s,max) by which intercalated-Na fraction is normalized was taken as 14,687 mol-Na/m³ and 8,095 mol-Na/m³ for NTP and NMO respectively. These concentrations were chosen to produce theoretical capacities of 133 mAh/g-NTP and 50 mAh/g-NMO reported previously in the literature.³⁸ The reaction current-density i_n for intercalation at the surface of electroactive particles is modeled using the Butler-Volmer equation:³⁹

where R_g, T, and F are the universal gas constant, temperature (here, 298 K), and Faraday's constant, respectively. The exchange-current density i₀ for intercalation reactions⁴⁰ depends on salt concentration in the electrolyte c_e, the number of cations formed from dissociation of one salt molecule s₊, the fraction x_Na and the terminal concentration c_s,max of intercalated Na in electroactive particles, and the kinetic rate-constant k:

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Since no direct measurements of Na-ion intercalation kinetics exist in the literature for NTP and NMO (to our knowledge), we fit experimental polarization data at several C-rates (from Ref. 38) to the predictions of our model by adjusting the rate constant for NTP and NMO independently, the outcomes of which are described in the next sub-section. From this procedure we found NTP and NMO to have Na-intercalation rate-constants of 6.31 × 10⁻¹³ and 2.12 × 10⁻¹¹ mol/m²-s per (mol/m³)^1.5, respectively.

The fraction of intercalated Na inside solid electroactive-particles increases when intercalation reactions occur with the electrolyte. The intercalated-Na fraction can vary throughout the electroactive particles. However, due to the large Na-ion diffusivity in NMO and NTP (greater than 10⁻¹⁶ m²/s in NTP³⁸ and between 1 × 10⁻¹⁷ and 9 × 10⁻¹⁶ m²/s in NMO⁴¹), this variation is less than 10% in both NTP and NMO (based on formulas derived in Ref. 42). Consequently, we neglect this effect. Intercalated-Na fraction x_Na (at a particular point inside a given electrode) evolves with time (according to mass conservation) as a result of intercalation current-density i_n at electroactive-particle surfaces:

where a and ν_s are the volumetric surface area and volume fraction of electroactive particles in the porous electrode, respectively. The volumetric surface area of electroactive particles is 5.73 × 10⁶ m⁻¹ and 2.14 × 10⁷ m⁻¹ for NMO and NTP respectively (assuming spherical particles whose size was measured in Ref. 38).

Na ions are removed from flowing electrolyte as Na intercalates into electroactive particles. Thus, balance of species must be accounted for in a desalination device to preserve solution-phase electroneutrality. We model flow of electrolyte through porous electrodes (containing electroactive particles, electronically conductive carbon, and polymeric binder) using Darcy's Law. When the fluid permeability of the separator/membrane is neglected, the superficial-velocity of electrolyte is uniform and one-dimensional (i.e.,). For the present binary electrolyte we model the transport of Na⁺ and Cl⁻ using dilute-solution theory that is cast in terms of the solution-phase potential ϕ_e of the cation (here, Na⁺) and salt concentration c_e. Salt conservation in the electrolyte can be described by a potential-independent equation for binary electrolytes³⁷:

Salt concentration inside of each electrode is assumed to be uniform initially with the same concentration as the influent solution (that is chosen based on the specific operating conditions investigated). The rightmost term in Eq. 4 is the local rate of salt accumulation (in mol/L-s) that is driven by exchange of Na ions between electroactive particles and electrolyte. This rate is affected by the Na-intercalation current-density i_n, volumetric surface area of electroactive particles a, their volume fraction ν_s, and the electrolyte's Na-ion transference number t₊. Equation 4 assumes a concentration-independent transference number for Na ions t₊ of 0.39 (from the dilute-limit value), which deviates by less than 10% in concentrated solutions.⁴³ D_eff is the effective salt diffusivity, which is reduced from the bulk value D₀ (taken as 1.61 × 10⁻⁹ m²/s for dilute NaCl³⁷) by a factor D_eff/D₀ = ε^1.5 (assuming Bruggeman scaling) that depends on porosity of the electrode ɛ. Here, we neglect the effect of pore-scale dispersion on apparent diffusion through the porous electrode (assuming small pore-scale Peclet number⁴⁴).

Charge transport in the electrolyte is governed by current conservation:³⁷

where κ_eff is the effective ionic conductivity that depends on the corresponding bulk value κ₀ and porosity as κ_eff = ε^1.5κ₀ (assuming Bruggeman scaling). Here, the local ionic current-density is . We use experimental data⁴⁵ to model the dependence of bulk ionic-conductivity on salt concentration.

When a Na-blocking membrane is used to isolate the two electrodes, boundary conditions must be expressed on its opposing sides (labeled here as + and – for cathode and anode sides, respectively). We impose a Neumann condition on the solution-phase potential at these boundaries (i.e., , where is the outward-pointing unit-normal for a given side of the membrane) to enforce null Na-ion transport through an ideal perm-selective membrane. This condition is equivalent to a null-flux condition on cation transport because the solution-phase potential is proportional to the electrochemical potential of the cation that is the driving force for diffusive and migrational transport of the cationic species (see Ref. 46).

Additionally, solution-phase potential is not continuous across the membrane because concentration polarization is produced by the difference in salt concentration inside electrolyte between opposing sides of the membrane. Assuming that the perm-selective membrane is close to equilibrium, the drop in electrostatic potential across the membrane (from cathode to anode) is given by⁴⁷ . The solution-phase potential ϕ_e represents the reduced electrochemical-potential of the cationic species Na⁺ (see Ref. 46), which is defined as³⁷ . Using this relationship we find that the solution-phase potential-drop across the membrane is twice that of the electrostatic potential . Accounting for potential drop in this way is necessary for accurate modeling of the thermodynamic limit of desalination for a device using a membrane. When a porous separator is used to isolate anode and cathode, we neglect the separator's thickness and permeability, in which case Eqs. 4 and 5 are continuously differentiable across the separator and do not require additional boundary conditions. Current-collector and outflow boundaries are modeled as impenetrable to ionic diffusion and migration, and consequently null-flux conditions are applied to diffusive salt transfer and ion current on these boundaries (i.e., and on the boundary whose outward-pointing unit-normal is ). At inflow boundaries a constant-concentration (i.e., Dirichlet) condition is imposed to account for salt diffusion between the cell and the reservoir that feeds saline water to it. We impose null ionic-current on such boundaries, as though an insulator were placed between the influent streams feeding the two electrodes so as to eliminate shunt currents (see Ref. 48) between them.

Finally, electron conduction to and from the external circuit into the composite electrodes is necessary to make the cell operate. The current collectors that adjoin the respective electrodes are assumed to have uniform solid-phase potentials (ϕ_s,+ and ϕ_s,− are used to denote the solid-phase potential of cathode and anode current collectors, respectively). Cell voltage (defined as V_cell = ϕ_{s, +} − ϕ_{s, −}) is adjusted to maintain the average current-density i applied to the current collector at a specified time-independent value (i.e., galvanostatic conditions are imposed) while the anode solid-phase potential ϕ_s,− is grounded at 0 V. Separators and membranes are considered perfect electronic insulators. Solid-phase potential variations inside the porous electrode are governed by current conservation:

where σ_s is the effective electronic conductivity of the heterogeneous electrode. Here, we take a value of 100 S/m that results in small solid-phase potential variations relative to that of the solution-phase potential in the electrolyte. This value is reasonable, considering that Li-ion porous-electrode films can be fabricated with effective electronic conductivity of 20–400 S/m depending on binder and carbon content.⁴⁹ Further, ohmic polarization inside the cell is dominated by bulk ionic-transport resistance when effective electronic-conductivity exceeds the effective ionic-conductivity (here, 1.6 S/m at 700 mol/m³ salt concentration with 40% porosity). We show later that the results are insensitive to the choice of effective electronic-conductivity.

The coupled modeling equations were discretized using the finite-volume method. A first-order implicit scheme was used to integrate in time, while a second-order central-difference scheme was used for non-convective flux and a first-order upwind-difference scheme was used for convective flux. An algorithm used previously to simulate suspension flow-batteries⁵⁰ was modified for use in desalination simulations by (1) incorporating electrolyte potential and concentration variations, (2) anchoring solid parts of the porous electrode, and (3) incorporating continuous flow of the liquid-phase electrolyte. We note that non-linearity of the discrete governing equations requires iteration to obtain a converged solution at a given time step. Here, we employ a sequence of iteration loops to solve the non-linear system of equations (as described in Ref. 51), where electrolyte conductivity is resolved in the outer loop, electrochemical kinetics are resolved in the inner loop, and the aggregation-based algebraic multigrid method^52–55 is used as a linear solver.

The present model builds on a Li-ion battery model without electrolyte flow and whose solution-phase potential, solid-phase potential, and salt concentration fields were validated⁵¹ by comparison with those predicted by the Dualfoil 5 program (Refs. 56,57). The present model captures additional transport processes, including (1) anion-selective membrane transport and membrane polarization, (2) continuously flowing electrolyte, and (3) Na-ion intercalation. To verify implementation and confirm the physical consistency of these additional model features, output predictions were compared with idealized, analytical models. Firstly, a symmetric-NMO cell with a membrane was simulated with stationary electrolyte disconnected from electrolyte reservoirs. The electrodes for these cells were chosen with 1 mm thickness, 20 mm length, 40 vol.% porosity, and 50 vol.% NMO loading. Under these conditions, transient salt-depletion occurs in one electrode while the opposing electrode concentrates simultaneously. Salt concentration and average membrane-polarization simulated at 5 × 10⁻⁵ C differed respectively by 10⁻⁶% and 0.2% from the values expected based on average concentrations predicted by Faraday's Law (, where c⁰_e is the initial concentration). After 5 hours of charging at 5 × 10⁻³ C the concentration profile in the x-direction differed by 10⁻⁵% from that expected from a pseudo-steady solution of Eq. 4 with uniform intercalation current-density (). The same cell was simulated with electrolyte connected to a stationary electrolyte-reservoir at its inlet. When cycled at 5 × 10⁻⁶ C, both electrodes showed salt-concentration distributions in the y-direction that differed by less than 0.04% from those expected from a thickness-averaged, steady solution of Eq. 4 with uniform intercalation current-density and with membrane current-density localized at the inlet (, where cⁱⁿ_e is the inlet concentration). With flowing electrolyte at a superficial velocity of 22 μm/s, the time-averaged effluent-concentration in the desalinated electrode differed by 1% from that predicted by Faraday's Law when cycled at C/2 rate ().

We fitted the model predictions of individual-electrode polarization at 6 C and 12 C for an asymmetric NMO/NTP cell with aqueous Na₂SO₄ electrolyte to experimental data³⁸ to estimate the kinetic rate-constants of NTP and NMO Na-ion intercalation. No other adjustable parameters were fitted. As Table I shows, the model produced polarization levels within 20% of the experimental values between 6 C and 12 C on both electrodes, i.e., this accuracy was obtained by fitting four polarization values with two kinetic parameters. Experimental polarization at 0.6 C (from Ref. 38) was not compared to the model because cycling at that rate produced low experimental coulombic efficiency (∼90%) relative to that of 6 C and 12 C (>98%), suggesting that side reactions affect cycling substantially at that rate. We note also that the charge capacities predicted between 6 C and 12 C exceed experimental values by 20–30 mAh/g-NTP (the experimental capacity was 104 mAh/g-NTP at 6 C versus a modeled capacity of 126 mAh/g-NTP). This discrepancy is not unexpected considering that the present model neglects the effects of side reactions and electroactive-material decomposition that have been reported for experimental NIBs using aqueous electrolyte.^20,38 Holding to an optimistic outlook for the development of aqueous NIBs with long cycle-life (in which side and decomposition reactions have been mitigated), the present model optimistically predicts the performance of NID cells.

Three types of symmetric Na-intercalation cells were simulated here with an average applied current density of 51 A/m²: (1) NMO electroactive material with a Na-blocking membrane, (2) NTP electroactive material with a Na-blocking membrane, and (3) NMO electroactive material with a porous separator that is not ion-selective. The targeted degree-of-desalination is 68%. A charge balance on the cell shows that achieving the 68% desalination target requires a pore-scale mean-velocity (i.e., the product of porosity and superficial velocity) of 56 μm/s, which corresponds to an intra-cell water residence-time of 360 seconds. Hence NMO cells, with a C-rate of 0.5 hr⁻¹, theoretically allow a net desalinated-water output that is 20 times the open-pore cell volume during each charge/discharge cycle. The same flow velocity was used in both electrodes to maintain 50% water recovery.

Figure 3 shows the variation of cell voltage for the three cells of interest. The distribution of intercalated Na is shown at nine instants in time as well. Initially, electroactive particles in the cathode and anode, respectively, are concentrated with and depleted in intercalated Na (snapshot i in Figs. 3a–3c). This produces a negative cell voltage initially because equilibrium potential of a given electroactive material decreases with increasing intercalated-Na fraction, which requires each cell to operate in a mode where energy is recovered from the cell rather than delivered to it. As we show later, the net energy consumed will depend on the efficiency with which energy is recovered. Cell voltage increases during charging, becoming positive after a certain period of time that depends on the type of electroactive material used (NMO or NTP here). Charging stops once the cutoff cell-voltage is reached (chosen as 0.455 V for NMO and 0.200 V for NTP). For the NMO cells (Figs. 3a and 3c) nearly complete utilization of electroactive-material capacity is obtained (snapshot v), while electroactive material at the outlet of the NTP cell is underutilized at the end of the charge cycle (Fig. 3b, snapshot v). Despite this, the NTP cell cycles for a longer period of time (2.8 hr for NTP versus 1.9 hr for NMO) because NTP produces a higher areal capacity (16.1 mAh/cm²) than NMO (10.2 mAh/cm²).

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When the direction of current is reversed, the cell begins to discharge current from the cathode and cell voltage decreases. Discharging stops once the low-potential cutoff-voltage is reached (−0.455 V for NMO and −0.200 V for NTP). The gap between the charge (black) and discharge (red) cell-voltage curves is equal to twice the cell polarization, which is a measure of the net amount of energy required to desalinate (without energy recovery losses). Among the three cells simulated, the NTP-Membrane cell shows the highest cell-level polarization, followed by the NMO-Membrane cell. The NTP-Membrane cell has highest polarization due to the low reaction rate-constant of NTP. Snapshots i-ix in Figs. 3a and 3b show that a reaction zone propagates from the inlet to the outlet in the two cells having Na-blocking membranes, while a separator allows for more uniform intercalation of Na.

Though the NMO-Separator cell shows lowest polarization and highly uniform reactions, these improvements occur with low degree-of-desalination. These results show that a Na-blocking membrane must be used to achieve high degree-of-desalination. Figure 4 shows the time variation of effluent salt-concentration (or salinity in mol/L) in the concentrated and diluted electrodes. Also, the distribution of salinity inside each electrode is shown at several instants in time. Initially salt concentration changes uniformly inside each electrode (before snapshot ii), but eventually the membrane-based cells in Figs. 4a and 4b (using NMO and NTP, respectively) show steady effluent salt-concentrations and exhibit near-theoretical degree of desalination based on charge counting (65%). The Na-blocking membrane enables highly concentrated salt-solution to persist in the outlet of the cell without diffusing across the membrane (Figs. 4a and 4b, snapshots ii and iii). In contrast, the NMO-Separator cell (Fig. 4c) shows only 25% degree-of-desalination. The separator-based cell achieves less than half the theoretical degree-of-desalination because the separator allows salt to diffuse from the concentrated electrode into the diluted electrode.

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The robustness of the present results was confirmed by simulating the sensitivity of desalination performance to the values assumed for the effective electronic conductivity σ_s of the porous electrodes and the kinetic rate-constant k for Na intercalation. Table II shows that the energy consumed during desalination (with lossless energy recovery) by NTP and NMO cells (using membranes) increases by 2% when electronic conductivity is reduced one order of magnitude from 100 S/m to 10 S/m. Also, electroactive-material utilization decreases by less than 1% from 100 S/m to 10 S/m. These results indicate that the predictability of the present simulations is not limited by variations in effective electronic-conductivity that could result from electrode processing. Table III shows that desalination energy for both cells decreases by 4–9% when the kinetic rate-constant increases by one order of magnitude from the value fitted to experimental polarization. Thus, the present results are indicative of electroactive materials with facile intercalation kinetics. Reducing the rate constant by an order of magnitude produces 43% and 98% higher desalination energy for NMO and NTP cells, respectively. While the NMO cell is able to achieve substantial utilization (74%), the NTP cell does not function effectively with such low rate constants, achieving less than 1% utilization. Thus, facile intercalation kinetics are needed for NID cells to function.

A critical consideration in the design of NID cells is the electrochemical stability window of water, because a stable electrolyte will enable efficient, reversible cycling. Reduction potentials range between 0.35 to 0.80 V vs. SHE inside NMO-Membrane cells and −0.62 to −0.40 V vs. SHE inside NTP-Membrane cells for the cases shown in Figs. 3 and 4. O₂ evolution occurs for neutral pH at potentials higher than 0.81 V vs. SHE, and H₂ evolution occurs at potentials less than −0.41 V vs. SHE. Seawater pH is slightly basic, with a pH of about 8.⁵⁸ Therefore, O₂/H₂ evolution is unlikely to occur during cycling of NMO-based cells when seawater is desalinated. In contrast, NTP-based cells will likely evolve H₂ in seawater unless pH exceeds 11.5. Also, we note that previous experiments²⁰ suggest that NTP decomposition occurs in alkaline solutions with pH > 9, and this effect may require use of NTP cells outside of the electrolyte stability-window. We note that the present model does not account for the mechanism of electroactive-material mediated hydrolysis believed to occur with other aqueous electroactive materials,³⁸ and experimental investigation is needed to assess these effects. Also, the concentration and dilution of salt could induce a shift in pH inside the cell and that effect has not been accounted for presently.

When a Na-ion cell with a Na-blocking membrane is used to desalinate water, the rate at which desalination occurs is affected by a variety of parameters. High current densities can be used to desalinate water at high throughput rates by increasing flow rate in proportion to current density. Also, the cost of a Na-ion cell decreases as electrode thickness increases because the fraction of cost from inactive cell components (e.g., membranes and current collectors) is reduced with cells having thick electrodes. Therefore, understanding the effects of current density and electrode thickness on desalination performance will enable the design and operation of low-cost, efficient desalination devices.

We first investigate the effect of applied current-density on desalination performance (Fig. 5). Among the different current densities tested, the water flow-rate was varied in proportion with current density, so as to maintain the same theoretical degree-of-desalination expected from Faraday's Law. For influent seawater with 700 mM NaCl, the present cases are expected to achieve 224 mM salinity in the depleted stream (68% degree-of-desalination). For both NTP and NMO cells, current densities below 77 A/m² produce salinity levels less than 260 mM on average in the depleted stream (63% degree-of-desalination), which is near the 68% theoretical degree-of-desalination (Fig. 5b). For the lowest current density simulated (5 A/m²) the desalination energy needed with lossless energy recovery exceeds the thermodynamic minimum (0.49 kWh/m³, see Ref. 59) by 32% for NMO and 58% for NTP. While energy recovery has negligible impact on NTP cells (due to the large range of intercalation over which equilibrium potential is constant as shown in Fig. 2), NMO cells cycled at low current density require twice as much energy when energy recovery is not utilized. We also observe that the utilization of electroactive-material charge-capacity decreases with increasing current density (Fig. 5c). At high current densities polarization (Fig. 5a) causes cell voltage to reach the pre-specified voltage cutoff value earlier in both the charge and discharge processes. When this occurs, a smaller amount of water is desalinated during a complete cycle and is manifested as decreased utilization (less than 50%) at current densities higher than 77 A/m². Also, the desalination energy (in kWh/m³, Fig. 5d) varies dramatically for current densities in this range: 0.7–0.9 kWh/m³ is needed to desalinate at 13 A/m², while 2.0–2.2 kWh/m³ is needed to desalinate at 77 A/m². High current densities enable rapid production of desalinated water (since flow velocity is proportional to the current density), but the energy required to produce it may dictate use of lower current densities. Also, NTP cells show higher polarization than NMO cells at the same current density, which is consistent with the results in Figs. 3 and 4. NTP cells show higher kinetic polarization due to its lower kinetic rate-constant compared to NMO (6.31 × 10⁻¹³ versus 2.12 × 10⁻¹¹ mol/m²-s per (mol/m³)^1.5).

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As mentioned previously, electrode thickness will affect the cost of an NID cell. Therefore, it is desirable to understand how design changes to this dimension will affect cell performance. When the C-rate is held constant among different electrode thicknesses, the theoretical time-scale for charge and discharge is held constant among all cases. Figure 6 shows the desalination performance for various electrode thicknesses simulated at 0.25 C, 0.50 C, and 1.00 C rate (corresponding to theoretical charge times of 4 hr, 2 hr, and 1 hr, respectively). In addition the pore-scale mean-velocities were set to 28 μm/s, 56 μm/s and 111 μm/s for 0.25 C, 0.50 C, and 1.00 C, so as to achieve 68% theoretical degree-of-desalination in all cases.

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For a given C-rate, cell polarization increases quadratically with electrode thickness (Fig. 6a) due to the dominant role of ohmic drop through the electrolyte, limiting the degree-of-desalination, utilization, and efficiency of thick electrodes (Figs. 6b–6d). At 0.25 C, 0.50 C, and 1.00 C, thicknesses of 0.75 mm, 1.25 mm, and 1.75 mm (respectively) operate with 70% utilization and 60% degree-of-desalination, but require three times the minimum desalination-energy expected from thermodynamics even with lossless energy-recovery. In contrast, thicknesses of 250 μm, 500 μm, and 750 μm can desalinate within 50% of the thermodynamic minimum (0.72–0.74 kWh/m³) when cycled at 0.25 C, 0.50 C, and 1.00 C with lossless energy-recovery, respectively.

The specific application for which a desalination device is used dictates the salt concentration of the influent and the amount of effluent concentrate that is tolerable. Therefore, we predict the desalination performance that can be achieved when using water sources with influent salinity ranging from levels of typical brackish water (70 mM) up to those of typical seawater (700 mM). Though moderate levels of water recovery are acceptable when potable water is processed from abundant sources, higher water-recovery is required where influent feed-water is scarce. Accordingly, we predict that Na-ion cells can achieve high water-recovery by simulating desalination performance for 50–95% water-recovery levels.

Figure 7 shows desalination performance as a function of influent salt-concentration. In each cell, C-rate was reduced in proportion with influent salt-concentration (Fig. 7a) because brackish water has lower ionic conductivity than seawater and, consequently, cells operating with brackish influent will have lower rate capability than those operating with seawater. Among the influent concentrations simulated, influent velocity was varied so as to achieve 68% theoretical degree-of-desalination. Both 1.00 mm- and 0.50 mm-thick electrodes were simulated at each influent concentration. For a given thickness, polarization, degree-of-desalination, and electroactive-material utilization varied by less than 5 mV, 5%, and 1%, respectively, among the different influent concentrations (Figs. 7b–7d). While the aforementioned performance metrics are invariant with influent salinity, desalination energy decreases as influent concentration decreases (Fig. 7e) because the amount of charge-transfer (in coulombs per m³ of solution) required to achieve a certain degree-of-desalination decreases with influent salinity. For 1.00 mm-thick electrodes desalination energy was 2.5 times the thermodynamic minimum-energy with lossless energy-recovery (Fig. 7e), but desalination energy decreases to within 40% of the thermodynamic minimum for 0.50 mm-thick electrodes (Fig. 7e).

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Water recovery r is defined as the fraction of desalinated water-volume relative to influent water-volume. The symmetric cells simulated here are designed with equal cathode and anode electrode thicknesses (i.e., w_a = w_c). Therefore, water recovery r increases as superficial velocity u_s,d in the depleting electrode (anode on charge, cathode on discharge) increases and as superficial velocity u_s,a in the accumulating electrode (cathode on charge, anode on discharge) decreases, according to the following relationship r = u_{s, d}/(u_{s, d} + u_{s, a}). To adjust water recovery we reduce the superficial velocity in the accumulating electrode and keep the superficial velocity in the depleting electrode constant (Fig. 8a). When charging stops and discharging starts, the direction of current is switched. Consequently, superficial velocities in the respective electrodes are swapped. In an experimental device, appropriate plumbing will be required to achieve these flow conditions. Cell performance is shown in Fig. 8 as a function of water recovery for 700 mM and 70 mM influent concentrations. For both influents, polarization and desalination energy (Figs. 8b,8d) increase mildly with water recovery due to the increase in Donnan potential resulting from the high salt-concentrations produced at high water recovery. We note also that the desalination energy with lossless energy-recovery for high water-recovery levels exceeds the thermodynamic minimum by a lesser amount (84%) than with 50% water-recovery.

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With 700 mM influent the degree-of-desalination decreased from 65% to 30% between water-recovery levels of 80% and 95% (Fig. 8c). This effect results from the discharge process beginning with residual concentrated solution from the charge process. This effect is illustrated in Fig. 9a, which shows the variation of effluent concentration with time for 700 mM influent with 80% water recovery. Immediately following completion of the charge process (1.75 hr), the discharge process commences with an effluent concentration (2 mol/L) far in excess of the inlet concentration (0.7 mol/L). Desalination of residual NaCl delays production of desalinated effluent with the target concentration (224 mM) by 30 minutes, which constitutes a substantial fraction of the entire discharge process (1.5 hours). For higher water-recovery levels this effect becomes more significant. In addition effluent concentrations in excess of the NaCl solubility limit were predicted at 90% water-recovery. For lower levels of water recovery the degree-of-desalination can be enhanced by purging the cell with fresh effluent immediately following a charge or discharge step.

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In contrast, cells cycled with 70 mM influent sustained degree-of-desalination levels of greater than 64% for water recovery levels as high as 95%. The transient effluent-concentration in 70 mM cells exhibits similar effects due to residual NaCl that are similar to those in 700 mM cells, but the time delay to production of target concentrations is very short (<30 min) relative to the total discharge time (15 hr). Furthermore, the maximum effluent-concentration reached during the entire charge/discharge process at 95% water recovery (∼1 mol/L) was below the NaCl solubility limit. Thus, high water recovery levels are possible for NID devices using 70 mM influent.