Electrodialysis of Lithium Sulphate Solution: Model Development and Validation

The ions concentrations and velocity of the electrolyte solutions vary along the channels due to the fluxes of ions and water through the IEMs, which can be determined using material balance relations. Due to the transfer of ions and water molecules from the dilute channel to the concentrate channels, the concentration and velocity of the solution increase at the concentrate channels, while these quantities decrease in the dilute channel downstream. Therefore, the mass conservation equation for the solution and ions are presented in Eqs. 1 and 2, respectively, for all compartments as: ³²

$$\begin{eqnarray}&&\left|{J}_{w}\right|={ab}\left|{u}_{{out}}{-u}_{{in}}\right|\end{eqnarray} \tag{ 1 }$$

where $a$ and $b,$ shown in Fig. 1, are the width, and length of channels, respectively, ${J}_{w},$ ${u}_{{out}}$ and ${u}_{{in}}$ are the transmembrane water flux through IEMs, outlet and inlet velocities, respectively. Transmembrane water flux is written as ${J}_{w}={J}_{w}^{{AEM}},$ and ${J}_{w}={J}_{w}^{{CEM}},$ for the anode and cathode channels, respectively. For the dilute channel, ${J}_{w}$ consists of the flux of water molecules moving through both anion and cation exchange membranes, i.e., ${{J}_{w}=J}_{w}^{{CEM}}+{J}_{w}^{{AEM}}.$

The relationship between species flux across the membrane direction and net flow through each compartment can be expressed as:

$$\begin{eqnarray}&&{S}_{m}\left|{J}_{i}\right|={ab}\left|{{c}_{i,\,{out}}u}_{{out}}-{c}_{i,\,{in}}{u}_{{in}}\right|\end{eqnarray} \tag{ 2 }$$

with

$$\begin{eqnarray}&&{S}_{m}={bh}\end{eqnarray} \tag{ 3 }$$

where ${S}_{m},$ and $h$ are the membrane surface area and channels height, ${J}_{i}$ is the flux of species $i$ through an IEM, ${c}_{i,\,in}$ and ${c}_{i,\,out}$ are the concentration of specie $i$ at the inlet and outlet of channels, respectively.

According to Eqs. 1 and 2, in order to find the outlet velocities and outlet concentration of lithium and sulphate ions, the flux of water and ions moving across IEMs must be determined first. Since the current remains constant during the ED operation and the flux of co-ions through IEMs is assumed to be zero, the flux of lithium and sulphate ions across IEMs are determined using Faraday's law as follows: ³³

$$\begin{eqnarray}&&I={z}_{i}F{J}_{i}\end{eqnarray} \tag{ 4 }$$

where $F$ is Faraday's constant, ${z}_{i}$ is the valence of ion $i,$ and $I$ is current density defined as current divided per membrane area.

Water molecules tend to penetrate a membrane due to osmotic and electro-osmotic mechanisms. In the ED process, the osmotic water transport, i.e., the transfer of water molecules caused by the osmotic pressure or concentration difference across the membrane is negligible in comparison with the electro-osmotic phenomenon established by the transport of hydrated ions under the influence of an electrical potential gradient. ^22,33–35 Hence, to determine the velocity of water molecules across a membrane, transmembrane water velocity, it is assumed that water molecules are dragged into IEMs by electro-osmotic force, which is a function of the applied electrical potential and the friction force that the ions encounter in the solution. Since these forces are equal in a steady-state condition, the water velocity through an IEM, ${u}_{m},$ can be calculated as: ^22,35

$$\begin{eqnarray}&&{u}_{m}=\sigma \displaystyle \frac{{r}_{i}}{\eta }\displaystyle \frac{{\rm{\Delta }}{\phi }_{m}}{{l}_{m}}\end{eqnarray} \tag{ 5 }$$

where ${l}_{m}$ is the thickness of the membrane, ${\rm{\Delta }}{\varphi }_{m}$ is potential difference across the membrane, $\eta$ is dynamic viscosity, ${r}_{i}$ is the radius of counter-ion $i$ moving within the IEM, ²² and $\sigma$ is the membrane surface charge density, i.e., the surface charge divided per the membrane area, that can be evaluated using the Helmholtz double-layer model:

$$\begin{eqnarray}&&\sigma =\displaystyle \frac{\,q}{{S}_{m}}\end{eqnarray} \tag{ 6 }$$

where $q$ is the surface charge. According to the Helmholtz model, the ions arranged at an IEM surface and the oppositely signed ions appealed to that membrane surface can be considered as capacitor plates at the distance of ${r}_{i},$ as depicted in Fig. 2(a). ³⁵ Therefore, based on the capacity equation of capacitors, ³⁶ Eq. 7 was proposed to evaluate membrane surface charge density. ³⁵

$$\begin{eqnarray}&&\sigma =\displaystyle \frac{\,\varepsilon \zeta }{{r}_{i}}\end{eqnarray} \tag{ 7 }$$

where $\zeta$ is called zeta potential representing the Helmholtz double-layer potential, as depicted in Fig. 2(b). ³⁷ In the present study, the following relation is proposed to calculate zeta potential based on the surface charge and Eqs. 6 and 7.

$$\begin{eqnarray}&&\zeta ={z}_{i}{d}_{i}F{c}_{i,ave}^{m}\displaystyle \frac{{r}_{i}}{\varepsilon }\end{eqnarray} \tag{ 8 }$$

where ${c}_{i,ave}^{m}$ is the average concentration of species $i$ in a membrane, and ${d}_{i}$ is the diameter of counter-ion $i$ transferring through the membrane. In addition, $\varepsilon$ is the permittivity of solvent and is defined as multiplying of the relative permittivity (dielectric constant), ${\varepsilon }_{r},$ and the vacuum permittivity, ${\varepsilon }_{0},$ as follows: ³⁸

$$\begin{eqnarray}&&\varepsilon ={\varepsilon }_{r}{\varepsilon }_{0}\end{eqnarray} \tag{ 9 }$$

where ${\varepsilon }_{0}$ is a constant value equal to 8.854 × 10⁻¹² F·m⁻¹ and ${\varepsilon }_{r}$ is calculated using Eq. 10 for dilute solutions: ^21,39,40

$$\begin{eqnarray}&&{\varepsilon }_{r}=185.765-0.35963T\end{eqnarray} \tag{ 10 }$$

Finally, substituting Eq. 7 into Eq. 5 leads to:

$$\begin{eqnarray}&&{u}_{m}=\displaystyle \frac{\varepsilon \zeta }{{\eta }_{m}}\displaystyle \frac{{\rm{\Delta }}{\varphi }_{m}}{{l}_{m}}\end{eqnarray} \tag{ 11 }$$

and the flux of water is determined by:

$$\begin{eqnarray}&&{J}_{w}={S}_{m}{u}_{m}\end{eqnarray} \tag{ 12 }$$

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In order to estimate the concentration of species at the solution adjacent to an IEM, Tanaka firstly developed Nernst-Planck equations for co- and counter-ions transferring at diffusion layer adjacent to the IEM. ²⁸ Based on his methodology, Eq. 13 is developed to evaluate the concentration of ions at the solution side of an IEM-solution interface (${c}_{i}^{s}$) as a function of the bulk concentration, transmembrane water velocity, and diffusion coefficient of mobile ions in a single-salt solution: ²⁸

$$\begin{eqnarray}&&{c}_{i}^{s}=\left[\displaystyle \frac{{\beta }I}{{\alpha }{u}_{m}}+\left({c}_{i}^{bulk}-\displaystyle \frac{{\beta }I}{{\alpha }{u}_{m}}\right){e}^{-{\alpha }{\delta }| {u}_{m}| }\right]\end{eqnarray} \tag{ 13 }$$

where $\delta$ is the thickness of diffusion layer, and superscript $bulk$ refers to the bulk solution. The following relations were introduced by Tanaka to define $\alpha$ and $\beta$ coefficients for monovalent salts, e.g., NaCl: ²⁸

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{{D}_{+}^{s}+{D}_{-}^{s}}{2{D}_{+}^{s}{D}_{-}^{s}}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\beta =\displaystyle \frac{{t}_{+}{D}_{-}^{s}-{t}_{-}{D}_{+}^{s}}{2F{D}_{+}^{s}{D}_{-}^{s}}\end{eqnarray} \tag{ 15 }$$

where subscripts $+$ and − refer to cations and anions, respectively, $t$ is the transport numbers of ions in a membrane, and ${D}^{s}$ demonstrates the diffusion coefficient of ions at the solution. The present study proposes the equations listed in Table I to determine $\alpha$ and $\beta$ for divalent salts such as Li₂SO₄ as well as H₂SO₄ based on the development of Nernst-Plank equations at the diffusion boundary layers of IEMs vicinity.

Table I.  $\alpha$ and $\beta$ coefficients for all solution-membrane interfaces.

solution-membrane interface	Solution	$\alpha ,$ s·m−2	$\beta ,$ mol·A−1·m−2
Base channel-CEM	$LiOH$	$\tfrac{{D}_{Li}^{s}+{D}_{OH}^{s}}{2{D}_{Li}^{s}{D}_{OH}^{s}}$	$\beta =\tfrac{{t}_{Li}{D}_{OH}^{s}-{t}_{OH}{D}_{Li}^{s}}{2F{D}_{Li}^{s}{D}_{OH}^{s}}$
Dilute channel-CEM	$L{i}_{2}S{O}_{4}$	$\tfrac{{D}_{Li}^{s}+2{D}_{SO4}^{s}}{3{D}_{Li}^{s}{D}_{SO4}^{s}}$	$\beta =\tfrac{2{t}_{Li}{D}_{SO4}^{s}-{t}_{So4}{D}_{Li}^{s}}{6F{D}_{Li}^{s}{D}_{SO4}^{s}}$
Dilute channel-AEM	$L{i}_{2}S{O}_{4}$	$\tfrac{{D}_{Li}^{s}+2{D}_{SO4}^{s}}{3{D}_{Li}^{s}{D}_{SO4}^{s}}$	$\beta =\tfrac{2{t}_{Li}{D}_{SO4}^{s}-{t}_{So4}{D}_{Li}}{6F{D}_{Li}^{s}{D}_{SO4}^{s}}$
Acid channel-AEM	${H}_{2}S{O}_{4}$	$\tfrac{{D}_{H}^{s}+2{D}_{SO4}}{3{D}_{H}^{s}{D}_{SO4}}$	$\beta =\tfrac{2{t}_{H}{D}_{SO4}^{s}-{t}_{So4}{D}_{H}^{s}}{6F{D}_{H}^{s}{D}_{SO4}^{s}}$

It is noted that according to the assumption of zero co-ions flux through the IEM, the transport number of co-ions through IEMs, i.e., sulphate and lithium ions through cation- and anion-exchange membrane, respectively, is considered zero. In addition, water dissociation, i.e., H⁺ and OH⁻ production, occurs in the over-limiting current densities. Therefore, the transport number of H⁺ and OH⁻ species at the concentrate interfaces of IEMs would be negligible when the applied current density is lower than the limiting amount.

To predict the concentration of ions in the membrane side of solution-membrane interfaces, the Donnan potential theory is used in the present study. According to the Donnan potential equation, this potential difference of an IEM-solution interface is a function of pressure difference across the IEM and ion concentration at the solution and IEM sides of the IEM-solution interface. Recall the aforementioned assumption, the effect of pressure difference is neglected, the Donnan potential equation can be written as follows: ^18,41,42

$$\begin{eqnarray}&&{\phi }^{m}-{\phi }^{s}=\displaystyle \frac{-RT}{{z}_{i}F}\,\mathrm{ln}\left(\displaystyle \frac{{c}_{i}^{S}}{{c}_{i}^{m}}\right)\end{eqnarray} \tag{ 16 }$$

in which superscripts $m$ and $s$ refer to IEM and solution sides of IEM-solution interfaces.

On the other hand, to calculate the potential difference that occurs at the IEM-solution interfaces, Eq. 17 is developed based on the Helmholtz double-layer model:

$$\begin{eqnarray}&&{\phi }^{m}-{\phi }^{s}=\pm \left(\displaystyle \frac{{l}_{i}^{d}}{\varepsilon }F{d}_{i}{c}_{i}^{S}\right)\end{eqnarray} \tag{ 17 }$$

where positive and negative signs allocate to CEM and AEM, respectively, and ${l}_{i}^{d}$ is the thickness of the double layer and is considered equal to the hydrated ion radius (${r}_{i}^{H}$) or Debye length (${r}_{D}$) based on the Helmholtz and Gouy-Chapman double layer models, respectively. The Debye length is given as: ⁴³

$$\begin{eqnarray}&&{r}_{D}={\left(\displaystyle \frac{\in RT}{\displaystyle {\sum }_{i=1}^{N}{c}_{i}{z}_{i}^{2}{F}^{2}}\right)}^{0.5}\end{eqnarray} \tag{ 18 }$$

Finally, ${c}_{i}^{m}$ is evaluated by substituting Eq. 17 into Eq. 16:

$$\begin{eqnarray}&&{c}_{i}^{m}={c}_{i}^{S}\exp \left(\pm \displaystyle \frac{{z}_{i}{l}_{i}^{d}}{\varepsilon RT}{F}^{2}{d}_{i}{c}_{i}^{S}\right)\end{eqnarray} \tag{ 19 }$$

where positive and negative signs allocate to CEM and AEM, respectively

The Nernst-Planck equation is considered for cation- and anion-exchange membranes to calculate the voltage difference across IEMs as follows: ^35,44

$$\begin{eqnarray}&&{J}_{i}=-\displaystyle \frac{{z}_{i}F{D}_{i}{c}_{i,ave}^{m}}{RT}\displaystyle \frac{{\rm{\Delta }}{\varphi }_{m}}{{l}_{m}}-{D}_{i}^{m}\left(\displaystyle \frac{d{c}_{i}^{m}}{{l}_{m}}\right)+{u}_{m}{c}_{i,ave}^{m}\end{eqnarray} \tag{ 20 }$$

where the first, second and third terms on the right-hand side are the migration, diffusion, and convection fluxes, respectively. The convection term is caused by osmotic pressure and electro-osmotic phenomena, and as explained before, the effect of osmotic pressure can be neglected in electrodialysis processes. ⁴⁵ In this term, $R$ is the universal gas constant, ${D}_{i}^{m}$ is the diffusion coefficient of species $i$ at membrane phase, $d{c}_{i}^{m}$ indicates the concentration difference of species $i$ across the IEM, and ${c}_{i,ave}^{m}$ is the average concentration of ion $i$ at the IEM. Due to computing ${c}_{i}^{m}$ from Eq. 19 and ${u}_{m}$ from Eq. 11, the potential difference across a membrane can be evaluated using Eq. 20.

The diffusion coefficient of ions in a solution at 298 K can be calculated based on the limiting (zero concentration) ionic conductance using the following relation: ^46,47

$$\begin{eqnarray}&&{D}_{i}^{s}=\displaystyle \frac{RT}{{z}_{i}^{2}{F}^{2}}{\lambda }_{i}^{T}\end{eqnarray} \tag{ 21 }$$

here ${\lambda }_{i}^{T},$ which represents the limiting ionic conductance of species $i$ at the temperatures T, is calculated as follow: ⁴⁸

$$\begin{eqnarray}&&{\lambda }_{i}^{T}=\displaystyle \frac{T}{334\eta }{\lambda }_{i}^{0}\end{eqnarray} \tag{ 22 }$$

in which $\eta$ is the water viscosity at the desired temperature, in centipoises, and ${\lambda }_{i}^{0}$ is the limiting ionic conductance of species $i$ at 298 K and is available for various ions in literature. ^48,49

On the other hand, the following relation is used to predict the amount of diffusion coefficient in an IEM based on the diffusivity of ions in water. ^19,50

$$\begin{eqnarray}&&{D}_{i}^{m}={\left(\displaystyle \frac{p}{2-p}\right)}^{2}{D}_{i}^{s}\end{eqnarray} \tag{ 23 }$$

in which $p$ is the volume fraction of liquid water in the IEM. Also, Eq. 25 is proposed to calculate $p$ from membrane properties, namely water uptake, ⁵¹ presented in Eq. 24, and dry membrane density.

$$\begin{eqnarray}&&wu=\displaystyle \frac{{m}_{w}-{m}_{d}}{{m}_{d}}\times 100\end{eqnarray} \tag{ 24 }$$

where ${m}_{d},$ and ${m}_{w}$ demonstrate the dry and wet weight of membrane, respectively, and $wu$ is water uptake. Using Eq. 24, we have: ⁵²

$$\begin{eqnarray}&&p=\displaystyle \frac{wu}{wu+\displaystyle \frac{{\rho }_{w}}{{\rho }_{p}}}\end{eqnarray} \tag{ 25 }$$

where ${\rho }_{p}$ and ${\rho }_{w}$ are dry membrane and water densities, respectively. The parameters for the IEMs used in the present study are presented in Table III.

In the batch configuration, material balance equations are employed to consider the effect of electrolyte circulation between an ED and external tanks, which are filled by LiOH, H₂SO₄, and Li₂SO₄ solutions. As shown in Fig. 1, these solutions enter their corresponding compartments in the ED cell, and after flowing along these compartments and completing the electrodialysis process, they are discharged from the ED and recirculate back to their own tanks.

First of all, the number of cycles (${n}_{{\rm{\max }}}$) is determined from the total operation time of the ED (${t}_{total}$) and the resident time of solutions in the ED (${\rm{\Delta }}t$), i.e., the time which lasts until the solutions are discharged from the ED cell:

$$\begin{eqnarray}&&{n}_{{\rm{\max }}}=round\left(\displaystyle \frac{{t}_{total}}{{\rm{\Delta }}t}\right)\end{eqnarray} \tag{ 26 }$$

The investigation of the changes of various parameters in ED compartments and tanks is done cycle by cycle. To find the concentration of solutions in each tank, Eq. 27 is proposed based on the mass conservation equation. ³²

$$\begin{eqnarray}&&{c}_{i,tank}^{n}{V}_{tank}^{n}={c}_{i,tank}^{n-1}{V}_{tank}^{n-1}+{{\rm{c}}}_{i,\,out}^{n-1}{V}_{out}^{n-1}-{{\rm{c}}}_{i,\,in}^{n-1}{V}_{in}^{n-1}\end{eqnarray} \tag{ 27 }$$

where superscript $n$ is the cycle number and varies from one to ${n}_{{\rm{\max }}},$ and ${c}_{i,tank},$ ${c}_{i,out},$ and ${c}_{i,in}$ are the concentration of species $i$ at the tank, and at the outlet and inlet of the ED, respectively. Since the solutions entering the ED has the same concentration as tanks solutions, ${{\rm{c}}}_{i,\,in}^{n-1}={c}_{i,tank}^{n-1},$ ${{\rm{c}}}_{i,\,out}^{n-1}$ is calculated from Eq. 27.

Furthermore, ${V}_{tank}$ in Eq. 27 is the solution volume in each tank, and ${V}_{{\rm{out}}}^{n-1}$ and ${V}_{in}^{n-1}$ are the volume of the solutions which are respectively discharged and entered the ED at the n-1th cycle. To determine ${V}_{tank},$ the following relations is proposed based on the mass balance of solutions: ³²

$$\begin{eqnarray}&&{V}_{tank}^{n}={V}_{tank}^{n-1}+{V}_{out}^{n-1}-{V}_{in}^{n-1}\end{eqnarray} \tag{ 28 }$$

where

$$\begin{eqnarray}&&{V}_{out}^{n-1}={u}_{out}^{n-1}ab{\rm{\Delta }}t\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&{V}_{in}^{n-1}={u}_{in}^{n-1}ab{\rm{\Delta }}t\end{eqnarray} \tag{ 30 }$$

To calculate ${u}_{out}$ and ${c}_{i,\,out}$ for all compartments, the parameters including the initial concentration (${c}_{i,tank}^{0}$) initial volume (${V}_{tank}^{0}$) for all tanks containing LiOH, H₂SO₄, and Li₂SO₄ solutions, and the inlet velocity of solutions to ED, should be prescribed at the first step of ED model. Parameters ${u}_{out}$ and ${c}_{i,\,out}$ are determined by simultaneously solving of Eqs. 1–25, using the equation Engineering Solver (EES). ⁵³ EES software is also used to evaluate water properties, e.g., water viscosity and water density, as a function of temperature and concentration of water. Then, ${V}_{tank}^{1}$ and ${c}_{i,tank}^{1}$ are estimated for all tanks based on Eqs. 27 and 28. Finally, the output parameters obtained at each cycle are input as the initial values of the next time step, and this process goes on until $n={n}_{{\rm{\max }}}.$

To investigate the accuracy of the present model, the model results are compared with the experimental data from two ED systems, i.e., one that was operated in single-pass mode and the other in batch configuration. The experimental data of the single-pass electrodialysis was extracted from the study done by Jung et al., ³¹ while the experimental study of the batch configuration ED is done in the present work. In the single-pass ED, the concentrations of Li⁺ or SO₄ ²⁻ ions do not follow the ratio of 2:1 based on the chemical dissociation reaction of Li₂SO₄ because the solution fed into the dilute channel is the remained solution from the lithium-ion recycling process and was not prepared by dissolving Li₂SO₄ in water. During the recycling process, crushed lithium-ion batteries are firstly dissolved in LiOH solution and then pass through several leaching and filtering steps, in which some sulphate ions would be removed during the extraction of other valuable materials, i.e., Fe, Ni, Co and Mn.

All the input parameters required for the present model, including applied current, inlet concentrations, the initial electrolyte volume in tanks, test cell dimensions as well as experimental conditions, are listed in Table II. In addition, the properties of the membranes used in the ED systems are shown in Table III.

Table IV shows the experimental results and the results obtained by the present model to investigate the accuracy of the present model in both single-pass and batch configurations. As indicated in Table IV, this comparison was done for batch configuration after 2, 4, and 6 h of the ED operation. One can see that the data is in agreement between the model results and reported experimental data in both configurations. In the single-pass mode, the mean error, i.e., the average error among the outlet concentrations listed in Table IV, between modeling and experimental outputs is 5.5% after 6 h of the ED operation. During 2, 4, and 6 h of the batch-mode ED operation, errors reach 3.5, 2.5, and 6.25%, respectively. It should be noted that, due to the steady-state assumption in the present model and the unstable condition at the beginning of the batch-configurated experiment, the mean error of the present model at the first time step (2 hr) is higher than in the second time step (4 hr).

Table IV. Comparison between experimental data and present model results for the single-pass and batch configurations.

Configuration	Time ($hr$)		Concentration, mol·m−3				Mean Error, %
			Li at base channel	Li at dilute channel	SO4 at dilute channel	SO4 at acid channel
Single-pass	6	Experimental date	565	828	375	297
		Present modeling results	614	902	387	301
		Error (%)	8.6	8.9	3.2	1.3	5.5
Batch	2	Experimental value	877	1002	518	473
		Present modeling results	935	1017	509	495
		Error (%)	6.6	1.4	1.7	4.6	3.5
	4	Experimental value	1335	874	465	830
		Present modeling results	1346	891	445	802
		Error (%)	0.8	1.9	4.3	3.3	2.5
	6	Experimental value	1622	742	405	1137
		Present modeling results	1553	713	357	1078
		Error (%)	4.2	3.9	11.8	5.1	6.3

Figure 3 shows the concentrations of lithium and sulphate ions, obtained by the present model and measured experimentally. One can see that both the lithium and sulphate concentrations increase at the concentrate tanks, while they decrease at the dilute tank as can be expected.

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Figures 4a and 4b respectively show the concentration distributions of lithium, and sulphate ions in the CEM, AEM and their vicinity diffusion boundary layers versus the dimensionless length. The inlet concentration of lithium ions at the dilute and concentrate base channels, which have been set 800 mol·m⁻³, and the sulphate inlet concentration of the dilute and concentrate acid channels, considered as 400 mol·m⁻³, are respectively shown as straight lines in Figs. 4a and 4b. As indicated in these figures, the ions concentration in the dilute compartment decreases toward the IEMs, while at the concentrate sides it rises toward the IEMs. In addition, in the membrane phase, the ions concentration at the concentrate channel side is greater than that at the dilute channel side. Furthermore, increasing the current leads to higher concentration gradients at IEMs and their adjacent solutions.

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Figure 5 shows the magnitude of transmembrane velocity of water molecules passing through the AEM and CEM as a function of current magnitude. It is well known that in ED systems, water molecules penetrate through the membranes due to the electro-osmotic mechanism, which is related to concentration of counter-ions at IEMs as well as potential gradient across the membrane, as described by Eq. 5. Therefore, it is expected that increasing current leads to the enhancement of water velocity through IEMs, as presented in Fig. 5. The water velocity through CEM is greater than that of AEM due to higher voltage drop across CEM. Considering the average concentration of counter-ions at the membrane phase of solution-IEM interfaces as the concentration of counter-ions, which is directly dependent on transmembrane water velocity, the variation of transmembrane water velocity would follow a non-uniform trend.

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Figure 6a compares the concentration variations of lithium and sulphate species stored respectively at the LiOH and H₂SO₄ tanks with respect to different initial concentrations of the Li₂SO₄ tank, cLi₂SO₄ ⁰. Two initial concentrations of the Li₂SO₄ solution, i.e., 500 and 1000 mol·m⁻³ were considered, while the concentration of Li⁺ and SO₄ ²⁻ were 100 mol·m⁻³ at the beginning of the experiments. As indicated in this figure, which shows a good agreement between experimental data and model results, increasing the cLi₂SO₄ ⁰ results in a higher concentration of the LiOH and H₂SO₄ solutions, and this variation is more obvious for the LiOH solution.

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Since the present model is developed assuming constant current condition during the ED operation, the concentration of concentrate compartments is a function of water velocity through the IEMs. Increasing the initial concentration of Li₂SO₄ solution would leads to the reduction of the transmembrane water velocity and enhancement of the concentration of concentrate compartments. As evidenced in Fig. 6b, increasing the initial concentration of the Li2SO4 solution reduces the volume of the LiOH tank due to the reduction of water velocity through the CEM. Therefore, increasing the inlet concentration of species at the dilute channel improves the function of ED.

The effect of membrane characteristics, e.g., its thickness and water volume fraction, on the transmembrane water velocity across the CEM as well as outlet lithium concentration of the concentrate base channel are interest to practical application of ED. Figures 7a and 7b show the transmembrane water velocity and the outlet lithium concentration of the concentrate base channel as functions of membrane thickness and water volume fraction, respectively. As illustrated in these figures, the lithium concentration of the concentrate base channel increases by raising the thickness and water volume fraction of the CEM due to reducing the water velocity through the CEM. Moreover, these membrane parameters are more influential at lower values.

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The effect of counter-ion's size on the transmembrane water velocity and the outlet concentration of the concentrate base channel is shown in Fig. 8. One can see that larger particles results higher transmembrane water velocity and lower concentration in the concentrate channels. This phenomenon shows that larger species carry more water molecules through the IEMs in comparison with the smaller ones. Therefore, among lithium, sodium, potassium, rubidium, and caesium ions, the lithium ions have the lowest bare ion radius and can provide the highest membrane performance. It should be noted that these results are taken at the conditions presented in Table V.

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In statistics, uncertainty propagation refers to the effect of input parameters uncertainty on the uncertainty of objective output parameters. In other words, to study how uncertainty in a calculated variable, which is known as an output of a numerical model, might be attributed to different input parameters. Considering amount of input parameters and their relative uncertainties, the uncertainty of the output objectives can be calculated as follows: ⁵⁹

$$\begin{eqnarray}&&{U}_{y}=\sqrt{{\displaystyle \sum \left(\displaystyle \frac{\partial y}{\partial x}\right)}_{i}^{2}{U}_{x}^{2}}\end{eqnarray} \tag{ 31 }$$

in which ${U}_{y},$ ${U}_{x},$ and $\partial y/\partial x$ are the uncertainty of output objectives, uncertainty of input parameters, and partial derivative of each parameter, respectively. The uncertainties of output objectives are obtained in percentage, and a higher uncertainty represents a greater influence of the corresponding input variable on the output parameter.

In the present study, uncertainty propagation is used to investigate the importance of studied input parameters, i.e., current, inlet concentration of the dilute solution, counter-ion radius, the thickness and porosity of CEM, on output objectives including water velocity through CEM and outlet concentration of the concentrate base compartment. Table VI presents the effect of input parameters on the base channel's outlet concentration and water velocity through CEM. As indicated in this table, the relative uncertainty of all input parameters is considered 50% to provide a better comparison among input parameters.

Based on Table VI, current is the most effective parameter among studied input parameters on both outlet concentration of the base channel and transmembrane water transport. However, the influence of current on the outlet concentration of base channel is significantly higher than its influence on the water velocity. Besides the current, the size of counter-ion, CEM water volume fraction and concentration of the dilute channel are the most important parameters affecting the objective parameters' uncertainty, respectively. Also, the effect of the counter-ion radius has higher impact on the transmembrane water velocity than concentration.