Calculating limiting current densityIn the scenario where the diluate solution encompasses n ions, each ion j has a corresponding bulk concentration denoted as Cj,b (mol m−3) and an ionic charge represented by zj. The first nc ions are categorized as cations, while the subsequent na ions are identified as anions. It is posited that ionic transport takes place within a film layer of effective thickness δeff, which can be calculated utilizing film theory principles. This approach facilitates the understanding and modeling of ion transport dynamics across the film layer adjacent to the ion exchange membranes, essential for optimizing electrodialysis processes involving complex ionic solutions.$${\delta }_{eff}=\frac{{D}_{eff}}{{k}_{c,eff}}.$$
(1)
In this context, Deff represents the effective diffusivity of the multi-ionic solution, and kc,eff is the mass-transfer coefficient, which is identified using an appropriate mass-transfer correlation that incorporates Deff. The utilization of film theory is justified when the electric current density remains beneath the threshold of the limiting current density, ensuring the absence of electro-convection within the stagnant film layer10. This scenario ensures that ionic transport can be adequately modeled without the complications introduced by convective movements.Given the presence of more than two ions, it is hypothesized that the concentration ratio between different ions within the film layer remains constant. This assumption holds under limiting current conditions, where the concentrations of all ions at the solution/membrane interface drop to zero, provided that the distribution of ionic concentrations maintains an approximately linear profile across the interface.In the diluate film layer close to an ion exchange membrane, the total flux of ion j, represented as Nj (mol m−2 s−1), is determined by the Nernst–Planck equation. This equation accounts for the movement of ions under the influence of both concentration gradients and electric fields within the layer.$${N}_{j}=-{D}_{j}\frac{d{C}_{j}}{dx}-{z}_{j}{C}_{j}{D}_{j}\frac{F}{RT}\frac{d\upphi }{dx}.$$
(2)
Within the context of the diluate film layer adjacent to an ion exchange membrane, the variable x represents the distance from the edge of the film layer (in contact with the bulk diluate) towards the membrane. The Nernst–Planck equation, which describes the total flux of ion j, Nj, in mol m−2 s−1, incorporates several factors: Dj is the diffusivity of ion j, F stands for the Faraday constant, R is the molar gas constant, T denotes the absolute temperature, and \(\phi\) is the electrical potential. This equation, however, does not consider the friction between different ions and is thus applicable primarily to dilute multi-ionic solutions, where interactions are mainly between the solute and solvent, as reflected in the individual ionic diffusivities.In the field of electrodialysis, it is typically assumed that the concentration of counterions within an ion exchange membrane remains almost constant. This assumption implies that ionic diffusion across the membrane is minimal, and that ionic transport primarily occurs through electro-migration. Consequently, the total molar flux of ion j within the membrane can be linked to the electrical current density, i, highlighting the direct relationship between ionic movement and the applied electrical force in the process.$${N}_{j}^{m}=\frac{{t}_{j}^{m}i}{{z}_{j}F},$$
(3)
where \({t}_{j}^{m}\) represents the transport number of ion j in the membrane. In a steady-state condition, the total molar flux of ion j within the membrane (\({N}_{j}^{m}\)) and Nj (as described by Eq. (2)) are equivalent (Eq. (4)), meaning that the flux of ions through the membrane, as driven by electro-migration, matches the flux described by the Nernst–Planck equation. This equality ensures consistency in the description of ion transport within the system, bridging the gap between electrochemical and physical transport mechanisms.$$\frac{{t}_{j}^{m}i}{{z}_{j}F}=-{D}_{j}\frac{d{C}_{j}}{dx}-{z}_{j}{C}_{j}{D}_{j}\frac{F}{RT}\frac{d\upphi }{dx}.$$
(4)
For electrical current densities that are below the limiting current density, ilim, the principle of electroneutrality is maintained within the film layer. This means that the sum of the positive charges carried by the cations equals the sum of the negative charges carried by the anions throughout the layer, ensuring charge balance and stability in the electrochemical system.$$\sum_{j=1}^{n}{z}_{j}{C}_{j}=0.$$
(5)
By integrating the set of explicit differential equations (specifically, Eq. (4) for each ion and Eq. (5)), we can calculate the distributions of ionic concentrations and electrical potential within the film layer. For a specified set of parameters, the current density at which the concentration of ion j drops to zero at the diluate/membrane interface can be determined iteratively by adjusting the electrical current density, i.Assuming negligible steric exclusion, it is expected that the electrical current can be increased as long as there are counterions present at the diluate/membrane interface to facilitate electrical transport. The limiting current density is reached when the concentration of all counterions (and, due to the electroneutrality condition, co-ions as well) at the interface drops to zero. As the system approaches the limiting current density, the transport numbers of ions within the membrane adjust in a manner that ensures the limiting current density is achieved only when the concentrations of all counterions and co-ions simultaneously become null at the diluate/membrane interface.Examining the degrees of freedom in this system indicates that at the diluate/membrane interface, all ionic concentrations drop to zero at specific values of the electrical current density and the transport numbers of counterions within the membrane. Identifying this unique set of limiting transport numbers poses a significant challenge, as solving the differential equations necessitates an iterative approach. In this process, both the electrical current density and the counterions’ transport numbers are adjusted iteratively until the limiting current density is reached. Consequently, calculating the limiting current density using the Nernst–Planck equations can be labor-intensive, and the complexity of this calculation escalates with the increase in the number of different ions present in the multi-ionic solution.However, by assuming that steric exclusion is negligible, there is no need for a mass-transport model within the membrane to determine the transport numbers of counterions for calculating the limiting current density. This simplification is due to the existence of a unique set of counterions’ transport numbers at which the concentrations of all counterions and co-ions simultaneously become null, facilitating the computation process despite the inherent complexity of the underlying differential equations11.To streamline the process of solving this complex issue, we opted to linearize the Nernst–Planck equations, starting from the premise that the distributions of ionic concentrations within the film layer approximate a linear pattern. This linearization approach is justifiable given the inherent uncertainties associated with film theory. Furthermore, with the understanding that electrodialysis membranes possess a strong charge, it was postulated that the transport numbers of co-ions in the membranes are effectively zero. Consequently, Eq. (4) can be reformulated under these considerations, simplifying the computational effort required to predict ionic transport and facilitating a more efficient determination of the limiting current density.$$\frac{{t}_{j}^{m}i}{{z}_{j}F}=\frac{{D}_{j}}{{\delta }_{eff}}\left({C}_{j,b}-{C}_{j,m}\right)-{z}_{j}{C}_{j,m}{D}_{j}\frac{F}{RT}{\xi }_{m},$$
(6)
where Cj,m is the concentration of the ion j next to the ion exchange membrane and \({\xi }_{m}\) is the electrical potential gradient at the membrane/diluate interface. Combining Eqs. (1), (6):$$\frac{{t}_{j}^{m}i}{{z}_{j}F}=\frac{{D}_{j}}{{D}_{eff}}{k}_{c,eff}\left({C}_{j,b}-{C}_{j,m}\right)-{z}_{j}{C}_{j,m}{D}_{j}\frac{F}{RT}{\xi }_{m}.$$
(7)
Even when the electrical current densities are below the limiting current density (ilim), the principle of electroneutrality remains applicable at the diluate/membrane interface. This means that the total positive charge carried by the cations is balanced by the total negative charge carried by the anions, ensuring charge balance across this critical junction in the electrodialysis process.$$\sum_{j=1}^{n}{z}_{j}{C}_{j,m}=0.$$
(8)
Combining Eqs. (7), (8) and rearranging them yield:$${\xi }_{m}=\frac{RT}{F}\frac{\left(\frac{{k}_{c,eff}}{{D}_{eff}}\right)[\sum_{j=1}^{nc}\left({C}_{j,b}-{C}_{j,m}\right)-\sum_{j=1+nc}^{nc+na}\left({C}_{j,b}-{C}_{j,m}\right)]-(\frac{i}{F})[{\sum }_{j=1}^{nc}\left(\frac{{t}_{j}^{m}}{{z}_{j}{D}_{j}}\right)-{\sum }_{j=1+nc}^{nc+na}\left(\frac{{t}_{j}^{m}}{{z}_{j}{D}_{j}}\right)]}{2{\sum }_{j=1}^{nc}{z}_{j}{C}_{j,m}}.$$
(9)
This equation demonstrates that as the concentrations of cations at the diluate/membrane interface drop to zero (and consequently, the anions’ concentrations do as well), the potential difference across the membrane, denoted as \({\xi }_{m}\), approaches infinity (\(-\infty\) for a cation exchange membrane and \(+\infty\) for an anion exchange membrane), signifying that the limiting current density has been reached.Consider a scenario where, for a specific multi-ionic solution and a given set of ionic transport numbers within an ion exchange membrane, there exists a “critical” current density at which the concentration of at least one ion becomes zero at the diluate/membrane interface. If not, all ionic concentrations reach zero at the interface simultaneously, then \({\xi }_{m}\) remains finite. Accordingly, for each ion j, the critical current density that leads to a zero concentration at the interface (Cj,m = 0) can be calculated using Eq. (7):$${i}_{crit,j}=F\frac{{z}_{j}}{{t}_{j}^{m}}\frac{{D}_{j}}{{D}_{eff}}{k}_{c,eff}{C}_{j,b}.$$
(10)
For a specified multi-ionic solution and a predetermined set of ionic transport numbers within an ion exchange membrane, the critical current density is identified as the minimum value within the set of icrit,j values, where each icrit,j corresponds to the critical current density for ion j. This critical current density is fundamentally a function of the ionic transport numbers, indicating the lowest current density at which at least one ion’s concentration at the diluate/membrane interface drops to zero. This concept underscores the importance of understanding the specific transport dynamics of each ion in the solution to accurately predict the conditions under which the limiting current density is approached.In the specific scenario involving a strongly charged cation exchange membrane, the dynamics of ion transport and the resulting critical current density exhibit unique characteristics. Due to the membrane’s strong cationic selectivity, it preferentially facilitates the passage of cations while significantly hindering or completely blocking the transport of anions. However, it should be mentioned that in practice, there is no membrane that is 100% selective to cations. Co-ions also pass through the membrane to some extent. This selective transport behavior plays a crucial role in determining the operational limits and effectiveness of electrodialysis processes, especially when aiming to achieve optimal separation and concentration of ions from a multi-ionic solution.$${\sum }_{j=1}^{nc}{t}_{j}^{m}=1.$$
(11)
When reaching the limiting current density at which the concentration of every cation at the diluate/cation exchange membrane interface drops to zero, the critical current density becomes the same across all cations. This scenario allows for the derivation of an expression from combining Eq. (10) for each cation with Eq. (11), which articulates the specific transport numbers of cations needed for the concentrations of all ions to simultaneously reduce to zero at the interface. This expression is crucial for understanding the conditions under which electrodialysis processes achieve their maximum efficiency, particularly in systems utilizing strongly charged cation exchange membranes. It serves as a fundamental basis for optimizing the transport dynamics and overall performance of electrodialysis systems in separating and concentrating ions from multi-ionic solutions.$${t}_{j,lim}^{m}=\frac{{z}_{j}{D}_{j}{C}_{j,b}}{{\sum }_{j=1}^{nc}{z}_{j}{D}_{j}{C}_{j,b}}.$$
(12)
A similar approach can be taken for a strongly charged anion exchange membrane:$${t}_{j,lim}^{m}=\frac{\left|{z}_{j}\right|{D}_{j}{C}_{j,b}}{{\sum }_{j=1+nc}^{nc+na}\left|{z}_{j}\right|{D}_{j}{C}_{j,b}}.$$
(13)
The limiting transport numbers of the counterions, which indicate the point at which the limiting current density is achieved for both cation and anion exchange membranes, are described by Eqs. (12), (13), following the linearized Nernst–Planck methodology. It is important to emphasize that this specific set of limiting transport numbers is distinctive and remains consistent across any mass-transport model utilized within the membranes.These limiting transport numbers are derived under an asymptotic condition where all ionic concentrations approach zero but do not exactly reach nullity. When the concentrations of all ions are zero, Eq. (10) becomes inapplicable for calculating ilim because, under such circumstances, the product \({{C}_{j,m}\xi }_{m}\) does not equate to zero in Eq. (7). Instead, with Cj,m = 0 inserted into Eq. (7), a new formula is derived for \({{C}_{j,m}\xi }_{m}\), providing a basis for understanding the behavior of ionic transport at the threshold of the limiting current density, where traditional calculations based on concentration gradients and potential differences need to be re-evaluated.$${z}_{j}{C}_{j,m}{\xi }_{m}=\frac{RT}{F}\frac{\left(\frac{{D}_{j}}{{D}_{eff}}\right){k}_{c,eff}{C}_{j,b}-(\frac{{t}_{j,lim}^{m}{i}_{lim}}{{z}_{j}F})}{{D}_{j}}.$$
(14)
By combining Eq. (14) with Eq. (8) (the electroneutrality) and rearranging, we can obtain the equation for ilim:$${i}_{lim}=F\frac{{k}_{c,eff}}{{D}_{eff}}\frac{{\sum }_{j=1}^{n}{C}_{j,b}}{{\sum }_{j=1}^{n}(\frac{{t}_{j,lim}^{m}}{{z}_{j}{D}_{j}})}.$$
(15)
This equation results in different limiting current densities for cation and anion exchange membranes, with the outcome being positive for the cation exchange membrane and negative for the anion exchange membrane, in alignment with the established coordinate system (x). In practical applications, the limiting current density is determined by selecting the lowest among the absolute values calculated for each type of membrane. This approach ensures the identification of the operational threshold that prevents excessive ion depletion or concentration polarization, thereby optimizing the effectiveness and stability of the electrodialysis process.Calculation of system resistances and system level current densityOhm’s law is used for calculating the current of the system:$$I=\frac{V}{{R}_{total}},$$
(16)
where i is current in A, V is applied voltage in V, and Rtotal is the total resistance of the system (Ω). Current density (i, A m−2) is calculate using the following equation:The effective surface area (S) of the cationic exchange membranes is determined by multiplying the number of cationic exchange membranes by the surface area of each membrane. It is assumed that the surface areas for both cationic and anionic membranes are identical.The total resistance of the system is composed of the resistances from the feed compartments (Rf), concentrate compartments (Rc), rinse compartments (Rrc for cationic rinse and Rra for anionic rinse), all cationic and anionic exchange membranes (Rcm and Ram), and the electrodes (Relectrodes) as expressed in the following equation:$${R}_{total}=n{R}_{f}+m{R}_{c}+{R}_{rc}+{R}_{ra}+p{R}_{cm}+q{R}_{am}+{R}_{electrodes},$$
(18)
where n, m, p, and q denote the number of feed compartments, concentrate compartments, cationic exchange membranes and anionic exchange membranes, respectively. The user has the freedom to select any number of compartments with the criteria shown in Table 2.Table 2 The criteria for the number of feed and concentrate compartments and cation and anion exchange membranes.To calculate the resistance of the feed and concentrate compartments, the specific conductivity is utilized. This measure is linked to ion mobility, which can be determined from the ion diffusion coefficient using the expression below:$${R}_{c} or {R}_{f}=\frac{h}{\kappa S{\epsilon }^{2}},$$
(19)
where h is the distance between membranes (compartment thickness or spacer width) in m, k is the specific conductivity in S/m, S is the active area of the ion exchange membranes (unmasked by the spacer) and ε is the porosity of the spacer (the porosity is squared to reflect the tortuous ion transport)12. Tortuosity is defined as the average ratio of the actual path length that a particle or fluid must travel within a porous medium compared to the straight-line distance between two points.$$\tau =\frac{actual\, path\, length}{straight\, distance}.$$
(20)
In electrodialysis, the compartment is treated as a porous medium due to the structure of the spacer. As a result, the actual path length that cations traverse before reaching the membrane is longer than the actual thickness of the compartment13.$$\uptau =\frac{1}{{\varepsilon }^{2}}.$$
(21)
Spacer porosity, a characteristic of the spacer, is typically provided by the manufacturer. In this study, we assumed a porosity of 0.7, which corresponds to a tortuosity of 2.The specific conductivity can be expressed as:$$\kappa =\sum {n}_{i}{q}_{i}{\mu }_{i},$$
(22)
where \({n}_{i}\) (m−3) is the number concentration of ion i (both cations and anions), \({q}_{i}\) (C) is the charge of ion i, and \({\mu }_{i}\) (m2 V−1 s−1) is the ion mobility of the ion i. The ion mobility can be obtained from the Nernst Einstein (NE) equation as follows:$${\mu }_{i}=\frac{{z}_{i}e{D}_{i}}{{k}_{B}T},$$
(23)
where zi is the charge of ion i, e is elementary charge (1.602 × 10−19 C), Di is the diffusion coefficient of ion i, kB is the Boltzmann constant (1.380649 × 10−23 m2 kg s−2 K−1) and T is temperature (K).The values of ni and qi in Eq. (22) are calculated as follows:$${n}_{i}={N}_{A}{C}_{i},$$
(24)
$${q}_{i}={z}_{i}e,$$
(25)
where NA is the Avogadro’s number (\({N}_{A}= 6.022\times {10}^{23}\) mol−1) and Ci is the concentration of ions (cations and anions) in each compartment.As the electrodialysis process progresses, the concentration of ions in the solution, denoted by Ci, changes over time. Consequently, the resistances of the feed (Rf) and concentrate (Rc) compartments also vary as a function of time, given their dependence on the ion concentration.The resistance of the rinse compartments can be determined using the same methodology described above, specifically referenced in Eq. (26). The resistances of the rinse compartments are considered constant because the concentration within these compartments is assumed to remain stable:$${R}_{rc} or {R}_{ra}=\frac{{d}_{r}}{\kappa S{\epsilon }^{2}}.$$
(26)
The resistance of the rinse compartments is calculated using the distance between the electrode and the first membrane (dr), which represents the thickness of the rinse compartment or spacer width in meters (m). The specific conductivity (k) is measured in S m−1, and S denotes the active area of the ion exchange membranes that is exposed and not covered by the spacer. Additionally, ε represents the porosity of the spacer12.The resistances of the cationic and anionic membranes (Rcm and Ram, respectively) are provided by the manufacturer. These values are directly incorporated into Eq. (18) to calculate the overall system resistance.System key performance indicators (KPI)To assess the efficacy of the electrodialysis system under specific conditions of HEDTA/Nd molar ratio and pH, two primary parameters are established and utilized: the purity percentage, the yield percentage and separation factor. These parameters are determined through the following calculations:$$Purity \left(\%\right)=\frac{{{(C}_{Nd, total}+{C}_{Pr, total})}_{feed,final}}{{\left(\sum_{i=1}^{n}{C}_{i,total}\right)}_{feed, final}},$$
(27)
$$Yield \left(\%\right)=\frac{{{(C}_{Nd, total}+{C}_{Pr, total})}_{feed,final}}{{{(C}_{Nd, total}+{C}_{Pr, total})}_{feed, initial}},$$
(28)
$$SF=\frac{\frac{{({C}_{La(III)}+{C}_{Ce(III)})}_{concentrate}}{{({C}_{La(III)}+{C}_{Ce(III)})}_{total,feed,final}}}{\frac{{({C}_{Nd(III)}+{C}_{\text{Pr}(III)})}_{concentrate}}{{({C}_{Nd(III)}+{C}_{\text{Pr}(III)})}_{total,feed,final}}}.$$
(29)
The rationale behind these definitions stems from the primary aim of electrodialysis: to retain Nd + Pr within the feed compartment by forming complexes with HEDTA, while La + Ce are moved to the concentrate compartment as free ions. Purity is defined as the combined concentration of Nd + Pr, both as free ions and as complexes, within the feed compartment, divided by the total concentration of all REEs, again considering both free and complexed forms. The target is to reach a Nd + Pr purity level of 99.9%. Yield, on the other hand, is calculated by dividing the total concentration of Nd + Pr, in both their free and complexed states in the feed compartment, by their initial total concentration. The goal here is to attain a yield exceeding 95%. Achieving both these objectives simultaneously is challenging. The approach involves opting for a lower HEDTA/Nd molar ratio to maximize the complexation of Nd(III) and Pr(III) ions while minimizing the complexation of La(III) and Ce(III) with HEDTA, which typically results in a relatively lower yield.
Class II (three-layer system) phenomenological model based on limiting current density and dynamic chelation chemistry for separation of rare earth elements using electrodialysis
