Research on the Mechanism of Ion Exchange Based on Thermodynamic Equilibrium

Abstract

Referencing ion exchange applied in the extraction of ion-adsorption minerals, a mathematical model based on thermodynamic equilibrium has been researched and established to reveal and represent the chemical reaction between minerals and the leaching solution. Meanwhile, a Python-based numerical solver has been developed and programmed to solve the equations of the established model in order to achieve computational efficiency and obtain an accurate solution. Based on the simulation and computation with the established mathematical model, the effects of leaching solution concentration and mineral grade distribution on the extraction of ion-adsorption minerals can be evaluated, validating the set value of 20 g/L for the initial leaching solution concentration and the extraction rate of above 90% for the cut-off grade. Furthermore, the established mathematical model can be integrated into the simulation system for the overall multi-field extraction of ion-adsorption minerals, illustrating the relationship between extracted metal ion concentration in the aqueous phase over time and providing theoretical support for the engineering project.

1. Introduction

Ion-adsorption minerals, such as copper oxide, rare earth, and uranium, are typically found in the weathered crust and are primarily extracted using hydrometallurgical methods, such as in situ extraction, bioleaching, and electrokinetic mining [1,2,3]. So far, in situ extraction is still the primary extraction method for ion-adsorption minerals, considering its engineering applications.

During the in situ extraction process, the leaching solution seeps into the ore body through its internal pore structure in all directions, resulting in the migration of cations dissolved in the leaching solution toward the solid surface of ore particles due to convection and concentration gradients. Furthermore, an ion exchange reaction occurs between the dissolved cations and the metal ions adsorbed on the surface of the ore particles, enabling metal extraction [4]. As a result, the extraction process of ion-adsorption minerals can be divided into three main steps, including the seepage flow of the leaching solution in the porous ore body, the migration of dissolved ions, and the ion exchange reaction [5].

In the engineering project, the seepage and contact of the leaching solution is carried out on the surface of ore particles, triggering further reactions that facilitate the extraction of metal ions. The primary objective of this method is to enhance the ion exchange reaction between metal ions adsorbed on the surface of ore particles and cations present in the leaching solution, thereby enabling the displacement and extraction of metal ions [6]. However, most of the projects are located in remote mountain areas and are conducted based on hands-on experience, especially for process control and parameter adjustment, such as the concentration and injection rate of the leaching solution, resulting in excessive injection in order to achieve a high extraction rate of metal ions. Meanwhile, the available research on the extraction mechanism has not been fully promoted and applied, leading to a lack of efficient theoretical support for practical operations, particularly concerning the ion exchange reaction.

Ion exchange can be considered a reaction occurring at the interface between two immiscible phases, namely, the interactions among individual ions from the solid phase and the aqueous phase, which could be the most critical and key aspect to consider in the whole extraction process [7,8]. For example, a typical ion-adsorption mineral, such as rare earth ions, which demonstrate trivalent metal cations, has been adopted in this study.

3N⁺(l) + [Ore Deposit]·Me³⁺(s)↔Me³⁺(l) + [Ore Deposit]·3N⁺(s)

(1)

Here, N⁺ represents the cations present in the leaching solution, and Me³⁺ represents the metal ions extracted from the ore body.

For the mechanism of the ion exchange reaction, a general equilibrium model can be constructed based on the equation of the thermodynamic equilibrium constant, referring to the instantaneous equilibrium [9,10], as shown in Equation (2) as follows:

$$K={\displaystyle \frac{a_{\left({\mathrm{M}\mathrm{e}}^{3+}\right)}{a}_{\left[{\mathrm{N}}^{+}\right]}^3}{a_{\left[{\mathrm{M}\mathrm{e}}^{3+}\right]}{a}_{\left({\mathrm{N}}^{+}\right)}^3}}$$

(2)

where K is the thermodynamic equilibrium constant; a is the activity of components involved in the ion exchange reaction; [] represents the solid phase; () represents the aqueous phase; Me³⁺ represents the metal ion; and N⁺ represents the cations present in the leaching solution.

Furthermore, different empirical models could be obtained via the modification of Equation (2) in combination with different assumptions, including the Kerr model, Vanselow model, and Gapon model [11,12,13,14,15].

Long et al. [16] compared the performance of the commonly used Kerr model, Vanselow model, and Gapon model, indicating that the application of the Kerr model demonstrated a better fit to the experimental data due to its simple structure and could be reasonably adopted to describe the process of ion exchange reactions.

On the other hand, Sobri et al. [17] have reviewed the ion exchange leaching method for extracting rare earth elements from ion-adsorption clay, with a particular focus on the shrinking core model related to the ion exchange reaction. They analyzed the effects of various parameters on the determination of the rate-controlling step to summarize the selection of different mathematical equations. In conclusion, they projected that the Kerr model could be used to describe the ion exchange reaction between the solid and liquid phases.

Tian et al. [18] also established the unreacted shrinking core model to study the non-equilibrium kinetics of ion exchange. Xiao et al. [19] further applied double-layer theory to explain the unreacted shrinking core model, demonstrating that the ion exchange process could be controlled by internal diffusion. Cao et al. [20] found that during the extraction process of ion-adsorption minerals, the ore particle size could be assumed to remain unchanged; namely, the shrinking process of the unreacted core did not fully occur, which means that internal diffusion in the ash layer could be neglected.

Zeng et al. [21,22] applied the Computational Fluid Dynamics (CFD) method to simulate the leaching process for ion-absorbed rare earth ore, incorporating the shrinking unreacted core model. Yan et al. [23] studied the fractal kinetics of the leaching process for ion-absorbed rare earth ore and established a corresponding model. However, these models were built and simplified based on a 2D column leaching process. Moreover, a unified interpretation could not be achieved due to differing understandings and specific assumptions. Nevertheless, some fundamental conditions can be concluded, including that the ore particle size can be assumed to remain unchanged and that the ion exchange reaction can be treated as a reversible reaction.

In conclusion, steady-state equilibrium of the ion exchange reaction can be achieved and maintained throughout the entire extraction process in the case of a saturated leaching solution. Therefore, based on the above considerations and analysis, a mechanism model based on thermodynamic equilibrium will be constructed to represent the ion exchange reaction. In this research, the Kerr model has been adopted as the foundational model, in combination with modifications that incorporate assumptions derived from the prominent features that emerged during the engineering project.

2. Mechanism and Mathematical Model of Ion Exchange

2.1. Basic Assumption

The ion exchange process involved in the extraction of ion-adsorption minerals can be divided into the following steps, specifically based on corresponding assumptions:

(1) The migration of cations, N⁺, from the bulk of the leaching solution to the solid–aqueous interface.

For the extraction process of ion-adsorption minerals, the concentration of cations present in the leaching solution can generally be maintained at a high and stable level. This indicates that the saturation of N⁺ in the leaching region can be easily and quickly achieved, favoring the rapid diffusion of N⁺ from the leaching solution to the solid–aqueous interface.

(2) The ion exchange reaction between metal ions, Me³⁺, and cations, N⁺, and chemical equilibrium at the solid–aqueous interface.

Considering the physical properties of ore particles, which are typically classified as clay minerals with small particle sizes, the adsorption of Me³⁺ on their surface has been well-characterized and confirmed, indicating that a corresponding reaction between Me³⁺ and N⁺ upon contact and steady-state chemical equilibrium can be achieved.

(3) The diffusion of metal ions, Me³⁺, generated by ion exchange reactions at the solid–aqueous interface into the bulk of leaching solution.

Due to the steady seepage flow and saturation of the leaching solution in the boundary layer, it can be assumed that the diffusion in the boundary layer can also be ignored. Namely, the mass transfer process involved in the ion exchange process, including the migration of metal ions, Me³⁺, and cations, N⁺, can occur instantaneously.

As a result, the above step (2) can be identified as the critical determination for the ion exchange reaction to construct the corresponding rate-controlling equations, including the equation of thermodynamic equilibrium and mass balance for the chemical reaction at the solid–aqueous interface. In addition, it further establishes the mathematical model of the ion exchange reaction at the equilibrium state.

2.2. Equation of Thermodynamic Equilibrium

The equation of thermodynamic equilibrium can be established primarily according to Equation (2) when the steady-state equilibrium of the reaction has been maintained, as shown in Equation (3) as follows:

$$K_{\mathrm{k}}={\displaystyle \frac{\left(C_{\mathrm{a}\mathrm{q}}^{\mathrm{M}}{|}_{\mathrm{E}}\right)(C_{\mathrm{S}}^{\mathrm{N}}{|}_{\mathrm{E}}{)}^3}{\left(C_{\mathrm{S}}^{\mathrm{M}}{|}_{\mathrm{E}}\right)(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}{)}^3}}$$

(3)

where K_k is the thermodynamic equilibrium constant of the ion exchange reaction, L²/g²; $C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}$ is the mass concentration of cations, N⁺, present in the leaching solution, g/L; $C_{\mathrm{a}\mathrm{q}}^{\mathrm{M}}$ is the mass concentration of metal ions, Me³⁺, dissolved in the leaching solution, g/L; $C_{\mathrm{S}}^{\mathrm{N}}$ is the mass concentration of cations, N⁺, adsorbed on the surface of ore particles, g/g; $C_{\mathrm{S}}^{\mathrm{M}}$ is the mass concentration of metal ions, Me³⁺, adsorbed on the surface of ore particles, g/g; and |_E represents the equilibrium state after the occurrence of the ion exchange reaction.

2.3. Equation of Mass Balance

The equation of mass balance can be conducted primarily according to the analysis of thermodynamic mass balance, as shown in

Table 1. Analysis of thermodynamic mass balance for ion exchange reaction.

	3N+ (l)	Me3+ (s)	Me3+ (l)	3N+ (s)
Initial (mol)	$C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}$|I × Vaq/MN	$C_{\mathrm{S}}^{\mathrm{M}}$|I × ms/MM	$C_{\mathrm{a}\mathrm{q}}^{\mathrm{M}}$|I × Vaq/MM	$C_{\mathrm{S}}^{\mathrm{N}}$|I × ms/MN
Reacted (mol)	($C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}$|I − $C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}$|E) × Vaq/MN	($C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}$|I − $C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}$|E) × Vaq/MN×(1/3)	($C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}$|I − $C_{\mathrm{a}}^{\mathrm{N}}$|E) × Vaq/MN×(1/3)	($C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}$|I − $C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}$|E) × Vaq/MN
Equilibrium (mol)	($C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}$|E × Vaq/MN)	($C_{\mathrm{S}}^{\mathrm{M}}$|E × ms/MM)	($C_{\mathrm{a}\mathrm{q}}^{\mathrm{M}}$|E × Vaq/MM)	($C_{\mathrm{S}}^{\mathrm{N}}$|E × ms/MN)

.

Here, V_aq is the volume of the aqueous phase; m_s is the mass of the solid phase; M_M and M_N are the molecular weights of metal ions, Me³,⁺ and cations, N⁺, respectively; aq represents the aqueous phase, referring to the leaching solution and the mother liquor after extraction; s represents the solid phase, referring to the ore particles; |_I represents the initial state before the occurrence of the ion exchange reaction; and |_E represents the equilibrium state after the occurrence of the ion exchange reaction.

The concentration of metal ions, Me³⁺, and cations, N⁺, dissolved in the aqueous phase, namely in the leaching solution, before the occurrence of the ion exchange reaction can be determined by the numerical results of the mathematical model for the solute transport process. It can be treated as a pre-condition for the ion exchange reaction in the extraction process and can be established based on the Convection-Dispersion Equation, which is solved by applying the finite element method. This approach demonstrates the concentration distribution of metal ions, Me³⁺, and cations, N⁺, dissolved in the leaching solution solely via the ongoing seepage and solute transport. Consequently, the concentration of each involved component can be regarded as given parameters prior to the occurrence of the ion exchange reaction.

According to the above analysis of thermodynamic mass balance and assuming that $C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}$|_E is the only independent variable, the following equations can be derived:

$$C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}=C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}$$

(4)

$$C_{\mathrm{S}}^{\mathrm{M}}{|}_{\mathrm{E}}=C_{\mathrm{S}}^{\mathrm{M}}{|}_{\mathrm{I}}-{\displaystyle \frac{1}{3}}{\displaystyle \frac{M_{\mathrm{M}}}{M_{\mathrm{N}}}}{\displaystyle \frac{V_{\mathrm{a}\mathrm{q}}}{m_{\mathrm{s}}}}(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}-C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}})$$

(5)

$$C_{\mathrm{a}\mathrm{q}}^{\mathrm{M}}{|}_{\mathrm{E}}=C_{\mathrm{a}\mathrm{q}}^{\mathrm{M}}{|}_{\mathrm{I}}+{\displaystyle \frac{1}{3}}{\displaystyle \frac{M_{\mathrm{M}}}{M_{\mathrm{N}}}}(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}-C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}})$$

(6)

$$C_{\mathrm{S}}^{\mathrm{N}}{|}_{\mathrm{E}}=C_{\mathrm{S}}^{\mathrm{N}}{|}_{\mathrm{I}}+{\displaystyle \frac{V_{\mathrm{a}\mathrm{q}}}{m_{\mathrm{s}}}}(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}-C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}})$$

(7)

$${\displaystyle \frac{V_{\mathrm{a}\mathrm{q}}}{m_{\mathrm{s}}}}={\displaystyle \frac{V\times n}{V\times (1-n)\times {\rho }_{\mathrm{b}}}}={\displaystyle \frac{n}{(1-n)\times {\rho }_{\mathrm{b}}}}$$

(8)

where V is the total volume; n is the porosity; and ρ_b is the density of the ore body.

Assuming that V_aq/m_s = a and M_N/M_M = b, Equations (5) to (7) can be further transformed into the following equations:

$$C_{\mathrm{S}}^{\mathrm{M}}{|}_{\mathrm{E}}=C_{\mathrm{S}}^{\mathrm{M}}{|}_{\mathrm{I}}-{\displaystyle \frac{1}{3}}{\displaystyle \frac{a}{b}}(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}-C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}})$$

(9)

$$C_{\mathrm{a}\mathrm{q}}^{\mathrm{M}}{|}_{\mathrm{E}}=C_{\mathrm{a}\mathrm{q}}^{\mathrm{M}}{|}_{\mathrm{I}}+{\displaystyle \frac{1}{3}}{\displaystyle \frac{1}{b}}(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}-C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}})$$

(10)

$$C_{\mathrm{S}}^{\mathrm{N}}{|}_{\mathrm{E}}=C_{\mathrm{S}}^{\mathrm{N}}{|}_{\mathrm{I}}+a(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}-C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}})$$

(11)

Then, the final mathematical model can be obtained by substituting Equations (9) to (11) into Equation (3), as shown in Equation (12) as follows:

$$K_{\mathrm{k}}={\displaystyle \frac{(C_{\mathrm{a}\mathrm{q}}^{\mathrm{M}}{|}_{\mathrm{I}}+\frac{1}{3b}(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}-C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}\left)\right)(C_{\mathrm{S}}^{\mathrm{N}}{|}_{\mathrm{I}}+a(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}-C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}){)}^3}{(C_{\mathrm{S}}^{\mathrm{M}}{|}_{\mathrm{I}}-\frac{a}{3b}(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}-C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}\left)\right)(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}{)}^3}}$$

(12)

where the parameters related to the concentration of each component before the occurrence of the ion exchange reaction, such as $C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}$|_I, $C_{\mathrm{a}\mathrm{q}}^{\mathrm{M}}$|_I, $C_{\mathrm{S}}^{\mathrm{M}}$|_I, and $C_{\mathrm{S}}^{\mathrm{N}}$|_I, can be regarded as given parameters based on the computation and solution of the mathematical model for the solute transport process and the initial data on mineral grade distribution; the thermodynamic equilibrium constant, K_k, can also be regarded as a given parameter, which normally can be calculated using the experimental data from laboratory extraction trials or referenced from the verified data available in the related literature [24].

Assuming the following:

$$C_{\mathrm{S}}^{\mathrm{M}}{|}_{\mathrm{I}}-{\displaystyle \frac{1}{3}}{\displaystyle \frac{a}{b}}{C}_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}=f$$

(13)

$$C_{\mathrm{a}\mathrm{q}}^{\mathrm{M}}{|}_{\mathrm{I}}+{\displaystyle \frac{1}{3}}{\displaystyle \frac{1}{b}}{C}_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}=d$$

(14)

$$C_{\mathrm{S}}^{\mathrm{N}}{|}_{\mathrm{I}}+a{C}_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}=e$$

(15)

Equation (12) can be further expressed as follows:

$$K_{\mathrm{k}}={\displaystyle \frac{(d-\frac{1}{3b}{C}_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}})(e-a{C}_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}{)}^3}{(f+\frac{a}{3b}{C}_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}})(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}{)}^3}}$$

(16)

Then, Equation (16) can be further manipulated as follows:

$$\begin{gathered} (K_{\mathrm{k}}{\displaystyle \frac{a}{3b}}-{\displaystyle \frac{a^3}{3b}})(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}{)}^4+(K_{k}f+a^{3}d+{\displaystyle \frac{a^{2}e}{b}}\left)\right(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}{)}^3+ \\ (-3a^{2}de-{\displaystyle \frac{a{e}^2}{b}})(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}{)}^2+(3ad{e}^2+{\displaystyle \frac{e^3}{3b}})C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}+(-d{e}^3)=0 \end{gathered}$$

(17)

Assuming the following:

$$K_{\mathrm{k}}{\displaystyle \frac{a}{3b}}-{\displaystyle \frac{a^3}{3b}}=A$$

(18)

$$K_{\mathrm{k}}f+a^{3}d+{\displaystyle \frac{a^{2}e}{b}}=B$$

(19)

$$-3a^{2}de-{\displaystyle \frac{a{e}^2}{b}}=D$$

(20)

$$3ad{e}^2+{\displaystyle \frac{e^3}{3b}}=E$$

(21)

$$-d{e}^3=F$$

(22)

Therefore, the determination of the mathematical equation can be demonstrated as follows in terms of just one unknown variable:

$$A(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}{)}^4+B(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}{)}^3+D(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}{)}^2+E{C}_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}}+F=0$$

(23)

As a result, the exact solution, which indicates the concentration of cations, N⁺, dissolved in the aqueous phase after the occurrence of the ion exchange reaction, can be obtained by solving Equation (23). Then, based on the Equations (24) to (26), the concentrations of other components involved in the ion exchange reaction can also be determined, including not only the metal ions, Me³⁺, extracted and dissolved in the aqueous phase but also the metal ions, Me³⁺, and cations, N⁺, adsorbed on the surface of ore particles.

$$C_{\mathrm{S}}^{\mathrm{M}}{|}_{\mathrm{E}}=C_{\mathrm{S}}^{\mathrm{M}}{|}_{\mathrm{I}}-{\displaystyle \frac{1}{3}}{\displaystyle \frac{M_{\mathrm{M}}}{M_{\mathrm{N}}}}{\displaystyle \frac{V_{\mathrm{a}\mathrm{q}}}{m_{\mathrm{s}}}}(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}-C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}})$$

(24)

$$C_{\mathrm{a}\mathrm{q}}^{\mathrm{M}}{|}_{\mathrm{E}}=C_{\mathrm{a}\mathrm{q}}^{\mathrm{M}}{|}_{\mathrm{I}}+{\displaystyle \frac{1}{3}}{\displaystyle \frac{M_{\mathrm{M}}}{M_{\mathrm{N}}}}(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}-C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}})$$

(25)

$$C_{\mathrm{S}}^{\mathrm{N}}{|}_{\mathrm{E}}=C_{\mathrm{S}}^{\mathrm{N}}{|}_{\mathrm{I}}+{\displaystyle \frac{V_{\mathrm{a}\mathrm{q}}}{m_{\mathrm{s}}}}(C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{I}}-C_{\mathrm{a}\mathrm{q}}^{\mathrm{N}}{|}_{\mathrm{E}})$$

(26)

In conclusion, Equation (23) can be regarded as the mechanistic model of thermodynamic equilibrium corresponding to the ion exchange reaction in the extraction process of ion-adsorption minerals.

3. Solver of the Mathematical Model

In this research, an initial Python-based numerical solver for Equation (23) has been developed and programmed. Python (Version 3.11.0) has become a powerful programming language for developing mathematical algorithms and corresponding solvers for various mathematical equations and numerical problems. Some built-in packages have been adopted during the development of the numerical solver, such as NumPy and SciPy. Based on the established mechanistic model of thermodynamic equilibrium corresponding to the ion exchange reaction, modifications and optimizations have been made to the numerical solver in order to eliminate the effects of floating-point precision and the existence of complex roots, in combination with the engineering conditions adopted in the extraction process, indicating good numerical precision and an acceptable execution period.

Finally, the customized numerical solver for Equation (23) could be implemented by introducing the initial concentration data of the leaching solution and the initial mineral grade distribution data, ensuring that an exact solution for the corresponding concentration value can be obtained.

4. Results and Discussion

Taking a specific ion-adsorption mineral as an example, which demonstrates trivalent metal cations, the initial values for the following parameters could be set based on the completed geological exploration, as shown in

Table 2. Initial value for given parameters in the model.

No.	Parameters	Value	Unit
1	n	0.17	dimensionless
2	ρb	1.56 × 103	kg/m3
3	MM	0.13673	kg/mol
4	MN	0.018	kg/mol

.

Also, a Two-dimensional (2D) geometric model was built to represent the cross-section of the ore body along the x-z direction concisely. Then, mesh processing with a triangle mesh was adopted to generate nodes and elements in order to facilitate the calculation of the mathematical model. In accordance with the available geological exploration, the 2D geometric model was set and designed to present the majority of the ore-bearing layers in the ore body, including the fully weathered layer, strongly weathered layer, moderately weathered layer, and slightly weathered layer in a top-down order. As highlighted in Figure 1, marked by the characteristic points, the initial average mineral grade distribution for each zone in the 2D geometric model was set respectively. Meanwhile, the initial concentration of metal ion Me³⁺ dissolved in the aqueous phase was set to 0 g/L, and the initial concentration of cations N⁺ adsorbed on the surface of the solid phase was set to 0 g/g in accordance with the engineering conditions of the extraction process.

4.1. Effect of Leaching Solution Concentration on the Ion Exchange Process

For a specific chemical reaction, the thermodynamic equilibrium constant has been applied to reveal its thermodynamic potential, as shown in Equation (27) as follows:

$$∆G_0=-RT\mathrm{I}\mathrm{n}K$$

(27)

where ΔG₀ is the Gibbs free energy of the reaction in the standard state, R is the molar gas constant, T is the temperature, and K is the thermodynamic equilibrium constant of the reaction [25].

Theoretically, the thermodynamic equilibrium constant of a chemical reaction can be treated as a function of temperature at the standard state, which means that the value of the thermodynamic equilibrium constant can only be related to the temperature. The greater the thermodynamic equilibrium constant, the more negative the Gibbs free energy of the reaction, indicating a greater thermodynamic potential for the forwarding of the reaction. Therefore, the ongoing ion exchange reaction at the standard state will be adopted in this research, and the value of K_k, applied as the thermodynamic equilibrium constant of the ion exchange reaction, can be treated as constant.

As demonstrated by Equation (27), ΔG₀, which is generally determined by the thermodynamic data of each component involved in the reaction, can be used to calculate the value of K_k for the ion exchange reaction at standard state. However, because of the fact that special beneficiation of ion-adsorption minerals can be observed, it is difficult to acquire accurate thermodynamic data for the metal ion, Me³⁺, and directly calculate ΔG₀, leading to the impossibility of obtaining the value of K_k through Equation (27). As a result, the value of K_k can only be calculated based on the experimental data for the ion exchange reaction under equilibrium conditions and further processed in accordance with Equation (3).

According to the grade distribution data of ion-adsorption minerals, as demonstrated in Figure 2, available experimental data in the literature [26,27,28,29] with similar mineralogy have been compared and analyzed to finalize the value of K_k. In conclusion, the thermodynamic equilibrium constant of the ion exchange reaction, in terms of the standard state, was set to 6.73 × 10⁻⁹ L²/g² in this research [26].

As a result, based on the set values of the given parameters, the numerical solver was run to simulate the effect of leaching solution concentration on the ion exchange process by setting the initial concentration of cations, N⁺, dissolved in the leaching solution as the independent variable, with a range of 5 to 40 g/L. Then, the grade distribution of the ore body and the concentration of the metal ion, Me³⁺, after the reaction could be obtained, as shown in Figure 2, with a leaching solution concentration of 20 g/L.

For the characteristic points A, B, C, and D marked on Figure 2, the corresponding concentration of metal ions, Me³⁺, relative to the initial leaching solution concentration is illustrated in Figure 3b, indicating that more metal ions, Me³⁺, could be extracted from the ore body with the increase in the initial leaching solution concentration.

As illustrated by Figure 2 and Figure 3, the average concentration of metal ions, Me³⁺, dissolved in the mother liquor after the occurrence of the ion exchange reaction increased from 2.78 g/L to 3.94 g/L, and the average grade distribution of the ore body decreased from 0.00052 g/g to 7.34 × 10⁻⁷ g/g, in accordance with the increase in the initial concentration of cations, N⁺, dissolved in the leaching solution from 5 g/L to 40 g/L, as demonstrated by Figure 4. Also, the average extraction rate increased from 70.2% to 99.8%.

In conclusion, the overall extraction rate could be improved by increasing the initial concentration of cations, N⁺, dissolved in the leaching solution. However, the growth trend would gradually become steady. As shown in Figure 3, the overall extent of reaction tended to be stable, with the average extraction rate of metal ions, Me³⁺, reaching 98.6% when the initial leaching solution concentration was 20 g/L. Although a higher concentration of leaching solution would be preferred to improve the extraction rate of metal ions Me³⁺ from the ore body, a negative impact on the internal structure of the ore body might be expected. Therefore, the initial leaching solution concentration was set to 20 g/L for the subsequent computation and analysis, which is also in accordance with the concentration of leaching solution applied under the engineering conditions.

4.2. Effect of the Initial Grade Distribution on the Ion Exchange Process

Referring to the completed geological exploration, the minimum cut-off grade during extraction is 0.10%, namely, 0.001 g/g. Therefore, in order to verify whether the studied and applied values of the initial leaching solution concentration and thermodynamic equilibrium constant can be adopted to achieve the extraction of ion-adsorption minerals at this cut-off grade, a numerical solver was run to simulate the effect of initial grade distribution on the ion exchange process using the same 2D geometric model. As shown in Figure 5, the distribution of the extraction rate was obtained using this computation.

As illustrated by Figure 5, the average extraction rate for the ion-adsorption minerals was greater than 90%, even for the cut-off grade, indicating that the set values of the given parameters could serve as a theoretical reference for the extraction process, and the ion exchange reaction demonstrated a better thermodynamic potential and state.

4.3. Simulation of the Integrated Mathematical Model of the Overall Extraction Process

In order to further verify the accuracy of the numerical solver programmed for the ion exchange reaction, the mechanism model of thermodynamic equilibrium has been integrated into the mathematical model for the entire process of extraction, and a corresponding solver has been developed in order to achieve a mathematical simulation and provide predictive guidance and analysis for the engineering process. The specific logic for solving the integrated mathematical model of the overall extraction process is demonstrated in Figure 6. Firstly, the time-dependent and space-dependent distribution of water potential in the ore body can be obtained by solving the equation of seepage flow. Then, based on the correlated parameters passed from the seepage flow to solute transport, the time-dependent and space-dependent concentration distribution of cations N⁺ and metal ions Me³⁺ dissolved in the aqueous phase can be obtained by solving the equation of solute transport without considering the occurrence of ion exchange reactions. Finally, based on the correlated parameters passed from solute transport to the ion exchange reaction, the solution of the equation for the ion exchange reaction can be achieved, indicating the simulation of the overall extraction process.

Taking specific ion-adsorption minerals as an example, with the same parameters and corresponding set of values applied to the above 2D geometric model, a Three-dimensional (3D) geometric model was built to represent the surface of the ore body in accordance with the real topography. Then, mesh processing using tetrahedron mesh was adopted to generate nodes and elements in order to facilitate the calculation of the integrated mathematical model. In accordance with the available geological exploration, the 3D geometric model was set and designed to demonstrate the major fully weathered layer. The overall average grade of the corresponding ore-bearing layers was set to 0.0011 g/g, as highlighted in Figure 7. According to the analysis of the effects of the initial leaching solution concentration and the thermodynamic equilibrium constant on the ion exchange reaction, the initial concentration of cations, N⁺, in the leaching solution was set to 20 g/L, and the numerical value of the thermodynamic equilibrium constant was set to 6.73 × 10⁻⁹ L²/g².

As a result, based on the set values of the given parameters, the numerical solver for the integrated model has been run to simulate the time-dependent concentration profile of metal ions (Me³⁺) extracted and dissolved in the mother liquor after the occurrence of the ion exchange reaction, referring to the predetermined leaching area, as shown in Figure 8.

The average concentration of metal ions (Me³⁺) dissolved in the mother liquor increased overall following the ion exchange reaction over time, as indicated by the color change. Points A and B were marked to represent the southern collecting area and northern collecting area of the mother liquor solution, respectively. Point C was marked to represent the regular area for extraction. As shown in Figure 9, trends with high concentrations can be observed due to the convergence of metal ions, Me³⁺, toward the collection system. Meanwhile, fluctuations in the corresponding concentration can be observed due to the continuous collection of the mother liquor solution.

5. Conclusions

In this research, an ion exchange mechanism model centered on thermodynamic equilibrium has been proposed to overcome the complexity of traditional kinetic models. By simplifying the extraction process as an instantaneous equilibrium reaction at the solid-liquid interface and incorporating the mass balance equation, a quartic equation (Equation (23)) was successfully established, along with the development of an efficient Python-based numerical solver.

Compared with kinetic models such as the shrinking core model or the electrical double-layer theory, the proposed model significantly reduces computational complexity. Furthermore, simulation results confirm a strong correlation between leaching rate and initial leaching solution concentration, providing a theoretical framework for the rapid prediction of leaching behavior.

Based on the simulation with the 2D model, a nonlinear relationship between initial leaching solution concentration and leaching rate was revealed. This indicates that a concentration of 20 g/L could be the economically optimal choice in engineering processes, as it effectively balances the leaching efficiency while minimizing the environmental risks associated with excessive injection and reducing reagent costs. Additionally, the model achieved a leaching efficiency of over 90% even for low-grade regions, providing technical support for the development of marginal resources.

Furthermore, the established model could be integrated into the mathematical model for the overall extraction process, and a corresponding solver has been developed to simulate the concentration profile of the extracted metal ion in the aqueous phase over time and validate the applicability of the thermodynamic equilibrium model for the ion exchange reaction. As a result, the concentration peaks of the extracted metal ion in the collection system occurred on the 10th day, suggesting the need to optimize the layout of the collection system to align with the seepage flow patterns.

Through the mechanism study of the ion exchange reaction and the integrated model, a fundamental understanding can be developed to provide digital support for the actual extraction process. Additionally, the relationship between extracted metal ion concentrations in the aqueous phase over time can be illustrated to provide theoretical support for the engineering project.