Numerical Study of Percolation and Seepage Behaviors in Ion-Adsorption-Type Rare Earth Ore Leaching Process

Ionic rare earth ore is a type of featured rare earth ore in China. Its mining process sufers from a long leaching cycle and considerable consumption of leaching agents. Improving mining efciency requires a sound physical understanding of the leaching process. In this study, the CFD-based numerical model is used to analyze the physical process of leaching through porous media formed by particles. Te simulation results indicate that a lower packing porosity and smaller particles packed granular porous medium result in much larger energy dissipation during seepage, and the energy dissipation increases with seepage velocity. It is found that when the seepage velocity increases to a certain high value, the energy dissipation exceeds the value predicted by Darcy’s law, which is mainly caused by liquid turbulence. Additionally, the efect of particle shape is examined. Te results show that the granular medium composed of prolate particles causes larger energy dissipation than oblate particles, and spherical particles play the least role. Tis phenomenon may result from the particle shape afecting the area of the frontal contact surface between particles and liquid. Te results provide new insights into the fundamental understanding of percolation and seepage behaviors in the ion-adsorption-type rare earth ore leaching process.


Introduction
Ionic rare earth ore, also known as weathering crust leaching rare earth ore, is a type of featured rare earth ore in China.It plays a signifcant role in high-tech and new material industries due to its various advantages, e.g., diverse sources, abundant reserves, and including all the light, medium and heavy rare earth, particularly rich in terbium, dysprosium, and europium [1].Ionic rare earth ore is formed by the adsorption of rare earth elements in the form of ions on clay minerals [2].Tere are considerable reserves of rare earth elements in the earth's crust.However, due to its unique properties and the limitation of the technical level, although lots of rare earth elements have been explored in many areas, they cannot be well exploited and utilized.In the mining process, several key issues are a concern, such as the long leaching cycle and the large consumption of leaching agents.Terefore, it is of great signifcance to improve the understanding and optimize the extraction process of rare earth minerals.
Te in-situ leaching process incorporates a typical fuid seepage in a porous medium accompanied by an ion exchange reaction, which is infuenced by the hydrodynamics in the porous medium and the leaching reaction kinetics.
Many researchers have conducted extensive experiments to improve the leaching efciency [3,4].To date, previous experimental work mainly generates input and output data but lacks analyses of fuid fow, mass transfer, pressure, and velocity distributions.In terms of numerical studies, the lattice Boltzmann method is mostly adopted in previous numerical research, which is regarded as a more suitable method for a small number of lattices but not for engineering calculation [4].Zhao et al. [5] used Fluent software to simulate the migration of the water phase in three diferent flling structures under diferent initial water velocities and dynamic viscosities and indicated between the fxed particles (soil skeleton) in the flling structure, the looser structure, and the greater infuence of gravity on the water phase seepage.Wang et al. [6] study the grouting seepage of microfractures under diferent grouting pressures and fracture opening conditions, and the grouting pressure and fracture opening changes are nonlinearly attenuated along the grout seepage direction.Li et al. [7] examined the efects of particle size on fow regimes and the validity of Darcy law in the seepage process.Luo et al. [8] carried out column leaching experiments to consider the impact of permeability in the rare earth leaching process and analyzed the mechanism of the change of permeability coefcient.Dixson and Hendrix [9] proposed a mathematical model for the leaching of various solid reactants from ore pellets heap assuming that the size, density, and packing porosity of the spherical particles were the same and the leaching process was isothermal and irreversible.Ma et al. [10,11] established a onedimensional radial three-phase fow model of water-rock-silt and obtained the temporal-spatial distribution of hydraulic properties by numerical simulation.Li et al. [12] established a fow model for the seepage process under gravity in the insitu leaching process and simulated the vertical infltration process of the leachate, which provided knowledge for further exploration of the fow mechanism of the leachate.Li et al. [13] quantitatively investigated the pore structure and hydraulic properties of the broken rock mass by the nuclear magnetic resonance technology and the non-Darcy models.Additionally, a logistic regression model was proposed to characterize the efects of compressive stress and GSD on the permeability and the characteristic parameter of nonlinear fow in the broken rock mass.Huang et al. [14] investigated the efect of surface roughness on laminar fow in macro tubes and conducted head drop experiments in six organic glass tubes with diferent relative roughness.In general, although the applicable critical value of Darcy's law has been measured in previous studies, the composition of experimental materials is relatively simple, and the research is limited to simplifed situations, which are difcult to represent the most practical scenarios.Te related work is mainly based on experiments, which can qualitatively analyze the process, while the relevant mathematical models mostly focus on the efect of packing particle size, ignoring some factors such as particle profle.
Tis study would apply a single-phase fow model and the Euler-Euler two-fuid model to simulate the seepage process of the fuid through a porous medium formed by particle packing.Te efects of most related factors, such as seepage velocity and particle properties, on seepage efciency will be analyzed, aiming to provide a fundamental understanding of this leaching process.

Model Description
In this work, simulations were conducted using the commercialized Fluent software package.Te efects of porosity, particle size, and particle shape on energy loss during seepage were investigated.More specifcally, the Euler-Euler two-fuid model, which is widely used to describe the fuid fow in porous media, was adopted in the seepage process simulations, while the single-phase fow model was employed to examine the efect of particle size and shape for better capturing pore scale information.In the two-fuid model, the solid and liquid phases are considered fully interpenetrating continua coupled with an interaction term.Since the volume occupied by one phase can no longer be occupied by other phases, the concept of phase volume fraction is introduced.Te volume fraction is a continuous function of time and space, and the sum of the volume fractions of all phases equals 1.0.Te two phases of transport phenomena are resolved with similar mass and momentum conservation equations.For the liquid phase l, the mass conservation equation is given as follows: where α l , ρ l , and v l represent the volume fraction, density, and velocity of the liquid phase, respectively; _ m ls stands for the mass transfer from the liquid phase to the solid phase; _ m sl denotes the mass transfer from the solid phase to the liquid phase; and S l is the source term.Te momentum conservation equation of the liquid phase can be expressed as follows: where p is pressure, τ l is the stress-strain tensor of the liquid phase, and v ls is the relative velocity of the liquid and solid phases.
Similarly, the mass and momentum conservation equations of the solid phase are given as follows: 2 Mathematical Problems in Engineering (3) As for the interaction force, namely, the drag force, the Gidaspow model is adopted.It is based on the Wen-Yu model and combines it with the Ergun equation [15].When α l > 0.8, the liquid-solid exchange coefcient equation is described as follows: When α l < 0.8, the liquid-solid exchange coefcient equation is given as follows: Te standard k-epsilon model, which is suitable for fuids with fully developed turbulence [16], is adopted for turbulence description.Te expression of the turbulence equations, including the turbulent kinetic energy k and the turbulent dissipation rate ε are given as follows: where μ t is the turbulent viscosity; G k is the turbulent kinetic energy; G b is the turbulent kinetic energy caused by buoyancy; Y M is the change in the total dissipation rate caused by pulsating expansion; μ is the fuid viscosity; σ k and σ ε are the turbulent Prandtl number of k and ε, respectively; and u i is the average fuid velocity.In general, C 1ε � 1.44, C 2ε � 1.92, C 3ε � 0.99, σ k � 1.0, and σ ε � 1.3 as reported elsewhere [17,18].

Simulation Conditions
3.1.Geometric Structure and Meshing.In this study, the simulation simplifes the physical model (a tube, referring to a previous experimental study [19]) with a 2D cross-section to improve the computing efciency in considering the efect of pack bed porosity.Te calculation domain is in the dimensions of 350 mm (length) × 100 mm (width, tube diameter) as shown in Figure 1.To speed up the calculation convergence and eliminate the inlet and outlet efects, two bufer sections with a length of 50 mm were set at the inlet and outlet, respectively.Note that the size of the established numerical model is slightly shorter than that of the experimental model.Terefore, the sufcient and stable fow of the fuid phase should not be afected.Additionally, the corresponding 3D geometry is given in Figure 2. In particular, the tube diameter is 47 mm and the length of the middle section of the porous medium is 100 mm.Te inlet and outlet bufer sections were set up with a length of 150 mm.Te CFD mesh is critical to the reliability and accuracy of the simulation results.Figure 3 shows the efect of grid number on the simulated outlet velocity.According to this grid independence test, the grid numbers with the values of 8,750 and 371,110 were selected in the 2D and 3D simulations, respectively, to study the efect of porosity, particle size, and shape.

Boundary Condition.
Te "Velocity Inlet" boundary condition was adopted, with fuid speed values of 0.001-0.011m/s.Te outlet adopted the widely used "Pressure Outlet" boundary condition.Additionally, the operating temperature was ambient temperature, set to 298 K. Te pressure reference algorithm was selected, and the implicit solver was used for the transient solution.Te time step size was 10 −3 s.Pressure-speed coupling chose the SIMPLE algorithm, and the diference scheme adopted second-order upwind; the wall was processed according to the "no-slip" boundary condition.Te Gidaspow model was adopted as drag force calculation, and the residuals for simulation calculation were less than 10 −3 .

Model Validation.
To validate the numerical model, the variation of the hydraulic gradient with seepage velocity was depicted as shown in Figure 4. Te hydraulic gradient indicates the ratio of the head drops along the seepage path to its length, given as follows:

Mathematical Problems in Engineering
where ∆H is the head drop of the fuid fowing through the porous media region, and ∆L is the length of the porous media region.It can be seen that the simulated results are in good agreement with the experimental results [14], demonstrating the efectiveness of the present CFD model.

Efect of Porous Medium Porosity.
Ionic rare earth ore is normally packed as a porous medium.Normally, its porosity varies according to its diferent categories.To investigate its infuence, the porosity of 0.37, 0.40, 0.43, 0.46, and 0.49 is adopted in the simulation.Te pressure contours under diferent porosity are depicted in Figure 5. Te results show that the pressure distribution varies with porosity under a constant seepage velocity of 0.001 m/s. Figure 6 shows the hydraulic gradient increases with seepage velocity, which results from the small vortices formed during the fuid fowing through the porous medium.As the seepage velocity increases, the resulting vortex increases correspondingly, which is a key factor for energy dissipation.Additionally,   4 Mathematical Problems in Engineering under the same seepage velocity, a larger porosity results in a smaller hydraulic gradient.Tis is because porosity afects the resistance of fuid fow.More specifcally, the fow resistance is larger with a smaller porosity, which requires a larger hydraulic gradient to achieve a certain fow rate.

Efect of Packed Bed Particle Size.
For the size of most ionic rare earth ores, around two-thirds of the ore particles' diameters are smaller than 7.75 mm [1].In this study, fve diferent values of spherical particle diameter are adopted for comparison analyses, namely, 1.0 mm, 1.5 mm, 2.0 mm, 2.5 mm, and 3.0 mm.Te pressure contours of diferent particle-size packed beds are shown in Figure 7. Te smaller the particle size, the larger the pressure drop.Te variation of the hydraulic gradient with seepage velocity is shown in Figure 8. Te results indicate that under the same seepage velocity, the smaller the particle size, the larger the corresponding hydraulic gradient, which indicates that the fuid fow in a medium packed with fne particles is more difcult than that packed with coarse particles.
According to Darcy's law, the cross-sectional area A, the seepage fow Q through A, the head drop (h 1 − h 2 ), and the seepage path length L are in the following relationship [19]:    Mathematical Problems in Engineering which is simplifed as follows: where K is the permeability coefcient, determined by the solid and fuid materials; v s is the seepage velocity; and J is the hydraulic gradient.Te corresponding curves based on Darcy's law are shown in Figure 8.With the increase in seepage velocity, the seepage phenomenon for each of the fve types of particles gradually deviates from Darcy's law.Tis is because the seepage process can be regarded as laminar fuid fow under a small seepage velocity.With a larger seepage velocity, the seepage velocity and hydraulic gradient are no longer linear, and the seepage process changes into a turbulent fuid fow.A similar deviation from Darcy's law was also observed at high velocity [20,21].
4.4.Efect of Packed Bed Particle Shape.Since the ore particles are generally nonspherical ones in practical industries, this work also examines the efect of particle shape on the seepage process, such as prolate particles and oblate particles.Note that each particle with diferent shapes (ellipsoid aspect ratio) has the same volume in this study.Te aspect ratio of a spherical particle is 1.0, while it is 1.25 for a prolate particle and 0.82 for an oblate particle as shown in Figure 9. Figure 9(a) represents spherical particles with a radius of 0.57 mm.Te other ellipsoidal particles' projections are perpendicular to the inlet direction presenting a circle with a radius of 0.61 mm and an ellipsoid with a long/short semiaxis of 0.75 and 0.5 mm, respectively.Figure 10 shows the variation of energy dissipation with seepage velocity for the systems packed with diferently shaped particles.It is shown that the energy dissipation is not signifcantly infuenced by the particle shape under a small seepage velocity.However, the seepage resistance for a spherical particle system is smaller than that of a nonspherical particle system at a larger seepage velocity.For the nonspherical systems, the prolate particle system has larger seepage resistance and energy dissipation than that of the oblate particle system.Te diference can also be observed in the contours of fuid pressure on the particle surfaces shown in Figure 11 and the corresponding reasons are discussed later in the article.
Te mean curvature, referring to the ratio of the angle between the tangents at the ends of the arc to the length of the arc, and the frontal contact surface area of the particle are calculated for diferent particle shapes as shown in Table 1.Since the selected arc corresponds to the long axis, the angle between the ends of the arc is 180 °.Te frontal contact surface area in the situation of this work equals half of the particle surface area.As listed in Table 1, the oblate and spherical particles have almost the same curvature.In contrast, the energy dissipation under larger seepage velocities for oblate particles is larger than that for spherical particles.Tis may reveal that the curvature does not determine the energy dissipation during leaching.Te diference in frontal contact surface area corresponds better with the diference in energy dissipation, indicating that the frontal contact surface area may be the main factor determining the energy dissipation.

Conclusions
Tis work applies a validated CFD model to fundamentally study the seepage features in the ionic rare earth ore leaching process.Te results indicate that a smaller packing porosity and a smaller particle size of the granular porous medium lead to larger energy dissipation during seepage.Te energy dissipation increases with seepage velocity.Due to turbulence, the energy dissipation becomes larger than that predicted by Darcy's law when the seepage velocity increases to a large value.Additionally, the granular porous medium packed with prolate particles causes more energy dissipation, followed by the oblate and spherical particles, which may be mainly caused by the varied particle frontal contact surface areas.

Figure 2 :
Figure 2: (a) 3D geometry of the simulation domain and (b) grid arrangement for the study on the efect of the particle size and shape.

Figure 3 :
Figure 3: Te efects of diferent grid numbers on the simulated outlet velocities: (a) 2D simulation to study the efect of porosity; (b) 3D simulation to study the efect of particle size and shape.

Figure 1 :
Figure 1: (a) 2D geometry of the simulation domain and (b) grid arrangement for the study on the efect of porosity.

Figure 5 :
Figure 5: Contours of the pressure under diferent porosity.Te black lines close to the inlet and outlet are the boundaries between the bufer section and the porous medium.

Figure 6 :
Figure 6: Variation of energy dissipation as a function of seepage velocity under diferent porosity.

Table 1 :
Curvature and frontal contact surface area for each particle shape.