2.1

Mathematical model of flow field

The physical equations primarily utilized in this simulation include the continuity equation, the momentum equation, and the k-ε turbulent two-equation model. These three equations are derived and integrated into a unified equation for the purpose of this simulation:

In the equation, ρ represents density; x corresponds to the horizontal coordinate; and u signifies velocity.

2.2

Mathematical model of volume of fluid method(VOF)

The VOF (Volume of Fluid) method is an Eulerian-based interface tracking technique. In the VOF model, various fluid components share a unified set of momentum equations, and the volume fraction or volume fraction of each phase is recorded within every computational cell across the entire flow domain. This method can effectively visualize the dynamics of interfaces in multiphase flows and is well-suited for scenarios involving free surfaces and stratified flows. In the VOF model, a common set of momentum equations is employed for mutually immiscible fluid components, with the introduction of phase volume fractions α.

VOF kinetic equation

In the equation, ρ is the density, p is the pressure, g is the gravitational acceleration, U is the fluid velocity, μ is the effective viscosity, and F is the volume force.

VOF phase equation

In the equation, p, q are the phases, ρ is the density, α is the volume fraction, and 𝑆_𝛼𝑞 is the source phase. , are the mass transport between the two phases.

2.3

Mathematical model of discrete phase model(DPM)

In multiphase flow, the fluid phase is considered to be continuous and the dispersed phase is considered to be discrete; this model is called the dispersed phase (DPM) model and is of the Eulerian-Lagrangian type. This model is used to describe dilute concentration particle particles, which generally have a total volume fraction of discrete particles of less than 10-12 %.

The main forces acting on the particles during the discharging process are: gravity, traction, traction^[12]. For the purpose of this study, neodymium oxide particles are equated to smooth spheres of uniform particle size. he force balance equation can be written as:

In the equation, 𝑚_𝑝 is the quality of particles; is the velocity of the continuous phase; is the particle velocity; 𝜌 is the density of the continuous phase; 𝜌_𝑝 is the density of particles; is the additional force; is the traction of particles; 𝜏_𝑟 is the relaxation time 1 of particles.

3.1

Assumption

The following assumptions have been applied to the flow within the rare earth electrolytic cell: (1) The fluid within the cell is divided into three phases: the electrolyte, bubbles, and rare earth metals; (2) The gas generated at the anode is CO₂, treated as an incompressible fluid; (3) Rare earth metals precipitate at the cathode, and bubbles form on the inner surface of the anode; (4) Wall slip velocity is not considered.

3.2

Geometry model

Figureure1 (a) and (b) depicts a schematic illustration of the rare earth electrolytic cell’s geometry, while Figureure 1(c) and (d) illustrates the computational fluid domain simulation model. Given that the cathode and anode are excluded from the analysis, the simulation specifically targets the electrolyte portion within the rare earth electrolytic cell.

Figure. 1

Physical model and mesh

3.3

Boundary conditions

Figure. 2

Boundary conditions in electrolytic:➀mass-flow inlet,➁velocity inlet.

4.1

Results of flow field simulations

4.1.1

Three-phase distribution

According to the numerical simulation, the three-phase distribution under the joint action of metal monomer-electrolyte-gas was obtained, as shown in Figure. 3. Bubbles are generated at the anode and rise to the surface of the electrolyte, while liquid metal is produced at the cathode, collecting and dripping at the bottom.

Figure. 3

Transient changes in three-phase distribution

Figure. 4

(a) velocity distribution; (b) velocity distribution in the reference [14]; (c) velocity distribution in the reference [16]

Subsequently, this paper compared the calculated velocity field with the research findings of other scholars[14,16]. Given that the velocity fields from the other two scholars solely considered the impact of bubbles without accounting for the settling of liquid metal, the comparison was confined to the upper region of the velocity field influenced by the gas in the electrolyte. From Figureure 4 (a), it can be observed that gas is primarily concentrated near the inner wall of the anode, resulting in the maximum velocity near the inner wall of the anode. The maximum upward velocity is 0.9 m/s, and this value aligns with the maximum velocity values in Figures 4 (b) and (c). Interestingly, the velocity field in this paper combines the characteristics of both Figure 4 (b) and (c). In the upper region of the velocity field in this study, characteristics akin to (a) are observed. An ‘island-like’ high-velocity flow area emerges between the inner wall of the anode and the intermediate region of the cathode, significantly influencing the surrounding flow, resulting in denser streamlines in that particular area. Simultaneously, the lower region of the velocity field in this paper exhibits similarities to (b), manifesting a distinct banded stratification within the low-velocity flow area.

4.1.2

Flow structure

To analyze the impact of multiphase flow on flow structure, the streamlines are combined with the three-phase distribution, as shown in Figureure 5. Flow within the rare earth electrolytic cell forms two pairs of vortices, distributed between the anode and cathode as well as at the bottom. In the subsequent sections, the factors contributing to the formation of vortex flow structures and their implications will be discussed.

Figure. 5

Velocity streamlines

4.2

Analysis of vortex

4.2.1

Formation of vortex

In this section, the method of controlling variables is employed to investigate the causes behind the formation of vortex flow. Four different conditions were established, as illustrated in Figure. 6. In the absence of both CO₂ gas generation and metal sedimentation, vortex structures do not form within the interior of the electrolytic cell (Figure. 6 (a)). The introduction of only CO₂ gas results in the formation of a pair of vortices in the upper region of the cell (Figure. 6 (b)), consistent with the observations in reference ^[16]. Conversely, the presence of only metal sedimentation leads to the formation of a pair of vortices in the lower region of the cell (Figure. 6 (c)). When both factors operate simultaneously, as depicted in Figureure 5, two pairs of vortices are formed(Figure. 6 (d)).

Figure. 6

Streamlines in different conditions (a) none; (b) CO2 generate; (c) metal settlement; (d) CO2 generate with metal settlement

In summary, the vortices in the upper region of the electrolytic cell’s interior are driven by the upward motion of gases, which in turn propels the flow of the electrolyte. Conversely, in the lower region of the cell, the flow is induced by the settling of liquid metal, creating a circulation pattern in the bottom electrolyte.

4.2.2

Factors affecting vortex flow

From the previous discussion, it is evident that both the upward motion of gases and the settling of liquid metal collectively influence the flow within the rare earth electrolytic cell. However, it remains undetermined which of the two factors is the primary influencing element, and how their variations affect the flow. Therefore, this section delves into the analysis and discussion of metal sedimentation at different mass flow rates and CO₂ gas at varying velocities.

To begin, we established varying mass flow rates with substantial intervals, specifically qm = 29.980 g/s, qm=69.980 g/s, and qm=99.980 g/s. Based on the streamline results(Figure. 7), the flow field structure did not exhibit significant changes under the three different mass flow rate conditions. The overall structure of the velocity field also remained relatively consistent(Figure. 8), with no prominent alterations. The maximum flow velocity consistently maintained at 0.9 m/s. Hence, it can be deduced that the sedimentation of liquid metal exclusively leads to the formation of vortex structures at the bottom. Once established, these structures exhibit remarkable stability, and variations in sedimentation rates exert negligible influence on the internal flow dynamics of the rare earth electrolytic cell.

Figure. 7

Streamlines for different mass flow rates

Figure. 8

Velocity field at different mass flow rates

However, the velocity of gas generation on the inner wall of the anode significantly influences the flow field within the rare earth electrolytic cell. We similarly established three sets of different gas velocities, namely 0.0011 m/s, 0.0034 m/s, and 0.0082 m/s. By examining the streamlines(Figure. 9), a clear trend emerges wherein the vortex structures undergo changes with varying gas velocities. Specifically, in the upper portion of the electrolytic cell, the longitudinal range of vortices progressively expands and extends downwards, while the lower vortices experience compression, leading to a narrowing of their range and a denser arrangement of streamlines.

Figure. 9

Streamlines for different gas velocity

The distribution of the velocity field shows the same evolving trend, as illustrated in the Figure. 10. The high-velocity zone consistently expands with increasing gas velocity and gradually extends downward from the inner wall of the anode, significantly affecting the flow of the electrolyte. The maximum velocity in the flow field also undergoes changes, with values of 0.65 m/s, 0.9 m/s, and 1.3 m/s, respectively. In summary, the gas velocity significantly impacts the flow within the rare earth electrolytic cell, with gas upwelling serving as the determining factor in driving the circulation of the electrolyte. The internal flow dynamics within the electrolytic cell are predominantly fueled by the movement of gases.

Figure. 10

Velocity field at different gas velocity

For a more precise characterization of this variation, a rigorous scientific approach was employed, known as the Ω vortex identification method[17]. Ω is a nondimensional scalar from 0 to 1 and it can be used to capture both strong and weak vortices simultaneously[18]. Its expression is as follows:

A is a symmetric rate-of-strain tensor, and B is an antisymmetric spin tensor.It is generally recommended in the reference literature to use Ω=0.52 as the isosurface value to visualize vortices[19].

The vortex identification using the omega method is depicted in Figure. 11. Its variation at different gas velocities follows a similar trend as the streamlines, with the vortex flow region continually expanding and extending downward. It is noteworthy that under lower gas velocities (0.0011 m/s, 0.0034 m/s), the omega vortex identification closely corresponds to the streamlines. However, at higher gas velocities (0.0062 m/s, 0.0082 m/s), the turbulence level within the flow field intensifies. In these conditions, the streamlines fail to accurately identify vortices, as depicted in Figure. 11 (c) and (d). Notably, certain streamlines exhibit vortex-like patterns with considerable intensity, even in regions where the streamlines do not form closed loops. This observation aligns with the findings in reference [20], affirming the validity of the omega method and substantiating the accuracy of the vortex trends expounded in this study.

Figure. 11

Ω method for vortex identification

4.3

Raw materials particle movement

In this section, we employ the DPM model to investigate the movement of Nd2O3 particles of different sizes upon their introduction from the top into the flow field.

On the whole, as the particle size increases, the range of particle descent decreases. This is primarily due to the particles experiencing a stronger gravitational force while encountering less substantial drag forces from the flow. Larger particles exhibit a greater tendency to break free from vortices, as depicted in Figure. 12. Smaller particles tend to cluster within vortices at the bottom, whereas larger particles descend nearly vertically at the bottom. This is further corroborated in Figure. 13. The x-coordinate in Figure. 13 represents the model’s coordinate in the negative direction of x, where smaller numerical values indicate closer proximity to the inner wall of the anode, signifying a smaller range of movement. It is apparent that under all circumstances, the x-coordinate of the landing point for larger particles is consistently smaller than that for smaller particles. This indicates that larger particles are more effective at evading the influences on the outer edges of vortices, resulting in a smaller range of movement.

Figure. 12

Trajectories of oxide particles

Figure. 13

Different particle descent positions for various particle sizes

On the other hand, Figure. 12 illustrates that for particles of the same size, as gas velocity increases, particles become more susceptible to the influence of the flow field. This is due to the substantial impact of gas velocity on the electrolytic cell’s flow field, serving as the primary driver of the circulatory motion. Higher gas velocities generate more strength vortices within the flow field(Figure.14), resulting in greater fluid drag forces that affect particle motion. This can be observed in the movement trajectories of particles with diameters of 1mm and 1.5mm at a gas velocity of v=0.0082 m/s in Figure 12. At this gas velocity, the vortices formed exhibit high strength (1400), approximately double the intensity value under normal conditions (800). Smaller particles in the range of 0.4 mm to 0.8 mm, due to their lower gravitational and inertial forces, are directly entrained by the outer vortices and directed towards the bottom. In contrast, larger particles with diameters of 1 mm and 1.5 mm, influenced by their larger gravitational and inertial forces, remain relatively unaffected by the outer vortices and engage with the inner vortices, where they participate in the circulation of the flow. Figure. 13 serves to reinforce the aforementioned points. It illustrates that with increasing gas velocity, the x-coordinate values of particle impact points shift further away from the origin, signifying a broader range of particle motion. In essence, higher gas velocities correlate with a more pronounced influence of the flow on particle behavior.

Figure. 14

Variations in vortex intensity with changes in gas velocity

We also monitored the residence time of particles. Figure. 15 illustrates the trend of particle residence time as a function of gas velocity, revealing some intriguing observations. The residence time of large particles (diameter ≥ 1 mm) and small particles (0.4 mm and 0.6 mm) exhibits two distinct trends with varying gas velocity. When the particle diameter is equal to or greater than 1 mm, the residence time monotonically decreases with increasing gas velocity(from 1100-1200 s to 330 s). However, for particles with a diameter less than 1 mm, the residence time initially decreases and then increases with the growing gas velocity, with the inflection point occurring at v = 0.0062 m/s. For particles with a diameter of 0.8 mm, the trend changes from a initial decrease to a nearly flat profile.

Figure. 15

The relationship between particle residence time and gas velocity magnitude

Hence, it is postulated that around a particle size of approximately 0.8 mm, there might be a critical threshold for particle motion. When the particle size is below this threshold, higher gas velocities can re-entrain the already settled oxide particles, causing a reverse increase in the residence time of small particles. In practical production, given the low solubility of oxide particles, typically around 2%–5%, this phenomenon prolongs the particles’ movement within the electrolytic cell, enabling them to participate more extensively in reactions. This is advantageous for enhancing oxide particle solubility and, consequently, improving yield.