Mathematical Model of Ilmenite Separation Efficiency Using a High Gradient Plate Magnetic Separator

Abstract

High gradient magnetic separation is widely used in magnetic minerals upgrading, and its separation performance is significant depending on the parameters. In this investigation, the Mathematical model of the plate high gradient magnetic separator is established, the magnetic induction and the flow field distribution are investigated based on the COMSOL multi-physical simulation, and then the separation efficiency and TiO₂ grade are analyzed using the plate high gradient magnetic separator. Additionally, the key factors affecting the efficiency of mineral separation are detailed in the experimental separation, the separation efficiency is demonstrated and its feasibility is verified by experiments. It is founded that the mathematical model and simulation results are basically validated by the experimental separation process, and the TiO₂ grade can be effectively upgraded from 5.2% to 11.5% with the rinsing water consumption 9.5 L/min and the belt rotating speed 2 r/min. It is thus concluded that plate high gradient magnetic separator has provided an effective way in upgrading ilmenite quality.

1. Introduction

Ilmenite is one of the main titanium minerals, whose chemical composition is FeTiO₃ [1,2], it is also an important industrial metal for aerospace and chemical engineering. However, ilmenite is also mixed with impurities, it must be separated to obtain the high grade ilmenite. At present, the main separation methods include flotation [3,4], gravity [5], magnetic separation [6,7], and so on. For example, decanoic acid was found to be an effective collector, to float ilmenite in the strongly acidic pulp. The combination of decanoic acid and oleate proved to be significant in improving ilmenite floatability within a broad pH range, but the discharge of wastewater from the beneficiation industry has caused environmental pollution. Although gravity separation is suitable for ilmenite with coarse particle size, the beneficiation efficiency is too low, and it is often necessary to combine gravity separation with other processes in separating. High gradient magnetic separation has been an effective method for the concentration or removal of fine paramagnetic particles from suspension, but its powerful magnetic capture to magnetic particles results in the mechanical entrainment of non-magnetic particles in magnetic product, and thus reduces the separation selectivity. High mineral recovery is obtained by high gradient magnetic separation, but there is still a technical problem of low selectivity. Therefore, the research and development of a new weak magnetic ore separation process have important research significance and application value to improve the existing technology.

For nearly 20 years, the high gradient magnetic separation method is widely used in magnetic minerals separating process as its green characteristics advantage [8,9], and its principle has been detailed by Xiong and Chen [10]. For example, the effect of removing iron and extracting lithium from spodumene ore was investigated by SLon high gradient magnetic separator, the Li₂O grade was increased from 1.51% to 5.56%, and the Fe₂O₃ content was successfully reduced from 4.98% to 3.2% [11]. Moreover, the plate high gradient magnetic separator was used to separate quartz ore, the Fe₂O₃ content significantly declined from 0.062% to 0.02%. Further, the key operating parameters of plate high gradient magnetic separator were investigated [12]. However, the details of the magnetic induction and the fluid distribution were not analyzed, and the upgrade of TiO₂ grade using plate high gradient magnetic separator was not reported. In this investigation, to better understand the separation performance of plate high gradient magnetic separator, the whole plate high gradient magnetic separation process is further analyzed and simulated by finite element method (FEM) and COMSOL software, which is an effective numerical analysis method to solve structural nonlinear, hydrodynamic, and coupled field problems [13]. The separation performance under the different rinsing water consumptions and belt rotating speeds are detailed, also the separation efficiency is demonstrated and its feasibility is verified by experiments.

2. Plate High Gradient Magnetic Separator

As shown in Figure 1, the plate high gradient magnetic separator, whose main components are the feeding box, belt wheel, frame, water tank, inclination adjustment mechanism, and so on. A permanent magnet magnetic system is alternatively installed under the belt of the magnetic separator, which is made of High-quality NdFeB material and fixed on the magnetic pure iron. During the separating process, the slurry sample flows into the belt surface from the feed box, the strong magnetic mineral particles are adsorbed on the belt surface and transported to the magnetic products recovery box with the belt rotating; meanwhile, the particles of low magnetic susceptibility and some impurity conjoined particles are separated downward with the rinsing water, equipment size 2300 mm × 500 mm × 1500 mm.

3. Mathematical Analysis

A geometric model of a magnetic particle is assumed to be a spherical particle under ideal conditions in the flowing process along the belt surface, regardless of its size and shape [12]. It can be supposed that the particle diameter is d, volume is V, density is δ, and the magnetic susceptibility is K. When the magnetic particle locates in the belt center, its force analysis is shown in Figure 2. The particle is affected by gravity F_g, buoyancy F_f, friction F_μ, magnetic force F_m, and fluid force F₁, respectively.

The particle is subjected to gravity F_g, which is shown:

$$F_{\mathrm{g}}=\frac{\mathsf{\pi }{d}^3}{6}\delta g$$

(1)

Gravity is decomposed into F_g1 parallel to the belt direction and F_g2 perpendicular to the belt direction, which can be written as:

$$F_{\mathrm{g}}{}_1=\frac{\mathsf{\pi }{d}^3}{6}\delta g\cdot \sin \alpha$$

(2)

$$F_{\mathrm{g}}{}_2=\frac{\mathsf{\pi }{d}^3}{6}\delta g\cdot \cos \alpha$$

(3)

where g is the acceleration of gravity, and α is the dip angle of the magnetic system.

The particle is subjected to magnetic field force F_m, which can be calculated as:

$$F_{\mathrm{m}}=\frac{1}{{\mu }_0}\mathrm{K}VH\cdot \mathrm{grad}H$$

(4)

where μ₀ is the vacuum permeability, H is the magnetic field intensity, and grad H is the magnetic field gradient.

Particle buoyancy F_f can be calculated:

$$F_{\mathrm{f}}=\frac{\mathsf{\pi }{d}^3}{6}\rho \mathrm{g}$$

(5)

where ρ is the slurry density.

The magnetic particle moves upward with the belt under the friction force F_μ:

$$F_{\mathsf{\mu }}={\mu }_{\mathrm{p}}\cdot F_{\mathrm{N}}$$

(6)

The belt provides the supporting force F_N in the vertical direction for particles:

$$F_{\mathrm{N}}=F_{{\mathrm{g}}_2}+F_{\mathrm{m}}-F_{\mathrm{f}}\cdot \cos \alpha$$

(7)

Therefore, the F_μ can be rewritten as:

$$F_{\mathsf{\mu }}={\mu }_{\mathrm{p}}\left(F_{\mathrm{m}}+F_{\mathrm{g}}\cos \alpha -F_{\mathrm{f}}\cos \alpha \right)={\mu }_{\mathrm{p}}\left[\frac{1}{{\mu }_0}\mathrm{K}VH\mathrm{grad}H+\cos \alpha \cdot \frac{\mathsf{\pi }{d}^3}{6}\left(\delta -\rho \right)g\right]$$

(8)

where μ_p is the belt friction coefficient.

The magnetic separation process is consistent with the principle of thin film flow beneficiation. When the slurry flows on the surface of the conveyor belt, the F_l of the magnetic particle is subjected to the fluid force, which can be deduced from the Stokes formula:

$$F_{\mathrm{l}}=3\mathsf{\pi }\mu \left(u+v\right)d$$

(9)

where v is the belt speed, μ is the slurry viscosity, and u is the water flow speed.

When the resultant force of the particle upward along the belt is greater than other forces, the magnetic particle can be captured and move upward with the belt. Due to the small belt speed, and the centrifugal force on the particles can be ignored, thus the critical conditions for the effective capture of particles are as follows:

$$F_{\mathsf{\mu }}+F_{\mathrm{f}}\cdot \sin \alpha \ge F_{{\mathrm{g}}_1}+F_{\mathrm{l}}$$

(10)

Hence,

$$F_{\mathrm{m}}\ge \frac{1}{{\mu }_{\mathrm{p}}}\left[3\mathsf{\pi }\mu \left(u+v\right)d+\frac{\mathsf{\pi }{d}^3}{6}\left(\delta -\rho \right)g\cdot \left(\sin \alpha -{\mu }_{\mathrm{p}}\cos \alpha \right)\right]$$

(11)

After that, the magnetic field force required for magnetic particles to be effectively captured can be expressed as:

$$H\mathrm{grade}H\ge \frac{{\mu }_0}{{\mu }_{\mathrm{p}}\mathrm{K}}\left[18\frac{\mu \left(u+v\right)}{d^2}+\left(\delta -\rho \right)g\cdot \left(\sin \alpha -{\mu }_{\mathrm{p}}\cos \alpha \right)\right]$$

(12)

Whether the particles can be effectively captured is mainly determined by the magnetic field force generated from the magnetic system. From Equation (12), it can be seen that the magnetic field force is related to the specific magnetization coefficient, particle density, belt speed, slurry flow rate, magnetic system inclination angle, belt surface friction coefficient, and other factors.

When the belt speed increases to a certain value, the centrifugal force on the particle should not be ignored. Therefore, the critical conditions for the effective capture of the magnetic particle are as follows:

$$F_{\mathsf{\mu }}+F_{\mathrm{f}}\cdot \sin \alpha +F_{\mathrm{r}}\ge F_{{\mathrm{g}}_1}+F_{\mathrm{l}}$$

(13)

The magnetic field force can be expressed as:

$$H \mathrm{grade} H\ge \frac{{\mu }_0}{{\mu }_{\mathrm{p}}\mathrm{K}}\left[18\frac{\mu \left(u+v\right)}{d^2}+\right.\left(\delta -\rho \right)\mathrm{g}\cdot \left(\sin \alpha -{\mu }_{\mathrm{p}}\cos \alpha \right)\left.-\rho \cdot \frac{v^2}{R}\right]$$

(14)

According to Equations (13) and (14), the belt speed can be appropriately increased to separate particles of low magnetic susceptibility when the magnetic field force of the equipment is not large enough. However, it is worth noting that the effect of centrifugal force is greater than magnetic force when the rotating speed is too large, and the separating performance is not obvious.

4. Simulation and Experiment Conditions

4.1. Simulation Model

The fluid characteristics and the magnetic distribution of the belt surface are difficult to obtain, which are related to whether the magnetic particles are captured. In order to simulate a real physical system with mathematical approximation, a finite number of unknowns can be used to approximate the real system with infinite unknowns by using simple and interactive elements. Thus, the two physical fields of fluid flow particle tracking and non-current magnetic field are described by finite element method (FEM) and the COMSOL software [14,15], the particle statistical method is consistent with the literature [16]. It is shown in Figure 3 that the whole simulation model is in the air domain, in which the separation region is the fluid domain. The stop strip on the belt is arranged in an elliptical convex shape and located at the lower boundary of the separation area. The magnetic system is set below the separating area, the permanent magnet iron blocks are arranged in a rectangular shape and spliced together, and the simulation model is consistent with the experimental equipment.

The free triangular mesh is adopted, and the common area is meshing with the refined mesh specification. The maximum cell is 88.8 mm, the minimum cell is 0.3 mm, the maximum cell growth rate is 1.25, and the curvature factor is 0.25. Extremely fine mesh specifications are used in critical areas, with a maximum cell size of 5mm and a minimum cell size of 0.04 mm, a maximum cell growth rate of 1.05, and a curvature factor of 0.2, as shown in Figure 3. The main parameters of simulation are shown in

Table 1. Main parameters of simulation.

Parameter	Value	Parameter	Value
Water density (kg/m3)	997	Particle size (mm)	≤2
Water Viscosity (kg/m)	8.9 × 10−4	Magnetite density (kg/m3)	0.75
Magnetic field intensity (T)	0.8–1.4	Non-magnetic density (kg/m3)	2800
Magnetite relative permeability	6.19	Belt rotating speed (r/min)	2

.

4.2. Experiment Conditions

As shown in Figure 4, the two side walls of the magnetic separator belt are shallow U-shaped with a slight bulge and a thickness of 2.0 mm, which has the function of transporting slurry. Raised baffles on the surface of the belt are distributed at equal intervals. The SPBC-0315 plate high gradient magnetic separator is used in this experiment, magnetic system length is a fixed value of 1 m, whose inclination angle is 5° [17,18], and the separation experiment content includes rinsing water consumption and belt speed two factors on the separation performance. The rinsing water consumption is set to 9.5–29.9 L/min, and the belt speed is set to 2–6.5 r/min. The main technical parameters of the magnetic separator are shown in

Table 2. Main parameters of plate high gradient magnetic separator.

Parameter	Value	Parameter	Value
Magnetic system area (mm)	1000 × 200	Particle size (mm)	≤2
Processing power (kg/h)	5–500	Motor power (kW)	0.75
Magnetic field intensity (T)	1.0	Machine size (mm)	2300 × 500 × 1500
Rising water (m3/h)	0.57–1.62	Belt rotating speed (r/min)	2–6.5

. The sample of mineral particles containing ilmenite is used in this experiment, which is taken from the Yunnan factory in China, the raw ore TiO₂ grade is about 7.3%. The elements in mineral samples are determined by X-ray fluorescence spectral analysis technology, and then use chemical analysis to analyze the content of each element in the sample according to the national standard. The chemical composition analysis results are shown in

Table 3. Chemical composition analysis results of the ore sample (wt.%).

Composition	Fe2O3	TiO2	MgO	Al2O3	CaO	SiO2	K2O	ZnO
Content	16.828	7.327	10.41	10.381	12.77	37.545	0.879	0.014

.

5. Simulation Result Analysis

5.1. Magnetic Distribution

From Figure 5, it can be seen that the magnetic induction lines are regularly distributed on the surface of the belt, and the magnetic field polarity is alternately distributed equidistantly down the belt. It is also shown that the magnetic field intensity at the magnetic pole connection is 1.0 T, and the magnetic field intensity in the middle of the magnetic block is 0.5 T. From Figure 6, the simulation results are consistent with the actual measurement results, and the error range is from 0.16% to 4.69%. The numerical value from the simulation results is consistent with the characteristics of the magnetic system in the actual separating condition.

5.2. Particle Trajectory

In this study, the generated particle size is set to be less than 2 mm, the proportion of magnetizable particles is set to 20% of the total number of particles, 300 particles are released for every second, and the simulation results reach a stable state after 8.0 s. Figure 7 shows the separation performance under the different magnetic induction, at rinsing water consumption 9.5 L/min, belt rotating speed 2 r/min. With the increase of magnetic induction, the concentrate TiO₂ grade increases, but the recovery decreases gradually. The concentrate TiO₂ grade and recovery are 10.3% and 74.6%, respectively, when the magnetic induction is 1.0 T. Meanwhile, it can be imagined that the inclusion of non-magnetic particles becomes obvious combined with the theoretical equation. It can also be seen that the prediction results of the simulation are similar to the changing trend of experimental data, especially in terms of concentrate TiO₂ grade. Moreover, compared with the experimental results, the correctness of the recovery and grade prediction results are verified.

Figure 8 shows the particle trajectory at different times with the magnetic induction 1.0 T. The blue particles represent non-magnetic mineral particles and the red ones represent magnetic mineral particles. Combining Equations (12) and (14), once the magnetic force acting on the particles should overcome other competitiveness, the magnetic particles can be captured in turn. Theoretically, coarser magnetic particles can be trapped in the region of high magnetic field intensity, while finer particles cannot.

5.3. Flow Field Distribution

From Figure 9, it is obviously seen that the slurry spreads on the surface of the belt in a thin film flow after the slurry is released from the feed port; meanwhile, the peristaltic flow is formed near the baffle, which fluctuates in a wave shape. It is also can be seen that the slurry collides with the belt surface, part of the slurry vector changes irregularly, and the velocity distribution of the fluid is uneven. Under the action of magnetic force and gravity, most magnetic particles are transported to the magnetic particle collection area with the belt, and most non-magnetic particles are moved with the fluid forming a stable laminar flow under the fluid action. Similarly, the laminar flow is also found in the experiment process, as shown in Figure 10.

6. Simulation Result Analysis

6.1. Effect of Rinsing Water Consumption on Separation Performance

In the separating process, the rinsing water consumption affects the separation performance of separator, and the captured particles are washed away by the large rinsing water consumption, as the magnetic product recovery is limited. Therefore, the effect of rinsing water consumption on the separation performance is firstly investigated, and this is achieved at a magnetic induction of 1.0 T, and with a belt rotating speed 2 r/min.

As can be seen from Figure 11, as the rinsing water consumption is increased from 9.5 L/min to 27 L/min, with the yield of magnetic products decreases gradually. With an increase in the rinsing water consumption, the titanium grade is also decreased, due to the fact that the captured magnetic force is less than the fluid force from the large rinsing water consumption. In such case, only the more sufficiently magnetic products are preferentially captured onto the belt surface and prevented from being washed, increasing the ilmenite content in the magnetic productivity, as mathematical analyzed before.

6.2. Effect of Belt Rotating Speed on Separation Performance

The belt rotating speed, which conveys particles to the different position, can be adjusted to separating particles. In practice, the increase of belt velocity leads to the increase of particle flow resistance in slurry. Thus, the effect of belt rotating speed on separation performance is investigated, and this achieved at a magnetic induction of 1.0 T and rinsing water consumption 9.5 L/min, with the results as shown in Figure 12.

It can be seen that the belt rotating speed has a significant influence on the plate high gradient magnetic separator performance. With increase in the belt rotating speed, the magnetic products are increased, meanwhile reducing the titanium grade after the belt rotating speed is large than 2 r/min. It is consistent with the mathematical model, the separation performance should be worsened when the centrifugal force on the magnetic particles exceed the critical value, and then the particles slip on the belt, resulting in relative stillness.

7. Conclusions

It can be seen from the results and discussion above that:

(1)

The belt rotating speed should be appropriately increased to separate magnetic materials when the magnetic field force of the equipment is not large enough, and it is worth noting that the separating performance is not obvious when the belt rotating speed is too large.

(2)

The capture of magnetic belt to magnetic particles could be improved in a rising water consumption, but the inclusion of non-magnetic particles cannot be excluded from the adsorbed particles on the belt surface, which leads the high magnetic products content. The plate high gradient magnetic separator may effectively upgrade the low-grade ilmenite minerals from 5.2% to 11.5% with the rinsing water consumption 9.5 L/min and the belt rotating speed 2 r/min.