Particle capture of special cross-section matrices in axial high gradient magnetic separation: A 3D simulation

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Particle capture of special cross-section matrices in axial high gradient magnetic separation: A 3D simulation ⁎

Zixing Xue, Yuhua Wang, Xiayu Zheng , Dongfang Lu, Xudong Li School of Minerals Processing & Bioengineering, Central South University, Changsha 410083, China Key Laboratory of Hunan Province for Clean and Efficient Utilization of Strategic Calcium-containing Mineral Resources, Central South University, Changsha 410083, China

A R T I C LE I N FO

A B S T R A C T

Keywords: High gradient magnetic separation Magnetic matrices Numerical simulation Particle capture Particle trajectory

Enhancing the recovery of ultrafine magnetic particles in high gradient magnetic separation (HGMS) is a tough issue in industrial practice. Mathematic modeling has been adopted to describe particle capture in HGMS and many analytical models for regular shape (only circular and elliptic) matrices under ideal conditions (ideal potential flow) have been derived. Irregular shape matrices have better magnetic characteristics and can enhance particle recovery in HGMS. However, analytical models cannot be derived for irregular shape matrices and generally qualitative analyses were conducted, which can hardly be rigorous and convincing. It is essential to develop methods to conduct quantitative analyses for irregular shape matrices. In this paper, 3D numerical simulation was adopted to study particle capture by irregular matrices in axial HGMS. A simulation model consisting of a particle group and magnetic matrices (circular, elliptic, square and diamond cross-section) for axial HGMS was established. Evolution of particle group in the HGMS system employing the four kinds of matrices were detailly demonstrated. Particle motion trajectories were depicted and particle capture cross section area were calculated and compared quantitatively. Elliptic and diamond matrices present better particle capture performance than circular and square matrices under moderate induction range and can be applied to enhance recovery of ultrafine particles. The numerical simulation results are consistent with our previous theoretical studies using analytical models and experimental results on the matrices. The present study also indicates that there should exist optimal aspect ratio for diamond matrices and optimal tooth angle for grooved plates used in HGMS.

1. Introduction High gradient magnetic separation (HGMS) is effective in removing or recovering weakly magnetic materials from slurry and has been applied in many industrial fields [1–9]. Magnetic matrices are the core component of the HGMS system and play a decisive role in the performance of the system. Rod matrices of high magnetic susceptibility are commonly adopted in high gradient magnetic separators [10]. The widest application field of HGMS is in mineral processing for the processing of weakly magnetic minerals. Generally, a pulsating flow is applied in the separation space to eliminate the blockage of particles in the vicinity of matrices [11,12]. As the fluid drag force acting on particles increase with increase of relative velocity of fluid and particle, introduction of pulsating flow (usually has relatively high peak velocity) will enlarge the fluid drag force in HGMS. Due to the enlarged fluid drag force and the limited magnetic force produced by rod

⁎

matrices, existing industrial high gradient magnetic separators still showed low recovery for ultrafine minerals in processing weakly magnetic ores [13,14]. Enhancing the capture of ultrafine particles in HGMS is of great significance for the clean and efficient utilization of magnetic minerals. The matrices’ shape is an important parameter influencing magnetic characteristics of matrices and consequently matrices’ performance in HGMS. Special shape matrices may have better magnetic and fluidic characteristics and present better performance in HGMS. Many researchers have studied performance of special shape matrices in HGMS, theoretically or experimentally [15–18]. For the study of special shape matrices in HGMS, numerical simulation of magnetic and fluid field around matrices were usually adopted [19–22]. Although these simulations could present distribution of magnetic and fluid field around matrices, only qualitative comparisons and analyses of the matrices’ characteristics were conducted, which is quite empiricist and unsatisfactory. Mathematic modeling is widely

Corresponding author at: School of Minerals Processing & Bioengineering, Central South University, Changsha 410083, China. E-mail address: [email protected] (X. Zheng).

https://doi.org/10.1016/j.seppur.2019.116375 Received 4 September 2019; Received in revised form 26 November 2019; Accepted 1 December 2019 1383-5866/ © 2019 Elsevier B.V. All rights reserved.

Please cite this article as: Zixing Xue, et al., Separation and Purification Technology, https://doi.org/10.1016/j.seppur.2019.116375

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incompressible Navier Stokes.

used to describe or explain a general process or phenomenon [23–26]. Using mathematic modeling, quantitative analyses and study of the process or phenomenon can be conducted. In HGMS, mathematic modeling has been used in the mechanism study of capture of magnetic particles by matrices [27–31] and buildup of magnetic particles on matrices [32–34]. Analytic models describing particle capture and buildup in HGMS have been derived. We have also extended particle capture models of circular and elliptic cross-section matrices in HGMS, considering both the case that matrices are unsaturated and saturated by applied magnetic field [35,36]. With these analytic equations, influence of many configuration and operation parameters on the performance of HGMS system can be investigated and predicted. However, it should be noted that only analytic equations of particle capture by very regular shape (circular and elliptic cross-section) matrices under quite ideal fluid conditions (ideal potential flow) can be derived. For more general conditions (such as more generalized or complex shape matrices or under more generalized fluid conditions), particle capture motion equations cannot be analytically expressed. In our previous study, experiments were conducted to investigate performance of four kinds of cross-section (circular, elliptic, square and diamond) matrices in axial HGMS [37]. For square and diamond matrices, analytic expression of magnetic and fluid field around matrices cannot be obtained. Thus in the theoretical analyses section, only qualitative comparison of magnetic field around the four kinds of matrices were analyzed. It will be more rigorous and convincing if quantitative comparison of particle capture performance of the four kinds of matrices can be conducted. Moreover, researchers are studying the performance of matrices with more complex shapes under more generalized fluid conditions in HGMS. It is quite necessary to develop method which can be applied to study particle capture performance of matrices for more generalized conditions in HGMS. Numerical simulation has been widely used in the theoretical study of particulate systems [38–41]. HGMS is a particulate separation system in compound force fields. In this paper, we proposed numerical simulation of particle capture in axial HGMS using finite element package COMSOL Multiphysics. 3D simulation of particle capture by the four kinds of shape (circular, elliptic, square and diamond) matrices in axial HGMS were conducted. Physical models of HGMS system applying the four kinds of matrices were built. Magnetic field and flow field around matrices were numerically solved. With the magnetic and flow field distribution, particle tracing in the HGMS system employing the four kinds of matrices were conducted. Particle capture performance of the four kinds of matrices were compared using the particle tracing data. The results of numerical simulation are consistent with previous experimental results as well as theoretical results using extended particle capture models. Additionally, the theoretical analyses also indicate that there should exist an optimal aspect ratio for diamond shape matrices and optimal tooth angle for grooved plates in axial HGMS. The proposed numerical simulation method can be applied to study particle capture performance of HGMS system with more generalized and complex conditions.

2.2. Magnetic field Generally, magnetic field strength H and magnetic induction B are used to describe a magnetic field. The relation between B and H in vacuum:

B = μ0 H where μ0 is vacuum permeability, 4π × 10 medium, the relation between B and H:

B = μ0 (H + M )

(1) −7

H/m. In a magnetic (2)

where M is magnetization of medium. In the simulation model, Eq. (2) applies to the matrix domain and Eq. (1) applies to the domain outside the matrix (supposing the permeability of fluid is equal to vacuum permeability). Magnetic field belongs to conservative field and no electric current is available in the simulation system, the following equations can apply:

∇·B = 0

(3)

∇×H=0

(4)

Scalar magnetic potential Vm is used to calculate magnetic field:

H = −∇Vm

(5)

Thus Eq. (3) can be expressed as:

− ∇ ·(μ0 ∇Vm − μ0 M ) = 0

(6)

For matrix domain, magnetization M is determined by B-H curve of matrix material. The matrix is pure iron and B-H curve can be found in our previously published paper [45]. 2.3. Flow field The flow is incompressible viscous liquids described by NavierStokes equation and continuity equation:

ρ

∂V = −∇p + ρF + ηΔV ∂t

∇ · ρV = 0

(7) (8)

where ρ is density of fluid, p is pressure, V is fluid velocity, η is fluid viscosity. In the present study, a velocity normal to the inlet surface was specified as V0 and pressure on the outlet surface was set as zero. A slip boundary was specified on interior wall of the channel and surface of matrix. 2.4. Particle tracing A time-dependent study is applied for particle tracing. Particles were released with an initial velocity equal to the fluid. The particle motion is governed by Newton’s second law. As has been illustrated in our previous papers, for HGMS of particles in the size range of several to dozens of microns, the dominating forces are magnetic force and fluid drag force [27,29]. Thus particle trajectory can be traced by the following equation:

2. Modeling methodology 2.1. The simulation model The numerical simulation package COMSOL Multiphysics is widely used in coupled field simulation analyses [42–44]. In axial HGMS, the flow is parallel to matrix axis and both are perpendicular to applied magnetic field. Fig. 1 shows the computational domain of axial HGMS: the matrix is placed axially along z axis, the fluid flows into the channel (fluid domain) along matrix’s axis with initial velocity V0. A uniform magnetic field H0 is applied along x axis. To eliminate influence of tip effect of magnetic field on reliability of simulation results, both ends of matrix stretch 20 mm out of the flow domain. The implemented simulation model consists of two coupled modules: electromagnetic and

m

dv = Fm + Fd dt

(9)

where m is particle mass, v is particle velocity. Magnetic force acting on a particle of radius R can be expressed as follow:

Fm =

4 3 πR μ0 κH ∇H 3

(10)

where κ and R are susceptibility and radius of particle. The fluid drag force acing on a particle is calculated with Eq. (11). 2

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Fig. 1. Schematic diagram of simulation model of axial HGMS.

Fd = −6πηR (v − Vf )

3. Results and discussion

(11)

where Vf is velocity of fluid. For particles arriving at matrix surface (captured by the matrix), they are set to be trapped there. In axial HGMS, particle capture performance of matrix is evaluated by particle capture cross section. Particle capture cross section is the area at inlet within which particles can all be captured by matrix. Particles with initial location along the boundary of capture cross section can just be captured at the end of matrix and those with initial location outside the boundary will escape. More detailed introduction of particle capture cross section can be consulted in previous publications [36,45]. For calculation and comparison of particle capture cross section, detailed values of configuration and operation parameters are specified in Table 1. The matrix material is pure iron. Circular, elliptic, square and diamond cross-section matrices (short for circular, elliptic, square and diamond matrices hereafter) were involved. Cross-section area of the four kinds of matrices were all π mm2 (under the same crosssection area, packing fraction of matrices was the same). Ratio of long axis to short axis of elliptic and diamond matrices were 2:1.

3.1. Magnetic force distribution Magnetic field strength H0, magnetic flux density B0 can be exported directly from the computation results. But the main fraction of magnetic force HgdadH can only be derived with existing parameters. This can be realized with the user define formula function, as shown by Eqs. (12)–(14).

2H gradHx =

2H gradHy = H gradH =

Channel Cross section (mm2) Length L (mm) Inlet velocity V0 (m/s) Particles Particle radius R (μm) Particle density ρp (kg/m3) Particle magnetic susceptibility κ (dimensionless) Fluid Fluid density ρ (kg/m3) Fluid dynamic viscosity η (Pa‧s) Fluid relative permeability μf (dimensionless) Matrices Cross section area (mm2) Circular matrix radius (mm) Elliptic matrix long axis vs short axis (mm; mm) Square matrix side length (mm) Diamond matrix side length (mm) Magnetic field Magnetic induction B0 (T)

(12)

∂H 2 ∂y (H gradHx )2 + (H gradHy)2

(13) (14)

where H gradHx is x component of H gradH , and H gradHy is y component of H gradH . To present an intuitive observation of magnetic force distribution around matrices, vector plot of normalized H gradH and contour plot of H gradH at plane 1 (shown in Fig. 1) were displayed in Figs. 2 and 3 respectively. As shown in Fig. 2, attractive and repulsive area exist around the four kinds of matrices and they follow the same pattern: magnetic forces around the left and right sides of matrix is attractive while that around the upper and lower sides of matrix is repulsive. Magnetic particles will be captured at the left and right sides. Basing on the vector path of HgradH, it is also predictable that particles with initial location in the repulsive area can be pushed away first and then be attracted back and captured by the matrix. Contour plot of H gradH can intuitively reflect magnetic force magnitude distribution. As demonstrated in Fig. 3, strong magnetic force mainly distributed in attracting area around matrix (left and right sides of matrices) and peak value lies in the end points in direction parallel to H0. To conduct a deeper comparison of magnetic force generated by the four kinds of matrices, the areas surrounded by isolines of HgradH = 1 × 1014 A2/m3 around matrices were calculated to evaluate the magnetic force effect depth. Statistic results under applied induction of 0.2 T, 0.5 T, 0.9 T, 1.2 T and 1.5 T are exhibited in Table 2. It can be seen that magnetic force effect depth increases with increase of applied induction. The isoline surrounded area of elliptic matrix is nearly equivalent to that of diamond matrix and both are larger than

Table 1 Specification of the axial high gradient magnetic separation system. Parameters

∂H 2 ∂x

size

40 × 40 100;200;300;400;500 0.1 5;10;20;30 5 × 103 2.5 × 10−3

1 × 103 1 × 10−3 1 π 1 1.414; 0.707 1.772 1.981 0.1;0.2;0.3;0.5;0.7;0.9;1.2;1.5

3

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Fig. 2. Normalized H gradH vector distribution around matrices.

those of circular and square matrices when magnetic induction is below 0.9 T. Under the induction of 1.2 T, isoline surrounded area of the four kinds of matrices are equivalent. When magnetic induction is increased to 1.5 T, isoline surrounded area of circular and elliptic matrices are larger than those of elliptic and diamond matrices.

Table 2 Surrounded area by isoline of HgradH = 1 × 1014 A2/m3 (mm2). Matrix shape

circular elliptic square diamond

3.2. Evolution of particle group and particle trajectories 3.2.1. Evolution of particle group In our previously published papers [35–37], analytic particle motion equations of circular and elliptic matrices were derived and particle motion trajectories could be calculated with the equations. The analytic particle motion equations of square and diamond matrices

Magnetic induction (T) 0.2

0.5

0.9

1.2

1.5

0.040 0.493 0.324 0.588

3.453 4.377 3.003 4.140

11.523 12.516 11.429 12.263

16.854 16.446 16.387 16.329

20.208 19.835 20.041 19.744

Fig. 3. Contour plot of H gradH around matrices under magnetic induction B0 = 0.3 T. 4

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Fig. 4. Capture process of particles by diamond matrix when moving with the fluid: (a) t = 0 s; (b) t = 0.25 s; (c) t = 0.75 s; (d) t = 1 s.

others continue to travel with fluid. For particles of specific radius captured exactly at the outlet, locus of their initial location at the inlet is defined as capture cross-section boundary. Particles with initial location inside the boundary when released will all be captured by matrix. The dotted yellow and green arcs in Fig. 6 show a quarter of the particle capture cross section of diamond and elliptic matrices, respectively. P1, P3, P5 and P7 denote trajectories of particles around elliptic matrix while P2, P4, P6 and P8 denote trajectories of particles around diamond matrix. As the four kinds of matrices considered in this study are all axisymmetric, particle capture cross section in the other three quadrants can all be obtained through symmetry operation with respect to x and y axes. Particles captured by the four kinds of matrices are monitored to get their initial location at the inlet, the particle motion trajectories and particle capture cross section area can all be determined with the help of MATLAB.

cannot be derived. With the help of finite element analyses tool COMSOL Multiphysics, particle capture performance of irregular shape matrices can also be investigated and presented. In this simulation, a group of particles (2000 particles) were released at the inlet (z = 0 plane). Being subjected to magnetic and fluid drag force, particles with different initial position would move in different trajectories. Fig. 4 shows capture process of particles by diamond matrix. The group of particles are released at the z = 0 plane at t = 0 s. The particle group moves with the fluid along axial direction of matrix. At the same time, some particles will move in the direction perpendicular to matrix axis under action of magnetic force and can finally be captured by matrix. Initially, particles near to matrix will be captured first. As time goes on, the displacement of particle group in axial direction increases and more and more particles can be captured. At t = 1 s, the particle group arrives at the end of matrix and some particles still cannot be captured by matrix and will move out of the HGMS system with fluid. Additionally, it can also be seen from Fig. 4 that overall shape of particle group changes as particle group moves along the axial direction. The dimension of particle group along the direction of x axis shrinks while that along the direction of y axis expands. This variation of particle group shape is ascribed to the distribution of magnetic force around the matrices, as shown in Figs. 2 and 3. The magnetic force around the matrix ends along x axis is attractive while that around the matrix ends along y axis is repulsive. Animation of particle capture process of circular, elliptic, square and diamond matrices can be visually consulted in Figs. S1, S2, S3 and S4, respectively. Motion of particles in the plane perpendicular to matrices’ axis can be seen in Figs. S5, S6, S7 and S8. (Fig. S1 to Fig. S8 are linked in Appendix A) Final shape of particle group arriving at the outlet of HGMS system employing the four kinds of matrices are shown in Fig. 5.

3.3. Comparison of particle capture performance of the matrices 3.3.1. Variation of particle capture cross section Fig. 7 shows variation of capture cross section area of 10 μm particles with increasing applied induction in the range of 0.1 T–1.5 T. Overall, for all the four kinds of matrices considered, particle capture cross section enlarges with increase of applied induction, as a consequence of that magnetic force acting on particles increases. Additionally, it can also be seen that, under relatively low magnetic induction, the shape of particle capture cross section of matrices are quite different and are in accordance with the HgradH contour in Fig. 3. Under relatively high magnetic induction, particle capture cross section of the four kinds of matrices are quite similar. Fig. 8 shows variation of particle capture cross section with increasing particle size. The same regularity as that in Fig. 7 can be observed. Particle capture cross section enlarges with increasing particle size. For small size particles, the shape of particle capture cross section is quite similar to the HgradH contour in Fig. 3. For particles of relatively large size, quite slight difference of particle capture cross section among the four kinds of matrices can be observed.

3.2.2. Particle motion trajectories and particle capture cross section Although Fig. 4 shows evolution of particle group in the HGMS system, particle motion trajectories are still not shown. We monitored the motion of individual particles and obtained the location coordinate data. Then particle motion trajectories in the HGMS system can be depicted. Fig. 6 is a family of particle trajectories in axial HGMS employing diamond and elliptic matrices in the first quadrant. xin and yin are the initial coordinates of particles when entering the HGMS system. In axial HGMS, capture performance of matrices is evaluated by particle capture cross section. For particles released from the inlet (z = 0 mm), some particles are captured by matrix soon after released while some

3.3.2. Comparison of particle capture cross section area Particle capture cross section area of matrices under different circumstances were calculated using MATLAB and then particle capture performance of the four kinds of matrices could be quantitatively compared. Fig. 9 shows comparison of capture cross section area of 5

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Fig. 5. Final shape of particle group arriving at the outlet of HGMS system employing the four kinds of matrices: Particle radius 30 μm, matrix length 0.1 m, applied induction 0.3 T.

5 μm, 10 μm, 20 μm and 30 μm particles by the four kinds of matrices in induction range of 0.1 T–1.5 T. For all the circumstances considered, particle capture cross section area increases rapidly first and then slowly with increasing applied induction. This regularity is related to the variation of magnetization state (unsaturated or saturated) of matrices with increasing applied induction. The induction corresponding to the inflection point of the

lines in Fig. 9 is the induction under which matrices just reach magnetization saturation. In our previously published papers, we have thoroughly studied magnetization process of matrices in magnetic field and derived judgement method of determining magnetization state (unsaturated and saturated) of circular and elliptic matrices in HGMS [46,47]. The demarcation of applied induction for judging magnetization state of circular and elliptic matrices can be determined by the

Fig. 6. Motion trajectories of particles around the diamond and elliptic matrices in the first quadrant (10 μm particles under magnetic induction B0 = 0.9 T): P1 and P2, P3 and P4 share the same initial location; P5 and P6, P7 and P8 are released from the cross section boundary at θ = π/6 and θ = 0, respectively. 6

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Fig. 7. Variation of particle capture cross section of four kinds of matrices with increasing applied magnetic induction: Particle radius 10 μm, matrix length 0.1 m.

Fig. 8. Variation of particle capture cross section with increasing particle size: Applied induction 0.3 T, matrix length 0.1 m. 7

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Fig. 9. Comparison of particle capture cross section area of the four kinds of matrices: (a) Particle diameter R = 5 μm; (b) R = 10 μm; (c) R = 20 μm; (d) R = 30 μm.

capture cross section area of circular and elliptic matrices are slightly larger than those of elliptic and diamond matrices. The results are quite consistent with the variation of surrounded area of HgradH isoline in Table 2. The most widely applied field of HGMS is the processing of fine weakly magnetic minerals in mineral processing. To ensure a magnetic product with high recovery and grade of valuable metal, a medium applied induction (0.2 T–0.6 T) is usually adopted. In one of our previously published paper [37], magnetic field characteristics of the four kinds of matrices are analyzed and separation experiments were conducted on circular, elliptic and square matrices (more experimental details can be found in the previous paper [37]) with a refractory specularite. The sample mainly contained three iron minerals: specularite, siderite and limonite. The specific susceptibility χ of the iron minerals were in the range of (64.51–80.82) × 10−9 m3/kg. More detailed information on mineralogical properties of the ore can be consulted in the published paper [48]. The experimental results of the −38 μm fraction are shown in Fig. 10. It can be seen that the grade of magnetic products obtained by the three kinds of matrices are nearly equivalent. The recovery of magnetic products obtained by elliptic matrices is the highest, followed by square matrices and that by circular matrices is the lowest. On the whole, the experimental results are consistent with the results of particle capture cross section area in Fig. 9, which also confirms the reliability and validity of the numerical simulation. The above results indicate that adopting elliptic or diamond matrices can enhance the capture of ultrafine magnetic particles in

following equation:

B0 = Ms /(γ + 1)

(15)

where γ is aspect ratio of matrix (for circular matrix, γ = 1). In this study, the matrix material is pure iron and saturation magnetization Ms = 2.14 T. According to Eq. (15), circular and elliptic matrices (γ = 2 in this study) will respectively reach saturation under applied induction of approximately 1.07 T and 0.71 T. It can be seen from Fig. 9 that the applied induction of inflection point of the lines for circular and elliptic matrices are approximately 1.0 T and 0.7 T. The results are also consistent with the study results of effect of matrix saturation magnetization on particle capture in HGMS [45]. These consistencies with our previously published papers reveal the accuracy and validity of numerical simulation in the present study. Interestingly, it can also be seen from Fig. 9 that demarcation of applied magnetic induction for judging magnetization state of square and diamond matrix are quite close to that of circular and elliptic matrices, respectively. Comparing particle capture cross section areas among the four kinds of matrices, within the whole induction range considered, particle capture cross section area of square matrix is quite equivalent to that of circular matrix while cross section area of diamond and elliptic matrices are quite similar. For all the particles considered, within applied induction range of approximately 0.1–1.1 T, particle capture cross section area of elliptic and diamond matrices are larger than those of circular and square matrices and the sequence is diamond > elliptic > square > circular. When applied induction is above 1.1 T, particle 8

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length. Within the whole matrix length range of 0.1–0.5 m, particle capture cross section area of diamond matrix is the largest, followed by elliptic and square matrix and that of circular matrix is the smallest. Additionally, absolute difference of particle capture cross section area among the matrices increases the increase of matrix length. It has been seen from the analyses above that particle capture performance of circular, elliptic, square and diamond matrices are studied and compared quantitatively. The theoretical analyses correspond well with previous theoretical and experimental results. A very interesting result that should be given adequate attention is that, for all the circumstances considered in this study, particle capture performance of diamond matrix is the best (largest particle capture cross section). For the processing of weakly magnetic minerals in mineral processing, two kinds of high gradient magnetic separators have been widely applied in industry: vertical ring pulsating high gradient magnetic separator and horizontal ring high gradient magnetic separator. Rod matrices are adopted in vertical ring pulsating high gradient magnetic separator while grooved plates are used in horizontal ring high gradient magnetic separator. In our previous study, it has been proved that, for elliptic matrix, there exists optimal aspect ratio for maximum particle capture under specific induction in axial HGMS [49]. The results above show that diamond matrix (aspect ratio 2) has larger particle capture cross section area than square matrix (aspect ratio 1). It can also be inferred that, for diamond matrix, there also exists optimal aspect ratio (with the same cross section area or the same matrix packing fraction) under which particle capture cross section area reaches the maximum in axial HGMS, as shown by the left part in Fig. 12. The aspect ratio of diamond matrix can be optimized. Grooved plates are used in horizontal ring high gradient magnetic separator [50]. The tooth angle is typically 90°. The horizontal ring high gradient magnetic separator shows superiority over other magnetic separators in the recovery of very fine magnetic minerals and has been used in hematite recovery from red mud [13]. Enhancing the capture of fine minerals by grooved plates is significantly important for the improvement of performance of horizontal ring high gradient magnetic separator. Analyses above show that diamond matrix has larger particle capture cross section than square matrix, it can also be inferred that tooth angle 90° of grooved plates is not the optimum scheme, grooved plates can be optimized further for higher particle capture efficiency. The right part in Fig. 12 shows optimization scheme for grooved plates, as shown by the dotted lines. The tooth angle can be decreased under the same cross section area (under the same packing fraction). The two optimization studies in Fig. 12 involves a great deal of research work. This is out the scope of this paper and will be conducted in the near future, both theoretically and experimentally.

Fig. 10. Separation results of −38 μm fraction of weakly magnetic minerals by circular, elliptic and square matrices [37].

Fig. 11. Variation of particle capture cross section area of the four kinds of matrices with increasing matrix length: Applied induction 0.3 T, particle diameter 10 μm.

HGMS and improve separation performance. To present a comprehensive analyses of particle capture performance of the four kinds of matrices in axial HGMS, the effect of matrix length on particle capture is also investigated. Fig. 11 shows variation of capture cross section area of 10 μm particles by matrices with increasing matrices’ length. Overall, particle capture cross section area of the four kinds of matrices increases with the increase of matrices

Fig. 12. Optimization scheme for diamond matrices (left) and grooved plates (right) adopted in HGMS. 9

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4. Conclusions

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In this paper, 3D numerical simulation of particle capture of circular, elliptic, square and diamond cross-section matrices in axial HGMS was conducted. Particle capture cross section of the matrices differ greatly for low applied induction and small size particles while for relatively high applied induction and large size particles, particle capture cross sections are almost identical. Particle capture cross section area of the four kinds of matrices increase rapidly and then slowly with increasing applied induction and the inflection point is exactly the induction under which matrices just reach magnetization saturation. Particle capture performance of elliptic and diamond matrices are equivalent while that of circular and elliptic matrices are quite similar. Particle capture cross section area of elliptic and diamond matrices are larger than those of circular and square matrices when applied induction is below approximately 1.1 T and the absolute difference increases with the increase of matrix length. The results of numerical simulation are consistent with our previous theoretical studies with the analytical models (circular and elliptic matrices) and experimental results on the matrices. Adopting elliptic or diamond matrices can enhance capture of ultrafine magnetic particles in HGMS and improve separation performance. The numerical simulation results also indicate that there exists optimal aspect ratio for diamond matrices and optimal tooth angle for grooved plates in HGMS. This will be theoretically and experimentally investigated in our follow-up work. CRediT authorship contribution statement Zixing Xue: Methodology, Software, Writing - original draft. Yuhua Wang: Supervision, Data curation. Xiayu Zheng: Conceptualization, Methodology, Writing - review & editing. Dongfang Lu: Visualization. Xudong Li: Software, Formal analysis. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment This research work was financially supported by the National Natural Science Foundation of China (Grant No. 51804341; No. 51674290; No. 51974366), Natural Science Foundation of Hunan Province (Grant No. 2019JJ50833; No. 2016JJ3150), Innovation Project for Postgraduates of Central South University (No. 2019zzts314). Key Laboratory of Hunan Province for Calcium-containing Mineral Resources (No. 2018TP1002) and Co-Innovation Center for Clean and Efficient Utilization of Strategic Metal Mineral Resources. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.seppur.2019.116375. References [1] S. Singh, H. Sahoo, S. Rath, A.K. Sahu, B. Das, Recovery of iron minerals from Indian iron ore slimes using colloidal magnetic coating, Powder Technol. 269 (2015) 38–45. [2] Z. Kheshti, K. Azodi Ghajar, Altatee Ali, M.R. Kheshti, High gradient magnetic separator (HGMS) combined with adsorption for nitrate removal from aqueous solution, Sep. Purif. Technol. 212 (2019) 650–659. [3] S.K. Tripathy, N. Suresh, Influence of particle size on dry high-intensity magnetic separation of paramagnetic mineral, Adv. Powder Technol. 28 (2017) 1092–1102. [4] L. Chen, G. Liao, Z. Qian, J. Chen, Vibrating high gradient magnetic separation for purification of iron impurities under dry condition, Int. J. Miner. Process. 102–103 (2012) 136–140.

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