Matching relation between matrix aspect ratio and applied induction for maximum particle capture in longitudinal high gradient magnetic separation

Separation and Purification Technology 241 (2020) 116687

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Matching relation between matrix aspect ratio and applied induction for maximum particle capture in longitudinal high gradient magnetic separation

T

Xiayu Zheng , Zixi Sun, Yuhua Wang , Dongfang Lu, Zixing Xue ⁎

⁎

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

ARTICLE INFO

ABSTRACT

Keywords: Longitudinal HGMS Magnetic matrices Aspect ratio Particle capture Matching relation

Longitudinal high gradient magnetic separation (HGMS) is the most widely applied in industry by virtue of to its high processing capacity and low magnetic leakage. However, due to introduction of pulsating flow and the limited magnetic forces induced by rod matrices, longitudinal high gradient magnetic separator still showed quite lower recovery compared with horizontal high intensity magnetic separator in recovering ultrafine magnetic minerals. Optimization of longitudinal HGMS system is of great significance for improving particle capture efficiency or reducing energy consumption. In this study, we proposed a new concept of matching relation between matrix aspect ratio and applied induction for maximum particle capture in longitudinal HGMS. Particle capture performance of matrices with aspect ratio of 1/5–6 in applied induction of 0.2–2T were investigated using extended particle capture models for unsaturated and saturated matrices as well as model selection methodology. The matching relation under the two cases of equal matrix cross-section area and equal horizontal axis were demonstrated. Particle motion trajectories around matrices were depicted and particle capture radius was determined. It was proved that there existed matching relation between matrix aspect ratio and applied induction for maximum particle capture for both the two cases and the relation was independent of matrix and particle size. The matching relation can be adopted to configure the HGMS system for improving particle capture efficiency or reducing energy consumption. Based on results of this study, it was also inferred that there should exist matching relation between matrix aspect ratio and applied induction for maximum separation efficiency.

1. Introduction High gradient magnetic separation (HGMS) has witnessed widespread application in mineral processing [1–7], water treatment [8–10], waste management [11,12], bioengineering [13,14] etc. There are three configurations in HGMS: axial, transversal and longitudinal [15,16]. In axial HGMS, direction of flow is parallel to matrix axis and both are perpendicular to applied field. In transversal HGMS, direction of flow, matrix axis and applied field are mutually perpendicular. In longitudinal configuration, direction of flow is parallel to applied field and both are perpendicular to matrix axis. The most widely applied field of HGMS is in mineral processing and different configurations correspond to respective industrial high gradient magnetic separators: Vertical ring high gradient magnetic separator with horizontal magnetic system (Transversal) [17,18], vertical ring high gradient magnetic separator with vertical magnetic system (Longitudinal) [19] and ⁎

horizontal ring high intensity magnetic separator (Axial) [20]. In industrial practice, rod matrices are used in transversal and longitudinal high gradient magnetic separators while grooved plates are adopted in horizontal ring high intensity magnetic separator. The magnets in HGMS is usually current energized and the energy consumption is relatively high. Initially, the greatest obstacle for large scale application of high gradient magnetic separator in mineral processing is the serious blockage of materials in the vicinity of matrices. In longitudinal and transversal HGMS, a pulsating flow is generally applied in the separation space to eliminate matrix blockage [21]. As fluid drag force acing on magnetic particle is proportional to relative velocity between particle and fluid, increasing flow velocity (the peak velocity of pulsating flow is relatively high) will lead to decrease in recovery of ultrafine minerals [22,23]. Additionally, the magnetic force induced by rod matrices is limited. Consequently, horizontal ring high intensity

Corresponding author. E-mail addresses: [email protected] (X. Zheng), [email protected] (Y. Wang).

https://doi.org/10.1016/j.seppur.2020.116687 Received 9 December 2019; Received in revised form 30 January 2020; Accepted 8 February 2020 Available online 10 February 2020 1383-5866/ © 2020 Elsevier B.V. All rights reserved.

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Nomenclature μ0 κ V H H0 B0 R Rc θ r α β e i

Ms η Vp Vl V0 a b c d lh lv λ γ

vacuum permeability, 4π × 10−7 H/m magnetic susceptibility of particle volume of particle, m3 magnetic field, A/m the applied magnetic field strength, A/m the applied magnetic induction, T radius of particle, m particle capture radius, m polar angle coordinate polar coordinate, m angle of applied field with respect to matrix long axis angle of fluid with respect to matrix long axis base of natural logarithm imaginary symbol

u v

magnetic separator present superiority (much higher recovery) over vertical ring high gradient magnetic separator in recovering ultrafine magnetic minerals from red mud [20]. However, as the spacing between magnetic poles in longitudinal high gradient magnetic separator is relatively shorter, the magnetic leakage is lower than that in transversal and axial high gradient magnetic separators. The magnetic field produced in longitudinal configuration is much higher, along with higher capacity and lower energy consumption [17]. With the exploitation of mineral resources, more and more refractory magnetic minerals are being processed and large amount of ultrafine minerals exist in the grinding products. Improving the capture of ultrafine weakly magnetic minerals will greatly promotes the industrial application of longitudinal high gradient magnetic separators as well as the clean and efficient utilization of weakly magnetic minerals. Simulation is widely adopted in process study [24–27] and modeling of particle capture is commonly used in the study of HGMS [28–31]. In our previous publications [32,33], we have studied the effect of matrix aspect ratio on particle capture using extended particle capture models in axial HGMS. It was found that there existed matching

the saturation magnetic of the matrix, T viscosity of the fluid, Pa·s velocity of particle, m/s velocity of fluid, m/s initial velocity of fluid, m/s half the long axis of elliptic matrix, m half the short axis of elliptic matrix, m half the focal length of elliptic matrix, m radius of circular matrix, m horizontal axis of elliptic matrix, m vertical axis of circular matrix, m ratio of matrix long axis to short axis ratio of axis along magnetization to that perpendicular to magnetization elliptic angle coordinate elliptic coordinate

relation between matrix aspect ratio and applied induction for maximum particle capture. The matching relation can be used to optimize the HGMS system. In axial HGMS, the fluid is parallel to matrices’ axis, the flow field around matrices with different aspect ratio is the same. In longitudinal HGMS, magnetic field and flow field around matrices will change simultaneously with varying matrix aspect ratio [15]. The matching relations between optimal aspect ratio and applied induction in longitudinal and axial HGMS are different. In this paper, extended particle capture models (for unsaturated and saturated matrices) in longitudinal HGMS were derived. Particle capture performance of matrices with various aspect ratio under a variety of applied induction were investigated with the extended models and model selection methodology. The matching relation between optimal aspect ratio and applied induction in longitudinal HGMS is expected to be obtained and then longitudinal high gradient magnetic separators can be configured using the matching relation for improving particle capture or reducing energy consumption.

Fig. 1. Variation of matrix aspect ratio: Case A-under the same cross section area; Case B- under the same horizontal axis. 2

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(6)

2. Extended particle capture models in longitudinal HGMS.

µHv1 = µ 0 Hv1

2.1. Variation of matrix aspect ratio

With the magnetic potentials in Eqs. (3) and (4), the boundary conditions can be expressed as follow:

The matrix shape has significant influence on matrix magnetic characteristics and its performance in HGMS [34]. As shown in Fig. 1, effect of matrix aspect ratio on particle capture in longitudinal HGMS were investigated under two cases: under the same cross-section area (Case A) and under the same horizontal axis (Case B). The aspect ratio γ was ratio of the axis parallel and perpendicular to applied field and a wide γ range of 1/5–6 was considered. Fig. 2 showed contour plot of magnetic field around matrices with aspect ratio of 2, 1 and 1/2 under the two cases considered. It can be seen that magnetic field around matrices with the same aspect ratio are the same and the maximum magnetic induction is independent on matrix size. High magnetic induction area is around the upper and lower sides of matrices. Induced magnetic induction increases with increase of matrix aspect ratio.

1

1 H0 {e 2

i

c 2)1/2] + (C1 cos

=

2

v

c sh2v + sin2 u 1

)

(7)

2

c sh2v + sin2 u

(8)

u

1

Hout1v =

2

c sh2v + sin2 u 1

Hout1u =

v 2

c sh2v + sin2 u

u

H0 cos u

=

2 sh2v + sin2 u

(e v

H0 sin u

=

2 sh2v + sin2 u

C1 e v )

(e v + C1 e v )

(9) (10)

For unsaturated and saturated matrices, C1 is determined by Eqs. (11) and (12), respectively. More detailed calculation of the coefficients can be consulted in our previously published paper [32]. (11)

e 2v 0

C1

2Ms shv0 chv0 µ 0 H0

C1 = 1

(12)

where Ms is saturation magnetization of matrix. With the magnetic field derived, the magnetic forces acting on a paramagnetic particle of volume V (V = 4πR3/3, R is particle radius):

Fv = µ 0 V (Hv = µ0 V

H02

+ Hu

e 2v

C12e 2v [ c 4(sh2v + sin2 u)3/2

Fu = µ 0 V (Hv = µ0 V

Hv v

H02 c

[

Hv u

C1 sin 2u

+ Hu

2(sh2v + sin2 u)3/2

Hu )/(c v

sh2v + sin2 u )

shvchv (e 2v + C12e 2v 2C1 cos 2u) ] 4(sh2v + sin2 u)5/2 Hu )/(c u

(13)

sh2v + sin2 u )

sin 2u (e 2v + C12e 2v

2C1 cos 2u)

8(sh2v + sin2 u)5/2

]

(14)

For unsaturated and saturated matrices, C1 in the equations is determined by Eqs. (11) and (12), respectively. For the case of γ < 1, substituting α = 270° into Eqs. (1) and (2) and the magnetic potentials can also be obtained. 3

= A2 H0 c sin ushv

4

=

1 H0 c sin u (e v + C2 e v ) 2

(15) (16)

(1)

iA2 sin ) z

[z + (z 2

1

u

1

) = µ0 (

where μ0 and μ are permeability of vacuum and matrix. With the coefficients C1, the magnetic field around matrices can be determined:

For the HGMS of paramagnetic particle in the size range of several microns to dozens of microns, the Van Der Waals force, the electric force, the Brownian motion and inertial force are negligible compared with magnetic and fluid drag forces. Particle capture can be modeled by balancing magnetic and fluid drag forces [22]. Initially, particle capture models in HGMS were established based on the premise that matrix was saturated by applied field [35–37]. In our recent work, we extended particle capture models in axal and transversal HGMS, considering both the cases that matrix was saturated and unsaturated by applied field. The validity of the extended particle capture models was confirmed by experimental results [38,39]. Fig. 3 showed cross-section of circular and elliptic matrices in longitudinal HGMS. Radius of circular matrix was d. long and short axes of elliptic matrix were 2a and 2b, the focal length was 2c. With aspect ratio varying, the matrix shape transformed between circle and ellipse. λ was the ratio of long axis to short axis, λ = a/b. λ = γ for matrix with γ > 1 and λ = 1/γ for matrix with λ < 1. α and β were the angle of matrix long axis with respect to direction of magnetic and flow field, respectively. For longitudinal configuration, two circumstances were involved: α = β = 270° and α = β = 0°, as shown in Fig. 3. Magnetic field and electric field belong to conservative field and have many similarities. Magnetic field around elliptic matrix can be calculated by analogy with the calculation of electric field around an elliptic dielectric. The complex potential functions of electric field around a elliptic dielectric has been presented by Smythe [40]. Based on the results of Smythe, complex potential functions of magnetic field inside and outside elliptic matrix can be expressed by Eqs. (1) and (2):

w2 =

1

v

c sh2v + sin2 u c sh2v + sin2 u

2.2. Magnetic force

w1 = H0 (A1 cos

1

µ(

+ iC2 sin )[z

(z 2

c 2 )1/2]}

(2)

where H0 is applied magnetic field, z = x + iy, i is imaginary symbol, A1, A2, C1 and C2 are coefficients to be determined. For the case of γ > 1, substituting α = 0° into Eqs. (1) and (2), the real part in complex potential functions are the magnetic potentials, as shown by Eqs. (3) and (4). 1

= A1 H0 c cos uchv

2

=

1 H0 c cos u (e v + C1 e v ) 2

(3) (4)

The following boundary conditions can be applied to calculate the coefficients A1 and C1 at the matrix boundary v = v0:

Hu1 = Hu 2

Fig. 2. Contour plot of magnetic field around matrices with aspect ratio γ of 2, 1 and 1/2: (a) under the same cross-section area; (b) under the same horizontal axis.

(5) 3

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H0 d

H0

V0

y

y

r

b x

θ

V0

-c (α=270°, β=270°)

a

c

x

(α=0°, β=0°) Fig. 3. Cross-section of circular and elliptic matrices in longitudinal HGMS.

The same boundary conditions as Eqs. (5) and (6) can be applied to calculate the coefficients A2 and C2. For unsaturated and saturated matrices, C2 can be determined by Eqs. (17) and (18), respectively.

C2

e 2v0

C2' =

1

=

Vv = (18)

The magnetic force acting on paramagnetic particle can be calculated:

Fv = µ 0 V (Hv = µ0 V

+ Hu

e 2v

C22 e 2v [ c 4(sh2v + sin2 u)3/2

H02

Fu = µ 0 V = µ0 V

Hv v

H02 c

[

H (Hv uv

Hu )/(c v

+

C2 sin 2u

(19)

Fd =

+(

M2 i e N2

e

i

)[

1 V0 c [cos u (e v + 2

z 2

z ( )2 2

N 2 ]}

+1 v e ) + i sin u (e v 1

1 V0 c cos u (e v + 2

(21)

+1 v e )] 1

=

(22)

+1 v e ) 1

Vu =

1 sin2 u

v

c sh2v + sin2 u

u

c

sh2v

+ 1

=

=

(23)

=

V0 cos u 2

1 V0 c [ sin u (e v + 2

2 sh2v + sin2 u

+1 v e ) 1

(28)

(e v +

+1 v e ) 1

(29)

6

R (Vp

(30)

Vf )

sh2v

+

sin2 u

(e v

V0 sin u 2 sh2v + sin2 u

(e v +

+1 v e ) 1 +1 v e ) 1

+1 v e ) + i cos u (e v 1

+1 v e )] 1

dv dt V0 cos u

sh2v

sin2 u

(e v

e 2v C12 e 2v +1 v e ) + Vm ( 1 4(sh2v + sin2 u)3/2

+ shvchv (e 2v + C12 e 2v 2C1 cos 2u) ) 4(sh2v + sin2 u)5/2

(31)

du dt V0 sin u

+1 v C1 sin 2u (e v + e ) + Vm ( 1 2(sh2v + sin2 u)3/2 sh2v + sin2 u sin 2u (e 2v + C12 e 2v 2C1 cos 2u) ) 8(sh2v + sin2 u)5/2

(32)

c ) ,C1 is determined by Eqs. (11) and (12) where Vm = for unsaturated and saturated matrix, respectively. For the case of γ < 1, α = β = 270°, particle capture models can be derived with Eqs. (19), (20), (28) and (29). 2µ 0 H02 R2 /(9

(24) (25)

dv dt V0 sin u

c sh2v + sin2 u

For the case of λ < 1, substituting β = 270° into Eq. (21), the complex potential function in the elliptic coordinate system:

w3 =

V0 cos u

=

(e v

c sh2v + sin2 u

The flow velocity around matrix can be calculated with the potential and the components are shown by Eqs. (24) and (25).

Vv =

u

2 sh2v + sin2 u

c sh2v + sin2 u

The real part is flow velocity potential:

=

1 c sh2v + sin2 u

V0 sin u

=

Having obtained the magnetic and fluid drag forces, extended particle capture models in longitudinal HGMS can be established (for unsaturated and saturated matrices). For the case of γ > 1, α = β = 0°, particle capture models can be derived with Eqs. (13), (14), (24) and (25), as shown of Eqs. (31) and (32).

where V0 is fluid initial velocity, M=(a + b)/2, N = c/2. For the case of γ > 1, substituting β = 0° into Eq. (21), the complex potential function can be obtained. In the elliptic coordinate system:

w3 =

v

2.4. Extended particle capture models in longitudinal HGMS

(20)

Flow field around matrices can be calculated using the complex potential function of flow [41]: i

1 c sh2v + sin2 u

sh2v + sin2 u )

2.3. Fluid drag force

w3 = V0 {ze

(27)

With the components of flow velocity, the fluid drag force can then be calculate with Eq. (30).

sin 2u (e 2v + C22 e 2v + 2C2 cos 2u) ] 8(sh2v + sin2 u)5/2

2(sh2v + sin2 u)3/2

Vu =

sh2v + sin2 u )

shvchv (e 2v + C22 e 2v + 2C2 cos 2u) ] 4(sh2v + sin2 u)5/2

H Hu uu )/(c

+1 v e ) 1

Components of flow velocity are as follow:

(17)

2Ms shv0 chv0 µ0 H0

1 V0 c sin u (e v + 2

=

(26)

The velocity potential: 4

sh2v

sin2 u

(e v

e 2v C22 e 2v +1 v e ) + Vm ( 1 4(sh2v + sin2 u)3/2

+ shvchv (e 2v + C22 e 2v + 2C2 cos 2u) ) 4(sh2v + sin2 u)5/2

(33)

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du dt V0 cos u

c sh2v + sin2 u =

C2 sin 2u +1 v (e v + e ) + Vm ( 1 2(sh2v + sin2 u)3/2 sh2v + sin2 u sin 2u (e 2v + C22 e 2v + 2C2 cos 2u) ) 8(sh2v + sin2 u)5/2

(34)

For unsaturated and saturated matrices, C2 is determined by Eqs. (17) and (18), respectively. For the case of γ = 1, the matrix is circular cylinder. Based on particle capture models derived by Watson for saturated circular matrix [42], extended particle capture model in longitudinal HGMS can be expressed by Eqs. (35) and (36):

dr = V0 cos (1 dt r

d = dt

4µ 0 R2 BH0 cos 2 B2 ( + 5) 9 r3 r

d2 ) r2

V0 sin (1 +

d2 ) r2

(35)

4µ 0 R2 BH0 sin 2 9 r3

Fig. 4. Selection methodology of particle capture models when varying matrix aspect ratio under a variety of applied induction: the matrix is pure iron and saturation magnetization Ms = 2.14T.

(36)

For unsaturated and saturated matrices, B is determined by Eqs. (37) and (38), respectively [32].

B = H0 d 2

(37)

B = Ms d 2/(2µ 0 )

(38)

The green lines are the trajectories of particles captured by the matrices while the red lines are trajectories of particles escaping. The blue lines denote particles which can just be captured by matrices and determine the particle capture radius Rc. Particles with initial location xin ≤ Rc will all be captured and those with initial location xin > Rc will escape. The horizontal distance of starting point of blue lines is 2Rc. In longitudinal HGMS, particles are mainly captured at the upper and lower sides of matrices, around which high magnetic field mainly exists, as shown in Fig. 2. Fig. 6 showed arrangement of matrices in longitudinal HGMS. The horizontal distance between adjacent matrices was 2n. For a specific paramagnetic particle, the capture efficiency can be determined by Eq. (40). In mineral processing, ultrafine weakly magnetic minerals are difficult to be recovered as these particles have quite small particle capture radius. Eq. (40) shows that, under the same matrix’s arrangement, capture efficiency of fine weakly particles is only a function of Rc. Thus effect of matrix aspect ratio on particle capture efficiency can be investigated by comparing particle capture radii under various circumstances.

2.5. Selection methodology of particle capture models In this study, applied induction of 0.2T, 0.4T, 0.6T, 0.8T, 1T, 1.5T, 2T and matrix aspect ratio of 1/5, 1/2, 1, 2, 4, 6 were considered. With increasing aspect ratio, matrix magnetization state changes and can reach saturation even under a low induction. Rapid determination of magnetization state of matrix is quite necessary. We have investigated magnetization process of circular and elliptic matrices with increasing applied induction [43,44]. Demarcation of applied induction for judging magnetization state of matrix can be determined by the simple equation: (39)

B0 = Ms /( + 1)

Table 1 was a summary of particle capture models and their attributes in longitudinal HGMS. The matrix materials was pure iron, with saturation magnetization Ms = 2.14T [45]. Fig. 4 showed selection methodology of particle capture models with varying matrix aspect ratio under a variety of applied induction. The red line was demarcation of applied induction to determine matrix magnetization state. Each symbol at the intersection of applied induction and matrix aspect ratio represented a corresponding model in Table 1. Magnetic state of matrices under all conditions could be determined and particle capture models could be selected.

Ec =

Rc n

100% 100%

. ...............................

(R c < n) (R c n )

(40)

Fig. 7 showed variation of capture radius of particles (R = 2 μm and R = 10 μm) with increasing aspect ratio under the same cross-section area. It can be seen that, for particles of R = 2 μm and 10 μm, there exists optimal aspect ratio at which particle capture radius reaches the maximum under all applied induction except 0.2T. The optimal matrix aspect ratio decreases with the increase of applied induction. Although the optimal aspect ratio under 0.2T is not shown, it can be inferred that there also exists optimal ratio considering the curves’ change regularity. The optimal aspect ratio under 0.2T will be larger than 6 and is not considered in this study. For particles of R = 2 μm and R = 10 μm, the optimal aspect ratio is almost the same under specific induction. Additionally, the optimal aspect ratio under this case are distributed in the whole range of 1/5–6, i.e., the optimal aspect ratio can be lower and

3. Particle motion trajectories and particle capture radius 3.1. Under the same cross-section area Fig. 5 showed particle motion trajectories around matrices with aspect ratio of 1/2, 1 and 2 under the same cross-section area (π mm2). Table 1 Particle capture models in longitudinal HGMS for various cases. Model 1

Model 2

Model 3

Model 4

Model 5

Model 6

γ=1 (35)–(37)

γ=1 (35),(36),(38)

γ > 1 (11),(31),(32)

γ > 1 (12),(31),(32)

B =

C1

C1 = 1

γ Equations Coefficients

γ < 1 (17),(33),(34)

γ < 1 (18),(33),(34)

C2

C2 = - 1

State Identification

unsaturated ●

e 2v 0

saturated ◆

2Ms shv0 chv0 µ 0 H0

B = H0 d2

unsaturated ■

5

Ms d2 2µ 0

saturated ▲

e 2v 0

unsaturated ★

saturated ▾

2Ms shv0 chv0 µ0 H0

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2Rc

2Rc

2Rc

Fig. 5. Particle motion trajectories around matrix with aspect ratio 1/2, 1 and 2 under the same cross-section area: Applied induction B0 = 0.6T and particle radius R = 10 μm.

and applied induction for maximum particle capture is independent of matrix and particle sizes. 3.2. Under the same horizontal axis Also, matching relation between matrix aspect ratio and applied induction for maximum particle capture was investigated under the same horizontal axis (2 mm). Fig. 9 showed particle motion trajectories around matrices with aspect ratio of 1/2, 1 and 2. Particle motion trajectories are quite similar to those in Fig. 5. Particles are mainly captured at the upper and lower sides of matrices. Fig. 10 showed variation of particle capture radius with increasing matrix aspect ratio under the same horizontal axis (the same lh in Fig. 6). There also exists an optimal aspect ratio at which particle capture radius reached the maximum. The optimal aspect ratio decreases with increase of applied induction. For all circumstances considered, the optimal aspect ratio is larger than 1. This is different from the case of the same cross-section area, in which optimal aspect ratio is lower than 1 under induction of 1.5T and 2T.

Fig. 6. Arrangement of matrices with the same cross-section area in longitudinal HGMS.

higher than 1. Another interesting phenomenon which can be observed in Fig. 6(a) is that particle capture radius at certain aspect ratio coincide with each other under some applied induction (1T, 1.5T and 2T). For matrices with aspect ratio of 2 and 4, capture radius of particles of R = 2 μm under applied induction of 1T, 1.5T and 2T are almost the same. This regularity cannot be seen for particles of R = 10 μm in Fig. 6(b). These results indicate the difficulty of recovering ultrafine weakly particles. Increasing applied induction will not always improve particle recovery, especially for ultrafine particles. Fig. 8 showed effect of matrix size on optimal aspect ratio for maximum particle capture. For matrices with different size, variation regularity of particle capture radius with increasing aspect ratio is almost the same, particle capture radius reaches the maximum at the same aspect ratio. The matching relation between matrix aspect ratio

4. Discussion and maximum separation efficiency matching Longitudinal high gradient magnetic separators are the most widely applied in mineral processing. Fig. 11 showed appearance and magnetic system of longitudinal high gradient magnetic separator. A pulsating flow is applied to eliminate the blocking of materials in the vicinity of matrices. As fluid drag force is proportional to the first order of flow velocity, the pulsating flow (with relatively high peak velocity) will lead to decrease in recovery of ultrafine magnetic minerals. That’s why horizontal ring high intensity magnetic separator presents advantage over vertical ring high gradient magnetic separator in recovering fine hematite from red mud [20]. The analyses above showed that there existed matching relation between matrix aspect ratio and applied induction for maximum particle capture in longitudinal HGMS. For 6

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Fig. 7. Particle capture radius by matrices with increasing aspect ratio under the same cross-section area in longitudinal HGMS: (a) Particle radius R = 2 μm; (b) Particle radius R = 10 μm.

Fig. 8. Capture radius of particles of R = 2 μm by matrices with different size: (a) under applied induction B0 = 0.4T; (b) B0 = 1.5T.

specific longitudinal HGMS system, the matching relation can be used to enhance the recovery of weakly magnetic minerals. In mineral processing, a moderated applied induction within the range of 0.3–0.8T is usually adopted. The matching relation can be applied in two orientations: improving recovery of ultrafine minerals or reducing energy consumption under the same separation indexes. The matrices used in longitudinal magnetic separator are steel rods (γ = 1). For the two cases considered, as shown by Figs. 7 and 10, the optimal aspect ratio is in the range of 2–4 under applied induction of 0.3–0.8T. The particle capture radius at optimum aspect ratio can be much larger than that of circular matrix (γ = 1). On the other hand, as shown by Fig. 7, particle capture radius of circular matrix under applied induction of 0.6T is almost equal to that of matrix with γ = 2 under induction of 0.4T. The magnetic field is energized by current and the energy consumption is proportional to the square of magnetic induction induced [46]. Thus for obtaining the same particle capture efficiency, the energy consumption using matrices with γ = 2 (0.42 = 0.16) is nearly half of that adopting circular matrices (0.62 = 0.36). As has been mentioned above, in many practices of HGMS, capture efficiency of magnetic particles is not the only index, the quality of magnetic products is equally important. In mineral processing, many efforts have been made to improve the selectivity in HGMS of weakly magnetic ores [47,48]. Specially, for longitudinal HGMS, it was typically recognized that mechanical entrainment of non-magnetic minerals

was caused by the direct impingement of pulp flow on mineral deposit on the matrices [49]. With pulp flow impinging on the particle deposit, the following two entrainment mechanism existed: (1) nonmagnetic particles of large size would accumulate on the deposit with magnetic particles; (2) nonmagnetic particles of small size would penetrate into the blanks among particles of the deposit and stayed there. In HGMS, particles are attracted by matrices and then accumulate on matrices. Variation of particle aspect ratio will not only affect particle capture by matrices but also particle accumulation on matrices. As mechanical entrainment in longitudinal HGMS is mainly caused by direct impingement of pulp flow on particle deposit, deposit morphology will also affect impingement range of pulp flow and consequently mechanical entrainment of gangue minerals. Fig. 12 showed the schematic diagram of particle deposit on matrices with different aspect ratio in longitudinal HGMS. With the increase of aspect ratio, the deposit morphology transforms from “flat” to “narrow and long”. The direct impingement range of pulp flow decreases. Thus mechanical entrainment of gangue minerals can be decreased and separation performance can be improved. Combining the particle capture analyses, with increasing matrix aspect ratio in certain range, particle capture efficiency and particle selectivity can simultaneously be improved. But further increasing aspect ratio, particle capture efficiency will decrease. It can be inferred that, with increasing matrix aspect ratio, there should exist matching relation between matrix aspect ratio and applied 7

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Fig. 9. Particle motion trajectories around matrices with aspect ratio of 1/2, 1 and 2 under the same horizontal axis: Applied induction B0 = 0.6T and particle radius R = 10 μm.

(Front view) N H0

S

(Side view) Fig. 10. Variation of particle capture radius with increasing matrix aspect ratio under the same horizontal axis: Particle radius R = 10 μm.

Fig. 11. Appearance of longitudinal high gradient magnetic separator (left) and the magnetic system (right).

induction for maximum separation efficiency and this may apply to all the three configurations of HGMS. The matching relation of maximum separation efficiency with respect to matrix aspect ratio and applied induction is exactly our follow-up study subject and will be conducted before long.

adopted in mineral processing. The matching relation can be used to configure the HGMS system for improving recovery of ultrafine magnetic minerals or reducing energy consumption under the same separation indexes. It is also believed that there should exists matching relation between matrix aspect ratio and applied induction for maximum separation efficiency in three configurations of HGMS, which will be investigated systematically in our follow-up studies.

5. Conclusions In this paper, a new concept of matching relation between matrix aspect ratio and applied induction for maximum particle capture in longitudinal HGMS was proposed. The matching relation exists for both the cases of equal matrix cross-section area and horizontal axis, and is independent of particle and matrix sizes. Particle capture radius of matrices with optimal aspect ratio can be much larger than that of conventional circular matrices within induction range commonly

CRediT authorship contribution statement Xiayu Zheng: Conceptualization, Methodology, Writing - original draft. Zixi Sun: Data curation, Formal analysis. Yuhua Wang: Supervision, Data curation, Writing - review & editing. Dongfang Lu: Supervision, Data curation. Zixing Xue: Visualization. 8

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Magnetic minerals

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Fig. 12. Schematic diagram of particle accumulation on matrices with different aspect ratio in longitudinal HGMS.

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