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Model of particle accumulation on matrices in transverse field pulsating high gradient magnetic separator
Minerals Engineering 146 (2020) 106105
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Model of particle accumulation on matrices in transverse field pulsating high gradient magnetic separator ⁎
Zhicheng Hua, Jun Zhangb,c,d, Jianguo Liub,c,d, Yuhe Tangb,c,d, , Xiayu Zhenga,
T
⁎
a
School of Minerals Processing & Bioengineering, Central South University, Changsha 410083, China Guangdong Institute of Resources Comprehensive Utilization, Guangzhou 510650, China c State Key Laboratory of Rare Metals Separation and Comprehensive Utilization, Guangzhou 510650, China d Guangdong Province Key Laboratory of Mineral Resources Development and Comprehensive Utilization, Guangzhou 510650, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Transverse configuration Pulsating slurry Double wires Particle buildup model
The static buildup model of particle on single wires (SBM-sw), illustrating the effect of these influencing parameters such as applied magnetic induction, fluid velocity and wires size on particle accumulation, plays a dominant role in instructing the design of a high gradient magnetic separation (HGMS) device. However, in a real separation chamber of HGMS filled with piles of matrices, the changes of hydrodynamic due to gap shrinkage of matrices disturb particle accumulation. When slurry within the matrix is exposed to pulsation, this disturbance is intensified. Consequently, compared to SBM-sw, the model of magnetic particle accumulation on double wires (SBM-dw) is supposed more suitable to reveal the processing of magnetic particle accumulation on matrices. In this article, SBM-sw and SBM-dw of a transverse field pulsating HGMS separator (TPHGMS) was presented based on force equilibrium. The effect of pulsating slurry, wire spacing and applied magnetic induction on particle accumulation of the TPHGMS was investigated and discussed with calculating the buildup profiles under specific parameters. Pulsating slurry directly impinges the regions of paramagnetic particle capture and leads to sharp decrease of particle buildup under high frequency. Enlarging wire spacing or decreasing pulsating frequency can weaken the impact of slurry and slow down the decrease of particle buildup. In addition, increasing applied magnetic induction within a certain range can also significantly increase particle buildup. Finally the SBM-dw model was compared with a series of experiments. Results indicated that experimental particle buildup weights agreed with the calculated particle buildup weights of SBM-dw which provides better authenticity to understand and predict the performance of the TPHGMS.
1. Introduction High gradient magnetic separation (HGMS) has been widely used in the mineral processing industry to separate small particles of weakly magnetic materials (Chen et al., 2017; Iranmanesh and Hulliger, 2017; Xu et al., 2018). A HGMS separator consists of a region of a high and approximately uniform magnetic field and a ferromagnetic matrix of fine wires or rods that distort the field and produce large local gradients (Bean, 1971; Stekly and Minervini, 1976). Poor selectivity and matrix blockage are often responsible for the declining reputation of HGMS until a slurry within the matrix is exposed to pulsation (Svoboda and Fujita, 2003). In industrial practice, two types of pulsating HGMS separators are usually used: longitudinal field pulsating HGMS separators (LPHGMS) and transverse field pulsating HGMS separators (TPHGMS) (Xiong, 2004; Tang, 2009; Chen, 2011), as illustrated in Fig. 1. Magnet of the LPHGMS possesses much shorter magnet spacing and produce ⁎
much higher magnetic induction than the TPHGMS. The disadvantage is that the magnets area easily corroded in weakly acidic slurry and need to be renew in about three years. Meanwhile, the transverse field pulsating HGMS separator solves this special case because its magnetic poles are not immersed in the slurry. Moreover, the transverse field pulsating HGMS separator has higher selectivity but lower magnetic induction, lower capacity and higher energy consumption. Table 1 gives separation index of LPHGMS and TPHGMS in Panzhihua Iron and Steel Co. Ltd. (Sichuan, China). Obviously, for ilmenite separation, under similar recovery, the LPHGMS has to yield to the TPHGMS in terms of concentrate grade, tails grade and separation efficiency only except ore throughput. So re-recognition of TPHGMS is essential at the theoretical level to find why ore throughput is difficult to increase. The theoretical analysis of a magnetic particle captured by the wire of an HGMS separator can be viewed as two separate, but related, problems. The first one is the motion of the particle in the separation
Corresponding authors at: Guangdong Institute of Resources Comprehensive Utilization, Guangzhou 510650, China (Y. Tang). E-mail addresses: [email protected] (Y. Tang), [email protected] (X. Zheng).
https://doi.org/10.1016/j.mineng.2019.106105 Received 30 April 2019; Received in revised form 19 October 2019; Accepted 25 October 2019 0892-6875/ © 2019 Elsevier Ltd. All rights reserved.
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Fig.1. Magnet of LPHGMS (up) and TPHGMS (down). Table 1 Separation results of LPHGMS and TPHGMS in Panzhihua Iron and Steel Co. Ltd. Types
Longitudinal field pulsating HGMS Transverse field pulsating HGMS
Ring diameter (mm)
Matrices volume (m3)
Applied magnetic induction (T)
Ore throughput (t/h)
Grade of ilmenite (TiO2%) Feed
Concentrate
tails
Recovery (%)
Separation efficiency (%)
2750
1.645
1.0
120
10.01
15.00
2.80
88.56
36.38
2750
1.602
0.45
70
9.89
17.54
2.65
86.23
46.31
particle buildup mechanism, the effect of hydrodynamic changes due to gap shrinkage of matrices on particle accumulation in a real HGMS is ignored. Moreover, pulsating slurry can inevitably exacerbate the effect on the behavior of accumulation. Thus, these models are unsuitable to predict the performance of a transverse field pulsating HGMS separator and provide precise theoretical guidance. The objective of this study is to extend SBM from single wires to double wires for the investigation of magnetic particle accumulation on the ferromagnetic rods of a TPHGMS separator and find the reasons why ore throughput is difficult to increase. In our model, we think friction between each buildup layer is nonzero and calculate the particle buildup profiles and weights under specific conditions to research the effect of pulsating slurry, wire spacing and applied magnetic induction on particle accumulation in TPHGMS separator. Then, to further confirm the accuracy of this double wires model, the capture experiments of the magnetic mineral monomers by the magnetized ferromagnetic collecting wire in the separator matrix of an TPHGMS unit are carried out to compare experimental results with the model calculated results.
region by analyzing its trajectory to determine under what conditions a particle will strike the wire. The second one is examination of whether the particle will stick to the wire and continue to buildup on the surface of the wire with other attracted particles, or the particle will be washed away by the fluid after the impact of a particle on the wire (Liu, 1983). The formula of motion trajectory of magnetic particle and the conception of magnetic velocity in a HGMS system were firstly introduced by Watson (1973,1975), and the dimensionless ratio of the magnetic velocity to the free stream fluid velocity was a dominant parameter for the particle capture. After that, Luborsky (1975), Lawson (1977), and Zheng (2018) extended the original particle trajectory model for the calculation of HGMS performance, to respectively include the buildup of multiple layers of particles on the wire, nonvanishing gravitational forces and a broad range of Stokes number, and the condition that the matrix is unsaturated. As of today, the application of magnetic separation is gradually broadened to sewage treatment, healthcare and solid waste recycling. The diversity of separator makes many researchers turn to computational simulations to study the trajectory (Belounis et al., 2015, 2016; Zhou et al., 2015; Wei et al., 2010; Haverkort et al., 2009; Misael et al., 2005). However the main factor for determining HGMS system throughput is still magnetic particle accumulation or buildup on a matrix. The static buildup model (SBM) and simulation of dynamic accumulation (SDA) of particles on single wires were developed to reveal the accumulation process (Luborsky and Drummond, 1976; Cowen et al., 1976; Nesset and Finch, 1980; Badescu et al., 1996; Hournkumnuard et al., 2011; Chen et al., 2012; Natthaphon et al., 2017). These researches on single wires focus on
2. Model 2.1. SBM on single wires in transverse-type HGMS The SBM successfully predicts the accumulation of particles on a single wire in a longitudinal-type HGMS (Nesset and Finch, 1981). The basic idea of SBM is the equilibrium of forces acting on a particle 2
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Fig. 2. Transverse HGMS configuration: (a) single wires; (b) double wires in pulsating slurry.
cylinder, these forces are split into radial and tangential components (Maass et al., 1983). Apparently, a magnetized ferromagnetic rod can be divided into attractive regions within which particles can be captured and repulsive regions within which particles cannot be captured. When the saturation buildup is attained, the sum of the radial forces and the sum of the tangential forces of the particles located in the boundaries equate to zero, as indicated in Eqs. (1) and (2).
Fmr + Fgr = 0
(1)
Fmθ + Fdθ + Fgθ = 0
(2)
The boundaries of the saturation buildup can be obtained by solving the above equations. Whether the force of static friction needs to be considered in the buildup process of particles or not is the debate. When this force is ignored in the SEM of a longitudinal-type HGMS, the boundary curve possesses satisfactory consistency with the saturated Mn2P2O7 buildup photographs taken from video recordings (Friedlaender et al.,1978; Nesset and Finch, 1981). In a transverse-type HGMS, given that all tangential components have the same sign and cannot provide the boundary curve, the force of static friction is worthy of being reconsidered, and the surface roughness of the HGMS-buildup has the same order of magnitude as the diameter of the particles. Thus, the fluctuations for the static friction coefficient between single particles and accumulation layer are a real existence. Because of the large number of particles at a saturation buildup, its magnitude and shape can be determined by averaging these fluctuating single forces (Maass et al., 1983). These forces in transverse-type SBM are presented in Fig. 3 and based on the above analysis, Eq. (2) is modified to Eq. (3).
Fig. 3. Force components acting on the particle of retention on the matrix.
Fmθ + Fdθ + Fgθ + Ff = 0
(3)
These components in Eqs. (1) and (3) were reasonably derived and modified by Watson (1973), Nesset and Finch (1980), Glew (1982), and Zheng et al. (2015, 2017a, 2018). The magnetic force solved by using polar coordinate function of Laplace equation is shown in Eqs. (4) and (5) when the matrices are assumed to be unsaturated (Zheng et al., 2017a, 2018). Fig. 4. Velocity curve of pulsating HGMS separator.
located on the surface of the buildup. In a transverse HGMS configuration shown in Fig. 2(a), the direction of the slurry, magnetic field, and cylindrical matrices’ axis are mutually perpendicular. Magnetic particles are adsorbed and accumulate on the sides of the cylindrical matrices determined by the forces acting on the particles, including the magnetic force Fm, fluid drag force Fd, and gravity force Fg. When the particles are spherical and the buildup takes the shape of a circular
Fmr = −
8πb3μ0 H02 d 2 d2 κ e (cos 2θ + 2 ) 3 3r r
(4)
Fmθ = −
8πb3μ0 H02 d 2 κ e sin 2θ 3r 3
(5)
where b is the radius of the particle, μ0 is vacuum permeability, H0 is the applied magnetic field, d is the radius of the matrix cross section, r is the distance between the particle and the matrix axis, and κ e is equivalent magnetic susceptibility of the particle around the matrix given by Eq. (6) (Nesset and Finch, 1980) 3
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Table 2 The practical values of the operating parameters. Parameter
Value
Parameter
Value
The Magnetic field induction B0 The applied magnetic field strength H0
0.2–0.8 T B0/ μ 0 1.55 T
Inlet velocity of the fluid v0 Kinematic viscosity of the fluid υ
0.05 m/s 1 mm2/s
Density of the fluid ρf
1 g/cm3
1.5 mm 4.5–10.5 mm 30 μm 0.0125–0.0049
Pulsating slurry frequency ω Permeability of vacuum μ 0 Gravitational acceleration g Static friction coefficient f Correction factor α
120–420 r/min 4π × 10−7 H/m 9.8 N/kg 1.2 0.8–1
The saturation magnetic of the matrix MS Radius of the matrix d Wire spacing D Radius of the magnetic particle b Equivalent magnetic susceptibility of the magnetic particle κ e Volume of the magnetic particle V
4 πb3 3
4650 kg/m3
Density of the magnetic particle ρp
4 Fgθ = − πb3 (ρp − ρf ) g cos θ 3
(9)
Where ρp is the particles density, andg is the gravitational acceleration. The force of static friction is given by Eq. (10).
Ff = f (Fmr + Fgr )
(10)
where f is the static friction coefficient. The buildup of particles on matrix rods is a fairly complex process and difficult to observe. The final boundary contour of buildup of particles was obtained by assuming that the particles show compact and layer-by-layer buildup in each layer r = b + d + (n − 1) 3 b (n = 1, 2, 3…). The θ satisfying Eqs. (1) and (3) is calculated for each layer and the smaller of the two angles θc is selected. Therefore, θc as the function of r gives the saturated buildup profile. 2.2. SBM on double wires in transverse-type pulsating HGMS We propose a extended SBM to include the double wires system in pulsating slurry depicted in Fig. 2(b), which is more accurate to predict the performance of a transverse field pulsating HGMS separator. Pulsating slurry can overcome matrix blockage and upgrade the selectivity of HGMS. The pulsating mechanism drives the slurry in the separating zone up and down, clearing the slurry channels and keeping a smooth traffic. This movement affects the accumulation of magnetic particles. In a double wire system in a pulsating slurry, fluid drag force changes with up-and-down pulsating slurry. The velocity curve of flow is illustrated in Fig. 4. Pulsating velocity, maximum pulsating velocity, and average pulsating velocity are respectively given by Eqs. (11)–(13) (Xiong, et al., 1998).
Fig. 5. Buildup profile of the ilmenite sample on the matrices: Magnetic induction B0 = 0.45 T; feed velocity v0 = 0.05 m/s; pulsating stroke S = 12 mm; pulsating frequency ω = 300 r/min; wire spacing D = 6.8 mm.
M0
κ e = κ∞ + H0 1+2
d2 r2
cos 2θ +
d4
≅ κ∞ +
M0 2.4H0
r4
(6)
The fluid drag force, only a force in the tangential direction, is calculated using the shear stressτ0 at the bottom of a boundary layer, which is accurately determined by the Blasius solution to the boundary layer equations. Consequently, the fluid drag force is given via Eq. (7) (Schlichting, 1968).
(12)
1 2π
∫0
2π
1 Sω Sω sin(ωt ) d (ωt ) = 2 π
v = v0 + v~max sin(ωt )
1
(13)
(14)
Simplification can be done so that the boundaries of buildup depend on the maximum velocity of alternating slurry. Thus, Eq. (13) can be rewritten as
(7)
where ρf , V0 , υare the fluid density, velocity (far from wire), and kinematic viscosity, respectively. The gravity force is given by Eqs. (8) and (9).
4 Fgr = − πb3 (ρp − ρf ) g sin θ 3
1 v~max = Sω 2
where S , ω , and t are the pulsating stroke, pulsating angle speed, and time variable, respectively. The velocity of the slurry is the sum of feeding velocity (v0 ) and pulsating velocity, shown in Eq. (14).
3 υ 2 π 2b2 ρf V02 ⎛ ⎞ (6.973θ − 2.732θ3 + 0.292θ5 − 0.0183θ7 4 ⎝r ⎠
+ 0.000043θ9 − 0.000115θ11)
(11)
v¯ =
Fdθ =
1 v~ = Sω sin(ωt ) 2
1
V=
⎧ 2 Sω + v0
(downflow )
⎨ 1 Sω ⎩2
(upflow )
− v0
(15)
In the gap between the matrices, presented in Fig. 2(b), the velocity of the slurry can be expressed by Eq. (16).
(8) 4
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Fig. 6. Particle buildup of different pulsating frequencies on double matrices in pulsating slurry: (a) buildup profile; (b) cumulative radius and weight; magnetic induction B0 = 0.45 T; feed velocity v0 = 0.05 m/s; pulsating stroke S = 12 mm; wire spacing D = 6.8 mm.
( V′ = ⎨ ( ⎩ ⎧
0
)( Sω − v )(
1 Sω 2 1 2
+ v0
D D − 2r cos θ
0
D D − 2r cos θ
) )
layer is completed when (θiup – θidown) < 100. Here, α is the correction factor. Given the possibility of noncompact accumulation between particles and the slurry is dragging the whole buildup region away from the surface of matrices, correction factor α is necessary for the calculated results. Through comprehensive consideration, the value of α is 0.8–1 related to the flow rate of slurry. Of course, an explanation is owed that, in longitudinal-type HGMS, α is not considered because the slurry is compacting the whole region of buildup close to the matrices’ surface.
(downflow ) (upflow )
(16)
where D is the center distance of adjacent wires. The fluid drag force of double wires in pulsating slurry can be solved through Eq. (3) by substituting V0 with V0′. Here we approximately treat V0′ as the free stream fluid velocity. That the pulsating pulp continues to maintain a laminar flow is assumed. The consistency of model calculation and experimental data is evaluated on the basis of the total weight of particle accumulation on the matrices. The calculated profile of saturation buildup can be converted to the weight via Eq. (17).
3. Calculated content and parameters We have already known the effect of these parameters such as magnetic induction, fluid velocity, matrices size on particle accumulation based on monofilament research. However, in double wires system, besides these we mainly focus on the particle buildup profiles and weights under different pulsating slurry and wire spacing because they can cause fluid changes. The magnetic particle is endued with these properties such as
m
(θ up − θidown ) r 4 MT = απNLρp b2 ∑ i 3 2b i=1
(17)
where N, L, and m are the total number of matrices, effective length of matrices, and the serial number of the final layer, respectively; θiup and θidown are the up θc and down θc of layer i. We assume that the final 5
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Fig. 7. Particle buildup of different wire spacing on double matrices in pulsating slurry: (a) buildup profile; (b) cumulative radius and weight; magnetic induction B0 = 0.45 T; feed velocity v0 = 0.05 m/s; pulsating frequency ω = 300 r/min; pulsating stroke S = 12 mm.
4. Calculated results
magnetism, size, and density of the ilmenite sample in Section 5. The matrix is the SUS430 wires with saturation magnetization Ms = 1.55 T (Xu, 1995) and radius d = 1.5 mm. For ilmenite capture, the applied magnetic induction B0 = 0.2 T, 0.3 T, 0.4 T, 0.5 T, 0.6 and 0.8 T, wire spacing D = 3d, 4d, 5d, 6d and 7d, pulsating slurry frequency ω = 120 r/min, 180 r/min, 240 r/min, 300 r/min and 420 r/min are respectively considered. The mean of maximum static friction coefficient is assumed to be 1.2, referring to the paper of Maass et al. (1983) and experimental particle buildup. The various parameters are detailedly listed in Table 2. Mathematical software (Matlab) are used to layer-by-layer solve the Eqs. (1) and (3) and get the values of θc. These coordinate points (r, θc) form saturation boundaries of the paricle buildup on the wires.
4.1. Pulsating slurry Fig. 5 reveals the saturated particle buildup on single wires and dual-wire pulsation when the applied magnetic induction is 0.45 T. The pulsating slurry shockingly affects the profile of particle buildup and causes sharp decrease of particle buildup with the consequent impact. Furthermore, compared with single wires, decrease of particle buildup on double wires obviously becomes more serious. Here is the most intuitive explanation why SBM-dw is supposed to be more suitable to reveal the magnetic particle accumulation on matrices in a TPHGMS. The effect of pulsating frequency on ilmenite accumulation in double wires system is illustrated in Fig. 6. The buildup profile of ilmenite, depicted in Fig. 6(a), decreases successively with the increase of the pulsating frequency because increasing the pulsating frequency must result in the increase of the fluid drag force and decrease the 6
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wire spacing, depicted in Fig. 7(b). However, in a TPHGMS unit, the separation space is filled with a large number of matrices. With the increase of wire spacing, there is usually a corresponding drop in the packed fraction of the matrices and consequently it is necessary to take a trade-off. The red dotted square in Fig. 8(a) is a representative element of the cross section of the separation space (Zheng et al., 2017b). The matrices packing fraction ξ can be calculated by using Eq. (18):
ξ=
πd 2 D2
(18)
Then we use the value of multiplying the buildup weight of each wire (MT) by the matrices packing fraction (ξ) to represent the element particle accumulation capacity of a TPHGMS unit. Fig. 8(b) shows the element particle accumulation capacity of a TPHGMS unit under different wire spacing. There is a certain wire spacing (D = 5.73 mm) at which particle accumulation capacity of a TPHGMS unit reaches maximum. 4.3. Applied magnetic induction
Fig. 8a. Arrangement of matrices.
The magnetic field is a decisive operation parameter in high gradient magnetic separation. The ability of paramagnetic particle capture can be effectively improved by increasing magnetic field strength. Fig. 9 depicts the effect of magnetic induction on ilmenite accumulation on the double wires in pulsating slurry. When magnetic induction is increased from 0.2 T to 0.8 T, the buildup profile enlarges continuously, but the span between adjacent buildup boundaries gradually decreases in a trend similar to wire spacing, especially after 0.4 T, as shown in Fig. 9(a). What’s more, the variation of buildup radius or weight in Fig. 9(b) displays the same information. Consequently, as the magnetic induction continues to increase, it can be inferred that, the profile eventually stops changing. This trend is essential to magnet selection, as the most suitable magnetic field for treating a certain mineral must be determined in the design period, particularly for a permanent magnet high gradient magnetic separator. Increasing the magnetic field strength on a small scale can result in the increase of the matrices’ surface area available for the buildup of the particles, shown in Fig. 9(a), which is absolutely contrary to longitudinal-type HGMS (Svoboda, 1994). The force of static friction is the main factor causing the difference. Normally, the magnetic force acting on particles is an increasing function of the magnetic field strength. The force of friction at the opposite direction of the tangential magnetic force increases with the radial magnetic force and enables the matrices’ surface area covered with particles to increase slowly. However, in longitudinal-type HGMS, the matrices’ surface area available for the buildup of the particles is determined by the perturbation term (A = 2πMw/H0 in CGS) (Nesset and Finch, 1980) for the cylinder matrix. When A decreases with the increase of magnetic field strength, the buildup area located on matrices’ surface decreases with the decrease of A. The difference of the two types of HGMS needs to be verified by visual test in the next paper. A new idea can be tried to prove the existence of friction by observing the real variety of buildup profile with the increase of magnetic field strength.
Fig. 8b. The element particle accumulation weight of a TPHGMS unit.
buildup profile. However, from the tendencies of cumulative radius and weight in Fig. 6(b), we can find the magnitude of the decrease tends to be gentle. The magnetic force acting on the particles located on the surface of the buildup rapidly increases with the decrease of accumulation radius and the particle is difficult to be dragged out when the pulsation frequency increases to a critical value. In addition, the matrices’ surface available for the buildup shows a small contraction given the increase of fluid drag force. The slurry settles the matrix blockage but affects the process of particle buildup especially for transverse-type HGMS. This effect clearly explains why the transverse field pulsating HGMS separator has low processing capability.
4.2. Wire spacing 5. Confirmed experiments The wire spacing is a parameter worth considering in a TPHGMS unit, but it has been ignored. Up to date, none has investigated whether it has a significant effect on the particle buildup. Here is the first discussion for it from theoretical level. Fig. 7(a) provides particle buildup boundaries, under the different wire spacing, distinguished by colorful curves. It can be seen that the wire spacing significantly affects the process of particle buildup. As the spacing is enlarged, for the crosssectional area of particle buildup, there is a quantitative increase, yet the upper limit of the area is the same as that of single wires accumulation. So we can increase the amount of particle buildup by increasing
5.1. Sample Ilmenite is prepared by the flowsheet depicted in Fig. 10(a) and concentrate with a particle size of −0.075 + 0.045 mm is selected as the experimental sample. The result of X-ray diffraction, revealed in Fig. 10(b) indicates that the purity of the experimental sample is more than 90%. The density of the experimental sample is 4650 kg/m3. In the mineral processing industry, the magnetic susceptibility of numerous minerals treated with HGMS is not constant and can be affected by the 7
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Fig. 9. Particle buildup profile of different magnetic induction on double matrices in pulsating slurry: (a) buildup profile; (b) cumulative radius and weight; feed velocity v0 = 0.05 m/s; pulsating frequency ω = 240 r/min; pulsating stroke S = 12 mm; wire spacing D = 6.8 mm.
applied magnetic field. Thus, the equivalent susceptibilitiesκ e of the experimental sample are measured by a vibrating sample magnetometer. When the applied magnetic field of 0.2 T, 0.3 T, 0.4 T, 0.5 T, and 0.6 T is considered, correspondingκ e are approximately 0.0125, 0.0092, 0.0076, 0.0065, and 0.0058 (SI), respectively.
5.3. Methods The sample and water were fully mixed in a stirred beaker and evenly fed to the separator within 30 s. The feed velocity, feed percentage of solids, feed weight, and pulsating stroke are 0.05 m/s, 10%, 400 g, and 12 mm, respectively, which are determined in advance as constants during this experiment. Optimizing tests of feed weight demonstrate that the weight of buildup on the matrix stops changing when the 400 g sample is fed. At this weight, we can approximately assume the accumulation reaches saturation. Magnetic induction of 0.2 T, 0.3 T, 0.4 T, 0.5 T, and 0.6 T and pulsating frequency of 180, 240, and 300 r/min are investigated as variables. When each experiment with the operating variables of the pilot separator is finished, the current of the energizing coils is turned off, and the magnetic particles captured on matrices are washed out with clean water, dried in an oven, weighed with an electronic scale, and analyzed.
5.2. HGMS separator The cyclic transverse field pulsating HGMS separator used for the experiment, illustrated in Fig. 11(a), is designed and manufactured by Guangzhou Research Institute of Non-ferrous Metals (China). The magnetic field between the magnetic poles is oriented perpendicular to the direction of the slurry. The particles that accumulate on both sides of the matrices are washed by pulsating slurry and obtain a higher grade than the longitudinal field pulsating HGMS separator but lower recoveries for finer magnetic particles. The unit of the matrix used in the experiment is designed as shown in Fig. 11(b). The effective length of a single matrix is 135 mm and per unit consists of 18 stainless steel rods with a diameter of 3 mm. A total of 5 units (total of 90 rods) are stacked in the region of the magnetic field to conduct the experiment.
5.4. Experiment results The SBM on double wires in TPHGMS seems to be more effective than SBM on single wires in transverse-type HGMS for practical 8
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Fig. 11a. Cyclic transverse field pulsating HGMS separator. Fig. 10a. Principle flowsheet of ilmenite sample preparation.
Fig. 11b. Designed matrices unit.
flow to turbulent flow as presented in the inner illustration of Fig. 12(a). Fig. 12(b) shows the calculated and experimental weights of buildup when the pulsating frequencies are 180 r/min, 240 r/min, 300 r/min, and 420 r/min. The pulsating frequency are usually used to improve the performance of HGMS. The weights also show that the model can predict the results of the pulsating frequency experiment. The decrease of theoretical and the experimental weights are simultaneous with the increase of the pulsation frequency. However, the trend of the two curves exist certain slight differences because the level of non-compact buildup show the change under different pulsating frequencies. It is consistent with the above paragraph analyzes.
Fig. 10b. XRD spectrum of prepared ilmenite sample.
instruction. A series of experiments are carried out to preliminarily assess the accuracy of the model. Fig. 12 is the comparison of calculations with experimental results. Fig. 12(a) reveals the calculated and experimental weights of buildup when magnetic induction is increased from 0 T to 0.6 T. In general, the experimental weights of buildup are in good agreement with the model. The growth trend of experiment results agrees well with calculation results, the disparity is less than 5% when the correction factor is 0.85 and magnetic induction is below 0.4 T. However, within the range of 0.4–0.6 T, the disparity is unexpectedly enlarged. The increase of magnetic field strength will lead to the shrinkage of the gap between the two matrices because of the increase of particle buildup. This will further cause the flow rate increase and influence the level of noncompact accumulation. Another reason possibly responsible for the change of disparity is that Reynolds number of flow located in the gap between two matrices is also continuously increasing from 1160 to 2440, surpassing the critical Reynolds number 2300 at 0.6 T (Schlichting, 1968) and resulting in the transformation from laminar
6. Conclusion The TPHGMS, characterized by high separation index and low capacity, is designed to circumvent the gradual corrosion of the magnets of LPHGMS in weakly acidic slurry. Re-recognition of the performance from a theoretical level is essential. The pulsating slurry directly impinges the regions of paramagnetic capture and leads to an astonishingly negative impact, being intuitional shown by the calculated buildup profiles of the model on the single wires in the transverse-type HGMS and the model on the double wires in the transverse-type pulsating HGMS. Comparing the two models, SBM of particle accumulation on double wires in transverse-type 9
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Acknowledgement We sincerely acknowledge the fundings to this research work from the Key Program for Applied Basic Research of Guangzhou (Grant No. 201904020021), Guangdong Academy of Sciences for Innovation Capacity Building (Grant No. 2018GDASCX-0109), the National Natural Science Funds of China (Grant No. 51804341), and Natural Science Foundation of Hunan Province (Grant No. 2019JJ50833). References Badescu, V., Murariu, V., Rotariu, O., Rezlescu, N., 1996. A new modeling of the initial buildup evolution on a wire in an axial HGMF filter. J. Magn. Magn. Mater. 163 (1–2), 225–231. Bean, C.P., 1971. Theory of magnetic filtration. Bull. Am. Phys. Soc. 16 (3), 350. Belounis, A., Mehasni, R., Ouil, M., Feliachi, M., Latreche, E.H., 2015. Design with optimization of a magnetic separator for turbulent flowing liquid purifying applications. IEEE Trans. Magn. 51 (8), 1–8. Belounis, A., Mehasni, R., Ouili, M., Feliachi, M., Latreche, E.H., Razek, A., 2016. Optimization of a dual capture element magnetic separator for the purification of high velocity water flow. EU. Phy. J. Appl. Phy. 73 (2), 20601. Cowen, C., Friedlaender, F.J., Jaluria, R., 1976. Single wire of high gradient magnetic separation processes Ⅰ. IEEE Trans. Magn. 12 (5), 466–470. Chen, F., Smith, K.A., Hatton, T.A., 2012. A dynamic buildup growth model for magnetic particle accumulation on single wires in high-gradient magnetic separation. AIChE J. 58 (9), 2865–2874. Chen, L., 2011. Effect of magnetic field orientation on high gradient magnetic separation performance. Miner. Eng. 24 (1), 88–90. Chen, L., Liu, W., Zeng, J., Ren, P., 2017. Quantitative investigation on magnetic capture of single wires in pulsating HGMS. Powder Technol. 313, 54–59. Friedlaender, F.J., Takayasu, M., John, B.R., Czeslaw, P.K., 1978. Particle flow and collection process in single wire HGMS studies. IEEE Trans. Magn. 14 (6), 1158–1164. Glew, J.P., Wong, M.K., Parker, M.R., Birss, R.R., 1982. Single-wire buildup of diamagnetic particles at low Reynolds numbers. IEEE Trans. Magn. 18 (6), 1656–1658. Haverkort, J.W., Kenjereš, S., Kleijn, C.R., 2009. Computational simulations of magnetic particle capture in arterial flows. Ann. Biomed. Eng. 37 (12), 2436–2448. Hournkumnuard, K., Chantrapornchai, C., 2011. Parallel simulation of concentration dynamics of nano-particles in high gradient magnetic separation. Simul. Modell. Pract. Theory 19 (2), 847–871. Iranmanesh, M., Hulliger, J., 2017. Magnetic separation: its application in mining, waste purification, medicine, biochemistry and chemistry. Chem. Soc. Rev. 46 (19), 5925–5934. Luborsky, F., Drummond, B., 1975. High gradient magnetic separation: theory versus experiment. IEEE Trans. Magn. 11 (6), 1696–1700. Luborsky, F.E., Drummond, B.J., 1976. Buildup of particles on fibers in a high field-high gradient separator. IEEE Trans. Magn. 12 (5), 463–465. Liu, Y.A., Oak, M.J., 1983. Studies in magnetochemical engineering. Part II: theoretical development of a practical model for high-gradient magnetic separation. AIChE J. 29 (5), 771–779. Lawson, W.F., Simons, W.H., Treat, R.P., 1977. The dynamics of a particle attracted by a magnetized wire. J. Appl. Phy. 48 (8), 3213–3224. Maass, W., Duschl, M.H., Hoffmann, H., Friedlaender, F.J., 1983. A new model for the explanation of the saturation buildup in the transverse HGMS-configuration. Appl. Phys. A 32, 79–85. Misael, O.A., Ebner, A.D., Chen, H., Rosengart, A.J., Kaminski, M.D., Ritter, J.A., 2005. Theoretical analysis of a transdermal ferromagnetic implant for retention of magnetic drug carrier particles. J. Magn. Magn. Mater. 293 (1), 605–615. Natthaphon, C., Ebner, A.D., Mayuree, N., James, A.R., 2017. Simulation of dynamic magnetic particle capture and accumulation around a ferromagnetic wire. J. Magn. Magn. Mater. 428, 493–505. Nesset, J.E., Finch, J.A., 1980. A loading equation for high gradient magnetic separators and application in identifying the fine size limit of recovery. In: Somasundaran, P. (Ed.), Fine Particles Processing. AIME, New York, pp. 1217–1241. Nesset, J.E., Finch, J.A., 1981. The static (buildup) model of particle accumulation on single wires in high gradient magnetic separation: experimental confirmation. IEEE Trans. Magn. 17 (4), 1506–1509. Schlichting, H., 1968. Boundary-Layer Theory. McGraw-Hill, New York, pp. 44. Svoboda, J., 1994. The effect of magnetic field strength on the efficiency of magnetic separation. Miner. Eng. 7 (5/6), 747–757. Svoboda, J., Fujita, T., 2003. Recent developments in magnetic methods of material separation. Miner. Eng. 16 (9), 785–792. Stekly, Z.J.J., Minervini, J.V., 1976. Shape effect of the matrix on the caputure cross section of particles in high gradient magnetic separation. IEEE Trans. Magn. 12 (5), 474–479. Tang, Y., 2009. Appfication of the high gradient magnetic separator with horizontal magnet system in meishan iron concentrator. Metal Mine 7, 86–89 (in Chinese). Watson, J.H.P., 1973. Magnetic filtration. J. Appl. Phys. 44 (9), 4209–4213. Watson, J.H.P., 1975. Theory of capture of particles in magnetic high-intensity filters. IEEE Trans. Magn. 11 (5), 1597–1599. Wei, Z.H., Lee, C.P., Lai, M.F., 2010. Magnetic particle separation using controllable magnetic force switches. J. Magn. Magn. Mater. 322 (1), 19–24. Xiong, D., Liu, S., Chen, J., 1998. New technology of pulsating high gradient magnetic
Fig. 12. Comparison of calculations to experimental results: (a) Magnetic induction; (b) pulsating frequency.
pulsating HGMS is more suitably used to accurately estimate the performance of the TPHGMS. Furthermore, the results of confirmed experiments indicate good agreement between experimental values and the calculated accumulation weights in double wires system. But with the flow turning into turbulence, a slight deviation will exist. The calculated results of double wires model in terms of pulsating slurry, wire spacing and applied magnetic induction give designer the direction of increasing ore throughput. Enlarging wire spacing or decreasing pulsating frequency can weaken the impact of slurry and increase particle buildup. In addition, increasing applied magnetic induction within a certain range is also a significant operation to increase particle buildup. However, in practice, the TPHGMS is difficult to increase magnetic induction unless the magnet spacing is reduced and as a result less amount of matrix can be filled. Besides, larger wire spacing also result in less amount of matrix, and very low pulsating frequency may cause matrix blockage. So it is essential to weigh and consider these parameters to optimize processing capacity. At last, the calculated buildup profile should be compared with the observed contour in the next paper to make further confirmation.
Declaration of Competing Interest We declare that we have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Zheng, X., Wang, Y., Lu, D., 2015. A realistic description of influence of the magnetic field strength on high gradient magnetic separation. Miner. Eng. 79, 94–101. Zheng, X., Wang, Y., Lu, D., Li, X., Li, S., Chu, H., 2017a. Comparative study on the performance of circular and elliptic cross-section matrices in axial high gradient magnetic separation: role of the applied magnetic induction. Miner. Eng. 110, 12–19. Zheng, X., Wang, Y., Lu, D., 2017b. Study on buildup of fine weakly magnetic minerals on matrices in high gradient magnetic separation. Physicochem. Probl. Miner. Process. 53 (1), 94–109. Zheng, X., Wang, Y., Gao, D., Lu, D., Li, X., 2018. Study on the demarcation of applied magnetic induction for determining magnetization state of matrices in high gradient magnetic separation. Miner. Eng. 127, 191–197.
separation. Int. J. Miner. Process. 54, 111–127. Xiong, Dahe, 2004. Study on comparison between vertical and horizontal magnetic fields in pulsating high gradient magnetic separation. Metal Mine 10, 24–27 (in Chinese). Xu, J., 1995. Effect of steel wool magnetized states on magnetic dressing. Nonferrous Met. 47, 38–42 (in Chinese). Xu, J., Xiong, D., Song, S., Chen, L., 2018. Superconducting pulsating high gradient magnetic separation for fine weakly magnetic ores: cases of kaolin and chalcopyrite. Results Phys. 10, 837–840. Zhou, Y., Kumar, D.T., Lu, X., Kale, A., Xuan, X., 2015. Simultaneous diamagnetic and magnetic particle trapping in ferrofluid microflows via a single permanent magnet. Biomicrofluidics 9 (4), 044102.
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Model of particle accumulation on matrices in transverse field pulsating high gradient magnetic separator ⁎
Zhicheng Hua, Jun Zhangb,c,d, Jianguo Liub,c,d, Yuhe Tangb,c,d, , Xiayu Zhenga,
T
⁎
a
School of Minerals Processing & Bioengineering, Central South University, Changsha 410083, China Guangdong Institute of Resources Comprehensive Utilization, Guangzhou 510650, China c State Key Laboratory of Rare Metals Separation and Comprehensive Utilization, Guangzhou 510650, China d Guangdong Province Key Laboratory of Mineral Resources Development and Comprehensive Utilization, Guangzhou 510650, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Transverse configuration Pulsating slurry Double wires Particle buildup model
The static buildup model of particle on single wires (SBM-sw), illustrating the effect of these influencing parameters such as applied magnetic induction, fluid velocity and wires size on particle accumulation, plays a dominant role in instructing the design of a high gradient magnetic separation (HGMS) device. However, in a real separation chamber of HGMS filled with piles of matrices, the changes of hydrodynamic due to gap shrinkage of matrices disturb particle accumulation. When slurry within the matrix is exposed to pulsation, this disturbance is intensified. Consequently, compared to SBM-sw, the model of magnetic particle accumulation on double wires (SBM-dw) is supposed more suitable to reveal the processing of magnetic particle accumulation on matrices. In this article, SBM-sw and SBM-dw of a transverse field pulsating HGMS separator (TPHGMS) was presented based on force equilibrium. The effect of pulsating slurry, wire spacing and applied magnetic induction on particle accumulation of the TPHGMS was investigated and discussed with calculating the buildup profiles under specific parameters. Pulsating slurry directly impinges the regions of paramagnetic particle capture and leads to sharp decrease of particle buildup under high frequency. Enlarging wire spacing or decreasing pulsating frequency can weaken the impact of slurry and slow down the decrease of particle buildup. In addition, increasing applied magnetic induction within a certain range can also significantly increase particle buildup. Finally the SBM-dw model was compared with a series of experiments. Results indicated that experimental particle buildup weights agreed with the calculated particle buildup weights of SBM-dw which provides better authenticity to understand and predict the performance of the TPHGMS.
1. Introduction High gradient magnetic separation (HGMS) has been widely used in the mineral processing industry to separate small particles of weakly magnetic materials (Chen et al., 2017; Iranmanesh and Hulliger, 2017; Xu et al., 2018). A HGMS separator consists of a region of a high and approximately uniform magnetic field and a ferromagnetic matrix of fine wires or rods that distort the field and produce large local gradients (Bean, 1971; Stekly and Minervini, 1976). Poor selectivity and matrix blockage are often responsible for the declining reputation of HGMS until a slurry within the matrix is exposed to pulsation (Svoboda and Fujita, 2003). In industrial practice, two types of pulsating HGMS separators are usually used: longitudinal field pulsating HGMS separators (LPHGMS) and transverse field pulsating HGMS separators (TPHGMS) (Xiong, 2004; Tang, 2009; Chen, 2011), as illustrated in Fig. 1. Magnet of the LPHGMS possesses much shorter magnet spacing and produce ⁎
much higher magnetic induction than the TPHGMS. The disadvantage is that the magnets area easily corroded in weakly acidic slurry and need to be renew in about three years. Meanwhile, the transverse field pulsating HGMS separator solves this special case because its magnetic poles are not immersed in the slurry. Moreover, the transverse field pulsating HGMS separator has higher selectivity but lower magnetic induction, lower capacity and higher energy consumption. Table 1 gives separation index of LPHGMS and TPHGMS in Panzhihua Iron and Steel Co. Ltd. (Sichuan, China). Obviously, for ilmenite separation, under similar recovery, the LPHGMS has to yield to the TPHGMS in terms of concentrate grade, tails grade and separation efficiency only except ore throughput. So re-recognition of TPHGMS is essential at the theoretical level to find why ore throughput is difficult to increase. The theoretical analysis of a magnetic particle captured by the wire of an HGMS separator can be viewed as two separate, but related, problems. The first one is the motion of the particle in the separation
Corresponding authors at: Guangdong Institute of Resources Comprehensive Utilization, Guangzhou 510650, China (Y. Tang). E-mail addresses: [email protected] (Y. Tang), [email protected] (X. Zheng).
https://doi.org/10.1016/j.mineng.2019.106105 Received 30 April 2019; Received in revised form 19 October 2019; Accepted 25 October 2019 0892-6875/ © 2019 Elsevier Ltd. All rights reserved.
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Fig.1. Magnet of LPHGMS (up) and TPHGMS (down). Table 1 Separation results of LPHGMS and TPHGMS in Panzhihua Iron and Steel Co. Ltd. Types
Longitudinal field pulsating HGMS Transverse field pulsating HGMS
Ring diameter (mm)
Matrices volume (m3)
Applied magnetic induction (T)
Ore throughput (t/h)
Grade of ilmenite (TiO2%) Feed
Concentrate
tails
Recovery (%)
Separation efficiency (%)
2750
1.645
1.0
120
10.01
15.00
2.80
88.56
36.38
2750
1.602
0.45
70
9.89
17.54
2.65
86.23
46.31
particle buildup mechanism, the effect of hydrodynamic changes due to gap shrinkage of matrices on particle accumulation in a real HGMS is ignored. Moreover, pulsating slurry can inevitably exacerbate the effect on the behavior of accumulation. Thus, these models are unsuitable to predict the performance of a transverse field pulsating HGMS separator and provide precise theoretical guidance. The objective of this study is to extend SBM from single wires to double wires for the investigation of magnetic particle accumulation on the ferromagnetic rods of a TPHGMS separator and find the reasons why ore throughput is difficult to increase. In our model, we think friction between each buildup layer is nonzero and calculate the particle buildup profiles and weights under specific conditions to research the effect of pulsating slurry, wire spacing and applied magnetic induction on particle accumulation in TPHGMS separator. Then, to further confirm the accuracy of this double wires model, the capture experiments of the magnetic mineral monomers by the magnetized ferromagnetic collecting wire in the separator matrix of an TPHGMS unit are carried out to compare experimental results with the model calculated results.
region by analyzing its trajectory to determine under what conditions a particle will strike the wire. The second one is examination of whether the particle will stick to the wire and continue to buildup on the surface of the wire with other attracted particles, or the particle will be washed away by the fluid after the impact of a particle on the wire (Liu, 1983). The formula of motion trajectory of magnetic particle and the conception of magnetic velocity in a HGMS system were firstly introduced by Watson (1973,1975), and the dimensionless ratio of the magnetic velocity to the free stream fluid velocity was a dominant parameter for the particle capture. After that, Luborsky (1975), Lawson (1977), and Zheng (2018) extended the original particle trajectory model for the calculation of HGMS performance, to respectively include the buildup of multiple layers of particles on the wire, nonvanishing gravitational forces and a broad range of Stokes number, and the condition that the matrix is unsaturated. As of today, the application of magnetic separation is gradually broadened to sewage treatment, healthcare and solid waste recycling. The diversity of separator makes many researchers turn to computational simulations to study the trajectory (Belounis et al., 2015, 2016; Zhou et al., 2015; Wei et al., 2010; Haverkort et al., 2009; Misael et al., 2005). However the main factor for determining HGMS system throughput is still magnetic particle accumulation or buildup on a matrix. The static buildup model (SBM) and simulation of dynamic accumulation (SDA) of particles on single wires were developed to reveal the accumulation process (Luborsky and Drummond, 1976; Cowen et al., 1976; Nesset and Finch, 1980; Badescu et al., 1996; Hournkumnuard et al., 2011; Chen et al., 2012; Natthaphon et al., 2017). These researches on single wires focus on
2. Model 2.1. SBM on single wires in transverse-type HGMS The SBM successfully predicts the accumulation of particles on a single wire in a longitudinal-type HGMS (Nesset and Finch, 1981). The basic idea of SBM is the equilibrium of forces acting on a particle 2
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Fig. 2. Transverse HGMS configuration: (a) single wires; (b) double wires in pulsating slurry.
cylinder, these forces are split into radial and tangential components (Maass et al., 1983). Apparently, a magnetized ferromagnetic rod can be divided into attractive regions within which particles can be captured and repulsive regions within which particles cannot be captured. When the saturation buildup is attained, the sum of the radial forces and the sum of the tangential forces of the particles located in the boundaries equate to zero, as indicated in Eqs. (1) and (2).
Fmr + Fgr = 0
(1)
Fmθ + Fdθ + Fgθ = 0
(2)
The boundaries of the saturation buildup can be obtained by solving the above equations. Whether the force of static friction needs to be considered in the buildup process of particles or not is the debate. When this force is ignored in the SEM of a longitudinal-type HGMS, the boundary curve possesses satisfactory consistency with the saturated Mn2P2O7 buildup photographs taken from video recordings (Friedlaender et al.,1978; Nesset and Finch, 1981). In a transverse-type HGMS, given that all tangential components have the same sign and cannot provide the boundary curve, the force of static friction is worthy of being reconsidered, and the surface roughness of the HGMS-buildup has the same order of magnitude as the diameter of the particles. Thus, the fluctuations for the static friction coefficient between single particles and accumulation layer are a real existence. Because of the large number of particles at a saturation buildup, its magnitude and shape can be determined by averaging these fluctuating single forces (Maass et al., 1983). These forces in transverse-type SBM are presented in Fig. 3 and based on the above analysis, Eq. (2) is modified to Eq. (3).
Fig. 3. Force components acting on the particle of retention on the matrix.
Fmθ + Fdθ + Fgθ + Ff = 0
(3)
These components in Eqs. (1) and (3) were reasonably derived and modified by Watson (1973), Nesset and Finch (1980), Glew (1982), and Zheng et al. (2015, 2017a, 2018). The magnetic force solved by using polar coordinate function of Laplace equation is shown in Eqs. (4) and (5) when the matrices are assumed to be unsaturated (Zheng et al., 2017a, 2018). Fig. 4. Velocity curve of pulsating HGMS separator.
located on the surface of the buildup. In a transverse HGMS configuration shown in Fig. 2(a), the direction of the slurry, magnetic field, and cylindrical matrices’ axis are mutually perpendicular. Magnetic particles are adsorbed and accumulate on the sides of the cylindrical matrices determined by the forces acting on the particles, including the magnetic force Fm, fluid drag force Fd, and gravity force Fg. When the particles are spherical and the buildup takes the shape of a circular
Fmr = −
8πb3μ0 H02 d 2 d2 κ e (cos 2θ + 2 ) 3 3r r
(4)
Fmθ = −
8πb3μ0 H02 d 2 κ e sin 2θ 3r 3
(5)
where b is the radius of the particle, μ0 is vacuum permeability, H0 is the applied magnetic field, d is the radius of the matrix cross section, r is the distance between the particle and the matrix axis, and κ e is equivalent magnetic susceptibility of the particle around the matrix given by Eq. (6) (Nesset and Finch, 1980) 3
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Table 2 The practical values of the operating parameters. Parameter
Value
Parameter
Value
The Magnetic field induction B0 The applied magnetic field strength H0
0.2–0.8 T B0/ μ 0 1.55 T
Inlet velocity of the fluid v0 Kinematic viscosity of the fluid υ
0.05 m/s 1 mm2/s
Density of the fluid ρf
1 g/cm3
1.5 mm 4.5–10.5 mm 30 μm 0.0125–0.0049
Pulsating slurry frequency ω Permeability of vacuum μ 0 Gravitational acceleration g Static friction coefficient f Correction factor α
120–420 r/min 4π × 10−7 H/m 9.8 N/kg 1.2 0.8–1
The saturation magnetic of the matrix MS Radius of the matrix d Wire spacing D Radius of the magnetic particle b Equivalent magnetic susceptibility of the magnetic particle κ e Volume of the magnetic particle V
4 πb3 3
4650 kg/m3
Density of the magnetic particle ρp
4 Fgθ = − πb3 (ρp − ρf ) g cos θ 3
(9)
Where ρp is the particles density, andg is the gravitational acceleration. The force of static friction is given by Eq. (10).
Ff = f (Fmr + Fgr )
(10)
where f is the static friction coefficient. The buildup of particles on matrix rods is a fairly complex process and difficult to observe. The final boundary contour of buildup of particles was obtained by assuming that the particles show compact and layer-by-layer buildup in each layer r = b + d + (n − 1) 3 b (n = 1, 2, 3…). The θ satisfying Eqs. (1) and (3) is calculated for each layer and the smaller of the two angles θc is selected. Therefore, θc as the function of r gives the saturated buildup profile. 2.2. SBM on double wires in transverse-type pulsating HGMS We propose a extended SBM to include the double wires system in pulsating slurry depicted in Fig. 2(b), which is more accurate to predict the performance of a transverse field pulsating HGMS separator. Pulsating slurry can overcome matrix blockage and upgrade the selectivity of HGMS. The pulsating mechanism drives the slurry in the separating zone up and down, clearing the slurry channels and keeping a smooth traffic. This movement affects the accumulation of magnetic particles. In a double wire system in a pulsating slurry, fluid drag force changes with up-and-down pulsating slurry. The velocity curve of flow is illustrated in Fig. 4. Pulsating velocity, maximum pulsating velocity, and average pulsating velocity are respectively given by Eqs. (11)–(13) (Xiong, et al., 1998).
Fig. 5. Buildup profile of the ilmenite sample on the matrices: Magnetic induction B0 = 0.45 T; feed velocity v0 = 0.05 m/s; pulsating stroke S = 12 mm; pulsating frequency ω = 300 r/min; wire spacing D = 6.8 mm.
M0
κ e = κ∞ + H0 1+2
d2 r2
cos 2θ +
d4
≅ κ∞ +
M0 2.4H0
r4
(6)
The fluid drag force, only a force in the tangential direction, is calculated using the shear stressτ0 at the bottom of a boundary layer, which is accurately determined by the Blasius solution to the boundary layer equations. Consequently, the fluid drag force is given via Eq. (7) (Schlichting, 1968).
(12)
1 2π
∫0
2π
1 Sω Sω sin(ωt ) d (ωt ) = 2 π
v = v0 + v~max sin(ωt )
1
(13)
(14)
Simplification can be done so that the boundaries of buildup depend on the maximum velocity of alternating slurry. Thus, Eq. (13) can be rewritten as
(7)
where ρf , V0 , υare the fluid density, velocity (far from wire), and kinematic viscosity, respectively. The gravity force is given by Eqs. (8) and (9).
4 Fgr = − πb3 (ρp − ρf ) g sin θ 3
1 v~max = Sω 2
where S , ω , and t are the pulsating stroke, pulsating angle speed, and time variable, respectively. The velocity of the slurry is the sum of feeding velocity (v0 ) and pulsating velocity, shown in Eq. (14).
3 υ 2 π 2b2 ρf V02 ⎛ ⎞ (6.973θ − 2.732θ3 + 0.292θ5 − 0.0183θ7 4 ⎝r ⎠
+ 0.000043θ9 − 0.000115θ11)
(11)
v¯ =
Fdθ =
1 v~ = Sω sin(ωt ) 2
1
V=
⎧ 2 Sω + v0
(downflow )
⎨ 1 Sω ⎩2
(upflow )
− v0
(15)
In the gap between the matrices, presented in Fig. 2(b), the velocity of the slurry can be expressed by Eq. (16).
(8) 4
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Fig. 6. Particle buildup of different pulsating frequencies on double matrices in pulsating slurry: (a) buildup profile; (b) cumulative radius and weight; magnetic induction B0 = 0.45 T; feed velocity v0 = 0.05 m/s; pulsating stroke S = 12 mm; wire spacing D = 6.8 mm.
( V′ = ⎨ ( ⎩ ⎧
0
)( Sω − v )(
1 Sω 2 1 2
+ v0
D D − 2r cos θ
0
D D − 2r cos θ
) )
layer is completed when (θiup – θidown) < 100. Here, α is the correction factor. Given the possibility of noncompact accumulation between particles and the slurry is dragging the whole buildup region away from the surface of matrices, correction factor α is necessary for the calculated results. Through comprehensive consideration, the value of α is 0.8–1 related to the flow rate of slurry. Of course, an explanation is owed that, in longitudinal-type HGMS, α is not considered because the slurry is compacting the whole region of buildup close to the matrices’ surface.
(downflow ) (upflow )
(16)
where D is the center distance of adjacent wires. The fluid drag force of double wires in pulsating slurry can be solved through Eq. (3) by substituting V0 with V0′. Here we approximately treat V0′ as the free stream fluid velocity. That the pulsating pulp continues to maintain a laminar flow is assumed. The consistency of model calculation and experimental data is evaluated on the basis of the total weight of particle accumulation on the matrices. The calculated profile of saturation buildup can be converted to the weight via Eq. (17).
3. Calculated content and parameters We have already known the effect of these parameters such as magnetic induction, fluid velocity, matrices size on particle accumulation based on monofilament research. However, in double wires system, besides these we mainly focus on the particle buildup profiles and weights under different pulsating slurry and wire spacing because they can cause fluid changes. The magnetic particle is endued with these properties such as
m
(θ up − θidown ) r 4 MT = απNLρp b2 ∑ i 3 2b i=1
(17)
where N, L, and m are the total number of matrices, effective length of matrices, and the serial number of the final layer, respectively; θiup and θidown are the up θc and down θc of layer i. We assume that the final 5
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Fig. 7. Particle buildup of different wire spacing on double matrices in pulsating slurry: (a) buildup profile; (b) cumulative radius and weight; magnetic induction B0 = 0.45 T; feed velocity v0 = 0.05 m/s; pulsating frequency ω = 300 r/min; pulsating stroke S = 12 mm.
4. Calculated results
magnetism, size, and density of the ilmenite sample in Section 5. The matrix is the SUS430 wires with saturation magnetization Ms = 1.55 T (Xu, 1995) and radius d = 1.5 mm. For ilmenite capture, the applied magnetic induction B0 = 0.2 T, 0.3 T, 0.4 T, 0.5 T, 0.6 and 0.8 T, wire spacing D = 3d, 4d, 5d, 6d and 7d, pulsating slurry frequency ω = 120 r/min, 180 r/min, 240 r/min, 300 r/min and 420 r/min are respectively considered. The mean of maximum static friction coefficient is assumed to be 1.2, referring to the paper of Maass et al. (1983) and experimental particle buildup. The various parameters are detailedly listed in Table 2. Mathematical software (Matlab) are used to layer-by-layer solve the Eqs. (1) and (3) and get the values of θc. These coordinate points (r, θc) form saturation boundaries of the paricle buildup on the wires.
4.1. Pulsating slurry Fig. 5 reveals the saturated particle buildup on single wires and dual-wire pulsation when the applied magnetic induction is 0.45 T. The pulsating slurry shockingly affects the profile of particle buildup and causes sharp decrease of particle buildup with the consequent impact. Furthermore, compared with single wires, decrease of particle buildup on double wires obviously becomes more serious. Here is the most intuitive explanation why SBM-dw is supposed to be more suitable to reveal the magnetic particle accumulation on matrices in a TPHGMS. The effect of pulsating frequency on ilmenite accumulation in double wires system is illustrated in Fig. 6. The buildup profile of ilmenite, depicted in Fig. 6(a), decreases successively with the increase of the pulsating frequency because increasing the pulsating frequency must result in the increase of the fluid drag force and decrease the 6
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wire spacing, depicted in Fig. 7(b). However, in a TPHGMS unit, the separation space is filled with a large number of matrices. With the increase of wire spacing, there is usually a corresponding drop in the packed fraction of the matrices and consequently it is necessary to take a trade-off. The red dotted square in Fig. 8(a) is a representative element of the cross section of the separation space (Zheng et al., 2017b). The matrices packing fraction ξ can be calculated by using Eq. (18):
ξ=
πd 2 D2
(18)
Then we use the value of multiplying the buildup weight of each wire (MT) by the matrices packing fraction (ξ) to represent the element particle accumulation capacity of a TPHGMS unit. Fig. 8(b) shows the element particle accumulation capacity of a TPHGMS unit under different wire spacing. There is a certain wire spacing (D = 5.73 mm) at which particle accumulation capacity of a TPHGMS unit reaches maximum. 4.3. Applied magnetic induction
Fig. 8a. Arrangement of matrices.
The magnetic field is a decisive operation parameter in high gradient magnetic separation. The ability of paramagnetic particle capture can be effectively improved by increasing magnetic field strength. Fig. 9 depicts the effect of magnetic induction on ilmenite accumulation on the double wires in pulsating slurry. When magnetic induction is increased from 0.2 T to 0.8 T, the buildup profile enlarges continuously, but the span between adjacent buildup boundaries gradually decreases in a trend similar to wire spacing, especially after 0.4 T, as shown in Fig. 9(a). What’s more, the variation of buildup radius or weight in Fig. 9(b) displays the same information. Consequently, as the magnetic induction continues to increase, it can be inferred that, the profile eventually stops changing. This trend is essential to magnet selection, as the most suitable magnetic field for treating a certain mineral must be determined in the design period, particularly for a permanent magnet high gradient magnetic separator. Increasing the magnetic field strength on a small scale can result in the increase of the matrices’ surface area available for the buildup of the particles, shown in Fig. 9(a), which is absolutely contrary to longitudinal-type HGMS (Svoboda, 1994). The force of static friction is the main factor causing the difference. Normally, the magnetic force acting on particles is an increasing function of the magnetic field strength. The force of friction at the opposite direction of the tangential magnetic force increases with the radial magnetic force and enables the matrices’ surface area covered with particles to increase slowly. However, in longitudinal-type HGMS, the matrices’ surface area available for the buildup of the particles is determined by the perturbation term (A = 2πMw/H0 in CGS) (Nesset and Finch, 1980) for the cylinder matrix. When A decreases with the increase of magnetic field strength, the buildup area located on matrices’ surface decreases with the decrease of A. The difference of the two types of HGMS needs to be verified by visual test in the next paper. A new idea can be tried to prove the existence of friction by observing the real variety of buildup profile with the increase of magnetic field strength.
Fig. 8b. The element particle accumulation weight of a TPHGMS unit.
buildup profile. However, from the tendencies of cumulative radius and weight in Fig. 6(b), we can find the magnitude of the decrease tends to be gentle. The magnetic force acting on the particles located on the surface of the buildup rapidly increases with the decrease of accumulation radius and the particle is difficult to be dragged out when the pulsation frequency increases to a critical value. In addition, the matrices’ surface available for the buildup shows a small contraction given the increase of fluid drag force. The slurry settles the matrix blockage but affects the process of particle buildup especially for transverse-type HGMS. This effect clearly explains why the transverse field pulsating HGMS separator has low processing capability.
4.2. Wire spacing 5. Confirmed experiments The wire spacing is a parameter worth considering in a TPHGMS unit, but it has been ignored. Up to date, none has investigated whether it has a significant effect on the particle buildup. Here is the first discussion for it from theoretical level. Fig. 7(a) provides particle buildup boundaries, under the different wire spacing, distinguished by colorful curves. It can be seen that the wire spacing significantly affects the process of particle buildup. As the spacing is enlarged, for the crosssectional area of particle buildup, there is a quantitative increase, yet the upper limit of the area is the same as that of single wires accumulation. So we can increase the amount of particle buildup by increasing
5.1. Sample Ilmenite is prepared by the flowsheet depicted in Fig. 10(a) and concentrate with a particle size of −0.075 + 0.045 mm is selected as the experimental sample. The result of X-ray diffraction, revealed in Fig. 10(b) indicates that the purity of the experimental sample is more than 90%. The density of the experimental sample is 4650 kg/m3. In the mineral processing industry, the magnetic susceptibility of numerous minerals treated with HGMS is not constant and can be affected by the 7
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Fig. 9. Particle buildup profile of different magnetic induction on double matrices in pulsating slurry: (a) buildup profile; (b) cumulative radius and weight; feed velocity v0 = 0.05 m/s; pulsating frequency ω = 240 r/min; pulsating stroke S = 12 mm; wire spacing D = 6.8 mm.
applied magnetic field. Thus, the equivalent susceptibilitiesκ e of the experimental sample are measured by a vibrating sample magnetometer. When the applied magnetic field of 0.2 T, 0.3 T, 0.4 T, 0.5 T, and 0.6 T is considered, correspondingκ e are approximately 0.0125, 0.0092, 0.0076, 0.0065, and 0.0058 (SI), respectively.
5.3. Methods The sample and water were fully mixed in a stirred beaker and evenly fed to the separator within 30 s. The feed velocity, feed percentage of solids, feed weight, and pulsating stroke are 0.05 m/s, 10%, 400 g, and 12 mm, respectively, which are determined in advance as constants during this experiment. Optimizing tests of feed weight demonstrate that the weight of buildup on the matrix stops changing when the 400 g sample is fed. At this weight, we can approximately assume the accumulation reaches saturation. Magnetic induction of 0.2 T, 0.3 T, 0.4 T, 0.5 T, and 0.6 T and pulsating frequency of 180, 240, and 300 r/min are investigated as variables. When each experiment with the operating variables of the pilot separator is finished, the current of the energizing coils is turned off, and the magnetic particles captured on matrices are washed out with clean water, dried in an oven, weighed with an electronic scale, and analyzed.
5.2. HGMS separator The cyclic transverse field pulsating HGMS separator used for the experiment, illustrated in Fig. 11(a), is designed and manufactured by Guangzhou Research Institute of Non-ferrous Metals (China). The magnetic field between the magnetic poles is oriented perpendicular to the direction of the slurry. The particles that accumulate on both sides of the matrices are washed by pulsating slurry and obtain a higher grade than the longitudinal field pulsating HGMS separator but lower recoveries for finer magnetic particles. The unit of the matrix used in the experiment is designed as shown in Fig. 11(b). The effective length of a single matrix is 135 mm and per unit consists of 18 stainless steel rods with a diameter of 3 mm. A total of 5 units (total of 90 rods) are stacked in the region of the magnetic field to conduct the experiment.
5.4. Experiment results The SBM on double wires in TPHGMS seems to be more effective than SBM on single wires in transverse-type HGMS for practical 8
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Fig. 11a. Cyclic transverse field pulsating HGMS separator. Fig. 10a. Principle flowsheet of ilmenite sample preparation.
Fig. 11b. Designed matrices unit.
flow to turbulent flow as presented in the inner illustration of Fig. 12(a). Fig. 12(b) shows the calculated and experimental weights of buildup when the pulsating frequencies are 180 r/min, 240 r/min, 300 r/min, and 420 r/min. The pulsating frequency are usually used to improve the performance of HGMS. The weights also show that the model can predict the results of the pulsating frequency experiment. The decrease of theoretical and the experimental weights are simultaneous with the increase of the pulsation frequency. However, the trend of the two curves exist certain slight differences because the level of non-compact buildup show the change under different pulsating frequencies. It is consistent with the above paragraph analyzes.
Fig. 10b. XRD spectrum of prepared ilmenite sample.
instruction. A series of experiments are carried out to preliminarily assess the accuracy of the model. Fig. 12 is the comparison of calculations with experimental results. Fig. 12(a) reveals the calculated and experimental weights of buildup when magnetic induction is increased from 0 T to 0.6 T. In general, the experimental weights of buildup are in good agreement with the model. The growth trend of experiment results agrees well with calculation results, the disparity is less than 5% when the correction factor is 0.85 and magnetic induction is below 0.4 T. However, within the range of 0.4–0.6 T, the disparity is unexpectedly enlarged. The increase of magnetic field strength will lead to the shrinkage of the gap between the two matrices because of the increase of particle buildup. This will further cause the flow rate increase and influence the level of noncompact accumulation. Another reason possibly responsible for the change of disparity is that Reynolds number of flow located in the gap between two matrices is also continuously increasing from 1160 to 2440, surpassing the critical Reynolds number 2300 at 0.6 T (Schlichting, 1968) and resulting in the transformation from laminar
6. Conclusion The TPHGMS, characterized by high separation index and low capacity, is designed to circumvent the gradual corrosion of the magnets of LPHGMS in weakly acidic slurry. Re-recognition of the performance from a theoretical level is essential. The pulsating slurry directly impinges the regions of paramagnetic capture and leads to an astonishingly negative impact, being intuitional shown by the calculated buildup profiles of the model on the single wires in the transverse-type HGMS and the model on the double wires in the transverse-type pulsating HGMS. Comparing the two models, SBM of particle accumulation on double wires in transverse-type 9
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Acknowledgement We sincerely acknowledge the fundings to this research work from the Key Program for Applied Basic Research of Guangzhou (Grant No. 201904020021), Guangdong Academy of Sciences for Innovation Capacity Building (Grant No. 2018GDASCX-0109), the National Natural Science Funds of China (Grant No. 51804341), and Natural Science Foundation of Hunan Province (Grant No. 2019JJ50833). References Badescu, V., Murariu, V., Rotariu, O., Rezlescu, N., 1996. A new modeling of the initial buildup evolution on a wire in an axial HGMF filter. J. Magn. Magn. Mater. 163 (1–2), 225–231. Bean, C.P., 1971. Theory of magnetic filtration. Bull. Am. Phys. Soc. 16 (3), 350. Belounis, A., Mehasni, R., Ouil, M., Feliachi, M., Latreche, E.H., 2015. Design with optimization of a magnetic separator for turbulent flowing liquid purifying applications. IEEE Trans. Magn. 51 (8), 1–8. Belounis, A., Mehasni, R., Ouili, M., Feliachi, M., Latreche, E.H., Razek, A., 2016. Optimization of a dual capture element magnetic separator for the purification of high velocity water flow. EU. Phy. J. Appl. Phy. 73 (2), 20601. Cowen, C., Friedlaender, F.J., Jaluria, R., 1976. Single wire of high gradient magnetic separation processes Ⅰ. IEEE Trans. Magn. 12 (5), 466–470. Chen, F., Smith, K.A., Hatton, T.A., 2012. A dynamic buildup growth model for magnetic particle accumulation on single wires in high-gradient magnetic separation. AIChE J. 58 (9), 2865–2874. Chen, L., 2011. Effect of magnetic field orientation on high gradient magnetic separation performance. Miner. Eng. 24 (1), 88–90. Chen, L., Liu, W., Zeng, J., Ren, P., 2017. Quantitative investigation on magnetic capture of single wires in pulsating HGMS. Powder Technol. 313, 54–59. Friedlaender, F.J., Takayasu, M., John, B.R., Czeslaw, P.K., 1978. Particle flow and collection process in single wire HGMS studies. IEEE Trans. Magn. 14 (6), 1158–1164. Glew, J.P., Wong, M.K., Parker, M.R., Birss, R.R., 1982. 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Fig. 12. Comparison of calculations to experimental results: (a) Magnetic induction; (b) pulsating frequency.
pulsating HGMS is more suitably used to accurately estimate the performance of the TPHGMS. Furthermore, the results of confirmed experiments indicate good agreement between experimental values and the calculated accumulation weights in double wires system. But with the flow turning into turbulence, a slight deviation will exist. The calculated results of double wires model in terms of pulsating slurry, wire spacing and applied magnetic induction give designer the direction of increasing ore throughput. Enlarging wire spacing or decreasing pulsating frequency can weaken the impact of slurry and increase particle buildup. In addition, increasing applied magnetic induction within a certain range is also a significant operation to increase particle buildup. However, in practice, the TPHGMS is difficult to increase magnetic induction unless the magnet spacing is reduced and as a result less amount of matrix can be filled. Besides, larger wire spacing also result in less amount of matrix, and very low pulsating frequency may cause matrix blockage. So it is essential to weigh and consider these parameters to optimize processing capacity. At last, the calculated buildup profile should be compared with the observed contour in the next paper to make further confirmation.
Declaration of Competing Interest We declare that we have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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