Jan Svoboda. Magnetic Techniques for the Treatment of Materials

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190 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

Up st r e a m

build-up

Downstream

build-up

Figure 3.18: Build-up of paramagnetic particles on a magnetized wire in the

longitudinal conﬁguration.

Particle build-up

Previous analyses of particle trajectories in the vicinity of a magnetized wire

considered the capture process in the initial stage, i.e. with no loading of the

wire with the captured particles. The determination of the exact process of the

particle build-up is a complicated problem. As the suspension ﬂows through the

matrix, some of the particles, under the inﬂuence of several forces, are deposited

onto the matrix elements or onto already deposited particles. Consequently the

geometry, the structure and the surface characteristics of the matrix may be

signiﬁcantly modiﬁed. These changes, in turn, aect the ﬂow of the suspension

through the matrix and the subsequent capture. Further complications may

also arise as a result of dislodgment of the captured particles and their possible

re-capture. As a result, the dynamic behaviour of the separation process varies

with the time as well as with other variables.

The force that is principally responsible for the particle capture is the mag-

netic force. Particles are attracted towards the wire if the radial component of

the magnetic force I

? 0= Thus the condition I

=0deﬁnes the critical

angle 

given by eq. (3.66). The stability of a particle on the wire surface

is determined by the balance of the tangential components mainly of the mag-

netic and hydrodynamic forces. Gerber found that the angle 

,forwhicha

paramagnetic particle will be retained on the wire, is given as [G1]:

| 

|6 arcsin

1  (

)

1@2

(3.92)

3.4. HIGH-GRADIENT MAGNETIC SEPARATION 191

Figure 3.19: Build-up of paramagnetic particles on a magnetized wire in the

transverse conﬁguration.

It transpires from eq. (3.92) that retention will occur only if y

A 1 [W8,

W9]. However, the experimental results show that a much less stringent condi-

tion can still lead to a reasonably high degree of capture.

Examples of the build-up of paramagnetic particles on a magnetized wire

in three principal conﬁgurations are shown in Figs. 3.18, 3.19 and 3.20. Dia-

magnetic particles are retained in the complementary regions as follows from

Fig. 3.16. The experimentally determined build-up of paramagnetic kimberlite

particles in the transverse conﬁguration is shown in Fig. 3.21.

Essentially all theoretical descriptions of particle capture by a magnetized

wire assumed the idealized situation in which the collection wire axis was ori-

ented orthogonally to the magnetic ﬁeld direction. This assumption resulted

from the belief that only these conﬁgurations would create sizable deposits.

Birss et al. [B14, S26] considered the collection process on a cylinder with its

axis oriented at an angle with the magnetic ﬁeld direction. They found that

the eect of the non-orthogonal wire orientation can be taken into account by

replacing the coe!cients y

and N by more general terms. These terms are

the functions of the oset angle , deﬁned in Fig. 3.22.

Experimental observation by Svoboda et al. [S27], of particle collection at

various oset angles , demonstrated that even at the oset angle as large as

there is considerable collection of particles, as shown in Fig. 3.23. The

length of the section of the wire where the diamagnetic capture takes place is a

function of the oset angle. For the oset angle equal to @2 (the magnetic ﬁeld

parallel with the wire axis) the paramagnetic capture takes place only at the

end of the wire while the entire length of the wire is repulsive for paramagnetic

particles, as shown in Fig. 3.24.

192 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

Figure 3.20: Build-up of paramagnetic particles on a magnetized wire in the

axial conﬁguration.

Figure 3.21: Build-up of paramagnetic particles of kimberlite (mean size 4.7 m,

magnetic susceptibility 1.7×10

6

/kg) on a magnetized nickel

wire in transverse conﬁguration (E

= 1.7 T, y

= 0.25 cm/s)

[S27].

3.4. HIGH-GRADIENT MAGNETIC SEPARATION 193

Figure 3.22: The plane view of the generalized transverse orientation with the

magnetic ﬁeld making an angle (@2  ) with the wire axis.

Figure 3.23: Collection of kimberlite particles on a nickel wire (diameter 125 m)

in the transverse conﬁguration, with the magnetic ﬁeld making an

angle of 10

with the wire axis (oset angle  =80

), (B

=1.7

T, v

= 0.25 cm/s) [S27].

194 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

Figure 3.24: Collection of paramagnetic kimberlite particles on a nickel wire

magnetized along its axis (oset angle  =90

) [S27].

Nesset and Finch [N2, N5] developed a static model of particle capture in

HGMS, in which the ﬂuid drag force included the changing boundary layer

thickness over the front portion of the wire. They assumed that for a particle

to remain attached to the wire, two conditions would have to be met. First, the

net radial force acting on a particle must be attractive, that is

+ I

6 0 (3.93)

where I

and I

are given by eq. (3.62) and (3.74), respectively.

The second condition is that the net tangential force must push particles

towards the region of a greater attractive radial force. In the case of the longi-

tudinal paramagnetic build-up, this condition can be expressed as:

p

+ I

j

+ I

g

6 0 (3.94)

where I

g

is the ﬂuid drag force, which is proportional to the shear stress 

the boundary layer. Thus [N2]

g





(3.95)

By solving equations (3.93) and (3.94), the outline of the build-up proﬁle can

be found computationally as intersecting loci of points {u> } satisfying these

equations. The two retention criteria are satisﬁed only in the cross-hatched

region shown in Fig. 3.25.

3.4. HIGH-GRADIENT MAGNETIC SEPARATION 195

Figure 3.25: Loci of zero radial and azimuthal forces around the wire in the

longitudinal orientation (adapted from [N5]).

Capture of diamagnetic particles

It transpires from eq. (1.8) that if the susceptibility dierence 

 

is pos-

itive, then the magnetic force between the wire and the particle is attractive,

provided that the radial component I

is negative. This is the case of capture

of paramagnetic particles, under the condition that eq. (3.65) is satisﬁed.

On the other hand, there are two symmetrical curved volumes, the bound-

aries of which are determined by the critical angle 

, given by eq. (3.66),

in which the radial force acting on paramagnetic particles is repulsive, while

the radial force acting on diamagnetic particles is attractive. These regions, in

which collection of diamagnetic particles takes place, are complementary to the

regions of collection of paramagnetic particles, as illustrated in Figs. 3.18, 3.19

and 3.20.

The value of 

depends on both P

(or M

)andK

(or E

)asshownin

Fig. 3.26. It can be seen that as the strength of the external magnetic ﬁeld

increases, the area of the wire collector available for the capture of diamagnetic

particle also increases. The eective area for the capture of paramagnetic par-

ticles is, therefore, correspondingly reduced. This phenomenon indicates that

an increase in the operating magnetic ﬁeld can negatively aect the loading

capacity of the matrix, the selectivity of separation and the overall performance

of a magnetic separator. The ﬁrst experimental conﬁrmation of the diamagnetic

capture was reported by Friedlaender et al. [F9]. They showed that by dissolv-

ing a paramagnetic salt in water, 

canbemadepositiveandsu!ciently large,

so that  =| 

 

| is large and the attractive magnetic force is su!ciently

strong to recover particles of aluminium oxide. A theory of diamagnetic capture

was developed and published by Birss and Parker [B15], simultaneously with

196 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

Figure 3.26: The critical angle 

as a function of the magnetic induction (for

= 1.7 T) [S1].

Friedlaender’s experimental paper [F9].

Although the recovery of diamagnetic particles by magnetic separation is

feasible in principle, simultaneous capture of paramagnetic particles prevents a

magnetic separator from being a selective device for separation of diamagnetic

materials. This limitation can be, however, overcome by magnetizing the wire

collectors along their long axes (the oset angle  = @2). At this orientation,

the paramagnetic particles are captured only at the end tip of the wire, as is

shown in Fig. 3.24. The entire length of the wire is then attractive for diamag-

netic particles. Collection of diamagnetic bismuth particles in this conﬁguration

was observed by Svoboda et al. [S27] and is shown in Fig. 3.27.

Magnetic separation at moderate Reynolds numbers

Experimental investigation by Friedlaender et al. into the particle capture on a

single wire in the longitudinal conﬁguration indicated [F10] that for moderate

values of Reynolds number for the wire (Re 6 to 40), the material can be cap-

tured at the rear of the wire. Theoretical treatment by Watson [W10] showed

that at higher velocities of the suspension a pair of stable standing vortices,

which appear on the downstream side of a wire, can induce capture of particles

on the trailing side of a matrix collector. This situation is illustrated in Fig.

3.28. The shaded areas on the upstream and downstream sides of the wire are

attractive for paramagnetic particles. The unshaded portions around the | axis

are repulsive, and particles arriving from the front part of the wire towards the

wake must, therefore, pass through the repulsive regions.

It was shown in subsequent studies that the magnitude of the Reynolds num-

3.4. HIGH-GRADIENT MAGNETIC SEPARATION 197

Figure 3.27: Collection of diamagnetic particles of bismuth (" = - 16.3×10

9

/kg) on the upstream and downstream sides of the nickel wire

magnetized parallel to its axis ( E

=1.9T,v

= 0.13 cm/s) [S27].

Figure 3.28: The downstream capture into a stationary wake formed by twin

vortices.

198 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

ber is not the only condition for the vortex capture [W11]. The thickness of the

boundary layer, the particle size and the ﬁeld strength were found to be the

main factors aecting the vortex capture. In particular it was determined that,

for vortex capture to occur, it is necessary that y

? 1. The separation

process, taking place under such conditions, for which the downstream particle

capture takes place, was termed vortex magnetic separation by Watson [W11].

The applicability of this phenomenon, investigated mainly in single-wire exper-

iments, to real operational situations, has been a subject of controversy [S28,

W12].

It was argued by Watson et al. [L8] that if the conditions for the downstream

capture are met, the recovery, the grade of the magnetic concentrate and the

throughput would be very high. However, it was shown by Svoboda [S28] that

such conditions can be possibly met only when very ﬁne (e.g. 125 m) steel

wool matrix is used and when the ﬂow rate of the slurry through the matrix

is close to the upper limit possible in such a matrix [S29]. Such conditions are

usually used to remove ﬁne weakly magnetic impurities from industrial minerals,

such as kaolin. In those applications, however, the magnetic fraction is a waste

and its grade is of no relevance to the e!ciency of the process. On the other

hand, in those applications where the magnetic fraction is the useful product,

such as beneﬁciation of ferrous and non-ferrous ores, much coarser matrix (e.g.

expanded metal or woven mesh) and much higher slurry velocities must be used

in order to maintain meaningful economically viable throughput. Under these

circumstances the conditions for the vortex formation are not met since the

Reynolds number is at least an order of magnitude greater than the upper value

required for downstream capture to take place.

3.4.2 Capture of particles in multi-collector matrices

The matrix of a real magnetic separator consists of a large number of individual

collectors. There are essentially two possible ways of developing a theory of

multi-collector magnetic ﬁlters: one way is to build the theory from ﬁrst prin-

ciples and the other is to incorporate the single-collector model into a multi-

collector phenomenological description.

In the ﬁrst approach, the particle trajectories and the capture e!ciencies

were investigated for the case of wire collectors arranged in regular arrays. Con-

ﬁgurations of single and double layers of wires were analyzed and reported in

numerous publications [H14, T4, E1, S30, G7, R12, R13]. As a result of consid-

erable mathematical complexity of the problem the applicability of the theory

to real matrix is limited.

Particle capture by ferromagnetic spheres. In spite of high e!ciency of

a ball matrix in some industrial applications of magnetic separation [S1], only

a limited attention has been paid to the theoretical analysis of particle capture

by spherical collectors. The character of the particle capture and of the build-

up is inevitably dierent for spheres compared to the wires. The dierence

follows from the fact that the cross-sectional capture area of a sphere increases

3.4. HIGH-GRADIENT MAGNETIC SEPARATION 199

Figure 3.29: The e!ciency of magnetic ﬁltration of Fe

particles (diameter

0.08 m) in a bed of steel spheres, as a function of the applied

magnetic ﬁeld (adapted from [A16]).

quadratically with the capture radius U

, whereas that of a wire increases

only linearly. Friedlaender et al. [F11, F12] solved numerically the equation

of motion (eq. 3.58) in transverse conﬁguration and found that the capture

of paramagnetic particles occurred in conically shaped surfaces deﬁned by the

critical angle

| 

|= arcsin

;

1+2N

(

)

1+N

(

)

(3.96)

which is equivalent to eq. (3.66) derived for a cylindrical wire. The parameter

given by

(3.97)

is equivalent to eq. (3.61) for the wire. Capture of diamagnetic particles then

takes place on conical surfaces complementary to those for paramagnetic cap-

ture.

It transpired from theoretical analysis of particle capture in a bed of spheres

by Watson and Watson [W13] that the probability of particle capture increased

with increasing background magnetic ﬁeld until the ﬁeld was high enough to

saturate magnetically the ferromagnetic spheres.

However, Moyer et al. [M15, M16] argued that the observed saturation

of the capture probability occurred at the magnetic ﬁeld strengths far below

those required to saturate magnetically the spheres comprising the matrix bed.