Jan Svoboda. Magnetic Techniques for the Treatment of Materials

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200 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

Figure 3.30: Schematic view of the normalized build-up Q@Q

in a ﬁlter, as a

function of the ﬁlter length.

This theoretical prediction was conﬁrmed experimentally by Arajs et al. [A16].

As Fig. 3.29 illustrates, the ﬁltration e!ciency of submicrometer particles of

Fe

in a bed of steel spheres levelled o at the magnetic induction of about

0.1 T, far below the saturation polarization of 1 T of the sphere material. Such

a behaviour was not observed for an isolated collecting sphere [F12]. This is a

very signiﬁcant observation, since it indicates that the mechanisms of particle

capture on a single collector and on an assembly of collectors in a packed bed

are governed by dierent laws. It becomes clear again that it is not permissible

to extend single-collector theoretical models (eqs. (3.83) and (3.84)) to multi-

collector assemblies of matrix collectors.

Phenomenological model of HGMS

The alternative approach to multi-collector magnetic separation is a phenom-

enological model [W14, G1], which is less rigorous but at the same time less

complicated and closer to practical needs of high-gradient magnetic ﬁltration.

It cannot, however, be easily applied to the process of separation of multicom-

ponent mixtures of materials.

The fundamental equations describing the particle retention in a ﬁlter and

the time-dependent performance of a ﬁlter are the macroscopic conservation

equations and the rate equation. Let us consider particles travelling through

an absorption ﬁlter of length O, as illustrated in Fig. 3.30. By equating the

dierence between the numbers of particles entering and leaving the volume

{ per unit time, to the time rate of increase of the number of particles, the

conservation equation is obtained as

CQ({> )

C

= y

Cf({> )

(3.98)

3.4. HIGH-GRADIENT MAGNETIC SEPARATION 201

out

Figure 3.31: A breakthrough curve of a ﬁlter with a constant absorption coe!-

cient .AttimeW the ﬁlter becomes fully saturated.

where Q({> ) is the number of captured particles in the ﬁlter between { and

{ + {,attime= f({> ) is the number of suspended particles in a volume of

the ﬂuid {,  is the porosity of the ﬁlter matrix, y

the superﬁcial velocity

and  the displacement time deﬁned as  = w  {@y

The rate equation can be expressed in the form of the Iwasaki relation [R14]

(

)

= f (3.99)

where  is the ﬁlter coe!cient having the dimension of the reciprocal length.

Equation 3.99 shows that the proﬁle of the particulate concentrate in the liquid

phase throughout a ﬁlter is logarithmic.

The solution of the rate equation can thus be written as

f({)

= exp({) (3.100)

rxw

= exp(O) (3.101)

where f

and f

rxw

are the inﬂuent and e"uent concentrations.

Since nothing is known, in the phenomenological approach, about the phys-

ical processes taking place inside the ﬁlter, all information about the ﬁlter per-

formance must be derived from its breakthrough curve, i.e. from the output

concentration as a function of time, as shown in Fig. 3.31.

By incorporating the single-wire magnetic separation model into the above

eqs. (3.100) and (3.101) Watson found [W8], the e!ciency of the magnetic ﬁlter

to be

rxw

=exp(

iOU

d

) (3.102)

202 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

where  = 2 for ordered ﬁlters and  = 4/3 for random ﬁlters. U

is the

normalized capture radius given, for example, by eqs. (3.82) to (3.84), while i

is the ﬁlling factor of the ﬁlter matrix (i =1 ).

Equation (3.102) clearly illustrates that it is not possible to extend the single-

wire capture approach to multi-collector situation, as already demonstrated by

Fig. 3.29. According to eq. (3.102), the e!ciency of the ﬁlter depends on the

capture radius calculated for the single-collector scenario. In the single-collector

situation the particle capture is indeed determined by the capture radius and,

therefore, by the magnetic velocity or by the strength of the applied magnetic

ﬁeld. In a multi-collector situation, there are several additional mechanisms that

lead to particle collisions with the collectors and ultimately to capture. As will

be shown in the next section, under suitable conditions, usually present in real

operational situations, these mechanisms can be signiﬁcantly more important

than the magnetic traction mechanism, the only mechanism of particle capture

considered in the single-collector approach. By inclusion of the single-collector

capture radius U

, eq. (3.102), therefore, underestimates, by a wide margin,

the e!ciency of the ﬁlter and, at the same time, overestimates the role of the

magnetic ﬁeld in particle capture.

Particle capture in a real matrix

It transpires from the above discussion that it is not permissible to incorporate

the single-collector model into a multiple-collector situation. In a single-collector

situation the probability of a particle colliding with the matrix element is very

small and the particle thus must be attracted to the collector by the magnetic

traction force. On the other hand, in a real situation, where the matrix is

essentially a bed of matrix elements, the e!ciency of particle collision with the

matrix elements is determined, to a great extent, by mechanical, i.e. ”non-

magnetic” mechanisms of collision, as has been argued by Svoboda [S31, S32].

It can, therefore, be assumed that HGMS is essentially a deep bed ﬁlter with

an externally applied magnetic ﬁeld.

High-gradient magnetic separation can thus be viewed as a stochastic process

in which the chance of a particle reporting into the magnetic fraction is the

product of two probability terms:

uhfryhu|

= S

froolvlrq

×S

uhwhqwlrq

(3.103)

where S

uhfryhu|

is the probability of recovery of a particle into the magnetic

fraction, S

froolvlrq

is the probability of collision between a particle and the matrix

and S

uhwhqwlrq

is the probability of retention of a particle on the matrix.

While the ﬁrst term on the right-hand side of eq. (3.103) is controlled by

the hydrodynamic and geometrical conditions in the matrix, the second term

depends on the magnetic and surface forces between the particle and the matrix

element, and on the hydrodynamic regime.

Collision of particles with the matrix. Figure 3.32 illustrates typical colli-

sion mechanisms in a deep bed ﬁlter. These mechanisms also play a decisive role

3.4. HIGH-GRADIENT MAGNETIC SEPARATION 203

Figure 3.32: Collision mechanisms in deep-bed ﬁltration.

in multi-collector HGMS. A particle following a streamline may come in con-

tact with the matrix element by virtue of its ﬁnite size (interception or straining,

case A), and as a result of inertia (case B), sedimentation (case C) or diusion

(case D).

The velocity of the liquid can be also greater on one side of the particle than

on the other, causing the particle to rotate. As a result, a force that develops

laterally to the direction of ﬂow will drive particles across the ﬂow ﬁeld. This

hydrodynamic eect also contributes to the collision of particles with collectors.

The collision probability can then be written, according to eq. (3.101):

fro

=1 exp(O) (3.104)

where f

fro

is the concentration of particles which collided with the matrix col-

lectors. The ﬁlter coe!cient  can be written in terms of the collision e!ciency

 of a single collector as [M17]

 = (

3

)

1@3

ln(1  ) (3.105)

where s =(1 )

1@3

The total ﬁlter coe!cient  can be obtained from eq. (3.105) by calculating

the collision e!ciencies 

of individual processes (e.g. [W15, I4, C12, H15])

that lead to collision. It was found [S33] that, for example, for steel wool and

ball matrices, with element diameter of 200 m and 6 mm, respectively, the

dominant mechanism of collision is sedimentation. The hydrodynamic eect

also contributes, particularly for small particles. On the other hand, wedging in

the crevices of the matrix becomes signiﬁcant for larger particles.

Equation (3.104) then can be used to calculate the ratio f

fro

, i.e. the

ratio of the concentration of particles that collided with the matrix, to the

concentration of particles that entered the matrix bed. Figure 3.33 documents

that all particles as small as 4 m, moving with superﬁcial velocity of 0.07 m/s

204 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

Figure 3.33: The proportion of particles colliding with steel wool matrix as a

function of of the matrix length. y =0.07m/s,2d = 400 m,  =

0.95 (adapted from Svoboda [S33]).

Figure 3.34: The proportion of particles colliding with a steel ball matrix, as a

function of the matrix depth. y = 0.07 m/s, 2d =6mm, =0.4

(adapted from Svoboda [S33]).

3.4. HIGH-GRADIENT MAGNETIC SEPARATION 205

Figure 3.35: The proportion of particles colliding with steel wool matrix, as a

function of superﬁcial velocity. O = 200 mm, 2d = 400 m,  =

0.95 (adapted from [S33]).

or smaller, will have collided with the steel wool matrix, with the matrix bed

depth of 200 mm. A similar trend can be observed for the steel ball matrix,

as illustrated in Fig. 3.34. With decreasing velocity the probability of collision

rapidly increases, as shown in Fig. 3.35.

The retention of particles For a particle to be retained on the matrix

after a collision, the total interaction between the particles and the matrix

and the ﬂuid medium must be attractive. For su!ciently large particles the

surface forces can be neglected and the magnetic dipolar and the hydrodynamic

interactions are decisive. The mean free magnetic energy for the interaction of a

spherical paramagnetic particle of susceptibility  with a ferromagnetic sphere

of saturation magnetization P

> placed in an external magnetic ﬁeld E can be

written [S1] as

= 

4E

3



32

e

(3.106)

where u is the distance between the interacting bodies.

Since dAAeand u = d + e, eq. (3.106) gives [S31]

= 



e

(3.107)

The hydrodynamic potential energy of a particle captured on the surface of a

206 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

collector is assumed to be Stokesian:

=6yde (3.108)

A particle will remain attached to the matrix if Y

+ Y

? 0. Therefore, for a

particle to be retained by the matrix, the magnetic induction must be greater

than the threshold magnetic induction that can be determined from the steady-

state condition Y

+ Y

= 0. Therefore, [S33]:

27yd

16e

(3.109)

It is evident from eq. (3.109) that, for given conditions of separation, i.e. particle

size and susceptibility, ﬂow velocity and size of matrix elements, an increase

in strength of the applied magnetic ﬁeld beyond the threshold value will not

increase the retention of particles on the matrix.

It might appear from eq. (3.109) that by increasing the magnetic ﬁeld

strength it is possible to increase the ﬂow velocity and thus the throughput

of a separator, and still maintain the required rate of particle retention. How-

ever, if the particle velocity is too high, the particles will never collide with

the matrix bed and will thus not be available for retention. This scenario is

illustrated in Fig. 3.35.

Furthermore, the selectivity of the separation process is be determined by the

e!ciency with which the "non-magnetic" particles are being swept away from

the matrix by the erosive force of the ﬂuid. Consequently, for particles that are

supposed to report into the non-magnetic fraction, the inequality inverse to that

given by eq. (3.109) should apply. This is in contrast to the trajectory models,

in which the non-magnetic particles never reach the matrix surface.

3.5 Linear open-gradient magnetic separation

The concept of separation in a magnetic ﬁeld with open gradient of the ﬁeld

is widely practised in drum and roll magnetic separators. Paradoxically, it is

the linear multipole open-gradient magnetic separation (OGMS) that received

considerable attention from the theoretical point of view, in spite of its fail-

ure to prove its advantages on the production scale. It has been explained in

Section 2.5.3, that the linear multipole OGMS consists of an array of verti-

cally positioned coaxial coils energized in opposite directions. The concept is

schematically shown in Figs. 2.85(a) and (b) and in Fig. 3.36.

The magnetic force directed radially inwards diverts the free-falling magnetic

particles inwards while the feebly magnetic particles fall vertically, unaected

by the magnetic force.

The main advantage of open-gradient separators, in comparison with HGMS,

is the absence of matrix which allows the elimination of the blockage problem.

Another advantage of OGMS is the use of superconducting magnets which lend

themselves easily to scale-up. This is because in resistive magnetic systems the

3.5. LINEAR OPEN-GRADIENT MAGNETIC SEPARATION 207

Feeder

Splitter

ostat with superconductin

coils

Figure 3.36: Schematic diagram of a linear multipole open-gradient magnetic

separator.

magnetic ﬁeld in the separation space is limited by the non-linear saturation

eects in steel [G5]. On the other hand, systems consisting of superconducting

magnets are linear as far as the magnetic ﬁeld generation is concerned, and

the electric power consumption, which is spent mostly for cryogenic purposes,

increases only moderately with their size. It was also found by Gerber et al.

[G8] that the metallurgical e!ciency of the superconducting OGMS is superior

to the conventional OGMS as a result of considerably larger particle deﬂections

achievable in various designs of coil conﬁgurations.

A detailed analysis of particle trajectories in dry OGMS was made by Collan

et al. [C13], while Kopp [K10, K11] developed a simple kinetic model of the

separation process. These models were subsequently reﬁned by numerous au-

thors [B16 , G9, J3, B17, P6, G5]. Particle trajectories and particle deﬂections

as a function of particle properties, particle collisions, electric current densities

in the coils and the separator geometry were determined. Diamagnetic mode of

OGMS was also described theoretically and conﬁrmed experimentally by Krist

et al. [K12]. Considerable eort was also devoted to the optimization of the

design parameters of the superconducting coils for OGMS [G10, G11, K13, J4,

P7].

Although theoretical analyses of particle motion in OGMS are of consider-

able mathematical elegance, their applicability to practical problems of material

separation is limited. In view of a somewhat esoteric character of the subject

and as a result of the failure of several eorts to industrialize the concept of

208 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

OGMS [G3, K9, A8, S17] and to achieve commercial success, detailed descrip-

tion is beyond the scope of this monograph. The interested reader is referred

to the literature listed above.

3.6 Magnetic ﬂocculation

The demand for mineral products is subject to two factors aecting the e!-

ciency of beneﬁciation: a decrease in the quality of economically accessible ores

and an increase in the degree of ﬁneness of their valuable components. Since

the recovery of ﬁne particles is still very ine!cient, vast quantities of valuable

minerals are often lost in the form of slimes. Unless new technologies are de-

veloped, the increasing amount of ﬁnely-dispersed minerals that being mined

and processed, will lead to increased volumes of waste. This will result in a

substantial loss of valuable components and in environmental problems.

Methods for the treatment of ﬁne particles are still in the early stages of de-

velopment, and conventional methods are usually ine!cient for particles smaller

than 20 m. Particles smaller than 5 m can be manipulated only by HGMS,

a rather specialized technique applicable to a limited range of materials. A

more general solution to the problem may lie in the formation of aggregates

comprising ﬁne particles.

In the past, ﬂocs were successfully formed by the addition of inorganic elec-

trolytes or polymers, for instance for the removal of colloidal particles from

e"uents. Flocculation can also be induced by the application of external forces

to a system of particles, e.g. by stirring, or by the application of magnetic or

electrical ﬁelds.

Magnetic ﬂocculation has been used for some time to create ﬂocs of ferro-

magnetic particles. To induce this process, a magnetic induction of the order

of 0.01 T is required. Experimental observations of magnetic ﬂocculation of

strongly magnetic minerals and its application to the mining industry have

been described in numerous publications [B18, B19, B20, B21, L9, K14, K2].

The theoretical description of mutual coagulation of ferromagnetic particles is

also well established [D9, J5, L10, L9].

Although investigations into the magnetic ﬂocculation of paramagnetic and

diamagnetic particles started much later [H16, W16, S34], systematic studies of

the theoretical and experimental aspects [S35, P8, S36, P9, S37, S38, W17, W18,

W19, P10, T5, S39, S40, S41] show that ﬂocculation of ﬁne, feebly magnetic,

particles can be induced in magnetic systems that are presently available.

In the following sections the present status of the theory of magnetic ﬂoccu-

lation of strongly and weakly magnetic materials will be brieﬂy outlined.

3.6.1 Magnetic ﬂocculation of strongly magnetic materials

The tendency of an assembly of ferromagnetic particles in the magnetic ﬁeld

of a magnetic separator to reduce its magnetostatic energy often results in the

formation of aggregates of the particles. The eect of magnetic ﬂocculation

3.6. MAGNETIC FLOCCULATION 209

is the inherent part of any process of manipulation of ﬁnely ground strongly

magnetic materials and directly aects the e!ciency of material separation.

In magnetic separation the process of magnetic ﬂocculation can often play a

positive role by increasing the recovery and/or throughput. In other processes,

such as recovery of dense medium, magnetic ﬂocculation is detrimental to the

e!ciency of dense medium separation.

Theory of magnetic ﬂocculation

We shall consider a suspension of identical spherical ferromagnetic particles

in a non-magnetic ﬂuid. Under the inﬂuence of a su!ciently strong external

magnetic ﬁeld such particles tend to associate in long chains. Each particle

of volume Y

and magnetization P

carries a magnetic moment 

= Y



The energy of magnetic interaction between particles (1) and (2) separated by

a distance u is



{



(1)

(2)



3[

(1)u][

(2)u]

} (3.110)

If magnetization of the particles is su!ciently large, the electrical double

layer interaction and the van der Waals interaction can be neglected. Total

interaction energy, therefore, consists of the magnetic dipolar term only (eq.

(3.110)). Eective magnetic interaction in the suspension can be obtained by

averaging Y

over all orientations of the colloidal particles. Thus [C14, C15]

= nW ln ? exp(

) A (3.111)

where n is the Boltzmann constant and W is the absolute temperature and

? exp(

) A=

64

exp(

)g

g

(3.112)

where g

(l =1,2,3) represents an element of a solid angle describing the ori-

entation of each of the dipoles (1, 2) and that of the external magnetic ﬁeld (3).

It was shown by Chan et al. [C14, C15] and by Scholten et al. [S42] that in

the weak interaction limit, i.e. when 

@

?nW, the average interaction

energy can be written as

= 



3

(3.113)

while in the strong interaction limit (

@

AnW)

= 

2



(3.114)

It can thus be seen that although the magnetic dipole-dipole interaction (eq.

(3.110)) can be either attractive or repulsive, depending on the relative orien-

tation of the dipoles, the orientational averaging process gives the energetically

favourable (attractive orientation) heavy weighing.