Jan Svoboda. Magnetic Techniques for the Treatment of Materials

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220 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

Figure 3.46: The potential energy curves for hydrophobic and magnetic ﬂoccu-

lation of rhodochrosite (2e

=2.25m) hydrophobized by sodium

oleate, at a magnetic induction of 0.9 T (adapted from [L11]).

by introduction of plethora of hydrophobic phenomena observed in the stability

studies can, in principle, be explained by capillary condensation, i.e. thermo-

dynamics.

Several authors have shown that the stability of a colloidal suspension can be

controlled, in addition to the electrostatic surface potential and the magnetic

interaction, by such a hydrophobic interaction. Lu et al. [L11] included the

energy of hydrophobic interaction Y

as an additive term in the expression of

the total interaction energy (eq. (3.123):

= Y

+ Y

(3.127)

The hydrophobic interaction energy was expressed by two terms, each represent-

ing dierent mechanism of the hydrophobic interaction. The potential energy

curves of three possible ﬂocculation mechanisms for paramagnetic rhodochrosite

were calculated and are shown in Fig. 3.46.

It can been seen that the potential energy of magnetic interaction exhibits

a deep secondary minimum and a primary minimum barrier. In hydrophobic

aggregation, a relatively low potential barrier partially prevents particles from

rapid ﬂocculation into the primary minimum. On the other hand, the potential

energy curve for combined magnetic and hydrophobic ﬂocculation shows deep

primary and secondary minima, with no potential barrier.

The experimentally determined rates of ﬂocculation in these three regimes

are illustrated in Fig. 3.47. It is clear that the combined ﬂocculation by a

3.7. MAGNETIC SEPARATION BY PARTICLE ROTATION 221

Figure 3.47: Flocculation of rhodochrosite by magnetic (B = 0.85 T), hy-

dropohobic (sodium oleate + kerosene) and combined mechanisms

(adapted from [L11]).

combination of magnetic and hydrophobic processes was more eective than

aggregation induced by sodium oleate or magnetic ﬁeld alone.

Škvarla and Zele

nák [S41] developed a simple model of magnetic-hydrophobic

ﬂocculation, as a variation of Lu’s model [L11]. They assumed that the potential

of the hydrophobic interaction was of the short range and it was approximated

by a single exponential

= 2e

 exp(

 k



) (3.128)

where 

is the solid/water interfacial energy of the spherical particles, k

the contact separation distance of the spheres (



0.2 nm), and  is the empir-

ical constant (



1 nm). The model was veriﬁed by experimental observation

of magnetic-hydrophobic ﬂocculation of siderite. A satisfactory correlation be-

tween the measured and calculated potential energy maxima was observed for

the hydrophobic process. The prediction of the model of magnetic ﬂocculation

alone was not satisfactory.

3.7 Magnetic separation by particle rotation

Motion of magnetizable particles in alternating or rotating magnetic ﬁelds is a

well established research ﬁeld. In 1925 Koizumi [K15] designed a magnetic sep-

arator with a rotating magnetic ﬁeld, to eliminate entrapment of non-magnetic

222 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

Figure 3.48: Schematic diagram of a separator with rotating magnetic ﬁeld

(courtesy of N. Allen [A17]).

particles in the magnetic concentrate. Similar approach was taken more recently

by other researchers to improve the quality of magnetic concentration [T3, D1,

F1, G12, G13, L9]. Although the results of these investigations had been posi-

tive, the technique did not appear to be accepted and developed further.

Recently, Allen [A9] employed the concept of the eect of a rotating magnetic

ﬁeld on magnetizable particles and developed rotating magnetic ﬁeld (RMF)

separators, described in Section 2.6.3. The rotation separation is based on mag-

netic anisotropy of particles, which is present in ferromagnetic as well as para-

magnetic materials. However, in paramagnetic minerals the magnetic anisotropy

is generally too weak to be of practical use for particle separation. In general,

ferromagnetic particles will rotate if they are placed in a rotating external mag-

netic ﬁeld, while paramagnetic particles will not.

An RMF is generated by a rotating magnetic drum ﬁtted, around its circum-

ference, with magnets, with alternate polarity. As is schematically illustrated

in Fig. 2.95, the external magnetic ﬁeld exerts a torque on a magnetically

anisotropic particle and makes it rotate and move it in the direction opposite

to that of the rotating ﬁeld. A schematic diagram of such an RMF separator is

shown in Fig. 3.48.

3.7.1 RMF rotation separation

In the RMF separator a particle is placed on a non-magnetic surface in a rotating

magnetic ﬁeld, as shown in Fig. 3.49. If the particle is capable of rotation in the

3.7. MAGNETIC SEPARATION BY PARTICLE ROTATION 223

Fg, Fm

Particle

Magnet

Drum

Direction of

drum rotation

Direction of particle

rotation

Figure 3.49: Schematic diagram of a particle placed on a rotating magnetic

drum.

ﬁeld, it will experience a torque in order to align its axis of easy magnetization

with the direction of the external magnetic ﬁeld. The torque, which thus arises

when the direction of the magnetization of a particle and the external magnetic

ﬁeld do not coincide, will force the particle to roll. The rolling is opposed by

the force of gravity and magnetic traction force acting on the particle.

The particles to be separated by rotation will, in general, contain ferro-

magnetic and paramagnetic regions. While the paramagnetic regions will make

limited contribution to the particle rotation, they will contribute to the mag-

netic traction (holding) force. This traction force can, therefore, be written

[A17] as:





EuE + P

uE (3.129)

where Y

and Y

are the volumes of the paramagnetic and ferromagnetic re-

gions of the particle, respectively. P

is the magnetization of the ferromagnetic

components while 

is the volume magnetic susceptibility of the paramagnetic

contribution. The magnitude of the torque of the force I , required to rotate a

particle, is the product of the magnitude of the force and the moment arm g;

that is

W = Ig (3.130)

The particle will not start rotating unless the initial magnetic rotational torque

exceeds the initial gravitational and magnetic traction force torques (case (a) in

Fig. 3.50). The cubic shape of the particle is usually used to demonstrate the

224 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

(a)

(b)

, F

( c)

Figure 3.50: Forces acting on a cubic particle in traction and rotating magnetic

ﬁelds.

case [A17, A18] and we have

lqlw

+ I

) (3.131)

where O is the length of the cube edge and the force of gravity I

is given by

eq. (1.14).

For a cubic ferromagnetic particle, under the assumption that the ﬁrst term

on the right-hand side of eq. (3.129) can be neglected, the initial gravitational

+ magnetic holding torques (that is, when the particle is resting ﬂat on the

surface) can be written

lqlw

= O



+ P

(3.132)

3.7.2 Magnetic torque on a particle and condition for sep-

aration by rotation

The application of an external magnetic ﬁeld to a ferromagnetic crystal, at a

certain angle  +  to a direction of easy magnetization, causes a deviation of

the magnetization, as indicated in Fig. 3.51. The external magnetic induction

E applies a torque to the particle magnetization P

and causes it to deviate

from the direction of easy magnetization {. As a result, the magnetocrystalline

forces apply a torque on the magnetization to return it to the direction of easy

3.7. MAGNETIC SEPARATION BY PARTICLE ROTATION 225

Figure 3.51: The torque experienced by an anisotropic magnetizable particle

placed in an external magnetic ﬁeld.

magnetization. This situation can be described by a relationship

E sin  = P

D sin  (3.133)

where D represents the anisotropy ﬁeld. If the external ﬁeld is very low, then

the angle  is small. If the external ﬁeld is very high, then  is small [A17].

For a cubic particle, the magnetic torque given by eq. (3.48), can be re-

written as:

pdj

= Y

E sin  = O

E sin  (3.134)

where  is the angle between the directions of the magnetic ﬁeld and particle

magnetization, as can be seen in Figs. 3.50(b) and 3.51. Therefore, a particle

will start rotating when

pdj

lqlw

(3.135)

or, explicitly,

E sin A

(



+ uE) (3.136)

Coercive forces are very low for many ferromagnetics (s.l.), such as mag-

netite and steel, while anisotropy forces are high. If the external ﬁeld is smaller

than the coercive force (E?

), angular deviations from directions of easy

magnetization are also small. Maximum torque is then applied when the exter-

nal ﬁeld is very close to 90

to the particle magnetization. [A17]. Although the

angle between the magnetization and the external ﬁeld has been maximized, the

external ﬁeld is very low for most materials and, therefore, the torque is small.

However, for materials of very high coercive force, such as roasted ilmenite, this

is an advantageous situation for magnetic separation by rotation.

226 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

In a magnetic ﬁeld that exceeds the coercive force K

of the material, the

movement of domain walls within the solid substitutes for the movement of par-

ticles. The overall magnetization of a ferromagnetic particle is able to change

direction in response to the direction of the external ﬁeld. However, while the

changes of the direction of the domain magnetization may occur freely between

directions of easy magnetization within a crystal, they are restrained for inter-

mediate angles by magnetocrystalline anisotropy. In this case the maximum

torque angle is approximately one quarter of the angle between directions of the

external ﬁeld and the particle magnetization [A20, A17].

3.7.3 Rotation of particles in a liquid

It has been assumed, so far, that the particles to be rotated are suspended in

air. If the particles are suspended in water, they are exposed, in addition to

the force of gravity and the magnetic traction force, to hydrodynamic forces.

These hydrodynamic forces are associated with a torque acting in the opposite

direction [L9]. It was shown by Lantto [L9] that the hydrodynamic torque at

a high Reynolds number is approximately given by

k|gur



(3.137)

where $ is the frequency of the rotating magnetic ﬁeld, e the radius of the

cylindrical particle, the length of which is fe (where fAA1).

The magnetic torque is

pdj

=2fe

E sin  (3.138)

Therefore, a cylindrical rod-like body will be brought into rotation when W

pdj

k|gur

. This condition is met when



4P

sin 

(3.139)

It can be seen that rotation is more easily accomplished for small particles with

large values of magnetization and large anisotropy angle . On the other hand,

an increase in frequency of the rotating ﬁeld hampers the ability of particles

to rotate. It has to be stressed, however, that if the value of the magnetic

induction given by eq. (3.139) exceeds that corresponding to the coercive force

(EA

) for the particle material, the particles will either cease to rotate

and may become demagnetized. If the particles are still rotating, the angle 

will be reduced.

In laminar ﬂow, with small Reynolds numbers, the viscosity of the liquid

will play an important role; otherwise the general character of the phenomenon

is the same as that at high Reynolds numbers [L9].

3.8. SEPARATION IN MAGNETIC FLUIDS 227

3.8 Separation in magnetic ﬂuids

The sink-and-ﬂoat technique of gravity separation relies on selective levitation

and sinking of materials based on their relative densities and that of the sep-

arating medium. The scope of the sink-and-ﬂoat technique is restricted by a

limited range of densities of the separating medium, the upper limit of ferrosili-

con medium being approximately 3500 kg/m

and that of Clerici solution about

4200 kg/m

at room temperature.

In the mid sixties of the last century the application of magnetic ﬂuids as

a heavy medium, had been investigated, and it was established that by expos-

ing a magnetic ﬂuid to a non-homogeneous external magnetic ﬁeld the ﬂuid

exhibited an apparent density exceeding densities obtainable with conventional

heavy liquids. Two broad classes of magnetic ﬂuids were investigated at that

time, namely paramagnetic liquids [A1, A12, A19] and ferroﬂuids [R1, R7, R8,

R16, G4, K16, F8].

Paramagnetic liquids are solutions of paramagnetic salts, such as MnCl

and

Mn(NO

)

. These liquids are paramagnetic in their behaviour: their magnetiza-

tion increases linearly with an increasing magnetic ﬁeld and their susceptibility

is low, of the order of 6×10

7

/kg. On the other hand, a ferroﬂuid is a stable

colloidal suspension of sub-domain ferromagnetic (s.l.) particles, for example

magnetite, in a liquid carrier. The behaviour of ferroﬂuids is superparamagnetic,

characterized by the absence of hysteresis and coercivity. Ferroﬂuids dier from

paramagnets in having quite large magnetic susceptibilities and being able to

become saturated in moderate magnetic ﬁelds. Saturation polarization of a

ferroﬂuid can be as high as 0.1 T (1000 G)

The concept of separation of non-magnetic particles suspended in a magnetic

ﬂuid is based on the generalized Archimedes law whereby, in addition to the

conventional force of gravity acting on the ﬂuid, also a magnetically induced

force acts on the ﬂuid. This additional magnetic pull creates a magnetically

induced buoyancy force on a particles immersed in the ﬂuid. This buoyancy

force can be controlled in a wide range of values and materials as dense as 20

000 kg/m

or higher can ﬂoat in such a ﬂuid.

A schematic diagram of the process of separation in magnetic ﬂuids is shown

in Fig. 3.52, while a general diagram of a magnetic circuit used in separation in

magnetic ﬂuids in shown in Fig. 3.53. Permanent magnets or electromagnets are

used to generate a non-homogeneous magnetic ﬁeld in the separation gap. The

desired pattern of the magnetic ﬁeld and its gradient is achieved by shaping the

pole tips. A separation chamber placed between the pole-pieces of the magnetic

circuit is ﬁlled with a magnetic ﬂuid.

3.8.1 Apparent density of a magnetic ﬂuid

There are two dominant forces acting on a volume of the magnetic ﬂuid, placed

in an external non-homogeneous magnetic ﬁeld, namely the force of gravity and

the magnetic traction force. The situation is schematically depicted in Fig. 3.52.

228 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

Floating

particles

Magnet

Sinking

particles

Element of

ferrofluid

Ferrofluid

Figure 3.52: Schematic diagram of the process of separation in magnetic ﬂuids.

Figure 3.53: Schematic diagram of a magnetic circuit used in separators with

magnetic ﬂuids.

3.8. SEPARATION IN MAGNETIC FLUIDS 229

The total force on the magnetic ﬂuid of volume Y

can be written as:



= 

j +





EuE (3.140)

where 

and 

are the physical density and volume magnetic susceptibility,

respectively, of the magnetic ﬂuid.

For a ferromagnetic (s.l.) ferroﬂuid, the magnetic force can be expressed as





uE (3.141)

where M

is the magnetic polarization of the ferroﬂuid.

When the ﬁeld gradient is parallel with the gravitational force and of the

same sense, eqs. (3.140) and (3.141) can be rearranged, as



= Y

j(





EuE) (3.142)

and



= Y

j(



uE) (3.143)

The expressions in parentheses can be viewed as an apparent density 

of a

magnetic medium exposed to an external non-homogeneous magnetic ﬁeld:



= 





EuE (3.144)

and



= 



uE (3.145)

In a more general case, when the ﬁeld gradient makes an angle  with gravity,

eq. (3.145) becomes



= 



uE cos  (3.146)

Equations (3.143) and (3.145) form the basis of ferrohydrostatic separation

(FHS).

In a typical case, a water-based ferroﬂuid with polarization of 0.02 T (200

G), placed in a magnetic ﬁeld of gradient of 2 T/m (200 G/cm) will exhibit

an apparent density of approximately 4200 kg/m

. Figure 3.54 illustrates the

apparent density as a function of the ﬁeld gradient for two types of kerosene-

based ferroﬂuid.

3.8.2 A particle suspended in a magnetic ﬂuid

A particle suspended in a magnetic ﬂuid is acted upon by several forces illus-

trated in Fig. 3.55. The force of gravity is given by



= 

j (3.147)