Jan Svoboda. Magnetic Techniques for the Treatment of Materials
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180 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION
Figure 3.11: The static force index as a function of the depth of the burden,
for various sizes of a ferromagnetic sphere (
s
= 1000>
s
= 7800
kg/m
3
,
e
= 800 kg/m
3
) (adapted from [S25]).
In practice, however, a ferromagnetic particle is usually buried in a layer
of loosely packed material, e.g. coal. With the body at rest, the minimum
magnetic force required to move the object is determined by its weight and by
the force of pressure on the upper hemisphere of the body:
I
s
=
e
Y
e
j (3.55)
where Y
e
is the volume of the burden above the body. In equilibrium
I
p
=
I
j
+
I
s
(3.56)
From eq. (3.56), taking into account that
s
AA
e
> a modified version of
eq. (3.47) can be written [S25]:
i
p
=
0
£
1+Q(
s
1)
¤
s
1
(
s
+
e
Y
e
Y
s
)j (3.57)
It can be seen from eq. (3.57) that the static magnetic force density (or force
index in the presence of the burden depends not only on the shape of the object,
but also, in contrast to the situation when the burden is absent (eq. (3.47)),
on the size of the object. A typical dependence of the static force index on the
depth of the burden, for various sizes of a ferromagnetic sphere, is shown in Fig.
3.11.
3.4. HIGH-GRADIENT MAGNETIC SEPARATION 181
Similar diagrams can be constructed, using eq. (3.57), for other shapes. It
can be seen that the static force index increases steeply with increasing depth of
the burden for small particles, while for larger objects the dependence is mild.
3.4 High-gradient magnetic separation
During the last three decades, high-gradient magnetic separation (HGMS) has
attracted considerable attention, both experimentally and theoretically. It has
established itself as a powerful technique for the manipulation of fine weakly
magnetic particles. HGMS has also become a beloved subject of theoretical
modelling and a notable number of publications dealing with the problem of
particle capture led to an understanding of the interaction between a magneti-
zable particle and a matrix element.
The problem of particle capture in a matrix of a high-gradient magnetic
separator can be looked upon from both the macroscopic and microscopic points
of view. The macroscopic studies are aimed at the phenomenological description
of the separation process and the prediction of its dynamic behaviour. The
microscopic theories are intended to provide insight into the mechanisms of
particle capture by the matrix. The complex problems facing the microscopic
theories were considerably simplified by considering, in the first instance, the
interaction between a single particle and a single matrix element. The theory of
multi-particle and multi-collector magnetic separation was then developed as a
direct extension of the single element model. Validity of such an approach has
been questioned and theoretical analysis and experimental data confirm only
limited applicability of such an extension to real-life multi-element scenario.
A detailed description of the theoretical models of high-gradient magnetic
separation is given in a book by Gerber and Birss [G1] and other publications
[B10, S1]. A brief survey of the theory of HGMS is given in the following section.
3.4.1 Theory of particle capture by a single-collector
The main emphasis in the early models of high-gradient magnetic separation
was on the analysis of particle trajectories in the vicinity of a single magne-
tized matrix collector, usually a wire [W8, W9, L6, C10]. These studies were
inspired by the early analysis by Zabel [Z6] who studied the mechanisms of
dielectrophoretic capture on fiber matrix.
Watson [W8] considered a ferromagnetic wire of radius d placed along the }
-axis in a diamagnetic fluid, as shown in Fig. 3.12. A uniform magnetic field
K
0
, strong enough to saturate the wire, is applied in the { direction. Spherical
paramagnetic particles of radius e are carried by a fluid of viscosity that flows
with uniform velocity y
0
in the negative {-direction.
This orientation is one of the three possible arrangements between the di-
rections of the magnetic field and particle flow velocity with respect to the
ferromagnetic wire. It is always assumed that the wire is orthogonal to the
external magnetic field; the fluid flow can be either parallel (longitudinal - L-
182 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION
Y
H
0
v
0
Z
a
b
X
Figure 3.12: A ferromagnetic wire and a paramagnetic particle in a magnetic
field.
Figure 3.13: Graphic representation of three geometrical configurations in
HGMS and of the profiles of the particle build-up on the ferro-
magnetic wire.
3.4. HIGH-GRADIENT MAGNETIC SEPARATION 183
a
H
0
b
x
y
ș
ȕ
g
Ȗ
r
v
0
Figure 3.14: The configuration of particle capture in a single-wire approxima-
tion.
orientation) or perpendicular (transverse - T - orientation) to the direction of
magnetic field. The third configuration, when the initial fluid flow is parallel
to the wire, is known as the axial (A) configuration. These three geometrical
configurations are illustrated in Fig. 3.13.
In order to describe the motion of a small paramagnetic particle travelling in
a fluid stream in the presence of a magnetized wire, one assumes that the particle
experiences magnetic, hydrodynamic and gravitational forces. The balance of
forces which describes the particle motion is given by eq. (3.1), or in a more
explicit form by eq. (3.58):
I
l
=
I
p
+
I
g
+
I
j
(3.58)
where I
l
is the inertial force.
The general configuration of the particle motion problem in a single-wire
approximation is shown in Fig. 3.14.
In order to obtain the magnetic force acting on a small paramagnetic particle
in the vicinity of a magnetized wire, the magnetic field, external to and generated
by the magnetized wire must be expressed as a function of the wire radius. The
components of the magnetic field outside the wire, in polar coordinates, as
defined in Figure 3.14, can be written [G1, S1] as:
K
u
= K
0
(1 + N
d
2
u
2
)cos (3.59)
K
= K
0
(1 N
d
2
u
2
)sin (3.60)
184 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION
where
N =
P
2K
0
(3.61)
For K
0
?K
v
N =1while for K
0
AK
v
N = P
v
@2K
0
. K
v
and P
v
are
the saturation magnetic field strength and saturation magnetization of the wire
material, respectively.
The magnetic force
By inserting eqs. (3.59) and (3.60) into eq. (1.8), components of the magnetic
traction force on a magnetizable particle are obtained:
I
pu
=
0
Y
s
P
v
d
2
u
3
(
P
v
d
2
u
2
+ K
0
cos 2) (3.62)
I
p
=
0
d
2
Y
s
P
v
K
0
u
3
sin 2 (3.63)
I
p}
=0 (3.64)
If the net volume magnetic susceptibility =
s
i
is positive, then the
radial component of the magnetic traction force between the particle and the
wire is attractive provided that
P
v
d
2
u
2
+ K
0
cos 2A0 (3.65)
The component I
pu
becomes repulsive for
s
A
i
in two symmetrical regions,
the boundaries of which are determined by a critical angle
f
[B10]
f
= arctan(
1+N
d
2
u
2
1 N
d
2
u
2
) (3.66)
As shown in Fig. 3.15, the angle
f
determines sections of the wire surface
that are attractive or repulsive to paramagnetic or diamagnetic particles. It
is apparent from eq. (3.65) that paramagnetic particles will be collected at
angles smaller than
f
and repelled at angles greater than
f
= On the other
hand, diamagnetic particles, with negative susceptibilities, will be attracted to
the wire collector at angles A
f
and repelled in regions where ?
f
=
The situation is summarized in Fig. 3.16, which shows the contours of the
magnetic force around a magnetized cylindrical wire. The hyperbolae indicate
the lines of zero radial force. It can be seen from eq. (3.66) that the angle
f
decreases with increasing magnetic field strength. Therefore, the area on the
matrix collector, available for the capture of paramagnetic particles decreases
with increasing field strength.
3.4. HIGH-GRADIENT MAGNETIC SEPARATION 185
Figure 3.15: Attractive and repulsive regions around a wire collector (r
d
= u@d)=
Figure 3.16: Pattern of the magnetic force around a magnetized cylindrical wire.
186 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION
The fluid flow distribution and the viscous drag
The hydrodynamic drag acting on a particle travelling in a fluid stream is eval-
uated by calculating the flow field around a circular cylinder. The simplest
approach to this di!cult problem is to assume that the fluid is inviscid and
incompressible. Then the fluid velocity can be written as the gradient of the
velocity potential and by solving Laplace’s equation it is possible to obtain the
fluid velocity in polar coordinates [G1]:
y
u
= y
0
(1
d
2
u
2
)cos( ) (3.67)
y
= y
0
(1 +
d
2
u
2
)sin( ) (3.68)
The potential flow gives good description of the flow conditions on the front
side of wire for high Reynolds number, i.e. for Re A 1, where Re is given by
Re =
2y
0
i
d
(3.69)
For high Reynolds numbers the inertial forces exceed the viscous forces every-
where in the flow field except within a thin boundary layer at the wire surface
and at the rear side of the wire.
For low Reynolds numbers, Re?1, the inertial forces are smaller than the
viscous forces and the flow has a viscous (or creeping or laminar) character. The
velocity distribution can then be obtained by Lamb’s method [B11, G1]:
y
u
= y
0
ln(
u
d
) 0=5
h
1 (
d
u
)
2
i
2=002 ln Re
cos( ) (3.70)
y
= y
0
ln(
u
d
)+0=5
h
1 (
d
u
)
2
i
2=002 ln Re
sin( ) (3.71)
As has been mentioned above, flow velocity at the wire surface becomes
zero in the viscous flow model, while in the potential flow model it can reach
values up to 2y
0
. These dierences between the two models are particularly
noticeable in the collection process. It was shown by Cummings et al. [C11]
that particles in viscous (laminar) flow begin to move away from the wire much
further upstream than those in potential flow. Because the viscous flow field
”sees” the wire much further upstream, particle transport to the region of the
highest magnetic force is greatly diminished.
Viscous drag force is obtained from Stokes’s equation (eq. (1.16)) and the
components of this drag force can be written:
I
gu
=6e(y
u
gu
gw
) (3.72)
I
g
=6e(y
u
g
gw
) (3.73)
The components of the fluid velocity y
u
and y
are given by eqs. (3.67) and
(3.68) or eqs. (3.70) and (3.71), depending on the value of the Reynolds number.
3.4. HIGH-GRADIENT MAGNETIC SEPARATION 187
The gravitational force
The gravitational force is given by eq. (1.14) and its components, in polar
coordinates, can be written [G1, G5]:
I
ju
=(
s
i
)jY
s
cos( ) (3.74)
I
j
= (
s
i
)jY
s
sin( ) (3.75)
Equations of particle motion
The dynamic behaviour of the particle in a single-wire approximation of high-
gradient magnetic separation is described by eq. (3.58). The complete equations
of motion in polar coordinates were obtained by a number of authors, for in-
stance [W8, G1, G5, C11, L6, L7]. If the analysis is limited to the motion of
small particles (e.g. 2e?75 m), for which the gravitational force is much
smaller than the hydrodynamic force, and the condition
2
s
e
2
9
?? 1 (3.76)
is satisfied, the inertial and gravitational terms in eq. (3.58) can be neglected.
The radial and azimuthal components of the equation of motion can be written
[B10] as :
gu
d
gw
=
y
0
d
(1
1
u
2
d
)cos( )
y
p
d
(
N
u
5
d
+
cos 2
u
3
d
) (3.77)
u
d
g
gw
=
y
d
d
(1 +
1
u
2
d
)sin( )
y
p
sin 2
du
3
d
(3.78)
where u
d
= u@d and
y
p
=
2
0
(
s
i
)PK
0
e
2
9d
(3.79)
is the parameter introduced by Watson [W8]. It has the dimension of [m/s] and
is termed magnetic velocity.
When = 0, eqs. (3.77) and (3.78) describe the motion of a particle in
the longitudinal configuration, while when = @2 they describe the motion
in transverse configuration. By combining eqs. (3.77) and (3.78) the time-
independent equation is obtained [W8]:
1
u
d
gu
d
g
=
(1
1
u
2
d
)cos( )
y
p
y
0
(
N
u
5
d
+
cos 2
u
3
d
)
(1 +
1
u
2
d
)sin( )+
y
p
y
0
sin 2
u
3
d
(3.80)
Particle trajectories can thus be described by an equation of a general form
1
u
d
gu
d
g
= i(
y
p
y
0
>
P
2K
0
>u
d
>) (3.81)
188 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION
The path taken by the particle depends, therefore, only on y
p
@y
0
>P@2K
0
,
and the initial position and the initial velocity of the particle.
The complete equations of motion (eqs. (3.77) and (3.78)) can be solved
only numerically. The solutions (i.e. the particle trajectories) are of two types,
depending on whether the particle is captured by the wire or not. The critical
trajectory, for which the particle is just captured, is characterized by the capture
radius U
f
.
Approximate analytical solutions of the equations of motions in the case
of the potential flow, under the condition that | y
pd
@y
0d
|
s
2> yield the
following expression for the capture radius in the longitudinal and transverse
configurations [G1, G5]:
U
fd
'
3
s
3
4
(|
y
pd
y
0d
|)
1@3
·
1
2
3
(
y
pd
y
0d
)
2@3
¸
(3.82)
where the normalized quantities are defined as U
fd
= U
f
@d> y
pd
= y
p
@d and
y
0d
= y
0
@d= Thus, for a given | y
pd
@y
0d
| ratio, both configurations have the
same capture radius.
When | y
pd
@y
0d
|?
s
2> the critical capture radius in the longitudinal con-
figuration ( =0)is given by
U
fd
=
1
2
y
pd
y
0d
h
(1 N
2
)
1@2
+ N( arccos N)
i
(3.83)
and in the transverse configuration ( = @2) [G1]:
U
fd
=
1
2
y
pd
y
0d
h
(1 N
2
)
1@2
+ N(2 arccos N
i
(3.84)
It is apparent from eqs. (3.82), (3.83) and (3.84) that the dependence of
U
fd
upon the ratio y
p
@y
0
can be divided into two categories: at low values of
y
p
@y
0
, the capture radius increases linearly with this ratio
U
fd
y
p
y
0
(3.85)
while at large values it increases approximately as
U
fd
(
y
p
y
0
)
1@3
(3.86)
Axial configuration
The analysis of a particle motion in the axial (A) configuration, where the
wire axis and the fluid flow are mutually parallel, and the magnetic field is
perpendicular to them (Figs. 3.13 and 3.17), does not require, in contrast to
other cases, numerical computations. Since the results can be derived in an
analytical form, this configuration became a favourite object of the modelling of
particle capture. The dynamics of the particles and their build-up on the wire
3.4. HIGH-GRADIENT MAGNETIC SEPARATION 189
x
y
z
H
0
v
0
ș
Figure 3.17: The axial configuration.
in A-configuration have been described in numerous papers, for example [G9,
B12, U4, B13].
The equations of particle motion (eqs. (3.77) and (3.78)) are reduced in this
configuration to the simplified forms
gu
d
gw
=
y
p
d
(
N
u
5
d
+
cos 2
u
3
d
) (3.87)
g
gw
=
y
p
d
sin 2
u
4
d
(3.88)
g}
d
gw
=
y
0
d
(3.89)
The elimination of time from eqs. (3.87), (3.88) and (3.89) produces a set of
dierential equations
gu
d
g
=
N
u
d
sin 2
+ u
d
cot 2 (3.90)
g}
d
g
=
y
0
y
p
u
4
d
sin 2
(3.91)
From these equations the particle trajectories can be determined analytically
[G1,G9, B12].