Jan Svoboda. Magnetic Techniques for the Treatment of Materials
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230 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION
Figure 3.54: The apparent density as a function of the gradient of the magnetic
field for two types of kerosene-based ferrofluid, of dierent densities
and magnetic polarizations.
while the magnetic traction force is
I
sp
=
1
0
s
Y
s
EuE (3.148)
where
s
is the volume magnetic susceptibility of the particle.
Such a particle experiences a loss of its weight as a result of two buoyancy
forces acting on the particle. The first buoyancy force is the classical Archimedes
gravity-related force:
I
sje
=
i
Y
s
j (3.149)
The other force is the magnetically induced buoyancy force due to the magnetic
traction force acting on the ferrofluid:
I
spe
=
1
0
Y
s
M
i
uE (3.150)
The net vertical force on a particle suspended in a ferrofluid and acted upon by
a non-homogeneous magnetic field with vertical gradient can thus be written:
I
s
=
I
sj
+
I
sp
+
I
sje
+
I
spe
(3.151)
or, in an explicit form:
I
s
= Y
s
(
s
i
)j +
uE
0
Y
s
(
s
E M
i
) (3.152)
3.8. SEPARATION IN MAGNETIC FLUIDS 231
F
pg
F
pm
F
pd
(2)
F
pgb
F
pmb
F
pd
(1)
Magnetic field
Gradient of
the magnetic
field
Pool of
f
e
rr
o
fl
u
i
d
Solid particle
Figure 3.55: Forces acting on a particle in a stationary ferrofluid, placed in a
non-homogeneous magnetic field.
Defining the eective cut-point density
fs
for separation as that particle density
s
for which I
s
= 0 (i.e. equilibrium of the forces acting on the particle), eq.
(3.152) yields:
fs
=
i
+
uE
j
(M
i
s
E)cos (3.153)
For non-magnetic particles
s
= 0 and eq. (3.153) reduces to
fs
=
i
+
M
i
0
j
uE cos (3.154)
where is the angle between the vectors of the gravitational force and the field
gradient. When the force of gravity and the gradient of the magnetic field are
parallel, eq. (3.154) becomes
fs
=
i
+
M
i
0
j
uE (3.155)
which is equivalent to eq. (3.145).
It can be seen that for non-magnetic particles, the density cut-point is equal
to the apparent density of the ferrofluid. Particles whose densities
s
are smaller
than the apparent density of the ferrofluid (
s
?
d
) will float in ferrofluid,
while particles with density greater than the apparent density of the ferrofluid
(
s
A
d
) will sink. Trajectories of sink and float particles in a stationary
ferrofluid placed in a non-homogeneous magnetic field are shown in Fig. 3.56.
232 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION
Sink particles
Float particles
Particles enter a pool of ferrofluid
Į
Field
gradient
Pool of ferrofluid
Figure 3.56: Trajectories of float and sink particles in a stationary ferrofluid
placed in a non-homogeneous magnetic field.
Figure 3.57: The eect of hydrodynamic drag on cut-point density, for two par-
ticle velocities. The nominal "large-particle" (eq. 3.155) cut-point
density
fs
= 3500 kg/m
3
.
3.8. SEPARATION IN MAGNETIC FLUIDS 233
Figure 3.58: The inaccuracy in cut-point density introduced by ignoring hydro-
dynamic drag, for two dierent particle velocities. The nominal
cut-point density
fs
= 3500 kg/m
3
.
3.8.3 The eect of the hydrodynamic drag
In many applications of ferrohydrostatic separation it is legitimate to ignore the
eect of the hydrodynamic drag. For small particles, however, the influence
of the drag on the cut-point density can be significant. If we assume that the
Stokes law applies, the hydrodynamic drag is given by eq. (1.16). By including
this drag in the expression of the total force (eq. (3.151)), the equilibrium of
forces on a non-magnetic particle yields
fs
=
i
+
M
i
0
j
uE +
9y
2je
2
(3.156)
Equation (3.156) applies to a particle that is moving vertically downwards,
i.e. the hydrodynamic drag I
sg
(1) acts on the particle. Such a particle is either
a sink particle, i.e. a particle whose density is greater than
d
, or a float particle
that is temporarily moving downwards because of the non-zero initial velocity
with which it entered the fluid, as shown in Fig. 3.56.
Once a float particle reaches its terminal velocity (equal to zero), it will
reverse the direction of its motion and will start moving upwards. At that point
theoppositehydrodynamicdragI
sg
(2) will be acting on the particle and the
particle will experience an apparent density given by
d
=
i
+
M
i
0
j
uE
9y
2je
2
(3.157)
234 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION
Figure 3.59: Density increment, for paramagnetic particles, as a function of mag-
netic susceptibility, at three dierent values of the applied magnetic
field. uE =2T/m.
Therefore, it becomes clear that, in general terms, the presence of the
hydrodynamic drag increases the cut-point density for sinking particles, i.e. sink
particles or float particles in their initial phase of motion within the ferrofluid.
For particles reporting to the float fraction, however, the hydrodynamic drag
decreases the cut-point density.
It can be found from eq. (3.156) that for particles greater than, for example,
1 mm the hydrodynamic drag can be neglected and the cut-point density is
independent of particle size, and is given by eq. (3.155). However, by inclusion
of the hydrodynamic drag, the cut-point density begins to deviate, for particles
smaller than 1 mm, from its "large particle value", as is illustrated in Fig. 3.57.
Figure 3.58 shows the inaccuracy in the cut-point density introduced by ignoring
the hydrodynamic drag.
3.8.4 The eect of particle magnetic susceptibility
A magnetic particle (
s
6= 0) will exhibit an eective density
hii
dierent from
its natural (true) density
s
, as a result of its attraction (or repulsion) in the
direction of the field gradient. This eective density can be written:
hii
=
s
+
s
E
0
j
uE cos (3.158)
It can be seen that the dierence between the eective and true densities
s
=
hii
s
increases, for paramagnetic particles, with increasing magnetic
3.8. SEPARATION IN MAGNETIC FLUIDS 235
susceptibility, magnetic field strength and the field gradient. A relative density
increment as a result of non-zero magnetic susceptibility of a particle, follows
from eq. (3.158)
hii
s
s
=
"
s
E
0
j
uE (3.159)
where "
s
is the mass magnetic susceptibility of the particle. A typical de-
pendence of the density increment as a function of magnetic susceptibility of
paramagnetic particles is shown in Fig. 3.59. It is evident that the density
increment increases with increasing magnetic susceptibility of the particle and
the field strength. This is a very important implication that aects the accuracy
of separation of materials of non-zero magnetic susceptibility.
On the other hand, the eective density of diamagnetic particles decreases,
as a result of their negative magnetic susceptibility. In view of the low values of
diamagnetic susceptibility, the density decrease is very small; at the magnetic
induction of 2 T, the decrease is approximately 0.2%. For example, the eective
density of quartz (" =-6.0×10
9
m
3
/kg) is reduced, at 2 T, from its physical
density of = 2700 kg/m
3
to approximately 2695 kg/m
3
.
The validity of eq. (3.159) has been tested using magnetic tracers, parti-
cles of constant density and of dierent values of magnetic susceptibility [R31].
Magnetic tracers, 5 mm in size, and of density 3550 kg/m
3
, with mass magnetic
susceptibility ranging from 2.5×10
7
m
3
/kg (20×10
6
cm
3
/g) to 1.25×10
5
m
3
/kg (1000×10
6
cm
3
/g), were separated in a ferrofluid the apparent density
of which was varied from 3600 kg/m
3
to 4200 kg/m
3
. The magnetic induction
to which the ferrofluid was exposed in the separation region was 0.2 T and the
field gradient was 1.5 T/m. Partition curves thus obtained are shown in Fig.
3.60.
It can be seen that 50% of tracers with density of 3550 kg/m
3
sank in a
ferrofluid of apparent density of 3700 kg/m
3
only when their susceptibility was
greater than 1.25×10
6
m
3
/kg. When the density of the ferrofluid was set to
4200 kg/m
3
, 50% of magnetic tracers of density 3550 kg/m
3
sank when their
magnetic susceptibility was greater than 8.2×10
6
m
3
/kg.
This experimental observation can be compared with the theoretical predic-
tion expressed by eq. (3.159). Using the above values of particle susceptibility,
and taking into account that E =0.2TanduE = 1.5 T/m, the right-hand side
of eq. (3.159) is equal to 0.19. This value agrees well with the relative density
increment (4200 - 3550)/3550, obtained from the left-hand side of eq. (3.159).
3.8.5 Motion of particles in a ferrofluid
The equation of motion of non-magnetic particles
The equations of motion of a non-magnetic particle suspended in a stationary
ferrofluid, as depicted in Fig. 3.61, can be expressed [M18] as:
g
2
{
gw
2
=
gy
{
gw
=
1
s
Y
s
(I
{
se
+ I
{
sp
+ I
{
sg
) (3.160)
236 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION
Figure 3.60: The partition curves for magnetic tracers of density 3550 kg/m
3
,
as a function of their mass magnetic susceptibility, separated in a
ferrofluid of dierent apparent densities [R31].
h
z
x
Į
v
0
F
z
pmb,
F
pgb
F
pg
,
į
g
rad B
F
x
pmb
F
pmb
Figure 3.61: Section through a pool of ferrofluid of depth , inclined at an angle
=
3.8. SEPARATION IN MAGNETIC FLUIDS 237
g
2
}
gw
2
=
gy
}
gw
=
1
s
Y
s
(I
}
se
+ I
}
sg
+ I
}
sp
I
j
) (3.161)
These equations can be re-written
gy
{
gw
+ Ny
{
= Z (3.162)
gy
}
gw
+ Ny
}
= Z
0
(3.163)
where parameters N> Z and Z
0
have the following forms:
N =
6e
s
Y
s
>Z = j
d
i
s
sin > Z
0
= j
d
cos
s
+2
i
sin
2
(@2)
s
(3.164)
The initial conditions for eqs. (3.162) and (3.163) are:
for w =0 y
{
=0 and y
}
= y
0
(3.165)
The initial velocity of a particle that is being dropped into the ferrofluid from
height k is y
0
=
s
2jk.
Dierential equations (3.162) and (3.163) can be solved analytically and the
solutions are [M18]:
g{
gw
= y
{
=
Z
N
(1 h
Nw
) (3.166)
g}
gw
= y
}
= y
0
h
Nw
+
Z ”
N
(1 h
Nw
) (3.167)
The initial conditions for eqs. (3.166) and (3.167) are
for w =0 { =0and} = (3.168)
Equations (3.166) and (3.167), with these initial conditions, yield the follow-
ing solutions:
{ =
Z
N
w
Z
N
2
(1 h
Nw
) (3.169)
} = +(
y
0
N
Z ”
N
2
)(1 h
Nw
)+
Z
0
N
w (3.170)
Particle trajectories can be obtained from these parametric equations by
calculating pairs of coordinates ({> }). Typical trajectories of non-magnetic
particles of dierent densities are shown in Fig. 3.62.
It can be seen that when k =0theny
0
= 0 and the trajectory of a particle
is a straight line.
238 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION
-5
0
5
10
0 5 10 15 20
x (cm)
3300
3400
3500
3600
Figure 3.62: Trajectories of non-magnetic particles of various densities:
d
=
3500 kg/m
3
, k =3mm,
i
= 990 kg/m
3
, =1.65×10
3
Pa×s,
=2
0
. The straight dashed lines are trajectories of particles of
densities of 3500 kg/m
3
and 3550 kg/m
3
for k = 0 (adapted from
[M18]).
-5
0
5
10
0 5 10 15
x (cm)
z (cm
)
0
40
60
70
Figure 3.63: Trajectories of magnetic particles of various mass magnetic sus-
ceptibilities (×10
8
m
3
/kg):
d
= 3500 kg/m
3
,
s
= 3300 kg/m
3
(adapted from [M18]).
z (cm
)
3.8. SEPARATION IN MAGNETIC FLUIDS 239
N
S
N
S
N
S
N
S
Attraction
Repulsion
Magnetic fluid
Magnet
Particle
Figure 3.64: Interaction between non-magnetic particles in a magnetized mag-
netic fluid (adapted from Fujita et al. [F14]).
The equation of motion of a magnetic particle
As has been discussed in Section 3.8.4, a magnetic particle suspended in a
ferrofluid placed in a non-homogeneous magnetic field will exhibit an eective
density
hii
(given by eq. (3.158)) dierent from its natural density. Trajecto-
ries of magnetic particles can be obtained by replacing the natural density
s
of
the particles by eective density. As a result of the non-uniformity of the mag-
netic field, the eective density is dierent at dierent positions of the particle
along the vertical axis, being the greatest where the pole tips of the magnet
are closest to one another, as shown in Fig. 3.52. The mean eective density
can be calculated, for example, by replacing E in eq. (3.158) with the mean
magnetic induction E
mean
= E
min
+ E
max
. In this situation the equations of
motion remain the same as parametric equations (3.169) and (3.170). Typical
trajectories of magnetic particles are shown in Fig. 3.63.
Interaction of particles during separation
The analysis of the particle motion in a ferrofluid assumed that particles were
non-interacting. This simplification ignores, however, the fact that non-magnetic
particles suspended in a magnetic fluid possess surface magnetic charges as a
result of magnetization of the surrounding fluid. Fujita and Mamiya [F14] de-
termined, theoretically and experimentally, that non-magnetic particles, placed
in a magnetized ferrofluid, attracted one another in the direction parallel to the
direction of magnetic field and repelled one another in the perpendicular direc-
tion. The situation is depicted in Fig. 3.64. The attractive force increases with
increasing density of the ferrofluid and magnetic field strength. The attractive
interaction was found to reduce the selectivity of separation of non-magnetic
particles in a magnetic fluid.