Biomedical Engineering Reference
In-Depth Information
From Rosensweig (1985), we know that the magnetic force,
F
m
,is
expressed by
1
4
π
M
F
m
=
∇
(
µ
0
H
)
(12.5)
where
µ
0
is the magnetic permeability of the vacuum,
H
is the magnetic field
intensity and
M
is the magnetization. If all particles are oriented according
to the direction of the magnetic field, for the
x
direction, equation (12.5)
reduces to
4
π
χ
m
µ
0
H
dH
1
F
m
=
(12.6)
dx
where
χ
m
is the mass magnetic susceptibility of the particles, and
dH/dx
is
the magnetic field gradient over the
x
direction.
12.4.1.2
Van der Waals Forces
These forces are not only present for atoms and molecules, but also for solid
particles. If we assume the particles to be spherical, the Van der Waals forces,
F
vw,
may be expressed by (Hamaker 1937)
F
vw
=
A
12
a
6
l
2
(12.7)
where
A
12
is the Hamaker's constant,
l
is the distance between the surfaces
of the two solid particles, and
a
is the relative radius defined as
1
a
=
1
a
1
1
a
2
+
(12.8)
where
a
1
and
a
2
are the radii of the spherical solid particles.
12.4.1.3
Electrostatic Forces
Electrostatic effects are always present, but they are more significant when
working with a gas-solid two-phase flow. The simple contact or collision of
the particles with each other, or with other surfaces, is enough to trigger the
electrostatic charging of the particles. This force,
F
e
, acts along a straight line
from one charged object to the other, and is expressed by
q
1
q
2
4
πε r
2
F
e
=
(12.9)
where
q
1
and
q
2
are the charges of the solid particles,
ε
is the permittivity of
the surrounding medium, and
r
is the distance between the particles.
12.4.1.4
Collisional Forces
Collisions between the particles in the bed exist when they move, even if
they quickly stabilize. For elastic spheres, the maximum collisional force in a
collinear impact between them is expressed by
3
E
√
a
15
mU
2
5
F
col
=
4
16
E
√
a
(12.10)
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