Biomedical Engineering Reference
In-Depth Information
where
U
is the relative velocity of the two particles,
a
is the relative radius
as defined by equation (12.8),
E
is the contact module, and
m
is the relative
mass of the particles, defined as
1
m
=
1
m
1
1
m
2
+
(12.11)
12.4.1.5
Force Balances and Parameters Computation
The equilibrium in a MSFB is reached (i.e., the bed particles do not move,
and therefore the bed “gets stabilized”) if the resultant force acting on the
particles is zero:
F
b
+
F
d
+
F
ip
−
F
g
= 0
(12.12)
with
F
ip
=
F
m
+
F
vw
+
F
e
+
F
col
(12.13)
Examples of the application of this balance may be found in literature,
and we may point out as an example (Augusto et al. submitted).
In the majority of the cases the magnetic force in Equation 12.13 is much
higher than other interparticle forces and, in these cases we obtain the follow-
ing equation from the force balance (Estevez et al. 1995):
H
g
=
S
u
2
ε
2
n
−
R
(12.14)
where
H
g
=
H
H
,
ε
is the void fraction (bed porosity),
n
is the Richardson
and Zaki index,
u
is the superficial velocity, and
S
and
R
are constants defined
as
S
=
3
f
D
∞
ρ
f
4
d
p
χµ
0
∇
and
R
=
g
(
ρ
p
−
ρ
f
)
χµ
0
, and
f
D
∞
is the individual drag coecient for
immersed bodies. This equation means that if we represent H
g
versus
u
2
we
obtain a straight line in the area where the magnetic bed behaves as fluidized,
as may be seen in the practical experiments depicted in Figure 12.4.
The minimum fluidization velocity,
U
mf
, may be computed by using the
general equations like (Hristov 2002):
U
mf
=Re
mf o
ν
d
p
e
aB−c
(12.15)
where Re
mfo
is the Reynolds number of the minimum fluidization velocity
(and a function of the Arquimedes number), a and c are constants,
υ
is the
kinematic viscosity of the fluid, and B is the acting magnetic field inside the
particles.
Elutriation velocity may be computed using the formula (Rodriguez et al.
2000):
K
e,i
d
pi
g
µ
(
u
o
−
10
−
9
Re
t
0
.
6
+2
.
5
10
−
5
Re
t
1
.
2
u
ti
)
2
=1
.
5
×
×
(12.16)
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