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bubble flow depends on the hydrodynamic parameter
ij
that has to be experimentally
identified (or can be roughly estimated at 0.8).
b
b
b
u
V
=
ε
with:
(
)
mf
b
uu
V
− ⋅ =
ϕ
(8)
According to Equation 9, the bubble rise velocity
u
b
depends on the visible bubble flow
V
լ
b
, on the bubble size
d
v
and also on the diameter of the fluidized-bed riser, expressed
by the geometry parameter
ׇ
(Eq. 10) [8].
v
b
b
gd
V u
ϑ
71.0
+ =
(9)
m d
m dm
m d
d
r
r
r
r
0.1
0.1
1.0
1.0
0.2
0.2
63.0
>
< ≤
<
°
¯
°
®
=
ϑ
(10)
As Equations 9 and 10 illustrate, the bubble velocity increases with increasing bubble
size. Furthermore, it can be seen from the geometry parameter
ׇ
that, for smaller bed
diameters, the bubbles rise at a slower rate due to the dominance of wall effects. With
increasing bed diameter those wall effects are steadily reduced until they are negligible
at diameters greater than one meter. So the model can also be used for lab-scale and
pilot-plant fluidized beds with smaller diameters.
Since the distribution of the gaseous phase for each discretized height element,
ǻ
h
i
, is
known from the Equations 7 – 10, the solids volume concentration
c
v
is accessible so
that the bed´s overall pressure drop,
ǻ
p
fb
can be calculated by the summation of the
pressure drop of each element according to Equation 11.
(
) (
)
[
]
{
}
¦
=
Δ⋅ ⋅
⋅
− + ⋅
= Δ
N
i
i
ig
iv
s
iv
fb
h g
c
c
p
0
,
,
,
1
ρ
ρ
with:
(
)
vd
ib
iv
c
c
⋅
−=
,
,
1
ε
(11)
The first term here represents the pressure drop caused by the solids, and the second
one the gas phase. However the gas phase is almost negligible due to the large
difference between solids and gas densities,
ȡ
s
and
ȡ
g,i
. c
vd
is a correction value, when
calculating the solids volume concentration from gas bubble fraction, depending on gas
velocities and material properties, taken from [8]. Based on the pressure drop
calculations, information on the total height and the overall solids mass content of the
dense bed is available. On the other hand, the required bed mass can also be calculated
from the model when a certain pressure drop or bed height is desired.
Modeling the influence of internals on the fluid dynamics
As mentioned at the beginning, the Mueller-Rochow synthesis is an exothermic
reaction which means that the heat of reaction has to be removed in order to keep the
temperature, and therefore yield and selectivity, constant. For this purpose, different
types of vertical or horizontal heat-exchange internals, connected to a cooling system,
as shown exemplarily in Figure 5, are commonly used [13].
163