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210 3. Heterogeneous Processes and Reactor Analysis
experiments showed that ranges between 0.25 and 1. the fraction of the bed occu- Then,
pied by clouds is
3 u
fm (3.497)
c
u u
fm b s fm
enspiel,
v
v
Furthermore, Leenspiel defined the following parameters (Le 1972; F , ogler
1999):
3 u
) fm u s fm u ) fm c (3.498)
(1
(1
c
fm b
fm
(1 )(1 )
fm ( ) (3.499)
e
c b
where:
the volume of solids dispersed in bubbles per volume of bubbles
b
the volume of solids within clouds and wes per volume of b ak ubbles
c
the volume of solids in emulsion per volume of b ubbles.
e
v
Typical values of , , and are 0.005, 0.2, and 1.5, respectiely (Le 1972; enspiel, v
b c e
Fogler, 1999). A weakness of this model is that there does not yet exist a reliable method
or correlation for determining the v alue of .
b
In Figure 3.59, the flow pattern in a fluidized bed is sho wn.
Bubble velocity and diameter In the context of the bubbling bed theory, the rise v eloc-
ity of a single bubble in a fluidized bed is gien by (W 1984) en, v
u 0.711( gd) 0.5 (3.500)
bs bub
ubble, whereas the rise velocity of b cloud, and we is ak
u bub u s u 0.711( gd bub ) 0.5 (3.501)
fm
The values of the rise velocity of the bubbles estimated by the equation above are generally
higher than the ones predicted by the two-phase-model of Davidson. This difference can be
explained as follows. The single-bubble velocity can be affected by various other f actors,
especially when many bubbles are present. Specifically, an increase in the number of bubbles
ubbles carry each other up leads to a lower drag force on each bubble. In other words, the b
through the bed. The number of bubbles formed depends on the gas velocity. The greater the
o gas velocity, the greater the flow rate and thus, the more the number of b T ubbles formed.
elocity results in an increase in the real bubble velocity. The sum up, an increase in the gas v
bubble velocity is also affected by the viscosity of the gas and the size and density of the solids
indirectly, since all these parameters affect the minimum fluidized velocity. Specifically, a low
viscosity and a large size of particles with high density increases the minimum fluidization
velocity, which in turn leads to a decrease in the velocity of the rising bubble.