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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.
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