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3.8 T Fluid–Solid Fluidized Bed Reactors o-Phase, w 209

• The concentration of solids in the wake is the same as the concentration of solids in the
emulsion phase, and therefore, the gaseous void fraction in the we is also the same ak
as in the emulsion phase. Because the emulsion phase is at the minimum fluidizing
condition, the v e is equal to oid fraction in the w ak . hois quite tur- The w e, ak er , we v
fm
bulent, and the aelocities of both the solid and gas in the we are assumed to v erage v ak
be the same and equal to the upward velocity of the b ubbles.

Applying the appropriate material balances for the solids and the gas, the fraction of the
bed occupied by the bubbles and wakes can be estimated using the Kunii–Levenspiel model.
The fraction of the bed occupied by that part of the bubbles which does not include the wake,
is represented by the parameter whereas the v ubble olume of the b e per v ak , olume of the w
is represented by es is ak , . Consequently the bed fraction in the w and the bed fraction in
the emulsion phase (which includes the clouds) is 1 – – 1999) , (F . Then ogler

u
u ss bub (3.492)
1

The velocity of the gas in the emulsion phase is

u fm
u u (3.493)
e ss
fm

The is present in this equation because u fm elocity icial v is the superf, i.e., based on the
fm
cross section of the empty tube. The velocity of the moving solids, u ss , is positie in the v
downward direction here as in most of the fluidization literature. The velocity of the gas
in the emulsion u e is taken as positie in the upward direction, but note that it can be ne g-
v
ative under some conditions. The fraction of the bed occupied by bubbles is gien by v
(Fogler, 1999)

u u
s fm (3.494)
u bub u fm (1 )

Kunii and Levenspiel assumed that the last equation could be simplified to (F 1999) , ogler

u
u
s fm (3.495)
u bub

which is v alid if u bub >> u fm . The total mass of solids in the bed (solids holdup) is

)(1
M s AZ f (1 ) fm p (3.496)
ak
where is the volume of we per volume of bAccording to Leenspiel (1972), ubble. v