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3.6 T w ed Beds o-Phase Fix 147

concentration of the reacting species. Then, as analyzed in detail in Section 5.3.4, the pres-
sure drop equation is coupled with the material and energy balance equations:
d P P i T i
( fx , , ) (3.303)
d z A P T

and the reactor model becomes a system of three, coupled, differential equations, which
should be solved simultaneously.
In liquid-phase systems, or generally in incompressible flothe ef , , w fect of pressure drop
on the concentration of solutes (on the density of the fluid) is negligible (F 1999). , ogler
Thus, when designing a f the pressure drop is excluded from the calcula- ix ed-bed reactor ,
we,
ed,
olv
v
er
tions. Ho in the case where a gas phase is in or more generally in a com-
v
pressible flow, the concentration of the gas species (the gas density) is proportional to the
total pressure and thus, the pressure drop may play a significant role in the f ix ed-bed
design. In general, a compressible flo w where the change in density is more w is a fluid flo
than 5 –10% (Perry and Green, 1999).
As can be proved (see Section 5.3.4), the pressure drop becomes independent of the con-
version and thus from the material balance if the expansion factor is near zero, and then
the two differential equations are decoupled. Furthermore, for nearly isothermal operation,
the pressure drop is not a function of temperature. Under these conditions, the f ed-bed ix
ied. model is greatly simplif
Considering most environmental applications, for catalytic as well as for adsorption
operations, the gas species to be remoed are in such low concentrations (large excess of v
inerts) that the expansion factor is practically zero and the temperature is nearly constant
throughout the reactor v olume.

The case of incomplete filling of the fixed bed with the flowing fluid
In the analysis above, the v olume of the fixed bed ( oid v V R ) is considered to be fully filled
with the fluid phase, i.e. the fluid holdup based on the total volume of the bed h e,t is equal
.
to the bed v oidage While this is expected in the case of a gas as fluid, it is not al ays w
true in the case of a liquid, especially in the case of dow operation. In the case of wnflo
incomplete filling of the bed with the fluid phase, the acti bed v and thus e v , wer olume is lo
a portion of the solid phase is not in contact with the fluid, or in other w is inacti ords, v e in
terms of reaction.
The void bed volume is V R while the total volume occupied by solid is
V R V R (1 V) R (3.304)
However, in the case of incomplete filling of the fed bed with the flo the v ix wing fluid, ol-
ume of the bed occupied by the fluid is h e,t V R . Then, the fraction of the void bed v olume
occupied by the fluid is equal to the fluid void bed v i.e. olume, olume/v
hV e,t R h e,t
h (3.305)
v,t
V
R