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144 3. Heterogeneous Processes and Reactor Analysis

For an ideal plug-flo the material balance (3.286) is greatly simplif w operation, ied:

C C
u R ) (3.287)
(
s
z t
This equation holds for adsorption, ion exchange as well as for catalytic systems, which
are in a transient operating condition, e.g. during se ation. F v ere catalyst deacti or a steady v
ix ed-bed operation, state catalytic f eq. (3.287) becomes
C d
0
u s ( R ) (3.288)
z d
It should be noted here that while in catalytic systems the rate is based on the moles dis-
xchange,
appearing from the fluid phase –r m , in adsorption and ion e the rate is normally
based on the moles accumulated in the solid phase r , and the rate is expressed per unit
m
mass of the solid phase as
q d
r (3.289)
m
t d
where q is the moles per unit mass of the solid phase (solid loading).
Note that the material balances for fed beds are valid for the case of constant-density ix
(constant volume) systems. The important term here is the one including the fluid v elo-
city, i.e. the term u ØC / Øz . For a variable volume system,
s
d F d x d( QC ) d Q d C (3.290)
F C Q
d V d V d V d V d V

For a constant volume system,
d Q
0 (3.291)
d V

e v and only then we ha

d F d C d C Q d C d C
Q Q u (3.292)
d V d V d( zA ) A d z s d z

Thus, the term u ØC / Øz comes from the derivative ØF / ØV , which should be used instead in
s
ariable-v eq. (3.287) for a volume system. For catalytic f in the typical case of a ix ed beds,
steady-state operation, eq. (3.288) takes the familiar form

d F
( ) R 0 (3.293)
d V