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68 3. Heterogeneous Processes and Reactor Analysis

The reciprocal of this coeficient is referred to as total resistance: f

1 1 1
(3.35)
k ov k g s s k

It should be noted that the o f icient ( v erall coef k ov ) has been deried under the assumption v
that the reaction is taking place mainly in the internal surface of the catalyst. See Chapter
5 for a more rigorous analysis on the subject.

ate-contr The concept of the r olling step
v
When a process consists of seeral individual steps in series, the concept of the rate-
controlling step simply states that one of the seeral steps ined provides the major v olv
v
v
resistance to the oerall process. In such a case, this slow step is termed the “rate-
controlling step” and can be considered as acting alone (Le 1972). v enspiel,
wing,
Instead of molecules flo consider water flowing through a pipe with a number of
alv partially opened vw of the water in the pipe will be determined by the v The flo es. e alv
that offers the largest resistance to the flow. Actually, we could come up with a fairly accu-
rate estimate of the flow by calculating the resistance to the flow in this v ne e, alv glecting
all the others.
Consider the first-order reaction analyzed in the previous paragraph. In the limiting case
where k →∞ or k k , the resistance to the oerall rate is due to the gas film around v
g
s
s
the catalyst and C s → 0. The rate-controlling step is the diffusion in the gas film and the
overall rate is
( r ) C k g G (3.36)
s
On the other hand, if k g →∞ or k g k , the resistance to the oerall rate is owing to the v
s
intrinsic reaction rate and C G → C s . The rate-controlling step is the reaction rate and the
overall rate is

( r ) k C s s G (3.37)
s
vidual rates, Note that due to the equality of the indi if k →∞ then C s → 0, and if k g →∞
s
then ( C G – C ) → 0 or C G → C ; and so, the indi er- v inite and equal to the o vidual rates are f
s
s
all rate. It is the resistance of the individual step and not the corresponding rate that could
be zero under certain operating conditions .
The concept of the rate-controlling step is much more useful in complex kinetic expres-
sions, where the o v erall rate is nonlinear and cannot be obtained by following a simple pro-
cedure as presented aboe for the case of a first-order reaction. v
For example, consider a second order reaction. In this case, the intrinsic reaction rate is

( r s ) C k s s 2 (3.38)

while the rate of mass transfer from the bulk of the gas to the catalyst surface is
r k C ( G C s ) (3.39)
g
g