Biomedical Engineering Reference
In-Depth Information
where
S
V
a ¼
(17.7)
is the specific external surface area of the catalyst particles.
One can infer from Chapters 8, 9, and 11 that the catalytic reaction rates are usually of the
Langmuir form. For simplicity, let use the following rate form
r
max
C
A
K
A
þ C
A
r
A
¼
(17.8)
as the base of our discussions. Here,
r
A
is the rate of disappearance of A based on the total
volume of catalysts, and
C
A
is the concentration of A in the fluid phase. Eqn
(17.8)
can be
unimolecular catalytic reaction rate, Michaelis
e
Menten equation, or Monod equation.
When different kinetics is involved, one can replace the rate form and follow with the anal-
yses as we will proceed.
In our analyses prior to this chapter, we have been assuming that the reaction rates can be
calculated with the concentration in the bulk fluid phase. This can be true if only the mass
transfer effect is negligible. There is a difference between the convenient rate form (based
on the bulk fluid phase concentration) and the intrinsic kinetic rate form. Let us define an
external effectiveness factor
r
AS
r
A
h
e
¼
(17.9)
where
r
AS
is the rate calculated based on the concentrations available to the reaction (i.e. at
the catalyst surface) and
r
A
is the rate calculated based on the bulk concentrations. It is
thus intuitive that if mass transfer effects are negligible, the effectiveness factor
h
should
be unity.
Let us consider the kinetics as given by
Eqn (17.8)
, the effectiveness factor can be obtained
if we know the value of
C
AS
for a given value of bulk concentration
C
Ab
. Substituting
Eqn (17.8)
into
Eqn (17.6)
, we obtain
r
max
C
AS
K
A
þ C
AS
k
c
aðC
Ab
C
AS
Þ¼r
AS
¼
(17.10)
which can be reduced to a quadratic equation. There are two roots or solutions for the
concentration at the surface. However, only one of them is physical, i.e. greater than zero
and less than the bulk concentration. The reactant concentration of A on the catalyst surface
can be solved to give
s
1
4
C
Ab
K
A
C
Ab
K
A
2
C
AS
¼
1
2
r
max
k
c
a
r
max
k
c
a
þ
þ C
Ab
K
A
(17.11)
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