Biomedical Engineering Reference
In-Depth Information
With kinetics given by
Eqn (17.8)
, the generalized Thiele modulus is given by
r
r
max
2D
eA
C
AS
Þ
2
½1 K
b
lnð1 þ K
1
¼
a
b
f
(17.31)
1 þ K
b
17.3.2. Isothermal Effectiveness Factor for
0
K
A
[
0, the reaction rate becomes constant and
Eqn (17.18)
can be integrated more
easily. Although the reaction rate is constant, one must be aware that the reaction rate would
be zero if there were no reactants available to react.
When
K
b
/
17.3.2.1. Effectiveness Factor for a Zeroth-Order Reaction in an Isothermal
Porous Slab
For a slab (
Fig. 17.4
a), or catalyst grown (or attached) to a flat wall,
Eqn (17.22)
can be
integrated to give
D
eA
d
C
A
d
x
Z
Z
x
d
C
A
d
D
eA
d
x
¼
r
max
d
x
(17.32)
x
0
0
which yields
d
C
A
d
x
¼ r
max
ðx x
0
Þ
D
eA
(17.33)
Integrate once more
Z
C
A
Z
x
D
eA
d
C
A
¼
r
max
ðx x
0
Þ
d
x
(17.34)
x
0
0
Thus, the concentration inside the porous catalyst is given by
r
max
2D
eA
ðx x
0
Þ
2
C
A
¼
(17.35)
At the outer surface, the concentration is known and
r
max
2D
eA
ð
d
p
x
0
Þ
2
C
AS
¼
(17.36)
which can be solved to obtain the value of
x
0
. There are two roots, but only one of them is
physical (i.e. less than
d
p
):
s
2D
eA
C
AS
r
max
x
0
¼
d
p
(17.37)
Clearly, when
x
0
¼
0, the effectiveness factor is unit (no mass transfer effect). Because the
reaction rate is constant, the effectiveness factor can be computed from the ratio of the active
portion of the catalyst. That is,
h ¼ 1;
for
f 1
(17.38a)
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