Biomedical Engineering Reference
In-Depth Information
17.3.3.2. Effectiveness Factor for a First-Order Reaction in an Isothermal
Porous Sphere
For a spherical geometry, the solution to
Eqn (17.22)
is given by
sinhðr
p
R
p
r
sinhð3
fr
=
R
p
Þ
sinh ð3
C
A
C
AS
¼
R
r
Þ
sinhðR
p
r
max
=K
A
=D
eA
p
Þ
¼
(17.51)
f
Þ
r
max
=K
A
=D
eA
and the effectiveness factor is given by
f
coth ð3
f
Þ1=3
h
¼
(17.52)
f
2
17.3.4. Effectiveness Factor for Isothermal Porous Catalyst
In the previous sections, we have learned that the effectiveness factor can be determined
fairly easily for slab and spherical geometries at the extreme values of
K
A
. For a finite value of
K
A
,
Eqn (17.22)
is nonlinear and closed form solution is not as easy to obtain. Therefore,
numerical solution is generally used.
17.3.4.1. Isothermal Effectiveness Factor in a Porous Slab
For “thin” film and surface attachment geometries as shown in
Fig. 17.4
a,
Eqn (17.22)
is
reduced to
d
2
C
A
d
x
2
þ r
A
¼ 0
D
eA
(17.53)
since the cross-sectional area perpendicular to the mass transport is constant and so is the
diffusivity. Again, let us consider the kinetics described by
Eqn (17.8)
, i.e.,
d
2
C
A
C
A
D
eA
d
x
2
r
max
K
A
þ C
A
¼ 0
(17.54)
Letting
C
A
þ
¼ C
A
=C
AS
and
x
þ
¼ x=
d
p
, one can reduce
Eqn (17.54)
to give
d
2
C
A
þ
d
x
2
þ
C
A
þ
f
2
ð1 þ K
b
Þ
2
½1 K
b
lnð1 þ K
1
2
Þ
K
b
þ C
A
þ
¼ 0
(17.55)
b
Central difference scheme can be applied to solve
Eqn (17.55)
efficiently. The effectiveness
factor can be obtained by integrating the reaction rate over the porous catalyst,
x¼d
ðr
AS
ÞV
d
C
A
d
x
D
eA
S
r
A
;
obs
r
AS
¼
h
¼
(17.56)
R
1
ðr
A
Þ
d
x
þ
Z
d
1
ðr
AS
ÞV
0
¼
ðr
A
ÞS
d
x
¼
r
AS
0
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