Biomedical Engineering Reference
In-Depth Information
the reaction rate is constant, the effectiveness factor can be computed from the ratio of the
active portion of the catalyst. That is,
r
R
p
3
R
p
r
0
R
p
h
¼
¼ 1
(17.45)
For a spherical coordinate, the generalized Thiele modulus is given by
r
r
max
2D
eA
C
AS
r
r
max
2D
eA
C
AS
Þ
2
½1 K
b
lnð1 þ K
1
a
R
p
3
b
f ¼ lim
K
b
/
0
¼
(17.46)
1 þ K
b
Thus, the effectiveness factor for spherical particles is given by
p
3
h
¼ 1;
for
f
(17.47a)
3
p
3
1
2
þ cos
4
2
3f
2
1
3
3
þ
1
p
h
¼ 1
3
arccos
;
for
f
>
(17.47b)
3
17.3.3. Isothermal Effectiveness Factor for
/
N
K
A
, the reaction rate becomes first order (with a rate constant being
r
max
/
K
A
)
and
Eqn (17.22)
is a linear differential equation. The linear differential equation can be solved
analytically to obtain closed form expressions.
The generalized Thiele modulus is given by
When
K
b
/
N
r
r
max
2D
eA
C
AS
r
r
max
D
eA
K
A
Þ
2
½1 K
b
ln ð1 þ K
1
a
¼
a
b
f
¼ lim
K
b
/
N
(17.48)
1 þ K
b
In this case, the value of
r
max
/
K
A
is finite, representing the rate constant for the first-order
reaction.
17.3.3.1. Effectiveness Factor for a First-Order Reaction in an Isothermal
Porous Slab
For a slab geometry, the solution to
Eqn (17.22)
is given by
C
AS
¼
coshð
x
p
Þ
¼
coshð
xf
d
p
Þ
sinhðfÞ
=
C
A
r
max
=K
A
=D
eA
Þ
d
p
p
(17.49)
sinhð
r
max
=K
A
=D
eA
and the effectiveness factor is given by
h ¼
tanh
f
(17.50)
f
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