Biomedical Engineering Reference
In-Depth Information
Eqn
(17.60)
can be rewritten as
d
d
r
þ
d
C
Aþ
d
r
þ
3
C
A
þ
K
b
þ C
A
þ
r
2
þ
f
2
ð1 þ K
b
Þ
2
½1 K
b
lnð1 þ K
1
r
2
þ
¼ 0
Þ
(17.62)
b
Again, central difference scheme can be applied to solve
Eqn (17.62)
efficiently. The effective-
ness factor can be obtained by integrating the reaction rate over the porous catalyst,
x¼d
p
ðr
AS
ÞV
d
C
A
d
x
3
R
1
0
D
eA
S
ðr
A
Þr
2
þ
R
p
d
r
þ
Z
r
A
;
obs
r
AS
¼
1
ðr
AS
ÞV
h ¼
¼
ðr
A
ÞS
d
r
¼
(17.63)
r
AS
0
Integration can be obtained via either the Trapezoidal or the Simpson's rule. Because central
difference method as well as the Trapezoidal rule or Simpson's rule all have symmetrical
error sequences, an extrapolation technique can be employed to speedup the convergence.
Table 17.4
shows the converged solutions obtained from Excel.
Fig. 17.7
shows the variation of the effectiveness factor with Thiele modulus for diffusion
and reaction inside porous spheres. One can observe that the overall behavior in a spherical
geometry is quite similar to that in a porous slab.
Similar to
Eqn (17.58)
, the effectiveness factor for a sphere can be computed by,
0
½0:186K
1
þ 0:306 lnð1 þ K
1
b
h
Þ þ
h
1
b
h
¼
(17.64)
1 þ 0:186 K
1
b
þ 0:306 lnð1 þ K
1
b
Þ
with
h
0
the effectiveness factor for
K
b
¼
0 and
h
1
is the effectiveness factor for
K
b
/
N
. The
maximum error from
Eqn (17.64)
is within 1%.
17.4. MASS TRANSFER EFFECTS IN NONISOTHERMAL
POROUS PARTICLES
We have learned that diffusion and reaction in a porous catalyst is governed by
Eqn
(17.22)
,
d
d
x
D
eA
S
d
C
A
d
x
þ r
A
S ¼ 0
(17.22)
which was obtained through mass balance in a differential volume depicted by
Fig. 17.5
.
Under nonisothermal conditions, the effect of temperature must be considered as the rate
of reaction is a function of temperature. Similar to
Eqn (17.22)
, energy balance, i.e. Eqn
(3.113) applied to the differential volume in
Fig. 17.5
, leads to
d
d
x
k
eT
S
d
T
d
x
þðDH
R
;
A
Þr
A
S ¼ 0
(17.65)
where
k
eT
is the effective thermal conductivity in the porous catalyst,
DH
R,A
is the heat of
reaction per unit amount of A (consumed).
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