Biomedical Engineering Reference
In-Depth Information
TABLE 17.6
Internal Mass Transfer Effectiveness Factor
Geometry
Slab
Sphere
s
r
max
d
p
2D
eA
C
AS
r
r
max
2D
eA
C
AS
Þ
2
Þ
2
½1 K
b
lnð1 þ K
1
b
½1 K
b
lnð1 þ K
1
b
R
p
3
f ¼
Thiele
modulus,
f
f ¼
1 þ K
b
1 þ K
b
p
3
h
0
h
0
¼
1, for
f
1
h
0
¼ 1;
for
f
(
)
3
4p
2
3f
2
1
h
0
¼
f
, for
f
1
2
þ cos
3
þ
1
1
h
0
¼ 1
3
arccos
;
p
3
for
f
h
1
¼
tanh
f
f
f cothð3fÞ1=3
f
2
h
1
h
1
¼
h
0
½0:15K
1
þ 0:3lnð1 þ K
1
b
h
0
½0:186 K
1
þ 0:306 lnð1 þ K
1
b
Þ þ h
1
Þ þ h
1
b
b
h
h ¼
h ¼
1 þ 0:15K
1
b
þ 0:3lnð1 þ K
1
b
1 þ 0:186 K
1
b
þ 0:306 lnð1 þ K
1
b
Þ
Þ
1:553 K
b
0:52 þ K
b
0:16K
b
lnð1 þ K
1
b
1:897 K
b
0:605 þ K
b
0:193K
b
lnð1 þ K
1
b
f
1
> 0:95 þ
f
1
> 1:442 þ
h >
0.95
Þ
Þ
and
E
a
RT
S
g
¼
(17.75)
where
DH
R,A
is the heat of reaction per unit A reacted,
k
eT
is the effective thermal conduc-
tivity in the particle,
T
S
is the temperature at the external surface,
E
a
is the activation energy
(governing
r
max
). The temperature change is linearly related to the concentration change if
the parameter
b
is constant. That is,
b
"
#
T
T
S
¼ 1 þ
C
A
C
AS
1
¼ 1 þ
b
ð1 C
A
þ
Þ
(17.71)
The parameter
b
can be viewed as the ratio of maximum possible temperature difference
divided by the external surface temperature. Since the maximum temperature deviation
from the external surface is at the location where concentration of A is zero. The rate of reac-
tion for a nonisothermal system is given by
T
g
!
"
#
r
max;0
C
A
K
A
þ C
A
exp
T
S
r
max
;0
C
A
þ
K
b
þ C
A
þ
exp
g
r
A
¼
¼
(17.76)
1 þ
b
bC
A
þ
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