Biomedical Engineering Reference
In-Depth Information
That is, the solution we obtained so far is directly applicable since the derivative of a constant
is zero. By definition, the generalized Thiele modulus is given by
"
# 2
Z
C AS
f ¼ ð r AS Þ
a
2
ðr A ÞD eA d C A
(17.29)
C Ae ;0
Noting that
C 0
1 þ K C ¼ 0
C 0 Ae ;0 ¼ C Ae ;0
(E17-4.10a)
C 0
1 þ K C
C Ae ;0 ¼
(E17-4.10b)
Eqn (17.29) is integrated out to give
s
r max
2D eA C 0 AS
Þ 2
½1 K b lnð1 þ K 1
f ¼ a
b
(E17-4.11)
1 þ K b
with K b given by Eqn (17.30) , r max and K A given by Eqn (E17-4.8) , and
C 0
1 þ K C
C 0 AS ¼ C AS
(E17-4.12)
Indeed, the equation for the Thiele modulus is directly applicable, Eqn (E17-4.11) is identical
to (17.31) .
This last example shows that when different kinetics is observed, the same approach to
effectiveness factor can be applied. The key to the analysis is to convert all the dependent
variables (i.e. concentrations) into one single dependent variable (concentration). While the
final forms of equations may or may not the same, one can solve the equations in the same
manner. We also learned in this section that nonisothermal reaction systems can be treated
in the same way, with one additional (energy balance) equation.
17.5. E XTERNAL AND INTERNAL MASS TRANSFER E FFECTS
When both internal and external mass transfer effects are important, the rate of the reac-
tion system is governed by
at surface ¼ðr AS Þ
d C A
d x
k c aðC Ab C AS Þ¼aN A j To surface ¼D eA a
h
(17.79)
That is, mass transfer from the bulk fluid phase to the external catalyst surface is the same as
the mass transfer flux into the catalyst particle and the same as the observed reaction rate (or
effective reaction rate).
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