Mass Transfer
463
For a first order reaction, Eq (4) is replaced by Eq (9).
4 a
=
(9)
and we get
^A,eff d f 2 & A
Ç2
d
^
- k xs A
= 0
Eq. (10) can be rearranged into
d
[V
? k isA
0
d £ j
D A:ejr
which can be solved by making the substitution
y
=
s A%
giving
d 2y
dÇ2
K
D
A,eff
y =
0
(
10
)
(
11
)
(
12
)
The general solution to Eq. (12) is given by
y =
C, sinh
+ C2
cosh
K
K
V
D A,eff j
(13)
With the appropriate boundary conditions, the specific solution becomes
sinh
5. =
S A , s ^ p
K
\
V
D A,eff )
sinh
R.
D
(14)
A .e ff
J
The calculated concentration profiles give an indication of the relation between reaction rate and diffusion
rate. If the concentration difference between center and surface of the particle is small, the maximum
diffusion rate is large compared to the reaction rate. However, if there is a large concentration difference
between the center and the surface, the maximum reaction rate is larger than the diffusion rate. The
concentration profiles suggest if starvation for substrate is likely to occur within the pellet._______________
The observed volumetric reaction rate,
qAobs
is obtained by integration of the reaction rate over the
pellet volume and division by the pellet volume i.e.
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