Biomedical Engineering Reference
In-Depth Information
Thus, the asymptotic behavior can be obtained via the generalized Thiele modulus,
"
Z
1
#
2
r
r
max;0
2D
eA
C
AS
d
C
A
þ
e
g
1 þ K
b
a
C
A
þ
K
b
þ C
A
þ
exp
g
f
¼
(17.77)
1 þ
b
bC
A
þ
0
For exothermic reactions, the effectiveness factor can be greater than unity due to the higher
reaction rate at internal sites due to the slow heat removal.
Overall effectiveness factor
r
A
;
obs
r
A
;
b
¼
h
o
¼
h
e
h
(17.80)
The use of
r
A,b
to define the overall effectiveness factor is for the convenience of reactor
design calculations. Since the bulk phase concentrations are easily obtained and/or
measured, reaction rate based on bulk concentrations are preferred.
For biocatalyst immobilized system via encapsulation, capsule size, biocatalyst loading
(inside the capsule), capsule shell material or permeability, and the capsule shell thickness,
all play important roles on the reaction system as a whole. Smaller (capsule) particle size leads
to highermass transfer rate andhigher effectiveness factor.When capsule shell permeability
k
p
is increased and/or the shell thickness is decreased, mass transfer rate increases and thus
favorable to the reaction system as whole, however, there is a limitation on the capsule shell
material: containing the biocatalyst inside the shell while sustaining the reaction environment.
For enzyme (or biocatalyst) loading, the situation can be more complicated. Enzyme
loading affects maximum rate
r
max
as well as the diffusivity inside the shell
D
eA
. Since
maximum reaction rate is proportional to the enzyme loading, higher biocatalyst loading
leads to higher maximum rate. Higher biocatalyst loading decreases the effective diffusivity.
Therefore, higher biocatalyst loading results in a higher Thiele modulus and lower effective-
ness factor. If the effect of biocatalyst loading on the diffusivity is negligible, the overall effect
of higher biocatalyst loading is higher net reaction rate. However, there is a physical limit on
how much biocatalyst loading can be achieved.
When external surface area is important, the overall surface effectiveness factor is given by
a
e
a
i
h þ
h
þ a
r
1 þ a
r
h
s
¼
¼
(17.91)
a
e
a
i
1 þ
which can employed to substitute the effectiveness factor for internal mass transfer as if there
is no external surface area.
The overall effectiveness factor with internal and external surface area effects can then be
defined as
r
A
;
obs
r
A
;
b
¼
h
þ a
r
1 þ a
r
h
o
¼
h
e
h
s
¼
h
e
(17.92)
where
a
r
is the ratio of external surface area to the internal surface area.
In general, iterative solutions are necessary when both internal and external mass transfer
effects are important. Reactor analysis including the mass transfer effects can be incorporated
by including the effectiveness factor with the reaction rate as one would.
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