Environmental Engineering Reference
In-Depth Information
(a spherical pellet) is balanced by the rate of conversion to products. If we consider a
spherical shell of thickness d
r
, the following mass balance for substrate [S] will hold:
rate of input at
(r
+
dr)
−
rate of output at
r
r
2
d
r
.
=
rate of consumption in the volume 4
π
r
2
∂
[
1
r
2
∂
∂r
S
]
D
s
=
r
S
,
(6.256)
∂r
where
D
s
is the diffusivity of substrate in the fluid within the pellet and
r
is the radius.
r
S
is given by the Michaelis-Menten kinetic rate law.
r
2
∂
[
1
r
2
∂
∂r
S
]
V
max
[
S
]
D
s
=
.
(6.257)
∂r
K
m
+[
S
]
The above second-order differential equation can be cast into a nondimensional
form by using the following dimensionless variables:
[
S
]
β =
[
S
]
∞
r
R
,
]
∗
=
K
m
;
r
∗
=
[
S
;
[
S
]
∞
where [S]
∞
is the substrate concentration in the bulk solution and
R
is the radius of
the pellet. The resulting equation is
(r
∗
)
2
∂
.
]
∗
∂r
∗
R
2
V
max
D
s
K
m
]
∗
1
(r
∗
)
2
∂
∂r
∗
[
S
[
S
=
(6.258)
]
∗
1
+ β[
S
,as
R
2
V
max
/
9
D
s
K
m
1
/
2
so that the observed
overall rate is expressed in moles per pellet volume per unit time, that is,
r
obs
S
We now define a
Thiele modulus
,
Φ
=
r
∗
=
1
]
∗
∂r
∗
3
R
D
s
∂
[
S
.
The above equation can now be written as
(r
∗
)
2
∂
2
)
.
]
∗
∂r
∗
]
∗
1
(r
∗
)
2
∂
∂r
∗
[
S
[
S
=
(
9
Φ
(6.259)
]
∗
1
+ β[
S
This equation can be solved numerically to get [S]
∗
as a function of
r
∗
, and further
to obtain the effectiveness factor
ω
as defined by
r
obs
S
r
s
]
∗
/∂r
∗
)
[
|
r
∗
=
1
(
3
/R)D
s
(∂
S
ω =
=
(6.260)
(V
max
[
S
]
/K
m
+[
S
]
)
Solutions to
are given in sources on biochemical engineering
(Lee, 1992; Bailey and Ollis, 1986), to which the reader is referred for further details.
ω
as a function of
Φ
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