Environmental Engineering Reference
In-Depth Information
dimensionless variables:
[
S
]
β =
[
S
]
∞
K
m
V
max
k
L
a
v
[
]
∗
=
[
S
;
;
Θ =
.
[
S
]
∞
S
]
∞
The last one,
, is called the
Dämkohler number
. It is defined as the ratio of the
maximum rate of enzyme catalysis to the maximum rate of diffusion across the
boundary layer. The above equation can now be rewritten as
Θ
]
∗
)
1
.
1
Θ =
(
1
−[
S
+
(6.252)
]
∗
β[
S
If
Θ
1, the rate is controlled entirely by the reaction at the surface and is given by
V
max
[
1, the rate is controlled by mass transfer across
the boundary layer and is given by
k
L
a
v
[S]
∞
. From the above equation one obtains
the following quadratic in [S]
∗
:
S
]
/K
m
+[
S
]
, whereas if
Θ
1
β
=
]
∗
)
2
]
∗
−
(
[
S
+ α[
S
0,
(6.253)
where
α = Θ −
1
+
(
1
/
β
)
The solution to the above quadratic is
.
]
∗
=
2
4
βα
[
S
−
1
±
1
+
(6.254)
2
If
α
>
0, one chooses the positive root, whereas for
α
<
0, one has to choose the
negative root for a physically realistic value of [S]
∗
.As
α →
0, it is easy to show that
)
1
/
2
.
To analyze the lowering of reaction rate as a result of diffusional resistance to mass
transfer, we can define an
effectiveness factor
,
]
∗
→
[
β
S
(
1
/
ω
. The definition is
actual rate of reaction
rate of reaction unaffected by diffusional resistance
.
ω =
If the reaction is unaffected by diffusional resistance to mass transfer, the rate is given
by the Michaelis-Menten kinetics where the concentration at the surface is the same
as the concentration in the bulk solution. In other words, no gradient in concentration
exists in the boundary layer. Thus,
ω =
(V
max
[
S
]
/(K
m
+[
S
]
))/(V
max
[
S
]
∞
/(K
m
+[
S
]
∞
))
=
(
1
+ β
)
[
S
]
∗
/(
1
+ β[
S
]
∗
)
.
(6.255)
ω
varies between the limits of 0 and 1.
Let us now turn to the case where the resistance to mass transfer is within the
enzyme-grafted solid surface. In this case the diffusion of substrate within the particle
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