Geoscience Reference
In-Depth Information
In the 3-D model,
DC
/
Dt
is
ε
DC
Dt
=
∂
C
∂
+
∂(
u
x
C
)
+
∂(
u
y
C
)
+
∂(
u
z
C
)
−
∂
∂
x
∂
C
t
∂
x
∂
y
∂
z
x
∂
x
ε
−
∂
∂
y
∂
C
−
∂
∂
z
∂
C
ε
(12.47)
y
∂
y
z
∂
z
ε
i
(
=
)
where
C
is the local constituent concentration, and
i
x
,
y
,
z
are the turbulent
diffusivities.
A common kinetic model that is adequate for many processes is the first-order
(linear) kinetics:
DC
Dt
=−
KC
(12.48)
where
K
is the first-order rate coefficient (day
−
1
).
A more complex formulation is the Michaelis-Menten or Monod kinetics:
DC
Dt
k
s
C
k
1
/
2
+
=−
(12.49)
C
where
k
s
is the limiting reaction rate when
C
k
1
/
2
; and
k
1
/
2
is called the half-
saturation constant, because
DC
k
1
/
2
.
The Michaelis-Menten kinetics may be written as Eq. (12.48), with the rate
coefficient:
/
Dt
is half the limiting value when
C
=
k
s
k
1
/
2
k
1
/
2
k
1
/
2
K
=
C
=
K
0
(12.50)
+
+
C
where
K
0
is the first-order rate coefficient, defined as
K
0
k
1
/
2
. Both linear and
Michaelis-Menten kinetics are depicted in Fig. 12.2. One can see that when
C
=
k
s
/
k
1
/
2
,
the Michaelis-Menten kinetics becomes the first-order kinetics.
Eq. (12.49) is the general formulation of the Michaelis-Menten kinetics. Its variants
for different species can be found in the next subsection.
The rate coefficient
K
usually depends on temperature. This is often described with
reference to the rate at 20
◦
C:
T
−
20
K
(
T
)
=
K
(
20
)θ
(12.51)
where
T
is in degree Celsius, and
θ
is a coefficient that is typically in the range of 1.01
to 1.10.
Eq. (12.51) implies that the reaction rate increases with temperature, as shown in the
dashed line (Theta) in Fig. 12.3. However, for many species, such as phytoplankton,
the temperature dependence is zero at a minimum temperature, increases to a peak
