Environmental Engineering Reference
In-Depth Information
a large value, the specific growth rate plateaus out. However, this is not always the
case. In spite of this, the Monod kinetics is a simplified expression for growth kinetics
at low substrate concentrations.
Introducing the Monod equation in the overall expression we get
d
[
d
t
=
ζ
max
[
X
]
S
]
r
g
=
]
[
X
]
.
(5.205)
K
s
+[
S
When
[
S
]
K
s
,wehave
r
g
=
ζ
max
K
s
[
S
][
X
]=
k
[
S
][
X
]
,
(5.206)
where
k
is a pseudo second-order rate constant. This expression is an adequate
representation for the growth rate of microorganisms in the soil and groundwater
environments (McCarty et al., 1992).
k
has units of L/mol h (or dm
3
/mol h). Note
that although Monod kinetics for cell growth and the Michaelis-Menten kinetics for
enzyme catalysis give rise to similar expressions, they do differ in the added term [X]
in the expression for Monod kinetics.
The Michaelis-Menten expression gives the rate at which a chemical transfor-
mation occurs if the enzyme was separated from the organisms and the reaction
conducted in solution. If we represent the rate
r
as the
change in moles of substrate
per unit organism in a give volume per time
, then
R
]
0
is the
moles of chemical
transformed per unit volume per unit time
. [X]
0
is the initial number of organisms
per volume of solution. If
r
is given by the Michaelis-Menten expression with
V
max
in moles per organism per time, we have
=
r
[
X
d
[
]
d
t
=
S
V
max
[
S
]
R
=−
]
[
X
]
0
.
(5.207)
K
s
+[
S
The cell yield
y
c
is defined as the ratio of mass of new organisms produced to the
mass of chemical degraded:
(
d
[
X
]
/
d
t)
y
c
=
/
d
t)
.
(5.208)
−
(
d
[
S
]
Note that the maximum specific growth rate
ζ
max
is the product of the maximum rate
of degradation (
V
max
)
and
y
c
:
ζ
max
=
V
max
y
c
.
(5.209)
Hence we can write the Monod equation as
d
[
]
d
t
=
S
V
max
[
S
]
−
]
[
X
]
.
(5.210)
K
s
+[
S
If we identify
K
m
with
K
s
and [X] with [X]
0
, one can infer the Monod kinetics expres-
sion directly from the Michaelis-Menten kinetics expression. This corroborates the
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