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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