Environmental Engineering Reference
In-Depth Information
into account that phosphorus and nitrate are produced by the degradation of organic
matter. Algae growth on the other hand consumes nutrients, nitrate-NO
3
and
phosphorus-HPO
4
, and produce oxygen.
Even more complex models may consider anoxic and anaerobic degradation as
well as growth and respiration of different bacteria populations (Vanrolleghem
et al.
2000
).
9.2 Details of Michaelis-Menten or Monod Kinetics
The Michaelis-Menten kinetics for the description of rate limited chemical and
biochemical degradation processes was introduced in Chap. 7. It can be written as
@
s
@t
¼r :¼
k
1
s
k
2
þ s
(9.3)
with maximum rate
k
1
and half-degradation concentration
k
2
.
s
denotes the concen-
tration of a chemical species, here called the substrate. It will be demonstrated that
the kinetics (
9.3
) may result from a sequence of reactions, in which aside from the
reactant and the final reaction product an enzyme
e
and an intermediate product
i
are involved. The system of chemical reactions can be noted as:
s þe
!
i ! pþe
The substrate
s
and enzyme
e
are connected with an intermediate species
i
by an
equilibrium reaction. Moreover, the intermediate
i
is broken down into the final
reaction product
p
with the enzyme
e
as a by-product in an irreversible reaction.
According to the law of mass action, the set of differential equations for the reaction
system is:
@s
@t
¼k
1
seþk
3
i
@
e
@t
¼k
1
se þk
3
i þk
2
i
@
i
@t
¼ k
1
se k
3
i k
2
i
@
p
@t
¼ k
2
i
(9.4)
with rate coefficients
k
1
,
k
2
and
k
3
. Transport is neglected in this system. It can be
shown (Fife
1979
; Morel and Hering
1993
) that for long times the behavior of the
four-species system can simpler be described by the following Michaelis-Menton
kinetics:
@
s
@t
¼
k
1
k
2
ði
0
þe
0
Þ
k
2
þk
3
þk
1
s
s
(9.5)
