Environmental Engineering Reference
In-Depth Information
E
XAMPLE
5.5 S
ERIES
R
EACTIONS
The consumption of oxygen (oxygen deficit) in natural streams occurs due to biological
oxidation of organic matter. This is called the biochemical oxygen demand (BOD). The
oxygen deficit in water is alleviated by dissolution of oxygen from the atmosphere.
These two processes can be characterized by a series reaction of the form
k
1
−−−−−→
oxidation
k
2
−−−−−−−→
oxygenation
organic matter
oxygen deficit
oxygen restoration,
(5.45)
A
k
1
−→
B
k
2
−→
C.
In this case, the initial conditions are somewhat different from what was discussed
earlier.
Here at
t
=
0,
[
A
]=[
A
]
0
,
[
B
]=[
B
]
0
, and
[
C
]=[
C
]
0
. The solution to the problem
is given by
]
0
e
−
k
1
t
,
[
A
]=[
A
k
1
k
2
−
k
1
[
A
]
0
(
e
−
k
1
t
−
e
−
k
2
t
)
+[
B
]
0
e
−
k
2
t
,
[
B
]=
(5.46)
1
−
(k
2
e
−
k
1
t
+
k
1
e
−
k
2
t
)
(k
2
−
k
1
)
+[
B
]
0
(
1
−
e
−
k
2
t
)
+[
C
]
0
.
[
C
]=[
C
]
0
Given a value of
k
1
=
0.1 d
−
1
,
k
2
=
0.5 d
−
1
, initial organic matter concentration
[A]
0
=
10 mg/L,andinitialoxygendeficit[B]
0
=
3 mg/L,after24 h(1day),theoxygen
deficit will be
[
B
]=
(
0.1
)
(10/0.4)(e
−
0.1
e
−
0.5
)
+
(
3
)
e
−
0.5
−
=
2.56 mg/L.
A frequently encountered reaction type in environmental engineering is one in
whichapre-equilibriumstepprecedestheproductformation.Thesteady-stateconcept
is particularly useful in analyzing such a reaction.
k
f1
k
b1
B
k
2
A
−→
C.
(5.47)
Most enzyme reactions follow this scheme. It is also of interest in many homogeneous
and heterogeneous reactions (in both the soil and sediment environments), that is,
those that occur at interfaces. For the above reaction
d
[
]
d
t
=
B
k
f1
[
A
]−
k
b1
[
B
]−
k
2
[
B
]
.
(5.48)
Using the pseudo-steady-state approximation we can set d[B]/d
t
=
0. Hence
k
f1
k
b1
+
[
B
∗
]=
[
A
]
.
(5.49)
k
2
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