Environmental Engineering Reference
In-Depth Information
where
C
O
2
is the saturation concentration of oxygen in water (mg/L or kg/m
3
)
and
C
O
2
is the existing oxygen concentration in water (mg/L or kg/m
3
)
. The following
example will clarify the relationship between
L
o
and
Δ
for a flowing stream such as
a river.
E
XAMPLE
6.10 O
XYGEN
D
EFICIT IN A
P
OLLUTED
N
ATURAL
S
TREAM
Bioorganisms decompose organic molecules by consuming oxygen from water in the
process. At the same time, dissolution of oxygen from air into water tends to restore
the oxygen level in a polluted natural stream. The latter process is called
reaeration
,
and the former process is termed
deoxygenation
. The rate constant for deoxygenation
is
k
d
and that for reaeration is
k
r
, both being first-order rate constants. Assume that the
stream is in plug flow. The rate of increase in oxygen deficit
Δ
is given by
d
d
t
=
k
d
L
o
e
−
k
d
t
−
k
r
Δ
.
(6.86)
Integrating the above equation with the initial condition,
Δ = Δ
o
at
t
=
0, we obtain
k
d
L
o
k
r
−
k
d
e
−
k
d
t
e
−
k
r
t
.
Δ = Δ
o
e
−
k
r
t
+
−
(6.87)
This is the well-known
Streeter-Phelps oxygen-sag equation
, which describes the
oxygen deficit in a polluted stream.
t
is the time of travel for a pollutant from its
discharge point to the point downstream. Thus it is related to the velocity of the stream
as
t
=
y/U
, where
y
is the downstream distance from the outfall and
U
is the stream
velocity.
Subtracting the value of
Δ
as given above from the saturated value
C
O
2
gives the
oxygen concentration
C
O
2
at any location below the discharge point.
The above equation can be used to obtain the
critical oxygen deficit
(
Δ
c
) at which
point the rate of deoxygenation exactly balances the rate of reaeration, that is, d
Δ
/
d
t
=
0. At this point, we have the following equation, which gives
Δ
c
.
k
d
k
r
L
o
e
−
k
d
t
c
.
Δ
c
=
(6.88)
Thisgivesthe
minimumdissolvedoxygenconcentration
inthepollutedstream.Toobtain
the value of
t
c
at which the value of d
Δ
/
d
t
=
0, we can differentiate the expression
obtained earlier for
Δ
with respect to
t
and set it equal to zero. This gives a relationship
for
t
c
solely in terms of the initial oxygen deficit (
Δ
o
) as follows:
1
k
r
−
k
d
·
ln
k
r
k
d
1
−
.
(k
r
−
k
d
)
k
d
Δ
o
L
o
t
c
=
(6.89)
Figure 6.18 is a typical profile for
C
O
2
as a function of time
t
(or distance
y
from the
discharge point) in a polluted stream. In any given stream,
L
o
is given by the BOD of
the stream water plus that of the wastewater at the discharge point.
continued
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