Environmental Engineering Reference
In-Depth Information
4.4.4.3  Distributed  Sources  of  BOD.  Spatially dis-
tributed sources of BoD (without addition of flow) are
typically associated degradation products from sedi-
ment decomposition that diffuse into the overlying
water. In cases where there is a distributed source of
BoD and atmospheric reaeration is the only source of
oxygen to the river, the governing equations for Do are
given by
Background deficit concentrations are typically on the
order of 0.5-2 mg/L, and are typically assumed as a
factor of safety is assessing the assimilative capacity of
streams.
EXAMPLE 4.18
A wastewater treatment plant discharges effluent into a
river that has a mean velocity of 3 cm/s and a tempera-
ture of 20°C. After initial mixing across the river, the
Do concentration of the mixed river water is 6 mg/L,
the ultimate BoD of this water is 20 mg/L, the labora-
tory rate constant for BoD decay in the river water is
estimated to be 0.5 d −1 , and the reaeration constant is
estimated to be 0.7 d −1 . Lateral distributed BoD sources
extend 4 km downstream of the wastewater discharge,
and these distributed sources are estimated to have an
average ultimate BoD of 3 mg/L. BoD removal by
sedimentation is estimated to increase the BoD decay
rate by 20%. Assess the impact of the distributed BoD
input on the oxygen concentration 4 km downstream of
the wastewater discharge. What is the asymptotic oxygen
deficit caused by the distributed BoD source?
c
x
(4.122)
V
= −
k L k c
+
(
c
)
d
a
s
dL
dt
= −
k L S
r
+
(4.123)
L
where S L is the BoD exerted by the distributed source
(ML −3 T −1 ), and the addition of the S L term in Equations
(4.122) and (4.123) is the only difference with the con-
ventional Streeter-Phelps formulation. Solving Equa-
tions (4.122) and (4.123) gives the oxygen deficit in the
river, D ( x ), as
D x D x
( )
=
( )
+
D
( )
x
(4.124)
SP
BOD
Solution
where D SP ( x ) is the oxygen deficit predicted by the
Streeter-Phelps equation (Eq. 4.71) for a point source
at x = 0 (if one exists), and Δ D BoD ( x ) is the additional
oxygen deficit caused by the distributed BoD source,
where (Thomann and Mueller, 1987)
At 20°C, the saturation concentration of Do is 9.1 mg/L
(see Table 2.2), and from the data given, L 0 = 20 mg/L,
k d = 0.5 d −1 , k s = 0.20(0.5 d −1 ) = 0.1 d −1 , k a = 0.7 d −1 , V =
3 cm/s = 2592 m/d, x = 4 km = 4000 m, and D 0 = 9.1
mg/L − 6 mg/L = 3.1 mg/L. The overall BoD removal
rate, k r , is given by
k S
k k
k x
V
k S
d
L
a
d
L
1
exp
(
k
k k
)
r
a
a
r
r
k
=
k
+
k
=
0.5
d
1
+
0.1
d
1
=
0.6
d
1
k
k
r
d
s
a
r
k x
V
k x
V
r
a
exp
exp
According to the Streeter-Phelps equation (Eq. 4.71),
in the absence of a distributed input of BoD, the Do
deficit 4 km downstream of the wastewater discharge is
given by
D
=
BOD
k S
k k
k x
V
d
L
a
1
exp
r
a
k
=
k
a
r
k
S
k
k x
V
d
L exp −
a
k L
k
k x
V
k x
V
r
d
0
r
a
D x
SP ( )
=
exp
exp
k
(4.125)
a
r
k x
V
exp
a
+
D
0
As a result of distributed BoD input, even in the
absence of discrete wastewater discharges (i.e.,
D SP ( x ) = 0), there is usually a deficit in Do in streams
and rivers that can be accounted for by distributed
BoD sources. Equation (4.125) indicates that the
oxygen deficit generated by a distributed source/sink
asymptotically (as x → ∞) approaches k d S L / k r k a since
0.5(20)
0.7 0.6
0.6 4000
2592
×
D SP (4000)
=
exp
0.7 40
×
00
0.7 4000
2592
×
exp
+
3.1
exp
2592
=
6.7
mg/L
k S
k k
which corresponds to a Do concentration of 9.1 mg/L −
6.7 mg/L = 2.4 mg/L. For a distributed source of BoD
d
L
lim
x
D
( )
x
= −
(4.126)
BOD
→∞
r
a
 
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