Environmental Engineering Reference
In-Depth Information
When the 4.6 hours travel time (over a distance of
500 m) occurs entirely during daylight hours, respiration
is nonexistent, and both photosynthesis and sediment
oxygen demand occur. In this case, the dissolved oxygen
deficit is given by Equations (4.127) and (4.116) as
is at t = f /2, and simple integration of Equation (4.129)
over a 24-hour period gives the daily average produc-
tion rate as
2
π
f
T
=
P
P
(4.130)
av
m
D x
( )
=
[
D x
( )
+
D
( )]
x
+
D x
( )
SF
BOD
S
where T = 24 hours. Chapra and Di Toro (1991) devel-
oped piecewise-periodic analytical solutions to Equa-
tions (4.128) and (4.129) by requiring that the Do
deficit at the beginning of the day is equal to its value
at the end of the day, and by also requiring that the
solutions be continuous at sunset. The analytical solu-
tion to Equations (4.128) and (4.129) during the photo-
period is
=
[
D x
( )
+
D
( )]
x
SF
BOD
S
+
S
+
S
x
V
p
r
b
1−
exp
k
a
k
a
3 0 5
0.7
.
0.7 500
2592
=
D (500)
=
[4.1]
1
exp
3.6
mg/L
which corresponds to a dissolved oxygen concentration
of 9.1 mg/L − 3.6 mg/L = 5.5 mg/L. Therefore, the dis-
solved oxygen concentration 500 m downstream of the
waste discharge is expected to fluctuate over the range
3.9-5.5 mg/L within 24 hours. As you might expect, this
would cause significant stress to the indigenous aquatic
life.
R
k
π
t
+
k t
a
(4.131)
D t
1 ( )
=
σ
sin
θ
γ
e
, 0
≤ ≤
t
f
f
a
and during the dark period, the solution is
R
k
k f
k t
(
f
)
D t
( )
=
σ
[
sin
( )
θ
+
γ
e
]
e
,
f
≤ ≤
t T
a
a
2
a
The analytical solutions presented here assume that
the distributed sources/sinks of oxygen and BoD do not
add flow to the stream, and therefore the flow rate, Q
and the average velocity, V , can be assumed constant for
the analysis. Any significant deviation from this assump-
tion will require that a numerical model be used.
(4.132)
where the parameter groups are
π
1
+
e
k T f
(
)
a
θ
=
tan
1
,
γ
=
sin
( )
θ
,
k f
1
e
k T
a
a
P
(4.133)
m
4.4.5 Chapra-Di Toro Model
σ
=
2
π
f
+
Chapra and Di Toro (1991) considered the case of a
stagnant water body in which respiration is the only
oxygen sink, and the oxygen production (photosynthe-
sis) rate is a function of the time of day, in which case
the oxygen deficit, D (ML −3 ), is given by
2
k
a
The solutions given by Equations (4.131) and (4.132)
satisfy the necessary periodic conditions that D 1 ( f ) =
D 2 ( f ) and D 1 (0) = D 2 ( T ), θ and γ are dimensionless, and
σ has units of M/L 3 . A typical solution is shown in Figure
4.10, along with the instantaneous production and Do
profiles. Figure 4.10 shows that the minimum and
maximum Do deficits both occur during the photope-
riod, with the maximum deficit occurring shortly after
sunrise and the minimum deficit occurring between
solar noon and sunset. This is always true for this model,
as can be demonstrated mathematically by differentiat-
ing Equations (4.131) and (4.132) with respect to t and
seeking their extrema, of which D 1 has two and D 2 has
none. The utility of Equations (4.131) and (4.132) is that
the observed oxygen deficit as a function of time can be
compared with these solutions, and the model parame-
ters k a , P av , and R , can be extracted by the best fit
between the observed data and the theoretical variation
of the of the Do deficit. The time of day, t min , at which
dD
dt
− ( )
(4.128)
+
k D R P t
a
=
where k a is the reaeration coefficient (T −1 ), R is the
(constant) respiration rate (ML−3T−1), −3 T −1 ), and P ( t ) is the
primary production (photosynthesis) rate (ML−3T−1). −3 T −1 ).
Defining sunrise as occurring at t = 0 and the photope-
riod (i.e., duration of sunlight) as f (T), the oxygen
production rate due to photosynthesis can be repre-
sented by
π
t
P t
( )
=
P
m max sin
, 0
(4.129)
f
where P m is the maximum instantaneous production
rate (ML −3 T −1 ). Equation (4.129) implies that solar noon
 
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