Environmental Engineering Reference
In-Depth Information
photoperiod,
f
photoperiod,
f
D
max
photosynthesis
respiration
P
m
solar
noon
P
(
t
)
R
∆
P
ave
t
min
solar
noon
D
min
0
4
8
12
16
20
24
0
4
8
12
16
20
24
Time (h)
Time (h)
Figure 4.10.
Dissolved oxygen versus time for a time-varying production rate.
the dissolved oxygen deficit is a minimum, is obtained
by setting the first derivative of Equation (4.131) to
zero, which yields the following requirement:
π
t
f
f
−
min
π
cos
−
θ
(
k f
)
γ
e
−
k t
a
=
0,
t
>
2
(4.138)
min
a
min
π
t
f
−
f
max
π
cos
−
θ
(
k f
)
γ
e
−
k t
a
=
0,
t
<
2
(4.139)
max
a
max
1
2
φ
−
−
−
k
(
φ
+
f
/
2)
π
cos
π
+
θ
(
k f
)
γ
e
=
0,
φ
>
0
a
a
f
and
δ
(in Eq. 4.136) is defined by the relation
(4.134)
π
t
f
+
max
δ
= −
sin
−
θ
γ
e
−
k t
a
max
where
ϕ
(T) is the time between solar noon and the time
of minimum Do deficit, given by the relation
(4.140)
π
t
f
+
min
+
sin
−
θ
γ
e
−
k
a
min
f
(4.135)
φ=
t
−
min
The respiration rate,
R
, can be estimated from
k
a
and
P
av
by integrating Equations (4.131) and (4.132), which
yields
2
Equation (4.134) shows that
k
a
is expressed in terms of
ϕ
and
f
, which can be estimated from observations of
the Do versus time (
t
min
) and sunrise/sunset times (
f
).
To calculate the average production rate,
P
av
, using
Equations (4.131) and (4.132), the following relation
can be derived (Chapra and Di Toro, 1991):
(4.141)
R P
=
+
k D
a
av
where
D
is the diurnal average Do deficit obtained
from the Do measurements. Utilization of Equations
(4.134), (4.136), and (4.141) to est
i
mate
k
a
,
P
av
, and
R
from measurements of
ϕ
, Δ, and
D
is called the
delta
method
. It is important to keep in mind that the delta
method does not limit the lower bound of the Do, and
parameters that yield negative Do values should be
discounted. Furthermore, in cases where observations
indicate a minimum Do before sunrise and a maximum
Do before noon, application of the Chapra and Di Toro,
(1991) Do model is questionable.
Although modern programmable calculators can
solve Equation (4.134) for
k
a
with minimal effort, the
following approximation was proposed by McBride and
Chapra (2005) to estimate
k
a
in lieu of using Equation
(4.134),
∆
P
πδ
=
(4.136)
2 (
k f
a
)
2
+
π
2
av
where Δ is the difference between the maximum and
minimum oxygen deficit as defined by the relation
∆ =
D t
( )
|
−
D t
( )
|
(4.137)
1
t
=
t
1
t
=
t
max
min
where
t
max
and
t
min
are derived by finding the extrema of
D
1
in Equation (4.131), which
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