Environmental Engineering Reference
In-Depth Information
Solution
Substituting the given data and the calculated deriva-
tives into the mass balance equation, Equation (7.62)
yields
From the given data:
A
= 1 ha = 10
4
m
2
, Δ
z
= 1 m,
S
m
= 1.25 (mg/L)/d,
K
z
= 0.35 cm
2
/s = 3.0 m
2
/d,
Q
in
=
Q
out
= 0 m
3
/s, Δ
t
= 1 day,
w
= 0.5,
c
U,
t
= 8.0 mg/L,
c
L,
t
= 5.0 mg/L,
c
t
= 7.0 mg/L,
c
U,
t
+Δ
t
= 7.5 mg/L,
c
L,
t
+Δ
t
=
4.2 mg/L, and
c
t
+ Δ
t
= 6.2 mg/L. The volume of the layer
of interest is given by Equation (7.64) as
∂
∂
c
t
∂
∂
2
c
z
∂
∂
c
z
V
=
AK
2
∆
z Q
+
∆
z Q c Q c VS
+
−
+
z
z
in in
out
m
(
10
4
)(
c
t
−
7 0
. )
+
∆ new
t
,
=
(
10
4
)( . )(
3 0
−
1 25 1
.
)( )
+
( )( .
0 1 78 1
)( )
V A z
=
∆
=
(
10
4
)( )
1
=
10
4
m
3
+ −
0 0
+ (
10
4
)( .
1 25
)
At times
t
and
t
+ Δ
t
, the derivatives of O
2
concentration
distribution are given by Equations (7.65-7.67) as
which yields
c
t
+Δ
t
,new
= 4.5 mg/L. Hence, on the next iter-
ation, the O
2
concentration in the current layer at time
t
+ Δ
t
is to be taken as 4.5 mg/L. It should be apparent
that a numerical scheme that uses
c
t
+Δ
t
,new
in calculating
the spatial derivatives is likely to lead to a more rapid
convergence of the concentrations distribution at time
t
+ Δ
t
.
∂
∂
2
c
c
−
c
c
−
+
c
1
U
,
t
t
t
L
,
t
≈
−
z
2
(
∆
z
+
∆
z
)
/
2
(
∆
z
∆
z
)
/
2
∆
z
U
L
t
8 0 7
.
−
.
0
7 0 5 0
1
.
−
.
1
1
=
−
1
2
= −
1 0
. (
mg/L /m
)
c
−
c
c
−
+
c
∂
∂
c
z
1
2
U
,
t
t
t
L
,
t
≈
+
(
∆
z
+
∆
z
)
/
2
(
∆
z
∆
z
)
/
2
7.5.2.2 Conservation of Energy Model.
One-dimen-
sional heat models are mostly concerned with predict-
ing vertical variations in temperature in large reservoirs.
Perhaps the simplest one-dimensional steady-state heat
model is one in which an isothermal well-mixed epilim-
nion and an isothermal well-mixed hypolimnion both
exist, there is steady-state heat exchange between the
epilimnion and hypolimnion, and there is steady-state
heat exchange with the atmosphere at the reservoir
surface. The governing equations for this circumstance
are as follows:
t
U
L
8 0 7 0
1
.
−
.
7 0 5 0
1
.
−
.
1
2
=
+
=
1 5
. (
mg/L /m
)
∂
∂
c
z
2
c
−
+
c
c
−
+
c
1
U t
,
+
∆
t
t
+
∆
t
t
+
∆
t
L
,
t
+
∆
t
≈
−
2
(
∆
z
∆
z
)
/
2
(
∆
z
∆
z
)
/
2
∆
z
U
L
t
+
∆
t
7 5 6 2
1
.
−
.
7 0 4 2
1
.
−
.
1
1
=
−
2
= −
1 5
. (
mg/L /m
)
c
−
c
c
−
+
c
∂
∂
c
z
1
2
dT
dt
U
,
t
+
∆
t
t
+
∆
t
t
+
∆
t
L
,
t
+
∆
t
≈
+
e
V c
ρ
=
Q c T t Qpc T
ρ
( )
−
+
JA v A c T T
+
ρ
(
−
)
(
∆
z
+
∆
z
)
/
2
(
∆
z
∆
z
)
/
2
e
p
p in
p e
s
t
t
p
h
e
t
+
∆
t
U
L
(7.68)
(7.69)
7 5 6 2
1
.
−
.
7 0 4 2
1
.
−
.
1
2
dT
dt
=
+
h
V c
ρ
=
v A c T T
t
ρ (
−
)
h
p
t
p
e
h
=
2 05
.
(
mg/L /m
)
where
V
e
and
V
h
are the volumes of the epilimnion and
hypolimnion, respectively (L
3
),
ρ
is the density of water
(ML
−3
),
c
p
is the specific heat of water (EM
−1
·°C
−1
),
T
e
and
T
h
are the temperatures of the epilimnion and hypo-
limnion, respectively (°C),
t
is time (T),
Q
is the flow
rate into the epilimnion (L3T−1),
3
T
−1
),
T
in
is the temperature
of the inflow (°C),
J
is the surface heat flux into
the epilimnion (EL−2T−1),
−2
T
−1
),
A
s
is the surface area of the
reservoir,
ν
t
is the thermocline heat transfer coefficient
(LT
−1
), and
A
t
is the surface area of the thermocline
interface (L
2
). The thermocline heat transfer coefficient,
ν
t
, is related to the vertical diffusion coefficient in the
thermocline,
E
t
(L
2
T
−1
) by the relation
∂
∂
c
t
c
−
c
c
−
7 0
.
t
+
∆
t
,
new
t
t
+
∆
t
,
new
≈
=
=
c
−
7 0
. (
mg/L /d
)
t
+
∆
t
,
new
∆
t
1
where
c
t
+Δ
t
,new
is the new iterative estimate of
c
t
+Δ
t
. Using
a weight of
w
= 0.5, the spatial derivatives of the con-
centration distribution are estimated as
∂
∂
2
c
z
∂
∂
2
c
+
∂
∂
2
c
≈
w
(
1
−
w
)
2
2
2
z
z
t
t
+
∆
t
=
( . )(
0 5
−
1 0
. )
+
(
1
−
0 5
. )(
−
1 5
. )
= −
1 25
.
(
mg/L /m
)
2
∂
∂
c
z
∂
∂
c
z
∂
∂
c
z
+
≈
w
(
1
−
w
)
t
t
+
∆
t
=
E
=
( . )( . )
0 5 1 5
+
(
1 0 5 2
−
. )(
.
05
)
t
ν
t
H
(7.70)
=
1 78
.
(
mg/L /m
)
t
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