Environmental Engineering Reference
In-Depth Information
c
U
∆
z
U
A
T
Q
z
c
Q
out
,
c
out
S
m
∆
z
Q
in
,
c
in
layer of interest
A
B
z
c
L
∆
z
L
y
x
Figure 7.11.
One-dimensional conservation of mass in lake.
water quality are commonly a result of thermal stratifi-
cation of the lake or reservoir. One-dimensional water-
quality models in lakes typically discretize the water
body into homogeneous (completely mixed) layers, in
which case the governing advection-dispersion equa-
tion can be put in the form
2
∂
∂
c
z
c
−
+
c
c
−
c
1
U
L
≈
−
(7.65)
2
(
∆
z
∆
z
)
/
2
(
∆
z
+
∆
z
)
/
2
∆
z
U
L
∂
∂
c
z
c
−
c
c
−
+
c
1
2
U
L
≈
+
(7.66)
(
∆
z
+
∆
z
)
/
2
(
∆
z
∆
z
)
/
2
U
L
∂
∂
c
t
c t
(
+
∆
∆
t
)
−
c
(7.67)
≈
t
∂
∂
c
t
∂
∂
2
c
∂
∂
c
z
V
=
AK
2
∆
z Q
+
∆
z Q c Q c VS
+
−
+
z
z
in in
out
m
z
Substituting Equations (7.63-7.67) into Equation
(7.62) yields a single equation, with
c
(
t
+ Δ
t
) being the
only unknown. These equations are then applied sequen-
tially to each layer until the estimated concentrations in
each layer at time
t
+ Δ
t
converge. This approach gener-
ally requires that the partial derivatives ∂
2
c
/∂
z
2
and
∂
c
/∂
z
be taken as the weighted average of their values
at
t
and
t
+ Δ
t
.
(7.62)
where
V
is the volume of a layer (L
3
),
c
is the tracer
concentration (ML
−3
), Δ
z
is the thickness of a layer (L),
A
is the plan area of a layer (L
2
),
K
z
is the vertical dif-
fusion coefficient (L2T−1),
2
T
−1
),
Q
in
in the volumetric inflow
rate (L
3
T
−1
),
Q
out
is the volumetric outflow rate (L3T−1),
3
T
−1
),
Q
z
is a vertical advection (L3T−1),
3
T
−1
),
c
in
is the concentration
of the tracer in the inflow (ML
−3
), and
S
m
are the sources
and/or sinks of tracer mass within the layer (ML−3T−1).
−3
T
−1
).
Evaluation of Equation (7.62) is typically done
numerically with a discretization, such as that shown in
Figure 7.11. For any layer of interest, values of
Q
in
,
c
in
,
Q
out
, Q
z
,
S
m
, and Δ
z
are specified, along with the thick-
ness of the layers above and below the layer of interest,
Δ
z
U
and Δ
z
L
, respectively, and the top and bottom plan
areas of the layer of interest,
A
T
and
A
B
, respectively.
For any given initial time,
t
, the concentrations of the
tracer in the above and below layers,
c
U
and
c
L
, as well
as the concentration in the layer of interest,
c
, are
assumed to be known, and the concentration of
the tracer at a time Δ
t
later,
c
(
t
+ Δ
t
), in the layer of
interest is to be found. Using the given discrete vari-
ables, the following terms in Equation (7.62) can be
approximated,
EXAMPLE 7.16
A 1-ha lake is discretized into 1-m layers to estimate the
distribution of oxygen (O
2
) in the upper portion of the
lake, where phytoplankton are estimated to generate O
2
at a rate of 1.25 (mg/L)/day and the diffusion coefficient
of O
2
is estimated as 0.35 cm
2
is During the season of
interest, there is negligible lateral inflow and outflow
from the lake, and vertical advection in the lake is neg-
ligible. Time steps of 1 day are to be used in the simula-
tion, and a weight of 0.5 is to be used in calculating the
partial derivatives within each time step. Under initial
conditions, the O
2
concentration in the layer of interest
is 7 mg/L, and the concentrations above and below this
layer are 8 and 5 mg/L respectively. After the first itera-
tion, the estimated O
2
concentration in the layer of
interest is 6.2 mg/L, and the concentrations above and
below this layer are 7.5 and 4.2 mg/L, respectively.
Determine the next iterative estimate of the concentra-
tion of O
2
in the layer of interest.
1
2
(
)
(7.63)
A
≈
A A
+
T
B
≈ ∆
(7.64)
V A z
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