Environmental Engineering Reference
In-Depth Information
Columns 1-3 contain the given data, where
T
1
and
T
2
are the temperatures in October and November, respec-
tively. Using these given data, the computations proceed
from Column 4 to Column 10 as follows:
Column 10.
This is the vertical mixing coefficient,
K
z
,
calculated for each layer based on Equation (7.81) and
using the corresponding layer data in Columns 6 and 9
such that
Column 4.
This is the vertical temperature gradient
between each October measurement point and is calcu-
lated using the relation
∂
∂
∂
∂
T
t
∑
∆
z
K z
z
( ) =
T
z
∂
∂
T
z
T z
(
)
−
−
T z
(
)
1
1
U
1
L
=
z
z
U
L
The tabulated results shows that
K
z
varies in the range
of 0.30-2.31 m
2
/d with the higher mixing coefficients
occurring around mid-depth.
where
z
U
and
z
L
are the elevations of the upper and
lower elevations of each layer, respectively.
Column 5.
This is the vertical temperature gradient
between each November measurement point and is cal-
culated for each layer using the relation
7.5.3 Two-Dimensional Models
Two-dimensional water-quality models have been
developed for long-deep reservoirs in which significant
vertical water-quality gradients are coupled with hori-
zontal water-quality gradients.Typically, two-dimensional
models solve the advection-diffusion equation in a ver-
tical longitudinal plane through the reservoir. These
models are mostly used to predict the two-dimensional
temperature structure of deep reservoirs through the
annual stratification cycle. Depth-integrated two-
dimensional models have also been developed for
shallow wide lakes, where these models are typically
driven by wind shear and include no stratification effects.
A simplified two-dimensional model is the nearshore
mixing model, which is described below.
∂
∂
T
z
T z
(
)
−
−
T z
(
)
2
2
U
2
L
=
z
z
U
L
Column 6.
This is the average vertical temperature gra-
dient between October and November in each layer,
and is calculated from Columns 4 and 5 for each layer
using the relation
∂
∂
T
z
1
2
∂
∂
T
z
+
∂
∂
T
z
1
2
=
Column 7.
This is the rate of change of temperature at
each measurement elevation
z
between October and
November and is calculated using the relation
7.5.3.1 Nearshore Mixing Models.
Nearshore mixing
models are used to predict the distribution of
contaminants in the vicinity of waste discharges into
large bodies of water, such as lakes. Consider the waste-
water discharge illustrated in Figure 7.14, where the
advective currents are negligible and the steady-state
advection-diffusion equation with first-order decay is
given by
∂
∂
T
t
T z T z
t
( )
−
( )
2
1
( )
z
=
∆
Column 8.
This is an averaged quantity for each layer
that is calculated from the data in Column 7 using the
relation
∂
∂
T
t
1
2
∂
∂
T
t
+
∂
∂
T
t
∆
z
=
(
z
)
(
z
)
∆
z
U
L
y
Column 9.
This is the summation of the layer data in
Column 8 starting from the bottom layer of the lake
such that
Large water body
r
Mixing zone
boundary,
c
=
c
0
r
0
q
n
∂
∂
T
t
∂
∂
T
t
x
∑
∑
=
∆
z
=
∆
z
i
i
1
where the subscript
i
indicates the layer index with
i
= 1
being the bottom layer, and
n
is the number of layers
up to the current layer.
Waste discharge
Figure 7.14.
Wastewater discharge into a lake.
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