Geoscience Reference
In-Depth Information
Normally, there is very little heat transfer across the bed surface for deep-water
bodies. However, for shallow, transparent lakes and reservoirs, part of the heat flux
may penetrate into the sediment bed and should be excluded (Tsay
et al
., 1992).
A complete heat budget model in the sediment bed is preferable, which would consider
the absorption and reflection of the short-wave radiation by the sediment bed as well as
the heat exchange flux due to conduction, seepage flow, etc., at the wetted perimeter.
However, the following simple approach may be used (Hodges, 1998):
e
−
λ(
z
−
z
b
)
J
Bsw
(
z
)
=
β
r
J
Tsw
(
z
b
)
(12.38)
where
J
Bsw
(
z
)
is the heat flux returned to the water column at height
z
,
J
Tsw
(
z
b
)
is
the short-wave radiation that reaches the bottom boundary,
β
r
represents the fraction
of the short-wave radiation returned to the water column, and
z
b
is the bed surface
elevation.
For the heat source from short-wave radiation,
q
T
is determined by
=
∂
∂
q
T
z
[
J
Tsw
(
z
)
−
J
Bsw
(
z
)
]
(12.39)
where
J
Tsw
(
are determined using Eqs. (12.37) and (12.38), respectively.
Note that the heat fluxes
J
Tsw
z
)
and
J
Bsw
(
z
)
in Eq. (12.39) transfer in opposite direc-
tions. The determined
q
T
is used in the source terms of Eqs. (12.4) and (12.6) in the
3-D and width-averaged 2-D models.
The net heat flux absorbed in the water column is
(
z
)
and
J
Bsw
(
z
)
J
T
=
J
Tlw
+
J
Te
+
J
Ts
+
J
Tsw
(
z
s
)
−
(
1
−
β
)
J
Tsw
(
z
b
)
(12.40)
r
which is used as the heat source rate in Eqs. (12.8) and (12.10) in the depth-averaged
2-D and 1-D models. The last term on the right-hand side of Eq. (12.40) represents the
flux penetrating into the bed, which is excluded in the heat budget in the water column.
12.1.5 Numerical solutions
The numerical methods introduced in Chapters 4-7 can be extended to solve the afore-
mentioned flow, heat and salinity transport equations. For example, the SIMPLE(C)
algorithms described in Sections 6.1.3.1, 7.1.3.2 and 7.2.4 can be straightforwardly
applied to solve the 2-D and 3-D hydrodynamic equations here, since the flow density
has been considered in the formulations.
The heat and salinity transport equations are similar to the suspended sediment
transport equation. They are typical convection-diffusion equations and can be solved
easily. If the finite volume method is used, Eqs. (12.4) and (12.5) are discretized as
V
P
T
n
+
1
P
a
W
T
n
+
1
W
a
E
T
n
+
1
E
a
S
T
n
+
1
S
a
N
T
n
+
1
N
a
B
T
n
+
1
B
T
P
)
=
t
(
−
+
+
+
+
1
a
T
T
n
+
1
a
P
T
n
+
1
+
−
+
c
p
[
A
t
(
J
Tsw
,
t
−
J
Bsw
,
t
)
T
P
ρ
−
A
b
(
J
Tsw
,
b
−
J
Bsw
,
b
)
]
(12.41)
