Environmental Engineering Reference
In-Depth Information
diffusion equation describing the vertical transport of
heat in the water column is given by
stratified lakes are in the range of 10
−9
-10
−6
m
2
/s
(10
−8
-10
−6
ft
2
/s).
The flux gradient method yields an infinite diffusion
coefficient when ∂
T
/∂
z
approaches zero, and therefore
this method is not applicable under zero-gradient condi-
tions. The vertical diffusion coefficient is sometimes
expressed as a function of the stability effects in terms
of the Richardson number, where
∂
∂
T
t
∂
∂
∂
∂
T
z
−
=
z
K
S
T
(7.76)
z
where
T
is temperature and
S
T
is a heat source or sink
in the water column. An example of a heat source is
solar radiation. The first term on the right-hand side of
Equation (7.76) represents the diffusion transport of
heat in the vertical direction, and the diffusion coeffi-
cient
K
z
is a function of the thermal and current struc-
ture of the lake. The heat source term,
S
T
, can usually be
omitted in the water column except near the surface
because light extinction usually limits penetration of
solar radiation into deep water. A typical temperature
profile is shown in Figure 7.12, and integration of Equa-
tion (7.76) with respect to
z
from the bottom of the lake
to depth
z
yields
K
k
0
(7.79)
K z
( ) =
z
1
+
Ri
where
K
0
(L
2
T
−1
) is the vertical diffusion coefficient
without stratification (Ri = 0),
k
is an empirical constant
(dimensionless), and Ri is the Richardson number
(dimensionless), which is defined by Equation (7.28)
and can be expressed as
∂
∂
T
z
u
(7.80)
2
∂
∂
T
z
z
∂
∂
T
t
dz
Ri
= −
α
gz
∫
v
K
=
(7.77)
2
*
z
h
where
α
v
is the coefficient of volumetric thermal expan-
sion of water (°C
−1
) and
u
*
is the shear velocity (LT
−1
).
The advantages of using Equation (7.79) to estimate
K
z
are that it is easy to use and the computation procedure
is straightforward.
where the heat flux into the sediments and the radiation
absorbed by the sediments at
z
= −
h
is not included.
Equation (7.77) can be rearranged to give the following
expression for the vertical mixing coefficient:
∂
∂
∂
∂
T
t
dz
T
z
z
∫
h
K z
z
( ) =
(7.78)
EXAMPLE 7.18
A lake undergoes fairly rapid destratification between
the months of October and November as shown from
the measurements plotted in Figure 7.13. The actual
temperature measurements as a function of depth are
as shown in the following table.
where the numerator represents the accumulated rate
of change of stored heat between
z
and the bottom of
the lake, and the denominator is the temperature gradi-
ent at depth
z
. Equation (7.78) does not apply at the
surface because solar radiation at
z
= 0 is not included.
Accurate temperature readings are essential to success-
ful application of the flux gradient method, and negative
values of
K
z
, which do not have physical meaning, can
occur in computations due to errors in the measured
temperature gradient. Typical values of
K
z
in thermally
0
November pro
le
5
z
water surface
October pro
le
z
= 0
T
10
lake
h
temperature pro
le
15
10
15
20
z
= -
h
temperature (
o
C)
Figure 7.12.
Temperature profile in a stratified lake.
Figure 7.13.
Temperature profile in lake.
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