Environmental Engineering Reference
In-Depth Information
′
∂
c
x
∂
∂
∂ ′
∂
c
y
2
∂
∂
c
t
∂
∂
c
x
∂
∂
c
x
u
∂ ′
=
ε
(3.235)
+
u
=
K
(3.242)
y
x
Integrating Equation (3.235), noting that
c
is only a
function of
x
,
u
′ is only a function of
y
, and the zero-flux
boundaries require that ∂
c
′/∂
y
= 0 on the boundaries,
yields
which shows that longitudinal dispersion can be
described by the one-dimensional advection-diffusion
equation, and so all of the analytical solutions for the
fundamental solution are applicable in describing lon-
gitudinal dispersion.
The derived expression for the longitudinal disper-
sion coefficient given by Equation (3.240) is applicable
for the case in which the depth of the channel is constant
across the cross section. In cases where the depth varies
across the channel, the longitudinal dispersion coeffi-
cient can be derived similarly and is given by (Fischer
et al., 1979)
dc
dx
1
y
y
∫
∫
′
( )
=
+
′
( )
c y
u dydy c
0
′
(3.236)
′
ε
0
0
y
The average mass flux,
M
, across any section in the
longitudinal direction, is given by
M u u
(3.237)
=
(
+ ′
)(
c
+ ′ =
c
)
uc u c
+ ′
′
1
1
W
y
y
which is equal to the advective flux,
uc
,
plu
is the addi-
tional mass flux caused by dispersion,
∫
∫
∫
′
( )
′
( )
K
= −
u h y
u h y dydydy
(3.243)
x
( )
u c
. Equation
(3.236) can be used to estimate the mass flux caused by
dispersion, the dispersive mass flux, as
A
ε
h y
′
′
0
0
0
y
where
h
(
y
) is the depth of flow in the channel as a func-
tion of the transverse coordinate,
y
, and
A
is the total
cross-sectional area of the channel.
In applying the one-dimensional advection-diffusion
equation to describe longitudinal dispersion, the
assumptions that led to this equivalence much always
be kept in mind and validated when possible. Specifi-
cally, that longitudinal turbulent diffusion is negligible
compared with longitudinal dispersion (
K
x
>>
ε
x
), trans-
verse concentration perturbations are small rel
a
tive to
the transverse-averaged concentration (
′
c c
1
1
dc
dx
W
y
y
∫
∫
∫
u c
′
′ =
u
′
u dydydy
′
(3.238)
W
ε
′
0
0
0
y
The key r
esul
t here is that the longitudinal dispersive
mass flux,
u c
, is directly proportional to the longitudi-
nal concentration gradient, and can therefore be
described by Fick's law in terms of a longitudinal disper-
sion coefficient,
K
x
, where
′
′
), and
temporal variations in concentration when viewed from
a reference frame moving with the mean velocity are
small. In cases where these assumptions are not valid,
longitudinal dispersion is not Fickian.
The advection-dispersion equation given by Equa-
tion (3.242) is commonly used to describe dispersion
of contaminants in rivers and streams, with
K
x
calculated from either a measured velocity distribution
across the channel or an empirical formula based
on field measurements of tracer dispersion. In these
applications, transverse velocity variations occur in
both the vertical and horizontal-transverse directions,
and Equation (3.240) can be applied to calculate
K
x
using either of these velocity variations. In most
practical cases, horizontal-transverse velocity variations
in depth-averaged velocity yield much greater values of
K
x
than vertical-transverse variations in velocity, and
hence the longitudinal dispersion coefficient is most
often calculated using the horizontal-transverse velocity
variations.
dc
dx
1
1
dc
dx
W
y
y
∫
∫
∫
−
K
′
=
u
′
u dydydy
′
(3.239)
x
W
ε
′
0
0
0
y
and hence
K
x
is given by
1
1
W
y
y
∫
∫
∫
K
= −
u
′
u dydydy
′
(3.240)
x
W
ε
0
0
0
y
which shows how
K
x
can be determined directly from the
transverse velocity distribution,
u
′(
y
), and the transverse
turbulent diffusion coefficient,
ε
y
(
y
). The appropriate
advection-dispersion equation is therefore given by
∂
∂
c
t
∂
∂
′
2
c
(3.241)
=
K
x
x
2
Transforming back to the fixed (
x
,
y
) coordinates using
Equation (3.231) gives
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