Environmental Engineering Reference
In-Depth Information
(ii) If advection is larger than dispersion, that is,
D
∂
2
/∂x
2
[
]
A
[
]
U (∂
A
/∂x)
,wehave
U
∂
]
∂x
=
[
A
R
F
∂
[
A
]
−
(6.209)
∂t
or, alternatively,
(∂
[
A
]
/∂t)
U
R
F
−
/∂x)
=
,
(6.210)
(∂
[
A
]
where
U
is the Darcy velocity of the groundwater. The left-hand side is
interpreted as the velocity of the pollutant,
U
. Then
U
/U
=
1
/R
F
. In other
words, the retardation factor scales as the velocity of the groundwater. Large
R
F
means
U
< U
and the chemical movement is retarded.
A qualitative picture of the effects of dispersion, adsorption, and reaction on the
movement of pollutants is shown in Figure 6.54. Dispersion alone will make the
pollutant front less sharp. The addition of adsorption will delay the breakthrough
time; however, the concentration at the sampling point will eventually reach the feed
concentration at
x
0. Inclusion of reaction will further decrease the peak maxi-
mum and the feed concentration will never be attained at the downstream sampling
point. The relative magnitudes of dispersion and advection are assessed in terms of a
dimensionless Peclet number based on dispersivity,
Pe
=
UX
s
/D
.
Forsoils(porousmedia),thedispersioncoefficienthastwocontributions,onefrom
molecular diffusion and the other from fluid movement through the porous media. For
a porous medium, the tortuous path that a molecule has to take in traversing the pore
fluid reduces the effective diffusion coefficient.
D
w
ε
=
4
/
3
, where
ε
is the porosity and
D
w
is the molecular diffusivity of the pollutant in water, gives the effective diffusion
coefficient. The dispersion due to fluid motion is related to fluid velocity through a
saturated medium, which is given by Darcy's law:
d
h
d
x
,
U
D
=−κ
(6.211)
where
U
D
is the Darcy velocity (cm/s),
is the
hydraulic conductivity
of the medium
(cm/s), and d
h
/d
x
is the
hydraulic gradient
. In terms of volumetric flow,
U
D
=
κ
Q/A
,
where
Q
is the volumetric flow rate of fluid through a cross-sectional area
A
. The
negative sign for
U
D
indicates that the flow of fluid is in the direction of decreasing
hydraulic head. Table 6.13 lists typical values of
for representative media. It can
vary over many orders of magnitude depending on the media.
The dispersion constant is given by
D
κ
4
/
3
α
D
is termed
the
dispersivity
. It is a measure of the media heterogeneity and has units of length.
A proposed relationship for
α
D
in terms of the travel distance for a contaminant is
α
D
=
=
D
w
ε
+
U
D
α
D
, where
0.017
X
1.5
(Neumann, 1990), where
α
D
is in m and
X
is in m.
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