Environmental Engineering Reference
In-Depth Information
where
J
l
and
J
g
represent contaminant fluxes in the liquid phase and soil gas
[ML
−
2
T
−
1
], respectively. The three main transport processes that can be active in
both liquid phase and soil gas are molecular diffusion, hydrodynamic dispersion,
and advection (often also called convection). The contaminant fluxes in the water
and gas phases are then the sum of fluxes due to these different processes:
J
l
=
J
lc
+
J
ld
+
J
lh
(18.27)
J
g
=
J
gc
+
J
gd
+
J
gh
where the subscripts
c
,
d
, and
h
denote convection (or advection), molecular
diffusion, and hydrodynamic dispersion, respectively.
18.3.1.1 Diffusion
Diffusion is a result of the random motion of molecules. This process causes a con-
taminant to move from a location with a higher concentration to a location with a
lower concentration. Diffusive transport can be described using Fick's first law:
D
∂
c
)
D
o
∂
c
∂
z
J
d
=−
θ
∂
z
=−
θξ
(
θ
(18.28)
where
D
is liquid phase diffusion coefficient [L
2
T
−
1
],
D
o
is the diffusion coefficient
of the contaminant in free water [L
2
T
−
1
], and
is a tortuosity factor to account
for the increased path lengths and decreased cross-sectional areas of the diffusing
contaminant in both phases (Jury and Horton
2004
).
The parameter
D
is related to the effective diffusion coefficient
D
eff
(diffusion
through the pore space of the porous medium) following
D
eff
=
ξ
D
×
θ
. Values for
10
−
9
m
2
/s for
cations like Ca
2+
. Since contaminant diffusion in the liquid phase is severely ham-
pered by both air and solid particles, the tortuosity factor as defined here increases
strongly with water content. Many empirical models have been suggested in the
literature to account for the tortuosity (e.g., Moldrup et al.
1998
). Among these,
the most widely used model for the tortuosity factor is probably the equation of
Millington and Quirk (
1961
) given by:
10
−
9
m
2
/s for anions like Cl
−
and Br
−
and 0.78
D
o
vary between 2
×
×
7
/
3
=
θ
ξ
(
θ
)
(18.29)
s
θ
θ
s
is the saturated water content [L
3
L
−
3
].
Jin and Jury (
1996
) suggested that the following version of the Millington-Quirk
model may be in better agreement with experimental data:
where
θ
D
=
D
o
(18.30)
2
/
3
θ
s
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