Environmental Engineering Reference
In-Depth Information
concentration. Air and water in the voids of a soil are the
conducting media for diffusion processes. On the other
hand, the soil structure determines the path length and
cross-sectional area available for diffusion. The transport
of gases (e.g., O
2
and CO
2
), water vapor, and chemicals
provides examples of diffusion processes in soils.
There are two examples of air diffusion common to unsat-
urated soils. The first example of diffusion involves the
flow of air through the pore-water in a saturated or unsatu-
rated soil (Matyas, 1967; Barden and Sides, 1967). Another
example of air diffusion involves the passage of air through
water in a high-air-entry ceramic disk of a pressure plate
apparatus. This type of diffusion involves gases dissolving
into the water on one side of the ceramic disk and subse-
quently coming out of the water on the other side of the
ceramic disk.
The second type of diffusion involves the movement of
constituents through the water phase due to a chemical con-
centration gradient or an osmotic suction gradient. Chemical
diffusion comes under the topic of contaminant transport
phenomena and is not discussed in this topic.
where:
u
i
¯
=
partial pressure of the diffusing constituent,
∂C/∂
u
i
¯
=
change in concentration with respect
to a
change in partial pressure, and
∂
u
i
/∂y
¯
=
partial pressure gradient in the
y-
direction (or
similarly in the
x-
or
z-
direction).
The mass rate of the constituent diffusing across a unit
area of the soil voids (i.e.,
∂M/∂t
) can also be determined
by measuring the volume of the diffusing constituent under
constant-pressure conditions. The ideal gas law is applied
to the diffusing constituent
in order to obtain the mass
flow rate:
∂V
fi
∂t
∂M
∂t
ω
i
RT
K
=¯
u
fi
(9.36)
where:
u
fi
=
¯
absolute constant pressure used in the volume
measurement of the diffusing constituent,
ω
i
=
molecular mass of the diffusing constituent,
R
=
universal (molar) gas constant,
9.4.1 Air Diffusion through Water
Fick's law can be used to describe the diffusion process. The
concentration gradient provides the driving potential for the
diffusion process and can be expressed with respect to the
soil voids. In other words, the mass rate of diffusion and the
concentration gradient are expressed with respect to a unit
area and a unit volume of the soil voids. The formulation
of Fick's law for diffusion in the
y
-direction is as follows:
T
K
=
absolute temperature,
∂V
fi
/∂t
=
flow rate of the diffusing constituent across a
unit area of the soil voids, and
V
fi
=
volume of the diffusing constituent across a
unit area of the soil voids.
The change in concentration of the diffusing constituent rel-
ative to a change in partial pressure (i.e.,
∂C/∂
u
i
) is obtained
by considering the change in density of the dissolved con-
stituent in the pore-water. The density of the dissolved con-
stituent in the pore-water is the ratio of the mass of dissolved
constituent to the volume of water:
∂C
∂
¯
∂M
∂t
D
∂C
∂y
=−
(9.34)
where:
∂(M
di
/V
w
)
∂
∂M/∂t
=
mass rate of the air diffusing across a unit area
of the soil voids,
u
i
=
(9.37)
¯
u
i
¯
D
=
coefficient of diffusion,
where:
C
=
concentration of the diffusing air expressed in
terms of mass per unit volume of the soil voids,
and
M
di
=
mass of the dissolved constituent in the pore-water
and
∂C/∂y
=
concentration gradient in the
y
-direction (or
similarly in the
x
-or
z
-direction).
V
w
=
volume of water.
Applying the ideal gas law yields the following equation:
The diffusion equation can appear in several forms, sim-
ilar to the forms presented for the flow of air through a
porous medium. The concentration gradient for gases or
water vapor (i.e.,
∂C/∂y
) can be expressed in terms of
partial pressures. Consider a constituent diffusing through
the pore-water in a soil. Fick's law for diffusion can be
rewritten with respect to the partial pressure of the diffusing
constituent:
∂C
∂
∂
[
(V
di
/V
w
)
u
i
(ω
i
/
RT
)
]
¯
u
i
=
(9.38)
¯
∂
u
i
¯
where:
V
di
=
volume of the dissolved constituent in the pore-
water.
∂M
∂t
=−
D
∂C
∂
∂
u
i
∂y
¯
The ratio of the volume of dissolved constituent to the
volume of water (i.e.,
V
di
/V
w
) is referred to as the volumetric
(9.35)
u
i
¯
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