Environmental Engineering Reference
In-Depth Information
coefficient of solubility,
h
. Under isothermal conditions,
h
is essentially a constant:
The diffusion coefficient of permeability can then be writ-
ten as follows:
ρ
fi
g
¯
∂C
∂
ω
i
RT
K
k
fi
=
Dh
(9.43)
u
i
=
h
.
(9.39)
u
fi
¯
where:
Substituting the ideal gas law into Eq. 9.43 results in
another form for the diffusion coefficient of permeability:
h
=
volumetric coefficient of solubility for the constituent
in water.
D
hω
i
g
RT
K
k
fi
=
(9.44)
Substituting Eq. 9.36 into Eq. 9.39 results in the following
diffusion equation (van Amerongen, 1946):
The above equation indicates that under isothermal con-
ditions the coefficient of permeability (i.e., diffusion type)
is directly proportional to the coefficient of diffusion since
the term
hω
i
g/RT
K
is a constant.
The coefficients of diffusion for several gases through
water and for air through different materials were presented
in Chapter 2. The diffusion coefficients
D
for air through
pure water were computed in accordance with Eq. 9.42 (Bar-
den and Sides, 1967). The diffusion values for soils appear
to be significantly smaller than the diffusion values for free
water. The difference has been attributed to factors such as
the tortuosity within the soil and the higher viscosity of the
adsorbed water close to the soil particle clay surface. Mea-
sured diffusion coefficients have been observed to decrease
as the water content of the soil decreases.
Dh
¯
∂
u
i
∂y
¯
v
fi
=−
(9.40)
u
fi
where:
v
fi
=
flow rate of the diffusing constituent across a unit
area of the soil voids (i.e.,
∂V
fi
/∂t
).
The above equation can be applied to air or gas diffu-
sion through water in a soil or through other materials such
as a rubber membrane (Poulos, 1964). The partial pres-
sure term can be expressed in terms of the partial pressure
head
h
fi
(i.e.,
h
fi
=¯
u
i
/ρ
fi
g
) with respect to the density of
the constituent,
ρ
fi
. The constituent density corresponds to
the absolute constant pressure
u
fi
used in the measurement
of the diffusing constituent volume. The absolute constant
pressure
¯
9.5 OTHER COMPONENTS OF AIR FLOW
u
fi
is usually selected to correspond to atmospheric
conditions (i.e., 101.3 kPa), and
ρ
fi
is the constituent density
at the corresponding pressure:
¯
Air flow may take place by two other mechanisms: (i) air
flow within the liquid pore-water by diffusion and (ii) dis-
solved pore-air being carried by water flow. In the latter
case the air movement is referred to as advective flow. If air
flows through the water in the soil, it is driven by gradients
in the dissolved pore-air concentration and is referred to as
flow by diffusion.
The mass flux of free air can be described in terms of
Fick's law and Darcy's law. The flow of dissolved air is also
driven by gradients in dissolved air concentration. Dissolved
air flow can also be described using Fick's law:
ρ
fi
g
¯
∂h
fi
∂y
v
fi
=−
Dh
(9.41)
u
fi
where:
ρ
fi
=
constituent density at the constant pressure
u
fi
used
in the volume measurement of the diffusing con-
stituent and
¯
h
fi
=
partial pressure head (
u
fi
/ρ
fi
g
).
¯
D
ad
ρ
a
∂C
ad
∂y
v
a
y
=−
(9.45)
The above equation has a form similar to Darcy's law. The
equation can be considered as a modified form of Darcy's
law for air flow through an unsaturated soil with occluded
air bubbles where the air flow is of the diffusion form:
where:
v
a
y
=
dissolved air flow rate in the
y
-direction across a
unit area of the soil due to pore-air concentration
gradients, m/s,
∂h
fi
∂y
v
fi
=−
k
fi
(9.42)
D
ad
=
molecular diffusivity of dissolved air through
water, m
2
/s (also referred to as coefficient of
transmission of air),
where:
C
ad
k
fi
=
diffusion coefficient of permeability for air through
an unsaturated soil with occluded air bubbles.
=
concentration of dissolved air in terms of the mass
of vapor per unit volume of soil,
C
ad
=
ρ
a
Snh,
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