Environmental Engineering Reference
In-Depth Information
where:
applying an air pressure gradient across the soil specimen.
The amount of air flowing through the soil specimen is mea-
sured at the exit point. The flow rate must be measured
at a designated constant-pressure condition (i.e., usually at
101.3 kPa absolute or zero gauge pressure) since air is com-
pressible (Matyas, 1967).
The mass rate of the air flow is measured at a constant
air density
ρ
ma
. Equation 9.13 can be rewritten as a volume
rate of air flow for the case of constant air pressure:
M
a
=
mass of air in the soil,
V
a
=
volume of air in the soil,
S
=
degree of saturation, and
n
=
porosity of the soil.
Substituting the density-of-air term
ρ
a
for
M
a
/V
a
in the
above equation gives
C
=
ρ
a
(
1
−
S)n
(9.9)
∂V
a
∂t
∂u
a
∂y
D
a
ρ
ma
=−
(9.14)
Air density is related to the absolute air pressure
i
n accor-
dance with the universal gas law [i.e.,
ρ
a
=
(ω
a
u
a
)/
RT
K
,
or
where,
ω
a
=
molecular mass of air,
R
=
universal (molar)
1
ρ
ma
∂u
a
∂y
D
a
v
a
=−
(9.15)
gas constant, and
T
K
=
absolute temperature; see Eq. 2.19].
Therefore, the concentration gradient can also be expressed
with respect to a pressure gradient in the air. The gauge air
pressure can be used in reformulating Eq. 9.7:
where:
ρ
ma
=
constant air density corresponding to the pres-
sure used in the measurement of the mass rate
(i.e., at the exit point of flow) and
∂C
∂u
a
∂u
a
∂y
J
a
=−
D
a
(9.10)
∂V
a
/∂t
=
volume rate of the air flow across a unit area
of the soil at the exit point of flow, designated
as the flow rate
v
a
.
where:
u
a
=
pore-air pressure and
∂u
a
/∂y
=
pore-air pressure gradient in the
y
-direction (or
similarly in the
x
- and
z
-directions).
The pore-air pressure
u
a
can also be expressed in terms of
a pore-air pressure head
h
a
using a constant air density
ρ
ma
:
A modified form of Fick's law is obtained by defining a
coefficient of transmission for air flow through soils,
D
a
:
D
a
g
∂h
a
∂y
v
a
=−
(9.16)
∂C
∂u
a
where:
D
a
=
D
a
(9.11)
h
a
=
pore-air pressure head (i.e.,
u
a
/ρ
ma
g
) and
or
∂h
a
/∂y
=
pore-air
pressure
head
gradient
in
the
∂
[
ρ
a
(
1
−
S)n
]
D
a
=
D
a
(9.12)
y
-direction, designated as
i
ay
.
∂u
a
The coefficient of transmission
D
a
is a function of the
volume-mass properties of the soil (i.e.,
S
and
n
) and air
density. Substituting
D
a
into Eq. 9.10 results in an equation
for the mass of air flow:
Equation 9.16 has the same form as Darcy's equation for
the air phase and can be written as follows:
∂h
a
∂y
v
a
=−
k
a
(9.17)
∂u
a
∂y
D
a
J
a
=−
(9.13)
Equation 9.17 was earlier written for each of the Carte-
sian coordinate directions (i.e., Eqs. 9.4, 9.5, and 9.6). The
relationship between the air coefficient of transmission
D
a
and the air coefficient of permeability
k
a
is as follows:
A modified form of Fick's law has been used in geotech-
nical engineering to describe air flow through soils (Blight,
1971). The coefficient of transmission
D
a
can be related to
the air coefficient of permeability
k
a
, which can be measured
in the laboratory.
Steady-state air flow can be established through an unsat-
urated soil specimen that has a fixed degree of saturation (or
constant matric suction). The soil specimen is treated as an
element of soil having an air coefficient of permeability cor-
responding to the selected degree of saturation. The air coef-
ficient of permeability is assumed to be constant through-
out the soil specimen. Steady-state air flow is produced by
D
a
g
k
a
=
(9.18)
The hydraulic head gradient in Eq. 9.17 consists of the
pore-air pressure head gradient as the driving potential and
has been used to compute the air coefficient of permeability
k
a
(Barden, 1965; Matyas, 1967; Langfelder et al., 1968;
Barden et al., 1969b; Barden and Pavlakis, 1971).
Air permeability measurements can be performed at var-
ious applied matric suction values (or different degrees of
Search WWH ::
Custom Search