Environmental Engineering Reference
In-Depth Information
density of the bulk air phase, kg/m
3
,
ρ
a
=
100
u
a
/(
RT
), kg/m
3
,
ρ
a
=
W
a
¯
SWCC
W
v
=
molecular weight of water vapor, 18.016 kg/
kmol,
W
a
=
molecular weight of pore-air, kg/kmol,
u
air
v
=
partial pressure of water vapor, kPa,
u
a
¯
=
absolute pressure in the bulk air phase (i.e.,
u
a
+
0
0.1
u
atm
), kPa,
log (soil suction), kPa
10
6
u
a
=
pore-air pressure, kPa,
(a)
u
atm
=
atmospheric pressure, 101.325 kPa,
m
w
R
=
universal gas constant, 8.314 J/(mol
·
K),
D
v
∗
=
diffusion coefficient of vapor
through soil
Water storage,
slope of SWCC
S)nD
v
W
v
/RT
(kg m)/(kN s), equal
to
(
1
−
(kg
·
m
)/
kN s
)
, and
D
a
∗
S)nD
a
W
a
/RT
(kg
=
(
1
−
·
m
)/(
kN s
)
.
Equation 7.21 defines the diffusion coefficient of vapor
through soil,
D
v
∗
, based on Fick's law. However, an impor-
tant factor has been omitted, namely, the tortuosity vari-
able for the diffusion of water vapor through the soil pores
(Lai et al., 1976). The diffusion coefficient of water vapor
through soil,
D
v
∗
, can be written in the following form when
tortuosity is taken into account:
0.1
log (soil suction), kPa
10
6
(b)
k
max
k
min
permeability
function,
anisotropic soil
D
v
∗
=
αβD
v
W
v
/RT
(7.22)
where:
10
6
0.1
log (soil suction), kPa
(c)
β
2
/
3
(Lai et al.,
1976), where
β
is the cross-sectional area of soil
available for vapor flow per total area [i.e.,
β
α
=
tortuosity factor of the soil;
α
=
Figure 7.14
Typical permeability and water storage functions for
anisotropic unsaturated soil: (a) degree of saturation versus soil
suction; (b) water storage versus soil suction; (c) coefficient of
permeability versus soil suction .
=
(
1
−
S)n
].
Neglecting the gradient in atmospheric pressure and
assuming the air phase is continuous and in direct contact
with the atmosphere, gradients in the free air will be equal
to gradients in the partial pressure of water vapor,
u
ai
v
.
Therefore, Eq. 7.21 can be rewritten as follows:
where:
q
i
=
flux rate of mass of water vapor within the air
phase in the
i
-direction per unit of total area,
kg/m
2
/
s,
D
v
∗
∂u
air
u
air
v
¯
D
v
∗
∂u
air
u
air
v
D
v
∗
∂u
air
=−
¯
u
a
+
v
∂y
(7.23)
Equation 7.23 describes vapor flow based on partial vapor
pressure gradients. It is not possible to make a compari-
son between the liquid flow equation and the vapor flow
equation (i.e., Eq. 7.23) because different gradients exist in
each equation. Consequently, a direct comparison between
the values of
k
w
and
D
v
∗
is not meaningful.
Based on the thermodynamic theory of soil moisture
(Edlefsen and Anderson, 1943), the vapor pressure in the
air,
u
ai
v
, can be expressed as a function of the total potential
of the liquid pore-water and temperature. The amount of
vapor in the pore-air phase depends on total suction and soil
temperature. Changes in total suction or temperature result
in mass transfer between the gaseous and liquid phases.
q
y
=−
v
∂y
v
∂y
−
u
a
u
a
¯
D
v
=
molecular diffusivity of water vapor in air [i.e.,
D
v
10
−
4
(
1
T
K
/
273
.
15
)
1
.
75
,m
2
/
s
=
0
.
229
×
+
(Kimball et al., 1976)],
D
a
coefficient of diffusion of air,
D
a
D
v
(Wilson
=
≈
et al., 1997a),
T
K
=
temperature, K,
C
v
=
concentration of water vapor in terms of the
mass of vapor per unit volume of soil,
C
v
=
ρ
v
(
1
−
S)n
,
C
a
=
concentration of air in terms of the mass of air
per unit volume of soil,
C
a
=
ρ
a
(
1
−
S)n
,
S
=
degree of saturation,
n
=
porosity,
density of the water vapor, kg/m
3
,
ρ
v
=
W
v
u
ai
v
/(
RT
), kg/m
3
,
ρ
v
=
Search WWH ::
Custom Search