Environmental Engineering Reference
In-Depth Information
The soil tortuosity term
β
is dimensionless and can be
expressed in terms of the volume of air in the soil as
follows:
k
y
22
=
λ
y
+
Lk
vT
(6.55)
C
=
C
s
θ
s
+
C
w
θ
u
+
C
v
θ
v
+
C
i
θ
i
+
C
a
θ
a
(6.56)
=
θ
a
2
/
3
∂θ
u
∂T
β
(6.60)
m
i
2
=
(6.57)
The term for water vapor conductivity due to the temper-
ature gradient can also be expressed as follows:
and
thermal conductivity, J/s m
◦
C,
λ
y
=
ρ
w
ρ
sv
0
∂u
sv
0
∂T
k
vh
γ
w
u
sv
0
273
.
15
u
a
−
u
w
volumetric heat capacity of soil mixture, J/m
3
,
C
=
k
vT
=
−
+
+
T
273
.
15
+
T
C
s
=
volumetric heat capacity of solid phase of soil,
J/m
3
,
(6.61)
where:
C
w
=
volumetric heat capacity of liquid water of soil,
J/m
3
,
u
a
=
pore-air pressure of soil, kPa, with pore-air in
the soil equal
C
v
=
volumetric heat capacity of vapor phase of soil,
J/m
3
,
to zero if it
is connected to the
atmosphere.
C
a
=
volumetric heat capacity of dry air phase of soil,
J/m
3
,
The temperature dependence of saturated vapor pressure
is omitted in some earlier formulations. Including the term
∂u
sv
0
/∂T
in Eq. 6.61 takes the temperature dependence on
saturated vapor pressure into consideration.
Initial Water Content Conditions.
An analysis of
moisture movement must start by having an initial set of
designated values. The initial values may be designated on
the basis of (i) the location of the water table and assumed
hydrostatic conditions, (ii) equilibrium with a SWCC
for the soil, (iii) computed steady-state water contents,
(iv) measured water contents in situ, or (v) some other
selected procedure.
Initial Temperature Conditions.
The soil temperature
in the soil domain needs to be initialized to a specific value
or an expression,
T
0
. For example, the average air tempera-
ture
T
a
might be used for the initial soil temperature
T
:
volumetric content solid phase in soil, m
3
/m
3
,
θ
s
=
volumetric vapor content in soil, m
3
/m
3
,
θ
v
=
volumetric air content in soil, m
3
/m
3
,
θ
a
=
L
=
volumetric latent heat of water vaporization or
condensation, J/m
3
,
L
10
9
J/m
3
if
T > T
ef
,
=
2
.
5
×
otherwise
L
=
0
,
L
f
=
volumetric latent heat of water freezing or thawing,
J/m
3
,
10
8
J/m
3
=
L
f
=
3
.
34
×
if
T
ef
T
T
ep
, otherwise
L
f
=
0
,
temperature at soil freezing point,
◦
C,
T
ef
=
T
ep
=
temperature at the end of soil phase change during
freezing,
◦
C
,
and
m
i
2
=
slope of the SFCC (i.e., relationship between
unfrozen water content and soil temperature).
The term for pore-water vapor conductivity,
k
vh
, asso-
ciated with vapor diffusion within the air phase can be
expressed in terms of the universal gas law as follows:
T
=
T
0
(6.62)
Thermal Boundary Condition at Soil Surface.
The
thermal boundary condition at the ground surface can be
specified as a constant temperature or a temperature expres-
sion (i.e., Dirchlet condition):
βθ
a
gD
v
ω
v
u
sv
0
ρ
w
R
(
273
.
15
k
vh
=
(6.58)
+
T
)
where:
T
|
surface
=
T
soil
(6.63)
u
sv
0
=
saturation vapor pressure at the soil surface, kPa,
saturation vapor density in soil, kg/m
3
,
ρ
sv
0
=
Temperatures within the soil profile are required for the
solution of the vapor flow component in the moisture flow
equation. The temperature profile in the soil can be com-
puted by applying an estimated ground surface temperature
to the soil. The ground surface temperature can differ from
the air temperature.
The thermal boundary condition at the soil surface can
be specified as a heat flux expression (i.e., Neumann-type
boundary condition). The thermal flux into the soil ground
at the soil surface can be rewritten in accordance with the
thermal flux balance (Eq. 6.8):
R
g
surface
=
ω
v
=
molecular weight of vapor, 0.018016 kg/mol,
β
=
soil tortuosity,
volumetric air content in soil, m
3
/m
3
,
θ
a
=
R
=
universal gas constant, 8.3144 J/mol/K,
D
v
=
molecular diffusivity of vapor in soil pore air,
m
2
/s, and
gravitational acceleration, m/s
2
.
g
=
The vapor diffusion term
D
v
can be written as a function
of temperature:
10
−
5
1
1
.
75
T
273
.
15
D
v
=
2
.
29
×
+
(6.59)
R
n
−
R
h
−
R
l
(6.64)
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