Environmental Engineering Reference
In-Depth Information
for the water phase in the
y
-direction can be expressed as
follows:
16.7.3 Heat Flow Equation for Unsaturated Soils
A Fourier diffusion equation can be used to describe heat
transfer in soils as described in Chapter 10 (Jame and Norum,
1980; Fredlund and Dakshanamurthy, 1982; Wilson, 1990).
The following form of the heat flow equation was proposed
by Wilson (1990):
k
w
∂u
w
∂t
∂u
a
∂t
1
ρ
w
gm
2
∂
∂y
∂u
w
∂y
=−
C
w
+
D
v
u
a
+¯
¯
u
v
1
ρ
w
m
2
∂
∂y
∂
u
v
∂y
¯
+
(16.70)
u
a
¯
λ
∂T
∂t
L
v
¯
∂
∂y
D
v
ζ
∂T
∂t
∂
∂y
u
a
+¯
u
v
d
u
v
∂y
¯
=
−
where:
u
a
¯
(16.73)
where:
C
w
=
interaction coefficient associated with the water
phase partial differential equation [i.e.,
(
1
m
2
/
−
ζ
=
volumetric specific heat of the soil as a function of
water content, (J/m
3
/
◦
C),
m
1
k
)/(m
2
/m
1
k
)
]
,
u
v
=
partial pressure of water vapor in air, and
λ
=
thermal conductivity of the soil as a function of water
content, (W/m/
◦
C), and
D
v
=
diffusion coefficient of the water vapor through the
soil, kg
·
m/kN
·
s.
L
v
=
latent heat of vaporization of water (i.e., 2,418,000
J/kg).
The coefficient of water vapor diffusion
D
v
can be calcu-
lated as follows (Philip and de Vries, 1957; de Vries, 1975;
Dakshanamurthy and Fredlund, 1981; Wilson, 1990):
The above equation describes the heat flow due to con-
duction and the latent heat transfer caused by phase changes.
Convective heat flow is considered to be negligible (Jame
and Norum, 1980; Wilson, 1990). The volumetric specific
heat of the soil,
ζ
, can be calculated using the relationship
given by de Vries (1963):
αβ
D
vm
ω
v
RT
K
D
v
=
(16.71)
where:
ζ
=
ζ
s
θ
solid
+
ζ
w
θ
+
ζ
a
θ
a
(16.74)
β
2
/
3
),
α
=
tortuosity factor for the soil (i.e.,
ε
=
β
=
cross-sectional area of the soil available for water
vapor flow [i.e.,
(
1
where:
−
S) n
],
D
vm
=
molecular diffusivity of water vapor in air [i.e.,
0
.
229
1
ζ
s
=
volumetric specific heat capacity of the soil solids
(i.e., a typical value is 2
.
235
T
K
/
273
1
.
75
J/m
3
/
◦
C for fine
10
6
10
−
4
(m
2
/s)]
×
+
×
(from
sands; de Vries, 1963),
Kimball et al., 1976), and
θ
solid
=
volumetric solid content (i.e.,
V
s
/V
0
),
ω
v
=
molecular mass of water vapor (i.e., 18.016 kg/
kmol).
V
s
=
volume of soils solids in the soil,
V
0
=
total volume of the soil,
ζ
w
=
volumetric specific heat capacity for the water phase
(i.e., 4
.
154
The above equation indicates that the diffusion coefficient
D
v
is a function of soil properties (i.e.,
S
and
n
), which in
turn are a function of matric suction
u
a
−
J/m
3
/
◦
Cforwaterat35
◦
C; Wil-
10
6
×
son, 1990),
u
w
. In addition,
D
v
is also a function of temperature
T
K
. Similarly, the coef-
ficient of permeability
k
w
is also a function of soil properties
that may vary with respect to location in the soil mass. If the
variations in the
k
w
and
D
v
coefficients with respect to space
are considered to be negligible, Eq. 13.70 can be simplified
as follows:
θ
=
volumetric water content (i.e.,
V
w
/V
0
),
V
w
=
volume of water in the soil,
ζ
a
=
volumetric specific heat capacity for the air phase,
θ
a
=
volumetric air content (i.e.,
V
a
/V
0
), and
V
a
=
volume of air.
The third term on the right-hand side of Eq. 16.74 is small
and can be considered negligible (de Vries, 1963). The ther-
mal conductivity of the soil,
λ
, can be estimated using the
following expression (de Vries, 1963):
∂
2
u
w
∂y
2
∂
2
∂u
w
∂t
∂u
a
∂t
u
v
∂y
2
¯
c
v
c
wv
v
=−
C
w
+
+
(16.72)
where:
f
s
θ
s
λ
s
+
f
w
θλ
w
+
f
a
θ
a
λ
a
λ
=
(16.75)
c
v
f
s
θ
s
+
f
w
θ
+
f
a
θ
a
=
coefficient of consolidation with respect to the water
phase [i.e.,
k
w
/
ρ
w
gm
2
] and
where:
c
wv
v
=
coefficient of consolidation with respect to the water
¯
u
v
u
a
+¯
D
v
ρ
w
m
2
f
s
,f
w
,f
a
=
weighting factors for the solid, water, and
air phases, respectively, and
vapor phase (i.e.,
).
u
a
¯
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