Environmental Engineering Reference
In-Depth Information
where:
3
λ dry air =
0.025 W/m/K and
λ water vapor =
varies linearly between 0 and 0.0736
W/m/K for volumetric water contents, θ ,
ranging between 0 and 12.1%.
2
The weighting factors f are calculated using the assumption
that the soil particles are ellipsoidal in shape. The weighting
factor for a continuous medium (i.e., air or water) is equal to
1.0. Water can be selected as the continuous medium with a
weighting factor f w equal to 1.0. The weighting factors for
the solid particles and air can then be calculated in accordance
with the following relationship:
1
0
0
5
10
15
20
25
Gravimetric water content ( w ), %
1
λ
λ w
1 g i 1
3
Figure 10.1 Thermal conductivity versus water content for the
Beaver Creek sand (after Wilson, 1990).
1
3
f
=
+
(10.6)
i
=
1
35
where:
30
f
=
f a or f p ,
25
λ
=
λ a or λ p , and
g
=
depolarization factors for the ellipsoid (i.e., g 1 , g 2 , and
g 3 , where g 1 +
20
1); the values of g 1 , g 2 , and
g 3 are independent of particle size but dependent on
the ratio of the length of the ellipsoid axes.
g 2 +
g 3 =
15
10
5
Wilson (1990) used equal depolarization factors of 1 / 3
in computing the weighting factors for a sandy soil. The
sand particles were assumed to have spherical shapes. The
depolarization factors g 1 and g 2 for the air phase were
assumed to decrease linearly from 0.333 to 0.105 for volu-
metric water contents θ ranging from 23.6 to 12.1%, respec-
tively, and from 0.105 to 0.015 for volumetric water con-
tents ranging from 12.1 and 0%, respectively (Jame, 1977).
Figure 10.1 illustrates the thermal conductivity λ variation
for Beaver Creek sand with respect to gravimetric water
contents.
The thermal conductivity of an unsaturated soil is a func-
tion of the relative amounts of air, water, and solids in the
soil and therefore can be considered to be a function of the
SWCC (Aldrich, 1956). Figure 10.2 shows the shape of the
thermal conductivity-suction relationship with the SWCC.
A specific thermal conductivity value can be computed
for a soil when the degree of saturation is a constant value.
However, if the partial differential equations for moisture
flow and heat flow equations are solved as “combined” pro-
cesses, the thermal conductivity will change as the amount
of water in the soil changes.
The thermal conductivity of water bears a relationship to
temperature, as shown in Fig. 10.3. The thermal conductivity
of ice is different from that of water. It is as if an additional
phase is slowly added to the soil when part of the pore-water
becomes frozen. Not all water in the pores of the soil freezes
when the temperature is lowered to 0 C. The amount of water
0
10 6
10
100
1000
10,000
100,000
Soil suction, kPa
(a)
2.5
2.0
1.5
1.0
0.5
0.0
10 6
1
10
100
1000
10,000
100,000
Soil suction, kPa
(b)
Figure 10.2 Volumetric water content and thermal conductivity
versus soil suction for silica flour (from Jame, 1972): (a) volumetric
water content versus soil suction; (b) thermal conductivity versus
soil suction.
that freezes is a function of the temperature below 0 C. The
unfrozen water content in the soil can be calculated from
the SWCC and the Clapeyron equation (Newman, 1995). The
latent heat of fusion L must be taken into consideration when
the water in a soil either freezes or thaws.
 
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