Environmental Engineering Reference
In-Depth Information
Equation 10.54 can be rewritten as follows for an unfrozen
soil:
The thermal conductivity for natural soils that were
rounded and subrounded particles was estimated using a
derivation from the de Vries (1963) equation:
λ
u
=
(λ
s(u)
−
λ
dry
)λ
ru
+
λ
dry
(10.57)
0
.
135
ρ
d
+
64
.
7
where:
λ
dry
=
(10.63)
2700
−
0
.
947
ρ
d
λ
s(u)
=
thermal conductivity of the saturated, unfrozen
soil, W/m/
◦
C.
It was noted that crushed rocks (angular and subangular)
yielded a different equation for the thermal conductivity for
dry materials:
Equation 10.54 can also be rewritten as follows for a
frozen soil:
0
.
39
n
−
2
.
2
λ
dry
=
(10.64)
λ
f
=
(λ
s(f)
−
λ
dry
)λ
rf
+
λ
dry
(10.58)
There are also other predictive models that have been pro-
posed for the estimation of thermal conductivity (Farouki,
1981); however, the Johansen (1975) model has been shown
to perform quite well.
where:
λ
s(f)
=
thermal conductivity of the saturated, frozen soil,
W/m/
◦
C.
10.6.1.3 Cote and Konrad (2005) Model
Cote and Konrad (2005) undertook an examination of the
factors influencing the prediction of thermal conductivity of
coarse soils and have suggested a series of refinements to the
Johansen (1975) model. The first refinement took into con-
sideration the change in porosity that occurs when water in
a saturated soil freezes. The water in the voids expands by
9% upon freezing, causing a change in the porosity of a satu-
rated soil:
Johansen (1975) used the concept of a geometric mean for
the soil particles and obtained the following thermal conduc-
tivity equations for saturated, unfrozen soils,
λ
s(u)
:
0
◦
C
λ
1
−
n
p
λ
w
λ
s(u)
=
when
T
≥
(10.59)
The equivalent equation proposed for the thermal con-
ductivity equations for saturated, frozen soils,
λ
s(f)
, was as
follows:
1
.
09
n
u
n
f
=
(10.65)
1
+
0
.
09
n
u
when
T<
0
◦
C
λ
1
−
n
p
λ
i
λ
s(f)
=
(10.60)
where:
where:
n
u
=
porosity of the saturated unfrozen soil and
thermal conductivity of the soil solids, W/m/
◦
C,
λ
p
=
n
f
=
porosity of the saturated frozen soil.
thermal conductivity of water, W/m/
◦
C,
λ
w
=
thermal conductivity of ice, W/m/
◦
C, and
λ
i
=
No volume change occurs when a soil is dry. If the assump-
tion is made that volume change occurs linearly with the
amount of water in the soil, the above equation can then be
written in terms of the degree of saturation:
n
=
porosity of the soil in decimal form.
It was noted that the amount of quartz mineral in a soil has
a significant effect on the thermal conductivity of the soil
solids. Johansen (1975) proposed the following equations to
compensate for the quartz content.
1
.
09
S
u
S
f
=
(10.66)
1
+
0
.
09
S
u
2
.
0
1
−
q
7
.
7
q
when
q
>
0
.
2 (or 20%
)
λ
p
=
×
(10.61)
where:
3
.
0
1
−
q
7
.
7
q
λ
p
=
×
when
q
≤
0
.
2 (or 20%
)
(10.62)
S
f
=
degree of saturation of the frozen soil and
S
u
=
degree of saturation of the unfrozen soil.
where:
Cote and Konrad (2005) also suggested that the geometric
mean method of Sass et al., (1971) and Woodside and Mess-
mer (1961) be used to determine the thermal conductivity
of soil solids:
q
=
amount of quartz expressed as a volume ratio refer-
enced to the solids portion.
The Johansen (1975) model takes the amount of quartz
mineral into consideration in the estimation of the over-
all thermal conductivity. However, it does not take a wider
variety of possible mineral solids into consideration.
z
z
λ
x
mj
λ
p
=
x
j
=
1
(10.67)
j
=
1
j
=
1
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