Environmental Engineering Reference
In-Depth Information
The final quantity of heat,
H
fs
, associated with the soil
solids,
H
s
, can be written in a similar manner:
The thermal equilibrium equation can be written by substi-
tuting the individual heat components into Eq. 10.42 to give
M
s
C
p
(T
Kf
−
T
Ki
)
+
(M
w
−
M
uw
)C
i
T
Ki
H
fs
=
M
s
C
p
T
Kf
(10.37)
+
(M
w
−
M
uw
)L
+
(M
w
−
M
uw
)C
w
T
Kf
where:
+
M
uw
C
w
(T
Kf
−
T
Ki
)
=
(M
1
−
M
2
)C
w
T
1
(10.43)
T
Kf
=
final sample temperature, K (i.e., equilibrium tem-
perature).
Rearranging
Eq.
10.43
and
solving
for
the
mass
of
unfrozen water gives
The change in stored heat,
H
s
, in the soil solids between
the initial and final temperatures can be written as
⎡
⎤
M
w
(C
i
T
Ki
+
L
+
C
w
T
Kf
)
⎣
+
M
s
C
p
(T
Kf
−
T
Ki
)
−
(M
1
−
M
2
)C
w
T
1
⎦
M
uw
=
H
s
=
M
s
C
p
(T
Kf
−
T
Ki
)
(10.38)
C
i
T
Ki
+
L
+
C
w
T
Kf
−
C
w
(T
Kf
−
T
Ki
)
(10.44)
The unfrozen water content can be calculated as follows:
The quantity of heat associated with the frozen water (ice),
H
i
, has a stored heat term for ice between the initial tem-
perature and zero degrees as well as a latent-heat-of-fusion
term. The two components of change in heat for ice can be
written as
M
uw
M
s
w
uw
=
(10.45)
Soil samples tested for unfrozen water content must be
initially frozen to various temperatures (e.g.,
H
i
=
(M
w
−
M
uw
)C
i
(
273
.
15
−
T
Ki
)
+
(M
w
−
M
uw
)L
(10.39)
2
◦
C,
4
◦
C,
−
−
8
◦
C, etc.) in order to obtain a function describing the rela-
tionship between the temperature below 0
◦
C and the amount
of unfrozen water in the soil.
−
where:
M
w
=
mass of water, kg,
M
uw
=
mass of unfrozen water, kg,
specific heat of ice, J kg
−
1
K
−
1
, and
C
i
=
L
=
latent heat of fusion, J/kg.
10.6 ESTIMATION PROCEDURES FOR
THERMAL PROPERTIES
The change in the quantity of heat associated with water
in the soil specimen,
H
w
, can be written as
Estimation procedures are of particular interest when dealing
with heat flow problems since it is quite common to use esti-
mated thermal properties when performing thermal analyses
in geotechnical engineering. Some of the common proce-
dures used for the estimation of unsaturated soil thermal
properties are described in the following sections.
H
w
=
(M
w
−
M
uw
)C
w
T
Kf
+
M
uw
C
w
(T
Kf
−
T
Ki
)
(10.40)
where:
specific heat of water, J kg
−
1
K
−
1
, and
C
w
=
10.6.1 Thermal Conductivity of Soils
Estimation procedures for the determination of the thermal
conductivity of soil mixtures have been proposed by several
researchers. The predictive models take into consideration
soil properties such as water content, degree of saturation,
and dry density (or porosity). Three estimation procedures
are presented for thermal conductivity: (i) the Kersten (1949)
model, (ii) the Johansen (1975) model, and (iii) the C ote and
Konrad (2005) model. The Johansen (1975) model requires
that the thermal conductivity of the saturated material be
part of the input to the predictive model.
T
Kf
=
T
Kf
273
.
15, difference between the final tem-
perature and 273.15 K.
−
The quantity of heat associated with the calorimeter,
H
c
,
can be written as
H
c
=
(M
1
−
M
2
)C
w
T
1
(10.41)
where:
M
1
=
mass of water in the calorimeter, kg,
M
2
=
mass of calorimeter, kg, and
T
1
=
temperature change of the calorimeter and con-
tents during the test.
10.6.1.1 Kersten Estimation Model
Kersten (1949) undertook an extensive study involving the
measurement of thermal conductivity on more than 1000
soils. The soils included gravels, crushed rocks, sands, silts,
clays, and peat. The results from a portion of the mea-
sured values were used to develop an empirical relationship
The above-mentioned changes in heat can be combined
into one equation to give
H
s
+
H
i
+
H
w
=
H
c
(10.42)
Search WWH ::
Custom Search