Environmental Engineering Reference
In-Depth Information
conductivity is particularly useful when solving heat flow
problems where the degree of saturation of the soil varies
throughout the geometry or with elapsed time.
The relationship between thermal conductivity and soil
suction becomes obvious when coupling a thermal analysis
with a moisture flow analysis. The SWCC can be used for
the calculation of all unsaturated soil heat flow and moisture
flow properties.
10.6.3 Relationship of Thermal Properties to SWCC
The SWCC provides a relationship between the amount of
water in a soil and soil suction. The amount of water in the
soil can be designated in terms of gravimetric water content,
volumetric water content, or degree of saturation.
The heat capacity of a soil is usually proportioned accord-
ing to the proportion of the overall volume that is water and
the amount that is comprised of soil solids. Under saturated
soil conditions, the volumetric water content represents the
porosity of the soil which partitions the soil mass into solids
and water. At any other soil suction the volumetric water con-
tent designates the portion of the total volume that is occupied
by water.
Volumetric heat capacity of a soil will vary with soil suction.
The SWCC can be used to calculate the heat capacity as a
function of soil suction. The relationship between soil suction
and heat capacity is hysteretic in character. The need to use the
relationship between heat capacity and soil suction becomes
more apparent when solving problems where the amount of
water in the soil is changing with time (e.g., coupled heat and
moisture flow).
The thermal conductivity of a soil is also a function of
the proportion of the phase making up the soil mixture. The
amount of air in the soil also has a significant influence on
the thermal conductivity. The plot of degree of saturation ver-
sus soil suction can be used to write the thermal conductivity
of a soil mixture in terms of soil suction. The thermal con-
ductivity relationship is hysteretic, as is the SWCC. There are
distinct advantages in relating all soil properties to the SWCC
when solving problems that couple more than one physical
process.
The SWCC is also of value in estimating the amount of
frozen water in the freezing zone of a soil.
10.6.2 Volumetric Specific Heat of Soils
The volumetric heat capacity
ζ
of a soil consisting of solids,
water, and air can be calculated by proportioning the volu-
metric heat capacities of each phase according to the volume
of each phase, as shown in Eq. 10.7. Typical values for the
volumetric heat capacity of water, soil solids, and air have
been presented earlier in this chapter. The volumetric heat
capacity of air can be assumed to be negligible in compari-
son to that of water or soil solids. Therefore the volumetric
heat capacity of a soil mixture can be written as
ζ
=
ζ
p
θ
p
+
ζ
w
θ
(10.72)
where:
volumetric heat capacity of the soil solids, J/m
3
/
◦
C
,
ζ
p
=
θ
p
=
volumetric solid content (i.e.,
V
s
/V
),
volume of solid particles in a total soil element, m
3
,
V
s
=
ζ
w
=
volumetric
heat
capacity
for
the
water
phase,
J/m
3
/
◦
C
,
and
θ
=
volumetric water content (i.e.,
V
w
/V
), and
V
w
=
volume of water in the soil element.
A typical value for the volumetric heat capacity of soil
solidsis2
.
2
10
6
J/m
3
/
◦
C and a typical volumetric heat
capacity for water is 4
.
2
×
10
6
J/m
3
/
◦
C. Therefore, the vol-
umetric heat capacity of a soil mixture can be approximated
as follows:
×
10.7 APPLICATIONS TO THERMAL PROBLEMS
The application section illustrates some of the concepts asso-
ciated with modeling heat flow. A limited number of rela-
tively simple heat flow problems are used for this purpose.
The conditions associated with each problem are described
along with the soil properties needed to solve the problem.
=
2
.
2
10
6
θ
p
+
4
.
2
10
6
θ
ζ
×
×
(10.73)
where:
ζ
=
volumetric heat capacity for the overall soil mass,
J/m
3
/
◦
C.
10.7.1 Solution to Boundary Value Heat Flow Problem
The solution of a heat flow boundary value problem requires
the following information. First, it is necessary that the
ground surface geometry and the stratigraphic soil units be
defined. It must also be decided whether the problem at hand
should be solved as a one-dimensional, two-dimensional, or
three-dimensional geometry.
Second, the geotechnical engineer must be satisfied that a
partial differential equation can be written for a representa-
tive elemental volume. The partial differential equation must
adequately describe the physical processes that control soil
behavior.
The volumetric phase representations can also be written
in terms of other volume-mass soil properties:
=
2
.
2
10
6
G
s
+
4
.
2
10
6
w
ρ
d
ζ
×
×
(10.74)
where:
w
=
gravimetric water content,
ρ
d
=
dry density of the soil, and
G
s
=
specific gravity of the soil solids.
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