Environmental Engineering Reference
In-Depth Information
wetter as water is stored. The ability of a soil to store heat
(e.g., the rate at which heat can be taken on or given off) is
a function of its temperature. A soil is less likely to take on
more heat as it gets warmer.
There are also a number of key differences between the
physics associated with liquid water flow and heat flow. For
example, heat flow analyses quite commonly encompass water
in both the liquid and solid state. The analysis may also involve
a phase change as is the case when the soil water freezes or
thaws. There may also be a phase change as water evaporates
at ground surface. The phase changes involve a material prop-
erty referred to as the latent heat of fusion or the latent heat of
evaporation. Heat flow in unsaturated, unfrozen soils is pre-
sented as well as situations where freezing and thawing may
take place.
The assessment of soil properties for water flow prob-
lems (i.e., saturated or unsaturated seepage) is quite different
from the procedure used for heat flow analyses. Unsaturated
soil water flow problems generally require that the saturated
coefficient of permeability be measured and that the perme-
ability function be estimated from the SWCC. The water
storage function is also computed from the SWCC. How-
ever, for heat flow analyses the thermal conductivity and
heat storage soil parameters can generally be indirectly cal-
culated based on an understanding of the compositions of
the soil (i.e., amount of air and water) and the types of min-
erals present in the soil. Heat flow soil properties can be
measured in the laboratory, but the validity of such mea-
surements depends quite strongly on the skill of the person
performing the test and the quality of the equipment. Con-
sequently, estimation procedures are satisfactory for many
heat flow engineering problems.
It is noteworthy that while the hydraulic conductivity of a
soil may vary over several orders of magnitude, thermal con-
ductivity typically varies by about three times for all types
of soil solids (i.e., about one-third of one order of mag-
nitude). The estimation of heat flow parameters constitutes
an acceptable approach for many geotechnical engineering
problems. Sensitivity and parametric studies play an impor-
tant complimentary role when analyzing heat flow problems.
The SWCC curve also has a role to play in conductive
heat flow analyses since it is the relative components of
air, water, and solids that largely dictate thermal proper-
ties. However, the proportion of air and water in a soil
often remains essentially constant during a conductive heat
flow analysis. Therefore, the thermal properties can often be
assumed to be constant for a particular analysis.
Consideration is given to the case where the thermal prop-
erties remain constant during the heat flow analysis as well
as situations where the thermal properties are nonlinear func-
tions of the SWCC. Writing the thermal properties in terms
of the SWCC is particularly relevant when considering the
solution of coupled analyses where more than one physical
process is involved (e.g., simultaneous flow of water, air,
and heat).
10.2 THEORY OF HEAT FLOW
The constitutive relations associated with heat flow are de-
scribed for each of the thermal soil properties. Typical prop-
erties are given along with the units of measurement. The
freezing and thawing of soils involve the latent heat of fusion,
and modeling these conditions is more complex than model-
ing conductive heat flow through unfrozen soils. The theory
section discusses heat flow when the temperatures are above
freezing as well as thermal modeling when temperatures are
reduced below the phase change isotherm (i.e., 0 C).
10.2.1 Heat Flow Constitutive Relations
The state variable that creates the driving potential for con-
ductive heat flow is temperature T . Temperature is a state
variable. Instruments designed to measure temperature are
referred to as thermometers. Some of the devices for measur-
ing temperature are mercury thermometers, liquid-in-glass
thermometers, bimetallic thermometers, bourdon thermome-
ters, electrical resistance thermometers, and thermocouples.
Heat flow by conduction is the result of a difference in
temperature between two points or what is termed a thermal
gradient. Heat flow q h can be described using Fourier's law,
which has a form similar to Fick's law. The soil property
controlling conductive heat flow is thermal conductivity λ
λ dT
dx
q hx =−
(10.1)
λ dT
dy
q hy =−
(10.2)
λ dT
dz
q hz =−
(10.3)
where:
temperature, C, and
=
T
thermal conductivity of the soil (W/m/K or J/m/ C
since 1 W
λ
=
=
1J/s)
Standard SI units of measurement are used to represent each
of the thermal properties for soils and other materials. Thermal
conductivity is generally assumed to be an isotropic soil prop-
erty although particle shape effects can be taken into account.
10.2.2 Thermal Conductivity of Soils
Thermal conductivity governs the rate of flow of heat through
a material (e.g., soil with solids, water, and air). Conduction
is responsible for the flow of heat through solid materials such
as soil particles. Heat also flows by conduction through water
and air. It is also possible that the mechanisms of convection
and radiation may be involved. The emphasis of this chapter
is on heat transfer by conduction.
Table 10.1 shows thermal conductivity values for water
over a range of temperatures between freezing and 80 o C.
Table 10.2 shows typical thermal conductivity values for a
 
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