Environmental Engineering Reference
In-Depth Information
analysis to be combined with a saturated-unsaturated soil
seepage analysis. The results of the seepage analysis could
be used to calculate the thermal conductivity throughout the
soil mass.
Only one soil thermal property is required when solving a
steady-state heat flow analysis, namely, the thermal conduc-
tivity of the soil. In some cases the thermal conductivity can
be represented as a constant value while in other cases the
thermal conductivity may need to be considered as a function
of the stress state or one of the volume-mass soil properties.
The nonlinearity in the heat flow equation arises in the
second term of Eq. 10.14, which accounts for the variation
in thermal conductivity with respect to space. If the thermal
conductivity is a constant value and the soil is homogeneous,
then Eq. 10.14 becomes linear, as shown in the equation
to time. The volumetric heat capacity of the soil,
ζ
,mustbe
specified when performing a transient heat flow analysis.
One-Dimensional Heat Flow:
The conservation of ther-
mal energy requires that the net heat flux through an elemen-
tal volume be equal to the volumetric heat capacity of the
soil multiplied by the change in temperature with respect to
time. The heat flux equation satisfying transient conditions
can be written as follows:
∂q
hy
∂y
ζ
∂T
∂t
=
(10.18)
where:
volumetric specific heat, J/m
3
/
◦
C.
The volumetric heat capacity
ζ
(i.e., in units of J/m
3
/
◦
C)
can be written as the mass specific heat (i.e., in units of
J/g/
◦
C) multiplied by soil density
ρ
(i.e., in units of g/cm
3
).
The above equation applies as long as the temperature of
the soil is above freezing.
A transient analysis occurs when one or more of the bound-
ary conditions are changed. Consequently, there are changes
in temperature within the continuum with respect to time.
Fourier's heat flow equation can be substituted for the quan-
tity of heat flow to give the following differential equation:
ζ
=
λ
d
2
T
dy
2
=
0
(10.15)
Two-Dimensional Heat Flow:
The partial differential
equation for steady-state heat flow through a saturated-
unsaturated soil can readily be extended to a two-dimensional
continuum. One face of the soil continuum (i.e., the third
dimension) would have an insulated surface or else it is
possible that the soil adjacent to the soil being analyzed is
subjected to the same boundary conditions and the same soil
properties as the two-dimensional plane of soil. The partial
differential equation for two-dimensional steady-state heat
flow can be written as follows:
λ
∂
2
T
∂x
2
λ
∂
2
T
∂y
2
∂λ
∂y
∂T
∂y
ζ
∂T
∂t
+
=
(10.19)
The second term on the left-hand side of the equation
takes into account variations in thermal conductivity in the
y-
direction. For example, variations in the degree of saturation
of a soil would influence the thermal conductivity of the soil
and thereby affect the results of a heat flow analysis.
If there is no change in thermal conductivity with respect
to the
y
-dimension, then Eq. 10.19 takes on the standard
diffusion form
λ
∂
2
T
∂y
2
∂λ
∂x
∂T
∂x
∂λ
∂y
∂T
∂y
+
+
+
=
0
(10.16)
There might be a variation in the thermal conductivity of
the soil in two-dimensional space. In this case, the solution
of the PDE for heat flow would rely upon an independent
analysis for the determination of appropriate thermal con-
ductivity values. Thermal conductivity is most significantly
affected by the proportion of air and water in the soil voids.
The SWCC can be used to relate thermal conductivity to the
degree of saturation of the soil (or soil suction).
Three-Dimensional Heat Flow:
The three-dimensional,
steady-state heat flow equation can be written as follows:
λ
∂
2
T
∂y
2
ζ
∂T
∂t
=
(10.20)
The above equation can be applied to heat flow in either
an unfrozen soil or a frozen soil provided the respective soil
properties are inserted for thermal conductivity and volu-
metric heat capacity. However, the temperatures must not
pass through the zero isotherm during the analysis.
Two-Dimensional Heat Flow:
A two-dimensional
transient analysis assumes that there is no heat loss in the
third dimension. The continuum is subjected to a change
in boundary conditions (or an internal thermal source or
sink), and the temperatures throughout the remainder of the
continuum are calculated:
λ
∂
2
T
∂x
2
λ
∂
2
T
∂x
2
λ
∂
2
T
∂y
2
λ
∂
2
T
∂y
2
∂λ
∂x
∂T
∂x
∂λ
∂y
∂T
∂y
∂λ
∂y
∂T
∂y
+
+
+
+
+
=
0
(10.17)
There may be situations where the soil is not isotropic,
and this situation can lead to different thermal conductivity
values in different principal directions. It is also possible that
the principal directions for the thermal conductivity properties
may not coincide with the Cartesian coordinate directions.
λ
∂
2
T
∂y
2
∂λ
∂x
∂T
∂x
∂λ
∂y
∂T
∂y
ζ
∂T
∂t
10.4.2 Transient Heat Flow Partial Differential
Equation for Unfrozen Soils
A transient analysis requires that the net heat flux through an
element be equal to the change in heat storage with respect
+
+
+
=
(10.21)
Three-Dimensional Heat Flow:
A
three-dimensional
transient
thermal
analysis
involves
an
extension
of
the
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