Environmental Engineering Reference
In-Depth Information
previous partial differential equation into the third dimension.
Each dimension that is added must have the heat flux in that
direction taken into consideration. However, the right-hand
side of the equation involving the heat storage term remains
the same:
The modified volumetric heat capacity term applies only
as the soil is going through either the freezing process or
the thawing process.
One-Dimensional Heat Flow:
The transient differential
equation that includes the freezing process in a soil must
use the equation with the modified volumetric heat capacity
of the soil. The modified volumetric heat capacity equation
(i.e., Eq. 10.23) can be substituted into the transient heat
flow equations presented in the previous section to yield the
following equation:
λ
∂
2
T
∂x
2
λ
∂
2
T
∂y
2
∂λ
∂x
∂T
∂x
+
+
λ
∂
2
T
∂y
2
∂λ
∂y
∂T
∂y
∂λ
∂y
∂T
∂y
ζ
∂T
∂t
+
+
+
=
(10.22)
ζ
∂T
∂t
λ
∂
2
T
∂y
2
∂λ
∂y
∂T
∂y
∂θ
i
∂T
The above partial differential equation satisfies transient
and steady-state conditions as long as the soil temperature is
in the unfrozen soil range. The thermal soil properties have
also been assumed to be isotropic.
+
=
−
L
f
ρ
i
(10.24)
The terms on the right-hand side of Eq. 10.24 can be
expanded to separate the heat flow components that are due
to latent heat from the remainder of the heat flow equation:
10.4.3 Heat Flow Partial Differential Equations
Including Freezing and Thawing of Soils
The phase change from liquid water to ice produces what
appears to be a discontinuity in the heat flow partial dif-
ferential equations. The phase change can be modeled as a
process that occurs as the water in the soil gradually freezes
over a relatively small temperature range.
The heat storage portion of the formulation is influenced
by the freezing and thawing of a soil. There is no need to
write partial differential equations for steady-state conditions
involving freezing and thawing because it is only the transient
heat flow process that is affected by the latent heat of fusion.
As temperature is lowered to zero degrees (Celcius), the
latent heat of fusion
L
must be satisfied before the soil-water
temperature can decrease further. The reverse process occurs
as ice in the soil reverts to liquid water. The latent heat of
fusion behaves as a heat sink during the freezing process
and as a heat source during the thawing process.
The heat and mass transfer formulations proposed by Har-
lan (1973) have been extensively used for modeling heat flow
when passing through the freezing isotherm. Nixon (1975),
Taylor and Luthin (1978), and Fuch et al., (1978) have shown
that the convective heat term proposed by Harlan (1973) is
negligible and can be omitted from the formulation.
The volumetric heat capacity term
ζ
needs to be modified
to accommodate the latent heat of fusion, which can now
be written as follows:
λ
∂
2
T
∂y
2
∂λ
∂y
∂T
∂y
ζ
∂T
∂t
∂θ
i
∂T
∂T
∂t
+
=
−
Lρ
i
(10.25)
The term,
∂θ
i
∂T
, represents the slope of the SFCC,
m
i
2
.
Therefore, Eq. 10.25 can incorporate the soil-freezing func-
tion (i.e., the slope of the ice content versus temperature
below 0
◦
C):
λ
∂
2
T
∂y
2
∂λ
∂y
∂T
∂y
ζ
∂T
∂t
∂T
∂t
Lρ
i
m
i
2
+
=
−
(10.26)
The latent heat term applies when the temperature lies
within the zone of temperatures over which the water in the
soil freezes (or thaws). The zone of freezing temperatures
ranges from 0
◦
C to a lower temperature where all the water
in the soil becomes frozen.
The thermal conductivity
λ
and the volumetric heat capac-
ity
ζ
of the soil must correspond to the unfrozen soil when
the temperature is above 0
◦
C and the frozen soil properties
must be used when the temperature corresponds to condi-
tions where the pore-water is frozen.
Two-Dimensional Heat Flow:
The two-dimensional tran-
sient heat flow equation for freezing conditions can be writ-
ten as follows:
λ
∂
2
T
∂x
2
λ
∂
2
T
∂y
2
∂T
∂t
(10.27)
The soil properties must represent either the frozen or
unfrozen conditions in the soil.
Three-Dimensional Heat Flow:
The three-dimensional
transient heat flow equation for freezing conditions can be
written as follows:
λ
∂
2
T
∂x
2
∂λ
∂x
∂T
∂x
+
∂λ
∂y
∂T
∂y
=
ζ
∂T
L
f
ρ
i
m
i
2
+
+
∂t
−
∂θ
i
∂T
ζ
=
ζ
−
Lρ
i
(10.23)
where:
ζ
=
modified volumetric heat capacity as the soil freezes
or thaws, J/m
3
/
◦
C
,
λ
∂
2
T
∂y
2
λ
∂
2
T
∂z
2
∂λ
∂x
∂T
∂x
∂λ
∂y
∂T
∂y
∂λ
∂z
∂T
∂z
+
+
+
+
+
ζ
=
volumetric
heat
capacity
of
the
unfrozen
soil,
ζ
∂T
∂t
∂T
∂t
J/m
3
/
◦
C,
L
f
ρ
i
m
i
2
=
−
(10.28)
L
=
latent heat of fusion, J/kg,
density of ice, kg/m
3
, and
ρ
i
=
The soil properties must represent either the frozen or the
unfrozen conditions in the soil.
volumetric fraction of ice, m
3
/m
3
.
θ
i
=
Search WWH ::
Custom Search