Table 1. Thermal properties of the inclusion material (sand or sandy soil) and surrounding
soil for numerical simulations of heat flow.
cm soil (cal/cm ) unfrozen frozen
Sandy soil 3%
Latent heat: The quantity of heat released per unit volume of material frozen
Heat capacity: The quantity of heat required to increase the temperature of a unit quantity of material one degree
Thermal conductivity: Heat flow per unit time per unit temperature gradient across a cross-sectional area of material. The
transfer of heat occurs by conduction.
Thermal diffusivity: A measure of the transport of heat across a temperature gradient; equal to the thermal conductivi-
ty divided by the heat capacity per unit volume.
and wetter conditions, respectively, at the onset
where C = heat capacity
∅s = solid fraction (1 - porosity)
of winter. These three moisture contents are sug-
∅ = volumetric moisture content.
gestive of three different climates in terms of rel-
ative rainfall per year. The moisture contents are
This is based on a heat capacity of the mineral
multiplied by the material's dry density to con-
solids equal to 0.54 cal/cm3, of liquid water equal
vert them to moisture contents by volume.
to 1 cal/cm3, and of ice equal to 0.46 cal/cm3.
The latent heat released upon freezing a unit
The thermal conductivities of the soil for each
volume of soil or sand depends on volumetric
moisture content are taken from plots of the aver-
moisture content. For pure water, the latent heat
age frozen and unfrozen thermal conductivity of
released is the latent heat per unit mass of water
silt and clay soils as a function of water content
(80 cal/g) times the density of ice (0.917 g/cm3),
and dry density in Andersland and Anderson
or 73.36 cal per cm3 of ice. For a partially saturat-
(1978, Figures 3.8 and 3.9, respectively).
ed soil or sand, the latent heat released is 73.36
Two materials are used for the inclusion. The
cal/cm3 times the volumetric moisture content.
This assumes that all the water within the soil or
unfrozen quartz sand is taken from Farouki
sand freezes at 0C, and so ignores freezing
(1981, Fig. 53) who reports De Vries's (1974) data
point depression. The unfrozen moisture content
on thermal conductivity of quartz sand as a func-
of silt, however, is ≤ 5% by weight at tempera-
tion of the volume fraction of water. The thermal
tures less than 1C (Williams 1967, Anderson
conductivity of the sand when frozen is assumed
and Morgenstern 1973); for a coarser, silty soil,
to be 16% lower; this is based on the ratio of ther-
the unfrozen moisture content would be less.
mal conductivity of Lowell sand below freezing
The simplification of having all soil moisture
to that above freezing as a function of moisture
freeze at 0C instead of exhibiting a (below-
content, as reported by Farouki (1981, after Ker-
freezing) temperature dependence does not de-
sten 1963). The second inclusion material is a
tract from the usefulness of numerically investi-
sandy soil; thermal conductivities for this materi-
gating frost penetration in silty soils of distinctly
al frozen and unfrozen are taken from Figures 3.6
different moisture content (10, 17, or 25% by
and 3.7, respectively, of Andersland and Ander-
son (1978). The thermal conductivity of the soil
Heat capacities of the soil and sand depend
increases upon the soil freezing, but that of the
on their porosities and volumetric moisture con-
sand and sandy soil decreases. Farouki notes that
tents (eq 1).
saturated soils and soils with a high degree of
saturation have a higher thermal conductivity
C = 0.54 ∅s + 1.0 ∅ (unfrozen)
when frozen because the thermal conductivity of
ice is approximately four times that of water. At
C = 0.54 ∅s + 0.46 ∅ (frozen)
low degrees of saturation, however, heat conduc-