Finally, the ratio of thermal diffusivities is
αu
k
= α 21 = 21 .
(A10)
αf
C21
The latent heat is
L = 79.71 ε
(A11)
The Stefan Number is
0.4202 + 0.0388ε
C ∆T
(T1 - Ts ) = 1L 1 =
C1
ST =
∆T1.
(A12)
79.71ε
l
It is possible to present the results for soil systems, quite efficiently, since the property ratios can be de-
scribed as functions of the soil void ratio ε (Lunardini and Varotta 1981). Using the thermal conductivities
of Table A1, the property ratios used in the calculations are given in Table A2. The thermal conductivity
ratio will be representative of soil that is not too dry. Thus, eq A4 and A10 should be acceptable if ε ≥ 0.2
(Kersten 1949).
Table A1. Thermal conductivity of
Table A2. Calculated saturated granular
soil thawfreeze property ratios.
materials.
αu/αf
ρu/ρf
Soil
ku/kf
Cu/Cf
Thermal conductivity
porosity ε
W/(mC)
eq 21
eq 24
eq 26
eq 25
Substance
0.2
0.7448
1.2706
0.5862
1.008
Water
0.561
0.3
0.6484
1.3909
0.4662
1.0129
ice
2.281
0.379
0.5812
1.4847
0.3915
1.0174
air
0.0237
0.4
0.5645
1.5094
0.3740
1.0187
silicaceous soil solids*
4.295.87
0.5
0.4915
1.6265
0.3022
1.0256
*Lachenbruch et al. (1982)
0.021 W/m2 at 60800 m, anomalously low heat flow values.
Nonsaturated soil
For the nonsaturated soil, assuming that the porosity does not change during phase changes, the ratio of
thawed to frozen thermal conductivity
()
(1- ε)
ε
kwS ka(1- S)
ku = γkg
(A13)
= (k )
(
1- ε)
kiεSρwi ka(1-Sρwi )
ε
kf
(A14)
g
[
]
εS εS ρ -1
ka ( wi )
= γ (1- ε) kw /(ki ) wi
ρ
ku
(A15)
kf
where S is the thawed soil saturation level, ρwi = ρw /ρi; ρi/ρw = 0.91. Interestingly, the ratio k21 can have
the same values as for the saturated case if the saturation has certain values, e.g., S = 0.756, ε = 0.379.
24
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