Finally, the ratio of thermal diffusivities is

αu

= α 21 = 21 .

(A10)

αf

The latent heat is

(A11)

The Stefan Number is

0.4202 + 0.0388ε

(T1 - *T*s ) = 1L 1 =

∆*T*1.

(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).

αu/αf

ρu/ρf

ku/kf

Cu/Cf

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.

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

(A13)

= (k )

(

1- ε)

ε

(A14)

g

[

]

ε*S *εS ρ -1

= γ (1- ε) kw /(ki ) wi

ρ

(A15)

where *S *is the thawed soil saturation level, ρwi = ρw /ρi; ρi/ρw = 0.91. Interestingly, the ratio *k*21 can have

the same values as for the saturated case if the saturation has certain values, e.g., *S *= 0.756, ε = 0.379.

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