a5 = k21(β + 2)
m = β[σ(β + 2) + 2φ].
The problem has now been reduced to a simple numerical quadrature of eq 9 using the auxiliary relations
of eq 1012. The numerical solution of eq 9 can be obtained quite easily with a personal computer and any
standard numerical integration routine. (A FORTRAN program, PERM.FOR, to carry out the integration is
listed in Appendix E.)
The model requires only the ratios of the thawed to frozen values of thermal conductivity, specific heat ca-
pacity and density for the permafrost soils. These property ratios can be estimated with good accuracy for soil
systems as noted in Appendix A. The absolute values of the frozen and thawed soil properties are not needed
to carry out the solution of eq 912.
Table 1. Movement of Tf isotherm, homo-
Heterogenetic model verification
geneous soil for heterogenetic freezing.
It is possible to check the solution for a special case. Al-
ST =1000, = 0, Tf Ts = 10C,
though there is no exact solution for the phase-change case, a
φ = 0, G = 0.0286C/m.
solution was found for the transient location of the Tf isotherm
Time (years)
X
for a homogeneous soil with zero latent heat, i.e, infinite Ste-
(m)
Eq 9
Exact eq 13 % difference
fan number (Lunardini, in prep.). The relation is
27.97
2.08
2.15
3.6
69.93
22.63
25.30
10.6
X
GX
= 1-
.
erf
139.86
204.48
234.73
12.9
(13)
Tf - Ts
2 αt
314.69
53,053.2
48,141.3
10.2
332.17
242,852.4 238,469.1
1.8
If we let the Stefan number be large and hold the property
ratios to unity, the approximate solution can be compared to this exact relation. Table 1 notes the results for a
typical case.
The comparison indicates that the approximate technique gives good results, especially as the time in-
creases. The results also show that the Tf isotherm requires surprisingly long times to penetrate deeply, even
without phase change. This is explainable by the sensible-to-latent heat ratios to be examined below.
Sensible and latent heats
The total energy extracted from a unit area of soil is the sum of the latent and sensible energies.
QT = QL + QS .
(14)
The latent energy is
QL = XL
(15)
while the sensible heat flow is
X+δ
X
X
∫ [T - T ( x, t)]dx + C ∫ (T - T ) dx + C ∫ [T - T ( x, t)] dx
QS = Cf
(16)
f
1
u
i
f
u
i
2
X
0
0
where Ti (x) is the original temperature before the freeze starts. Using the temperature relations (eq 6 and 7)
leads to
QS = Cf ∆T1 X{C21 [(σ/2) + φ + β(σ+ φ) / 3] + (0.5/g + 1) / 3} .
(17)
The ratio of the sensible to the latent heat is
QS /QL = ST {C21 [(σ/2) + φ + β(σ + φ) / 3] + (0.5/g + 1)/3}.
(18)
This ratio is quite large even for small Stefan numbers and tends to increase as the freeze depth increases.
10
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