age over a space volume, instead of at each point of space (Goodman 1958). The concept is identical to the
well-known momentum integral technique of fluid mechanics, sometimes referred as the Karman-Pohlhaus-
en method (Schlichting 1968). The method has been applied successfully to constant initial temperature
problems of the semi-infinite slab (Lunardini and Varotta 1981) as well as the cylindrical geometry (Lunar-
dini 1980). A modification of the integral method utilizing a single integration over an entire nonconstant
property volume has yielded accurate solutions (Yuen 1980, Lunardini 1981b,1982).
The integral solution has been used for a problem with variable initial temperature distributions, but the
results were limited to shallow freeze depths (Lunardini 1984). This report presents an approximate solution
to the modified Neumann problem for which a linear initial temperature distribution exists. Such an initial
temperature is common for soil systems with a geothermal temperature gradient and is directly applicable to
the question of permafrost formation rates.
Heterogenetic freeze relations
Figure 9 shows the case of an infinite layer of soil with a linear initial temperature distribution (G repre-
sents a geothermal gradient). The soil is assumed to be homogeneous and conduction is the only mode of
energy transfer. At zero time the surface temperature drops to Ts and is held constant while freezing com-
mences. At any time t > 0, there is a frozen layer, called layer 1 (0 < x < X) and a thawed layer (x > X). The
thawed layer is further divided into layer 2 (X < x < X + δ) where temperature changes occur and layer 3 (x
> X + δ) where thermal effects are not discernible. We ignore the finite time it takes for the surface temper-
ature to drop to the freezing point, Tf. This time will be small compared to the formation time and a realistic
scenario prior to the onset of freezing is To = Tf .
1
2
3
Thawed
Frozen
Thawed
Ti = Gx + To
To
Tf
Ts
δ
X
x
Figure 9. Freeze of a semi-infinite region with linear initial temperature.
The governing equations are the conduction energy equations with appropriate boundary and initial con-
ditions; see the Nomenclature for definition of symbols not defined in the text.
For the frozen zone
2
T1
T1
α1
2 =
0≤x≤ X
(1)
x
t
T1 (X, t) = Tf
(1a)
T1 (0, t) = Ts .
(1b)
For the thawed zone
2
T2
T
α2
= 2
0≤ x ≤ X +δ
(2)
x2
t
7
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