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.

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 *T*s 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, *T*f. This time will be small compared to the formation time and a realistic

scenario prior to the onset of freezing is *T*o = *T*f .

1

2

3

Thawed

Frozen

Thawed

Ti = Gx + To

To

Tf

Ts

δ

X

x

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

α1

2 =

0≤*x*≤ *X*

(1)

(1a)

(1b)

For the thawed zone

2

α2

= 2

0≤ *x *≤ *X *+δ

(2)

7