where

1n = the partial freezing or thawing index

required to penetrate the nth layer,

EFhr

Ln = volumetric latent heat in the nth layer,

BTU/ft3

dn = thickness of the nth layer, ft

8 = the coefficient based on the weighted

average values for down to and

including the nth layer (see figure 12-

13)

layer, hr ft EF/BTU.

(1) Equation 12-4 is the Stefan solution for a

homogeneous material with a step change in surface

temperature (see example 12-9a). This is modified

(2) The solution for multi-layered systems is

in equation 12-5 to account for the temperature

facilitated by tabular arrangement of the interme-

change in the freezing or thawing soil. Equation 12-

diate values. The penetration into the last layer must

6 is a two-layer solution of the Stefan equation that

be solved by trial and error to match the total

is useful for calculations involving snow cover, a

freezing or thawing index at the site. It is necessary

gravel pad or a board of thermal insulation, in which

to determine the temperature condition at the

the surface layer has no latent heat and the equation

ground surface to determine subsurface thermal

is simplified (see para 12-9*b *and *c*). Equation 12-7

effects, including the depth of freezing and thawing.

is a close approximation of the Neumann solution

Since air temperatures are readily available, but

when the ground temperatures are near freezing.

surface temperatures are not, a correlation factor

Equation 12-8, the modified Berggren equation, is

which combines the effects of radiation, and

perhaps the most commonly used approach for

convective and conductive heat exchange at the air-

determining thermal responses of soils. When the

ground surface is used:

soil has a high moisture content the *X *coefficient

approaches unity, and the equation is identical to the

Ig = n Ia

(eq 12-10)

Stefan approach (equation 12-4). In climates where

the mean annual temperature is near or below

where

freezing, the thermal ratio approaches zero and the

Ig = ground surface freezing or thawing

index, EF hr

the soil warming or cooling can be significant and

Ia = air freezing or thawing index, EF hr

should be included. Multilayered soil systems can be

solved by determining that portion of the surface

n = n-factor, ratio of the surface and air

freezing or thawing index required to penetrate each

temperature indices.

layer. The sum of the thicknesses of the frozen or

thawed layers whose indices equal the total index is

The n-factor is very significant in analytical ground

equal to the depth of freeze or thaw. The partial

thermal considerations. It is highly variable and is

freezing or thawing index to penetrate the nth layer

usually estimated from published observations such

is

as the values listed in Table 12-3.

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