solutions, such as assuming a step change in surface

where

temperature or neglecting the soil temperature

k

= thermal conductivity of the material

changes, they generally overestimate the maximum

above the freezing isotherm, kf for frost

freezing isotherm depths for the given conditions

penetration and kt for thawing calcula-

and are, therefore, conservative for engineering

tions, BTU/hrftEF.

applications. They are generally Neumann or Stefan-

based solutions which have the basic form:

L = volumetric latent heat of the material

undergoing phase change, BTU/cf (for

X = m(Ig)

(eq 12-3)

water L = [(144 BTU/lb) (62.4

pcf) ] = 8986 BTU/cf)

where

C = volumetric heat capacity of the material

above the freezing isotherm, Cf or C1.

X = depth of freezing or thawing, feet

For thawed soil: Ct = ( [ Cs + Cw

(w/100) ] and for frozen soil: Cf = ([ Cs

m = coefficient of proportionality

+ Ci(w/100)]

Ig = ground surface freezing (If) or

( = dry unit weight of soil, pcf

thawing (It) index, EFhr

Cs = mass heat capacity of mineral matter in

The following equations incorporate various as-

soil; assume a value of 0.2 BTU/lb

sumptions, and are useful for specific conditions:

Cw = mass heat capacity of water = 1.0

BTU/lb

Ci = mass heat capacity of ice, assumed value

of 0.5 BTU/lb

w = moisture content of soil, %

Tm = mean annual site temperature, EF

t

= freezing or thawing period, consistent

units

To = freezing point, 32EF for water

d

= thickness of layer of material, feet

8 = a correction coefficient which takes into

consideration the effect of temperature

change in the soil, and primarily

accounts for the volumetric specific heat

effects. It is a function of two

parameters. the thermal ratio (a) and the

fusion parameter (), and is determined

from figure 12-11:

Ts = Ig/t, surface freezing or thawing index

divided by the time period t, EF.

Subscripts f and t refer to freezing and thawing, and

subscripts 1 and 2 refer to the surface layer and the

underlying material (all symbols are also defined in

table 12-1).

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