Frost Terminology

Some equations and variables in this section are expressed in English units only due to their source. 

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Density (g_d or g_dry)

Typical densities for commonly encountered dry soils.  Ice rich soils can have substantially smaller dry densities.

Table 1: Typical Soil Densities

Normally, dry densities are used in most calculations (g_d); however, the total density of a soil (including moisture) can be calculated as follows:

g_total = g_d (1 + w/100)

 

 

For example, if a gravel has g_d = 130 lb/ft³ and w = 5%, then g_t = 130(1 + 5/100) = 136.5 lb/ft³ 

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Moisture Content (w)

Soil moisture content can be calculated as follows:

 

Table 2: Typical Moisture Contents of Non-Ice Rich Soils

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Thermal Conductivity(k)

The thermal conductivity is the rate of heat flow through a unit area under a thermal gradient (recall that 1 BTU is the energy (heat) required to raise the temperature of 1 lb of water 1 °F):

Units:    BTU/hr • ft² • °F/ft or BTU/hr • ft • °F

 Further, k_{snow (loose)} = 0.06,   k_{snow (compact)} = 0.20.

In the range of water contents (5 to 10%) and dry densities (125-135 lb/ft³) commonly encountered in embankments and pavement base courses, thermal conductivity is very sensitive to moisture content and soil type.  Soil thermal conductivities can be obtained from Figures 1 through 3 (from Kirsten, 1949 and Air Force, 1966):

 For example, using Figures 1 through 3:

Base Course (granular)

gd = 135 lb/ft3 w = 5%

kfrozen = 1.8 BTU/hr • ft2 •  °F kunfrozen= 1.65 BTU/hr • ft2 • °F

 Subgrade (silt)

gd = 100 lb/ft3 w = 15%

kfrozen = 0.8 BTU/hr • ft2 •  °F kunfrozen= 0.72 BTU/hr • ft2 • °F

Figure 1: Average Thermal Conductivity for Silt and Clay Soils,
Frozen and Unfrozen (redrawn from Kersten, 1949 and Air Force, 1966)
 

Figure 2: Average Thermal Conductivity for Granular Soils, Frozen and Unfrozen
(redrawn from Kersten, 1949 and Air Force, 1966)
 

Figure 3: Average Thermal Conductivity for Peat, Frozen and Unfrozen
(redrawn from Kersten, 1949 and Air Force, 1966)

The equations used to develop Figures 1 and 2 follow.  There are separate equations for frozen (tests conducted at -4 °C) and unfrozen (tests conducted at +4 °C) conditions. 

• Unfrozen silt–clay soils
  
• Frozen silt–clay soils
  
• Unfrozen granular soils (sands and gravels)
  
• Frozen granular soils (sands and gravels)
  

The equations for silt–clay were based on five soils and are valid for moisture contents of seven percent or higher.  The equations for granular soils were based on four soils (two sands, a sandy loam, and a gravel) and are valid for moisture contents of one percent or higher.  Farouki (1986) noted that Kersten's equations do not apply to dry soils or to crushed rocks.  Use of the above equations to estimate k is more accurate than use of Figures 1 and 2. 

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Volumetric Specific Heat(C)

The volumetric specific heat expresses the change in thermal energy in a unit volume of soil per unit change in temperature (units are in BTU/ft³ • °F). 

Volumetric heat is derived from specific heat.  (Recall:  specific heat is the change in thermal energy per unit weight per unit change in temperature.  If objects of the same weight but of different materials receive the same amount of energy (heat), they will come to equilibrium at different temperatures.  Or, stated another way, how much each object's temperature changes depends on the specific heat of the material, if the mass and energy inputs are identical).

• Typical values of specific heat (c)
• c_water = 1.0 BTU/lb • °F
• c_ice = 0.5 BTU/lb • °F
• c_rock = 0.17 BTU/lb • °F (soil minerals)
• Volumetric specific heat relationship (C)
• Unfrozen soil
  
• Frozen soil
  

Calculated values for a gravel with gd = 130 lb/ft and w = 5%:

Cu = 130 (0.17 + 5/100) = 28.6 BTU/ft3 • °F Cf = 130 (0.17 + (0.5) (5/100)) = 25.4 BTU/ft3 • °F Cavg = 130 (0.17 + (0.75) (5/100)) = 27.0 BTU/ft3 • °F

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Latent Heat(L)

All objects have energy (heat).  A portion of this thermal energy (stored heat) is released when the object cools.  The sketch below represents a volume of soil with some moisture as it freezes:

As water freezes, thermal energy equal to L is released while the temperature of the soil remains nearly constant.  Thus, the latent heat is the energy required to transform 1 lb of a pure substance from one phase to another at constant temperature.  Further, 1 lb. of water gives off 144 BTU as it freezes.  The latent heat of a soil can be represented by:

An embankment material with w = 5% and g_d = 140 lb/ft³,

 

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Freezing and Thawing Indices

Depth of freezing and thawing depends in part on the magnitude and duration of the temperature differential below or above freezing (32 °F) at the ground surface.  The freezing or thawing index is therefore given by the summation of the degree-days for a freezing or thawing season.

       

 

Calculations for Freezing Index (FI) or Thawing Index (TI)
 

  

 

Example FI/TI Calculations

 Notes:

1. Assume Day 1 start of freezing season. The negative sign in this case indicates freezing degree-days (normally omitted).
2. For the purpose of assessing spring load restrictions, use 29 °F in lieu of 32 °F. This accounts for the "dark" bituminous surface

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Air and Surface Indexes

Normally, data are only available for air freezing and thawing indexes (@ 1 meter in air above ground).  However there is still a need to establish potential heat flow at the air-ground interface.  No simple correlation exists between air and surface indexes.  Differences between air and surface temperatures are influenced by:

 

However, designers generally use "n-factor" for purposes of correlation.

n-Factor for Freezing Conditions

"n" increases with increases in latitude and wind speed.  Snow covered surfaces reflect large portion of incoming solar radiation with a resulting larger surface freezing index.           

Table 3: Typical “n” Values for Freezing Conditions

n-Factor for Thawing Conditions

"n" decreases with increases in latitude and wind speed. 

Table 3: Typical “n” Values for Thawing Conditions

 Design Freezing and Thawing Indexes

For design purposes, generally use freezing (or thawing) index based on three coldest winters (or warmest summers) in last 30 years of record.  If not available, use air-freezing index for the coldest winter in last 10 years.

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