The modified Berggren formula was developed in the early 1950s to address the shortcomings of the Stefan formula. The modified Berggren formula assumes that the soil is a semi-infinite mass with uniform properties and existing initially at a uniform temperature (T_{i}). It is further assumed that surface temperature is suddenly changed from T_{i} to T_{s} (below freezing). The modified Berggren formula is simply the Stefan formula corrected for the effects of temperature changes in the soil mass:

where: | x | equals | depth of freeze or thaw, (ft)) | |

λ | equals | dimensionless coefficient which takes into consideration the effect of temperature changes in the soil mass (i.e., a fudge factor). Corrects the Stefan formula for the neglected effects of volumetric heats (accounts for "sensible heat" changes) | ||

k_{avg} |
equals | thermal conductivity of soil, average of frozen and unfrozen (BTU/hr • ft • °F) |
||

n | equals | conversion factor for air freezing (or thawing) index to surface freezing (or thawing) index | ||

FI | equals | air freezing index (°F • days) | ||

TI | equals | air thawing index (°F • days) | ||

L | equals | latent heat (BTU/ft^{3}) |

## Determination of λ

l can be determined by chart (Figure 1) based on inputs of a (thermal ratio) and m (fusion parameter).

λ | equals | f (FI (or TI), mean annual air or ground temperature, thermal properties of soil) |

equals | f(m,a) and can be read from Figure 1 | |

μ | equals | fusion parameter |

C | equals | average volumetric heat capacity of a soil (BTU/ft^{3} • °F) |

L | equals | latent heat (BTU/ft^{3}) |

equals | surface freezing (or thawing) index, nFI (or nTI) divided by length of freezing (or thawing) season. Represents temperature differential between average surface temperature and 32 °F taken over the entire freeze (or thaw) season. | |

equals | ||

d | equals | length of freezing or thawing duration. For example, if the winter freezing season is December through February, then the duration of freezing (d) equals about 90 days. |

T_{f} |
equals | 32 °F |

T_{s} |
equals | average surface temperature for the freezing (or thawing) period |

α | equals | thermal ratio |

T | equals | average annual air or ground temperature |

|T - T_{f}| |
equals | represents the amount that the mean annual temperature exceeds (or is less than) the freezing point of the soil moisture (assumed to be 32 °F). |

## Modified Berggren Formula Example

Determine the depth of frost penetration into a homogeneous sandy silt for the following conditions:

Given:

- Mean annual temperature = 48 °F
- Surface freezing index = nFI = 750 °F • days
- Duration of freezing season = d = 100 days
- Soil properties:
- Dry density = g
_{d }= 100 lb/ft^{3} - Water content = w = 15%
*‘*‘

**Solution**

- Calculate soil thermal properties

Volumetric latent heat of fusion:

L = (144 BTU/lb) (100 lb/ft^{3}) (15/100) = 2160 BTU/ft^{3}

Average volumetric specific heat:

C_{avg}= 100 (lb/ft^{3}) (0.15) = 28.2 BTU/ft^{3 }• °F

Average thermal conductivity:

k_{f}@ 0.80 BTU/hr • ft • °F

k_{u}@ 0.72 BTU/hr • ft • °F

Therefore, k

_{avg}= 0.76 BTU/hr • ft • °F

- Calculate λ (recall λ = f(α, μ))

From the chart, λ= 0.74

- Calculate depth of freezing