Modified Berggren Formula

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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:

x = \lambda \sqrt{\frac{48 k_{avg}\, n\mathit{FI}}{L}}

**Table 1.**
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/ft³)

Determination of λ

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

**Table 2.**
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 $= \left( \left\| T_f - T_s \right\| \right) \left( \dfrac{C}{L} \right)$
C	equals	average volumetric heat capacity of a soil (BTU/ft³ • °F)
L	equals	latent heat (BTU/ft³)
	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	$\dfrac{n\mathit{FI}}{d} \text{ or } \dfrac{n\mathit{TI}}{d}$
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 $= \left\| \dfrac{\bar{T} - T_f}{T_f - T_s} \right\|$
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).

**Figure 1.** l coefficient in the Modified Berggren Formula.

**Figure 2.** Pictorial representation of variables involved with l.

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/ft³
Water content = w = 15% ''

Solution

Calculate soil thermal properties

Volumetric latent heat of fusion: L = (144 BTU/lb) (100 lb/ft³) (15/100) = 2160 BTU/ft³

Average volumetric specific heat: $C_{avg}$ = 100 (lb/ft³) (0.15) = 28.2 BTU/ft³• °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))

$\alpha = \left| \frac{\bar{T} - T_f}{T_f - T_s} \right| = \frac{\left| \bar{T} - T_f \right| (d)}{n\mathit{FI}} = \frac{\left| 48 - 32 \right| (100)}{750} = 2.13$

$\mu = \left( \left| T_f - T_s \right| \right) \left( \frac{C_{avg}}{L} \right) = \frac{n\mathit{FI}}{d} \left( \frac{C_{avg}}{L} \right) = \left( \frac{750}{100} \right) \left( \frac{28.2}{2160} \right) = 0.10$

From the chart, λ= 0.74

Calculate depth of freezing

$x = \lambda \sqrt{\frac{48 k_{avg}\, n\mathrm{FI}}{\mathrm{L}}} = (0.74) \sqrt{\frac{48\,(0.76)\,(750)}{2160}} = 2.7\ \mathrm{ft.}\ (0.8\ \mathrm{m})$

Calculator

Frost depth (Stefan and modified Berggren)

Stefan · modified Berggren (Aldrich and Paynter)

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Pavement Interactive. (2018, September 19). Modified berggren formula. HeadLight. Retrieved [the date you read it], from https://pavementinteractive.org/reference-desk/design/design-parameters/modified-berggren-formula/

ASCE

Pavement Interactive. (2018). “Modified Berggren Formula.” HeadLight, <https://pavementinteractive.org/reference-desk/design/design-parameters/modified-berggren-formula/> ([the date you read it]).

BibTeX

@misc{pavint-modified-berggren-formula,
  author = {{Pavement Interactive}},
  title = {Modified Berggren Formula},
  year = {2018},
  publisher = {HeadLight},
  howpublished = {\url{https://pavementinteractive.org/reference-desk/design/design-parameters/modified-berggren-formula/}},
  note = {Accessed [date]}
}

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