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1
FROST DEPTH - A GENERAL PREDICTION MODEL
94st Transportation Research Board Annual Meeting
Washington D.C.
January 2015
Pegah Rajaei, PhD student
Department of Civil and Environmental Engineering
Michigan State University
Tel: 517-614-3381; Email: [email protected]
Gilbert. Y. Baladi, Ph.D., PE, Professor (corresponding author)
Department of Civil and Environmental Engineering
Michigan State University, College of Engineering
428 S. Shaw Lane, Room 3546
East Lansing, MI 48824-1226
Tel: 517-355-5147; Fax: 517-432-1827; Email: [email protected]
Word count: 4,348 words; 1 table and 11 figures =7,348 words
Abstract: 227 words
Submission Date
November 12, 2014
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FROST DEPTH - A GENERAL PREDICTION MODEL
ABSTRACT
Accurate frost depth prediction is an important aspect in different engineering designs such as
pavement, building and bridge foundations and/or utility lines. The actual frost depth is affected by
the material type, soil thermal properties, soil water content, and climatic conditions such as
temperature, wind speed, precipitation, and solar radiation. Frost depth could be estimated using
numerical or analytical modeling techniques; however the required input data are either not
available or expensive to collect. Hence, using numerical and analytical models entails the
estimation of the missing data that affects the reliability of the results. In this paper, the accuracy of
different analytical and semi-empirical frost depth prediction models including Stefan model (1),
Modified Berggren model (2) and Chisholm and Phang empirical model (3) were tested using the
soil temperature data from Road Weather Information System (RWIS) in the state of Michigan.
Inasmuch as none of the models yielded accurate results. Therefore, revised empirical models for
different soil types were developed that require only daily high and low air temperatures as input.
The predicted frost depths were more accurate in comparison to the previous models. Finally, by
using thermal conductivity values for each soil type the models were combined into one general
model that requires thermal conductivity and average air temperature as inputs.
Keywords: Heat Transfer, Cumulative Freezing Index, Frost Depth, Stefan Equation, Modified
Berggren Equation, Empirical Model
Rajaei, Baladi
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BACKGROUND
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Frost depth and thaw are important factors that affect the design of all infrastructures including
pavements, building and bridge foundations and/or utility lines. In Geotechnical Engineering, the
effects of frost depth are neutralized by building the foundations below the frost line. For
pavements, most State Highway Agencies (SHAs) use non-frost susceptible soils (such as granular
materials) to decrease the effects of frost and heave. However, over time, the soil becomes frost
susceptible due to the migration of fines from the lower soils. In General, if the percent fine in sand
and gravel exceeds about seven percent, the soils become frost susceptible. The most frost
susceptible soil is silt. Silt has high water holding capacity and relatively low permeability.
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In the past decades, different numerical techniques (finite differences and finite elements)
software have been used for modeling transient heat flow in pavement layers (4, 5, 6, and 7). The
required inputs for these models could include thermal properties of materials, water content, air
temperature, solar radiation, precipitation and wind velocity. If the input data are available the
results of such models could be quite accurate. Otherwise the estimation of the missing data is
required which affects the accuracy and reliability of the results. In the case of missing data, higher
or similar accuracy was obtained from the simpler analytical and semi-imperial models presented
in the empirical model section in this paper.
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One of the first solutions to the heat transfer phase-change problem was proposed by
Neumann in the 1860’s (1). He assumed one-dimensional heat transfer in a semi-infinite region
which is initially at a temperature above the freezing temperature. The surface temperature
suddenly drops to a temperature below the freezing temperature and freezing starts to propagate
through the liquid phase as stated in Equation 1.
P = μ4α t(1)
The parameters and can be calculated using Equations 2 and 3.
exp
−
exp(− )
√$%
−
=
(2)
&' ( )
1 − erf " k
k0
α =
andα0 =
(3)
ρc,
ρc,
Where the subscripts u and f refer to unfrozen and frozen, respectively;
P= frost depth (m);
µ= constant obtained from Equation 2;
α= thermal diffusivity (m2/s) calculated using Equation 3;
t= time since the freezing starts (s);
k= thermal conductivity of the soil (W/ (oC.m);
T3 = Initial ground Temperature (oC.);
T4 = applied constant surface temperature (oC.);
erf = Gauss error function;
l= latent heat of fusion (J/Kg);
c, = specific heat at constant pressure (J/(Kg.oC)); and
ρ= density (Kg/m3).
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Further, in 1891, Stefan modified Neumann’s equation and solved the problem for a special
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case of which no heat transfer in liquid layer was assumed (1). His modified equation is stated
below.
P=
2k T4
t(4)
ρl
It was assumed that the applied constant surface temperature (T4 ) multiplied by the time (t)
is equivalent to the cumulative freezing index (CFI). He further introduced a dimensionless
multiplication parameter (n) to converts air temperature to surface temperature. After converting
metric units to English system Equation 4 becomes Equation 5 (10).
P=
48k ∗ n ∗ CFI
(5)
L
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L = 144wγ? (6)
Where P = frost depth (ft);
kf = thermal conductivity of frozen soil (Btu/(ft hr °F));
n = dimensionless parameter which converts air temperature to surface temperature;
CFI= cumulative freezing index (oF-day);
L = volumetric latent heat of fusion (Btu/ft3);
w = water content;
Ƴd = dry density (pcf); and
all others are as before.
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The resulting Stefan’s Equation 5 does not consider the volumetric heat capacity of the soil
and water and therefore it produces inaccurate results. Consequently, various studies have been
conducted to develop more accurate prediction of frost depth. These include; the modified
Berggren equation (2) and Nixon and McRoberts equation (9). Aldrich et al applied a correction
factor to the Berggren Equation (which is similar to the Neumann’s equation). Their correction
coefficient (B) accounts for the effects of temperature changes in the soil mass and it is a function
of two dimensionless parameters; thermal ratio and fusion parameters. The thermal ratio is a
function of the initial temperature differential (mean annual temperature -32 °F) and the average
temperature differential. The fusion parameter is a function of the average volumetric heat
capacity and volumetric heat fusion (8).
Equation 7 is their modified Berggren equation.
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48k ∗ n ∗ CFI
P = λ
(7)
L
Where k = the average thermal conductivity of frozen and unfrozen soil (Btu/(ft hr °F));
B = correction factor; and
All other factors are as before.
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The Pavement-Transportation Computer Assisted Structural Engineering (PCASE)
software was used to provide accurate numerical solution to the Modified Berggren equation (10).
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Using the data from different stations throughout Ontario, Chisholm and Phang developed
an empirical equation (Equation 8) correlating the calculated cumulative freezing index (CFI) and
the measured frost depths (3).
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P = 1.6968√CFI − 12.91(8)
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Where P = frost depth (in); and
all parameters are as before.
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It should be noted that since the local daily air temperature data were not available all the
time in Ontario, the CFI was calculated based on the monthly average temperature (14).Since then,
many State Highway Agencies (SHAs) used similar approach to generate their own equations or
simply calibrated Equation 8 using local frost depth data and freezing index.
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In this study the frost depth data in 2010 and 2011 from 10 Road Weather Information
System (RWIS) stations in the Upper Peninsula of the State of Michigan and 8 RWIS stations in
the Lower Peninsula were used to develop frost depth prediction model (11). First, the accuracy of
each of three analytical and semi-empirical models (Stefan Modified Berggren and Chisholm and
Phang equations) was evaluated. The results are presented in the next section. Second, an
empirical model regardless of the soil types was developed based on the data in the state of
Michigan. Further, the 2001 to 2012 frost depth data from 8 stations in the State of Minnesota were
used to evaluate the accuracy of the model. Third, the database was divided into two soil types;
sand and clay. Two empirical models were developed by considering the soil types and the
Michigan data. The accuracy of the two models was also evaluated using the Minnesota frost depth
data. Finally, the two frost prediction models were combined using the thermal conductivity data
of each soil type. The accuracy of the combined soil model was also checked using the Minnesota
frost depth data. The results are presented at a later section (empirical models) in this paper
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EXISTING MODELS
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Existing frost depth prediction models can be classified into mechanistic, empirical and
mechanistic-empirical models. The use of any of these models is a function of the types of
available and required input data. Some models require various material properties including heat
capacity of the soil and water. Others require only the cumulative freezing index. During this
study, disturbed soil samples from six stations in the State of Michigan were provided by the
Michigan Department of Transportation (MDOT). Their thermal properties were measured in the
laboratory using KD2 pro thermal properties analyzer. KD2 pro is a small and portable device with
the capability of measuring different thermal properties of almost any material. The device has
different sensors which could be used depending on the required thermal properties and types of
the material. KD2 pro complies fully with ASTM D5334-08 (12). Table 1 shows a summary of the
measured thermal properties of six different types of soil in saturated condition. It should be noted
that all soil samples were disturbed and were not compacted in the laboratory; water was added to
saturate the samples over 24 hour period.
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Stefan Equation
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As stated before, Stefan equation (Equation 5) is one of the first frost depth prediction equations
that was developed and still being used by some State Highway Agencies (SHAs). The cumulative
freezing index (CFI) was calculated using Equation 9 (13):
M
CFI = G DailyFreezingIndex (QR − QST) ≥ 0(9)
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NOP
TZ[\ + TZNM
^ degree − day(10)
2
= Maximum daily air temperature (°F);
TheDailyFreezingIndex = X32Y F −
Where TZ[\
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TZNM = Minimum daily air temperature (°F). It should be noted that in this method the
cumulative freezing index is reset on July 1 of each year.
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TABLE 1 Measured Thermal Properties of Different Types of Soil Using KD2 Pro Analyzer
Station Name
Houghton
Lake
Wolverine
Williamsburg
Rudyard
Material
SILTY fine SAND with trace of
GRAVEL
fine SAND
fine SAND with trace of GRAVEL
fine SAND
Soft CLAYEY SANDY , some SILT
& some GRAVEL
SILTY CLAY
SILTY CLAY
Moisture
Condition
Saturated
Thermal
Heat
conductivity
Capacity
(Btu/(ft.hr.oF)) (Btu/(ft3.oF))
1.49
39.84
1.48
1.44
1.40
42.37
40.13
40.13
1.01
44.46
0.88
0.65
46.25
47.74
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Stefan Equation was used to estimate the frost depths in the saturated condition in different
RWIS stations in the state of Michigan. Unfortunately the in-situ water content and dry density
data of the soils were not available. Therefore, they were estimated using graphs that were
developed by the U.S Army Corps of Engineers relating thermal conductivity to dry density and
moisture content of various soil types (8). Freezing index values for the years 2010 and 2011 were
calculated using the weather data. Further, the volumetric latent heat of fusion (L) was calculated
using Equation 6. Equation 5 was then used to calculate the frost depths for these two years using
the calculated CFI and L. Figure 1 depicts the maximum calculated versus the maximum measured
frost depth data for saturated condition. The straight line in the figure is the line of equality
between the measured and calculated frost depth data. It can be seen from the figure that, for all
soil types, the calculated maximum frost depths in saturated condition are much higher (more than
25-inch) than the measured values. Part of the discrepancy of Equation 5 between the measured
and the calculated maximum frost depths data could be related to two reasons:
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1. The volumetric heat capacity of soil and water was not considered in the equation.
2. Errors in estimating the in-situ water content, dry density using the soil thermal conductivity
and the Corps of Engineers chart.
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Modified Berggren Equation
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Aldrich et al (2) modified Berggren equation in 1953 (Equation 7), which is widely used by
various SHAs. In their modification and solution, they assumed:
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1. One-dimensional heat transfer and the soil is at its mean annual temperature prior to freezing
(8).
2. When the freezing season begins, the soil surface temperature decreases from the mean annual
temperature to a degrees below the freezing point equal to the daily freezing index averaged
over the freezing season, and remains constant at that temperature (10).
Rajaei, Baladi
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Maximum Calculated Frost depth (in)
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Line of Equality
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Sturated Condition
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Maximum Measured Frost Depth (in)
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FIGURE 1 Measured versus calculated maximum frost depths data using Equation 5
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The Maximum frost depths were calculated using Equation 7 and compared to the
maximum measured frost depth data in the state of Michigan. The results are shown in Figure 2. It
can be seen that the Modified Berggren equation (Equation 7) produced more accurate results
relative to the Stefan’s Equation (Equation 5, see Figure 1). However, the differences between the
calculated and measured values in some cases are more than 20 inches. The discrepancy between
the measured and calculated data could be:
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1. Equation 7 does not account the water movement in the soil.
2. The errors in the estimated thermal conductivity, water content and dry density.
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It should be noted that since thermal conductivity data are unavailable and/or expensive to
measure; most DOTs use approximate values, which could lead to inaccurate results.
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Chisholm and Phang (1980) developed Equation 8 for predicting frost depths under asphalt
pavements in Ontario, Canada. They showed that 80 % of the measured frost depth data in Ontario
fall within 12 inches of the predicted frost depths (3). Equation 8 was also used to predict the
maximum monthly frost depths measured at different stations in the state of Michigan. The results
are depicted in Figure 3. The figure indicates that the Chisholm and Phang equation
underestimates the maximum monthly frost depths in most cases. In fact the differences between
the predicted and the measured frost depths could be as high as 30 inches and for some cases, the
calculated frost depths could be negative when the CFI value is small. Given that the empirical
equation was developed based on the measured frost depth data in Ontario, one can conclude that
the equation is regional and should not be used in different regions without calibration.
Chisholm and Phang Equation
Rajaei, Baladi
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Maximum Calculated Frost depth (in)
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Line of Equality
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Sturated Condition
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Maximum Measured Frost Depth (in)
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FIGURE 2 Measured versus calculated maximum frost depths data using Equation 7
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Line of Equality
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Calculated Frost depth (in)
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80
90
-10
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Measured Frost Depth (in)
FIGURE 3 Measured versus calculated maximum frost depths data using Equation 8
To have a better understanding of the difference between the existing models (Equations
Rajaei, Baladi
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5, 7 and 8), the three equations were used to calculate the progression of the frost depth using the
daily calculated CFI during one winter season for one RWIS station located in the Upper Peninsula
in the State of Michigan. The resulting time dependent frost depth data were plotted versus the
measured time dependent frost depth data in Figure 4. It can be seen from the figure that Stefan’s
equation (Equation 5) produced the highest discrepancy between the measured and calculated time
dependent frost depth data followed by the modified Berggren and Chisholm and Phang equations.
It should be noted that the Modified Berggren Equation is based on the average seasonal correction
factor “B" and therefore, it is mostly used for calculating the maximum frost depth (8). Since none
of the existing models yielded accurate results, new empirical models were developed in this study
using RWIS data in the State of Michigan. These new models are presented below.
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NEW EMPERICAL MODELS
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First, the measured frost depths data in 2010 and 2011 in the State of Michigan and the calculated
cumulative freezing index values were used to develop one statistical prediction model for sandy
and clayey soils. The resulting equation is stated below and the measured data and the equation are
depicted in Figure 5.
P = 1.369(CFI)`.abbc (11)
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As can be seen, Equation 11 is parallel to Equation 12, which was developed by the U.S.
Corps of Engineers (15) (also depicted in Figure 5).
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Where: all parameters are as before.
P = 1.6575(CFI)`.def (12)
Line of Equality
Stefan Equation
Modified Berggren Equation
Chisholm and Phang Equation
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Calculated Frost Depth (in)
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Measured Frost Depth (in)
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FIGURE 4 Measured versus calculated time dependent frost depths data using Equation 5, 7, and 8.
Rajaei, Baladi
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U.S Corp of Eng. Prediction
Frost Depth (in)
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y = 1.369x0.5339
R² = 0.9135
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1000
o
Cumulative Freezing Index ( F-day)
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FIGURE 5 Frost depths versus freezing index in Michigan for clayey and sandy soils.
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The data in Figure 5 indicate that Equation 11 predicts the measured frost depths in
Michigan better than Equation 12. Nevertheless, Equation 11 was verified using frost depth data
obtained from the Minnesota Department of Transportation (MnDOT). The results are shown in
Figure 6. It can be seen that the equation over predict the measured frost depth in Minnesota. The
calculated coefficient of determination (R2) is 0.77 for Minnesota data which is much lower than
the 0.91 for the Michigan data.
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To improve the prediction capability of Equation 11 and to assess the impact of soil type on
the measured frost depth data, the Michigan data was divided into two groups per soil type (clay
and sand). Each group of data was then modeled using power equation forms. The results are
stated in Equations 13 and 14 for clayey and sandy soils, respectively, and shown in Figures 7 and
8.
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ForsandysoilsP = 1.3302(CFI)`.adb (14)
Where: all parameters are the same as before.
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The data in Figures 7 and 8 indicate that Equations 13 and 14 represent the data better than
Equation 11. Further, the data in Figure 7 indicates that the US Corps of Engineer equation
represents the measured frost depths data in clayey soils in Michigan much better than the sandy
soils as shown in Figure 8. It is important to note that the number of measured data points in
Figure 7 (clayey soils) is 29 whereas the number of measured data points in sandy soils (Figure 8)
is 129. Nevertheless, to verify Equations 13 and 14, they were used to predict the measured frost
depths in sandy and clayey soils in the State of Minnesota. The results are shown in Figure 9a and
9b. It should be noted that the number of data points in Figure 9a (clayey soil) is 374 while the
ForclayeysoilsP = 1.5901(CFI)`.dfci (13)
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number of data points in Figure 9b (sandy soils) is 247. It can be seen that the equations predict the
measured frost depths in clayey and sandy soils more accurately than Equation 11 (see Figure 6).
In fact, the coefficient of determination (R2) is 0.88 and 0.9 for clayey and sandy soil, respectively.
The high values of R2 indicate that Equations 13 and 14 can predict frost depths in Michigan and
Minnesota or perhaps at the regional level accurately.
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Calculated Frost Depth (in)
Line of Equality
Clay-Minnesota
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Sand-Minnesota
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Measured Frost Depth (in)
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FIGURE 6 Measured versus calculated frost depths data in Minnesota using Equation 11.
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In order to consolidate Equations 13 and 14 into one equation for all soil types, the average
measured thermal conductivity of clayey and sandy soil samples obtained from the State of
Michigan (see Table 1) were used to express the statistical parameters of Equations 13 and 14 as
functions of the thermal conductivity of the soils. This resulted in Equation 15.
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P = (−0.45k + 1.9614) ∗ CFI(.`cPbjk`.dPdb) (15)
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Where: k= thermal conductivity of the soil (Btu/(ft.hr.oF)); and
all other parameters are the same as before.
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Equation 15 was then used to predict the frost depths in the States of Michigan and
Minnesota. The results are shown in Figures 10a and 10b for Michigan and Minnesota data,
respectively. One important note, the frost depth data in Figure 10b were predicted using the
average thermal conductivity of the sandy and clayey soils samples obtained from the State of
Michigan. The reason is that no thermal conductivity data or soil samples were available from the
State of Minnesota. Nevertheless, The data in Figure 10a (for the State of Michigan) indicate that
the results for clayey soils were improved relative to those calculated using Equations 11 and 13
(see Figures 5 and 7). On the other hand, the calculated frost depths in sandy soils are statistically
similar to those calculated using Equation 14.
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Frost Depth in Clay (in)
U.S Corp of Eng. Prediction
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y = 1.5901x0.4896
R² = 0.9433
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600
800
1000
o
Cumulative Freezing Index ( F-day)
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FIGURE 7 Frost depths versus freezing index in Michigan for clayey soils showing Equations 12 and
13.
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Frost Depth in Sand (in)
U.S Corp of Eng.Prediction
y = 1.3302x0.5423
R² = 0.9119
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400
600
800
1000
Cumulative Freezing Index (oF-day)
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FIGURE 8 Frost depths versus freezing index in Michigan for sandy soils showing Equations 12 and
14.
Rajaei, Baladi
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Clay- Minnesota
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60
40
20
0
Line of Equality
Calculated Frost Depth (in)
Calculated Frost Depth (in)
Line of Equality
Sand- Minnesota
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60
40
20
0
0
10 20 30 40 50 60 70 80 90 100
Measured Frost Depth (in)
c
FIGURE 9a
0
10 20 30 40 50 60 70 80 90 100
Measured Frost Depth (in)
FIGURE 9b
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FIGURE 9 Measured versus calculated frost depths in clayey (a) and sandy (b) soils in Minnesota
using Equations 13 and 14
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The data in Figure 10b, on the other hand, indicate that the prediction of the measured frost
depths in Minnesota improved substantially compared to those calculated using Equation 11 (see
Figure 6). The calculated coefficients of determination (R2) are 0.9 in both clayey and sandy soil.
The high values of R2 indicate that Equation 15 could predict frost depths relatively accurate in
other states. In order to further evaluate the proposed model, the measured frost depths in clayey
and sandy soil in the State of Minnesota and the CFI values were statistically modeled using
mathematical power function. The measured frost depth data, the resulting power function and
equation 15 are plotted in Figure 12. As can be seen, the results of the statistical model and
Equation 15 are almost the same for clayey soil. For sandy soil the differences between the power
function and Equation 15 is not significant, although Equation 15 fit the data better..
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SUMMARY AND CONCLUSION
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Measured frost depth data in the State of Michigan were used in three mechanistic,
mechanistic-empirical and empirical models (Stefan, Modified Berggren and Chisholm and
Phang) in order to investigate the accuracy and reliability of the models in estimating the measured
data. Unfortunately, none of the models yielded good results. Therefore, different statistical
models were developed to predict frost depth.
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Firstly, a statistical model was developed using measured frost depth data in the state of
Michigan. Secondly, measured frost depth and freezing index data obtained from MnDOT were
used in order to validate the model. The results indicated that the model cannot be used in another
State without calibration. Hence, the Michigan frost depth data were divided into two groups per
soil type (clayey and sandy) and two statistical models were developed for each soil type. The two
models were also validated using the MnDOT data. Further, the two statistical models were
combined using the average measured thermal conductivity of soil samples obtained from MDOT.
Rajaei, Baladi
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Line of Equality
Clay- Michigan
80
Sand-Michigan
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40
20
0
0
10
20 30 40 50 60 70
Measured Frost Depth (in)
Clay-Minnesota
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Calculated Frost Depth (in)
Calculated Frost Depth (in)
Line of Equality
Sand-Minnesota
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40
20
0
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0
FIGURE 10a
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FIGURE 10 Measured versus calculated frost depths in Michigan (a) and Minnesota (b) using
Equations 15.
Statistical Power Function
Sand-Minnesota
Poposed Model(Equation 15)
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Frost Depth (in)
Frost Depth (in)
Statistical Power Function
Clay-Minnesota
Proposed Model (Equation 15)
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0
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800
1200 1600 2000 2400
Cumulative Freezing Index (oF-day)
FIGURE 11a
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20
0
0
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5
6
10 20 30 40 50 60 70 80 90 100
Measured Frost Depth (in)
FIGURE 10b
0
400
800 1200 1600 2000 2400
Cumulative Freezing Index (oF-day)
FIGURE 11b
FIGURE 11 Frost depths versus freezing index for clayey soil (a) and sandy soil (b) showing the best
fit and Equation 15 in the State of Minnesota.
Rajaei, Baladi
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Based on the results, the following conclusions were drawn:
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1) Existing frost depth prediction models do not accurately predict the measured frost depth in the
States of Michigan and Minnesota. The errors in the models could be related to the various
simplifying assumptions such as the exclusion of volumetric heat capacity in Stefan equation and
water movement in Modified Berggren equation.
2) The inclusion of the average soil thermal conductivity in the developed frost depth statistical
model increased the accuracy of the model, although, for each soil type, only one average thermal
conductivity values of loose and saturated soils were available and used in developing Equation
15.
3) The thermal conductivity of any soil type could vary depending on the soil grain size
distribution curve, water content and dry density. The thermal conductivity values used in
developing Equation 15 were measured in the laboratory using disturbed and saturated soils.
Therefore, the use of Equation 15 should be based on the thermal conductivity of disturbed and
saturated soils.
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ACKNOWLEDGEMENTS
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The Authors like to thank the Michigan Department of Transportation (MDOT) for the financial
support and the MDOT staff for their cooperation and for providing soil samples and the soil
temperature data. In addition the Authors gratefully acknowledge the Minnesota Department of
Transportation (MNDOT) for providing the soil temperature data and for its assistance.
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REFERENCES
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