# Comparison between Measured and Calculated Thermal Conductivities within Different Grain Size Classes and Their Related Depth Ranges

^{*}

## Abstract

**:**

## 1. Introduction

_{b}), total volumetric water content (θ

_{W}), and grain size distribution (Equations (1) and (2)). The granulometry, as third relevant parameter, is reflected within the differentiation of both formulas. Equation (1) has to be applied if soils contain more than 50% sand and Equation (2) if soils consist of more than 50% of silt and clay [24,30]. It has to be considered that the following formulas are only valid for unfrozen soil conditions. Although there are other approaches for determining thermal conductivity [19,31,32,33,34,35], to be able to compare the outcomes with other studies, the approach of Kersten (1949) was applied. Additionally, when compared to many other models, the approach used enables its application on all textural soil classes.

_{W}/ρ

_{b}) + 0.4) × 10

^{0.6243 × ρ}

_{b}

_{W}/ρ

_{b}) − 0.2) × 10

^{0.6243 × ρ}

_{b}

## 2. Materials and Methods

#### 2.1. Soil Parameter

_{b}, is represented by the ratio of mass of soil substances to occupied volume as given by Equation (3).

_{b}= m

_{d}/V

_{c}

_{d}gives the dry mass and V

_{c}is the volume of the TK04 cylinder containing the soil sample. The bulk density of the investigated samples was determined as described in the next section following DIN 18125-2 and classified according to Table 1.

_{b, eff}, was also determined and compared with the regular bulk density. The difference between the effective and normal bulk density depends on the amount of clay content n

_{c}(Equation (4)) [42].

_{b, eff}= ρ

_{b}+ 0.009 × n

_{c}

^{−}³ and the analysis temperature 35 °C. Porosity (ϕ) can be calculated from the bulk density and density of the soil components (ρ

_{s}) (Equation (5)):

_{b}/ρ

_{s})

_{s}, the same density as quartz (2.65 g cm

^{−}³) was assumed. To consider the computed porosity, the amount of saturated pore volume (S

_{p}) can be determined using the following equation (Equation (6)) [43].

_{p}= θ

_{w}/ϕ

#### 2.2. Thermal Conductivity Measurements

#### 2.3. Data Analysis

^{−}³, 1.5 g cm

^{−}³, and 1.8 g cm

^{−}³. By comparing the values according to the ThermoMap project with the measured thermal conductivities, the best fit between the calculated and measured thermal conductivity values can be assessed.

#### 2.4. Data Projection

## 3. Results

#### 3.1. Bulk Density and Porosity

^{−}³ up to 1.359 g cm

^{−}³ (Table 2) with an average of 1.127 g cm

^{−}³. Elevated bulk density values could be observed with an increasing proportion of sand (Figure 2). For sandy clay loam samples, high and marginally very high bulk densities were measured, reaching values of 1.836 g cm

^{−}³. However, the range of bulk density within each grain size class was quite large, as observable within the loamy sands or clays.

#### 3.2. Thermal Conductivity

#### 3.2.1. Thermal Conductivity Measurements

^{−1}and 2.435 W (m·K)

^{−1}(Table 2). In this range, clay possessed the lowest values, whereas loamy sand and sandy loam showed very high thermal conductivities. However, the spread of the thermal conductivity in the ‘loamy sand’ and ‘sandy loam’ section was relatively high, as they covered a range from 1.581 W (m·K)

^{−1}up to 2.435 W (m·K)

^{−1}. This was accompanied by a positive correlation with increasing bulk density (Figure 5).

#### 3.2.2. Thermal Conductivity Measurements vs. Kersten Formulas

^{−1}) but also for sandy clay (≤0.4 W (m·K)

^{−1}), and minor differences for sandy clay and loamy sand (≤0.35 W (m·K)

^{−1}). Higher contrasts in thermal conductivities, calculated by applying only Equation (1) (Figure 6c), appeared in the areas of sandy loam and silty clay (up to 0.6 W (m·K)

^{−1}). The measured thermal conductivities of clay loam, clay as well as sandy clay loam, on the contrary, hardly deviated from the calculated ones. The overall lowest deviations between the calculated and measured thermal conductivities (Figure 6e) occurred by only applying Equation (2). In this case, differences of ≤0.35 W (m·K)

^{−1}appeared within the loamy sand and silty clay.

^{−1}above the measured thermal conductivities. This relationship is also reflected in Figure 6g where a strong deviation between the measured and calculated thermal conductivities was implied. This observation was true for nearly all of the data points and classes, except for some areas within clay loam, loam, and loamy sand.

#### 3.2.3. Thermal Conductivity Measurements vs. ThermoMap Values

^{−1}) from the measured values. In contrast, clays were rather overestimated with a difference in thermal conductivity of up to 0.30 W (m·K)

^{−1}. By using a standardized bulk density of 1.5 g cm

^{−}³, representing soils at depths of 3–6 m, for calculating thermal conductivity after Kersten (1949), a better fit was achieved. The values generally deviated less from the best fit when compared to the ones calculated for a depth range of 0–3 m. However, a general overestimation of the calculated thermal conductivities calculated with a standardized bulk density of 1.5 g cm

^{−}³ was observed in Figure 9. A maximum positive deviation of 0.55 W (m·K)

^{−1}was reached for clay, whereas loamy sand, sandy loam, and sandy clay loam were still underestimated (down to −0.27 W (m·K)

^{−1}). The thermal conductivities calculated for soils at a depth of 6–9 m, with an expected bulk density of 1.8 g cm

^{−}³, generally showed an unfavorable correlation with the measured thermal conductivity values (Figure 7). This was reflected in the calculated thermal conductivities deviating at least ~0.4 W (m·K)

^{−1}from the highest and ~1.3 W (m·K)

^{−1}from the lowest measured thermal conductivities.

^{−1}(for clays) calculated from the measured thermal conductivities. The smallest deviations in thermal conductivities were determined for loamy sands, silt-poor sandy loams, and silt-poor sandy clay loams.

## 4. Discussion

^{−3}fits best. However, an even better result was achieved by distinguishing between certain grain size classes and applied bulk densities. Clay soils were represented best by using the bulk density of 1.3 g cm

^{−3}(Figure 8), whereas a bulk density of 1.8 g cm

^{−}³ only provided suitable outcomes for very pure sands (Figure 10). All of the other intermediate grain size classes such as loamy sand, sandy loam, or silts should be calculated with an assumed bulk density of 1.5 g cm

^{−3}(Figure 9). These results corresponded to the bulk density ranges stated by Chaudhari et al. (2013), although a high clay content could mask the effect of an increased organic content [1,2]. In contrast to coarse soils [47], clays possess a relatively high porosity, resulting in lower bulk densities.

_{s}) [21,28,30,49]. Additionally, mineral type and grain size distribution are decisive for the thermal conductivity of soils [50]. As all samples in this study were measured under saturated conditions, the degree of saturation and volume fraction of air could be neglected. Moreover, a mineralogical test was not conducted. Consequently, only grain size distribution was taken into account.

^{−1}[51] than most minerals with values >1.9 W (m·K)

^{−1}. Furthermore, the amount of quartz minerals that possess very high thermal properties of up to 7.7 W (m·K)

^{−1}[52,53] was lower in clayey/silty than in sandy soils. Altogether, this led to the decreased thermal conductivities of fine grained soils in contrast to sandy soils. The observations made within this study were thereby in agreement with the results of Brigaud and Vasseur (1989) [52]. Referring to the factors investigated in this study such as bulk density, grain size, and mineralogy, Zhang et al. (2017) reported similar results [50]. This regarded the positive correlation between increasing bulk density, particle size, and thermal conductivity.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

λ [W (m·K)^{−1}] | thermal conductivity (TC) |

ϕ [-] | porosity |

ρ_{b} [g cm^{−}³] | bulk density (BD) |

ρ_{s} [g cm^{−}³] | density, soil components |

θ_{W} [-] | water content (WC) |

S_{p} [-] | amount of saturated pore volume |

m_{d} [g] | mass, dry |

θ [-] | degree of saturation |

V_{c} [cm^{3}] | Volume, cylinder |

n [cm^{3}] | volume, fraction of air |

ρ_{b, eff} [g cm^{−3}] | bulk density, effective |

v_{s} [cm^{3}] | volume, fraction of solids |

n_{c} [-] | amount of clay content |

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**Figure 4.**Distribution of measured thermal conductivity [W (m·K)

^{−1}] for the different textural soil classes.

**Figure 6.**(

**a**) Deviation (%) of the thermal conductivity values using the Kersten formulas (Equations (1) and (2)) from the TK04 measurements; (

**b**) Plot of the measured versus the calculated thermal conductivities after Kersten (Equations (1) and (2)) (Linear regression line = black dashed line); (

**c**) Deviation (%) of the thermal conductivity values using the Kersten formula (Equation (1)) from the TK04 measurement; (

**d**) Plot of the measured versus the calculated thermal conductivities after Kersten (Equation (1)) (Linear regression line = black dashed line); (

**e**) Deviation (%) of the thermal conductivity values using the Kersten formula (Equation (2)) from the TK04 measurements; (

**f**) Plot of the measured versus the calculated thermal conductivities after Kersten (Equation (2)) (Linear regression line = black dashed line); (

**g**) Deviation (%) of the thermal conductivity values using the Kersten formulas (Equations (1) and (2)) and effective bulk density values from the TK04 measurements; (

**h**) Plot of the measured versus the calculated thermal conductivities after Kersten using the effective bulk densities instead of standard bulk densities. (Linear regression line = black dashed line).

**Figure 7.**The measured vs. calculated thermal conductivities according to Kersten (1949) for varying soil depths; i.e., bulk densities according to the ThermoMap project.

**Figure 8.**Positive and negative deviation of the ThermoMap values (BD = 1.3 g cm

^{−3}) from the TK04 measured thermal conductivities.

**Figure 9.**Positive and negative deviation of the ThermoMap values (BD = 1.5 g cm

^{−3}) from the TK04 measured thermal conductivities.

**Figure 10.**Positive deviation of the ThermoMap values (BD = 1.8 g cm

^{−3}) from the TK04 measured thermal conductivities.

**Figure 11.**Comparison between the measured (TK04 results) and calculated thermal conductivities. The calculations according to both Kersten (1949) equations are displayed (sand formula, Equation (1); clay formula, Equation (2)). The values were sorted after the calculated thermal conductivity corresponded to both soil dependent formulas (Equation (1) > 50% sand; Equation (2) < 50% sand).

**Table 1.**Classification of bulk density ranges according to [42].

Classification | Bulk Density [g/cm^{3}] |
---|---|

very low | <1.2 |

low | 1.2–1.4 |

medium | 1.4–1.6 |

high | 1.6–1.8 |

very high | >1.8 |

**Table 2.**Data table with the results of all samples investigated. Bulk density and one thermal conductivity (TK04-column) were measured by the TK04 device. Porosity and thermal conductivity according to Kersten (1949) were both computed based on the outcomes of the performed measurements of this study. The third calculated thermal conductivity was provided by the ThermoMap project. Textural soil classes were derived based on the samples granulometry.

Sample-ID | Textural Soil Class | Bulk Density [g/cm³] | Porosity [%] | Thermal Conductivity [W (m·K)^{−1}] | |
---|---|---|---|---|---|

TK04 | Kersten (1949) | ||||

I3 | Loamy Sand | 1.060 | 60.0 | 2.197 | 1.088 |

II2b | Sandy Loam | 1.561 | 41.0 | 1.834 | 2.080 |

II3a | Loamy Sand | 1.723 | 35.0 | 2.318 | 2.542 |

III2a | Sandy Clay Loam | 1.598 | 40.0 | 1.771 | 2.179 |

III2b | Sandy Clay Loam | 1.836 | 31.0 | 2.426 | 2.910 |

III3a | Sandy Clay Loam | 1.503 | 43.0 | 1.398 | 1.934 |

III3b | Clay Loam | 1.246 | 53.0 | 1.429 | 1.169 |

III4b | Sandy Clay Loam | 1.672 | 37.0 | 1.912 | 2.388 |

III5a | Clay | 1.359 | 49.0 | 0.959 | 1.341 |

III5b | Sandy Loam | 1.707 | 36.0 | 2.019 | 2.493 |

IV2b | Clay | 0.959 | 64.0 | 0.906 | 0.815 |

IV3a | Clay | 1.185 | 55.0 | 1.147 | 1.084 |

IV3b | Clay | 1.134 | 57.0 | 1.188 | 1.017 |

IV4 | Clay | 1.134 | 57.0 | 1.152 | 1.017 |

RD2 | Clay | 1.085 | 54.6 | 1.241 | 0.936 |

RD3 | Clay | 1.106 | 52.6 | 1.183 | 0.954 |

RD4 | Clay | 1.078 | 50.4 | 1.266 | 0.904 |

RD5 | Sandy Loam | 1.547 | 24.6 | 1.704 | 1.831 |

RD6 | Clay Loam | 1.071 | 56.2 | 1.169 | 0.924 |

RD7 | Loam | 1.209 | 45.2 | 1.293 | 1.057 |

RE1 | Sandy Clay Loam | 1.595 | 23.9 | 2.020 | 1.949 |

RE2 | Sandy Clay Loam | 1.532 | 26.6 | 2.065 | 1.823 |

RE3 | Sandy Clay Loam | 1.718 | 21.0 | 2.285 | 2.258 |

RE4 | Sandy Loam | 1.644 | 22.9 | 2.173 | 2.072 |

RE5 | Sandy Clay Loam | 1.624 | 23.6 | 1.873 | 2.025 |

RE6 | Sandy Loam | 1.629 | 23.6 | 2.070 | 2.041 |

RF1 | Sandy Clay Loam | 1.483 | 28.9 | 1.875 | 1.730 |

RF2 | Loamy Sand | 1.694 | 20.5 | 2.412 | 2.170 |

RF3 | Loamy Sand | 1.560 | 25.5 | 1.687 | 1.880 |

RF4 | Sandy Loam | 1.694 | 21.0 | 2.413 | 2.182 |

RF5 | Loamy Sand | 1.545 | 26.5 | 2.189 | 1.856 |

RF6 | Loamy Sand | 1.538 | 26.9 | 2.037 | 1.844 |

RF7 | Loamy Sand | 1.544 | 26.1 | 2.192 | 1.845 |

RI-2 | Silt Loam | 1.260 | 52.4 | 1.157 | 1.189 |

RI-5 | Sandy Clay Loam | 1.590 | 39.9 | 1.666 | 2.156 |

RIII-2 | Silty Clay | 1.450 | 45.1 | 1.138 | 1.494 |

RIII-5 | Clay Loam | 1.360 | 43.0 | 1.317 | 1.294 |

RIV-1b | Sandy Clay Loam | 1.280 | 51.6 | 1.355 | 1.452 |

RIV-2 | Clay Loam | 1.210 | 54.5 | 1.180 | 1.119 |

RIV-4 | Clay Loam | 1.440 | 45.6 | 1.281 | 1.478 |

RIX-1 | Clay | 1.100 | 58.4 | 1.098 | 0.974 |

RVIII-2 | Loamy Sand | 1.770 | 33.1 | 2.435 | 2.688 |

V2 | Sandy Loam | 1.667 | 37.0 | 1.921 | 2.373 |

VA2 | Clay Loam | 1.340 | 43.0 | 1.407 | 1.257 |

VA3 | Clay Loam | 1.560 | 43.0 | 1.762 | 1.725 |

VA4 | Clay Loam | 1.430 | 43.0 | 1.345 | 1.431 |

VA5 | Loam | 1.490 | 42.0 | 1.487 | 1.548 |

VA6 | Sandy Loam | 1.650 | 52.0 | 1.616 | 2.475 |

VA7 | Sandy Loam | 1.650 | 43.0 | 1.665 | 2.385 |

VA8 | Sandy Loam | 1.660 | 52.0 | 1.674 | 2.510 |

VA9 | Sandy Loam | 1.620 | 36.0 | 1.581 | 2.205 |

VB1 | Sandy Clay Loam | 1.470 | 35.0 | 1.230 | 1.767 |

VB10 | Sandy Loam | 1.590 | 54.0 | 1.591 | 2.286 |

VB2 | Clay Loam | 1.210 | 43.0 | 1.477 | 1.043 |

VB3 | Sandy Clay Loam | 1.360 | 54.0 | 1.517 | 1.384 |

VB5 | Loam | 1.300 | 36.0 | 1.408 | 1.122 |

VB6 | Loam | 1.270 | 42.0 | 1.352 | 1.129 |

VB7 | Silty Clay Loam | 1.360 | 48.8 | 1.257 | 1.344 |

VB8 | Loam | 1.430 | 43.0 | 1.466 | 1.431 |

VB9 | Loam | 1.410 | 54.0 | 1.426 | 1.488 |

VI3b | Sandy Loam | 1.723 | 35.0 | 2.378 | 2.542 |

**Table 3.**TK04-measured thermal conductivities (TC) for each classified soil sample and three calculated ThermoMap thermal conductivity values; each column has one single bulk density (BD) value. The three different bulk density values represent the depth layers 0–3 m, 3–6 m, and 6–10 m [7].

Sample-ID | Textural Soil Class | TC (TK04) [W/(m·K)] | TC ThermoMap [W (m·K)^{−1}] (BD = 1.3 g cm^{−}³) | TC ThermoMap [W (m·K)^{−1}] (BD = 1.5 g cm^{−}³) | TC ThermoMap [W (m·K)^{−1}] (BD = 1.8 g cm^{−}³) |
---|---|---|---|---|---|

I3 | Loamy Sand | 2.197 | 1.41 | 1.78 | 2.50 |

II2b | Sandy Loam | 1.834 | 1.42 | 1.77 | 2.51 |

II3a | Loamy Sand | 2.318 | 1.41 | 1.78 | 2.50 |

III2a | Sandy Clay Loam | 1.771 | 1.43 | 1.77 | 2.50 |

III2b | Sandy Clay Loam | 2.426 | 1.43 | 1.77 | 2.50 |

III3a | Sandy Clay Loam | 1.398 | 1.43 | 1.77 | 2.50 |

III3b | Clay Loam | 1.429 | 1.17 | 1.38 | 1.88 |

III4b | Sandy Clay Loam | 1.912 | 1.43 | 1.77 | 2.50 |

III5a | Clay | 0.959 | 1.17 | 1.41 | 1.90 |

III5b | Sandy Loam | 2.019 | 1.42 | 1.77 | 2.51 |

IV2b | Clay | 0.906 | 1.17 | 1.41 | 1.90 |

IV3a | Clay | 1.147 | 1.17 | 1.41 | 1.90 |

IV3b | Clay | 1.188 | 1.17 | 1.41 | 1.90 |

IV4 | Clay | 1.152 | 1.17 | 1.41 | 1.90 |

RD2 | Clay | 1.241 | 1.17 | 1.41 | 1.90 |

RD3 | Clay | 1.183 | 1.17 | 1.41 | 1.90 |

RD4 | Clay | 1.266 | 1.17 | 1.41 | 1.90 |

RD5 | Sandy Loam | 1.704 | 1.42 | 1.77 | 2.51 |

RD6 | Clay Loam | 1.169 | 1.17 | 1.38 | 1.88 |

RD7 | Loam | 1.293 | 1.17 | 1.37 | 1.86 |

RE1 | Sandy Clay Loam | 2.020 | 1.43 | 1.77 | 2.50 |

RE2 | Sandy Clay Loam | 2.065 | 1.43 | 1.77 | 2.50 |

RE3 | Sandy Clay Loam | 2.285 | 1.43 | 1.77 | 2.50 |

RE4 | Sandy Loam | 2.173 | 1.42 | 1.77 | 2.51 |

RE5 | Sandy Clay Loam | 1.873 | 1.43 | 1.77 | 2.50 |

RE6 | Sandy Loam | 2.070 | 1.42 | 1.77 | 2.51 |

RF1 | Sandy Clay Loam | 1.875 | 1.43 | 1.77 | 2.50 |

RF2 | Loamy Sand | 2.412 | 1.41 | 1.78 | 2.50 |

RF3 | Loamy Sand | 1.687 | 1.41 | 1.78 | 2.50 |

RF4 | Sandy Loam | 2.413 | 1.42 | 1.77 | 2.51 |

RF5 | Loamy Sand | 2.189 | 1.41 | 1.78 | 2.50 |

RF6 | Loamy Sand | 2.037 | 1.41 | 1.78 | 2.50 |

RF7 | Loamy Sand | 2.192 | 1.41 | 1.78 | 2.50 |

RI-2 | Silt Loam | 1.157 | 1.15 | 1.38 | 1.90 |

RI-5 | Sandy Clay Loam | 1.666 | 1.43 | 1.77 | 2.50 |

RIII-2 | Silty Clay | 1.138 | 1.16 | 1.42 | 1.92 |

RIII-5 | Clay Loam | 1.317 | 1.17 | 1.38 | 1.88 |

RIV-1b | Sandy Clay Loam | 1.355 | 1.43 | 1.77 | 2.50 |

RIV-2 | Clay Loam | 1.180 | 1.17 | 1.38 | 1.88 |

RIV-4 | Clay Loam | 1.281 | 1.17 | 1.38 | 1.88 |

RIX-1 | Clay | 1.098 | 1.17 | 1.41 | 1.90 |

RVIII-2 | Loamy Sand | 2.435 | 1.41 | 1.78 | 2.50 |

V2 | Sandy Loam | 1.921 | 1.42 | 1.77 | 2.51 |

VA2 | Clay Loam | 1.407 | 1.17 | 1.38 | 1.88 |

VA3 | Clay Loam | 1.762 | 1.17 | 1.38 | 1.88 |

VA4 | Clay Loam | 1.345 | 1.17 | 1.38 | 1.88 |

VA5 | Loam | 1.487 | 1.17 | 1.37 | 1.86 |

VA6 | Sandy Loam | 1.616 | 1.42 | 1.77 | 2.51 |

VA7 | Sandy Loam | 1.665 | 1.42 | 1.77 | 2.51 |

VA8 | Sandy Loam | 1.674 | 1.42 | 1.77 | 2.51 |

VA9 | Sandy Loam | 1.581 | 1.42 | 1.77 | 2.51 |

VB1 | Sandy Clay Loam | 1.230 | 1.43 | 1.77 | 2.50 |

VB10 | Sandy Loam | 1.591 | 1.42 | 1.77 | 2.51 |

VB2 | Clay Loam | 1.477 | 1.17 | 1.38 | 1.88 |

VB3 | Sandy Clay Loam | 1.517 | 1.43 | 1.77 | 2.50 |

VB5 | Loam | 1.408 | 1.17 | 1.37 | 1.86 |

VB6 | Loam | 1.352 | 1.17 | 1.37 | 1.86 |

VB7 | Silty Clay Loam | 1.257 | 1.17 | 1.40 | 1.90 |

VB8 | Loam | 1.466 | 1.17 | 1.37 | 1.86 |

VB9 | Loam | 1.426 | 1.17 | 1.37 | 1.86 |

VI3b | Sandy Loam | 2.378 | 1.42 | 1.77 | 2.51 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Bertermann, D.; Müller, J.; Freitag, S.; Schwarz, H.
Comparison between Measured and Calculated Thermal Conductivities within Different Grain Size Classes and Their Related Depth Ranges. *Soil Syst.* **2018**, *2*, 50.
https://doi.org/10.3390/soilsystems2030050

**AMA Style**

Bertermann D, Müller J, Freitag S, Schwarz H.
Comparison between Measured and Calculated Thermal Conductivities within Different Grain Size Classes and Their Related Depth Ranges. *Soil Systems*. 2018; 2(3):50.
https://doi.org/10.3390/soilsystems2030050

**Chicago/Turabian Style**

Bertermann, David, Johannes Müller, Simon Freitag, and Hans Schwarz.
2018. "Comparison between Measured and Calculated Thermal Conductivities within Different Grain Size Classes and Their Related Depth Ranges" *Soil Systems* 2, no. 3: 50.
https://doi.org/10.3390/soilsystems2030050