# Mathematical Modelling for Predicting Thermal Properties of Selected Limestone

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}/s), and specific heat (1.57–2.563) ((MJ)/(m

^{3}× K)). The results also suggest a direct relationship between conductivity and diffusivity and an inverse relationship between conductivity and specific heat. On the other hand, the results indicate the direct relationship between the conductivity and diffusivity, and the inverse relationship between the specific heat and density, hardness, sound velocity, and rock strength; the opposite happens when the rock’s porosity is considered. Simple regression, multivariate regression, and the backpropagation–artificial neural network (BP–ANN) approach were utilized to predict the thermal properties of limestone. Results indicated that the ANN model provided superior prediction performance compared to other models.

## 1. Introduction

^{2}), ΔT is the temperature difference (K), c is the specific heat (J/(kg. K)), q is the change in thermal energy (J), m is the mass of the specimen (kg), $\alpha $ is the thermal diffusivity (m

^{2}/s), and 𝜌 is the density of the material (kg/m

^{3}).

^{2}= 0.85. Additionally, thermal conductivity was linked with dry density, porosity, ultrasonic pulse velocity, and uniaxial compressive strength through a set of strong models by Yaşar et al. [1] in limestone, dolomite, marble, travertine, sandstone, siltstone, and basalt, and by Özkahraman et al. [17] in limestone and travertine. In clayey and chalky limestone, Çanakci et al. [9] concluded that the thermal conductivity could be estimated through strong models which link thermal conductivity with dry density, porosity, and water absorption.

## 2. Materials and Methods

#### 2.1. Rock Specimens Preparation

#### 2.2. Method and Laboratory Tests

#### 2.2.1. Thermal Properties Determination

#### 2.2.2. Physical and Engineering Properties Determination

^{3}) for each specimen by the ratio between dry weight and volume [28]. The Ultrasonic Pulse Velocity (V

_{p}) test was carried out according to the recommendations of ASTM D2845 [29] by using PUNDIT Pulse (Portable Ultrasonic Non-Destructive Digital Indicating Tester) (MATEST (S.r.l)/TREVIOLO 24048; ITALY) with two transducers having a frequency of 54 kHz and 6.4 mm wavelength. Five velocities for each specimen were calculated, and their averaging was taken. Schmidt Hammer Rebound (H

_{r}) test was performed on each core specimen according to the recommendations in [30] as a rock hardness index. Apparent specific gravity (Gs) and porosity (n) values for each specimen were determined according to procedures suggested by [28], where the oven-dried specimens were soaked in a water-filled tank. The weight of the specimen was monitored for 22 days until it began to demonstrate a constant weight to ensure that all specimens were fully saturated. In more detail, the weights of all samples began to stabilize after 17 days of soaking, and they were monitored for another five days to ensure that the constant weight had been achieved. Uniaxial Compressive Strength (UCS) was carried out on oven-dried specimens according to the ASTM D7012 [31] recommendations at a constant strain rate on the specimens with length-to-diameter equal to 2. Point load strength (Is

_{(50)}) on irregular rock fragments was also carried out according to ASTM D5731 [32].

## 3. Results and Discussion

#### 3.1. Thermal Properties of Limestone Rock

#### 3.2. Regression Analysis for Predicting Thermal Properties

_{d}, H

_{r}, V

_{p}, and n%, which can be found directly by easy, simple, inexpensive, and non-destructive tests, were considered as independent variables. Additionally, to present a more comprehensive study, engineering properties, including UCS and Is(50), were entered into the list of the independent variables.

#### 3.3. Simple Regression Analysis

_{p}is the predicted value, x is the input parameter, c is the regression constant, and b is the regression coefficient.

^{2}), Root Mean Square Error (RMSE), and Variance Accounted For, four predicted models were evaluated, and the strongest one was chosen (VAF). R

^{2}is a relative measure of the proportion of the dependent variable’s variance that the model explains (R

^{2}can range from 0 to 1); RMSE is an absolute measure of the average distance that data points fall from the regression line (RMSE is in the units of the dependent variable). VAF is frequently employed to validate the accuracy of a model by comparing the actual output to the estimated output (VAF can range from 0% to 100%). Therefore, a regression model was considered the strongest one if (R

^{2}) was close to 1, (VAF) was close to 100%, and (RMSE) was close to zero [36,37,38]. Equation of each statistical parameter is shown below:

_{m}is the measured value, y

_{p}is the predicted value, y

_{a}is the average of the measured value, and n is the total number of data.

_{d}, Gs, V

_{p}, H

_{r}, and n%, with strong relationships, and the graph for each model is shown in Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, respectively. There is a direct relationship between conductivity and diffusivity (i.e., an inverse relationship of the specific heat) with density, hardness, and pulse velocity. An inverse relationship between the conductivity and diffusivity (i.e., a direct relationship of the specific heat) was observed when the thermal properties were linked with the rock’s porosity.

#### 3.4. Multivariate Regression Analysis

^{2}and VAF and lowest RMSE [42]. Additionally, the rock’s thermal properties are a function of rocks’ physical, textural, and mineralogical composition (i.e., they do not depend only on a particular rock index). Finally, to learn more about the relationships between the thermal properties and other indices of rock, a multivariate regression analysis was applied in this study as the second stage of regression analysis by considering more than one input parameter.

#### 3.5. Artificial Neural Network Analysis

_{𝑖}: Node weight.

_{𝑖}: Sigmoid value for the weighted sum of the inputs to the hidden layer.

## 4. Conclusions

- There is a difference in the thermal properties of the included limestone, especially in the specific heat. This is due to the difference in physical and engineering properties, the variance in the abundance of minerals in it, and the variance in porosity.
- The results of an independent samples t-test analysis indicate a statistically significant difference in the thermal, physical, and engineering properties between rocks, where the calculated p-value for each property is close to zero (i.e., less than 0.05).
- The thermal conductivity increases with the increasing thermal diffusivity and decreases with the increasing heat capacity. Increasing conductivity means that further heat will be extracted from the source, which means the diffusivity of absorbed heat become faster. On the other hand, a small conductivity value means that the material mostly absorbs heat, and a small amount of the heat will be transmitted through the medium.
- The low thermal conductivity values for studied rocks compared to some other building stones mean that it can be considered as the best and perfect thermal conductor.
- There is a direct relationship between the conductivity and diffusivity (i.e., an inverse relationship to the specific heat) and the density, hardness, pulse velocity, and rock strength. When the thermal properties are linked with the porosity of the rock, an inverse relationship between the conductivity and diffusivity (i.e., a direct relationship to the specific heat) was observed with the previous properties.
- Indirectly, the thermal properties of the lime building stone can be predicted through a set of mathematical models that relate the thermal properties with the physical and engineering properties. These mathematical models were developed by using regression analysis.
- According to multivariate regression analysis, a strong correlation between thermal properties and rock’s compressive strength was observed, where the UCS was termed in all multivariate regression models; therefore, it can be considered the most important factor affecting the thermal properties of the rock.
- ANN models provide more accurate and better results in terms of R
^{2}, RMSE, and applicability on a holdout sample (validation data set), especially for specific heat prediction as opposed to multivariate regression.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**After preparation and marking, one hundred cylindrical core specimens (Height = 20 cm, Diameter = 10 cm).

**Figure 5.**Thermal conductivity versus thermal diffusivity and specific heat for limestone rocks in (

**a**) source 1 and (

**b**) source 2 ((k) in a unit of (W/(m × K)), (α) in a unit of (mm

^{2}/s), and (C) in a unit of (MJ)/(m

^{3}× K)).

**Figure 6.**Thermal properties vs. dry density in (

**a**) source 1 and (

**b**) source 2((k) in a unit of (W/(m × K)), (α) in a unit of (mm

^{2}/s), and (c) in a unit of (MJ)/(m

^{3}× K)).

**Figure 7.**Thermal properties vs. Specific gravity in (

**a**) source 1 and (

**b**) source 2 ((k) in a unit of (W/(m × K)), (α) in a unit of (mm

^{2}/s), and (c) in a unit of (MJ)/(m

^{3}× K)).

**Figure 8.**Thermal properties vs. pulse velocity in (

**a**) source 1 and (

**b**) source 2 ((k) in a unit of (W/(m × K)), (α) in a unit of (mm

^{2}/s), and (c) in a unit of (MJ)/(m

^{3}× K)).

**Figure 9.**Thermal properties vs. hardness in (

**a**) source 1 and (

**b**) source 2 ((k) in a unit of (W/(m × K)), (α) in a unit of (mm

^{2}/s), and (c) in a unit of (MJ)/(m

^{3}× K)).

**Figure 10.**Thermal properties vs. Porosity in (

**a**) source 1 and (

**b**) source 2 ((k) in a unit of (W/(m × K)), (α) in a unit of (mm

^{2}/s), and (c) in a unit of ((MJ)/(m

^{3}× K)).

**Figure 12.**Thermal properties vs. compressive strength in (

**a**) source 1 and (

**b**) source 2 ((k) in a unit of (W/(m × K)), (α) in a unit of (mm

^{2}/s), and (c) in a unit of (MJ)/(m

^{3}× K)).

**Figure 13.**Thermal properties vs. point load strength in (

**a**) source 1 and (

**b**) source 2 ((k) in a unit of (W/(m × K)), (α) in a unit of (mm

^{2}/s), and (c) in a unit of (MJ)/(m

^{3}× K)).

**Figure 14.**Some of the broken specimens for both included limestone ((

**a**) source 1, and (

**b**) source 2).

Property | Min. | Max. | Avg. | Standard Deviation | Coefficient of Variation |
---|---|---|---|---|---|

k (W/(m × k)) | 1.931 | 3.468 | 2.861 | 0.327 | 11.45 |

α (mm^{2}/s) | 1.032 | 1.810 | 1.323 | 0.123 | 9.31 |

c ((MJ)/(m^{3} × K)) | 1.570 | 2.563 | 2.157 | 0.192 | 8.91 |

Property | Unit | No. of Specimens | Min. | Max. | Avg. | Standard Deviation | Coefficient of Variation |
---|---|---|---|---|---|---|---|

ρ_{d} | g/cm^{3} | 100 | 2.521 | 2.692 | 2.521 | 0.113 | 4.492 |

Gs | - | 100 | 2.677 | 2.714 | 2.677 | 0.031 | 1.16 |

Hr | - | 100 | 34.423 | 43.65 | 34.422 | 4.912 | 14.269 |

UCS | MPa | 100 | 81.341 | 138.39 | 81.341 | 20.448 | 25.139 |

Is_{(50)} | MPa | 100 | 3.357 | 4.825 | 3.357 | 0.579 | 17.241 |

n% | - | 100 | 5.845 | 15.924 | 5.846 | 3.265 | 55.87 |

V_{p} | m/s | 100 | 5476.3 | 6366.3 | 5476.32 | 618.228 | 11.289 |

Source 1 | Source 2 | |||||
---|---|---|---|---|---|---|

DV | Statistic | Df * | Sig. | Statistic | Df * | Sig. |

k | 0.082 | 50 | 0.200 | 0.097 | 50 | 0.200 |

α | 0.118 | 50 | 0.082 | 0.111 | 50 | 0.172 |

c | 0.102 | 50 | 0.200 | 0.093 | 50 | 0.200 |

Independent Variable | Unit | Equation (Source 1) | R^{2} | VAF% | RMSE | Equation (Source 2) | R^{2} | VAF% | RMSE |
---|---|---|---|---|---|---|---|---|---|

ρ d | g/cm^{3} | k = 3.3716 ρ_{d} − 5.7177c = 207.92 ρ _{d}^{−4.719}α = 3.4696 ρ _{d} − 7.6889 | 0.66 0.64 0.64 | 66 62 64 | 0.07 0.09 0.07 | k = 3.1391 ρ_{d} − 4.9995c = −1.9003 ρ _{d} + 6.6864α = 0.9539 ρ _{d} − 1.0435 | 0.89 0.90 0.92 | 89 90 92 | 0.11 0.05 0.02 |

Gs | - | k = 9.0146 (Gs) − 21.314 c= −10.26(Gs) + 30.02 α = 8.8217(Gs) − 22.507 | 0.1 0.1 0.09 | 3.56 14.66 5 | 1.3 1.8 1.7 | k = 14.286 (Gs) − 35.191c = −9.9973( Gs) + 28.536α = 4.819 (Gs) − 11.483 | 0.38 0.51 0.48 | 4.51 14.53 3.67 | 19.94 0.87 0.81 |

Vp | m/s | k = 1.1302e^{0.0002 Vp}c = −0.0006 Vp + 5.791 α = 0.149e ^{0.0004 Vp} | 0.71 0.66 0.65 | 60 66 63 | 0.64 0.09 0.26 | k = 2.7834ln(Vp) - 21.055 c = −0.0003 Vp + 3.7657 α = 0.0002 Vp + 0.4315 | 0.82 0.87 0.87 | 82 86 85 | 0.13 0.21 0.15 |

Hr | - | k = 0.066 Hr + 0.518 c = −0.0749 Hr + 5.151 α = 0.0672 Hr − 1.2321 | 0.88 0.82 0.83 | 88 82 83 | 0.04 0.06 0.05 | k = 3.2977ln(Hr) − 8.577 c = −0.0659 Hr + 4.0479 α = 0.033 Hr + 0.2842 | 0.96 0.95 0.96 | 96 95 96 | 0.06 0.04 0.02 |

n% | - | k = −0.362ln(n) + 3.5343 c = 0.4067ln(n) + 1.7455 α = −0.361ln(n) + 1.8182 | 0.63 0.58 0.58 | 63 58 58 | 0.07 0.09 0.08 | k = −0.0925 n + 3.3856 c = 0.3897ln(n) + 1.2849 α = −0.0278 n + 1.5022 | 0.90 0.90 0.90 | 90 90 91 | 0.10 0.06 0.02 |

UCS | MPa | k = 0.0075 UCS + 2.3565 c = −0.0085 UCS + 3.0691 α = 0.0075 UCS + 0.6455 | 0.87 0.80 0.79 | 87 80 79 | 0.04 0.06 0.06 | k = 1.7287ln(UCS) − 4.5797 c = −0.0159 UCS + 3.1098 α = 0.0079 UCS + 0.7577 | 0.87 0.84 0.83 | 88 84 83 | 0.11 0.07 0.04 |

Is_{(50)} | MPa | k = 0.371 Is_{(50)} + 1.6507c = −0.4326 Is _{(50)} + 3.9225α = 0.3852 Is _{(50)} − 0.1196 | 0.82 0.82 0.81 | 82 82 81 | 0.05 0.06 0.05 | k = 0.6825 e^{0.4718 Is(50)}c = −0.6648 Is _{(50)} + 3.96α = 0.6056 Is _{(50)} ^{0.7124} | 0.85 0.72 0.72 | 85 72 71 | 0.12 0.09 0.05 |

Model | R^{2} | VAF% | RMSE |
---|---|---|---|

k = 2.932 − 0.073(n%) + 0.004 (UCS) | 0.94 | 88 | 0.04 |

α = 6.161 + 0.005 (UCS) − 4.56(G_{s})+2.620(ρ_{d}) + 0.056 (n%) | 0.80 | 84 | 0.05 |

c = −13.682+7.60(G_{s})+1.932(ρ_{d}) − 0.011(UCS) + 0.386 (Is_{(50)}) | 0.69 | 84 | 0.06 |

DV | Structure | Training Results | Validation Results | ||||
---|---|---|---|---|---|---|---|

R^{2} | MAE | RMSE | R^{2} | MAE | RMSE | ||

k | 2-2-1 | 96.07 | 0.050 | 0.065 | 98.03 | 0.032 | 0.041 |

α | 7-4-1 | 86.7 | 0.036 | 0.048 | 75.70 | 0.023 | 0.039 |

c | 7-4-2-1 | 92.16 | 0.043 | 0.043 | 92.08 | 0.049 | 0.056 |

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**MDPI and ACS Style**

Sharo, A.A.; Rabab'ah, S.R.; Taamneh, M.O.; Aldeeky, H.; Al Akhrass, H.
Mathematical Modelling for Predicting Thermal Properties of Selected Limestone. *Buildings* **2022**, *12*, 2063.
https://doi.org/10.3390/buildings12122063

**AMA Style**

Sharo AA, Rabab'ah SR, Taamneh MO, Aldeeky H, Al Akhrass H.
Mathematical Modelling for Predicting Thermal Properties of Selected Limestone. *Buildings*. 2022; 12(12):2063.
https://doi.org/10.3390/buildings12122063

**Chicago/Turabian Style**

Sharo, Abdulla A., Samer R. Rabab'ah, Mohammad O. Taamneh, Hussein Aldeeky, and Haneen Al Akhrass.
2022. "Mathematical Modelling for Predicting Thermal Properties of Selected Limestone" *Buildings* 12, no. 12: 2063.
https://doi.org/10.3390/buildings12122063