Mathematical Modelling for Predicting Thermal Properties of Selected Limestone

: Due to a lack of geotechnical and geothermal studies on Jordanian limestone, this paper aims to provide the thermal properties, including thermal conductivity, thermal diffusivity, and speciﬁc heat, using the Hot Disk Transient Plane Source (TPS) 2200 method. It also aims to provide a set of mathematical models through which the thermal properties can be indirectly predicted from the rocks’ physical and engineering properties. One hundred cylindrical rock specimens with a height of 20 cm and a diameter of 10 cm were extracted and prepared. The results showed that the thermal conductivity values ranged between (1.931–3.468) (W/(m × k)), thermal diffusivity (1.032–1.81) (mm 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.


Introduction
The knowledge of the thermal properties of building stones is important for building envelope thermal insulation, especially in very hot, very cold, and undeveloped countries, for energy saving [1]. Rock science's most prominent thermal properties are thermal conductivity, diffusivity, and specific heat [2]. Thermal conductivity is the ability of a material to conduct heat. It is defined as: "the rate of heat transfer through a unit thickness of the material per unit area per unit temperature difference" [3]. Thermal conductivity, denoted by (k), is the ability to transfer heat per unit of time and the unit surface area divided by the temperature difference [4]. Heat capacity is the ratio of heat transferred to an object and its temperature rise. In science and engineering, expressing heat capacity in terms of a unit mass is typically more practical. Therefore, the specific heat denoted by (c) is defined as the heat capacity per unit mass of material [4]. Thermal diffusivity, denoted by (α), is the thermal conductivity divided by density and specific heat; it represents how fast heat diffuses through a material [3]. Mathematically, thermal properties can be defined by the following equations: experiments, taking into account different joint arrangements and confining pressures. Bui et al. [20] predicted building energy consumption using the electromagnetism-based firefly algorithm-artificial neural network (EFA-ANN). Two datasets calculate heating and cooling loads. Façade energy consumption and building dimensions were collected from each dataset. Comparisons with other methods verified model performance. The results showed that EFA-ANN helped civil engineers and construction managers design energyefficient structures. Singh et al. [21] used the ANN and adaptive neuro-fuzzy inference system (ANFIS) to investigate the thermal conductivity and other fundamental properties of rocks. Six different ANN and ANFIS models are compared to assess their capabilities. P-wave velocity, porosity, bulk density, and the rock's uniaxial compressive strength were the input parameters that were used to predict thermal conductivity. Limestone is used in many engineering applications; the most important is its use as a construction material in buildings' facades and columns ( Figure 1). It possesses many desired properties, such as hardness, thermal insulation, aesthetic view, and indoor humidity prevention. Unfortunately, few studies have evaluated the physical engineering properties of lime building stone [22], especially studies related to heat flow and thermal properties of sedimentary rock [5]. This research paper aims to provide the thermal properties of lime building stone through the TPS 2200 method and explain how these properties are affected by the physical and engineering properties of the rocks. Despite the advantages of the Hot Disk Transient Plane Source (TPS) 2200 method, it is expensive. For this reason, the second goal of this research is to create a set of statistical models that enable the site engineer to indirectly predict the thermal properties of the rock from easy, simple, inexpensive, and non-destructive tests. To achieve the second goal, regression analysis, a well-known statistical analytical technique, and artificial neural networks (ANN) were implemented on a set of data that included the thermal, physical, and engineering properties of lime building stone to develop a set of statistical models to predict thermal properties from the physical (such as hardness, density, the velocity of sound waves, and porosity) and engineering (such as uniaxial compressive strength and point load strength) properties of the rock.

Rock Specimens Preparation
Limestone is found in huge quantities and at various depths in Jordan. This rock underlies 90% of Jordan's land area. According to the Natural Resources Authority [23], which is in charge of locating Jordan's raw minerals and natural resources, limestone stone is primarily sourced from Ajloun, the north-western part of Jordan (source 1), and Ruweished, the eastern part of Jordan (source 2). Due to its shallow depth in Ruweished, where it becomes superficial after removing the half-meter-thick earthen layer, and because it is exposed at the soil surface in Ajloun, lime building stone is extracted from these Buildings 2022, 12, 2063 4 of 17 two sites. In contrast, the existence of limestone at great depths in southern Jordan led to a major decline in its exploitation activity [23]. To obtain a representative specimen, several stone blocks were obtained from both areas, as shown in Figure 2. A total of 100 block specimens (50 specimens from each location) were collected to perform this study with a minimum size of approximately 50 cm. Cylindrical rock cores with a diameter of 10.0 cm and a length-to-diameter ratio of 2:1 were prepared from each block, according to ASTM D4543 standard [24], as shown in Figure 3.

Method and Laboratory Tests
After preparing the cylindrical rock core specimens, each specimen was dried in an oven at 105 • C for 24 h to extract the water absorbed by the rock specimens during the collection and preparation stage, which included polishing, coring, and diamond bit sawing. After that, a series of tests were conducted to achieve the objectives of this study. The tests conducted in this study were divided into two parts, the first is the tests related to the thermal properties of rock specimens, and the second is the tests related to rock specimens' physical and engineering properties. All the tests were performed on oven-dried specimens at room temperature and pressure.

Thermal Properties Determination
The TPS method was used in this investigation to determine the thermal characteristics of the limestone rock specimens. The hot disk transient technique is a non-destructive, high-accuracy method with easy-to-prepare specimens [13,25]. The main advantages of the hot disk technique include a wide range of measurable thermal conductivity, from 0.005 (W/(m × k)) to 500 (W/(m × k)), over a wide temperature range from −50 • C to 750 • C. The experiment setup is schematically shown in Figure 4. Nickel (Ni) is used for the TPS sensor because of its high thermal resistivity. The double-spiral-shaped sensor has four electrical connections: two for heat current and one for detecting. The sensor works as a heat source to raise the specimen temperature and a thermometer to measure the change. The thermal conductivity can be computed from the reported temperature rise [26]. The sensor is believed to be in infinity. Therefore, the measurement period must be managed so that thermal penetration depth does not exceed specimen thickness. There are several types of sensors that may be used in this method, the Kapton sensor (4922) is used in this study with 29.2 (mm) in diameter, 1000 (mW) injected power, 160 (s) measurement time, and (18-21) (mm) penetration depth [27].
The TPS sensor was positioned on a rock surface (a smooth surface without outcrops or cracks). The sensor was then insulated to avoid heat leakage. Figure 4 shows that the sensor was loaded with 2.56 kg to guarantee proper sensor-to-sample contact. After that, the Hot Desk Thermal Constant Analyzer software calculated the numerical results, which included thermal conductivity, thermal diffusivity, and specific heat.

Physical and Engineering Properties Determination
After drying the specimens in an oven at 105 • C for 24 h, dry density (ρd) was measured in (g/cm 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].

Thermal Properties of Limestone Rock
The statistical parameters for thermal conductivity (k), thermal diffusivity (α), and specific heat (c) for both rocks are shown in Tables 1 and 2. An independent specimens t-test analysis was used to determine whether there is a statistically significant difference between the means of both rocks' thermal, physical, and engineering properties. The results of the t-test indicate a statistically significant difference in some thermal, physical, and engineering properties between both sources, especially in porosity, and specific heat, where the calculated p-value for each property is close to zero (i.e., less than 0.05) and the higher porosity of the limestone (source 2) specimens had an important impact on obtaining lower thermal properties.
From the data found for rocks, it can be concluded that the thermal conductivity increases with increasing thermal diffusivity and decreases with increasing heat capacity ( Figure 5). This can be explained by the fact that the material with high thermal conductivity will have a large thermal diffusivity and a low heat capacity. The larger the thermal diffusivity, the faster the heat transmission through the medium, which is consistent with thermal conductivity. On the other hand, a small thermal conductivity value (i.e., a small value of thermal diffusivity) means that the material mostly absorbs heat, and a small amount of the heat will be transmitted through the medium [3,33]. Additionally, increasing thermal conductivity means that further heat will be extracted from the source, which also means the diffusivity of absorbed heat becomes quicker [5].
Cermak and Rybach [34] reported that the range of thermal conductivity for rock is (0.4-7) (W/)m × k)), and Çanakci et al. [9] also reported that the range of thermal conductivity for building stones is (1.163-8.6) (W/)m × k)). Thermal conductivity values found in this study are within these two ranges. Additionally, the thermal conductivity values found in this study occupy lower ranges than those specified by [9,34]. This can be attributed to the presence of quartz in very low percentages (less than 1%) in both types of rocks extracted from both regions, which is considered a good thermal conductor, where the range of thermal conductivity for quartz is (6.5-11.3) (W/)m × k)) [5]. Pechnig et al. [35] concluded that the thermal conductivity increased with quartz/feldspar ratios. The thermal conductivity of Kristallsandstein rock (90% quartz and 6% feldspar) is higher than the thermal conductivity of Schilfsandstein rock (57% quartz and 35% feldspar). . 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)).

Regression Analysis for Predicting Thermal Properties
In this study, thermal properties of rock, which require costly technology to estimate, were considered as dependent variables, and the physical properties, including ρ 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.
Statistical analysis methods, including hypothesis testing, analysis of variance (ANOVA), and regression analysis, require that the dependent variable is normally distributed. In this study, the Kolmogorov-Smirnov test was used by the SPSS program to check whether the dependent variables are normally distributed with a significant level (0.05). Table 3 shows the normality test results, where the calculated p-value for dependent variables in both rocks is more than (0.05), which means accepting the Null hypothesis and rejecting the alternative hypothesis. Therefore, the dependent variables are normally distributed, and parametric statistical tests can be used.

Simple Regression Analysis
In simple regression analysis, only one independent variable can predict the dependent variable. Four curve-fitting relationships were used: exponential, power, logarithmic, and linear. The mathematical formula for each model is shown in Table 4, where y p is the predicted value, x is the input parameter, c is the regression constant, and b is the regression coefficient. Based on several statistical indices, such as coefficient of determination (R 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: where y 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. Conductivity, diffusivity, and specific heat are linked with physical properties, including ρ d , Gs, V p , H r , and n%, with strong relationships, and the graph for each model is shown in Figures 6, 7, 8, 9, 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. Conductivity, diffusivity, and specific heat are linked with physical properties, including d, Gs, Vp, Hr, and n%, with strong relationships, and the graph for each model is shown in Figures 6-10, 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.   (a) (b) 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)).
Additionally, the relationship between thermal conductivity and porosity has been compared between the results of this study and with what some researchers have found, including Özkahraman et al. [17] on limestone and travertine and Çanakci et al. [9] on clayey and chalky limestone, Pichugin et al. [39] on different types of sedimentary rocks, and Zeb et al. [40] on porous limestone as shown in Figure 11. In general, it is clear from Figure 11 that there is an inverse relationship between the thermal conductivity and the porosity, but perhaps the reason for the incompatibility between the types of curves is that the relationships are more general and applied to more than one type of rock. Figure 11. Comparing the relationship between thermal conductivity and porosity based on the results of this study with some previous studies [9,17,35,40].
In the literature, the obtained mathematical models which linked the thermal properties with rock's indices were not specialized to one type of rock, such as limestone, for example, they were more generally for several types of rocks. Chekhonin et al. [41] reported that the trends significantly changed from one type of rock to another, so empirical equations must be unique and not generalized. Hence, the need to create a set of mathematical models specialized in finding the thermal properties of the physical and engineering properties of limestone in Jordan.
In terms of stresses, conductivity, diffusivity, and specific heat are linked with strong relationships with uniaxial compressive strength ( Figure 12) and point load strength index ( Figure 13). There is a direct relationship between the conductivity and diffusivity (i.e., an inverse relationship of the specific heat) with uniaxial compressive strength and point load strength index.  What can be noted is that the models are stronger for thermal conductivity prediction followed by thermal diffusivity, and are the weakest to the point that they almost do not describe the variance in the specific heat data. This is due to the huge difference in the specific heat properties between the two included rocks, making it very hard to predict this parameter for both. Additionally, the nature of limestone source 1 contains some bedding planes, as shown in Figure 14, which are located at different locations on the specimen; these bedding planes lead to the data being distributed more randomly in source 1.

Multivariate Regression Analysis
Multivariate regression analysis can be used to achieve stronger models for predicting the thermal properties of rocks, which means higher R 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.
Multivariate regression analysis requires a linear relationship between dependent and independent variables with a correlation coefficient larger than 0.3 [43]. Additionally, multicollinearity is a known problem in multivariate regression, where two or more independent variables correlate highly. Multicollinearity leads to a non-ideal model. The evaluation of the possibility of the existence of multicollinearity was based on the variation influence factor (VIF). A (VIF) value must be less than 3, 5, or 10 to verify the absence of multicollinearity [44]. By using SPSS, the aforementioned conditions were checked. The correlation coefficient values for linear relationships between the dependent and each category of independent variables are larger than 0.3. Additionally, (VIF) values were checked for each model, and (VIF) values were smaller than 10. Therefore, the multivariate regression analysis can be applied. SPSS program was used to build a multivariate regression model by stepwise method, which excludes the predictors added to the model without making it stronger. Table 5 shows the results of the multivariate regression. From Table 5, the strong correlation between thermal properties and the rock's compressive strength is observed, where the rock's UCS was termed in all models; therefore, the UCS parameter can be considered the most important factor affecting the thermal properties of the rock.

Artificial Neural Network Analysis
In this study, the multilayer perceptron, a kind of ANN, was implemented using the Waikato Environment for Knowledge Analysis (WEKA) software. MLP is a feed-forward network that uses a supervised learning algorithm without any brief assumption about the distribution and pattern in the data. It consists of at least three layers: input, hidden, and output. Each layer consists of several neurons or nodes fully connected to the next layer by weights (Gardner and Dorling, 1998). A weighted sum of the normalized parameter is the input to each node in the hidden layer that modified this sum by a certain activation function and transferred it to the output layer. That output layer signal is a function of the weighted sum signals from the previously hidden layer signals that can be calculated using the following equation.
Y : The predicted normalized value. n: Node's number in the hidden layer. ω i : Node weight. (α) i : Sigmoid value for the weighted sum of the inputs to the hidden layer. ϑ: The output bias. MLP learns through the training data and assigns the weight for each connection in the model. The predicted value is calculated using the previous equation, and the weight is then adjusted based on the error term between the actual and predicted value and using one of the backpropagation training algorithms to find the weights with the minimum error term [45]. In this study, the sigmoid activation function was used in the hidden layers to handle the nonlinearity in the data and its easy computations [46] compared with the linear activation function for the output layer.
The data were split into two subsets, training and validation, to test the model's performance based on unseen data. The number of nodes was determined using the process suggested by (Shahin et al., 2002). The process is based on starting with a small number of hidden layers and increasing it until no further improvement in the model is represented by the mean absolute error (MAE)) term and correlation coefficient. In addition, a stopping criterion was used to control the training time (epoch number) and avoid the overfitting problem. The number of epochs stops increasing when an adjacent increase in the error term within the training set occurs. Therefore, the final model and the final weights are the ones before the last increase in the error term. A review of the final results of all three models of predicting the rock's thermal properties was included in Table 6 for both training and validation sets. Two hidden layers were used to predict the specific heat values and handle the variation in the data instead of only one hidden layer for the thermal conductivity and diffusivity. Figure 15 represents the goodness of fit between the actual and predicted values of the thermal properties.

Conclusions
The thermal properties covered in this study are thermal conductivity, thermal diffusivity, and specific heat, which are considered the most superior thermal properties in rock science and measured by using the Hot Disk Transient Plane Source (TPS) 2200 as a method for rapid and accurate thermal properties measurement. Based on the results of this study, the following points can be concluded: 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.

2.
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).

3.
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.

4.
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.

5.
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. 6.
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. 7.
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. 8.
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. Funding: This research did not receive any specific grant from funding agencies.

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: All data of this research were presented in the article.

Conflicts of Interest:
The authors declare no conflict of interest.