Texture-Dependent Thermal Properties of Sandstone Rocks Examined by Scanning Electron Microscopy for Thermal Energy Storage Applications

As decarbonization efforts continue worldwide, prominent players in the energy industry, including major oil and gas companies, are shifting toward greater use of non-fossil fuel sources for energy generation. However, a major obstacle in the adoption of renewable energy is their intermittent nature [1]. To address this issue, energy storage systems are being employed to manage the seasonal variations in renewable energy generation. Thermal energy storage (TES) systems offer a solution to optimize the management of renewable energy by balancing energy generation and demand. For large-scale renewable energy projects, TES systems must have the capacity to store a significant amount of heat.

Since the 1970s, the concept of using sedimentary reservoirs to store heat has been explored as a way to mitigate the impact of seasonal variation on solar energy's capacity factor. Several sedimentary reservoir thermal energy storage (SRTES) projects have been developed since the early 1980s, demonstrating the feasibility of the concept, but further research is needed to move toward high-temperature, large-scale applications [2–6].

Subsurface rocks are ideal for thermal energy storage, as they can store energy generated by solar and wind plants. Sandstone rocks are believed to be suitable reservoirs for SRTES due to their thermal properties and abundance, and several authors have suggested their use in this application [7–11].

Sandstones offer multiple advantages as reservoir rocks for geothermal energy and underground thermal energy storage applications. Sandstones have high porosity which allows hot water or steam to flow through them easily [12]. Sandstones are permeable, meaning fluids can pass through the pore spaces between the grains [13]. The high permeability of sandstones allows water to be injected or withdrawn through wells drilled into the formation [14]. Sandstones form in large sedimentary basins which can store significant amounts of thermal energy [15–18]. The mineral composition of sandstones can withstand repeated heating and cooling cycles without degrading [19]. Sandstones often have an impermeable cap rock above them that traps geothermal fluids and prevents heat loss [20].

The production and injection cycles in thermal energy storage require analyzing the evolution of temperature and heat flux in the reservoir thermal battery. Temperature and heat are associated with rocks' thermal properties and the heat transfer fluid stored in the poral space. Volumetric heat capacity, thermal conductivity, and thermal diffusivity are critical thermal properties that impact the performance of thermal energy storage and geothermal reservoirs. These thermophysical properties describe how sedimentary rocks conduct and store heat.

The heat transfer between the subsurface and the surface is a critical component in the potential of a geothermal field or an SRTES reservoir. The amount of heat that can be extracted or stored is dependent on the properties of the rocks that make up the subsurface. The thermal properties of rocks control how efficiently heat can be transferred from the subsurface to the surface and vice versa [21]. The thermal properties analyzed in this study are thermal conductivity, thermal diffusivity, and volumetric heat capacity.

Thermal conductivity quantifies the rock's ability to conduct heat. It is expressed as the amount of heat that flows through a unit area of a material in a unit of time when a temperature difference exists across the rock. The thermal conductivity of sandstone rocks can range from 0.5 W/m · K to 6 W/m · K depending on factors like porosity, cementation, grain size/shape, and mineralogy [22].

Thermal diffusivity is a measure of how quickly heat can diffuse through a material. It is defined as the ratio of thermal conductivity to specific heat and density. The thermal diffusivity of sandstone rocks typically ranges from 0.5 mm²/s to 1.5 mm²/s [23]. For SRTES, the ability to store and release heat efficiently is crucial. A high thermal diffusivity enables faster heat transfer, allowing the sedimentary rock to absorb heat more quickly during the charging phase and release it more rapidly during the discharging phase. This characteristic is particularly important for applications where there are time constraints, such as in district heating or cooling systems.

Volumetric heat capacity is an important thermal property of rocks that influences their ability to store and transfer heat. It is defined as the measure of the amount of heat energy required to raise the temperature of a cubic meter of rock by 1 deg K. The units of volumetric heat capacity are typically joules per cubic meter per degree Kelvin (J/m³ · K). The volumetric heat capacity of sedimentary rocks is affected by mineral composition, porosity, and water content. In general, rocks with higher quartz content have lower volumetric heat capacities around 2.1–2.4 MJ/m³ · K, while those containing more feldspars and carbonates have higher values between 2.4 MJ/m³ · K and 2.9 MJ/m³ · K [22]. Porosity also affects the volumetric heat capacity, with higher porosity leading to lower values due to the lower heat capacity of pore fluids [24]. The volumetric heat capacity decreases approximately linearly as porosity increases from 0% to 30% [25]. The volumetric heat capacity is the product of specific heat and density. Specific heat is the amount of heat required to raise the temperature of a unit mass of a material by 1 °C. It is a measure of a material's ability to store heat. The specific heat capacity of sedimentary rocks is affected by the mineral composition of the rock. Different minerals have different specific heat capacities. Sedimentary rocks are typically made up of a combination of minerals, so the specific heat capacity of a given rock will depend on the relative proportions of these minerals. Rocks with higher amounts of quartz and feldspar will generally have higher specific heat capacities than those containing more calcite or dolomite [26]. The specific heat of sandstone rocks typically ranges from 800 J/kg · K to 2000 J/kg · K [23].

The surface roughness of sandstones is affected by several factors, including the size distribution of sand grains, their shape, and the amount of pore space in the rock. Well-sorted, fine-grained sandstones tend to have smoother surfaces compared to coarser-grained or poorly-sorted types [27]. This is because rounded, spherical sand grains pack together more smoothly than angular ones, which create rougher textures [28]. Another factor to consider is the pore space. Surface roughness increases with the amount of pore space in the rock. Higher porosity from interconnected pores and voids creates more surface irregularities [29]. The mineral composition also plays a role in surface roughness, with quartz-rich sandstones being more resistant to weathering processes that roughen surfaces over time. The surface roughness of sandstones, which is primarily a reflection of the size, shape, and arrangement of the individual grains within the rock, can also be influenced by the quartz content. Quartz grains tend to be rounded and typically exhibit a smooth surface texture. Therefore, sandstones that are composed almost entirely of quartz (i.e., quartz arenites) often have a relatively smooth surface compared to sandstones that contain a significant proportion of other minerals, such as feldspar or clay minerals, which can contribute to a rougher surface texture [30].

Scanning electron microscopy (SEM) can be used to analyze the thermal properties of rock samples by examining the mineralogical and textural features of the rock at high magnification. SEM can provide detailed images of the rock surface, allowing for the identification and measurement of mineral grain size and shape, porosity, and connectivity. Therefore, using SEM analysis to evaluate the thermal properties of rocks can provide more accurate and reliable results than relying on visual inspection with the naked eye. SEM images can provide insights into the microstructure and composition of the rocks, which can indirectly influence their thermal properties [31].

SEM enables the identification of different minerals present in sedimentary rocks. Since each mineral has specific thermal properties, differentiating those minerals can provide more insight into the thermal properties of the rocks examined by the SEM [32]. The thermal properties of rocks are influenced by the contact between individual grains. The SEM analysis can reveal the nature and quality of grain-to-grain contacts, including the presence of cementing materials or pore-filling minerals [33]. The effectiveness of heat transfer within the rock matrix is affected by these contacts.

Sedimentary rocks contain pore spaces that can significantly impact their thermal properties [34]. SEM analysis allows for the characterization of pore structure, including pore size distribution, connectivity, and shape [35]. The presence of open or closed pores, as well as the type of pore-filling materials, can influence the rock's thermal conductivity, heat capacity, and thermal diffusivity [36]. A number of researchers have used SEM as complementary tool, to analyze the thermal properties of sedimentary rocks [37,38]. SEM can indirectly contribute to understanding the thermal properties of sedimentary rocks by providing valuable information about their microstructure, mineral composition, pore characteristics, and thermal alteration effects.

The objective of this study is to establish a relationship between the thermal properties of 30 identified core samples and the corresponding characteristics observed in the SEM images. The core samples analyzed exhibit a range of thermal properties. Advanced image processing and analysis methods were utilized to identify and measure micro-structural attributes like grain size, pore size, and connectivity. The objective is to compare the identified thermal properties of the core samples to the observed image characteristics in order to identify connections between the core microstructures to the thermal properties of the samples.

The methodology involves the analysis of 30 core samples with diverse known thermal properties, preparing them for SEM analysis, and capturing high-resolution SEM images. Image processing and analysis techniques are employed to identify and quantify micro-structural features such as grain size, pore size, and connectivity. The known thermal properties of the core samples are then compared with the observed image characteristics to determine any significant correlations.

The rock samples of different sandstones were obtained from Berea sandstone, Boise sandstone, and Kentucky sandstone formations. Figure 1 depicts the workflow and the equipment used for measuring the petrophysical properties, thermal properties, and mineralogy of the samples.

The thermal properties of the 30 sandstone samples were obtained experimentally (Table 1). The thermal properties were measured using the light flash apparatus (LFA) Netzsch LFA 467. The equipment uses a short pulse of energy light to heat up the front surface of a flat sample, which is parallel to a plane. An infrared detector measures the temperature change of the back surface caused by the heat pulse. This measurement allows for the determination of thermal diffusivity and specific heat. When combined with the density of the sample, these thermophysical properties can be used to calculate thermal conductivity.

The one-inch core samples were cut to expose the surfaces that are going to be analyzed by the SEM. This is done to obtain more insight into sample texture. Once the samples were cut to the appropriate size, they were coated with a conductive metal layer to improve conductivity. A carbon coating was applied, to prevent the buildup of electrical charges during the SEM analysis. Before performing every experiment, the samples were stored in an oven at 65 °C for 24 h. The objective of this was to have the samples completely dry, prevent moisture from affecting the measurements, and have comparable results.

To quantitatively assess the morphological features of the rocks, the samples were prepared and examined using an FEI Quanta 200 scanning electron microscope. A total of 150 SEM images were captured and the magnification considered was 300 times. Micrographs with magnifications of 500, 1000, and 2000, were evaluated; however, the mentioned sizes were discarded since they did not allow to capture textural details that are lost for the higher resolution.

The SEM instrument was then used to scan the surface of the cut section, producing high-resolution images of the rock's texture and structure. To extract the information required to understand how textural features affect the thermal properties of the rocks, the images were analyzed with the Fiji version of the software imagej. imagej is a widely used open-source image processing software that provides a range of tools and plugins for analyzing and quantifying various properties from the SEM images. imagej has been used to analyze the SEM rock images for analyzing morphological features, particle size evaluation, or porosity [35,39–42]. In this study, the surface roughness and porosity were analyzed.

Once the SEM images were obtained, a pre-processing process helped to enhance the image features extraction process. First, the images were filtered using the “Median” filter in imagej software. This is a type of spatial filter used to reduce noise in digital images. It works by replacing each pixel in the image with the median value of the neighboring pixels within a given radius. For this study, the radius of two pixels was sufficient to enhance the images. After the filter, the enhance contrast option in imagej software was used to automatically adjust the brightness and contrast of SEM images to improve their visibility. Initially, the color scale of the image is normalized, which scales the pixel values in the image so that the minimum value becomes 0 and the maximum value becomes 255. This ensures that the full range of values is used in the image. Then, the pixels of the images were saturated, which refers to the percentage of pixels in the image that should be saturated (set to either 0 or 255). For this study, this was set to 0.5%, which means that the brightest and the darkest 0.5% of pixels in the image were set to 255 and 0, respectively. An example of SEM preparation result is presented in Fig. 2.

SEM images were examined to evaluate the feasibility of using them to assess their thermal properties based on textural features. Figure 3 shows SEM images of three rock samples; they represent samples that have, low, medium, and high values (Figs. 3(a)–3(c)) of thermal properties from the analyzed dataset.

The SEM images showed how different grain sizes, pores and voids, and grains separation affect the rocks' thermal properties. More spaces between grains allow air pockets to form, trapping air that acts as a thermal insulator. The samples with higher porosity, and thus more air pockets, show higher thermal resistance as represented by their higher thermal properties. The software imagej was used to extract the morphological characteristics of the rocks.

Surface roughness analysis involves quantifying and characterizing roughness features of a surface profile using various variables obtained from SEM images. These variables play a crucial role in understanding the topographic characteristics of the surface.

In Fig. 4, it is depicted the analysis performed in all images. The average surface roughness is a measure of the height variations in the surface of a material. It is calculated by averaging the absolute values of the deviations of the surface from its mean plane. The average surface roughness is calculated by first subtracting the mean plane from the surface height data. This produces a residual height map that represents the surface roughness (Fig. 5).

SEM images of rocks contain complex textures and structures that make it difficult to identify and measure the areas of interest manually. By using the threshold function in imagej, the pores and voids in every image were identified and segmented (Fig. 6). The threshold function in imagej was the tool that was used to segment the SEM image into foreground and background based on pixel intensity values. The tool was useful for separating the pores, voids, and cracks of the rock SEM images from the background. The threshold function applies the threshold to the image and generates a binary mask that separates the foreground and background pixels. The mask can then be used for measuring the area of objects in the image that represents the pores, voids, and cracks. Figure 7 shows the histogram of the calculated pore in µm².

Figure 8 presents the relationship between surface roughness (Ra) and thermal properties, specifically thermal diffusivity, thermal conductivity, and volumetric heat capacity for the 150 images. The plot of thermal diffusivity against Ra exposed a trend, indicating that as the Ra values increased, the thermal diffusivity consistently decreased. This suggests that higher surface roughness impedes the efficient conduction of heat, resulting in reduced thermal diffusivity. Similarly, in the plot of thermal conductivity against Ra, as the Ra values increased, indicating greater surface roughness, the thermal conductivity showed a corresponding decrease. When examining the relationship between volumetric heat capacity and Ra, no clear correlation was observed. The plot of volumetric heat capacity against Ra displayed scattered data points, indicating that volumetric heat capacity and Ra are uncorrelated.

The lack of correlation between volumetric heat capacity and surface roughness could be attributed to the fact that volumetric heat capacity is primarily influenced by the mineral composition and density of the sandstone rather than its surface characteristics. Volumetric heat capacity measures the amount of heat energy required to raise the temperature of a unit volume of material, and it depends on intrinsic factors such as mineralogy content. On the other hand, surface roughness pertains to the irregularities or texture of the outer surface of the sandstone and may not directly affect its volumetric heat capacity.

The surface roughness of the rock is influenced by the grain size and distribution. A smoother surface is indicative of smaller grains that are well-sorted, and possessing similar grain shapes. Smaller grains can pack tightly together, while similar grain shapes fit more harmoniously. Consequently, the result is a smoother surface. On the other hand, a rough and irregular surface indicates larger grains that are poorly sorted, with varying grain shapes. The larger grains cannot pack as tightly, and the differing shapes do not align perfectly, leading to a rough and uneven surface.

Additionally, the scale or wavelength of roughness features can provide clues about the grain size. Larger grains produce roughness features on a larger scale, while smaller grains result in smaller-scale roughness. The presence of isotropic or uniform roughness in all directions indicates well-sorted grains with similar sizes and shapes. Conversely, anisotropic roughness, where smoother and rougher areas align in different directions, signifies poorly-sorted grains with variable sizes and shapes. Lastly, the texture of roughness features, ranging from smooth to jagged, can offer insights into the interlocking of grains. Jagged and angular features indicate grains that are not well-interlocked, while smoother undulations suggest grains that are more interfit, demonstrating their interlocking nature. These observations enable a deeper understanding of the geological properties and composition of rocks based on their surface roughness characteristics.

These findings highlight the impact of surface roughness on thermal properties. Higher Ra values lead to decreased thermal diffusivity and thermal conductivity, indicating a hindered ability to conduct and transfer heat. In contrast, the volumetric heat capacity does not exhibit a significant dependence on surface roughness, as indicated by the lack of correlation between volumetric heat capacity and Ra. These findings contribute to our understanding of the intricate relationship between surface roughness and thermal properties, aiding in the design and optimization of materials for efficient heat transfer applications.

Figure 9 shows the relationships between the pore area and the thermal diffusivity, thermal conductivity, and volumetric heat capacity. The plot of pore area versus thermal diffusivity indicates an inverse relationship between the two variables. This means that as the pore area increases, the thermal diffusivity of rocks decreases. Thermal diffusivity is a measure of how quickly heat is conducted through a material, and a lower thermal diffusivity means that heat takes longer to travel through the material. This relationship makes intuitive sense since pores and cracks in rocks are essentially empty spaces that do not conduct heat as well as the solid material. Therefore, as the pore area increases, the proportion of non-conductive material in the rock increases, leading to a decrease in thermal diffusivity.

Similarly, the plot of pore area versus thermal conductivity also shows an inverse relationship between the two variables. This means that as the pore area increases, the thermal conductivity of rocks decreases. Thermal conductivity is a measure of how well a material conducts heat and a lower thermal conductivity means that the material is a poorer conductor of heat. This relationship is also understandable since pores and cracks in rocks act as thermal insulators, reducing the overall thermal conductivity of the rock.

The plot of pore area versus volumetric heat capacity, however, indicates that there is no correlation between the two variables. Volumetric heat capacity is a measure of how much heat a material can absorb before its temperature increases, and in this case, it seems that the presence of pores and cracks in rocks does not have a significant effect on their volumetric heat capacity. Since volumetric heat capacity is primarily influenced by the mineral composition and density of the sandstones, it is important to note that pore area alone may not be the sole determinant of heat capacity. While porosity and pore area can influence the thermal properties of a material, other factors such as pore shape, connectivity, and fluid properties inside the pores can also play significant roles. Therefore, it is possible that the variations in pore area alone were not sufficient to cause a noticeable correlation with volumetric heat capacity.

The experimental study surface roughness of rocks can be correlated with their thermal properties. Rocks with rougher surfaces generally exhibit larger, more poorly-sorted grain sizes. This leads to a reduced number of grain boundaries per unit volume and inefficient packing, ultimately decreasing the connectivity of the grain network. As a result, heat transfer through the rock becomes more challenging, resulting in lower thermal conductivity and diffusivity.

In contrast, rocks with smoother surfaces tend to have smaller, well-sorted grain sizes. The smaller grain size increases the number of grain boundaries per unit volume, and efficient sorting enables effective packing. This results in a well-connected grain network that facilitates heat transfer. Consequently, rocks with smoother surfaces exhibit higher thermal conductivity and diffusivity.

The presence of grain boundaries can scatter or impede heat transfer. Rocks with smaller, well-sorted grains have a higher density of grain boundaries, providing more opportunities for heat transfer between grains. On the other hand, rocks with fewer, larger grains possess fewer grain boundary sites, limiting heat transfer. Additionally, better packing and grain sorting in rocks minimize the presence of air spaces or pores. This reduces scattering or impedance of heat transfer since air does not conduct heat as efficiently as mineral grains. In contrast, poorer sorting and packing result in more air space, creating barriers to heat flow. Smooth grain shapes allow for increased surface contact between grains, enhancing heat transfer. In contrast, angular and irregular grain shapes lack sufficient surface contact, impeding the flow of heat. These insights highlight the significant influence of surface roughness and grain characteristics on the thermal properties of rocks, shedding light on their heat transfer behavior and conductivity.

Figure 10 displays the Pearson correlation between the textural features analyzed, average surface roughness, and the pore, cracks, and voids area extracted from the SEM images, with experimentally obtained thermal properties. Pearson correlation is a statistical measure that quantifies the strength and direction of the linear relationship between two continuous variables. It assesses how closely the data points of two variables align on a straight line. The Pearson correlation coefficient, often denoted as “r,” ranges from −1 to +1. The closer to 1 or −1 denotes a strong correlation. In contrast, the closer to 0 denotes that the variables are uncorrelated. Average surface roughness has a strong negative correlation, −0.59 and −0.6 with thermal conductivity and thermal diffusivity respectively. The magnitude of this strong negative correlation coefficient indicates a robust inverse linear relationship exists between the variables. As the absolute value of r approaches unity, the strength of the correlation increases. Here, r values of −0.59 and −0.6 represent a relatively high degree of negative correlation.

The negative sign of the correlation coefficient signifies that as surface roughness increases, thermal conductivity and diffusivity tend to decrease. Specifically, rougher surfaces in the sampled sandstones correlate with diminished ability to conduct and propagate heat. This inverse relationship may be attributed to surface irregularities introducing additional impediments to heat-flow pathways through the material.

The linear nature of the correlation implies changes in one variable will relate inversely to changes in the other in a linear fashion. As quantified by the coefficient of determination (r²), approximately 36% of the variance in one property can be explained by its linear correlation with the other.

While the strong correlation points to an apparent relationship between surface roughness and thermal properties, it does not prove direct causation. Other underlying factors may also influence the variables and further analysis would be needed to establish any causal mechanisms. Nonetheless, the correlation provides insight into potential influences of micro-structural characteristics on heat transport properties in sedimentary rocks. In contrast, the volumetric heat capacity and the average surface roughness are uncorrelated, confirming the results presented in the previous section.

Similar conclusions were obtained from other studies. Reference [43] simulated the thermal conductivity of porous media of Alberta oil sands, concluding that higher roughness resulted in lower effective thermal conductivity due to increased thermal contact resistance at grain boundaries. Reference [44] simulated fracture roughness influence on heat-flow coupling for percolated networks where fluid flow is continuous, finding that increased roughness impedes heat transfer by amplifying thermal resistance at interfaces between rough fractures and rock. According to the study, roughness only significantly impacts thermal transport under conditions where the fracture network is permeable and interconnected.

The pore, cracks, and void surface area have a less apparent negative correlation, −0.18 and −0.17, related to the thermal conductivity and thermal diffusivity. The reason that the pore, cracks, and voids area values have a less apparent negative correlation with thermal conductivity and thermal diffusivity compared to the average surface roughness could be that these properties affect heat transfer differently. The negative correlation between average surface roughness and thermal conductivity/diffusivity is stronger, as the surface roughness directly affects the connectivity of the grain network and packing efficiency, which are important factors in determining thermal conductivity and diffusivity. On the other hand, the effect of voids on heat transfer is dependent on various factors, and the correlation between pore, cracks, and voids area and thermal conductivity and diffusivity may be weaker due to the complexity of the effect of voids on heat transfer. Factors such as the size, shape, and distribution of the voids affect heat transfer. If the voids are interconnected, they can provide networks for heat to flow through, which can increase the thermal conductivity and thermal diffusivity of the sandstone. Additionally, the size and shape of the voids can affect the rate at which heat is transferred. Smaller voids, for example, can trap heat more effectively than larger voids. Similar conclusions were found in a theoretical and experimental study [45], where it was found that the effective thermal conductivity is most sensitive to the pore geometry and shape within porous rock materials, compared with porosity, which presented a lesser effect on effective thermal conductivity.

The thermal properties of rocks are influenced by several factors, including their texture, mineralogy, and porosity. These characteristics can be difficult to discern through visual inspection alone, especially when looking at a two-dimensional image. SEM provides high-resolution, detailed images of the rock's surface, allowing for a more accurate analysis of its texture and other characteristics that can impact its thermal properties.

The study aimed to analyze the porosity, permeability, density, and thermal properties of core samples and their correlation with mineral content. In this study, we present the results of the analysis of the SEM images obtained from rock samples to investigate various surface roughness properties. The objective was to quantitatively evaluate and characterize the roughness features of the rock surfaces. By utilizing these parameters, we aimed to gain insights into the micro-structural defects, crack types, and other surface characteristics that impact the thermal properties of sedimentary rocks. The analysis of these properties provides valuable information for optimizing the selection of rocks for geothermal energy systems and thermal energy storage applications.

The thermal properties of rocks are affected by their texture, mineralogy, and porosity, which can be difficult to discern through visual inspection alone. The use of SEM images allows for a more accurate analysis of these characteristics, providing insights into the micro-structural defects, crack types, and other surface characteristics that impact the thermal properties of sedimentary rocks. The study found that surface roughness plays a significant role in determining the thermal conductivity and diffusivity of rocks, with smoother surfaces exhibiting higher thermal conductivity and diffusivity. The correlation between pore, cracks, and voids area and thermal conductivity, and diffusivity was found to be weaker than that of surface roughness, likely due to the complexity of the effect of voids on heat transfer. The findings of this study provide valuable information for optimizing the selection of rocks for geothermal energy systems and thermal energy storage applications.

The authors of this paper would like to thank the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy (EERE) for funding this research, also thanks to Dr. Jivtesh Garg and the MPGE Department for allowing us to use laboratory equipment and the University of Oklahoma for the access to bibliographic material used in this study.

• This study was supported by funding from the United States Department of Energy (DOE) under Award Number DE-EE0009962. The opinions, findings, conclusions, or recommendations presented in this publication are solely those of the authors and do not necessarily represent the views or opinions of the United States Department of Energy (DOE).

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent was obtained for all individuals. Documentation is provided upon request. This article does not include any research in which animal participants were involved.

The authors attest that all data for this study are included in the paper.

m =

meter

s =

second

W =

Watt

K =

Kelvin degree

C_P =

specific heat capacity, J/kg · K

Ra =

average surface roughness, μm

MJ =

mega Joule

°C =

Celsius degree

κ =

thermal diffusivity, m²/s

λ =

thermal conductivity, W/m · K

ρ =

density, kg/m³