Effects of intrinsic properties, particle size, bulk density, and specific gravity on thermal properties of coal dusts

Thermal properties of pulverized coal govern the heat transfer and greatly influence the coal dust explosion and spontaneous combustion processes. This study measures the thermal properties of five coal samples at six distinct particle sizes using an advanced thermal property analyzer. The thermo-physical properties of coal dust positively correlated with the particle size. Thermal conductivity, diffusivity, and specific heat capacity increased with the ash percentage, bulk density, and specific gravity of coal dust. In contrast, they negatively correlated with the fixed carbon and volatile content of coal. Empirical relations between the thermo-physical properties were developed. The thermal conductivity, diffusivity, and specific heat capacity of coal dusts varied in the range of 0.091–0.147 W/mK, 0.125–0.164 mm2/s, and 0.715–0.945 MJ/m3K, respectively. With increase in particle size from < 38 to 500–1000 µm, thermal conductivity, thermal diffusivity, and specific heat capacity increased in the range of 25.60–32.89%, 9.76–22.11%, and 9.57–20.80%, respectively, for different coal samples.


Introduction
Coal is one of the major energy sources across the globe. The increasing coal production demand has posed a great concern for the mine management due to miners' health and safety. The generation of large amounts of coal dust with increasing mine mechanization has emerged as a serious cause of concern for miners' health (Sahu and Mishra 2022a). The coal dust explosion and spontaneous heating also pose severe threat to the mine safety (Deng et al. 2018b). The coal dust in industrial sites can undergo selfheating and increase the risks of fire and coal dust explosion (Li et al. 2021). Thermo-physical properties of coal dusts, such as thermal conductivity, diffusivity, and specific heat capacity, play a vital role in heat transfer. Therefore, the measurement of thermal properties of coal dust is essential for a better understanding of the heat transfer during coal dust explosion and spontaneous combustion processes. Thermal properties of coal depend on the temperature, density, composition, rank, and intrinsic properties (volatile, ash, carbon, and moisture content) of coal (Melchior and Luther 1982;Herrin and Demirig 1996;Wen et al. 2015;Deng et al. 2018a). The thermal conductivity of coal, which varies in the range of 0.22-0.55 W/mK, is much lesser than that of rock (varies in the range of 1-7 W/mK) (Herrin and Demirig 1996;Wen et al. 2015).
Thermal properties of coal particles play a vital role in coal dust explosion due to the involvement of the heat transfer process (Park et al. 2009;Sahu and Mishra 2022b). Knowledge of thermal properties is of paramount importance to understand the minimum ignition temperature (MIT) and temperature rise in coal dust explosions, as well as the development and spread of spontaneous heating and mine fire. Particle size greatly influences the thermal properties of coal dust. The thermal conductivity of coal dust decreases with decrease in the particle size or increase in the surface area of the coal particles (Azam and Mishra 2019). The thermal conductivity of powdered materials depends on different parameters, such as size, shape, temperature, compressional stress, packing structure, and porosity of the particles (Gan et al. 2017;Sakatani et al. 2017). The ignition temperature of coal dust particles increases with thermal conductivity. Reddy et al. (1998) studied the layer ignition temperature (LIT) of coal dust by adding inert material having higher thermal conductivity. They observed that the MIT and thermal conductivity of the mixture increased by 40 °C and 0.123 W/m K, respectively, due to the addition of 50% inert material. Few researchers have investigated the effect of particle size on the thermal conductivity of nanoparticles and found that coarser particles enhance the thermal conductivity of nanoparticles (Chopkar et al. 2008;Beck et al. 2009;Warrier and Teja 2011).
Thermal properties of the material are its ability to conduct, transfer, release, and store heat (Schön 2011). The relationship between thermal conductivity, thermal diffusivity, and specific heat is given by the formula: where ρ is density (kg/m 3 ), C p is specific heat (J/kg K), k is thermal conductivity (W/mK), and α is thermal diffusivity (mm 2 /s) of the material.
The specific heat capacity is defined as the amount of heat required to raise the temperature of a unit mass of material by 1 K, whereas the volumetric specific heat capacity is the heat required to raise the temperature of a unit volume of material by 1 K. The volumetric specific heat capacity (MJ/m 3 K) is the product of specific heat and density of a material, and it can be expressed as Herrin and Demirig (1996) measured the thermal conductivity of 55 different coal samples at room temperature and analyzed the effect of intrinsic properties of coal on thermal conductivity. Deng et al. (2018b) measured the thermo-physical properties of four different metamorphic grade coal samples using laser-flash apparatus and analyzed the effect of coal crystallinity on thermal diffusivity of coal samples. Deng et al. (2017a, b) also utilized the laser flash apparatus to study the thermophysical parameters of coal samples in the temperature ranges of 303-573 K and found that the thermal diffusivity decreases and thermal conductivity and specific heat increase with increase in the temperature. Krause et al. (2011) measured the thermal conductivity of porous dust aggregates and found a dependency of thermal conductivity on the volume fill factor of dust. Rahman et al. (2015) measured the thermal conductivity of insulating firebricks produced by coal dust and local clay by varying the coal percentage and particle size and observed decrease in thermal conductivity with increasing fineness of the coal dust particles. Cremers (1985) measured (2) Volumetric specif ic heat = C p = k the thermal diffusivity of powdered coal samples with two different particle sizes and found a small effect of particle size on thermal diffusivity. Gosset et al. (1996) measured the thermal diffusivity of powdered coal in different temperatures (293 K to 773 K) and found that the thermal diffusivity of coal decreases with increasing temperature up to 573 K. Wen et al. (2015) also found a similar trend of thermal diffusivity with increase in temperature. However, they did not notice a significant influence of temperature on the thermal diffusivity in the case of coal ash slags until the temperature of 673 K (Deng et al. 2018a). Melchior and Luther (1982) measured the specific heat capacity of bituminous coal and coke in the temperature ranges of 303-623 K and found that the specific heat increases with temperature. Zhumagulov (2013), Wen et al. (2015), and Deng et al. (2017a) also found increasing trend of specific heat capacity with temperature for bituminous coal. The temperature-dependent thermal conductivity also varies with the rock types. The thermal conductivity of coal increases with temperature (Zhumagulov 2013;Wen et al. 2015Wen et al. , 2016Deng et al. 2017a), whereas the thermal conductivity of granite and sandstone decreases with temperature (Abdulagatova et al. 2009;Miao et al. 2014;Wen et al. 2015). Chu et al. (2018) measured the thermo-physical properties of geo-materials, i.e., coal, sandstone, and concrete, utilizing the series and parallel model. Tanikawa et al. (2016) determined the thermal property of deep-water coalbed basins up to 2500-m depth below the sea floor and found an increase in thermal conductivity and diffusivity and a decrease in specific heat capacity with increasing depth.
Limited studies on the thermal behavior of coal dust are available. Hence, we measured the thermal properties of Indian coal dusts using Hot Disk TPS 2500S thermal constant analyzer at room temperature, and analyzed the effects of particle size, density, specific gravity, and intrinsic properties of coal on the thermal properties of coal dusts. Moreover, we developed empirical relations between the thermo-physical properties of coal dusts. The thermal properties data of varied size coal dusts generated through this research provided meaningful insights into the heat transfer process in coal dusts. This is to mention here that the thermal properties of a material generally include thermal conductivity, thermal diffusivity, and specific heat capacity. However, we have explicitly mentioned these thermal properties wherever applicable in the manuscript. The outcome of this study will help in better understanding of the heat transfer during spontaneous combustion and explosion processes involving coal dusts and further research in the related areas. This, in turn, will enable the estimation of appropriate quantity of inertant required to suppress the coal dust explosions in mining and allied industries.

Coal sample collection
For this study, four coal samples, viz., A, B, C, and D, were collected from Dhory colliery (Bermo seam), Dhori colliery (Karo seam), Jitpur colliery, and Moonidih colliery, respectively, of Jharia coalfield in Jharkhand, and one sample, i.e., sample E, was collected from Chinakuri colliery of Raniganj coalfields in West Bengal state of India. The locations of the collieries from where the samples were collected are shown in Fig. 1. All these five coal mines have a history of dust explosion and mine fire. The coal samples were obtained from the freshly exposed underground mining faces.

Sample preparation
The thermal properties of coal dust greatly depend on the particle size (Cremers 1985;Rahman et al. 2015). Therefore, the coal samples collected from the underground mines were prepared in six different size ranges  Coal dust samples prepared in different particle size ranges for analysis by crushing and sieving, such as < 38, 38-74, 74-125, 125-250, 250-500, and 500-1000 µm. Figure 2 shows the images of prepared coal dust samples of different particle size ranges of coal A.
The scanning electron microscope (SEM) images of different coal dust sizes were captured using the FE-SEM Supra 55 (Carl Zeiss, Germany). Figure 3 shows the SEM images of coal dust particles of different size ranges of coal A: First three finer size dusts, (a) < 38 µm, (b) 38-74 µm, and (c) 74-125 µm, were taken at 500X magnification, whereas the coarser size dusts, (d) 125-250 µm, (e) 250-500 µm, and (f) 500-1000 µm, were captured at 150 × magnification for better visibility of dust particles. The SEM images revealed that the coal dust particles are irregular and uneven in shape, which may significantly influence the packing density and, in turn, the thermal properties of the coal dust.

Measurement of intrinsic properties of coal
Since the thermal properties of coal are affected by its chemical composition, the ultimate and proximate analyses of coal samples were undertaken as per ASTM: D3176-15 and D7582-15, respectively. The ultimate analysis was carried out with "Flash 2000" Organic Elemental Analyzer (Thermo Scientific) for determining the carbon (C), hydrogen (H), nitrogen (N), sulfur (S), and oxygen (O) content, whereas the proximate analysis was done using Thermogravimetric Analyzer (TGA-2000A, Navas Instruments, USA) for determining the fixed carbon (FC), moisture (M), ash (A), and volatile matter (VM) content of the coal samples. The results of the ultimate and proximate analyses are presented in Table 1.

Measurement of bulk density and specific gravity of coal dust samples
The thermal properties of coal dust are influenced by the bulk density, which in turn affected by the coal particle size. The bulk density of coal dust samples at different particle sizes was measured with a fixed-volume stainless steel container as per ISO 23499:2013 to study the effect of bulk density on the thermal properties of coal dust. The steel container was filled with coal dust by applying a uniform pressure, and the weight of the sample was measured by a high-precision weighing machine. The bulk density of coal dust was determined by dividing the weight of coal dust by container volume. The specific gravity of coal dust of different particle sizes was determined by the pycnometer method as per IS: 2386 (Part III), 1963, equivalent to ISO 17892-3:2015 standard. The experiments for determining the bulk density and specific gravity of coal dust samples were repeated thrice, and the average value was considered. The bulk density and specific gravity of coal samples (coal A to coal E) of different particle sizes (< 38 to 500-1000 µm) varied in the range of 0.55-0.90 g/cc and 1.27-1.54, respectively. Variations of bulk density and specific gravity of coal dust with particle size are shown in Fig. 4. Both bulk density and specific gravity showed an increasing trend with particle size, which best fitted the polynomial trendline with R 2 values of 0.997 and 0.949, respectively. Mishra (2022) also observed an increasing trend of coal dust bulk density with increase in particle size. The average bulk density and specific gravity of different sizes of coal dust varied in the range of 0.56-0.86 g/ cc and 1.37-1.47, respectively, which increased by 34.14%, and 6.48%, respectively, with increase from the finest to coarsest particle size (Fig. 5).

Experimental setup
Several researchers have used laser flash apparatus to measure the thermal properties of coal and coal dusts (Wen et al. 2015;Deng et al. 2017bDeng et al. , a, 2018b. However, transient plane source (TPS) method is the most convenient and less time-consuming technique used nowadays to determine the thermal properties of materials (Ghosh et al. 2016). In this study, we used a Hot Disk (TPS 2500S) thermal property analyzer (Fig. 6) to measure the thermal properties of coal dusts. Hot Disk (TPS 2500S) measures the thermal conductivity, thermal diffusivity, and volumetric specific heat capacity of materials based on the transiently heated plane sensor as per ISO 22007-2. The Hot Disk sensor carved out of a thin nickel foil, consists of a double spiral shaped electrically conducting pattern (Hot Disk 2016). The nickel foil is supported between two thin layers of insulating materials, i.e., Kapton and mica. The Kapton insulation Hot Disk sensor is used to measure the thermal properties up to 300 °C, whereas the mica insulation sensor is used for higher temperatures up to 1000 °C. The mica insulation sensor is available without a cable connection and directly mounted on the high-temperature sample holder. The sensor is placed between two identical-sized samples having a plane surface facing the sensor. However, in case of powder material, the sample holder is designed to accommodate powders in a manner to control the mechanical pressure of the sample. The instrument also offers singlesided measurements with the help of an insulation cover. In case of single-side measurement, the insulation material is placed above the sensor instead of the sample. This instrument can measure the thermal properties of materials with a temperature range between -243 °C and 1000 °C.

Measurement method
In this study, the Kapton insulation Hot Disc sensor of 6.403mm radius was used for measuring the thermal properties of coal dust samples. The radius of the sensor depends on the thickness and diameter of the sample. For the analysis of isotropic materials, the sample thickness should be at least equal to the sensor radius. Selection of a sensor is very important for the precise measurement of thermal properties. The wrong sensor can over or under-rate the measurement results. As per the user manual, the largest possible sensor should be used for better measurement accuracy. The optimum sensor radius compared to the sample size should be as follows: where r is sensor radius, T is sample thickness, and D is sample diameter. The powder sample holder was used to measure the thermal properties of coal dusts of different particle sizes. Hot Disk sensor was placed between the equal volume coal dust samples with controlled mechanical pressure. For room temperature measurement, external PT100 sensor was used to determine the sample temperature. The thermal properties and other parameters were analyzed in the Hot Disk Thermal Analyzer software. Table 2 presents the experimental parameters used in this study for the measurement of the thermal properties of coal dusts.
The heating power and measuring time depend on the type of material. The low thermal conductive material requires low power and high measuring time and vice versa. Since coal dust is a very low conductive material, this study considered heating power of 30 mW and measuring time of 160 s. Table 3 presents the required output conditions for acceptable thermal properties results. All parameters should lie within these ranges; if any of the parameters are out of bounds, the relevant field will be marked yellow as a warning, and the result might be affected. However, the user can regulate the input parameters such as heating power, measurement time, and probing depth to fulfill the required output conditions.

Effect of ash content
Ash is the incombustible content of coal. The thermal conductivity of coal dust increases with the percentage of incombustible material content (Reddy et al. 1998). Also, the risk of coal dust explosion and spontaneous heating is reduced with increase in the ash percentage of coal dust (Ramlu 1991). The five coal samples used in this study have different ash contents ( Table 1). The ash content in Coal A, B, C, D, and E is found to be 25. 77, 19.65, 20.3, 7.51, and 11.05%, respectively. The variations of thermal conductivity of different sizes of coal dusts with ash content are shown in Fig. 7. It can be observed that the thermal conductivity of coal dusts increases with increase in the percentage of ash content as well as particle size. While the ash content of different coal samples varied in the range of 7.51-25.77%, the thermal conductivity of different coal dust samples of various particle sizes varied in the range of 0.0911-0.1465 W/ mK. The increase in the thermal conductivity with ash content may be attributed to the presence of greater amount of incombustible matter in the coal dust. The thermal conductivity of various sizes of coal dusts prepared from the coal with lowest ash content (7.51%) varied in the range of 0.092-0.122 W/mK, whereas in case of coal dusts prepared from the coal having highest ash content (25.77%), it varied in the range of 0.100-0.147 W/mK. From the figure, it is also evident that the thermal conductivity of coal dusts increases with increase in the particle size, irrespective of the ash content of coal. Figure 8 and 9, respectively, shows the variations of thermal diffusivity and volumetric specific heat capacity of different sizes of coal dusts with ash content. It can be observed that both thermal diffusivity and specific heat capacity of different sizes of coal dusts increase with increase in the coal ash percentage. The thermal diffusivity and specific heat capacity of various sizes coal dusts of lowest ash coal (7.51%) varied in the range of 0.125-0.135 mm 2 /s   Fig. 9 Variations of specific heat capacity of different sizes coal dusts with ash content and 0.737-0.898 MJ/m 3 K, respectively, whereas those of highest ash coal (25.77%), they varied in the range of 0.128-0.164 mm 2 /s and 0.785-0.895 MJ/m 3 K, respectively. Hence, it is apparent that the thermal properties or heat transfer capacity increases with the incombustible (ash) content of the coal dust.

Effect of FC and VM content
Both fixed carbon (FC) and volatile matter (VM) are the intrinsic properties of coal. The thermal conductivity of coal is correlated with the sum of FC and VM content of coal (Herrin and Demirig 1996). The FC + VM content in coals A, B, C, D, and E is determined 73.1, 79.76, 78.59, 91.31, and 86.69, respectively (Table 1). Figure 10 shows the variation of thermal conductivity of different sizes coal dusts with the sum of FC and VM content. It may be observed that the thermal conductivity of different coal dust sizes decreases with increase in the mass percentage of FC + VM. The sum of FC and VM of coal varied in the range of 73.1-91.31%. With increase in the FC + VM content from 73.1 to 91.31%, the thermal conductivity of smallest size (≤ 38 µm) and coarsest size (500-1000 µm) coal dust decreased from 0.100 to 0.091 W/mK and 0.147 to 0.122 W/mK, respectively. This phenomenon may be due to the decrease in ash content or increase in rank (carbon content) of coal. Previous studies also reported decrease in the thermal conductivity of coal with increase in the coal rank (Herrin and Demirig 1996;Shi et al. 2020). Figures 11 and 12, respectively, show the variations of thermal diffusivity and specific heat capacity of different sizes coal dusts with FC and VM content. From the figures, it is evident that both thermal diffusivity and specific heat capacity decrease with increase in the sum of FC and VM. While the thermal diffusivity of the highest and lowest FC and VM content coal varied in the ranges of 0.125-0.135 mm 2 /s and 0.128-0.164 mm 2 /s, the specific heat capacity varied in the ranges of 0.737-0.898 MJ/m 3 K and 0.785-0.895 MJ/m 3 K, respectively.

Effect of bulk density and specific gravity
The thermal properties of coal dust are greatly influenced by bulk density and specific gravity. Figure 13 shows the  where m is the mass of coal dust (g) and V is the volume (cc) of coal dust. The figure showed an increasing trend of thermal conductivity of coal dust with increasing bulk density and specific gravity. Also, the figure indicated that coal A has the highest and coal D has the lowest thermal conductivity among all the coal samples. The bulk density of different coal dust samples (coals A to E) varied with coal dust particle size. For coal dusts of particle size < 38 µm, 38-74 µm, 74-125 µm, 125-250 µm, 250-500 µm, and 500-1000 µm, the bulk density varied in the range of 0. 55-0.58, 0.63-0.70, 0.71-0.80, 0.77-0.83, 0.80-0.89, and 0.81-0.90 Figure 14 and 15 shows the variations of thermal diffusivity and specific heat capacity with bulk density and specific gravity of different coal dust samples, respectively. Thermal conductivity is directly proportional to the thermal diffusivity and specific heat capacity. Thermal diffusivity and specific heat capacity also increased with increase in the bulk density and specific gravity of coal dust. For coal dust samples A, B, C, D, and E having different bulk densities and specific gravity, the thermal diffusivity is measured in the range of 0. 128-0.164, 0.127-0.158, 0.125-0.151, 0.125-0.135, and 0.128-0.142 mm 2 /s, and the specific heat capacity measured in the range of 0. 785-0.895, 0.747-0.826, 0.754-0.945, 0.737-0.898, and 0.715-0.903 MJ/m 3 K, respectively. However, the average thermal diffusivity and specific heat capacity of    coal dusts varied in the range of 0.126-0.153 mm 2 /s and 0.747-0.902 MJ/m 3 K, respectively. The increase in the thermal properties of coal dust with bulk density and specific gravity may be attributed to increase in the particle sizes of coal dust.
Variations of average thermal conductivity, diffusivity, and specific heat capacity of coal dusts with average bulk density and specific gravity are shown in Fig. 16, 17, and 18, respectively. Regression analysis was performed, and the following empirical relations between the thermo-physical properties of coal dusts were obtained, which can be used to predict the thermal properties of the coal dusts of known bulk density and specific gravity: where k, α, and C p are thermal conductivity, diffusivity, and specific heat capacity, respectively, and ρ and x are bulk density and specific gravity, respectively, of the coal dusts.

Effect of particle size
Particle size of coal dust plays a vital role in coal dust explosion and spontaneous combustion (Mishra and Azam 2018;Rifella et al. 2019). The risks of explosion and spontaneous combustion increase with decrease in the particle size or increase in the exposed surface area of coal dust. Thermal properties of coal dust greatly influence the heat transfer process during the combustion and explosion of coal dust. The transient temperature curves of different coal dust sizes of a particular coal sample obtained from the thermal property analysis utilizing the transient plane source (TPS) method are shown in Fig. 19. The curves show a continuous rise in temperature with time, which is an indicator of the accuracy of the results of thermal properties measurement. As per the Hot Disk user manual, the total temperature increase should vary in the range of 2-5 K for better results output. Figure 20 shows the calculated graphs of temperature rise versus F(Tau) of different coal samples for a (7) α = 0.1161ρ 2 − 0.0707ρ + 0.1293 (8) C p = 1.3235ρ 2 − 1.4372ρ + 1.1415   Fig. 21 Variations of thermal properties of coal dusts with particle size for different coal samples certain particle size (125-250 µm) and different particle sizes of a particular coal sample (coal A). F(Tau) is a dimensionless time-dependent function of the thermal diffusivity and shape of the sensor, which is linearly related with the temperature. More details about F(Tau) may be found elsewhere (Gustafsson 1991;He 2005;Maeda et al. 2021). The figures depict the best fit straight line (continuously linear), which represents that the transient measurement is valid for all the data points. Moreover, both Figs. 19 and 20 epitomize that the measured thermal properties of different sizes of coal dusts are correct.

Effect of particle size on thermal conductivity
Particle size plays a vital role in the thermal conductivity of materials (Chopkar et al. 2008;Beck et al. 2009). The variations of thermal properties of coal dusts with particle size for five different coal samples are shown in Fig. 21. It can be observed that the thermal conductivity of coal dusts steadily increases with increase in the particle size. This illustrates that with increase in particle size, the heat conduction and heat transfer capacity of coal dusts increase. Reddy et al. (1998) observed increase in the ignition temperature of coal and rock dust mixture with increase in thermal conductivity. As per Eqs. (1) and (2), thermal conductivity is directly proportional to the density, specific heat, and thermal diffusivity. Thermal diffusivity and specific heat of coal dusts increase with increase in the thermal conductivity. With increase in particle size from < 38 to 500-1000 µm, the thermal conductivity of coal dust increased in the range of 0.100-0.147, 0.095-0.128, 0.094-0.141, 0.092-0.122, and 0.091-0.128 W/mK for coal A, B, C, D, and E, respectively. The average thermal conductivity of different sizes of coal dusts varied between 0.094 and 0.134 W/mK. However, the thermal conductivity of core-sized original coal samples varied in the range of 0.2416-0.3558 W/mK. The rate of increase of thermal conductivity with particle size is found to be higher as compared to the thermal diffusivity and specific heat.
The variation of average thermal conductivity with particle size shown in Fig. 22a depicts an increasing trend of best-fitted polynomial curves with R 2 = 0.992. The following empirical relation was developed between the thermal conductivity and particle size of coal dusts through regression analysis: where k and P s are the thermal conductivity (W/mK) and particle size (µm) of coal dust, respectively.

Effect of particle size on thermal diffusivity
Thermal diffusivity of a material is its ability to transfer heat from high to low temperature. It depends on the thermal conductivity, density, and specific heat of the material. Thermal diffusivity of the material decreases with temperature rise (Stanger et al. 2013;Deng et al. 2017b) and increases with increase in the particle size (Ramya et al. 2019). Material with high thermal diffusivity has the ability to rapid heat transfer. Thermal diffusivity of core-sized original coal samples varied in the range of 0.19-0.24 mm 2 /s. However, the thermal diffusivity of different coal dust samples varied in the range of 0. 125-0.128, 0.129-0.149, 0.135-0.151, 0.136-0.161, 0.142-0.163, and 0.135-0.164 mm 2 /s for the particle size of < 38 µm, 38-74 µm, 74-125 µm, 125-250 µm, 250-500 µm, and 500-1000 µm, respectively. The variation of thermal diffusivity of coal dust with particle size shown in Fig. 21 indicates increase in thermal diffusivity with particle size. With increase in particle size from finer to coarser size, i.e., < 38 to 500-1000 µm, the thermal diffusivity of coal dusts increased in the range of 9.76-22.11% for different coal samples. However, its increasing rate is relatively smaller as compared to the thermal conductivity. Coal E has the smallest, and coal A has the largest thermal diffusivity difference amongst the different particle sizes. The variation of average thermal diffusivity of coal dust with particle size is shown in Fig. 22b. The empirical relationship established between the thermal diffusivity and particle size of coal dust is given below: where α and P s represent the thermal diffusivity (mm 2 /s) and particle size (µm) of coal dust, respectively. This regression equation best fits the polynomial curve (R 2 = 0.9978).

Effect of particle size on volumetric specific heat
The volumetric specific heat capacity is the product of specific heat and density of the material. The specific heat of a material is characterized by its thermal storage capacity (Deng et al. 2017b). The material with higher specific heat has greater thermal storage capacity. The volumetric heat capacity of different coal dust samples (Coal A to Coal E) varied in the range of 0. 715-0.785, 0.757-0.807, 0.779-0.816, 0.788-0.857, 0.826-0.915, and 0.808-0.945 MJ/m 3 K for the particle size < 38 µm, 38-74 µm, 74-125 µm, 125-250 µm, 250-500 µm, and 500-1000 µm, respectively. However, the volumetric specific heat of core-sized coal samples varied in the range of 1.258-1.587 MJ/m 3 K. Figure 21 shows the variation of volumetric specific heat capacity of coal dust with particle size. It can be observed that the volumetric heat capacity of coal dust samples increases with increase in the particle size. However, the rate of increase of volumetric heat capacity is comparatively lesser than the thermal conductivity. With increase in particle size from < 38 to 500-1000 µm, the volumetric heat capacity and thermal conductivity increased in the range of 9.57-20.80% and 25.60-32.89%, respectively, for different coal dust samples. The variation of average volumetric specific heat with particle size of coal dust shown in Fig. 22c revealed an increasing trend that best fits the polynomial curve with R 2 = 0.9814.

Fig. 22
Variations of a average thermal conductivity, b average thermal diffusivity, and c average specific heat with particle size of coal dusts where C p and P s represent the volumetric specific heat (MJ/ m 3 K) and particle size (µm) of coal dust, respectively. The regression Eqs. (12), (13), and (14) can be used to predict the thermal properties, such as thermal conductivity, diffusivity, and volumetric specific heat, respectively, of coal dusts of known particle sizes up to 1000 µm.

Uncertainty analysis
Uncertainty analysis was conducted to assess the reliability of the measured values. Uncertainty estimates the dispersion in the results and their difference from the accurate value. As per the ISO GUM standards, the measurement uncertainties are categorized into types A and B (GUM 2008). Type A uncertainty is calculated by statistical analysis of the series of repeated measurements taken by the observer. Type B uncertainty is evaluated from the non-statistical source based on scientific judgment using all the relevant data available, such as manufacturer specifications, previous measurement data, data provided in the calibration, and reference data taken from the textbook and literatures. Due to the lack of thermal properties data of coal dusts in the literature, we calculated the uncertainty of thermal properties of coal dust samples of different particle sizes as per the Type A evaluation method (GUM 2008). The type A standard uncertainty is calculated in terms of mean standard deviation of the repeated observations as given in Eq. (15), and the percentage uncertainty is calculated using Eq. where n is the number of repeated observations and X m is the mean of the repeated observations. The uncertainty results are presented in Table 4. According to the Hot Disk user manual, the reproducibility of thermal conductivity, thermal diffusivity, and specific heat capacity measured by the Hot Disk thermal constant analyzer are ± 2%, ± 5%, and ± 7%, respectively. Irrespective of different dust samples and particle sizes used, the measurement uncertainty of thermal conductivity, thermal diffusivity, and specific heat capacity in this study varied in the range of 0.07-1.58%, 0.19-3.83%, and 0.13-2.56% respectively. The uncertainty of thermal diffusivity was found to be higher than the thermal conductivity and specific heat capacity.

Conclusions
Thermal properties of coal dust play a major role in heat transfer during the coal dust explosion and spontaneous combustion processes. In this study, transient plane source (TPS) method was utilized to measure the thermal properties of coal dusts at six distinct particle sizes. The effect of particle size on thermal properties of coal dust was examined. Moreover, the influence of intrinsic properties, bulk density, % uncertainty (type A) = Measured uncertainty X m × 100 and specific gravity on thermal properties was studied. The conclusions of the study are outlined as follows: 1. The bulk density and specific gravity of coal dusts increased with increase in the particle size. The thermophysical properties of coal dusts increased with increase in the bulk density and specific gravity of coal dust samples. 2. The ash content of coal influenced the thermal properties of coal dusts of different particle sizes. Thermal conductivity, thermal diffusivity, and specific heat capacity of coal dusts increased significantly with increase in the ash content. 3. Thermal properties of coal dust are also influenced by the FC and VM content of coal. Thermal properties of coal dusts of all the particle sizes decreased with increase in the sum of FC and VM content. 4. The thermal conductivity, diffusivity, and specific heat capacity of coal dust steadily increased with the coal dust particle size. With increase in particle size from finer to coarser size, i.e., < 38 to 500-1000 µm, the thermal conductivity, thermal diffusivity, and specific heat capacity of coal dusts increased in the range of 25.60-32. 89%, 9.76-22.11%, and 9.57-20.80%, respectively, for different coal samples. 5. Empirical relations between the thermo-physical properties and bulk density, specific gravity, and particle size of coal dusts are developed, which can be used to predict the thermal properties of coal dusts. 6. The measurement uncertainty of thermal conductivity, thermal diffusivity, and specific heat capacity of coal dusts varied in the range of 0.07-1.58%, 0.19-3.83%, and 0.13-2.56%, respectively. This is pertinent to mention here that the coal samples used in this study are categorized as bituminous coal. Moreover, the thermal properties of coal dusts were measured at room temperature. Hence, there is an ample scope for furthering the research on this important aspect. Future studies may consider the measurement of thermal properties of coal dusts of different coal ranks at different temperatures. Furthermore, the thermal properties of coal dusts may be correlated with the coal dust explosion characteristics and spontaneous combustion process.