The anthracite was acquired from Jincheng Lanyan Coal Mime Co. Ltd. The detailed information of anthracite were summarized in Tables 1–3.

The composition analysis of the anthracite sample was performed by a 5E- SDLA618 Automatic Industrial Analyzer (Sande Instruments, China). The ultimate analysis of the anthracite sample was performed by using an Elemental Analyzer (Kaiyuan Instrument, China). The Infrared thermal imaging data of anthracite sample was investigated via a T650sc thermal infrared imager (Teledyne FLIR, Wilsonville, OR, United States). The ignition temperature of the coal sample was tested by a RD-6000 Ignition Point Tester (Hongke Instrument, China).

All these procedures were performed three times for the specimen.

Figure 1 shows the change law and fitting curves of the coal temperature with time.

The program temperature was increased at an increase rate of 2°C/min. In the range of 30–454.2°C, the coal temperature has a linear relationship with the heating time (30–315°C) and an exponential function relationship (315–454.2°C).

(1) 30–315°C

The functional relationship corresponding to the fitting curve for the coal temperature is obtained as follows: y = A + B x (1) where y is the coal temperature (°C); x is the heating time of coal (min); A and B are dimensionless constants. For the anthracite sample selected in this experiment, A = 19.56621, and B = 1.77045. Therefore, the functional relationship between the coal temperature and time is given as follows: y = 1 .77045 x + 19.56621 (2)

Correlation coefficient: R = 0.99887.

(2) 315–454.2°C

The functional relationship corresponding to the fitting curve for the coal temperature is obtained as follows: y = A 1 e x t 1 + y 0 (3) where y is the coal temperature (°C); x is the heating time of coal (min); A ₁ and t ₁ are dimensionless constants; and y ₀ is the initial temperature of the coal (°C). For the anthracite sample selected in this experiment, A ₁ = 8.46606 × 10⁻¹¹, t ₁ = 8.59127, and y ₀ = 400.97887. Therefore, the functional relationship between the coal temperature and time is given as follows: y = 8.46606 × e x 8.59127 + 400.97887 (4)

Correlation coefficient: R = 0.97059.

Figure 2A is a curve showing the variations in coal temperature and R value with time. Figure 2B shows the changes in the coal temperature rise time and R value with coal temperature. Figure 2 clearly shows that the R value fluctuates in the process of coal oxidation and spontaneous combustion and shows obvious regularity.

To intuitively analyze the corresponding relationship between the coal temperature and R value, the coal temperature is taken as the independent variable for analyzing the R value.

In the process of coal oxidation and spontaneous combustion, there is a certain corresponding relationship between the R value and coal temperature. Through polynomial fitting, the changes in the fitting curve of the R value with the coal temperature are obtained for the process of coal oxidation and spontaneous combustion (see Figure 3).

The functional relationship between the R value and coal temperature is obtained as follows: y = A + B 1 x + B 2 x 2 + B 3 x 3 + B 4 x 4 + B 5 x 5 + B 6 x 6 (5) where y is the dimensionless R-value; x is the coal temperature (°C); A is a constant term; and B ₁, B ₂, B ₃, B ₄, B ₅, and B ₆ are coefficients.

For the anthracite sample selected in the experiment, the constant term is A = 497.3787; the coefficients are B ₁ = −15.47135, B ₂ = 0.2338, B ₃ = −0.00166, B ₄ = 5.90953 × 10⁻⁶, B ₅ = −1.01624 × 10⁻⁸, and B ₆ = 6.75969 × 10⁻¹². The R-value as a function of temperature is given as follows: y = 497.3787 − 15.47135 x + 0.2338 x 2 − 0.00166 x 3 + 5.90953 × 10 − 6 x 4 − 1.01624 × 10 − 8 x 5 + 6.75969 × 10 − 12 x 6 (6)

Correlation coefficient: R = 0.95807.

In summary, the change trend of the R value with the coal temperature can be divided into the following four stages:

Stage 1: 31.8–76.7°C.

In this stage, the R value decreases with the increase of coal temperature, from 218.82 to 88.99, thus reaching the first characteristic valley value, at which the corresponding temperature is 76.7°C.

Stage 2: 76.7–126.4°C.

In the 76.7–126.4°C range, the R value increases slightly and reaches the first characteristic peak value of 146.1 at 126.4°C.

Stage 3: 126.4–255.2°C.

In this stage, the R value shows a downward trend, from 146.1 to 41.11, thus reaching the second characteristic valley value, at which the corresponding temperature is 255.2°C.

Stage 4: 255.2–454.2°C.

The ignition temperature of the experimental coal sample is 398°C, which is within the 255.2–454.2°C range. In this stage, the R value shows an overall upward trend, but fluctuates slightly around 398°C. In other words, the R value first decreases from 131.62 (371.3°C) to 105.03 (379.8°C), then rapidly increases to 141.36 (389°C), then decreases to 129.47 (396.4°C), and finally rises to 186.86.

Figure 4A is a curve showing the variations in coal temperature and G value with time. Figure 4B shows the changes in the coal temperature rise time and G value with coal temperature.

During the whole experiment, the maximum, minimum, average and median values of G are 208.43, 160.36, 184.83 and 87.29, respectively. The maximum, minimum, mean and median values of R are 218.82, 41.11, 106.92 and 110.82, respectively. G value shows a similar change pattern to R value, but the change amplitude of G value curve is smaller than that of R value curve, and the average and median values of G are also higher than that of R.

Figure 5A is a curve showing the variations in coal temperature and B value with time. Figure 5B shows the changes in the coal temperature rise time and B value with coal temperature. Figure 5 clearly shows that the B value fluctuates in the process of coal oxidation and spontaneous combustion and shows obvious regularity.

To intuitively analyze the corresponding relationship between the coal temperature and B value, the coal temperature is taken as the independent variable for analyzing the B value.

In the process of coal oxidation and spontaneous combustion, there is a certain corresponding relationship between the B value and coal temperature. Through polynomial fitting, the changes in the fitting curve of the B value with the coal temperature are obtained for the process of coal oxidation and spontaneous combustion (see Figure 6).

The functional relationship between the B value and coal temperature is obtained as follows: y = A + B 1 x + B 2 x 2 + B 3 x 3 + B 4 x 4 + B 5 x 5 + B 6 x 6 (7) where y is the dimensionless B-value; x is the coal temperature (°C); A is a constant term; and B ₁, B ₂, B ₃, B ₄, B ₅, and B ₆ are coefficients.

For the coal sample selected in the experiment, the constant term is A = −146.85447; the coefficients are B ₁ = 8.00494, B ₂ = −0.13249, B ₃ = 9.92327 × 10⁻⁴, B ₄ = −3.60555 × 10⁻⁶, B ₅ = 6.23451 × 10⁻⁹, and B ₆ = −4.12402 × 10⁻¹². The B-value as a function of temperature is given as follows: y = − 146.85447 + 8.00494 x − 0.13249 x 2 + 9.9237 × 10 − 4 x 3 − 3.60555 × 10 − 6 x 4 + 6.23451 × 10 − 9 x 5 − 4.12402 × 10 − 12 x 6 (8)

Correlation coefficient: R = 0.9489.

In summary, the change trend of the B value with the coal temperature can be divided into the following four stages:

Stage 1: 31.8–76.7°C.

In this stage, the B value increases with the increase of coal temperature, from 8.31 to 42.29, thus reaching the first characteristic peak value, at which the corresponding temperature is 76.7°C.

Stage 2: 76.7–126.4°C.

In the 76.7–126.4°C range, the B value decreases slightly and reaches the first characteristic valley value of 10.52 at 126.4°C.

Stage 3: 126.4–255.2°C.

In this stage, the B value shows an upward trend, from 10.52 to 96.81, thus reaching the second characteristic peak value, at which the corresponding temperature is 255.2°C.

Stage 4: 255.2–454.2°C.

The ignition temperature of the experimental coal sample is 398°C, which is within the 255.2–454.2°C range. In this stage, the B value shows an overall downward trend, but fluctuates slightly around 398°C. In other words, the B value first increases from 15.32 (371.3°C) to 30.27 (379.8°C), then rapidly decreases to 11.76 (389°C), then increases to 20.89 (396.4°C), and finally drops to 3.04.

To sum up, B value basically shows a change rule contrary to R value.

Figure 7A is a curve showing the variations in coal temperature and RGB value with time. Figure 7B shows the changes in the coal temperature rise time and RGB value with coal temperature. Figure 7 clearly shows that the RGB value fluctuates in the process of coal oxidation and spontaneous combustion and shows some regularity.

During the whole experiment, the maximum, minimum, average and median values of RGB are 132.78, 99.30, 108.88, and 107.91, respectively. The maximum, minimum, mean and median values of G are 208.43, 160.36, 184.83, and 187.29, respectively. RGB value shows a similar change pattern to G value, but the change amplitude of RGB value curve is smaller than that of G value curve, and the average and median values of RGB are also higher than that of G.

Figure 8A shows a curve of the coal temperature and the R value, G value, B value, and RGB value changing with time. Figure 8B shows a curve of the coal reaction time and the R value, G value, B value, and RGB value changing with the coal temperature. From Figure 8, it is clear that the R value, G value, B value, and RGB value have obvious change rules.

In the whole experiment, from the viewpoint of temperature, there are one characteristic peak and two characteristic valleys in the R-value curve and G-value curve, respectively. However, the overall variation range of G-value curve is smaller than that of R-value. The B-value curve has two characteristic peaks and one characteristic valley. Considering the influence of the experimental instrument and environment, the RGB-value curve is deemed to have no peaks and valleys.

According to the characteristics of the R value, G value, B value, and RGB value changing with the coal temperature, the oxidation and spontaneous combustion process of anthracite is divided into the following five stages:

(1) Heat storage period: 30–92.2°C

The heat storage period is the initial preparation stage of coal oxidation and self-heating. The heat storage period of anthracite is longer. In the early stages of the heat storage, physical adsorption is the primary method, with chemical adsorption and chemical reaction serving as auxiliary methods. The gradual increase in temperature in the middle stage destroyed the van der Waals force between anthracite and the adsorbed gases, with some gases desorbed and released from the coal body. The chemisorption is obviously enhanced at a later stage, and unstable peroxides are produced (Wang and Liang, 2017; An et al., 2021). The experiment’s initial temperature is 30°C, so the corresponding temperature range of anthracite’s heat storage period is 30–92.2°C, and the critical characteristic temperature from the heat storage period to the self-heating critical period is 92.2°C.

(2) Critical period of heat storage and self-heating: 92.2–202.5°C

The critical period of heat storage and self-heating is the transition stage from the heat storage period to the self-heating period. The anthracite accelerates heat storage and self-heating during the critical period, and the R, G, and B values all change noticeably. (Wang et al., 2019; Wang et al., 2020a. The R value indicates a rapid decline, the G value indicates a slow decline, and the B value indicates a rapid upward trend. And then, at 202.5°C, the R and the B value curves intersect, and the system becomes self-heating. As a result, the temperature range corresponding to the critical period of anthracite heat storage and self-heating is 92.2–202.5°C. From the heat storage period to the heat storage-self-heating critical period, the critical characteristic temperature is 202.5°C.

(3) Self-heating period: 202.5–314.6°C

The self-heating period is a process in which the heat production of coal changes from being lower than the heat dissipation of the coal self-heating environment to being higher than the heat dissipation of the coal self-heating environment. This process can be regarded as an equilibrium process in which there is an overlap between the heat production process of coal and the heat dissipation process of the coal self-heating environment. Therefore, take the two intersection points of anthracite’s R and B value curves (corresponding to 202.5 and 314.6°C, respectively) as the temperature range of anthracite’s self-heating period, i.e., 202.5–314.6°C. During the self-heating period, the G value first decreased and then increased, revealing a characteristic valley value, and the corresponding anthracite temperature was 255.2°C. The heat production of coal is lower than the heat dissipation of the coal self-heating environment in the temperature range of 202.5–255.2°C, but the difference gradually decreases as the temperature rises. Near 255.2°C, the heat production and heat emission of coal are in balance, that is, the heat production of coal equals the heat dissipation of the coal self-heating environment; in the range of 255.2–314.6°C, the heat production of coal exceeds the heat dissipation of the coal self-heating environment, and the gap gradually widens with temperature increase (Liu et al., 2021b). As a result, the temperature range corresponding to anthracite’s self-heating period is 202.5–314.6°C, and the critical characteristic temperature of the self-heating period is 255.2°C.

(4) Critical period of self-heating and self-ignition: 314.6–379.8°C

The critical period of self-heating and spontaneous combustion is the transitional stage from the self-heating period to the spontaneous combustion period. At this stage, if the heat dissipation environment for coal oxidation and spontaneous combustion does not change, the coal will quickly enter the spontaneous combustion period. If the heat dissipation environment of coal oxidation and spontaneous combustion changes to a more conducive environment for heat dissipation, the coal will enter the cooling stage and then the weathering state. The spontaneous combustion tendency of coal is reduced, and it is difficult for self-heating to occur again. This paper deals only with the rapid entry of the coal into the spontaneous combustion period through this stage. During the critical period of self-heating and self-ignition, the R value rapidly increases from 62.5 to 131.6 (371.3°C) and then rapidly decreases to 105 (379.8°C). B value decreased from 71 to 15.3 (371.3°C) and then increased to 30.3 (379.8°C); G and RGB values change in the same way as the R values, but the change range is smaller. As a result, the temperature range for this process is 314.6–379.8°C. The temperature range is 314.6–371.3°C at the stage where of the R values rise and B values fall. Aromatic hydrocarbons, aliphatic hydrocarbons, oxygen-containing functional groups, and some small molecular substances in the anthracite molecular structure begin to crack, releasing small molecular volatiles, and promoting the spontaneous combustion process of coal. When the temperature rises above 371.3°C, the covalent bonds of macromolecular substances in anthracite begin to break and participate in the spontaneous combustion of coal. And lay a foundation for accelerating anthracite combustion in the next stage (Wang et al., 2020b). As a result, the temperature range corresponding to anthracite’s self-heating and self-ignition critical period is 314.6–379.8°C, and the critical characteristic temperature of the self-heating and self-ignition critical period is 371.3°C.

(5) Spontaneous combustion period: 379.8–454.2°C

The spontaneous combustion period is the stage of coal combustion (Wang et al., 2021; Xu et al., 2021). Under the experimental condition of 2°C/min temperature programmed heat source, the coal enters the spontaneous combustion period after 379.8°C; the coal temperature rises sharply and then begins to burn. From the beginning of the spontaneous combustion period to the end of the experiment, the coal temperature rises rapidly from 379.8 to 454.2°C. In this period, The R value curve fluctuates between 379.8 and 454.2°C, then drops to a minimum of 122.7 (396.4°C), and then rapidly rises. The B value curve shows a maximum value of 20.9 (396.4°C) after fluctuation, then rapidly decreases and approaches 0. The G, RGB, and R values show the same change rule, but the change range is smaller. Therefore, the critical temperature of anthracite spontaneous combustion period is 396.4°C.