*4.2. Fractal Law of Large-Scale Coal Spontaneous Combustion AE Count*

#### 4.2.1. Fractal Theory and the Determination of Phase Space Dimensions

In this paper, the associative dimension characteristics of AE counts are introduced, combined with the G-P algorithm [36], MATLAB software is used to calculate the bilogarithmic relationship under different experimental conditions [37], and the count of AEs is used as the basic parameter to determine a series set with a capacity of n:

$$\mathcal{X} = \{ \mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n \} \tag{3}$$

The first m number in Equation (3) constitutes the phase space (m < n) of the m dimension,

$$\mathbf{X}\_1 = \{ \mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_m \} \tag{4}$$

The following data bits are sequentially extended by 1 to obtain N = n − m + 1 vector,

$$\mathbb{X}\_2 = \{ \mathbf{x}\_2, \mathbf{x}\_3, \dots, \mathbf{x}\_{\mathbf{m}+1} \}, \mathbb{X}\_3 = \{ \mathbf{x}\_3, \mathbf{x}\_4, \dots, \mathbf{x}\_{\mathbf{m}+2} \} \tag{5}$$

The corresponding associated dimensions are:

$$\mathbb{C}(r(k)) = \frac{1}{N^2} \sum\_{i=1}^{N} \sum\_{i=1}^{N} h[r(k) - \left| X\_i - X\_j \right|] \tag{6}$$

where: *h* is the Riverside function; *r(k)* is a given scale function. Each determined scale *r(k)* corresponds to it. Each (lgr, lgC(r)) coordinate point is linearly fitted, and the slope after fitting is the associated dimension of the AE count. m is the phase space dimension, the size of m has a specific influence on the associated dimension, and the value of m is 1, 2, 3, 4, 5 ... 20. With the increase of m value, the correlation dimension gradually tends to be saturated and stable. At m = 10, the associative dimension gradually tends to saturate and stabilize, so the phase space dimension m = 10 is determined.

#### 4.2.2. Determination of Fractal Features of AE Counts

According to the introduction of literature [38–46], when the scale coefficient k ≤ 0.1, the fractal characteristics of the AE sequence are not prominent. This paper takes the scale factor k > 0.1 to calculate the associated dimension in the process of aqueous coal sample rupture. Moreover, the slope of the regression function is the associated fractal dimension. Figure 10 is a curve plot obtained by selecting 5 points with good lnr and lnC value conditions processed by MATLAB software, and after performing univariate linear regression analysis. It is found that the correlation coefficients R<sup>2</sup> of the AE counts of 30 ◦C, 100 ◦C, 200 ◦C, and 300 ◦C are 0.96, 0.97, 0.99, and 0.98, respectively, and the correlation coefficient R<sup>2</sup> is more significant than 0.90. The correlation between lnr and lnC is relatively high, indicating that the correlation between AE counts at different temperatures is high, and the AE count sequences of different temperatures during the spontaneous combustion of large-scale coals have obvious fractal characteristics.

**Figure 10.** Fit curve plot of double logarithmic relationship at different temperatures.

4.2.3. Fractal Characteristics of AE Counting

According to the fractal characteristics of the AE count determined above, the fractal dimension change of the large-scale coal spontaneous combustion AE count is plotted, as shown in Figure 11.

**Figure 11.** Fractal dimension change of AE counts at different temperatures.

The fitted slopes of the different fractal dimension change curves were 0.999, 0.987, 0.981, and 0.998, respectively. According to the fractal dimension change chart of the

AE count within 300 s before and after the temperature points of 30 ◦C, 100 ◦C, 200 ◦C, and 300 ◦C in the large-scale coal spontaneous combustion heating process, it can be seen that the fractal dimension of the sound emission count of the four temperature gradients continues to increase with the increase of time in the 300 s before and after the temperature, the linear law is very obvious, the slope is stable, and it is greater than 0.950, indicating that the AE events of the coal body continue to increase during the heating process, and the AE signal is continuously enhanced. The fractal dimension of the AE count at 30 ◦C increased from 0.0258 to 0.3487, the fractal dimension of AE at 100 ◦C increased from 0.0638 to 1.3735, the fractal dimension of the AE count at 200 ◦C increased from 0.0377 to 0.5630. The fractal dimension of AE at 300 ◦C increased from 0.0130 to 0.2937, indicating that the fractal dimension of the AE count was greatest at 100 ◦C and the smallest at 300 ◦C, showing a trend of first growing and then declining from 30 ◦C to 300 ◦C.

Figure 12 is the selection of m = 4, 8, 12, 16, 20 of the AE fractal dimensions with the temperature of the trend plot. The plot presents an inverted V-shaped feature; the fractal dimension of the AE count with the increase of temperature first shows an increase. Then it shows a downward trend; the m value is different, the change situation is the same, and the larger the m value, the greater the fractal dimension. According to the literature, the changing trend of fractal dimension can reflect the failure process of the coal body and the timing change of the AE signal. The larger the fractal dimension, the more disordered the AE event. Therefore, in the process of coal spontaneous combustion and heating of large-scale coal, the AE event of coal spontaneous combustion developed from orderly to disorderly with the increase of temperature and then from disorder to order. It further shows the increasing level of acoustic emission count and the increasing acoustic emission signal during the warming process of coal spontaneous combustion. The acoustic emission events show a change process from order to disorder and then from disorder to order.

**Figure 12.** Variation of fractal dimension of AE counts with different m values.

The above trend of fractal dimension of AE count shows that there is a certain regularity in the sound emission count during the spontaneous combustion heating of coal. The fractal dimension level of the AE count near a specific temperature is continuously improved, which further indicates that the sound emission count level of the coal spontaneous combustion heating process is continuously enhanced, and the AE signal is continuously enhanced. The AE event shows a change process from order to disorder and then from disorder to order.

#### *4.3. Discussion*

In this paper, by constructing a large-scale AE test system, we tested the change law of the AE signal during the warming process of coal. The obtained data were Fourier transformed to systematically test the AE signal generated during the warming process of coal and comprehensively compare and analyze the intensity, energy scale, and timing characteristics of the AE signal, which showed a gradual increase as the experiment proceeded. At the beginning of the experiment, the coal did not obtain enough energy, and the thermal stress was not enough to destroy the coal itself, and only more small cracks were produced; at the middle of the experiment, the thermal stress of the coal itself was enough to destroy the weaker structure, and the internal pore system of the coal was reconstructed, and at this stage, the primary cracks and new cracks inside the coal sample expanded and developed so more strong acoustics. At the later stage of the experiment, the surface temperature of the coal sample was increasing, so the destruction of the pore structure and crack expansion were also increasing, and the integrity of the coal body was seriously damaged. This indicates that the AE signal can effectively respond to the damage and crack evolution of the coal body and can realize the monitoring and early warning of coal spontaneous combustion. The passive monitoring of AE, combined with the corresponding active detection of the fire source, can effectively improve the efficiency of coal spontaneous combustion monitoring and early warning and realize the joint detection of coal spontaneous combustion fire, which provides theoretical guidance for the subsequent research.

In order to study the AE signal to realize the monitoring and early warning of coal spontaneous combustion, this paper analyzes the time-frequency characteristics of the AE signal in the process of coal spontaneous combustion and the change law of the main frequency of the AE signal in the process of coal spontaneous combustion is obtained by Fourier transform calculation. In addition, the characteristic frequency spectrum is divided into four types, and the fitted curve of the main frequency average frequency of coal under different temperature conditions is analyzed and obtained, which can effectively reflect the phase change of the coal spontaneous combustion process by the AE signal. Chai and Ambrosio [47,48] studied the structural health detection of objects by the peak frequency and center of mass frequency of AE signal, and the research has been applied in nondestructive testing, welding, and microseismic, which can effectively respond to the changes of AE signals. Therefore, it is important to deeply explore the spectral characteristics of AE signal during coal spontaneous combustion through peak frequency and center-of-mass frequency to determine the correlation between temperature and AE signal during coal spontaneous combustion, and it is needed to further explore the correlation between damage rupture and AE signal during coal spontaneous combustion so as to realize the efficiency of coal spontaneous combustion monitoring and early warning.

The fractal dimension law of the pore structure of low-temperature nitrogen-adsorbed coal during the warming process and the fractal dimension law of AE counting the two fractal dimension laws are obvious; the fractal dimension law of the pore structure of low-temperature nitrogen-adsorbed coal is similar to the fractal dimension law of AE counting, and the two fractal dimension change trends are the same. In order to analyze the characteristics of pore structure changes during the warming process of coal synchronously, our team has characterized the pore structure changes at different temperatures by SEM experiments in the previous study. The integrity is good, and the pore structure is single under the room temperature condition (30 ◦C); after treatment at 100 ◦C, the integrity of the coal sample is slightly damaged, and the number of surface pores increases. At 200 ◦C, the integrity of the coal sample is damaged, and the number of surface pores increases. The integrity of the coal sample was damaged after treatment at 200 ◦C, and a large number of regional pore clusters appeared on the surface, and the pore size increased significantly compared with that at room temperature. At 300 ◦C, the integrity of the coal sample was severely damaged, and some of the fissures were connected with each other, and the connectivity was good. Based on this, the fractal dimension law of coal pore structure and

the fractal dimension law of AE counting were investigated by low-temperature nitrogen adsorption experiments. The results of the study showed that from 30 ◦C to 100 ◦C, the fractal dimension D1 of the pore structure of the coal body and the fractal dimension of the AE count showed a significant upward trend, combined with the analysis of the AE count and the cumulative count of AE, the coal body was heated and ruptured by heating during the heating process, and the nitrogen adsorption of the coal body continued to increase. From 100 ◦C to 300 ◦C, the fractal dimension D1 of the pore structure of the coal body and the fractal dimension of the AE count showed a downward trend, which was due to the thermal expansion phenomenon caused by the evaporation of a large amount of water in the coal body, resulting in the extrusion and closure of the pores of the coal body, and the growth rate of nitrogen adsorption decreased. Similarly, the fractal dimension D2 of the pore surface of the coal body has the same change trend at 30 ◦C–200 ◦C and 200 ◦C–300 ◦C, which further indicates that the pore structure of the coal body produces thermal rupture with the heating of the coal body. The thermal expansion phenomenon leads to the closure of some pores of the coal body. With the increase in temperature, the level of AE count continues to increase. The fractal dimension of the AE count rises first and then declines, indicating that with the increase in temperature, the AE signal continues to increase, and the AE signal shows a change law from order to disorder and then from disorder to order.

The experiment verifies the law that the pore structure of the coal body continues to generate and gradually increases during the heating process of the coal body. The fractal law of the low-temperature nitrogen adsorption curve and the fractal law of the AE count are analyzed, which further illustrates that the pore structure of the coal body in the process of heating up is the root cause of the AE signal of the coal body. The number of pores and the pore size of the coal body in spontaneous combustion and heating is increasing, which in turn leads to the continuous enhancement of the AE signal. Therefore, AE signals can be used for monitoring and early warning of coal spontaneous combustion.

### **5. Conclusions**


complexity increases, which in turn leads to the continuous enhancement of the AE signal. Therefore, the AE signal provides favorable conditions for monitoring and early warning of coal spontaneous combustion.

**Author Contributions:** Writing—original draft, J.Y., B.K. and L.S.; Writing—review & editing, Z.L. and W.L.; Formal analysis, X.P.; Supervision, Z.Z.; Data curation, L.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was jointly supported by the National Natural Science Foundation of China (51904172), the China Postdoctoral Science Foundation (2020M682209), the Natural Science Foundation of Shandong Province (ZR2019QEE041).

**Data Availability Statement:** The data used and/or analyzed during the current study are available from the corresponding author on reasonable request.

**Conflicts of Interest:** No potential conflict of interest was reported by the author(s).

## **References**


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