Numerical Simulation of Microwave-Induced Cracking of Coal Containing Pyrite Powder Based on a Multi-Field Coupling Model

Abstract

Microwave irradiation has become a potential technical method for coalbed degassing, which can effectively improve its permeability. A coupled electromagnetic—thermomechanical damage (ETMD) model was established to study the damage characteristics of coal containing pyrites. Under microwave irradiation, the temperature increase rate of pyrite was significantly higher compared to the coal matrix. The coal matrix was cracked by expansion stress in high-temperature spots. In the rotational heating mode, the coal matrix was easier to damage than under the static heating mode. The coal matrix damage efficiency was also related to the position of the sample under the static heating mode. Uniform centripetal pressure could inhibit coal matrix damage. Moreover, the pressure distribution affected the damage paths. Compared to no confining pressure, the percentage of areas damaged in coal was lower under low parallel pressure and higher under high parallel pressure.

1. Introduction

With remarkable economic growth, coal consumption in China accounted for over 50 percent of the country’s total energy consumption in 2022 [1]. However, in coal mining, there can be a series of gas accidents, such as gas explosions and coal gas outbursts. These accidents are mainly due to inherent characteristics (such as large coal reservoir stress, low permeability, and high gas content) of coalbeds in China [2]. Moreover, coalbed methane is a highly valuable and clean energy source. Therefore, degassing coal seams could be a double-win strategy for coal mines’ safety and fulfilling energy consumption [3].

The permeability of coal seams can be improved by fracturing coal, which can increase gas flow and enhance the effectiveness of gas drainage [4]. Currently, hydraulic fracturing [5], hydraulic slotting [6], and loose blasting [7] are the main technologies used to increase the permeability of coal seams. These measures can improve gas recovery from coal seams, but they all have limitations. In recent years, the physical field excitation method (such as heat injection, acoustic vibration, and electromagnetic field excitation) has the object of many studies. Formation microwave heating treatment (FMHT) has been proposed to artificially fracture coals and improve gas extraction efficiency. Compared to traditional treatment methods, the interaction between coal and microwaves has the advantages of high efficiency, low energy consumption, and time savings [8]. Microwave energy can selectively heat receptive components (such as pyrite and water) in the coal matrix, resulting in differential thermal stresses, inducing fractures within the coal matrix, and then improving coalbed permeability. A systemic understanding of microwave energy on the petrophysical characterization of coal is essential for FMHT’s application in the field.

Liu et al. [9] found that coal samples dehydrate and shrink after microwave irradiation, making them more brittle and easier to fracture. Pickles et al. [10] found that low-rank sub-bituminous coal demonstrated more extensive fracturing after microwave radiation according to scanning electron microscopy. Hong et al. [11] found that the crack propagation of coal cores after microwave irradiation depends on the receptive component distribution and original crack distribution, using camera images and the ultrasonic method. Li et al. [12,13] found that the initial cracks in the coal extended and gave rise to secondary cracks after microwave radiation by scanning electron microscopy. In addition, the microwave fracturing effect on dried coal/pyrite mixtures was studied by camera image, leading to the finding that fracturing enhances as the pyrite proportion increases. Marland et al. [14] discovered that moisture within the coal was rapidly heated and gasified by microwave energy. This resulted in volume expansion, which caused the extension of pre-existing fissures and the emergence of new fractures. Kumar et al. [15] validated the effect of isotropic stress on cleat frequency and the distribution of coal cores after microwave irradiation by micro-focused X-ray computed tomography. Yang et al. [16] discovered that the comminution of coal samples increased with microwave power and finally tended towards an upper limit.

The above research has confirmed that the pores and fractures of coal develop significantly under microwave radiation. However, the variation in the spatiotemporal electromagnetic field and crack propagation inside the coal cores during microwave radiation are difficult to accurately measure in laboratory experiments.

Numerical simulations have been used to study the effect of microwave radiation on rock cores. Discrete element-based models can simulate the initiation and propagation of cracks, but they are not good at calculating the distribution of the electromagnetic field [17,18,19,20,21,22,23]. Therefore, the finite-element software COMSOL 6.0 has been used to study rock fracturing induced by microwave radiation. Li et al. [24] investigated the microscale stress–strain variability in pegmatite specimens during microwave heating by a fully coupled thermomechanical model. Pressacco et al. [25,26] studied the effect of microwave radiation on rock fracture strength by a damage viscoelasticity model. Xu et al. [27] proposed a electromagnetic–thermomechanical damage coupled model to investigate damage development in rock specimens caused by non-uniform microwave heating.

These simulation studies contributed to the study of rock damage, but the effect of mineral and confining pressures on coal fracturing is still unclear. Pyrite, a typical associated mineral in coal, has abundant presence and uneven distribution within the coal matrix [28]. It has a higher microwave energy absorption capacity than the coal matrix, meaning that is easier to induce thermal stress and coal fracturing in this mineral. Therefore, it is important to investigate the effect of pyrite on the damage mechanisms of coal samples. As widely known, the coal seam is always affected by ground stress. It is important to study the effect of confining pressure on coal fracturing during microwave irradiation.

Therefore, this manuscript establishes a coupled electromagnetic–thermomechanical damage model (ETMD) using COMSOL link with MATLAB2021b to study the effect of pyrite and confining pressure on coal fracturing. The goal of this study is to acquire a comprehensive understanding of the interaction between coal and microwave irradiation. It offers a theoretical foundation for the engineering applications of microwave-assisted coal seam gas extraction through heat injection. In addition, this study compares the differences between static and rotating radiation modes in terms of electric field distribution, temperature distribution, and damage distribution. The result can provide theoretical references for practical engineering applications to improve permeability enhancement efficiency and save energy.

2. Numerical Methodology

2.1. Model Simplifications

Although differences between an actual case and a numerical model cannot be avoided, a simulation can more intuitively show the influences of some factors, such as electric field distributions inside samples and the process of coal cracking. In order to reduce computational time and ensure model convergence, it was necessary to simply the numerical model in a reasonable way. Therefore, five assumptions were made, as follows:

Assumption 1.

The coal in question was dried and free of moisture.

Assumption 2.

Coal is a linear elastic material at the microscale, and its deformations obey the small-deformation hypothesis.

Assumption 3.

The coal matrix is a heterogeneous material, and its mechanical parameters follow the Weibull distribution.

Assumption 4.

Coal mechanics obey the continuum damage mechanics, and its destruction follows the maximum tensile and Mohr–Coulomb criterion.

Assumption 5.

The dielectric constants of both the coal matrix and pyrite remain constant.

2.2. Governing Equations

A coupled electromagnetic–thermomechanical model with damage judgment (ETMD) was established in this study. The electromagnetic field was determined by solving the Helmholtz equation, expressed as follows [29]:

$$\nabla \times {\mu }_r^{-1}\left(\nabla \times E\right)-k_0^2\left({\epsilon }_r-{\displaystyle \frac{j\sigma }{\omega {\epsilon }_0}}\right)E=0$$

(1)

where $E$ is the electric field (V/m); ${\mu }_r$ is the relative magnetic permeability; $\omega$ is the angular frequency; $k_0$ is the wave number in vacuum; $j$ is the imaginary unit; $\sigma$ is the electrical conductivity (S/m); and ${\epsilon }_r$ is the relative permittivity. The expression of the latter was given as follows (F/m):

$$\begin{array}{c}{\epsilon }_r={\epsilon }^{\prime }-{\displaystyle \frac{j{\epsilon }^{″}}{{\epsilon }_0}}\end{array}$$

(2)

where ${\epsilon }^{\prime }$ is the real part of the permittivity (F/m); ${\epsilon }^{″}$ is the imaginary part of the permittivity (F/m); and ${\epsilon }_0$ is the vacuum permittivity (F/m).

The waveguide and cavity boundaries were assumed to be ideal conductors. The expression for the boundary condition was as follows:

$$\begin{array}{c}{E}_{tangential}=0\end{array}$$

(3)

The volumetric heating term for coal ($Q_e\left(t\right)$) was the dielectric loss, expressed as follows [30,31]:

$$\begin{array}{c}{Q}_e\left(t\right)={\displaystyle \frac{1}{2}}\omega {\epsilon }_0{\epsilon }^{″}{\left|E\right|}^2\end{array}$$

(4)

The temperature distribution was obtained by solving the heat transfer equation, expressed as follows [32,33]:

$$\begin{array}{c}\rho C_p{\displaystyle \frac{\partial T}{\partial t}}-\nabla \cdot \left(k\nabla T\right)=Q_e\left(t\right)\end{array}$$

(5)

where $t$ is the time; $T$ is the temperature (°C); $\rho$ is the density (kg/m³); $C_p$ is the specific heat capacity (J/kg/K); and $k$ is the heat transfer coefficient (W/(m·K)).

The boundary between air and the material was set as an adiabatic boundary, expressed as follows:

$$\begin{array}{c}-n\cdot q=0\end{array}$$

(6)

The above temperature distribution results were used to calculate the mechanical response of the coal body under thermal loading, and the elastic strain tensor expression was expressed as follows [24]:

$$E_{th}={\alpha }_T\left(T-T_{ref}\right)$$

(7)

where ${\alpha }_T$ is the thermal expansion coefficient (K/1); and $T_{ref}$ is the reference temperature for strain (zero-strain temperature) (°C).

Considering the thermal stress constitutive relationship, the elemental deformation control equation was formulated by combining the equations of mechanical equilibrium, geometric deformation, and constitutive behavior [34,35]. According to the law of the conservation of momentum in a homogeneous isotropic medium, the static equilibrium equation was expressed as follows:

$${\sigma }_{ij,j}+F_i=0$$

(8)

where ${\sigma }_{ij}$ and $F_i$ are the stress tensor (N/m²) and volume force, respectively (N/m²).

According to the theory of elasticity, the relationship between strain and displacement was as follows:

$${\epsilon }_{ij}={\displaystyle \frac{1}{2}}\left(u_{i,j}+u_{j,i}\right)$$

(9)

where ${\epsilon }_{ij}$ is the strain tensor; and $u_{i,j}$ and $u_{j,i}$ are the displacement components ($i,j=1, 2, 3$).

The constitutive equation of coal deformation was expressed as follows:

$${\sigma }_{ij }=2G{\epsilon }_{ij}+2G{\displaystyle \frac{v}{1-2v}}{\epsilon }_{kk}{\delta }_{ij}-K{\alpha }_{T}T{\delta }_{ij}$$

(10)

$$G=E/2\left(1+v\right)$$

(11)

$${\epsilon }_{kk}={\epsilon }_{11}+{\epsilon }_{22}+{\epsilon }_{33}$$

(12)

$$K=E/3\left(1-v\right)$$

(13)

where $G$ is the shear modulus of coal (Pa); $E$ is the elastic modulus of coal (Pa); $v$ is Poisson’s ratio; ${\epsilon }_{kk}$ is the volumetric strain; ${\delta }_{ij}$ is the Kronecker delta function, defined as 1 for $i=j$ and 0 for $i\ne j$; and $K$ is the bulk modulus of coal. Substituting Equations (8) and (9) into Equation (10), the Navier-type governing equation of coal deformation was expressed as follows:

$$G_{ui,jj}+{\displaystyle \frac{G}{1-2v}}{u}_{j,ji}-K{\alpha }_T{T}_i+F_i=0$$

(14)

The inhomogeneity of coal under thermal stress has a significant effect on its failure and damage. In general, many properties of the coal follow the Weibull distribution [36,37]. Therefore, the mechanical properties of coal could be defined as follows [38,39]:

$$f\left(u\right)={\displaystyle \frac{m}{u_0}}{\left({\displaystyle \frac{u}{u_0}}\right)}^{m-1}\exp \left(-{\left({\displaystyle \frac{u}{u_0}}\right)}^m\right)$$

(15)

where $u$ is the mechanical parameter of each unit; $u_0$ is the scale parameter associated with the material parameter mean value; and $m$ is the shape parameter, reflecting the degree of material homogeneity. A higher value of $m$ indicated that more unit parameters were concentrated near $u_0$, which meant that the numerical specimen was more uniform. This distribution function was employed to characterize the non-uniformity of the coal matrix’s mechanical parameters, including Young’s modulus, tensile strength, and compressive strength (Figure 1).

In this model, the maximum tensile criterion and the Mohr–Coulomb criterion were employed to assess coal damage [40]:

$$F_1={\sigma }_1-f_T$$

(16)

$$F_2=-{\sigma }_3-f_C+{\displaystyle \frac{1+\sin \theta }{1-\sin \theta }}{\sigma }_1$$

(17)

where F₁ and F₂ are functions of tensile (N) and shear stress (N) states, respectively.; $f_T$ and $f_C$ are the tensile (N) and compressive strengths (N) of the rock, respectively; ${\sigma }_1$ and ${\sigma }_3$ are the maximum and minimum principal stresses (N), respectively; and $\theta$ is the angle of internal friction (°). The maximum tensile criterion was initially applied to assess the units’ damage under any stress condition. The Mohr–Coulomb criterion was employed to assess shear damage only without tensile damage in the units.

The damage variable (D) described the extent of crack propagation and damage development in the units. If the maximum tensile criterion was satisfied, the damage variable reflected the following relationship [41]:

$$D=\left\{\begin{array}{ccc}0& F_1<0,& F_2<0\\ 1-{\left|{\displaystyle \frac{{\epsilon }_{\mathrm{t}0}}{{\epsilon }_3}}\right|}^{\mathrm{n}}& F_1=0,& d{F}_1>0\\ 1-{\left|{\displaystyle \frac{{\epsilon }_{\mathrm{c}0}}{{\epsilon }_1}}\right|}^{\mathrm{n}}& F_2=0,& d{F}_2>0\end{array}\right.$$

(18)

where ${\epsilon }_{t0}$ and ${\epsilon }_{c0}$ are the maximum tensile principal strain and the maximum compressive principal strain when tensile and shear damage occurs, respectively; n is a constitutive coefficient for coal strength, specified as 2.0; and $d{F}_1>0$ and $d{F}_2>0$ are two states that coal still under loading conditions after damage, respectively, which can cause an increase in the damage variables. When $F_1<0$ or $F_2<0$, the stress did not satisfy the maximum tensile and Mohr–Coulomb criteria.

According to the elastic damage theory, the modulus of elasticity of the unit decreased monotonically with the development of damage, and the modulus of elasticity of the damaged material was expressed as follows [41]:

$$E=\left(1-D\right)E_0$$

(19)

where $E_0$ and $E$ are the modulus of elasticity before and after damage, respectively. It was assumed that the damage and its progression were isotropic. In other word, $E$, $E_0$, and $D$ were all scalars.

The logic relationship in this model is shown in Figure 2. The electric field (red cycles in the electromagnetic field) of the coal sample was calculated using the Helmholtz equation. The volumetric heating term for coal (red cycle in the heat field) was calculated using the dielectric loss equation. With the temperature increasing (blue cycles in heating field), the coal sample thermally expanded, possibly causing the displacement component to change (blue cycle in mechanical field). Furthermore, the thermal stress (purple cycle in the mechanical field) caused by said thermal expansion resulted in coal damage, determined by the maximum tensile damage criterion and the Mohr–Coulomb criterion (purple circles on purple arrow). In turn, Young’s modulus was reduced, according to the degree of damage, resulting in reduced shear modulus and bulk modulus (green cycles in the mechanical field).

2.3. Geometry

As mentioned earlier, this model overlooked the influence of the intermediate principal stress on the assessment of material fracture. Moreover, as microwave simulation requires an extremely fine mesh division (one-tenth of the wavelength or less), this resulted in the number of grids being huge. Furthermore, the parameters were iteratively calculated after each time step. This resulted in a huge amount of computational workload. Additionally, the 3D model calculations encountered severe convergence problems and, therefore, could not be analyzed reliably. Therefore, the 2D model was selected as a more suitable and stable approach for characterizing failure behaviors. It produced satisfactory results in this study. The microwave oven model was developed with reference to the Institute of Applied Electromagnetic Research at Sichuan University in the literature [42]. The model consisted of three parts (Figure 3): a rectangular waveguide, a rectangular microwave cavity, and a coal sample model. The rectangular microwave cavity and waveguide were filled with air.

The average sulfur content of anthracite from Guizhou (No. 5 coal seam of the Upper Permian Longtan Formation in the Xingyi Coalfield, Guizhou Province) was 5.75%, with pyritic sulfur accounting for 16.13% to 89.32%. In addition, the coal seam contained numerous aggregated pyrite blocks with particle sizes greater than 0.1 mm [43]. Therefore, a coal body model was created referencing this seam. Pyrite was randomly distributed in the coal, with a percentage of 1% and a radius of 0.8 mm (Figure 4). The coal sample was heated at the positions of 0°, 90°, 180°, and 270° in the static heating model (SM). The coal sample was heated with a rotational speed of $\pi /3$ rad/s in the rotational heating model (RM). The rotation of the coal sample was achieved using a discrete method, which had previously been verified for accuracy [42]. The accuracy of the discrete method depended on the degree of each iteration and the accuracy of the assignment of grid nodes. The lower the degree set for each iteration, the higher the number of iterations required, and the greater the amount of calculation. In previous studies on the discrete method, when the iteration degree was below 30°, the calculation results were accurate. In this study, a more refined 3° was used. The continuously rotating circle was divided into 60 discrete points: for instance, at 3°, the coal sample changed position 60 times in 1 rotation. These models were labeled S₀M, S₉₀M, S₁₈₀M, S₂₇₀M, and RM, respectively. The coal sample remained stationary at one position and did not affect the electromagnetic field in the SM. In contrast, the coal sample maintained a uniform speed of rotation and continuously stirred the electromagnetic field in RM. The pyrites were labeled A_d~J_d respectively, with subscript “d” indicating the position of the corresponding coal sample in each SM (Figure 4). The subscript was “R” in RM. The distance from the center of the coal matrix (Figure 4) to the center of the circle of each pyrite particle was r_A~r_J (fluorescent blue dotted line in Figure 4). The values of r_A~r_J are presented in

Table 1. The values of r_A~r_J.

Distance	Unit	Value
rA	cm	1.78
rB	cm	1.10
rC	cm	2.11
rD	cm	1.15
rE	cm	0.22
rF	cm	1.91
rG	cm	1.38
rH	cm	2.12
rI	cm	1.24
rJ	cm	1.76

.

2.4. Material Parameters

The microwave power was 500 W with a frequency of 2.45 GHz and a total heating time of 15 s. The simulation data were stored at 0.1 s intervals. The microwave heating objects were pyrite and coal matrix. The type of coal matrix in this research was anthracite. As the damage condition of the coal matrix was the key research content of this study and its composition could influence the damage results, an industrial analysis was carried out, as detailed in

Table 2. Industrial analysis data of coal matrix.

Material	Ro,max (%)	Proximate Analysis (%)
		M, ad	A, d	V, daf	FC, ad
Coal matrix	3.21	1.30	27.12	22.65	48.93

. The electrothermal and mechanical parameters of the coal matrix and pyrite are presented in detail in

Table 3. Electrothermal parameters.

Material	Relative Dielectric Constant	Density (kg/m3)	Thermal Conductivity [W/(m·K)]	Constant Pressure Heat Capacity [J·(kg·K)−1]
Pyrite	25.66–6.83j	5018	20.50	600
Coal matrix	1.9–0.1j	1250	0.487	1250

and

Table 4. Mechanical parameters.

Material	Young’s Modulus (GPa)	Angle of Internal Friction (°)	Tensile Strength (MPa)	Compressive Strength (MPa)	Poisson’s Ratio	Thermal Expansion Coefficient
Pyrite	292	35	12	120	0.16	2.93 × 10−5
Coal matrix	3	38	0.6	10	0.3	2.4 × 10−5

. The above parameters were based on the experimental data of team studies and authoritative papers [13]. Variations in moisture content within coal can affect its mechanical properties. However, the moisture of the anthracite selected for this study was only 1.3%, indicating an extremely low moisture content, which had a minimal impact. Therefore, to simplify the model, the influence of moisture on the mechanical properties of coal was neglected. The results under the assumption that the coal used was dry primarily aimed to explore the direct effects of microwaves on coal, providing a foundation for more complex simulation studies in the future. Subsequent research will incorporate moisture factors to achieve more accurate simulation results.

2.5. Solution Strategy

The ETMD model was solved by COMSOL LINK MATLAB. The mechanical parameters of the coal sample were updated during the heating process. The detailed solution strategy was as follows (Figure 5):

(1) We created a geometric model and set the model parameters, boundary conditions, and initial conditions;

(2) The coupled equations of the EHMT were computed by a highly accurate full coupling algorithm to obtain the distribution data of the multi-physics fields;

(3) Based on the principal stress and principal strain data of the coal sample after the calculation step, the amount of damage was judged according to the maximum tensile damage criterion and the Mohr–Coulomb criterion;

(4) We updated the material parameters of the coal sample after damage and rotated the geometry model if it was a rotated model;

(5) The next time step was then calculated, and the above steps (2)~(4) were repeated until the heating time was reached.

2.6. Mesh Qualities

Normalized power absorption (NPA) is used to confirm mesh independence, which is the ratio of the average simulated dissipated power in the loss medium to the effective input power [33]. If the NPA remains constant with the increase in the number of elements, it means that further increasing the mesh number or mesh density does not significantly affect the simulation results. Therefore, the model was considered to have achieved mesh independence. To ensure accurate results, the mesh size should be smaller than a tenth of the microwave wavelength. Therefore, the maximum mesh size was 0.012 m.

The relation between the NPA of the S₀M and the number of elements is shown in Figure 6. When the number of elements was larger than 12,566, the NPA barely changed with the increase in the number of elements. The NPA values at element counts of 12,566, 13,895, 15,243, 18,061, and 20,699 were 0.19468, 0.19411, 0.19406, 0.19411, and 0.19411, respectively. Their percent difference error was much smaller than 2%. This indicated that the simulation results were independent of the number of elements. In this study, the number of elements was 18,061.

2.7. Model Validation

In our previous work, we experimentally validated the accuracy and reliability of the electromagnetic heating of coal [29]. In order to demonstrate the reliability of thermal damage, the ETMD model was verified by the thermal damage experiment of Ghassemi [44]. In this experiment, a rock sample was heated to 800 °C and held for 2 h. The uniaxial compression strength of the rock specimen was compared before and after heat treatment. The settings of the simulations for heat and uniaxial compression (Figure 7) were as follows.

The material parameters (

Table 5. Mechanical and thermal parameters.

Parameters	Value	Unit
Homogeneity index, m	10	-
Elastic modulus, E	37.6	GPa
Uniaxial compressive strength, $f_c$	183.2	MPa
Uniaxial tensile strength, $f_t$	22.9	MPa
Density, $\rho$	2760	kg/m3
Poisson’s ratio, $\nu$	0.25	-
Thermal conductivity, $k$	0.1	W/(m·k)
Specific heat capacity, Cp	700	J/(kg·K)
Thermal expansion coefficient, ${\alpha }_T$	2.0 × 10−6	1/K

) were as follows [44,45].

The stress–strain curves of rocks before and after thermal damage calculated from this model were largely consistent with the experiment. The peak strength and elasticity modulus of the rock were reduced after thermal damage (Figure 8). The overall relative deviation in the untreated group was 12.41%, with a maximum relative deviation of 14.68%. In the heating group, the overall relative deviation was 10.13%, with the maximum relative deviation reaching 68.03%. The maximum deviations were concentrated in the elastic deformation stage, mainly due to the stronger inhomogeneous distribution of the experimental coal samples compared to the simulated coal models. The stronger random heterogeneity of rocks led to more significant upward fluctuations in the stress curve during the elastic stage. In addition, there were differences in the strength reduction after thermal damage. This was due to the strength distributions of the rock samples used in the experimental fracturing being inherently different, while the strength distribution of the rock samples used in the simulation was consistent. The overall deviations were relatively small. These discrepancies were kept within reasonable bounds. This indicated that the present thermal damage model accurately calculated and derived the damage extent of the coal sample by the thermal damage process. Therefore, the thermal damage in the ETMD model was reasonable and reliable.

In a previous work, a coupled electromagnetic–thermomechanical (ETM) model without damage judgment was used to study coal damage by microwave heating [24,46]. In the ETM model, Von Mises stresses were used to speculate on the rock damage caused by microwave radiation [24,47]. When the Von Mises stress was larger than its yield stress, the rock was considered to experience plastic failure. New cracks would form and increase with the microwave heating time. The coal sample was heated in the 0° position for 15 s by the ETM and ETMD models. It was thus found that the Von Mises stresses were not uniformly distributed around each pyrite particle (Figure 9a) due to the non-uniform distribution of the electromagnetic field intensity (Figure 10 S₀M). The peak value of Von Mises stresses occurred mainly near pyrite A₀ and D₀. Moreover, the Von Mises stresses of pyrite B₀ and E₀ were also noticeable. Therefore, it was predictable for the crack to first form near pyrite A₀ and D₀ and then propagate to pyrite B₀ and E₀. The damage distribution of the coal matrix obtained by the ETMD model under the same heating condition (Figure 9b) was clearer than that of the ETM model. The damage was also concentrated around A₀, D₀, B₀, and E₀. This was highly similar to the ETM model. As such, the ETMD model was deemed reliable and superior.

3. Results and Discussion

3.1. Electric Field Distribution in SM and RM

The electric field distribution on the surface of the coal sample was completely different between SM and RM (Figure 10). The electric field distribution in S₀M remained stationary. In contrast, the electric field distribution in RM changed with the heating time. Furthermore, the maximum value of the electric field distribution changed dramatically with the heating time. In addition, the surface electric field distribution of the coal sample was not uniform under all positions. This also caused a significant difference in the local dielectric loss density of the coal sample (Figure 11). The dielectric loss power density of pyrite was significantly higher than that of the coal matrix under similar electric field strengths. This was caused by the difference in their dielectric properties. As a result, the microwave energy was primarily absorbed by the pyrite powder.

3.2. Temperature and Damage Distribution in SM and RM

The temperature curve of S₀M was smooth, but the curve of RM fluctuated significantly (Figure 12). The temperature and heating time had a linear relationship in S₀M due to the constant electric field distribution. In contrast, the electric field distribution of RM varied significantly with the heating time, which resulted in the temperature being affected by the heating time and the electric field distribution. This led to fluctuation in the temperature increase rate. Briefly, S₀M presented more stable microwave energy adsorption.

At 15 s, the maximum temperature of S₀M was close to RM, but the minimum temperature of S₀M was significantly smaller than RM. In other words, the difference between the maximum and minimum temperatures in RM was smaller than in S₀M. Moreover, the average temperature in RM was larger than that in S₀M. This indicates that the temperature in RM was more uniform than that in S₀M. The electric field strength of each pyrite particle in S₀M was different, but it remained constant. This resulted in a linear relation between the temperature of the pyrite particles and the heating time in S₀M (solid line in Figure 12). In contrast, the electric field strength of each pyrite particle in RM varied with the heating time. This resulted in the temperature of pyrite particles fluctuating with the heating time (dotted line in Figure 12). Therefore, there was a significant disparity in the heating efficiency of pyrites in different models, which caused the uneven temperature distribution related to the heating models. Furthermore, the maximum-temperature pyrite in S₀M was always D₀, and the minimum-temperature pyrite was always C₀. Both the maximum- and minimum-temperature pyrites in RM changed over time (

Table 6. Maximum- and minimum-temperature pyrites in S₀M and RM.

Time	Maximum-Temperature Pyrite in S0M	Minimum-Temperature Pyrite in S0M	Maximum-Temperature Pyrite in RM	Minimum-Temperature Pyrite in RM
3	D0	C0	HR	JR
6	D0	C0	AR	GR
9	D0	C0	HR	GR
12	D0	C0	AR	GR
15	D0	C0	HR	GR

). This was because D₀ in S₀M was always in the hot spot of the electromagnetic field and C₀ was always in the cold spot of the electromagnetic field. However, the strength of the electromagnetic field of each pyrite particle in RM was constantly changing due to the rotation of the coal.

The significant difference in the temperature of pyrite in each static model at 15 s was due to the different electric field strengths of the coal sample at different positions (Figure 13). The average temperature in S₁₈₀M (93.32 °C) was 3.16 times higher than in S₉₀M (29.50 °C). Moreover, the minimum temperatures were all close to the initial temperature (20°), indicating some pyrite particles receiving very little microwave energy in SM. Among them, the minimum temperature in S₁₈₀M was higher than in other SMs, which was related to the distribution of the electric field. In S₁₈₀M, the relatively strong electric field covered all the pyrites, making the minimum electric field strength relatively high.

The maximum temperature in RM was smaller than in S₀M and S₁₈₀M. However, the minimum temperature of pyrites in RM was significantly higher than that in SM. Furthermore, the difference between the maximum and minimum temperatures in RM was the smallest out of all the models. So, the temperature distribution in RM was more homogeneous than in other models. This may indicate that rotary heating improves heating uniformity and effectively enhances microwave energy utilization.

Coal damage in S₀M gradually increased during the heating process (Figure 14). The evolution of damage in the coal matrix was closely correlated with the temperature distribution. At 3 s, a small amount of damage occurred around the three hottest pyrite particles (A₀, D₀, and E₀). The pyrites had higher temperatures and thermal expansion coefficients than the coal matrix. This resulted in the coal matrix suffering radial pressure and, thus, experiencing tensile damage, which is consistent with the work by Ali [19] and Wang et al. [48]. At 6 s, the damage areas around A₀, D₀, and E₀ extended towards the boundaries and other pyrite particles with the temperature increasing. The damage area was located around B₀ (temperature slightly lower than E₀) and was connected to E₀. Furthermore, the damaged areas around these four pyrite particles were interconnected with one other. This promoted the connectivity of pores and improved the ability of gas seepage. In addition, the stress concentrated on the ends of the fissure, easily extending along the ends until a connection was established. Then, a fracture network was formed and able to transport gas.

The temperature of the pyrite particles in RM was more uniform than in other models. There were more pyrite particles experiencing a rapid temperature increase and expansion, resulting in more damage to the coal matrix surrounding the pyrites (Figure 15). At 3 s, damage occurred around all pyrite particles, but the degree of damage varied. However, the damaged areas around A_R and D_R were not as serious as A₀ and D₀. The damaged areas were connected to the heating time. Finally, the damaged area connected all pyrite particles to form a fracture network, which could be used to promote gas seepage.

The temperature and damage distribution in S₀M, S₉₀M, S₁₈₀M, S₂₇₀M, and RM were significantly different at 15 s (Figure 16). The temperature of all pyrites in S₉₀M and S₂₇₀M was the lowest, which resulted in only one pyrite (H₉₀, H₂₇₀) incurring a small amount of damage. Although the average temperature of H₉₀ (42.23 °C) was lower than A₉₀ (45.85 °C) in S₉₀M, the area around A₉₀ remained undamaged. This might have been due to H₉₀ being closer to the boundary. The coal matrix near the boundary was less constrained, which resulted in the release of stress and made fissure formation easier. In S₁₈₀M, the temperature of five pyrite particles (A₁₈₀, D₁₈₀, F₁₈₀, G₁₈₀, and J₁₈₀) was high enough to damage the coal matrix. In RM, damage occurred near all pyrites. The fracture network was more abundant than in all other SMs. Considering these results, rotational heating was more conducive to the formation of fracture networks. In other words, rotational heating can be beneficial for promoting the formation of gas flow channels.

3.3. Damage Efficiency in SM and RM

In general, a unit can be regarded as a damaged area when its damage variable is higher than or equal to 0.2 [47]. In this study, the percentage of damaged area to the entire coal matrix area (PDA) was used to characterize the damage efficiency. The PDA values in S₀M, S₉₀M, S₁₈₀M, S₂₇₀M, and RM increased with the heating time (Figure 17). In general, the PDA values in RM were larger than in other models. This is consistent with the damage distribution shown in Figure 16. Interestingly, the PDA value nonlinearly increased with the heating time. This was because the distribution of the coal matrix’s mechanical parameters was non-uniform.

At 15 s, the PDA values were 7.49%, 0.14%, 12.83%, 0.16%, and 14.54% for S₀M, S₉₀M, S₁₈₀M, S₂₇₀M, and RM, respectively. From this point of view, S₀M, S₁₈₀M, and RM could be considered models with a high damage efficiency. Therefore, damage efficiency highly relied on the distribution of the electric field in the cavity, which depended on the heating model. In conclusion, the damage efficiency in RM was higher than in SM.

Parameter ΔD was used to describe the rate of the PDA value, which was defined as the increment in PDA value at 0.5 s intervals. The relation between the ΔD of models with a high damage efficiency and the heating time is shown in Figure 18. Moreover, the periods during which the damage area connected one pyrite particle to another pyrite particle, identified as the peak periods of damage area connectivity (PDACs), are shown via the pink histogram in Figure 18.

The PDAC durations for S₀M, S₁₈₀M, and RM ranged from 2.5 to 5.0 s, 1.5 to 5.0 s, and 3.0 to 7.5 s, respectively. The start time of the PDAC in RM was later than in the other two models. This was because the electric field strength was static in SM when the electric field in certain areas was large, which made it easier to heat and damage locally. In contrast, the electric field strength in RM changed with the heating time. Therefore, there were no constant hot spots in RM, which resulted in a notably rapid increase in pyrite temperature and induced the late-starting PDACs. The PDACs in RM lasted longer compared to S₀M and S₁₈₀M, which could be attributed to the higher number of pyrite particles involved in the damage in RM. This resulted in more damaged areas that were interconnected. In addition, the electric field strength of the coal sample was very low in SMs, such as in S₉₀ and S₂₇₀, which led to the slow development of damage and did not take PDAC into account.

3.4. Effect of Confining Pressure

Models of uniform centripetal pressure (UCP) and some parallel pressures were established to investigate the influence of confining pressure on damage development (Figure 19). In UCP, the coal was subjected to uniform centripetal pressure p_C. Models of parallel pressures included models of uniform pressures (uniform X-axis pressure (UXP), uniform Y-axis pressure (UYP)) and non-uniform pressures (two non-uniform X-axis pressures (NXPs) and two non-uniform Y-axis pressures (NYPs)). In NXP (a) and NYP (a), the maximum pressure (p_max) pointed to the center of the coal and decreased linearly along both sides to 0 MPa at the end. In NXP (b) and NYP (b), the pressure pointing to the center of the coal was 0 MPa and increased linearly along both sides to p_max at the end (

Table 7. Confining pressure table.

Model	Pressure Direction	Maximum Pressure Point	Minimum Pressure Point
UCP	Center	Uniform	Uniform
UXP	X-axis	Uniform	Uniform
UYP	Y-axis	Uniform	Uniform
NXP (a)	X-axis	Center	Sides
NXP (b)	X-axis	Sides	Center
NYP (a)	Y-axis	Center	Sides
NYP (b)	Y-axis	Sides	Center

).

The damage paths of S₀M and RM decreased with increasing p_C (0~1.8 MPa) by 15 s microwave irradiation under UCP (Figure 20). Tensile and shear stresses in the coal matrix were opposed by p_C, which limited the formation and development of damage. Under 1 MPa, all damage in the central region of the coal sample disappeared, leaving only the boundary still experiencing damage caused by close pyrites (A₀, A_R, C_R, and H_R). Under 1.4 MPa, the damaged area near A₀ was reduced, and the damaged area near A_R disappeared. This was due to the average temperature of A₀ (122.45 °C) being higher than A_R (104.64 °C), resulting in higher thermal stress, making the damage caused by A₀ more difficult to inhibit. Furthermore, although the average temperature of C_R (96.70 °C) was lower than that of A_R (104.64 °C), C_R was closer to the boundary, resulting in the coal matrix between it and the boundary still being damaged. Consequently, the damage caused by pyrites in some hot spots in static models was more difficult to inhibit by p_C compared to RM, and inhibition by p_C was more effective in relation to internal damage in the coal sample.

The damage percentage in S₀M and RM decreased with increasing p_C in UCP (Figure 21). The percentage of areas damaged in S₀M decreased from 6.54% without pressure to 0.31% under 1.8 MPa, representing a 95.25% percentage reduction. Similarly, the percentage of damaged areas in RM decreased from 13.97% to 0.38, representing a 97.30% percentage reduction. Therefore, p_C significantly inhibited the formation and development of damage, which was consistent with a previous study [15], in which increments in cuttings and crack volumes before and after microwave radiation of coal cores under isotropic stress were significantly lower than those under no stress.

The intermediate pressure (p_mid) was p_X and p_Y in the uniform parallel pressure model, while the p_mid was p_max/2 in the non-uniform parallel pressure model. When p_Mid increased to a certain level, the stress balance of the coal was disrupted due to damage, leading to collapse and the termination of the solution process. When comparing the stress models of the X-axis and Y-axis under the same pressure distribution, a significant difference in collapse intermediate pressure and time was observed (Figure 22). Under non-uniform pressure (A), the collapse intermediate pressure was the same, but there were slight differences in the collapse time. Additionally, under both uniform and non-uniform pressure (B), the Y-axis stress model exhibited significantly higher collapse intermediate pressure at collapse and a shorter collapse time. This was closely related to the heterogeneity of the coal body, including the distribution of pyrite and the Young modulus of the coal matrix.

In comparison to the model with no confined pressure, there were more damage paths along the X-axis and fewer along the Y-axis under UXP, while there were more damage paths along the Y-axis and fewer along the X-axis under UYP (Figure 23). The reason is that, after applying a parallel pressure, the coal matrix experiences compressive forces in the direction of the pressure. In other words, it acquires tensile stress resistance perpendicular to the pressure direction, which inhibits the development of damage paths along this direction. Therefore, the damage of the coal matrix caused by the thermal expansion of pyrites is more likely to occur in the direction parallel to the pressure, and the damage is prone to develop along the same direction. Compared to UXP, the damage fracture F_X1 disappeared in NXP (a), while F_X2 developed more extensively. In addition, in comparison to UYP, F_Y2 and F_Y3 disappeared in NYP (a), while F_Y1 developed more extensively. Conversely, in NYP (b), F_Y1 disappeared, and F_Y2 and F_Y3 developed more extensively. This was because of the different pressure distributions. In NXP (a) and NYP (a), the pressure mainly concentrated in the central part of the pressure direction, exerting a stronger inhibitory effect on damage in the central region and a weaker inhibitory effect on damage on both sides. In contrast, in NYP (b), the pressure concentrated on both sides, leading to a stronger inhibitory effect on damage on both sides.

As shown in Figure 24, the percentage of damaged area in the parallel pressure models under 0.2 MPa was lower than under no confining pressure (0 MPa). This indicated that the inhibitory effect on damage development was greater than the promoting effect. After the pressure increased to 0.4 MPa, the changes in the percentage of damaged areas under NXP, UYP, and NYP (a) were not obvious, while NYP (b) showed a significant increase due to localized crushing on both sides of the pressure direction. Under 0.6 MPa, NXP, UYP, and NYP (a) all exhibited significant increases in the percentage of damaged areas, surpassing the values recorded at 0 MPa. Under 0.8 MPa, the percentage of damaged areas in UYP further increased due to the local boundaries of the coal being crushed by a high pressure. The above observations indicate that, at low pressure levels, parallel pressure has a greater inhibitory effect on damage development, while, when the pressure increases to a certain level, the promoting effect on damage development becomes stronger.

4. Applications of Microwave Heating in Coal Gas Drainage

Microwave energy can be used to heat a coal seam through waveguides and antennas. Firstly, a borehole is drilled into the coal seam from the floor roadway. Secondly, the waveguide is connected to the antenna and inserted into the borehole along with the extraction pipe. A Teflon protective tube is installed between the antenna and the borehole wall. Finally, the borehole is sealed (Figure 25a) [11,15]. Gas extraction is conducted after the microwave generator is turned on. The microwaves pass sequentially through the rectangular waveguide, waveguide converter, coaxial waveguide, and antenna. The antenna radiates the microwaves into the coal seam and heats it.

Under microwave irradiation, pyrite heats up rapidly and conducts heat to the coal matrix. The increased coal temperature results in gas desorption. Simultaneously, the expansion of the hot area induces coal matrix fracturing, which significantly increases the coal’s porosity and permeability. This greatly enhances gas extraction.

According to this study, the rotating heating model can cause a more rapid temperature increase than the static heating model within a given time, resulting in more extensive damage to the coal matrix. Therefore, rotary joints are added to adjust the rotation speed (Figure 25b). By rotating the rectangular waveguide, the electromagnetic field changes with time. Compared to the static radiation method, this disturbance radiation method can potentially enhance the efficiency of coal seam gas desorption.

5. Conclusions

The ETMD model was used to investigate temperature changes and damage evolution in a coal sample (non-homogeneous coal matrix containing pyrite) under different microwave radiation modes, as well as the influence of the confining pressures on the damage effect. The main findings were the following:

(1) The electric field distributions of the coal sample vary at different positions in the microwave cavity, leading to different dielectric loss power densities of pyrite, which causes significant differences in temperature distribution. The constant dielectric loss density of each pyrite in the static heating mode results in the temperature between them being significantly different. The coal sample moves and changes position in the rotating heating mode, causing perturbations in the electromagnetic wave, which enhances the uniformity of the pyrites’ temperature and leads to a higher microwave-absorbed power.

(2) The damage of the coal matrix is closely linked to the pyrite temperature. Under the expansion stress of high-temperature pyrite, the surrounding coal matrix experiences damage. The damaged area extending outwards is centered around the high-temperature pyrite and tends to connect to the boundary and other nearby damaged areas. In the rotational heating mode, more pyrites capable of damaging the coal matrix are developed compared to the static heating mode, resulting in more damage propagation paths. Some specific pyrites in the hot spot of the electric field can achieve higher localized damage efficiency under static heating.

(3) The damage efficiency at different positions in the static heating mode is significantly different, and the damage area of S₁₈₀M is 91 times that of S₉₀M after 15 s of microwave heating. The rotational heating mode can achieve a higher damage efficiency and a wider damage coverage. The peak damage connectivity recorded under a higher electric field strength is faster than under other conditions, and the connectivity duration is shorter.

(4) The occurrence and development of coal matrix damage under microwave radiation are inhibited by a uniform centripetal pressure, and greater inhibition is caused by a higher pressure. Under a certain pressure, static heating with hot spots with a high electric field can achieve more stable damage in local hot spots, while rotational heating can better ensure the occurrence of damage in vulnerable regions.

(5) The intimidating pressure and heating time at which coal collapses are closely related to its heterogeneity. In addition, parallel pressure promotes damage paths parallel to the direction of pressure and inhibits damage paths perpendicular to the pressure direction. Compared to no confining pressure, the percentage of areas damaged in coal is lower under a low parallel pressure and higher under a high parallel pressure. Furthermore, under a non-uniform parallel pressure, the influence of high-pressure regions on the development of damage is more pronounced.

(6) The use of the rotating microwave radiation mode can cause more damage to the coal sample than the static microwave radiation mode. This can increase the speed of the seepage of gas to a greater extent, making it more convenient for gas extraction. The rotating microwave-irradiated coal seam device is thus designed for field applications. This research provides an important reference for the future development of microwave devices for irradiating coal seams.