**1. Introduction**

In the process of coal and gas outbursts, the direct damage to underground personnel and equipment in coal mines is the impact of airflow and pulverized coal that rushes with high-speed movement into the working space [1–3]. The damage of the impact airflow can be divided into high-pressure airflow shock wave overpressure damage and suffocation damage, among which the shock wave damage is the most direct and fastest [4–6]. The former Soviet Union scientist Savonko [7] studied the influence of roadway section reduction and expansion on the pressure of outburst shock waves and obtained the attenuation coefficient of air shock waves during movement. Zhang [8] pointed out that the outburst shock wave belongs to the weak shock wave and constructed the relationship between the propagation attenuation of the outburst shock wave and the initial energy, propagation distance and friction resistance of the roadway. Sun et al. [9] studied the phenomenon of gas accumulation in the roadway after the outburst airflow by using the outburst test device. According to the theory of aerodynamics, Cheng et al. [10,11] established the mathematical model of motion and dynamics of outburst shock wave propagation, and established the relationship between shock wave overpressure, impact airflow velocity and propagation distance, coal seam gas pressure, etc. Based on theoretical analysis and numerical simulation, Tang [12] studied the influence of gas pressure on outburst energy, outburst intensity and gas emission. Miao [13] studied the expansion characteristics of the

**Citation:** Sun, D.; Cao, J.; Dai, L.; Li, R.; Liu, Y. Investigation of Formation Process and Intensity of Coal and Gas Outburst Shockwave. *Processes* **2023**, *11*, 659. https://doi.org/10.3390/ pr11030659

Academic Editors: Feng Du, Aitao Zhou and Bo Li

Received: 9 February 2023 Revised: 21 February 2023 Accepted: 21 February 2023 Published: 22 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

outburst impact airflow and established the relationship between the expansion characteristics and the expansion failure strength. Hu [14] conducted a numerical study on the impact dynamic effect of the impact airflow on the road header and the outburst prevention door. Zhou et al. [15–17] studied the formation and propagation process of outburst shock waves and gas flow in different types of roadways. They analyzed the best roadway form to prevent the spread of outburst shock, studied the influence of gas outburst shock waves on airflow disturbance in mine roadways, and pointed out that the airflow disturbance caused by outburst shock waves was mainly controlled by shock wave overpressure and main fan wind pressure.

Although there have been many studies on the formation and development of outburst shock waves, most of them rely on physical experiments or numerical simulations. Unfortunately, the theoretical analysis of the underlying mechanism is not as comprehensive. Many of the expressions used in these studies are based on aerodynamic theory alone and do not take into account the effects of outburst pulverized coal on a shock wave's formation. It is difficult to reveal the theoretical mechanism in the formation and development of outburst shock waves.

#### **2. The Relationship between Outburst Shock Wave and General Shock Wave**

#### *2.1. The Formation of General Shock Waves*

In gases, liquids or solids, if the pressure, density and temperature change (nonlinear change) on a cross section, there is a shock wave inside the material [18–21]. The reason is generally that there is a wave source in the object whose moving speed exceeds the wave velocity. Shock waves are very common in nature and human society and may be called differently in different situations. For example, the shock wave formed during an explosion can be an explosive wave. The shock wave generated by a flying object when the propagation speed in the air exceeds the speed of sound is generally called sonic explosion. The water hammer effect formed by the sudden closure of water flow in a hard water pipe is also a shock wave. In aerodynamics, a strong shock wave is generally called a shock wave.

The formation process of the shock wave is shown in Figure 1. There is an ideal gas with pressure *p*1, density *ρ*<sup>1</sup> and temperature *T*<sup>1</sup> in a straight round tube. There is a piston on the left side of the ideal gas, and the piston and gas are static at the initial time, as shown in Figure 1a. In the parameter curve, only the change of pressure *p* is shown to represent the change of the gas state, so as to avoid repeating. In a very short period of time Δ*t* , the piston starts moving to the right at speed Δ*t* and squeezing the gas in the tube, as shown in Figure 1b. A weak compression wave is formed at the intersection of the gas and the tube, and the compression wave head *A*1-*A*<sup>1</sup> pushes to the right at the sound velocity *c*1. The sound velocity *c*<sup>1</sup> is the local sound velocity of stationary gas in the round tube. With the wave head as the boundary, the air parameters on the right side are the initial state in the tube, and the air parameters on the left side are pressure *p*<sup>1</sup> + Δ*p* , density *ρ*<sup>1</sup> + Δ*ρ* and temperature *T*<sup>1</sup> + Δ*T* . In addition, the velocity increases from 0 to Δ*v* , and the increments of the four parameters are all positive. The piston continues to move forward, and the piston speed reaches Δ*t*" after time Δ*v*". The gas behind the wave head formed in the previous stage is further compressed, and a new compression wave will be formed behind it. The compressed wave head *A*2-*A*<sup>2</sup> pushes the gas to the right at the sound velocity *c*<sup>1</sup> relative to wave *A*1-*A*1, as shown in Figure 1c. The sound velocity *c*<sup>2</sup> is the local sound velocity of the gas behind *A*2-*A*<sup>2</sup> in the round roadway. The absolute velocity of *A*2-*A*<sup>2</sup> is *c*1 *+* Δ*v* . With the wave head *A*2-*A*<sup>2</sup> as the boundary, the air parameters on the right side are pressure *p*<sup>1</sup> + Δ*p* , density *ρ*<sup>1</sup> + Δ*ρ* and temperature *T*<sup>1</sup> + Δ*T* . The air parameters on the left side are pressure *p*<sup>1</sup> + Δ*p*", density *ρ*<sup>1</sup> + Δ*ρ*" and temperature *T*<sup>1</sup> + Δ*T*". The velocity is increased from Δ*v* to Δ*v*". At this time, there are two wave heads, *A*1-*A*<sup>1</sup> and *A*2-*A*2, in the round roadway. As time goes on, the piston will form a series of wave heads in turn as it moves to the right.

(**c**) t = *̇TȞ*

**Figure 1.** Schematic diagram of shock wave formation.

The formula of sound velocity in gas is:

$$a = \sqrt{kRT} \tag{1}$$

In the formula, *k* is the specific heat ratio of air, *T* is the thermodynamic temperature, *R* is the universal gas constant of air and the universal gas constant is *R* = 8.31 J/(K·mol). The air-specific heat ratio is also constant *k* = 1.4 in non-extreme cases. It is easy to see that the sound velocity is only related to the gas temperature. The sound velocity is different at different temperatures in the flow field.

The moving speed of *A*1-*A*<sup>1</sup> is:

$$v\_1 = c\_1 = \sqrt{kRT\_1} \tag{2}$$

The moving speed of *A*2-*A*<sup>2</sup> is:

$$v\_2 = c\_2 + \Delta v = \sqrt{kR(T\_1 + \Delta T')} + \Delta v \tag{3}$$

Obviously, the wave head *A*2-*A*<sup>2</sup> will eventually catch up with the wave head *A*1-*A*<sup>1</sup> and merge with it after a certain period of time. By analogy, *A*3-*A*<sup>3</sup> and *A*4-*A*<sup>4</sup> after the wave head *A*2-*A*<sup>2</sup> will catch up and merge with the initial wave head *A*1-*A*1. Since the air compression wave is also a mechanical wave, the strength of the compression wave obeys the superposition principle, and the strength of the wave at the compression wave is strengthened. Due to the continuous advancement of the piston, new wave heads are continuously generated, and the compression wave is continuously strengthened. Finally, a compression wave with great intensity is formed, which is generally called a shock

wave. The continuously strengthened wave head is called the wave front. After the shock wave front passes through, the gas parameters change from the initial *p*1, *ρ*<sup>1</sup> and *T*<sup>1</sup> to *p*2, *ρ*<sup>2</sup> and *T*<sup>2</sup> (the values are calculated in the later chapters), and the gas velocity behind it changes from 0 to *vf*, which is called the accompanying velocity. It should be emphasized that the adjoint velocity is the velocity of the gas after the wave, and the velocity of the shock wave front is the propagation velocity of the mechanical wave. The two are completely independent concepts. The shock wave front velocity is greater than the accompanying velocity.

In summary, the shock wave is a strong disturbance wave. Its propagation velocity and wave front velocity are greater than the sound velocity. The gas state parameters of the shock wave change abruptly (pressure, temperature and density are all increased). The so-called mutation is because the gas state parameters have a large parameter gradient at the wave front and are macroscopically discontinuous. The greater the parameter gradient is, the more obvious the viscous dissipation effect of the gas is at the time of sudden change. The result is that the gas inertial force and viscous dissipation reach equilibrium, as shown in Figure 2a. The spatial thickness involved in this equilibrium stage is several molecular average free paths. The calculation formula of the average free path *λ* of gas molecules:

(**a**) The actual shock wave front (**b**) Simplified shock wave fronts

**Figure 2.** Wave front of a shockwave.

In the formula, *<sup>k</sup>* is the Boltzmann constant, *<sup>k</sup>* = 1.38 × <sup>10</sup>−<sup>23</sup> J/K; *<sup>T</sup>* is the absolute temperature, K, in the standard state, *T* = 273.15 K; *d*<sup>0</sup> is the molecular effective diameter, m; the effective average diameter of air molecules is 3.7 × <sup>10</sup>−<sup>10</sup> m; *<sup>p</sup>* is gas pressure, Pa; the atmospheric pressure is 1 × <sup>10</sup><sup>5</sup> Pa; and the average free path of air molecules in the standard state is 5.9 × <sup>10</sup>−<sup>8</sup> m. This length can be ignored on the macro scale. Therefore, the shock wave front can be completely described as a vertical plane, as shown in Figure 2b.

The shock wave front discussed above is perpendicular to the propagation direction of the wave front, which is called a normal shock wave. Because the axis direction of the roadway is perpendicular to the shock wave of a coal and gas outburst in the roadway, which is similar to the normal shock wave, the shock waves described in this paper are all normal shock waves.

#### *2.2. Coal and Gas Outburst Formation Characteristics*

*x*

After the formation of underground production and ventilation systems in a coal mine, although there are factors such as altitude and ventilation system, compared with the gas pressure in the coal seam (several atmospheric pressures), a strong gas pressure gradient is formed between the air pressure in the roadway and the gas pressure in the coal seam. Under certain combination conditions of gas pressure, ground stress and physical and mechanical properties of coal, a coal and gas outburst will occur.

The simplified coal and gas outburst scene is shown in Figure 3. Under the action of ground stress and gas pressure, the outburst is excited, and a large number of broken pulverized coal and gas migrate from the coal seam to the roadway at a high speed. From left to right, it can be divided into the stable coal seam area, coal-gas flow area, air compression area and roadway unaffected area. The location of these areas will change with the change of the outburst influence stage, and the interior of each area is also uneven (except for the temporarily unaffected area of the roadway).

**Figure 3.** Schematic diagram of a coal-gas outburst.

After the failure of the weak layer at the outburst mouth disappears, the pulverized coal-gas in the high-pressure space of the coal seam are released and freely migrate to the roadway space. The coal-gas flow will first form in the coal seam. The initial coal-gas flow first compresses the roadway air at the outburst port, and as the coal-gas flow moves, its interface with the air continues to move forward (right side). The compressed roadway air moves forward and compresses the more forward roadway air. The front of the compressed roadway air is called the outburst wave front. The range between the wave front and the coal-gas flow is the air compression zone. The roadway in front of the wave front has not been disturbed by the outburst, which is called the temporarily unaffected area. The air parameters in the area are stable and maintain initial parameters. The moving speed of the wave front is the development speed of the influence range of a coal and gas outburst. From the above discussion, it can be known that the forefront of the impact of a coal and gas outburst is the air compression zone, not coal-gas flow.

#### **3. Coal and Gas Outburst Shock Wave Theory**

#### *3.1. Overview of Shock Tube*

Shock wave is an important research topic in many fields of nature and laboratory. Its shock wave theory plays an important role in aerospace, aviation, explosion engineering and other fields. A shock tube can form stable and controllable shock waves in laboratory research, which is an important instrument for shock wave-related research. The shock wave process formed in coal and gas outbursts is highly similar to the shock wave process formed by a shock tube. Therefore, this section takes shock wave theory as the starting point to study the shock waves of coal and gas outbursts.

The world's first shock tube was born in France in 1861. The chemist P. Vieille used it to obtain a moving shock wave with a velocity of 600 m/s in the process of studying the detonation problem in combustion. With the development of shock tube technology and the increase in social development needs, a shock tube is of great significance in many fields. In addition to the basic theories of physics and chemistry, it has been widely used in the fields of electromagnetic fluid mechanics, pneumatic lasers, anti-explosion processes and combustion [22–26]. The shock tube is widely used. The main reason is that the shock tube itself has many advantages, such as a simple structure and strong controllability of shock wave parameters.

The shock tube in the laboratory can be in a variety of forms, but its core basic forms are consistent. The basic shock tube form is called the shock tube later. The shape of this shock tube is a straight tube with an equal cross-section, which is divided into two parts: a high-pressure section and low-pressure section, which are isolated by a bursting disc, as shown in Figure 4. When a shock wave is needed, the bursting disc breaks and the high-pressure gas in the high-pressure section suddenly rushes into the low-pressure section to form a shock wave. By changing the gas species in the high- and low-pressure sections of the shock tube and the gas pressure in the high- and low-pressure sections, the shock tube can form different shock waves [27–30].

**Figure 4.** Initial state of the shock tube.

In the shock tube experiment, when the pressure difference on both sides of the bursting disc exceeds its critical value, the bursting disc ruptures. Due to the huge pressure difference on the two sides of the bursting disc, an incident shock wave is generated from the bursting disc position to the right side, and a sparse wave is generated to the left side, as shown in Figure 5. In order to facilitate the later description, the original high-pressure section of the shock tube is called zone 4, and the low-pressure section is called zone 1. When the incident shock wave is formed, its power is the high-pressure gas on the left side, but after the incident shock wave begins to move, it begins to cause a sudden change in the parameters of the low-pressure section at the junction with it. The shock wave front is transferred to the original low-pressure section gas, and the gas parameters (including pressure) of the low-pressure section of this part are surged to form zone 2. With the passage of time, zone 2 gradually moves to zone 1, and the moving speed is the shock wave front speed. The influence range of the sparse wave moving to the left is called zone 3, and the gas parameters (including pressure) in zone 3 become smaller. One difference between rarefaction waves and shock waves is that rarefaction waves do not cause sudden changes in gas parameters, so rarefaction waves are generally not ignored as a line in the vertical direction in space. However, the spatial length of the sparse wave is very short compared with zone 3, and they are all formed by the original high-pressure gas, so this zone is merged into zone 3 and its parameters are regarded as zone 3. Therefore, after the rupture of the bursting disc, the shock tube is divided into zone 4, zone 3, zone 2 and zone 1 from left to right. The pressure in zone 4 is the largest, the pressure in zone 1 is the smallest, zone 3 and zone 2 are equal and between zone 4 and zone 1. Before the incident shock wave or rarefaction wave reaches the two ends of the shock tube, the range of zone 2 and zone 3 gradually increases, and the range of zone 1 and zone 4 gradually decreases. It

is easy to see that the gas in zone 1 and zone 2 is the original low-pressure gas, and the gas in zone 3 and zone 4 is the original high-pressure gas. Zone 3's position spans both sides of the original bursting disc position.

**Figure 5.** Region division and wave system diagram in shock tube.

The upper part of Figure 5 shows the shock tube wave diagram. In the figure, the ordinate represents the time, and the abscissa represents the spatial position of the shock tube axis. *L*1, *L*2, *L*<sup>3</sup> and *L*<sup>4</sup> represent the shock wave front, the original high-pressure and low-pressure gas interface, the sparse wave right end and the sparse wave left end, respectively. At time t1, these three sections move to *x*1, *x*2, *x*<sup>3</sup> and *x*4, respectively.

In Figure 6, the two figures are the pressure curves in the shock tube at the initial time and *t* = *t*<sup>1</sup> time, respectively.

**Figure 6.** Pressure curve in shock tube.

When the shock wave theory is applied to the theoretical analysis of a coal and gas outburst, zone 1 is the area where the roadway has not been affected, zone 2 is the roadway air affected by the shock wave, zone 3 is the coal-gas flow area and zone 4 is the outburst area in the coal seam.

#### *3.2. Derivation of Outburst Shock Wave Parameters*

Although most of the roadways in coal mines are curved, most of the roadways are also straight in a small range, and most of the coal and gas outbursts occur in the tunneling roadways; these roadways are also straight, which is similar to the geometry of the shock tube. Additionally, the space of the coal mine roadway is much larger than that of the ordinary shock tube. From this aspect, the coal mine roadway is more in line with the assumption of the ideal shock tube.

Some idealized assumptions are needed in the theoretical derivation of coal and gas outburst flow and fluid in shock tubes. Based on these assumptions, complex practical phenomena, in reality, can be transformed into strict formulas. Facts have proved that these assumptions are reasonable, and the calculated results only slightly deviate from the actual gas flow. The main assumption is that the gas in the shock tube is an ideal gas without viscosity. The shock tube wall is rigid. There is no heat exchange between the gas and the shock tube wall. The gas flow in the shock tube is a one-dimensional flow. The rupture of the bursting disc ends instantaneously and completely. In sparse waves, the gas is isentropic. The shock tube that satisfies the above assumptions is also called an ideal shock tube.

As shown in Figure 7a, the incident shock wave front passes through the gas in zone 1 with *v*s, and then the gas in this zone suddenly changes to the gas in zone 2. The gas changes from the stationary state to *v*b, and *v*<sup>b</sup> is the adjoint velocity. In order to facilitate the analysis, the coordinate transformation is carried out. The coordinate system is established on the shock wave front, and the incident shock wave is transformed into a stationary state to become a stationary shock wave. At this time, the gas in zone 1 enters the wave front at the speed of *v*<sup>1</sup> = *v*<sup>s</sup> (facilitate the calculation of the retention value size and ignore the speed direction, the same below). The gas after the parameter mutation moves to the left at the speed *v*<sup>2</sup> = *v*<sup>s</sup> − *v*b, as shown in Figure 7b. Gas thermodynamic static parameters such as pressure, density and temperature are independent of coordinate transformation.
