Turbulence Model

The stream of the air and methane mixture flowing through the longwall and longwall headings, as well as through the initial section of the goaves with caving, is of a turbulent nature.

The analyses related to the flow of gas mixtures, including air, use the Reynolds number as the criterion specifying the type of flow, whose critical value makes it possible to determine the critical state of the flow separating the area of static laminar flow from the turbulent flow.

Therefore, by taking a time average of the Navier–Stokes equations, the Reynolds-averaged Navier–Stokes (RANS) equations have the following form [46]:

$$\frac{\partial \rho}{\partial t} + \frac{\partial (\rho v\_i)}{\partial x\_i} = 0 \tag{12}$$

$$\frac{\partial(\rho v\_i)}{\partial t} + \frac{\partial(\rho v\_i v\_j)}{\partial \mathbf{x}\_j} = -\frac{\partial}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} \left[ \mu \left( \frac{\partial v\_i}{\partial \mathbf{x}\_j} + \frac{\partial v\_j}{\partial \mathbf{x}\_i} - \frac{2}{3} \delta\_{ij} \frac{\partial v\_l}{\partial u\_l} \right) \right] + \frac{\partial}{\partial \mathbf{x}\_j} \left( -\overline{\rho v\_i' v\_j'} \right) \tag{13}$$

As can be seen from Equation (11), a new variable, the Reynolds stress ρ*v i v j* is introduced to the equations and it must be solved to achieve the closure of the equations. Two approaches are adopted

to calculate the Reynolds stress, i.e., the Reynolds stress models (RSM) and the Boussinesq hypothesis. The Reynolds stresses are related to the mean velocity gradients in the Boussinesq hypothesis [46]:

$$-\sqrt{\rho v\_i' v\_j'} = \mu\_l \left(\frac{\partial v\_i}{\partial \mathbf{x}\_j} + \frac{\partial v\_j}{\partial \mathbf{x}\_i}\right) - \frac{2}{3} \left(\rho k + \mu\_l \frac{\partial v}{\partial \mathbf{x}\_k}\right) \delta\_{ij} \tag{14}$$

In the turbulence model *k* − ε, in the standard variation, the basic Navier–Stokes equation has been transformed into the Reynolds averaged equation. This equation includes an additional term in the form of the Reynolds stress tensor. Due to this term, the set of equations is not closed. To close the set of equations, it is necessary to introduce additional differential equations, which include the equation of kinetic turbulent energy and the equation of kinetic turbulent energy dissipation in the following form [46]:

$$
\rho \frac{\partial \mathbf{k}}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_i} (\rho k v\_i) = \frac{\partial}{\partial \mathbf{x}\_j} [ (\mu + \frac{\mu\_t}{\sigma\_k}) \frac{\partial \mathbf{k}}{\partial \mathbf{x}\_j} ) [ + \mathbf{G}\_\mathbf{k} + \mathbf{G}\_\mathbf{b} - \rho \varepsilon - \mathbf{Y}\_M + \mathbf{S}\_\mathbf{k} \tag{15}
$$

$$
\rho \frac{\partial \varepsilon}{\partial t} + \frac{\partial}{\partial \mathbf{x}^{\cdot}} (\rho \varepsilon v\_{\mathrm{i}}) = \frac{\partial}{\partial \mathbf{x}\_{\mathrm{j}}} [ (\mu + \frac{\mu\_{\mathrm{f}}}{\sigma\_{\mathrm{c}}}) \frac{\partial \varepsilon}{\partial \mathbf{x}\_{\mathrm{j}}} ) [ + \mathbb{C}\_{1\varepsilon} \frac{\varepsilon}{k} (\mathbb{G}\_{k} + \mathbb{C}\_{3\varepsilon} \mathbb{G}\_{\mathrm{b}}) - \mathbb{C}\_{2\varepsilon \rho} \frac{\varepsilon^{2}}{k} + \mathbb{S}\_{\varepsilon} \tag{16}
$$

where: *C*1<sup>ε</sup>*, <sup>C</sup>*2ερ*, C*3ε are constans, σ*k,* σε are turbulent Prandtl numbers for *k* and ε*, Gb* is the generation of turbulence kinetic energy due to buoyancy, *Gk* is the generation of turbulence kinetic energy due to the mean velocity gradients, *YM* is contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, *Sk*, *S*ε are user-defined source terms.

In the Ansys Fluent software, the porous medium (the goaves with caving) is represented as a fluid characterised by two additional parameters. These parameters include porosity and the permeability coefficient. In order for the tests to take into account the porous medium through which the flow occurs, it is required to consider the source term *Si* in the equation of momentum preservation. The additional source term assumes the following form [46]:

$$S\_i = -\left(\sum\_{j=1}^3 K\_{ij}\mu u\_i + \sum\_{j=1}^3 C\_{ij}\frac{1}{2}\rho v\_i^2\right) \tag{17}$$

where *Si* is the pressure loss items defined by Darcy's law and *C*2 is the inertial resistance factor.

#### *2.3. Problem Statement and Boundary Conditions*

The basis stage of the tests conducted was to develop a numerical model for the real-world (actual) region of the longwall under analysis. This area covers the goaves with caving, the longwall and the longwall headings. The actual tensile strength of the roof rocks forming the caving in this longwall amounted to 3.06 MPa. Additionally, tests were also conducted for the following strength values of these rocks: 2, 4, 5, 6 and 7 MPa.

The geometric parameters of the entire region under analysis were taken into consideration during the development of the model. The geometric model of the longwall under analysis, ventilated with the Y-type system along with the equivalence conditions adopted, is presented in Figure 2.

**Figure 2.** The geometric model with the assumed equivalence conditions for the longwall under analysis ventilated with the Y-type system.

The geometric model of the longwall area investigated takes into account the following:


In order to examine the air flow through the caving goaves of longwalls, it was also necessary to determine the vertical extent of this flow. Generally, it is assumed that the zone of air flow through the goaves with caving is equal to three to four times the thickness of the exploited seam (layer) [36].

The height of this flow is determined by the extent of the full caving and is equal to three times the thickness of the exploited layer. Taking into consideration the fact that the air also flows in the space (void) of the mined longwall, we obtain a flow zone measuring four times the thickness of the seam. On the other hand, taking into account the settlement of the basic roof by a value of 0.5 ÷ 0.6 of the seam thickness, it was assumed that the height of the air flow in the goaves amounts to 3.5 times the thickness of the exploited seam (layer).

Such geometrical models were subjected to the process of discretization. The selection of the right size of the numerical mesh elements was preceded by an analysis of its sensitivity to the calculation results obtained. Based on the analysis, it was concluded that, for model-based tests of air flow through goaves with caving, one may adopt a structural numerical mesh with the size of cubic elements equal to 0.05 m × 0.05 m × 0.05 m for a longwall and longwall headings, as well as a structural numerical mesh consisting of cuboidal elements (type of mesh: hexahedron) measuring 0.025 m × 0.025 m × 0.025 m for goaves with caving. Smaller elements significantly extend the time of calculations, without making any changes in the results obtained.

The "inlet" and "outlet" boundary conditions were defined in longwall galleries. The "inlet" boundary condition was set in the distance of 20.0 m from the longwall, in the maingate as well as in the tailgate. It was assumed that the length of longwall galleries amounting to 20.0 m would allow for full development of the speed profile for the air stream supplied to the longwall.

The volumetric flow rate of the air supplied to the longwall under analysis was equal to 14,250 m<sup>3</sup>/min. The volumetric flow rate of the air supplied through the tailgate amounted to 410 m<sup>3</sup>/min.

The "outlet" boundary condition (pressure-outlet) was defined in the tailgate (this reflects the actual condition in the longwall region).

For longwall-related interactions of the flow, standard functions and zero values were adopted for the flow speed in "wall-type" conditions (longwalls treated as the sidewalls of headings), whose surface roughness corresponded to the height of 0.1 m, and their temperature (treated as the temperature of the surrounding rock mass) amounted to 40 ◦C. The temperature of the air stream at the inlet to the headings was 23 ◦C. The oxygen concentration in the air stream at the inlet to the headings was assumed to be equal to 20.8% (the value registered by automatic gasometry sensors, oxygen metres).

The analysed systems of headings took into consideration the flow of methane in the goaves with caving, which was equal to 8 m<sup>3</sup>/min (according to actual measurements).

The computational domain consisted of two parts, with one mapping the longwall and longwall headings and the other the goaves with caving. In the domain reflecting the goaves with caving, a definition was provided for a change in the permeability coe fficient of the goaves with caving as a function of distance from the longwall front, by means of the created user definition function (*UDF*).

The geometrical models failed to incorporate the machinery and devices forming the equipment of longwall headings.

The simplifications adopted in the models developed, in relation to the real-world (actual) objects, arise out of their sizes and constitute a certain compromise between calculation precision and the time of finding a solution.

The ANSYS Fluent 18.2 software was employed for all numerical simulations. The pressure–velocity coupling and scheme-coupled algorithm, the second-order upwind discretization method and the algebraic multigrid method were used to solve the equation.

Such models, along with the adopted conditions of uniqueness, were subjected to numerical analysis.

The each calculation required approximately 1500–1800 iterations, with a convergence tolerance of 10−<sup>6</sup> for all variables (as per the "Fluent Theory Guide" support documentation).

### **3. Results and Discussion**

The analyses conducted helped to determine a series of physical and chemical parameters of the air stream and methane flowing through the region under investigation.

In order to illustrate the processes related to this flow, Figure 3 shows the trajectories of the mixture of air and methane flowing through the caving goaves of the longwall ventilated with the Y-type system with the air being discharged along the goaves. A preliminary analysis of the distribution obtained was enough to demonstrate its grea<sup>t</sup> di fference from the distributions concerning, for instance, the U-type ventilation system.

**Figure 3.** The trajectories of the mixture of air and methane flowing through the goaves with caving for Y-type ventilation system (**a**) and U-type ventilation system (**b**).

Figures 4–15 present the results of the analysis for the goaves with caving formed by roof rocks whose tensile strength is equal to 3.06 MPa (as is the case in a real-world system).

Figures 4–6 present, respectively, the distributions of the air speed and the dangerous speed due to the risk of endogenous fires, as well as the oxygen concentration levels in the goaves with caving at a distance of 0.5 m from the floor of the exploited seam.

**Figure 4.** The distribution of the air speed flowing through the goaves with caving at a distance of 0.5 m from the floor of the exploited seam.

**Figure 5.** The distribution of the air speed within the range from 0.02 m/s to 0.0015 m/s flowing through the goaves with caving at a distance of 0.5 m from the floor of the exploited seam.

**Figure 6.** The distribution of oxygen concentration in the air flowing through the goaves with caving at a distance of 0.5 m from the floor of the exploited seam.

Figures 7–9 present, respectively, the distributions of the air speed and the dangerous speed due to the risk of endogenous fires, as well as the oxygen concentration levels in the goaves with caving at a distance of 2.0 m from the floor of the exploited seam.

**Figure 7.** The distribution of the air speed flowing through the goaves with caving at a distance of 2.0 m from the floor of the exploited seam.

**Figure 8.** The distribution of the air speed within the range from 0.02 m/s to 0.0015 m/s flowing through the goaves with caving at a distance of 2.0 m from the floor of the exploited seam.

**Figure 9.** The distribution of oxygen concentration in the air flowing through the goaves with caving at a distance of 2.0 m from the floor of the exploited seam.

Figures 10–12 present, respectively, the distributions of the air speed and the dangerous speed due to the risk of endogenous fires, as well as the oxygen concentration levels in the goaves with caving at a distance of 7.0 m from the floor of the exploited seam.

**Figure 10.** The distribution of the air speed flowing through the goaves with caving at a distance of 7.0 m from the floor of the exploited seam.

**Figure 11.** The distribution of the air speed within the range from 0.02 m/s to 0.0015 m/s flowing through the goaves with caving at a distance of 7.0 m from the floor of the exploited seam.

**Figure 12.** The distribution of oxygen concentration in the air flowing through the goaves with caving at a distance of 7.0 m from the floor of the exploited seam.

Figures 13–15 present, respectively, the distributions of the air speed and the dangerous speed due to the risk of endogenous fires, as well as the oxygen concentration levels in the goaves with caving at a distance of 10.5 m from the floor of the exploited seam.

**Figure 13.** The distribution of the air speed flowing through the goaves with caving at a distance of 10.5 m from the floor of the exploited seam.

**Figure 14.** The distribution of the air speed within the range from 0.02 m/s to 0.0015 m/s flowing through the goaves with caving at a distance of 10.5 m from the floor of the exploited seam.

**Figure 15.** The distribution of oxygen concentration in the air flowing through the goaves with caving at a distance of 10.5 m from the floor of the exploited seam.

Based the results obtained, it may be concluded that the air speed value decreases along with an increase in the distance from the floor of the exploited seam. The air flowing through the goaves with caving reaches the highest speed value, irrespective of the flow height in the goaves, behind the caving line from the inlet side to the longwall, as well as along the tailgate maintained at the goaves. The highest speed value occurs at the flow height of 2.0 m from the floor of the exploited seam in the bottom corner of the longwall, and amounts to 0.36 m/s. The distribution of oxygen concentration for this ventilation system is also different than in the case of the U-type system [47]. It is clearly visible that the air stream moving along with tailgate leads to an increase in this concentration along this route.

Figure 16 presents the distribution of air speed values in the goaves with caving as a function of distance from the longwall front for the actual values of the tensile strength of roof rocks amounting to 3.06 MPa. Red horizontal lines were used to mark the range of dangerous speed values due to the risk of endogenous fires in these goaves.

**Figure 16.** The distribution of air speed values in the goaves with caving as a function of distance from the longwall front for the actual values of the tensile strength of roof rocks amounting to 3.06 MPa.

Based on the determined speed characteristics, it can be concluded that the air speed in goaves with caving decreases along with the increasing distance from the longwall front.

At a distance of up to 96.0 m from the caving line into the depths of the goaves, the speed of the flowing air reaches the critical value due to the risk of endogenous fires, i.e., the value from 0.0015 m/s to 0.02 m/s. After exceeding the distance of 96.0 m from the longwall front, the speed of the air flowing through goaves with caving reaches a value lower than 0.0015 m/s.

Figure 17 presents the distribution of oxygen concentration in the air flowing through the goaves with caving as a function of distance from the longwall front.

**Figure 17.** The distribution of oxygen concentration in the goaves with caving as a function of distance from the longwall front for the actual values of the tensile strength of roof rocks amounting to 3.06 MPa.

Based on the determined speed characteristics, it can be concluded that the concentration of oxygen in goaves with caving decreases along with the increasing distance from the longwall front.

At a distance of up to 335.0 m from the caving line inside the goaves, the oxygen concentration in the air flowing through the goaves with caving falls within the critical range due to the risk of endogenous fires, i.e., reaches a value higher than or equal to 8%.

The speed characteristics determined for the air flowing through the goaves with caving and for the oxygen concentration in this air served as the basis for demarcating the zone with a particularly high risk of spontaneous coal combustion (in which both of these conditions are met) (Figure 18).

**Figure 18.** The zone with a particularly high risk of endogenous fires in the goaves formed by rocks with tensile strength equal to 3.06 MPa.

Based on the tests and the results obtained, it was concluded that the zone with a particularly high risk of spontaneous combustion, for the longwall ventilated with the Y-type system, is formed immediately behind the longwall front, and reaches 100.0 m inside the goaves.

Therefore, it can be assumed that the goaves with caving formed by roof rocks whose tensile strength amounts to 3.06 MPa have no cooling zone in which the air (behind the powered roof support along the entire length of the longwall) reaches a flow value higher than 0.02 m/s. This value was exceeded only in the upper and bottom corner of the longwall at the flow height of up to 8.0 m from the floor of the exploited seam.

Behind the zone with a particularly high risk of spontaneous combustion, at a distance of more than 100.0 m from the longwall front to approximately 345.0 m, there forms a zone with insufficient air speed, ye<sup>t</sup> with sufficient oxygen concentration in the air, in terms of the risk of spontaneous coal combustion.

In order to determine the impact of the strength of the rocks forming the caving on the extent of the zone with a particularly high risk of spontaneous combustion for longwalls ventilated with the Y-type system with the air being discharged along the goaves and supplied along the tailgate, additional tests were conducted for different values of this strength (2, 4, 5, 6 and 7 MPa).

Figures 19–30 present the results of the analysis for the goaves with caving formed by roof rocks whose tensile strength is equal to 3.06 MPa (as is the case in a real-world system).

Figures 19–21 present, respectively, the distributions of the air speed and the dangerous speed due to the risk of endogenous fires, as well as the oxygen concentration levels in the goaves with caving at a distance of 0.5 m from the floor of the exploited seam.

**Figure 19.** The distribution of the air speed flowing through the goaves with caving at a distance of 0.5 m from the floor of the exploited seam.

**Figure 20.** The distribution of the air speed within the range from 0.02 m/s to 0.0015 m/s flowing through the goaves with caving at a distance of 0.5 m from the floor of the exploited seam.

**Figure 21.** The distribution of oxygen concentration in the air flowing through the goaves with caving at a distance of 0.5 m from the floor of the exploited seam.

**Figure 22.** The distribution of the air speed flowing through the goaves with caving at a distance of 2.0 m from the floor of the exploited seam.

**Figure 23.** The distribution of the air speed within the range from 0.02 m/s to 0.0015 m/s flowing through the goaves with caving at a distance of 2.0 m from the floor of the exploited seam.

**Figure 24.** The distribution of oxygen concentration in the air flowing through the goaves with caving at a distance of 2.0 m from the floor of the exploited seam.

**Figure 25.** The distribution of the air speed flowing through the goaves with caving at a distance of 7.0 m from the floor of the exploited seam.

**Figure 26.** The distribution of the air speed within the range from 0.02 m/s to 0.0015 m/s flowing through the goaves with caving at a distance of 7.0 m from the floor of the exploited seam.

**Figure 27.** The distribution of oxygen concentration in the air flowing through the goaves with caving at a distance of 7.0 m from the floor of the exploited seam.

**Figure 28.** The distribution of the air speed flowing through the goaves with caving at a distance of 10.5 m from the floor of the exploited seam.

**Figure 29.** The distribution of the air speed within the range from 0.02 m/s to 0.0015 m/s flowing through the goaves with caving at a distance of 10.5 m from the floor of the exploited seam.

**Figure 30.** The distribution of oxygen concentration in the air flowing through the goaves with caving at a distance of 10.5 m from the floor of the exploited seam.

Figure 31 presents the distribution of air speed values in the goaves with caving as a function of distance from the longwall front for the actual values of the tensile strength of roof rocks amounting to 6.0 MPa. Red horizontal lines were used to mark the range of dangerous speed values due to the risk of endogenous fires in these goaves.

**Figure 31.** The distribution of air speed values in the goaves with caving as a function of distance from the longwall front for the actual values of the tensile strength of roof rocks amounting to 6.0 MPa.

Based the results obtained, it may be concluded that the air speed value decreases along with an increase in the distance from the floor of the exploited seam.

At a distance of 25.0 m behind the longwall front to 160.0 m inside the goaves, the speed of the flowing air reaches the critical value due to the spontaneous combustion risk, i.e., the value from 0.0015 m/s to 0.02 m/s. After exceeding the distance of 160.0 m from the longwall front, the speed of the air flowing through goaves with caving reaches a value lower than 0.0015 m/s.

Figure 32 presents the distribution of oxygen concentration in the air flowing through the goaves with caving as a function of distance from the longwall front.

**Figure 32.** The distribution of oxygen concentration in the goaves with caving as a function of distance from the longwall front for the actual values of the tensile strength of roof rocks amounting to 6.0 MPa.

Based on the determined speed characteristics, it can be concluded that the concentration of oxygen in goaves with caving decreases along with the increasing distance from the longwall front.

At a distance of up to 440.0 m from the caving line inside the goaves, the oxygen concentration in the air flowing through the goaves with caving falls within the critical range due to the risk of endogenous fires, i.e., reaches a value higher than or equal to 8%.

The speed characteristics determined for the air flowing through the goaves with caving and for the oxygen concentration in this air served as the basis for demarcating the zone with a particularly high risk of spontaneous coal combustion (in which both of these conditions are met) (Figure 33).

**Figure 33.** The zone with a particularly high risk of endogenous fires in the goaves formed by rocks with tensile strength equal to 6.0 MPa.

Based on the tests and the results obtained, it was concluded that the zone with a particularly high risk of spontaneous combustion is formed immediately behind the longwall front, and reaches 160.0 m inside the goaves.

In the goaves with caving formed by roof rocks whose tensile strength amounts to 6.0 MPa, the cooling zone in which this air (behind the powered roof support along the entire length of the longwall) reaches a flow value higher than 0.02 m/s occurs at a distance of up to 25.0 m from the longwall front.

Behind the zone of a particularly high risk of spontaneous combustion, at a distance of over 160.0 m from the longwall front to approximately 460.0 m, there forms a zone with insufficient air speed, ye<sup>t</sup> with sufficient oxygen concentration in the air (due to the risk of spontaneous combustion).

The tests conducted made it possible to demarcate the zone with a particularly high risk of spontaneous combustion in the goaves with caving formed by rocks whose tensile strength was equal to 2, 3.06, 4, 5, 6 and 7 MPa. Table 2 summarises the extents of these zones for the actual conditions of the longwall in question as well as for the additional ones obtained from the analyses conducted.


**Table 2.** The zone with a particularly high risk of spontaneous combustion in the goaves with caving formed by rocks whose tensile strength was equal to 2, 3.06, 4, 5, 6 and 7 MPa.

The results obtained unambiguously indicate that the type of roof rocks (defined by their tensile strength) has a significant impact on the value of the air speed flowing through the goaves with caving, and on the value of oxygen concentration in this air. The more resistant the rocks, the greater the extent of the zones in which the air speed and oxygen concentration reach critical values due to the risk of spontaneous coal combustion.

The greater extent of the zone with a particularly high risk of endogenous fires in the caving goaves of the longwall ventilated with the Y-type system (compared, for example, to the U-type system) arises out of the necessity to maintain a tailgate along the goaves, through which the air can flow. Part of the ventilation air stream flowing through this tailgate migrates to the goaves through the sidewalls, thereby increasing the extent of this zone in the goaves.
