*2.2. Methods*

The objective of the tests conducted was to determine the impact of the type of roof rocks forming the goaves with caving on the extent of the zone with a particularly high risk of spontaneous combustion of coal in the goaves of a longwall ventilated with the Y-type system.

The analyses were conducted for a spatial model representing a real-world longwall along with longwall headings and goaves with caving, making use of Computational Fluid Dynamics (CFD). The Authors' experiences and the results obtained by other researchers indicate that this method may be used successfully for such analyses of the phenomena related with the flow of gases and the transfer of mass and heat [44].

The analyses were carried out by means of the ANSYS Fluent 18.2 commercial software. This software uses the finite volume method (FVM) for discretisation of the geometric model. The methodology for conducting tests by means of this programme involves development of a geometric model, a discrete model and a mathematical model of the phenomenon in question, as well as adoption of boundary conditions, performance of calculations and analysis of the results obtained. The most important stages of the methodology for the tests conducted are briefly discussed in the subsequent chapters of the article.

In the case at hand, this methodology is also supplemented with tests in real-world (actual) conditions. This is because the results of these tests serve as the basis for developing a geometric model for the region under analysis and for adopting the boundary conditions. The process of analysing the results of model-based tests also involves their verification with reference to real-world conditions.

## 2.2.1. Mathematical Models

The flow of air stream through the longwall and the longwall headings is deemed to be of a turbulent nature, while the flow of air stream through the goaves with caving of a laminar nature.

The mathematical mapping of a model of a longwall region and the goaves with caving takes the form of a set of equations which describe the aforementioned types of flows. These equations describe the flow of the mixture of air and mining gases released from the rock mass and generated as a result of the ongoing mining operations. Examining the three-dimensional flow of the air stream through region under analysis, encompassing the flow through the longwall, longwall headings and goaves with caving, one must consider the analytical models describing a turbulent and laminar flow towards the component parts of the Cartesian x, y and z coordinate system, in the particular calculation domains of the model. Modelling the flow of a multi-component mixture also requires solving additional equations for the transportation of the mixture components.

## Basic Flow Equations

The flow of the air stream mixture is described by means of constitutive equations, which include the equations of mass, momentum and energy conservation and species transport equation. Conservation equations for mass, momentum, and species can be expressed as [45]:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot \rho \mathbf{v} = 0 \tag{8}$$

$$\frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla \cdot \rho \mathbf{v} \mathbf{v} = -\nabla p \cdot \nabla \mathbf{r} + \rho \mathbf{g} \tag{9}$$

$$\frac{\partial}{\partial t}(pc\_pT) + \nabla \cdot \left(\rho c\_p \sigma T\right) = \nabla \cdot \left(k\_{eff} + \frac{c\_p \mu\_t}{Pr\_t}\right) \nabla T \tag{10}$$

$$\frac{\partial}{\partial t}(\rho \phi\_i) + \nabla \cdot (\rho \phi\_i \mathbf{U}) = \nabla \cdot \left(\rho D\_{i,eff} + \frac{\mu\_t}{\mathbf{S} \mathbf{c}\_t}\right) \nabla \phi\_i \tag{11}$$

where: ρ is the gas density (kg/m3), *v* is the gas velocity (m/s), *p* is pressure (Pa), τ is the viscous stress tensor (Pa), *g* is gravity acceleration (m·s<sup>−</sup>2), *cp* is the specific heat of the gas, *ke*ff is the e ffective gas thermal conductivity, *T* is the temperature (K), ω*i* is the mass fraction of species *i* (N2, O2 and CH4), μ*t* is turbulent viscosity (Pa·s), *Di,e*ff is the e ffective di ffusivity of species *i* (m<sup>2</sup>/s), *Sct* is the turbulent Schmidt number (0.7) and *Prt* is the turbulent Prandtl number.
