## 2 Methods
### Theoretical Study  
An experimental study on a laboratory bench was used to validate the mathematical model of heat and mass transfer in an inclined channel (a dimensionally reduced model of a mine working). This model was further needed to perform a series of calculations of the dynamics of air flows when varying the parameters of a mine working, as well as when considering more complex systems of several mine workings. The mathematical model was based on the Reynolds-averaged continuity equation, the Navier-Stokes equations, and the energy equation, written for a three-dimensional flow domain:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{u})=0 \tag{4}\]  
\[\frac{\partial(\rho\mathbf{u})}{\partial t}+\
abla\cdot(\rho\mathbf{u}\mathbf{u})=-\
abla p +\
abla\cdot[(\mu+\mu_{t})\
abla\mathbf{u}]+\mathbf{g}\rho \tag{5}\]  
\[\frac{\partial(\rho e)}{\partial t}+\
abla\cdot[\mathbf{u}(\rho e+p)]=\
abla\cdot[ (\lambda+\lambda_{t})\
abla T] \tag{6}\]  
where \(\rho\) is the air density (kg/m\({}^{3}\)); \(\mathbf{u}\) is the velocity vector (m/s); \(t\) is the time (s); \(p\) is the static pressure (Pa); \(\mu\) is the molecular viscosity (m\({}^{2}\)/s); \(\mu_{t}\) is the turbulent viscosity (m\({}^{2}\)/s); \(e\) is the specific air energy (J/(kg\({}^{\circ}\)C)); \(T\) is the temperature (\({}^{\circ}\)C); and \(\lambda\) and \(\lambda_{t}\) are, respectively, the molecular and turbulent thermal conductivities (W/(m\({}^{\circ}\)C)). Figure 4: The locations of measurement points. To close this system of equations, the Boussinesq hypothesis was used, which assumes that turbulent viscosity can be represented as a scalar quantity. In this case, turbulent thermal conductivity was assumed to be proportional to turbulent viscosity:  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\varepsilon},\ C_{\mu}=\frac{1}{A_{0}+A_{s}\frac {k\sqrt{2S\cdot S}}{\varepsilon}} \tag{7}\]  
\[\lambda_{t}=\frac{\mu_{t}}{Sc_{t}} \tag{8}\]  
where \(k\) is the turbulent kinetic energy (m\({}^{2}\)/s\({}^{2}\)); \(\varepsilon\) is the rate of dissipation of turbulent kinetic energy (m\({}^{2}\)/s\({}^{3}\)); \(A_{0}\) and \(A_{s}\) are the parameters of the realizable k-epsilon turbulence model [46]; \(S\) is the flow deformation rate tensor (1/s); and \(Sc_{t}=0.7\) is the turbulent Schmidt number. To determine the parameters \(k\) and \(\varepsilon\), we used the realizable k-epsilon turbulence model, which includes the following equations:  
\[\frac{\partial(\rho k)}{\partial t}+\
abla\cdot(\rho u)=\
abla\cdot\left[ \left(\mu_{m}+\frac{\mu_{t}}{\sigma_{\varepsilon}}\right)\
abla k\right]+G_{ k}+G_{b}-\rho\varepsilon \tag{9}\]  
\[\frac{\partial(\rho\varepsilon)}{\partial t}+\
abla\cdot(\rho u)=\
abla\cdot \left[\left(\mu_{m}+\frac{\mu_{t}}{\sigma_{\varepsilon}}\right)\
abla \varepsilon\right]+\rho C_{1}S\varepsilon-\rho C_{2}\frac{\varepsilon^{2}}{k+ \sqrt{\
u\varepsilon}}+C_{1\varepsilon}\frac{\varepsilon}{k}C_{3\varepsilon }G_{b} \tag{10}\]  
where \(\sigma_{k}\) and \(\sigma_{\varepsilon}\) are turbulent analogues of the Prandtl number; \(S=\sqrt{2S\cdot S}\) is the invariant of the tensor \(S\) of flow deformation rates; \(C_{1}\), \(C_{1\varepsilon}\), and \(C_{2}\) are empirical coefficients of the model; \(G_{k}\) is the generation of the turbulent kinetic energy due to velocity field inhomogeneities (kg/m/s\({}^{3}\)); and \(G_{b}\) is the generation of the turbulent kinetic energy due to buoyancy forces (kg/m/s\({}^{3}\)):  
\[G_{b}=\beta\frac{\mu_{t}}{Pr_{t}}g\cdot\
abla T \tag{11}\]  
where \(\beta\) is the coefficient of thermal expansion of air (1/\({}^{\circ}\)C) and \(Pr_{t}=0.85\) is the turbulent Prandtl number. The system of Equations (4)-(11) was supplemented with boundary conditions. At the inlet and outlet areas, a fixed pressure condition was set; on the channel walls, a no-slip condition and convective heat transfer condition were applied. A fixed heat flux was imposed on the wall of the heat source.