## 3 Results and Discussion
### Numerical Simulation of Air Flow in an Inclined Channel  
The validated mathematical model was used to conduct a series of numerical calculations of heat and mass transfer in the channel under several HRR values and fan operating modes. The fan modes were selected to maintain a specified average velocity in the channel, considering an HRR of zero from the heat source and with the valve fully open at the outlet. Six velocity points were chosen for the study. The pressure jump at the fan interface was determined for each velocity point (see Table 2). This information allowed for an assessment of the air resistance of the flow domain. Figure 8: Comparative analysis of calculated profiles and experimentally measured values of the air velocity (**a**) and temperature (**b**) in cross-section No. 3 at an inclination angle equal to \(-13^{\circ}\). It should be noted that the pressure differentials presented here are significantly lower than the characteristic fan pressures in Figure 3. There is no contradiction here for two main reasons: (1) the actual operating point of the fan during the experimental measurements was considerably shifted to the right, into a region of lower pressures, which also resulted in lower pressures during the experiments; (2) additionally, the model did not account for shock losses due to the narrowing of the flow entering the fan from the atmosphere, nor for the expansion of the flow leaving the channel through its outlet. The numerical simulation enabled the identification of the principal flow regimes, as illustrated in Figure 9, for a thermal power of 22 kW and a channel inclination angle of \(-13^{\circ}\). The figure depicts the velocity vector component aligned with the channel axis. The color scheme in this figure is designed such that red corresponds to downward air flows driven primarily by the fan pressure, while blue corresponds to return air flows influenced by thermal convection. At the maximum pressure drop (21.09 Pa) at the outlet section of the channel, a stable unidirectional flow of air was observed from left to right and downwards. As the pressure \begin{table}
\begin{tabular}{c c c c c c c} \hline Velocity, & & & & & & \\ m/s & 0.25 & 0.5 & 1 & 1.5 & 2 & 2.5 \\ \hline Pressure & & & & & & \\ jump, Pa & 0.41 & 1.06 & 3.61 & 7.79 & 13.62 & 21.09 \\ \hline \end{tabular}
\end{table} 
Table 2: Relationship between air velocity in the channel and pressure jump at the fan interface. Figure 9: Distributions of the longitudinal component of the velocity vector in an inclined channel under different fan operating modes: high performance mode, \(\Delta P=21.09\) Pa (**a**), intermediate mode, \(\Delta P=8.45\) Pa (**b**), and low performance mode \(\Delta P=1.06\) Pa (**c**). decreased, return flows began to form in the outlet section, becoming comparable to the forward flow. At a pressure drop of 8.45 Pa, the return flow in the upper part of the channel reached opening No. 1 and exited the channel through this boundary. In this scenario, the return flow of hot air did not propagate further up the channel towards the fan itself. Near the lower section of the channel, the average velocity in both directions over the cross-section approached zero. This indicates a situation resembling a thermal plug, where the inertial force of the flow is, on average, balanced by the buoyancy force. However, was is also a situation where the return flows reached the fan and completely occupied the cross-section of the domain. An example of this is illustrated by the calculated velocity field for a minimum pressure of 1.06 Pa (see Figure 9c). These three examples demonstrate that the fan pressure, which provides a given air velocity in an inclined channel at W = 0, is an important parameter that significantly influences the flow regime at a non-zero value of W (e.g., direct flow, reversed flow, and thermal slug). Since the prototype of our model is an inclined mine ventilated by the main fan of the mine, such an analysis is very important to us.