## 3 Results and Discussion
### Numerical Simulation of Air Flow in an Inclined Channel
Of course, in our case, the fan with the parameters from Table 2 should be interpreted as an equivalent, whose effect is equal to the effect of the mine fan minus the losses due to the friction resistance of the remaining network. At this stage of the study, our primary interest lay in determining the point of air flow reversal in an inclined channel under varying heat source powers, different degrees of ventilation valve opening near the channel exit, and various fan operation modes. During numerical calculations, it was assumed that valve at opening No. 1 (see Figure 5) was open. From a practical perspective, the system under consideration resembles parallel mine workings, with one potentially affected by a fire. By adjusting the ventilation valve opening, we varied the air resistance proportions between the two parallel workings, thereby investigating its influence on the likelihood of air flow reversal in the branch with the fire. Figure 10 depicts the dependencies of the air mass flow at the outlet on the fan operation mode, ventilation valve opening degree at the outlet section, and HRR from the heat source, which were derived from multiparameter simulations. Positive mass flow values indicate reversed air flow in the inclined channel (from bottom to top), while negative values correspond to air flow in the same direction as at an HRR of zero (from top to bottom). Figure 10 shows that at a minimum initial pressure drop, the air flow was reversed for all considered values of the HRR. As the initial pressure drop increased, the reversed air flow rate decreased in absolute value and eventually reached zero. The lower the HRR of the heat source, the lower the pressure at which the zero-flow point was reached. With a further increase in the initial pressure drop, the mass flow rate became negative, indicating that within this parameter range, the air flow did not globally change its direction after the heating source was turned on. An interesting feature of the dependencies presented in Figure 10 is the approximate coincidence of the intersection points of the curves characterizing the degree of opening of the valve of opening No. 3 with the ordinate axis at a fixed HRR from the source. This holds true for all considered HRR values (22, 33, 44, and 55 kW). Thus, there was a weak dependence of the moment of reversing the air flow on the air resistance of the subsequent ventilation network behind the outlet. That is, the reversal of the air flow depends, to a greater extent, on the power of the fire and the initial pressure drop in the inclined part of the channel. Based on the data from the multiparameter numerical simulations, we built a linear regression model to predict the mass air flow \(Q_{m}^{(out)}\) in the outlet section of the channel using three variable parameters: the initial pressure drop (\(\Delta P\)), the degree of opening of the ventilation valve at the outlet (\(\varphi\)), and the HRR (\(W\)). The form of the linear regression model with dimensional coefficients is presented below:  
\[Q_{m}^{(out)}=0.0664-0.0136\Delta P-0.000136\varphi+0.00185W. \tag{12}\]This model, despite its simplicity, has two advantages. Firstly, it allows for the analysis and comparison of the degree of influence of each model factor on the predicted value \(Q_{m}^{(out)}\). Secondly, linear regression addresses our main interest--studying the conditions for the reversal of an air flow in an inclined channel. In general, Figure 10 shows that the flow rate was a nonlinear function of the parameters \(\Delta P,\ \varphi,\ \text{and}\ W\). However, the proposed linear regression had an acceptable error and was applicable for quantitative analysis in a small region near the intersection of the curves with the y-axis. Considering the goodness-of-fit of the model (12), it should be noted that the linear regression (12) had an \(R^{2}\) value of 0.8 for the entire range of data points, which is not very high. The goodness-of-fit could be improved by considering nonlinear regression.