## 3 Results and Discussion
### Numerical Simulation of Air Flow in an Inclined Channel
However, determining a nonlinear regression that accurately predicts the air mass flow across a wide range of the parameters \(\Delta P,\ \varphi,\ \text{and}\ W\) is a challenging task. Moreover, it is not of primary interest from a practical standpoint, as our primary goal was to quantitatively capture the moment when the mass flow \(Q_{m}^{(out)}\) transitions from positive to negative values. We associate this transition with the reversal of the mass flow in the inclined channel. For this purpose, linear regression was sufficient. In this case, the mass flow rate was calculated  
Figure 10: Dependence of the air mass flow at the outlet on the fan operating mode, the valve opening degree at the outlet, and the HRR: W = 22 kW (**a**), W = 33 kW (**b**), W = 44 kW (**c**), and W = 55 kW (**d**). integrally over the entire cross-section of the channel. The moment of the onset of the reversal of the mass air flow, therefore, will be earlier than the moment of the appearance of partial return flows, which we did not study here. The regressors \(\Delta P,\;\varphi,\;\) and \(W\) take only positive values, suggesting that another option is to consider a log-linear regression model of the form:  
\[Q_{m}^{(out)}=0.0811-0.0658\text{ln}\left(\frac{\Delta P}{\Delta P_{0}}\right) -0.00619\text{ln}\left(\varphi\right)+0.00517\text{ln}\left(\frac{W}{W_{0}}\right) \tag{13}\]  
where \(\Delta P_{0}=1\) Pa and \(W_{0}=\)1 kW are reference values for the pressure drop and HRR, introduced to make the logarithmic expression dimensionless. The log-linear regression (13) had a lower \(R^{2}\) value \((0.67)\) compared to model (12). Therefore, we continued working with model (12). Given the condition \(Q_{m}^{(out)}=0\), we can express the \(W\) value from expression (12). This will be the critical HRR of the fire at which the air flow will reverse; it can be written as a function of the parameters \(\Delta P\) and \(\varphi\):  
\[W_{cr}=-35.8+7.37\Delta P-0.073\varphi \tag{14}\]  
From this dependence, it is clear that the degree of valve opening at the outlet affects the critical HRR less than the initial pressure drop in the channel. For example, with an initial pressure drop of 8 Pa, varying the valve opening degree from 10% to 60% changed the critical HRR from 22.3 kW to 18.7 kW. With an initial pressure drop of 13 Pa, varying the valve opening degree in the same range changed the critical HRR from 59.2 kW to 55.5 kW. Based on these quantitative estimates, we also observed good agreement between the predicted data of model (12) and the calculated data in Figure 10a,d. A similar linear regression was built for the air mass flow rate at the inlet zone:  
\[Q_{m}^{(in)}=0.117+0.0311\Delta P-0.000431\varphi+0.00181W \tag{15}\]  
Given the condition \(Q_{m}^{(in)}=0\), we can express the value of \(W\) from Equation (15). This represents the HRR of the heat source at which the air flow on the fan will reverse:  
\[W_{cr}^{*}=6.46+17.1\Delta P-0.238\varphi \tag{16}\]  
In this case, the value \(W_{cr}^{*}\) is significantly higher than \(W_{cr}\). At an initial pressure drop of 8 Pa, \(W_{cr}^{*}\) ranged from 129 to 141 kW as the valve opening degree varied from 10% to 60%. This confirms the intuitive fact that it is much more difficult to upset the air flow from a fan than in one of the branches of a parallel connection located downstream of the fan. An important consideration is that the obtained dependencies (14) and (16) were derived under test bench conditions and need to be scaled for real mine workings. According to dependencies (2) and (3), when changing the linear size of the object under study from \(L_{m}\) to \(L_{e}\), the HRR should be scaled by a factor of \(\left(\frac{L_{e}}{L_{m}}\right)^{2.5}\).