## 3 Results and Discussion
### Numerical Simulation of Air Flow in an Inclined Channel
Given that the pressure drop \(\Delta P\) is a quadratic function of the mean air velocity in the cross-section, \(\Delta P\) should be scaled by a factor of \(\  
These expressions can be used to estimate the critical HRR leading to the reversal of an air flow in a mine working with a given inclination angle of \(-13^{\circ}\) as a result of the action of a powerful heat source. A numerical simulation of air flow reversal in inclined channels with an increased characteristic channel size \(L\) confirmed the validity of formulas (17) and (18). For example, with a 10-fold increase in the characteristic channel size, formulas (17) and (18) predicted pressure drop values at a fixed HRR that differed from the numerical model by only 7.8%. The adaptation of this relationship in the calculation of air distribution in one-dimensional mine ventilation networks involves incorporating critical HRR values into the mass balance equation for each independent network circuit, especially those affected by fire. For instance, in the context of a simplified ventilation network comprising two circuits, one of which features a significant heat source (refer to Figure 11), the following system of equations should be formulated:  
\[Q_{1}=Q_{2}+Q_{3} \tag{19}\]  
\[R_{1}Q_{1}|Q_{1}|+R_{3}Q_{3}|Q_{3}|=H(Q_{1})+(\rho_{3}-\rho_{1})gh \tag{20}\]  
Figure 9:0  
where \(Q\) is the volumetric air flow (m\({}^{3}\)/s); \(R\) is the air resistance (N s\({}^{2}\)/m\({}^{8}\)); \(H\) is the fan pressure in branch No. 1 (Pa); \(h=z_{II}-z_{III}>0\) denotes the height difference between nodes II and III (m); and \(z\) is the vertical coordinate (m). Here, the indices for \(Q,~{}R\) and \(\rho\) indicate the branch number (see Figure 11). The orientation of the branches was chosen such that at an HRR of zero, the air flow rates in all three branches are positive. Nodes I and III are considered atmospheric, meaning that the outgoing flow maintains a specified atmospheric temperature \(T_{0}\) and density \(\rho_{0}\), regardless of the thermal characteristics of the inflowing air. As the HRR in branch No. 2 increases, the air density within this branch decreases due to heating. According to a simple advective transport model, the change in air temperature caused by the HRR from a source at the midpoint of the branch can be estimated as follows:  
Figure 9:1  
Figure 9:2 One-dimensional model of the ventilation network corresponding to the test bench under consideration. where \(Q_{2}\) is the volumetric air flow rate in branch No. 2, calculated upstream of the heating source (m\({}^{3}\)/s). As the flow approaches the point of thermal slug, preceding the reversal of the air flow, the parameter \(W\) will approach \(W_{cr}\). When \(W_{cr}\) is reached, the air flow in branch No. 2 should become zero, indicating that  
Figure 9:3  
Figure 9:4  
In Equation (24), we included a factor of \(\
u_{2}\) because only the lower half of branch No. 2, below the heat source, experiences an increase in air density (see Equation (22)). In our simple advective transport model, the second term on the right-hand side of Equation (24) tends towards zero as the denominator approaches zero. However, near the heating source, the air velocity can be significantly non-zero due to convection effects (see Figure 9b). Therefore, the accurate prediction of \(\rho_{2}\) requires three-dimensional numerical simulation. In this study, the average air density in an inclined channel was computed under various HRR values, fan operational modes, and outlet valve positions (see Figure 12). Figure 9:5 Dependencies of average air density 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**).