*3.1. Dead Zone Delimitation and Flow Field Patterns*

Figure 7 was obtained by plotting contours of the air velocity magnitude in a longitudinal plane of the tunnel, which intersects the duct. It can be observed that the air jet tends to adhere to the tunnel roof owing to the Coandă effect [27,28]. This effect increases friction and energy losses of the air jet and reduces the effective distance. Moreover, as the jet flow expands, the air velocity decreases down to values close to zero. In light of this, the first option would be to consider the effective distance as that in which the air velocity is almost zero.

**Figure 7.** Velocity contours of the air jet exiting the ventilation duct along the tunnel axis. The data correspond to scenario 4 and 20 m/s air velocity at the velocity inlet.

Figure 8 depicts the average velocity of the air jet along the tunnel, measured in control surfaces 1.25 time as long as the duct diameter and coaxial with it. In this way, a continuum decrease in the absolute value of the air jet velocity can be observed. This curve reaches a point when velocity is almost zero and, once again, this could be defined as the beginning of the dead zone.

**Figure 8.** Jet velocity for scenario 4 and a duct velocity of 20 m/s.

Figure 9 shows that most part of the air jet changes its direction, returning through the gallery before entering the dead zone. The airflow in the dead zone is also dragged by the air jet nearby and generates counter vortices due to the shear stresses. As a result, the air continually re-circulates inside the dead zone without leaving it. This phenomenon invalidates the premise of defining the dead zone as just a low-velocity region.

**Figure 9.** Counter vortices formation in the dead zone.

The analysis of the area-weighted average vertical air velocity (Vy) on control surfaces covering the entire transverse section of the tunnel is shown in Figure 10. The first phenomenon, observable at approximately 90 m from the face, was a zone of highly positive velocities. Upon analyzing the flow pattern, it can be perceived that judging from the simulation, the return flow is constricted in the lower part of the tunnel. However, at a certain moment the flow was free from the influence of the ventilation jet and filled the entire tunnel section. This phenomenon generates turbulences inside the tunnel and is responsible for this zone of mean positive vertical velocities in the simulation. Moreover, the negative mean vertical velocity observed in Figure 10 corresponds to the region where all the ventilation flow changes its direction towards the floor of the tunnel. This point is considered as the delimitation of the dead zone. In this way, we consider the limit of the dead zone where the negative vertical velocity reaches its maximum, although both approaches suggest very similar results.

**Figure 10.** Mean vertical velocity along the tunnel axis for scenario 4 and inlet velocity 20 m/s.

Figure 11a shows the influence of the flow rate over the effective ventilation distance. According to this simulation, this distance grows rapidly for flow rates less than 1 m3/s, which are not common for mine ventilation purposes and reach a horizontal asymptote at a flow rate of about 5 m3/s. Moreover, this figure also shows a poor correlation between our expression and the legal requirements (shaded zone) for the low airflow rates and a good correlation for the large ones (horizontal asymptote). Moreover, it can be seen that the effective ventilation distance has a logarithmic relation to the flow rate. The linearization of this function allows us to assume that for common ventilation conditions, the effective distance is proportional to the logarithm of the flow rate (Figure 11b).

**Figure 11.** (**a**) Effective zone (ez) versus flow rate (Q) for scenario 4. The shaded zone indicates the recommended distance (4 <sup>√</sup> *<sup>S</sup>* to 6 <sup>√</sup> *S*) between the duct end and the face; (**b**) the effective zone (ez) versus the log flow rate (Q) for scenario 4.
