*3.3. Mathematical Model Correction with the Mixture Model*

The concentration of NO2 along the tunnel axis for different ventilation times is shown in Figure 13. It can be seen that the majority of the gas dilutes in the first 12 min near the duct end (upper right part of the figure). This phenomenon corresponds to the aerodynamic sweep of NOx by the clean air jet. Moreover, it can be observed that the clearance times are longer than 12 min. The NOx concentration in the dead zone (left side of the figure) does not decrease significantly with the increasing ventilating time. This fact leads to the existence of a remaining concentration that can only be evacuated by diffusion. The minimum concentration remaining in the effective zone (bottom right of the figure) is the consequence of the existence of a dead zone, so the NOx that exits this area by diffusion must cross the effective zone to leave the tunnel dead end, provoking a residual concentration of NOx.

**Figure 13.** Concentrations of NO2 inside the dead end for different ventilating times. The data correspond to scenario 4 and an air velocity of 20 m/s at the duct inlet.

The diffusion mechanism is not reliable or fast enough to be taken into account for tunnel ventilation purposes, so only the distance swept by the air and not the distance cleaned by diffusion has been considered for the determination of the final effective ventilation distance corrected by the mixture model. A comparison of the data derived from the aerodynamic and the mixture model simulations is given in Figure 14. According to the mixture model, the recommended distance between the duct-end and the face is shorter than that in the aerodynamic model because in this case the time needed to evacuate the gases is taken into consideration. Moreover, if the ventilation arrangement is one that does not permit the formation of a dead zone, as could be possible in practice, the ventilating time would be shorter due to the delay imposed by NOx diffusion inside the dead zone in the evacuation of the gases. Thus, the mixture model dilution times shown in Figure 13 are longer than the real ones as no dead zone is expected in a real tunnel ventilation situation.

**Figure 14.** Maximum possible duct end-to-face distance for some ventilation of the dead zone, obtained with the aerodynamic and mixture models.

Finally, upon studying the relationship between the maximum distance recommended by the mixture and the aerodynamic models (Figure 14), it can be observed that this quotient is about 0.8. Thus, a coefficient, C1 = 0.8, was added to Equation (10) and thus Equation (11) is obtained:

$$d\_{\rm s} = C\_1 D\_h \left( 4.40 \, Ln \frac{Q \rho}{\mu D\_h} - 28.36 \right) \tag{11}$$

It is important to keep in mind that Equation (11) does not search for the optimum distance for mine ventilation. On the contrary, it establishes a maximum duct end-to-face distance over which forced ventilation can evacuate blasting gases in a maximum of 20 min. Thus, these results do not disagree with the conventional recommendations that this distance should not to exceed (4 <sup>√</sup> *<sup>S</sup>*–6 <sup>√</sup> *S*), but longer clearance times should probably be expected with this expression (e.g., [19]).

Equation (11) is plotted for practical reasons in Figure 15. This graph allows for the approximation of the maximum duct end-to-face distance for some ventilation in terms of the tunnel section, clean airflow rate and ventilation time, taking into account the mixture model. Firstly, it can be observed that the larger the tunnel section, the longer the effective distance. This fact agrees with legal requirements. Furthermore, our expression takes into consideration the clean airflow rate sweeping the face, which is not considered by legal specifications. In this way, our work indicates that the larger the airflow rate, the longer the effective distance. Better correlations between legal requirements and our expression were obtained for lower airflow rates, in the case of the smaller sections. On the contrary, less disparity was achieved in the case of higher airflow rates for the larger sections.

**Figure 15.** Maximum possible duct end-to-face distance for some ventilation of the dead zone, in terms of airflow rate and tunnel section.

#### **4. Conclusions**

The main aims of drivage ventilation are the evacuation of the blasting and machinery gases as well as the supply of the required air for the workers' breathing. This type of ventilation is usually performed by means of a forcing system. In this study, a full-scale three-dimensional CFDs model of a forcing system was created to assess the ventilation performance of a dead-end gallery. The simulation was performed for different parameters, namely, tunnel section, flow rate and position with regard to the tunnel axis.

The existence of a dead zone between the duct end and the face influences the ventilation system setup. This zone has been traditionally defined as a region of reduced velocity of the air current that hinders the mixture of gases with fresh air. Our study suggests that common approaches defining the dead zone as the region in which air velocity tends to zero are not accurate. This fact is consequence of the formation counter vortexes that recirculate gases inside this zone obstructing their evacuation. Thus, although very little air renovation takes place in this zone, the gases inside it are actually in motion.

As a result of our work, we obtained an expression to approximate the maximum effective ventilating distance for some ventilation as a function of the tunnel section and the ventilation flow rate, considering a CFDs analysis with a mixture model. The equation obtained should be valid for any tunnel of similar geometry. Our calculations indicate that the maximum effective distance is larger than that suggested by traditional legal requirements, thus confirming their level of conservativism. The results obtained may assist practicing engineers in improving ventilation productivity during tunnel construction works. Future research should also consider dust when establishing an effective duct end-to-face distance for dead-end tunnel ventilation.

**Author Contributions:** M.G.-D. improved the methodology, performed the simulations and co-wrote the original draft. C.S. conceptualized the problem, analyzed the results and co-wrote the original draft. C.M.-G. reviewed and edited the draft manuscript, improved visualization and validated the data. B.P. performed the formal analysis, provided the resources and acquired founding. All authors discussed the results and contributed to the final manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** M. G.-D. was supported by the Spanish "Ministerio de Educación, Cultura y Deporte" within the "FPU" Program (grant number FPU15/04375). C.M.G. was supported by the Spanish "Ministerio de Educación Cultura y Deporte" within the "Doctorados Industriales" Program (grant number DI-17-09596).

**Conflicts of Interest:** No potential conflict of interest was reported by the authors.
