*3.2. Mathematical Model for the E*ff*ective Distance*

Figure 12a shows the effective distance as a function of the flow rate for each scenario. Raw data were adimensionalized using the following coefficients, where the second coefficient was derived from the Reynolds number:

$$f\_1 = \frac{\varepsilon\_z}{D\_{\parallel}},$$

$$f\_2 = \frac{Q\rho}{D\_{\parallel}\mu},$$

where:

• *ez*: effective distance (m);


Figure 12b shows the transformed data according to the first dimensionless coefficient and the logarithm of the second. It was observed that if the curves were adimensionalized, they overlapped. That is to say, the solution is proportional if tunnel and duct sections remain constant. These results imply that this section alone should no longer be a parameter to study, and further work should be undertaken to study other parameters—For instance, the ratio between tunnel and duct sections. In general terms, it can be observed that the larger the tunnel section and the clean airflow rate, the longer the maximum effective distance. These results are in accordance with Feroze and Genc [29].

**Figure 12.** (**a**) Maximum effective distance (ez) versus flow rate (Q); (**b**) dimensionless effective distance versus the logarithm of the dimensionless effective flow rate, for the different scenarios.

From the data in Figure 12b, a linear least squares (LLS) approximation was performed so that Equation (10) was obtained. This equation allows for the approximation of the effective distance as a function of the ventilation flow rate and the hydraulic diameter, which is defined as:

$$c\_z = D\_h \left( 4.40 \operatorname{L\eta} \frac{Q \rho}{\mu D\_h} - 28.36 \right) \tag{10}$$

$$D\_h = \frac{4A}{P\_w}$$

where:

