# Temperature Prediction Model in the Main Ventilation System of an Underground Mine
## 4 Results and Discussion  
The generalized linear model (Equation (6)) and the model with interactions (Equation (7)) exhibited correlations of \(R^{2}\) = 0.858 and 0.887, respectively. However 5.96% of the data recorded by the sensors displayed irregular behavior compared to the dataset, i.e., extremely high temperatures (greater than \(\pm\)1.5 IQR (interquartile range)), probably due to the exhaust pipe of the diesel equipment passing close to the sensor during some specific maneuvers in the drift. The data analysis also suggested that some outliers could have been caused by erroneous sensor recording [38]. These outliers were avoided for the creation of the model. The model fully developed with only regular data is displayed in Equation (8). The \(R^{2}\) of the model with interactions was higher than that of the generalized linear model; thus, it was used moving forward, attaining \(R^{2}\) = 0.933 and an estimation error of \(\pm\)2 \({}^{\circ}\)C, with 100% of the residual values lying in the range of \(\pm\)2.5 \({}^{\circ}\)C (Figure 5), which can be considered acceptable for mine ventilation analysis according to the literature [27]. Furthermore, it can be modified according to the needs of the user and characteristics of the mine. \[ \begin{split} t_{f}=0.7142+0.9887t_{i}+0.004419D_{x}+0.02139D_{c} -1.509\times 10^{-4}t_{i}D_{x}-8.057\times 10^{-4}t_{i}D_{c}\\ +1.024\times 10^{-6}D_{x}D_{c}\end{split}  \tag{8}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Stretch** & **Nodes** & **Dx (m)** & **Dc (m\({}^{3}\)/s)** \\ \hline S1 & n\({}_{1}\)–n\({}_{2}\) & 1300 & \(-\)30 \\ S2 & n\({}_{2}\)–n\({}_{3}\) & 1700 & \(-\)127 \\ S3 & n\({}_{3}\)–n\({}_{4}\) & 670 & \(-\)3 \\ S4 & n\({}_{4}\)–n\({}_{5}\) & 1120 & \(-\)2 \\ S5 & n\({}_{5}\)–n\({}_{6}\) & 810 & \(-\)31 \\ S6 & n\({}_{6}\)–n\({}_{7}\) & 530 & \(-\)22 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of the stretches used in the model. Figure 4: Scheme of the prediction system created. Here, c\({}_{i}\) is the airflow at the beginning of the stretch (m\({}^{3}\)/s), c\({}_{f}\) is the airflow at the end of the stretch (m\({}^{3}\)/s), x\({}_{i}\) is the initial relative position (m), and x\({}_{f}\) is the final relative position (m). Figure 6 displays the stretches used in the analysis of the estimation error using the node-to-node and multi-node approaches. The \(x\)-axis includes the combination of nodes used in the multi-node approach, as detailed in Section 3.3. The highest error was concentrated in nodes 1 and 2, when the lag step was 1, 2 or 3. This was due to the ventilation circuit of the case study (a U-shaped system) having a high variability in temperature at the entrance of the intake, as well as potential problems with connections between the return and intake drifts through doors, and other disturbing factors in areas with a high pressure difference. This issue coincided mainly with nodes 1 and 2, which were within an area with a high airflow and pressure difference. If only nodes 3-6 were considered, the relative error was reduced to \(<\)10%. When the estimated temperatures were compared to the observed values in Figure 7 (i.e., those at the end of the stretch (t\({}_{\text{f}}\))), it can be seen that the estimation was quite good, with a deviation lower than 2 \({}^{\circ}\)C in most of the cases. Furthermore, the correlation was also good, whether using node-to-node or multi-node analysis with a lag step of 2 and 3 nodes (see Figure 8); there was important consistency between the temperatures measured and those estimated at the end of the stretches. The temperature  
Figure 5: Residual values from the linear model with interactions. Figure 6: Error distribution as a function of stretch, node-to-node (red), and multi-node lag steps (blue represents two steps and gray represents three steps). oscillations corresponded to the different seasons of the year, influencing the effective temperature at the bottom of the shaft and along the intake.