## 2 Calculation of the New Thermal Constitutive Matrix \([M_{\tau}]\)
### Numerical Simulation with \([M_{\tau}]\)  
In the geometric model, the boundary conditions were the same as in the previous case. Therefore, the numerical simulation has been carried out with the tube indicated in Figure 6, and the FEMM model used is shown in Figure 7. Similarly, the three-dimensional parametric section can be seen in Figure 6. However, in this case, the thermal conductivity matrix was \([M_{\tau}]\). The objective was to check the validity of the new thermal conductivity matrix \([M_{\tau}]\), and that it obtains better results than \([M_{\lambda}]\). On the one hand, a calculation with a 2D FEMM of 11,042 nodes has been made, and, on the other hand, five calculations was made with CM for an increasing number of nodes ranging from 102 to 13,136 nodes, using \([M_{\tau}]\). From 6428 nodes, the results are practically coincident between 3D CM and 2D FEMM, which we consider as a solution to the problem. The results obtained are shown in the graph Figure 10. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mesh [nodes]** & **108** & **182** & **4461** & **9679** & **13,136** & **References** \\ \hline R2 [0, +1]. Optimum: +1 & 0.9929 & 0.9982 & 1.0000 & 1.0000 & 1.0000 & [26] \\ RMSPE [\(-1\), +1]. Optimum: 0 & 0.0518 & 0.0122 & 0.0029 & 0.0013 & 0.0012 & [27] \\ MAEP [\(-1\), +1]. Optimum: 0 & 0.0471 & 0.0086 & 0.0025 & 0.0011 & 0.0011 & [27] \\ PBIAS [\(-1\), +1]. Optimum: 0 & 0.0450 & \(-0.0051\) & 0.0025 & 0.0011 & 0.0009 & [28] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Metrics for thermal simulation using \([M_{\lambda}]\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mesh [Nodes]** & **108** & **182** & **4461** & **9679** & **13,136** \\ \hline Temperature (\({}^{\circ}\)C) at 0.75 m & 67.8991 & 61.7719 & 64.0343 & 63.6987 & 63.6975 \\ (FEMM ref: 64.0653 \({}^{\circ}\)C) & 3.8338 & \(-2.2934\) & \(-0.0310\) & \(-0.3666\) & \(-0.3678\) \\ Difference of temperatures (\({}^{\circ}\)C) & 5.9842 & \(-3.5798\) & \(-0.0484\) & \(-0.5722\) & \(-0.5741\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Temperature at a point of the tube with two conductivities using \([M_{\lambda}]\). The distribution of errors when comparing CM with the temperatures obtained in FEM is shown in Figure 11. As can be seen, the bias has decreased to 0.05%. Hence, operating with a new thermal conduction constitutive matrix \([M_{\tau}]\) produces a smaller error than operating with \([M_{\lambda}]\). As we expected, the errors decrease as the mesh density increases, as can be seen in Table 5.  
The temperatures in Table 3 have been taken at one point placed radially at 0.75 m from the center of the tube, and axially at half its height (1.00 m). Applying the metrics R\({}^{2}\), RMSPE, MAEP, and PBIAS, the following values were obtained (see Table 6). Figure 11: Errors distribution between CM and FEM using \([M_{\tau}]\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mesh [nodes]** & **102** & **108** & **2328** & **6428** & **12,895** \\ \hline
1 & 68.4981 & 67.7963 & 63.7282 & 63.7047 & 63.6975 \\ Difference of temperatures [\({}^{\circ}\)C] & 4.8022 & 4.1004 & 0.0323 & 0.0088 & 0.0016 \\ Percentage difference [\%] & 7.5393 & 6.4375 & 0.0507 & 0.0138 & 0.0025 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Temperature at a point of the tube with two conductivities using \([M_{\tau}]\). Figure 10: Temperatures in the tube with two thermal conductivities using \([M_{\tau}]\). Observing Table 6, we conclude that the errors are lower when the mesh density is increased, and also that errors are very small for meshes with low densities. In any case, the errors quickly converge towards the optimum of the metrics. Observing Table 7, it is verified that the thermal constitutive matrix \([M_{\tau}]\) that we provide in this article behaves much better than with \([M_{\lambda}]\).