## 2 Calculation of the New Thermal Constitutive Matrix \([M_{\tau}]\)
### New \([M_{\tau}]\) Matrix  
In the previous section, the terms for \(A_{ij}\) have been obtained. Now we can build the matrix \([A_{\tau}]\),  
\[\begin{array}{lcl}[A_{\tau}]&=&\frac{\Delta}{\Delta}\left[\begin{array}{ ccc}(y_{3}z_{2}-y_{2}z_{3})&(y_{1}z_{3}-y_{3}z_{1})&(y_{2}z_{1}-y_{1}z_{2})&(y_{3}z_{ 0}-y_{0}z_{3})&(y_{1}z_{0}-y_{0}z_{1})&(y_{0}z_{2}-y_{2}z_{0})\\ (x_{2}z_{3}-x_{3}z_{2})&(x_{3}z_{1}-x_{1}z_{3})&(x_{1}z_{2}-x_{2}z_{1})&(x_{0} z_{3}-x_{3}z_{0})&(x_{0}z_{1}-x_{1}z_{0})&(x_{2}z_{0}-x_{0}z_{2})\\ (x_{3}y_This problem has the following analytical solution  
\[\tau_{x}\ =\ (\tau_{2}-\tau_{1})\frac{\ln\frac{r_{x}}{r_{1}}}{\ln\frac{r_{2}}{r_{1}}}+ \tau_{1} \tag{50}\]  
with \(r_{1}\) and \(r_{2}\) being the internal and external radius of the tube, respectively; \(r_{x}\) the radius of an intermediate point between \(r_{1}\) and \(r_{2}\); and \(\tau\) the temperature at the point considered. The profile of the temperatures of a parametric section made in the FEMM model has been compared with the values obtained from complete analytical Equation (50), as can be seen in Figure 3. In this parametric section, the temperatures were measured for two concrete meshes, one with 2517 nodes and the other with 1,013,144 nodes. The boundary conditions were the following. The highest temperature was on the outer surface of the cylinder (80 \({}^{\circ}\)C), and the lowest was on the inner face of the cylinder (30 \({}^{\circ}\)C). The top and bottom cylinder covers were considered perfect insulators. There was no heat source inside the cylinder. The initial and final conditions were the same, because it is a stationary process. Figure 4 shows the results obtained for the two meshes analyzed with FEMM and the analytical solution for the parametric cut shown in Figure 3. We observed that the three curves were coincident, which allowed us to conclude that the solution with 2517 nodes was already a very good solution. Figure 3: Temperature distribution in a section of a tube. Table 1 shows the temperature values at a characteristic point located 0.75 m from the cylinder axis, and compares the temperature obtained by the exact analytical equation with the result obtained by FEMM for 2517 and 1,013,144 nodes. As it can be seen, the error converged for 2517, and was almost coincident with the analytical value, with a 0.0223% difference percentage. Table 2 shows the metrics used to validate the proposed mathematical models. Metrics are statistics used to assess the errors between the standard and proposed model. As stated above, our standard model is the FEMM. We compared the results obtained with the proposed models, and, from the contrast of both results (the FEMM and studied model), we obtained the proposed metrics. R\({}^{2}\) is the coefficient of determination. RMSPE is the root mean square percentage error. MAEP is the mean absolute percentage error. PBIAS is the percentage bias. The metrics indicate the validity of FEMM as a reference standard, because the values obtained are very close to the optimum. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mesh [Nodes]** & **2517** & **1013144** \\ \hline Temperature (\({}^{\circ}\)C) a 0.75 m & 72.7585 & 72.7585 \\ (Analytical ref. 72.7422 \({}^{\circ}\)C) & 0.0162 & 0.0162 \\ Difference of temperatures (\({}^{\circ}\)C) & 0.0223 & 0.0223 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Temperatures for the validation of FEMM. Figure 4: Validation of FEMM vs. analytical solution. The temperatures obtained were close enough to the analytical solution to consider the FEMM a good benchmark. In the distribution of errors (Figure 5), there is a bias respect to the zero error. This indicates a certain systemic error in the FEMM, which we assumed (0%, 0.03%; see Figure 5). The simulations carried out were developed on a PC type Dell, Intel@Core (TM) i7-3820, 3.6 GHz, 32 GB RAM.