## 2 Materials and Methods
Exact solution (\(\varphi_{\text{exact}}\)) and its associated uncertainly (UC) can be calculated as [47]  
\[\varphi_{\text{exact}}=\ \varphi_{1}+\ \frac{\varphi_{1}-\varphi_{2}}{r_{21}^{p}-1} \tag{13}\]  
\[p=\ \frac{1}{\ln(r_{21})}\ast\left[\ln\left(\frac{\varepsilon_{32}}{\varepsilon _{21}}\right)+\ln\left(\frac{r_{21}^{p}-sign\left(\frac{\varepsilon_{32}}{ \varepsilon_{21}}\right)}{r_{32}^{p}-sign\left(\frac{\varepsilon_{32}}{ \varepsilon_{21}}\right)}\right)\right] \tag{14}\]  
\[\text{where }r_{21}=\ \frac{h_{2}}{h_{1}};\ r_{32}=\frac{h_{3}}{h_{2}};\ \varepsilon_{21}=\varphi_{2}-\varphi_{1}\ and\ \varepsilon_{32}=\varphi_{3}-\varphi_{2} \tag{15}\]  
\[UC=F_{s}\ *\ (\varphi_{\text{exact}}-\varphi_{1}) \tag{16}\]  
where \(\varphi_{i}\) are numerical solutions with i = 1,2 and 3 of the analyzed physical quantity (in this paper outlet flow temperature and outflows velocity) under three different mesh sizes \(h_{1},\ h_{2}\) and \(h_{3}\), respectively, where \(h_{1}<h_{2}<h_{3}\). [PERSON]'s analysis of the employed mesh is summed in Table 4.  
Exact temperature value is 31.801 \({}^{\circ}\)C what that entails a numerical uncertainly of temperature of 0.0109 \({}^{\circ}\)C. In the case of the velocity analysis, exact value achieve a value of 0.4299 m/s and its related uncertainly is 0.0002 m/s. These results reinforce the obtained results in the performed quality analysis. According to the obtained results, a structured mesh composed of 280.000 elements has been selected in this study. It allows the simulation of the installation with a reasonable computational cost and an adequate quality [46]. On the one hand, the orthogonal quality \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **1** & **2** & **3** \\ \hline \(h_{i}\) (cm) & 0.23 & 0.31 & 0.50 \\ Outlet flow temperature (\({}^{\circ}\)C) & 31.810 & 31.818 & 31.856 \\ Outflow velocity (m/s) & 0.4298 & 0.4296 & 0.4289 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters employed in the [PERSON]’s method. Figure 2: Quality mesh study. average value is 0.886 with the least value of 0.79. On the other hand, the skewness average has a value of 0.227 with a maximum value of 0.57. The mesh of the pipe is shown in Figure 3. These mesh elements have been colored according to their quality. The PVT and the PVT+TEG models have been validated through experimental data collected on the test bench. Figure 4 (conventional PVT panel) and Figure 5 (PVT + TEG panel) show the comparison between the numerical results obtained from the software Fluent-ANSYS and the experimental data of the outlet flow temperature of both panels. As can be observed, numerical results are nearly equal to the experimental ones. Figure 4: Comparison between experimental and numerical outlet flow temperature of the conventional PVT. Figure 5: Comparison between experimental and numerical outlet flow temperature of the PVT with integrated TEG modules. Figure 3: (**a**) Skewness and (**b**) orthogonal quality of the tube of the PVT panel simulated. The error between numerical and experimental data does not exceed 13.6%. Except for the test made at 10:45 AM. At this time, error achieves the value of 38.0% and 42.7% in the simulation of a conventional PVT and in that of a PVT + TEG panel, respectively. The mass flow of the working fluid at this time is 0.39 g/s. This value is low. Additionally, it is observed in both Figures 4 and 5 that the increment in the error at 15:15 a.m. At this time, low flow is not the source of error. Analyzing experimental data from both hybrid panels, it can be concluded that the flow is on a transient scenario because impulsion pump goes into operation few minutes before the capture of data. The finding suggests that the elaborated model has limited application with low mass flow experiments and transient situations. Besides, the electrical production of cells embedded on the PVT panels has been validated.