## 4 Verification and Validation
The system was simulated in RELAP5-3D by means of pipe components (P1, P2, and P3) and a branch (B). The three pipe components are characterized by 10 equally sized nodes. Similarly, the heat structures are assumed composed by 10 nodes and by the thickness divided into five meshes. Instead, in MELCOR 1.8.6 and 2.2, the three pipes and the branch are reproduced with four independent volumes, and the heat structures have only three meshes along the 5-mm thickness. Figure 4 presents an overview of the achieved results in the _TPE_ case for the RELAP5-3D code. This test was chosen, as it is the most usual utilization of the software. The two graphs in Figure 4 present four curves each: fictitious experimental data (black curve), best-estimate results obtained from the application of the best-estimate methodology (red curve), results of the default RELAP5-3D input deck (green curve), and results of the best-estimate RELAP5-3D input deck, i.e., an input deck adopting the best-estimated values of the parameters obtained from the methodology (blue curve). The good match between the best-estimate methodology results (red curve) and the fictitious experimental data supports the proper implementation of the methodology in the software. This conclusion is also supported by the smaller uncertainty of the best-estimate methodology results compared to that of the fictitious experimental data. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Parameters** & **Responses** & **Notes** \\ \hline
1 Resp 1 Par & K3 & Mass flow in J3 & Single time node \\
1 Resp 2 Par & K3, \(\varepsilon\) & Mass flow in J3 & Single time node \\
2 Resp 3 Par & K2, K3, \(\varepsilon\) & Mass flow in J2 and J3 & Single time node \\ Time & K2, K3, \(\varepsilon\) & Mass flow in J2 and J3 & Three time nodes \\ TPE (Time-Parameter-Evolve) & K2, K3, \(\varepsilon\), \(\dot{Q}_{3}\) & Mass flow in J2 and J3 & Three time nodes \\ TPE C\({}_{aa}\) full & K2, K3, \(\varepsilon\), \(\dot{Q}_{3}\) & Mass flow in J2 and J3 & As TPE plus a full size \(C_{aa}\) matrix \\ TPE C\({}_{aa}\) single & K2, K3, \(\varepsilon\), \(\dot{Q}_{3}\) & Mass flow in J2 and J3 & As TPE plus a single block \(C_{aa}\) matrix \\ TPE C\({}_{ar}\) full & K2, K3, \(\varepsilon\), \(\dot{Q}_{3}\) & Mass flow in J2 and J3 & As TPE plus a full size \(C_{ar}\) matrix \\ TPE C\({}_{ar}\) single & K2, K3, \(\varepsilon\), \(\dot{Q}_{3}\) & Mass flow in J2 and J3 & As TPE plus a single block \(C_{ar}\) matrix \\ TPE Sens full & K2, K3, \(\varepsilon\), \(\dot{Q}_{3}\) & Mass flow in J2 and J3 & As TPE plus a full size sensitivity matrix \\ TPE Sens single & K2, K3, \(\varepsilon\), \(\dot{Q}_{3}\) & Mass flow in J2 and J3 & As TPE plus a single block sensitivity matrix \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test cases matrix. Figure 4: Results of the TPE test case (RELAP5-3D code). Instead, the discrepancy between the best-estimate methodology results (red curve) and the best-estimate RELAP5-3D code results (blue curve) is due to the fact that the methodology predicts different best-estimated values for the various parameters at each time _node_, but usually only a single value can be used in the input decks of the system/thermal-hydraulics codes, such as RELAP5-3D and MELCOR. If only a value has to be used, the most straightforward way to define one is to calculate an average best-estimated value basing on those obtained from the methodology. Hence, the discrepancy between the two curves is a direct effect of this simplification. Nonetheless, in few specific scenarios, an additional source of discrepancy can arise from the usage of linearized sensitivity terms (Equation (5)). Depending on the scenario and on the model considered, one or more parameters may have an influence straying from linearity. In these cases, an iterative process (several calibrations one after another) may be required to reach a good agreement between experimental and computed results [4]. Nevertheless, this is not the case of the simple cases studied for the V&V of the BeSYC software.