## 3 Results and Discussion
### Additional Set of 30 Possible \(n\)-Tuples to Match the Experimental Results  
Thanks to the internal numerical optimization routine of the RSM software, it has been possible to generate further 30 \(n\)-tuples of values for the unknown parameters, which, according to the above-reported regression models, should be able to match the CFD-FVM model with the experimental results. These considered 30 \(n\)-tuples are only the \(n\)-tuples with the highest value of desirability (see Table 8). The same table also shows the numerical estimates for the three temperatures, calculated with the RSM models stated above. It is worth noting that, due to the linearization of the phenomena made by the RSM models, some estimates for the three temperatures can correspond to a physically impossible situation where \(T\leq T_{3}\) (cases nr. 11, 22, 23). As the CFD-FVM software operates instead according to rigorous physical principles, these \(n\)-tuples would originate for sure high prediction errors (in particular in correspondence with \(T_{3}\)), but, as a general rule, they should be, however, simulated to have some further points useful to correct the RSM models. These \(n\)-tuples were subsequently inquired via the CFD-FVM software. The results coming from the simulation have been collected in the following table (Table 9), together with the absolute and percentage errors (related to the experimental target values), and the above-defined metrics, based on these errors, used to evaluate each \(n\)-tuple. Figure 9 gives a graphical representation of the evaluation metrics. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Nr. **} & \multicolumn{8}{c}{**Factors (Unknown Independent Parameters)**} & \multicolumn{8}{c}{**RSM Optimization**} \\  & \(\mathbf{T}\) & \(\hat{m}_{ex}\) & \(\mathbf{H}_{h}\) & \(\mathbf{H}_{ex}\) & \(\mathbf{e}_{h}\) & \(\mathbf{e}_{ex}\) & \(\mathbf{C}_{h}\) & \(\mathbf{C}_{r}\) & **Desirability** & \(\mathbf{T}_{1}\) & \(\mathbf{T}_{2}\) & \(\mathbf{T}_{3}\) \\  & **K** & \(\mathbf{kg\cdot s^{-1}}\) & \(\mathbf{W\cdot m^{-2}\ K^{-1}}\) & \(\mathbf{W\cdot m^{-2}\ K^{-1}}\) & - & - & \(\mathbf{W\cdot m^{-1  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Simulation Outputs**} & \multicolumn{3}{c}{**Absolute Errors**} & \multicolumn{3}{c}{**Percentage Errors**} & \multicolumn{3}{c}{**Evaluation Metrics**} \\ \cline{2-13} \multicolumn{1}{c}{**Nr. **} & \multicolumn{1}{c}{\(\begin{array}{c}\textbf{\ \ \ \ \ \ \ \ \ \ \ \ \ \  
The positioning of the points in the Cartesian plane of Figure 9 (left) shows substantial parallelism with respect to the bisector of the quadrant (the slope of the interpolation line is approximately unitary) and an average deviation of about 2.5 percentage points. Only six points are below the threshold of 14.14 of the merit function, and only four points have both coordinates below the threshold of 5%, commonly considered the maximum acceptable deviation threshold for technical system simulations. More evident individuation of the best \(n\)-tuples can be seen in the histogram of Figure 9 on the right: the higher the bars, the higher the merit function, the closer the CFD-FVM results to the real case (P\({}_{0}\)), the better those \(n\)-tuples to tune the FVM model. The higher four bars in the represented graphic correspond to \(n\)-tuples 1, 2, 9, and 18. The best-ever combination of this set of 30 combinations of factors is the \(n\)-tuple number 9, achieving a value for \(M_{f}\) of 31.3, a mean of absolute errors of 1.8% and a maximum absolute error of 2.6% (the three temperatures are: 458, 535, 662 \({}^{\circ}\)C, hence, respectively, at +8, \(-\)5, and \(-\)18 \({}^{\circ}\)C from the target values of case P\({}_{0}\)). Although this result would be already more than acceptable, another iteration of the optimization procedure has been performed to better illustrate the method and demonstrate that it allows even a further improvement.