## 2 Materials and Methods
### Use of RSM to Tune the CFD-FVM Model (Step 2.1)
More in detail, every time the RSM was used in the above-illustrated general procedure, after analyzing the data, the software automatically selected for all the models a multilinear model of the 8 variables (\(x_{i}\) with \(i=1\) to 8) for each of the investigated responses \(y_{j}\), i.e., for the three temperatures \(T_{j}\) with \(j=1\) to 3 (\(a_{0}\) is the interception coefficient, and \(a_{i}\), are the coefficients of the linear terms) [74]:  
\[y_{j}=f(x_{i};i=1\text{ }to\text{ }8)=a_{0}+\sum_{i=1}^{8}a_{i}x_{i}=T_{j} \tag{4}\]  
The evaluation of the mathematical models of \(T_{1}\), \(T_{2}\), \(T_{3}\) (obtained via RSM in more than a point of the above-described procedure) was carried out, metrologically speaking, in terms of accuracy of the predicted values with respect to the experimental results. This means that the outputs of the associated CFD-FVM model were evaluated in terms of:  
* predictive capability as such (in terms of closeness of the CFD-FVM model outputs to the numerical predictions of the RSM in correspondence with each \(n\)-tuple of setup parameters);
* optimization capability (in terms of closeness of the CFD-FVM model output to the target experimental results in correspondence with each \(n\)-tuple of setup parameters). Figure 7: Detailed procedure followed to arrive at a tuned CFD-FVM model as regards the temperature at the three chosen points along the exhaust line (domains of pertinence: yellow—experimentation, blue—numerical simulation, red—numerical analysis/optimization). Metric to Evaluate the \(n\)-Tuples Proposed by RSM-Based Optimization on the Basis of the CFD-FVM Results  
The evaluation of the many \(n\)-tuples indicated by the RSM-based optimization procedure (and, in particular, the individuation of the best \(n\)-tuple) has been performed on the basis of the values of the temperatures resulting from the CFD-FVM simulations, after the CFD-FVM model has been run with the independent parameters set as the values of the \(n\)-tuples. At this purpose, a merit function \(M_{f}\), useful to rank the \(n\)-tuples using a single value, has been defined, starting from the following two quantities:  
* the mean of absolute percentage errors (\(\epsilon_{mean,abs}\)), defined as the arithmetic mean of the absolute value of relative errors (expressed as percentages) of the difference between each value obtained from the CFD-FVM simulations \(T_{i}\) from the target value \(T_{target,i}\) for the same temperature;
* the maximum absolute percentage error (\(\epsilon_{max,abs}\)), defined as the greatest among the absolute values of relative errors (expressed as percentages) of the difference between each value obtained from the CFD-FVM simulations \(T_{i}\) from the target value \(T_{target,i}\) for the same temperature;  
If each couple (\(\epsilon_{mean,abs};\epsilon_{max,abs}\)) is represented as a point on a proper Cartesian plane, a merit function, useful to evaluate the adherence of a CFD-FVM simulation to the reality by accounting both the described quantities at the same time, can be defined as the inverse of the Euclidean distance of these points from the origin:  
\[\left\{ \begin{array}{l}\varepsilon_{mean,abs}=\frac{1}{3}\cdot\sum\limits _{i=1}^{3}\left|100\cdot\frac{T_{i}-T_{target,i}}{T_{target,i}}\right|\\ \varepsilon_{max,abs}=\max\left\{i=1:3;\left|100\cdot\frac{T_{i}-T_{target,i} }{T_{target,i}}\right|\right\}\end{array} \right.\Rightarrow M_{f}=100\cdot \left(\epsilon_{mean,abs}^{2}+\epsilon_{max,abs}^{2}\right)^{-\frac{1}{2}} \tag{5}\]  
If the target values are the reference values from the experimentation as referred to the operative point P\({}_{0}\) (450, 540, and 680 \({}^{\circ}\)C), the merit function \(M_{f}\) is a valid aid to individuate the best \(n\)-tuple to tune the FVM model so as it is as close as possible to the reality in terms of outputs. The higher the value assumed by \(M_{f}\), the more suitable is the \(n\)-tuple for this purpose (i.e., the better the adherence of the associated simulation to reality).