## Appendix A Remarks on the RSM-Based Optimization  
The numerical optimization of multiple responses, proposed by the used statistical software, is based on the maximization of a (global) \"desirability\" function (\(D\)) [75]. This desirability function was used by the RSM software to identify the 30 \(n\)-tuples of factors that, if set in the CFD-FVM program, would most likely result in an output close to the experimental situation. The desirability function is an objective function that keeps into account the reaching of multiple goals together, each one formulated through a \"utility\" function \(d_{i}\), i.e., a dimensionless individual desirability, in this case having a value included in the variability range \([y_{\min,i};y_{\max,i}]\) and depending by the distance of each response \(x_{i}\) from its target value \(x_{i,i}\)[76]. More in detail, \(D\) ranges from zero to one (at the simultaneous matching of all goals \(d_{i}\)) and it is formulated as the geometric mean of all these utility functions, as follows:  
\[d_{i}=\left\{\begin{array}{l}1\text{ }for\ y_{i}=y_{t,i}\\ \frac{y_{t,i}-y_{t,i}}{y_{t,i}-y_{\min,i}}\text{ }for\ (y_{\min,i}\leq y_{i}\leq y _{t,i})\\ \frac{y_{t,i}-y_{t,i}}{y_{\max,i}-y_{t,i}}\text{ }for\ (y_{t,i}\leq y_{i}\leq y _{\max,i})\\ 0\text{ }for\ (y_{i}<y_{\min,i}\lor y_{i}>y_{\max,i})\end{array}\right. \tag{9}\] \[D=\left(\prod_{i=1}^{n}d_{i}^{r_{i}}\right)^{\sum_{i=1}^{r_{i}} \left(\text{if all weights are equal}\right)}\prod_{i=1}^{n}d_{i}\]  
where:  
* \(n\) is the number of simultaneously-optimized responses, i.e., for which a goal is formulated; in the present case \(n=3\) because a specific \(n\)-tuple is evaluated by the distance of its predicted outputs of \(T_{1}\), \(T_{2}\), \(T_{3}\) from the target values corresponding to the operative point P\({}_{0}\);
* \(d_{i}\) is the \"desirable range\" for a single generic response \(i\), i.e., a metric related to the closeness of an output from its target value; the desirable ranges are all from zero to one (i.e., from least to most desirable value, respectively);
* \(r\) is a weight accounting for the importance of that factor over the others (in the present case, all weights were set at the same value, meaning that no temperature of the three is more important than the others to be matched).  
If any of the responses \(y_{i}\) falls outside their desirability range, the overall function becomes zero. For a simultaneous optimization, each response must range within an assigned low and high variation value (not necessarily equally distant from the target value), and the individual desirability is formulated in this case as \"being equal to a target value\". If the weight is 1 for a specific response, the desirability values are linearly increasing from 0 (response lower or equal to the minimum value, or higher or equal to the maximum value for that response) to 1 (response perfectly equal to the target); see Figure 11. Finding an initial feasible region where all responses are within their ranges canbe difficult. So the RSM program used a penalty function in a downhill simplex (Nelder-Mead) multidimensional pattern search [77] that converges at either a stationary point or a design space boundary. Some example of applications can be found in [78; 79].