## 3 Results and Discussion
### Statistical Analysis
#### 3.2.2 Multi-Response Optimization Using Desirability Function  
The optimization analysis was performed to establish the best combination of input parameters (adjusting nozzle position, seat to valve distance) leading to the simultaneously expected evolution of output parameters (fluid velocity, shear stress and turbulent kinetic energy). In the case of fluid velocity, the aim is to maximize it, and the shear stress expected solution is to minimize it, while for turbulent kinetic energy, two situations were investigated since TKE provides a measure of turbulence intensity. Higher TKE levels could be necessary to dissipate excess energy or prevent pressure buildup within the system. In certain applications, such as safety relief valves in high-pressure systems, larger TKE values might indicate that the valve is effectively handling the release of energy and maintaining system safety. In some cases, is preferred to minimize TKE to reduce turbulence within the valve or the downstream flow. Lower TKE values can lead to more stable and controlled flow conditions, which can be advantageous in situations where excessive turbulence could lead to issues like noise, vibration, or erosion of valve components. Smaller TKE values might be desirable when the primary objective is to achieve a more laminar and less turbulent flow. Figure 31: Main effects plot for shear stress: (**a**) air; (**b**) water. Figure 32: Main effects plot for turbulent kinetic energy: (**a**) air; (**b**) water. The optimization was made utilizing the Desirability Function through Minitab 19 software. Desirability analysis assigns values on a scale from zero to one, where a score of one represents the utmost level of suitability. For the case when the importance is the same for each response, the composite desirability \(D\) is calculated with the formula [33; 34]:  
\[D=\left(d_{1}\cdot d_{2}\cdot\ldots\cdot d_{n}\right)^{1/n} \tag{5}\]  
where \(n\) is the number of responses, \(d_{i}\) represents the desirability for each individual response, calculated as [17]:  
* when the goal is to maximize the response desirability: \[ \begin{array}{l}d_{i}=0,\text{ if }y_{i}<L_{i}\\ d_{i}=\frac{(y_{i}-L_{i})\cdot r_{i}}{(T_{i}-L_{i})},\text{ if }L_{i}\leq y_{i} \leq T_{i}\\ d_{i}=1,\text{ if }y_{i}>T_{i}\end{array} \] (6)
* when the goal is to minimize the response desirability: \[ \begin{array}{l}d_{i}=0,\text{ if }y_{i}>U_{i}\\ d_{i}=\frac{(U_{i}-y_{i})\cdot r_{i}}{(T_{i}-L_{i})},\text{ if }T_{i}\leq y_{i} \leq U_{i}\\ d_{i}=1,\text{ if }y_{i}<T_{i}\end{array} \] (7)
* \(y_{i}\), \(T_{i}\), \(L_{i}\), \(U_{i}\) represent the predicted value, target value, lowest and highest value, respectively, of the analyzed response. Tables 6 and 7 present the rankings assigned to various configurations of input parameters for the air, respectively, and water environment. The configuration with the highest desirability score in the analysis (highlighted in Tables 6 and 7) is considered more suitable according to the criteria and objectives defined for the analysis. The plots for optimization (Figures 33 and 34) visually depict the impact of individual factors (presented in columns) on the responses or composite desirability (displayed in rows). In these graphs, the red lines indicate the existing factor settings, and the red numbers at the top of each column represent the current levels of these factors. The horizontal blue lines, along with their associated numbers, show the responses linked to the current factor levels. The optimization process indicated the following combination of input parameters to achieve the expected outcomes:  
\(\surd\) for the air environment: S-1/H2 for \"greater is better\" option of turbulent kinetic energy and S0/H1 for \"smaller is better\" option of turbulent kinetic energy;