*3.1. Baseline Computational Fluid Dynamics (CFD) Results*

Results obtained on the valve are shown in Figures 12–15. In particular, Figure 12 shows the pressure distribution on the walls of the fluid domain of the baseline geometry. It is clear that pressure was distributed according to the boundary conditions applied.

Other significative results are shown with two cross sectional views of the fluid domain: Figure 13 shows the pressure distribution, while Figure 14 is representative of the velocity behavior inside the domain.

Flux behavior inside the ports is also described with streamlines colored with the velocity magnitude in Figure 15.

The mass flux obtained with the baseline geometry (13.47 [kg/s]) was used as starting value for the optimization.

The objective was therefore to find the maximum possible mass flux compatible with the prescribed constraints.

**Figure 12.** Pressure distribution on Port A and P walls, baseline geometry.

**Figure 13.** Pressure distribution on sections of Port A and Port P, baseline geometry.

**Figure 14.** Velocity distribution on sections of Port A and Port P, baseline geometry.

**Figure 15.** Streamlines in the valve, baseline geometry.

#### *3.2. Optimizaion Results*

At the end of the DOE sequence solution process, CAESES® provides a detailed table of all the data used in the calculations. For each simulated design, the corresponding geometric characteristics as well as calculations results are provided. In this specific project, as previously mentioned, 90 design variants were tested. A chart mapping 67 solutions versus the obtained flow rate can be visualized in Figure 16.

**Figure 16.** Design of experiments (DOE) results, in terms of mass flow at the outlet of the valve.

The remaining 23 solutions are not shown in the chart because of calculations failure. This means either that the simulation didn't end correctly or that the geometry could not be built with the prescribed parameters. All the 90 ports geometries had volumes within the limits requested, so that they could be contained in the original valve compartment. Ports volumes were monitored during the optimization process, even if they were not considered as a strict constraint.

The best mass flux obtained was 14.38 [kg/s] that, compared with the baseline result of 13.47 [kg/s], provided a 6.8% flow-rate increment.

Table 2 sums up the geometric parameter values of the best solution, in comparison to the baseline geometry.


**Table 2.** Best design geometric values compared to the baseline geometry.

At the end of DOE sequence calculation, a parameter sensitivity analysis was performed to determine which parameter had the greatest influence on the mass flux.

Figure 17 shows the influence of the outer circle radius for Port A and P on the mass flux of the valve:

**Figure 17.** Outer circle radius impact on the optimization.

This parameter was the most effective in changing the mass flow value. In fact, small changes in the outer radius diameter provided a significant change in the mass flux: a 0.13 mm increment corresponded approximately a 1 kg/s mass flux increase.

The process continued with a 2-level "Tsearch" optimization, starting from the best Sobol sequence design. "Tsearch" optimization is an optimization method based on the local tangent minimum and is aimed at improving the solution within the neighborhood of the selected design.

The T-Search method was originally proposed by Hilleary in 1966 [18]. It combines smaller steps and larger moves through the design space (a pattern search) and directly handles inequality constraints (see [19] for an elaboration). Mathematically speaking, it is a gradient-free method, but it comes up with probing moves not dissimilar to gradient directions.

Results of the 2-level "Tsearch" optimization were very close to the best geometry obtained with the Sobol design of experiment sequence: they provided a further 2% increment in the mass flow rate of the valve.

**Figure 18.** 2-level T-Search result.

In Table 3, best design results against baseline are compared.

**Table 3.** T-Search Optimization Results.

