## 6 Optimization of Orifice Shape
### Optimization Results  
The numerical optimization based on RSM is carried out for the maximum volume flow rate using 15 cases. The same numerical methods as used for the flow and noise predictions of the existing fan system described in the previous sections are applied to each of these cases. Based on these results, the quadratic function of Equation (4) is obtained from the RSM using the Minitab software (version 17.0, Minitab Inc., United States of America,). From the quadratic function, the influence of the three parameters on the volume flow rate is obtained, as shown in Figure 15. Among the three parameters, R [mm] representing the inlet shape showed a tendency to include the maximum value of the volume flow rate within the selected design range. However, the distribution of H [mm] representing the neck shape and \(\theta\) [deg] representing the outlet shape has the maximum volume flow rate at the boundary of the set region. These results show that, as the length of the neck section is longer, and the discharge area of the outlet section is smaller, the flow rate is more increased. Note that the lower and upper limit of H and \(\theta\), respectively, is determined considering the allowable limit of orifice installation in the outdoor unit. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Factors**} & \multicolumn{3}{c}{**Level**} \\ \cline{2-4}  & \(-1\) & \(\mathbf{0}\) & \(\mathbf{1}\) \\ \hline R (mm) & 12.3 & 18 & 23.7 \\ H (mm) & 8 & 27.5 & 47 \\ \(\theta\) (deg) & 13.1 & 28.8 & 44.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Factors and level for optimization of orifice shape  
Figure 14: Design parameters for optimization of orifice shape. Figure 13: Shape of the orifice. Figure 16 shows the optimum shape of orifice, and the optimal design values for each section are (R, H, \(\theta\)) = (19.2, 55, 10). The numerical simulation of flow field driven by the axial-flow fan combined with the optimized orifice in the outdoor unit is carried out, and the flow rate of the optimized orifice is found to be higher than the original orifice by about 4.6%. The detailed flow field obtained from the numerical simulation using the optimized orifice is shown in Figures 17 and 18. Figure 17 shows the iso-contours of flow velocity magnitude on the horizontal cross-sectional plane. It can be found that the outlet flow direction of the outdoor unit using the optimized orifice is directed more downstream than that of the original one shown in Figure 8. It is emphasized that this dramatic change of flow pattern is achieved only by changing the orifice shape without changing the fan blade. Figure 18 compares the flow velocity vectors around the orifice between the original and the optimized ones. It can be seen that the vortex structure generated at the inlet section of the original orifice almost disappears in the optimized orifice and the discharging flow is developed more sufficiently at the neck section of the optimized orifice than the original one. In addition, in case of original orifice shape, the tip vortex formed and separated from the blade tip interacts with the inlet of orifice, whereas the tip vortex in the optimized orifice system was developed in the neck section. Figure 16: Optimized shape of orifice. Figure 15: The relation between flow rate and each design factor. As shown in Figure 17, there is a significant change in the outlet flow pattern between the optimized and original models. To find the reason for the difference, Figure 19 compares the flow velocity vectors between different outlet angles (\(\theta\)) at the fixed other design factors. Although the other two parameters are the same for each other, the tip vortex pattern changes significantly according to the variation of the outlet angle. In the case of the orifice with larger outlet angle, the tip vortex formed from the blade tip is convected downstream and interacts with the orifice vortex.