# Full Three-Dimensional Inverse Design Method for S-Ducts Using a New Dimensionless Flow Parameter
## 4 Design Example
Figure 19.7 Comparison between the cross-sections of the second and three-dimensionally-modified S-duct (AR = 0.41). Figure 19.8 Comparison between the pressure contour of the three-dimensionally-modified and the two other geometries on their symmetry plane. The total pressure loss decreased through the three-dimensionally-modified S-duct due to the modified cross-sectional profiles, which were obtained by 3-D inverse design. To show the capability of the proposed method, the performance of the three-dimensionally-modified S-shaped duct is compared with the results of [PERSON] et al. [18]. They performed a multi-objective optimization with two objective functions to quantify the performance of an S-Shaped duct during the optimization process. They used total pressure loss and swirl angle as the objective functions. The swirl angle is defined as follows:  
Figure 19.9  
Figure 19:0 Mach number contours at the outlet of the secondary and three-dimensionally-modified S-ducts. Figure 19:1 Streamlines of the three-dimensionally-modified 3D S-duct. where \(V_{\theta}\) as the tangential velocity is calculated as:  
Figure 19:2  
They obtained two optimized geometries considering the two objective functions. The solution with minimal total pressure loss was named \(opt_{CP}\) and the solution with minimal swirl angle was named \(opt_{\alpha}\). Table 2 compares the performance of the modified S-duct with that of the secondary duct and previous study. The modified S-duct had a more uniform flow and significantly lower DC (90) at the duct outlet. The modified S-duct also had a lower pressure loss, a higher pressure recovery coefficient and minimum swirl angle. The results also show that pressure recovery and outlet flow distortion increased by 30% and decreases by 69%, respectively. Although the aspect ratio of the three-dimensionally-modified S-shaped duct in the present work was 34% higher than that of the optimized S-shaped duct [18], it had a higher performance compared to the optimized S-shaped duct [18]. Furthermore, the S-shaped duct designed by the upgraded 3-D inverse design converged after only 50 geometry corrections while the optimized S-shaped duct in [18] converged after 1300 iterations. \begin{table}
\begin{tabular}{l l l l l} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{3}{c}{**Present Study**} & \multicolumn{2}{c}{**[PERSON] et al. [18]**} \\ \cline{2-5}  & **Secondary** & **3-D-Modified** & \(opt_{CP}\) & \(opt_{\alpha}\) \\ \hline Aspect ratio, AR = h/ L & 0.41 & 0.41 & 0.27 & 0.27 \\ \(DC(90)=\frac{Max|P_{0,max}-P_{0,max}|}{0.5 pt|V^{2}}\) & 0.264 & 0.085 & – & – \\ Pressure recovery coefficient & & & & \\ \(CPR=\frac{P_{0}-P_{1}}{P_{01}-P_{1}}\) & 0.393 & 0.473 & 0.491 & 0.491 \\ Total pressure loss \((P_{01}-P_{02})/P_{01}(\%)\) & 3.01 & 2.12 & 2.52 & 2.75 \\ Swirl angle \(\alpha=arctan(V_{\theta}/V_{x})\) & 1.096 & 0.861 & 3.256 & 1.411 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison between the performances of the secondary, three-dimensionally-modified (AR = 0.41), and the optimized S-duct [18] (AR = 0.27). Figure 35: Contours of stagnation pressure in various axial positions for the secondary and modified S-ducts. Every cross-section is perpendicular to x-direction and situated at x/L = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 from S-duct inlet.