# Blade Shape Optimization and Analysis of a Propeller for VTOL Based on an Inverse Method
## 6 Conclusions  
The inverse propeller design method based on the Betz optimal theory was successfully integrated into the rapid optimization framework for a VTOL propeller, which worked in a wide range of operation conditions.  
The proposed VTOL propeller optimization framework adopted the maximum thrust-to-weight ratio at hover and mission energy as the two objectives and strengthened the practicability of the optimization result. The maximum thrust-to-weight ratio at hover had a contradictory relationship with the mission energy efficiency. The maximum thrust-to-weight ratio optimal designs were not always those of the minimum hover power, which validated the necessity of taking the maximum thrust-to-weight ratio at hover as one of the objectives to capture maneuverability or agility in a vertical flight mode.  
Under certain BLDC power and torque constraints, a comprehensive optimal VPP and FPP were obtained for a Vahana A3 tilt-wing aircraft and were verified by the CFD  
Figure 30: Energy-saving advantage of VPP over FPP with different \(\xi\) and \(\kappa\). (**a**) 3D surface diagram of \(\delta\), \(\xi\) and \(\kappa\); (**b**) \(\delta-\xi\) relation curves with different \(\kappa\).  
Figure 29: Pareto-optimal front of VPP and FPP with different cruising speeds. (**a**) Pareto fronts for FPP; (**b**) Pareto fronts for VPP.  
numerical method. Using a VPP could save 17.05% of energy and improve by 4.57% the maximum thrust-to-weight ratio at hover than using an FPP under the given conditions. The advantages of a VPP in both energy consumption and maneuverability during takeoff were validated for a VTOL aircraft.  
Disk loading had a much larger impact on the maximum thrust-to-weight ratio than on the mission energy. Disk loading at hover should be as low as possible, especially for a manned VTOL aircraft, to ensure enough reliability and maneuverability during takeoff and landing. With an increase in the cruising speed, the minimum energy of the VPP had a slight increase whereas the maximum thrust-to-weight ratio at hover barely changed. However, the cruising speed had a heavier impact on the maximum thrust-to-weight ratio of the FPP at hover and minimum energy. This phenomenon validated the superior adaptability of the VPP.  
The quantitative rules on the energy-saving advantage of the VPP over the FPP were calculated in this article. When the maximum thrust-to-weight ratio at hover was larger than about 1.30, the energy-saving advantage of the VPP over the FPP increased as the level flight energy ratio grew. On the contrary, the energy advantage of the VPP over the FPP decreased as the level flight energy ratio grew when the maximum thrust-to-weight ratio at hover was less than about 1.30. Combining other design factors such as reliability and weight cost with the rules shown in Figure 30, a practical reference to choose a VPP or an FPP for a VTOL aircraft could be given.  
Conceptualization, X.X. and D.M.; methodology, X.X. and L.Z.; validation, X.L. and K.C.; formal analysis, L.Z. and D.M.; investigation, X.X.; resources, D.M.; data curation, X.X.; writing--original draft preparation, X.X.; writing--review and editing, X.X., D.M. and L.Z.; visualization, K.C.; supervision, L.Z.; project administration, D.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interest.