# A Conceptual Design Methodology for e-VTOL Aircraft for Urban Air Mobility
## 2 Design Methodology
### Motor Sizing  
This module evaluates the power required for the propulsion system, i.e., the electric motors, in each phase of the mission. The _TiltOne_ has 8 propulsion units (propeller, electric motor and control electronics), 4 installed on the aft wing and 4 on the rear wing, as depicted in Figure 5.  
The reason for setting the number of propulsors to 8 comes from the assumption that, in multi-rotor mode, the aircraft can be controlled as a quad-rotor with a \"X\" scheme, which sets the minimum number to 4. In order to increase the safety of the system, an additional group of 4 propulsors is introduced with a side-by-side distribution along the wingspan. This choice has been preferred to other possible alternatives (e.g., coaxial push-pull) since it provides the 4 half-wings with similar airflow conditions along the wingspan. Given the preliminary level of this study, a detailed motor failure analysis has not been carried out, the redundancy given by this double set of 4 propulsors allows to perform a safe landing manoeuvre in case of failure of 1 of the 8 motors, once a proper oversizing is considered. In fixed-wing mode the _TiltOne_ exploits all the propellers to generate the necessary thrust to the forward flight; however, this assumption can be easily modified. Moreover, as shown in Figure 6, the power requested at cruise is far lower than the take-off phase; accordingly, a variable pitch propeller has been assumed in order to avoid a reduction in propeller efficiency. Figure 5: Qualitative scheme of installation of propulsors on the _TiltOne_ (multi-rotor mode). The installed power is obtained by using the so-called matching chart: a specific power (_P_/_W_)-wing loading (_W_/_S_) chart where, for each wing loading, the corresponding specific power is evaluated [59; 68]. To use the matching chart, the following steps must be followed:  
1. Definition of the most power demanding phases;
2. Identification of regulations (e.g., FAR), which identify critical conditions that the propulsion system must satisfy, if any;
3. Estimation, for each phase, of the requested power;
4. Identification of the maximum requested specific power. According to the list above, four different flight conditions have been considered: take-off (_TO_), climb (_CL_), cruise (_CR_) and stall (_ST_). During take-off, the aircraft flies in multi-rotor mode; in the other phases the aircraft is in fixed-wing mode. For _TO_, _CL_ and _CR_, a mathematical expression of the specific power has been derived as reported in Equations (3)-(5), whereas the maximum wing loading of the configuration is obtained by Equation (6) (referred to the _ST_). The demonstrations for Equations (3)-(6), are reported in Appendix A.  
\[\left(\frac{P}{W}\right)_{TO}=\frac{k_{T}}{FoM\ \eta_{prop}}\ \sqrt{\frac{g}{2\rho} \frac{T}{A}} \tag{3}\]  
In Equation (3), \(k_{T}\) is the extra-thrust coefficient, defined as the ratio of the total thrust generated by the propellers to the weight of the aircraft. _FoM_ is a figure of merit of the rotors, defined as the ratio between the induced power (i.e., the power associated to the induced speed, evaluated by means of the disk actuator theory) and the total power given by induced speed, aerodynamic drag acting on the blades, tip losses and other secondary effects [67]. \(\eta_{prop}\) is the propeller efficiency, which has been set to 0.8 [67; 68; 69; 70; 71; 72] for all the flight phases, as it is assumed that variable pitch propellers are adopted. _T_/_A_ is the disk loading, already defined in Section 2; \(g\) is the acceleration of gravity; \(\rho\) is the air density. \[\left(\frac{P}{W}\right)_{CL}=\frac{1}{\eta_{prop}E_{CL}}\sqrt{\frac{2g}{\rho C _{CL}}\frac{W}{S}}\ +\frac{V_{z}}{\eta_{prop}} \tag{4}\]  
Figure 6: Example of matching chart of the _TiltOne_. In Equation (4), \(E_{CL}\) is the lift-to-drag ratio in climb; \(CL_{CL}\) is the lift coefficient, and \(V_{z}\) is the rate of climb. \(W/S\) is the wing loading, already defined in Section 2.