# A Conceptual Design Methodology for e-VTOL Aircraft for Urban Air Mobility
## 2 Design Methodology
### Motor Sizing
\[\left(\frac{P}{W}\right)_{CR}=\frac{1}{\eta_{prop}E_{CR}}V_{CR} \tag{5}\]  
\[\left(\frac{W}{S}\right)_{max}=\frac{\frac{1}{2}\rho SV_{stall}^{2}CLmax}{S} \tag{6}\]  
In Equation (5), \(E_{CR}\) and \(V_{CR}\) are the lift-to-drag ratio and lift coefficient in cruise, respectively. \(V_{stall}\) is the aircraft stall speed. \(CL_{max}\) is the maximum lift coefficient (with high lift devices deployed). The requested specific power is given by the Equation (7),  
\[\frac{P}{W}=max\bigg{(}\left(\frac{P}{W}\right)_{TO}\bigg{(}\frac{P}{W}\bigg{)} _{CL}\bigg{(}\frac{P}{W}\bigg{)}_{CR}\bigg{)} \tag{7}\]  
As shown in Equation (3), the take-off specific power does not depend on the wing loading \(W/S\) but on the disk loading \(T/A\); this is due to the fact that the during take-off the aircraft is in multi-rotor mode. The air density (\(\rho\)) affects hover performance, since lower density increases the requested power; in take-off phase, the air density has been evaluated at mean sea level according to the international standard atmosphere (ISA) [69]. The total take-off power is given by the sum of several components (e.g., power fraction due to induced speed and parasitic drag of the blade, tip losses, etc. ), and the following procedure has been adopted for its evaluation:  
1. evaluation of the power fraction due to induced drag by using the disk actuator model;
2. estimation of the figure of merit (\(FoM\)) according to literature data [67];
3. calculation of the total take-off power according to the Equation (3). The procedure described above is commonly used in the conceptual design phase [30; 41; 42]; it allows to estimate the total take-off power without designing a rotor blade. As said before, the figure of merit parameter allows to consider all the fraction of power contributions; in this paper, a value of 0.8 has been assumed according to [67]. An extra thrust factor (\(k_{T}\)) has been assigned in order to accomplish the phase also with a motor failure: \(k_{T}\) is set equal to 1.2 so, if a motor fail, the propeller thrust exceeds the MTOW of 5%. Equations (4) and (5) show a dependence of the required specific power for cruise and climb on wing loading, since the aircraft flies in fixed-wing mode. To estimate the maximum wing loading, the following hypotheses have been introduced:  
1. a stall speed of 30 m/s, in compliance with FAR23 [70];
2. a maximum lift coefficient (with flap deployed) equal to 2.0, that derives from literature data concerning wind tunnel test performed on the box-wing architecture [71]. FAR23 regulation has been considered since it is applied for aircraft with MTOW lower than 19,000 pounds and a maximum number of passengers of 19, which is in line with e-VTOL requirements. As detailed in [68], \(P/W\)-\(W/S\) matching charts can be used to estimate the specific power needed for each wing loading, taking the highest value given by the flight conditions considered. In the _TiltOne_ case, Figure 6 shows the plot of Equations (3)-(6) varying the wing loading; the take-off condition is the most power-demanding for any wing loading value, hence it drives the design of the propulsion system. As an example, the requested specific power for a single case (\(W/S=100\) kg/m\({}^{2}\), dashed black line) has been depicted.