# Preliminary Propulsion and Power System Design of a Tandem-Wing Long-Range eVTOL Aircraft
## 2 Propulsion System Design Methodology
### Noise Analysis  
One key aspect of aircraft that operate in urban environments is their noise emission. Noise can have a negative effect on social sustainability and thus on the acceptance of new urban air mobility solutions. Moreover, noise is regulated, and if the noise emissions of the aircraft are not within regulations, the aircraft cannot fly. In the particular case of the analysed configuration, with open rotors, the majority of the noise comes from the propellers. It is thus paramount to include a preliminary method to estimate the propeller noise in this design framework. The propeller noise under cruise conditions was analysed following a semi-empirical procedure obtained from [14]. This method is for isolated propellers, and hence only serves as a preliminary estimation for multi-rotor configurations in which the noise is affected by the interactions between the propellers (see Section 3.3 for further recommendations).  
The formula used to calculate the far-field noise (defined as noise at a distance further than one diameter of the propeller) for a single propeller is shown in Equation (12) [14]:  
\[SPL=L_{1}+20\;\log_{10}\biggl{(}\frac{4}{B}\biggr{)}+40\;\log_{10}\biggl{(} \frac{15.5}{D}\biggr{)}+C_{Mach}+C_{\theta}-20\;\log_{10}(r-1) \tag{12}\]  
In this equation, the first term \(L_{1}\) is the reference noise level; it is based on the motor power and it is obtained from Figure B.2 in [14]. The next two terms are corrections for the number of blades, \(B\), and propeller diameter, \(D\) (in ft). The next term, \(C_{Mach}\), is a correction for the tip Mach number, obtained from Figure B-3 in [14]. \(C_{\theta}\) is a correction which accounts for the direction in which the noise is being calculated and it is obtained from Figure B-8 in [14]. For this calculation, it was decided to use a correction of +4 dB, which is the maximum value of the average curve, to obtain the noise at the position in which it is at a maximum. Lastly, the last term, where r is the distance in ft at which the noise is to be calculated, accounts for the noise attenuation due to propagation from source to observer.  
As shown in Equation (12), a higher number of blades results in less noise. This is due to the fact that with more blades, the pressure increment to the flow is introduced in a more distributed manner, which favours selecting a high number of blades for the design of the propellers.  
Equation (12) holds for one propeller, so to calculate the combined noise from all propellers, Equation (13) is used, where \(L\) represents the noise in dB. This equation results in the sum of the noise by each propeller when calculated under isolated conditions and assuming incoherent sources, but it does not take into account the aerodynamic and aeroacoustic interactions between the propellers. These effects are difficult to model and thus not taken into account in this preliminary analysis; therefore, more accurate aeroacoustic simulations of the chosen architecture are needed and belong to the next steps towards a more detailed design:\[L_{tot}=10\log_{10}\left(\sum_{i=1}^{n}10^{(L_{prop,\ i}/10)}\right) \tag{13}\]