# Preliminary Propulsion and Power System Design of a Tandem-Wing Long-Range eVTOL Aircraft
## 2 Propulsion System Design Methodology
### Design of the Propeller Blades  
The main tool used in this framework for the preliminary design of the propeller blades is the blade element momentum theory (BEMT), a theory which analyses the performance of propeller blades by dividing each individual blade into a number of stations, coupling this to a momentum analysis for each individual annulus of the propeller streamtube. These stations, each with a different twist angle, were then analysed as 2D airfoils in order to compute their individual thrust and drag, and later added to obtain the overall performance of the propeller blade. Correction factors to account for 3D effects are also included in the process to improve the accuracy of the method. In order to achieve an efficient blade design, the procedure laid down by [PERSON] and [PERSON] [5] was used. This procedure was based on a similar work by [PERSON] [6] and modified [PERSON]'s equations to eliminate small angle approximations and extend the procedure from lightly to moderately and highly loaded propellers. The procedure yields the blade geometry that minimises the propeller induced and profile losses at a chosen design condition. The condition for which this happens is when the product of the local radius of a blade station times the tangent of the local flow angle at such station remains constant throughout the blade (\(r\tan(\Phi)=\text{const. }\)) [5], a condition which results in the vortex sheet shed by the propeller having the form of a regular screw surface. This condition was originally described by [PERSON] [10], and is thus referred to as the Betz condition. The original work by [PERSON] and [PERSON] includes further derivations to support the equations used such that only the final equations are herein presented. From [5], the term \(\tan(\Phi)\) can be rewritten as Equation (1), where \(\lambda\) is the tip speed ratio, defined as the ratio between the inflow speed and the tangential speed of the propeller tip (\(V/\Omega\ R\)) and \(\xi\) is the non-dimensional radius of the propeller, defined as the ratio between the local and the propeller radii (\(r/R\)). With this, Equation (1) can be rewritten as Equation (2):  
\[\tan(\Phi)=\left(1+\frac{\zeta}{2}\right)\frac{\lambda}{\xi} \tag{1}\]  
\[r\ \tan(\Phi)=\left(1+\frac{\zeta}{2}\right)R\lambda \tag{2}\]  
Since the displacement velocity ratio (\(\zeta\), defined as the ratio between the displacement velocity of the vortex sheet and the inflow speed), \(\lambda\) and \(R\) are constants, this agrees with the aforementioned Betz condition: \(r\tan(\Phi)\) is constant. Figure 3: Schematic of the 9 flight phases of a regular eVTOL flight mission. To define the chord of each blade station, Equation (3) [5] can be used:  
\[wc=\frac{4\pi\lambda R_{\zeta}VG}{BC_{l}} \tag{3}\]  
In this equation, \(G\) is a parameter defined in Equation (4) where \(F\) represents the momentum loss factor, defined as Equation (5) [5]:  
\[G=F\ \sin(\Phi)\cos(\Phi) \tag{4}\]  
\[F=\frac{2}{\pi}\arccos\left(e^{-\frac{\beta}{2}\frac{1-\zeta}{\sin(\Phi_{tip})}}\right) \tag{5}\]  
To define the blade geometry, the chord length and pitch angle of each station are needed. The local pitch angle of the blade, \(\beta\), is defined as the angle between the disk plane and the chord of the blade airfoil. As can be seen in Figure 4, it follows that the angle of attack of the section is defined as \(\alpha=\beta-\Phi\):  
Equation (2) allows one to calculate the required flow angle at each blade station, where \(R\) and \(\lambda\) are known. The angle of attack is then chosen such that the \(C_{l}/C_{d}\) of the blade is maximised for each station, while maintaining a reasonable value for \(C_{l}\) that allows the propeller to generate sufficient thrust. For this, the lift coefficient is first chosen, and the corresponding drag coefficient and angle of attack are then retrieved from known airfoil data.