*3.1. Propeller*

The propeller includes many parameters to consider, such as the material of the propeller, the diameter, the shape of the blades, the pitch, and the number of blades. Currently, a variety of materials are used for propeller manufacturing, including carbon fiber (CF), nylon, plastic, and wood. The material of the propeller greatly influences its aerodynamic performance. As the rotor blades spin, the angle of attack at each spanwise region can change with reference to the original blade design [22]. Carbon fiber propellers are known for their stiffness and for being lightweight, but their downside is their high cost. Increasing the pitch and the number of blades results in higher thrust production, but it leads to a decrease in propeller efficiency, necessitating increased electrical and mechanical power. Increasing the diameter will increase the efficiency, but the ability to handle the load of the motor must also increase. When the blades are larger, with other parameters constant, they will rotate at lower velocities to produce the same lift. At this point, the induced velocity will decrease. Thereby, the efficiency of the system will increase [22]. The propeller model is described by its thrust *T*(*N*) and its torque *M*(*Nm*) as given by

$$\begin{cases} T = \mathbb{C}\_T \cdot \rho \cdot \left(\frac{N}{60}\right)^2 \cdot D\_p{}^4\\ M = \mathbb{C}\_M \cdot \rho \cdot \left(\frac{N}{60}\right)^2 \cdot D\_p{}^5 \end{cases} \tag{1}$$

where *ρ* (kg/m3), *CT*, *CM*, *N* (rpm), and *Dp* (m) are respectively, the air density, the thrust coefficient, the torque coefficient, the propeller velocity, and the propeller diameter. The air density, *ρ*, is determined by both the local temperature *Tt* (unit: °C) and the air pressure *p*, which is further determined by altitude *hhover* (m). According to the international standard atmosphere model [23], we have the following expressions:

$$\begin{cases} \rho = \frac{273 \cdot p}{p\_0(273 + T\_l)} \cdot \rho\_0\\ p = p\_0 \cdot \left(1 - 0.0065 \cdot \frac{h}{273 + T\_l}\right)^{5.2561} \end{cases} \tag{2}$$

where *ρ*<sup>0</sup> is the standard air density, *ρ*<sup>0</sup> = 1.293 (kg/m3), at a temperature of 25 °C. The thrust and the torque coefficients are modeled using the blade element theory as presented in [22]:

$$\begin{cases} \mathsf{C}\_{T} = \frac{0.27 \pi^{3} \lambda\_{p}^{r2} K\_{0} \epsilon}{\pi A + K\_{0}} B\_{p}^{\;\ a\_{l}} \, \mathsf{p}\_{p} \\ \mathsf{C}\_{M} = \frac{1}{4A} \pi^{2} \lambda\_{5}^{r2} B\_{p} \left( \mathsf{C}\_{fd} + \frac{\pi A K\_{0}^{2} \epsilon^{2}}{\epsilon (\pi A + K\_{0})^{2}} \mathsf{p}\_{p} \, ^{2} \right), \end{cases} \tag{3}$$

where *Bp* and *ϕp*(*rad*) are, respectively, the propeller blade number and the pitch angle. They are defined using:

$$\varphi\_p = \arctan\left(\frac{H\_p}{\pi D\_p}\right),\tag{4}$$

*A*,,*λ*,*ζ*,*e*,*Cf d*,et *K*<sup>0</sup> are the blade parameters, which are directly related to the propeller blade airfoil shape. Their approximate values are presented in Table A1 in the Appendix A. Since the blade airfoil shapes are similar for certain series of propellers, especially those supplied by T-motor, the blade parameters are typically fixed. Figure 7a presents the regression model of the propeller's mass *Mprop* (g), which is based on data supplied by Tmotor [20] and Mejzlik [21]. The input of this model is the propeller diameter *Dp* optimized through the optimization methodology. Equation (5) presents the regression model of the propeller mass:

$$M\_{prop} = 0.303 \cdot D\_p^2 - 9.729 \cdot D\_p + 105.786. \tag{5}$$
