## 4 Propeller Theory for HAPSs
### Blade Element Momentum Theory  
The Blade Element Momentum Theory, well described in [105], operates under the same assumptions as Blade Element Theory, where the blade is divided into finite segments called blade elements, and the aerodynamic characteristics of each element are calculated using principles derived from airfoil theory [66], while also incorporating insights from Momentum Theory. Generally, BEMT accounts for the effect of propeller rotation on the induced angle of attack and provides an efficient approach for propeller design and analysis [106; 107], with minimal computational cost during the early design stages. However, it cannot accurately capture the three-dimensional aerodynamic effects of propeller rotation, which leads to relatively lower accuracy [68; 95; 108]. In [109], a method for designing an optimum propeller is presented, where the search for the optimal propeller involves solving a constrained minimization problem aimed at minimizing power consumption while maximizing thrust by applying BEMT. This optimization problem has been addressed by various authors in the literature [109; 110]. By combining the Momentum Theory equations with those from the BET, it is possible to derive the BEMT formulation, as thoroughly explained in [111]. From the BET, the thrust and torque distributed over each annular section \(dr\) are expressed in Equations (6) and (7). The fundamental forces acting on the airfoil are illustrated in Figure 7. Here, \(V_{\infty}\) represents the forward velocity of the propeller, and \(N_{b}\) is the number of blades. \(\Omega\) denotes the angular velocity of the propeller. The pitch angle \(\theta\) is defined relative to the zero-lift line of the airfoil section and varies with the radial position \(r\). The advance angle is \(\phi\), and \(V_{e}\) is the total relative velocity of the airfoil section, calculated from the velocity triangle in Figure 7 as \([V_{e}~{}=~{}\sqrt{\left(V_{\infty}(1+a)\right)^{2}+\left(\Omega r(1-a^{\prime })\right)^{2}}]\) where \(a\) is the axial induction factor and \(a^{\prime}\) is the tangential induction factor. The chord length of the propeller blade section is denoted by \(c\), and \(\Omega\) is the angular velocity. \[dT=N_{b}\frac{1}{2}\rho V_{e}^{2}(C_{l}\cos\phi-C_{d}\sin\phi)cdr \tag{6}\]  
\[dQ=N_{b}\frac{1}{2}\rho V_{e}^{2}(C_{l}\sin\phi+C_{d}\cos\phi)cdrdr \tag{7}\]  
By imposing that \(C_{n}=C_{l}\cos\phi-C_{d}\sin\phi\) (normal force coefficient) and \(C_{a}=C_{l}\sin\phi+C_{d}\cos\phi\) (axial force coefficient) and introducing solidity defined as \(\sigma=\frac{N_{c}}{2\pi r}\), the final expressions utilized in the BEMT for thrust and torque are obtained in Equations (8) and (9). \[\frac{dT}{dr} = \sigma\pi C_{n}r\rho V_{e}^{2} \tag{8}\]  
\[\frac{dQ}{dr} = \sigma\pi C_{a}r^{2}\rho V_{e}^{2} \tag{9}\]  
By equating the thrust and torque relations provided by the BET theory (Equations (6) and (7)) with those obtained from the Momentum Theory, \(dT~{}=~{}4\pi r\rho V_{\infty}^{2}(1+a)a\,dr\), \(dQ=4\pi r^{3}\rho V_{\infty}(1+a)a^{\prime}\Omega\,dr\), \(dP~{}=~{}4\pi\rho\big{[}V_{\infty}^{3}(1+a)^{2}ar\,dr+\Omega^{2}V_{\infty}(1 +a)a^{\prime 2}r^{3}\,dr\big{]}\) using the conservation of axial momentum to the stream tube swept out by each radial blade element [83; 112], and taking into account the relationships derived from the velocity triangle, \(V_{e}^{2}~{}=~{}\frac{V_{\infty}^{2}(1+a)^{2}}{\sin^{2}\phi}+\frac{\Omega^{2} r^{2}(1+a)^{2}}{\cos^{2}\phi}\), it is possible to derive the fundamental expressions for the axial and rotational induction factors, denoted as \(a\) and \(a^{\prime}\) (Equations (10) and (11)). \[a~{}=~{}\left[\frac{4\sin^{2}\phi}{\sigma C_{n}}-1\right]^{-1} \tag{10}\]  
\[a^{\prime}~{}=~{}\left[\frac{4\sin\phi\cos\phi}{\sigma C_{a}}+1\right]^{-1} \tag{11}\]  
The impact of induced velocity within the propeller plane is most significant near the blade tips.