## 4 Propeller Theory for HAPSs
### Vortex Theory  
Vortex Theory offers a more comprehensive framework for analyzing the aerodynamic performance of rotating propellers by explicitly modeling the circulation bound to the blades and the helical wake shed downstream. This approach accounts for the three-dimensional induced velocity field created by the rotation of the blades and the finite span effects, which play a critical role near the blade tips. Unlike classical BEMT, which combines simplified one-dimensional momentum balances with local two-dimensional blade element analysis, Vortex Theory directly relates the distribution of circulation along the blade to the induced velocities in the surrounding flow. This leads the determination of the induced angle of attack and, consequently, more accurate predictions of local aerodynamic forces. For its ability to capture the effects of bound circulation and trailing vortices, Vortex Theory remains a widely cited framework for preliminary propeller sizing and first-order performance estimates. However, at the low Reynolds numbers typical of high-altitude applications, it is usually complemented by CFD analyses during the design process. With reference to Figure 12, the induced angle of attack \(a_{i}\), results from the induced velocity \(w\), which is the vector sum of the tangential and axial components of induced velocity, \(w_{t}\) and \(w_{a}\) (Equation (18)). \begin{table}
\begin{tabular}{c c c c c} \hline
**Propeller** & **AF** & \(D\) **(m)** & \(N_{b}\) & \(\eta_{max}\) \\ \hline Propeller 1 & 43.4 & 0.85 & 2 & 0.87 \\ Propeller 2 & 70.0 & 0.85 & 2 & 0.85 \\ Propeller 3 & 96.6 & 0.85 & 2 & 0.83 \\ Propeller 4 & 123.2 & 0.85 & 2 & 0.83 \\ Propeller 5 & 149.8 & 0.85 & 2 & 0.85 \\ \hline \end{tabular}
\end{table} 
Table 2: Characteristics of different propellers designed for various HAPS operations [115; 116; 117; 118; 119; 120; 121]. Figure 12: Diagram of angles and speeds on the blade element considering the induced angle \(a_{i}\). \[a_{i}\ =\ \arctan\left(\frac{w_{t}}{w_{a}}\right)-\phi \tag{18}\]  
The Vortex Theory yields the expression for the total circulation around the propeller as given by Equation (19). \[N_{b}\Gamma\ =\ 4\pi rw_{t} \tag{19}\]  
By equating the expression derived from the Kutta-Jukowsky theorem for a blade element (Equation (20)) with the aforementioned expression (Equation (19)) and incorporating the solidity term \(\sigma\), the equation for the tangential velocity, denoted as \(w_{t}\), is derived (Equation (21)). \[l\ =\ \rho V_{\sigma}\Gamma \tag{20}\]  
\[w_{t}\ =\ \frac{\sigma}{4F}V_{\sigma}C_{l}(\theta-\phi-a_{i})\ =\ \frac{\sigma}{4F}V_{ \sigma}C_{l}\left(\theta-\arctan\left(\frac{w_{t}}{w_{a}}\right)\right) \tag{21}\]  
To determine the axial velocity as a function of the tangential velocity, it is necessary to equate the thrust derived from the BET with that from the general impulsive theory. This process involves obtaining the relationships between the axial and rotational interference factors (Equation (22)). \[\frac{1+a}{\chi^{2}}\ a\ =\ a^{\prime}(1-a^{\prime}) \tag{22}\]  
where \(\chi\), \(a\), and \(a^{\prime}\) are defined as \(\chi\ =\ \frac{\Omega r}{V_{\infty}}\), \(a\ =\ \frac{w_{t}}{V_{\infty}}\), and \(a^{\prime}\ =\ \frac{w_{t}}{\Omega r}\). By substituting these relations into Equation (22), a relation between the two components of the induced velocity is obtained in Equation (23), which can be solved for \(w_{a}\) as a function of \(w_{t}\), as defined in Equation (24). \[(V_{\infty}+w_{a})w_{a}\ =\ w_{t}(\Omega r-w_{t}) \tag{23}\]  
\[w_{a}\ =\ \frac{1}{2}\left[-V_{\infty}+\sqrt{\chi_{\infty}^{2}+4w_{t}( \Omega r-w_{t})}\right] \tag{24}\]  
The calculation process is discussed in [85]. After determining \(w_{a}\) and \(w_{t}\) through an iterative process (for example, solving the implicit equation given in Equation (21) by applying [PERSON]'s method to find the zeros of the function), it is possible to obtain the induced angle \(a_{i}\) using Equation (18).