# Performance Prediction and Design of Stratospheric Propeller
## 2 Methodology  
The blade element theory is typically used to predict the performance of a propeller. The blade is divided into a finite number of tiny segments called blade elements, and the aerodynamic performance of each blade element can be obtained according to the airfoil theory. The blade element is integrated along the radial direction to obtain the total aerodynamic performance of the blade. Figure 1 presents the forces and velocities acting on the blade element, where \(V_{\infty}\) is the free stream speed, \(V_{a}\) is the axial induction velocity, \(V_{t}\) is the tangential induction velocity, and \(V\) is the total velocity of the actual airflow; \(n_{s}\) is the angular velocity of the propeller, and \(r\) is the radial position of the blade element; \(\varphi_{0}\) is the induced angle of attack, \(\varphi\) is the angle of actual airflow, \(\theta\) is the pitch angle, and \(\beta\) is the interference angle, which is given by:  
\[\tan\beta=\frac{V_{i}}{V_{0}}=\frac{\sqrt{V_{a}^{2}+V_{t}^{2}}}{V_{0}} \tag{1}\]  
The axial and tangential induction factors are defined as follows:  
\[\left\{ \begin{array}{c}a=\frac{V_{a}}{V_{\varphi}}\\ a^{\prime}=\frac{V_{t}}{2\pi n_{s}r}\end{array} \right. \tag{2}\]  
The angle of actual airflow \(\varphi\) is as follows:  
\[\tan\varphi=\frac{V_{\infty}+V_{a}}{2\pi n_{s}r-V_{t}} \tag{3}\]  
The angle of attack \(\alpha\), the velocity of the actual airflow \(V\), and the local Reynolds number \(Re\) are given by:  
\[\alpha=\theta-\varphi \tag{4}\]\[V=\sqrt{\left(V_{\infty}+V_{a}\right)^{2}+\left(2\pi n_{s}r-V_{t}\right)^{2}}=V_{ \infty}\frac{\left(1+a\right)^{2}}{\sin\varphi} \tag{5}\]  
\[Re=\frac{\rho Vb}{\mu} \tag{6}\]  
The thrust and torque forces acting on the blade element are as follows:  
\[\left\{ \begin{array}{l}dT=\frac{1}{2}\rho V^{2}C_{T_{V}}bdr\\ dF=\frac{1}{2}\rho V^{2}C_{F_{V}}bdr\end{array} \right. \tag{7}\]  
where \(b\) is the chord length of the blade element. \(C_{T_{V}}\) and \(C_{F_{V}}\) are obtained from the geometric relationship shown in Figure 1; \(C_{L}\) and \(C_{L}\) can be determined by \(Re\) and \(\alpha\). \[\left\{ \begin{array}{l}C_{T_{V}}=C_{L}\cos\varphi-C_{D}\sin\varphi\\ C_{F_{V}}=C_{L}\sin\varphi+C_{D}\cos\varphi\end{array} \right. \tag{8}\]  
Equation (5) is integrated into Equation (7), and the lift and torque acting on the propeller are given by:  
\[\left\{ \begin{array}{l}dT=\frac{1}{2}\rho V_{\infty}^{2}C_{T_{V}}N_{b}b \frac{\left(1+a\right)^{2}}{\sin^{2}\varphi}dr\\ dM=rdF=\frac{1}{2}\rho V_{\infty}^{2}C_{F_{V}}N_{b}b\frac{\left(1+a\right)^{2} }{\sin^{2}\varphi}rdr\end{array} \right. \tag{9}\]  
The momentum theory equations are shown below:  
\[\left\{ \begin{array}{l}dT=4\pi r\rho V_{\infty}^{2}a(1+a)drF\\ dM=4\pi r^{2}\rho V_{\infty}(2\pi n_{s}r)a^{\prime}(1+a)drF\end{array} \right. \tag{10}\]  
Combining Equations (9) and (10), we obtained:  
\[\left\{ \begin{array}{l}a=\left(\frac{4F\sin^{2}\varphi}{\sigma C_{T_{V}}}-1 \right)^{-1}\\ a^{\prime}=\left(\frac{4F\sin\varphi\cos\varphi}{\sigma C_{F_{V}}}+1\right)^ {-1}\end{array} \right. \tag{11}\]  
where \(\sigma\) is the solidity ratio of the propeller, and \(F\) is the Prandlt momentum loss factor. \[\sigma=\frac{N_{b}b}{2\pi r} \tag{12}\]  
Figure 1: Forces and velocities acting on the blade element. \[F=\frac{2}{\pi}\arccos^{-f} \tag{13}\]  
where \(f\) is given by:  
\[f=\frac{N_{B}}{2}\frac{1-\overline{r}}{\sin\varphi_{t}} \tag{14}\]  
where \(\varphi_{t}\) is the flow angle of the blade tip, and \(\overline{r}=r/R\) is the dimensionless radius. At the beginning of the calculation, the value of \(\varphi_{t}\) is unknown, and the error is not too large to replace \(\varphi_{0}\).