## 4 Propeller Theory for HAPSs
### Blade Element Momentum Theory
However, classical BEMT does not account for the influence of vortices shed from the blade tips into the slipstream, which affect the induced velocity distribution. To address this limitation, a correction factor known as the momentum loss factor \(F\), originally introduced by [PERSON] [73], is commonly applied (Equation (12)), where \(f\) is defined by  
Figure 7: Schematic of local velocities and angle definitions on a blade element, used in BEMT. Equation (13). \(\phi_{t}\) represent the flow angle of the blade tip and R denotes the radius of the propeller. \[F = \frac{2}{\pi}\arccos\left(e^{-f}\right) \tag{12}\] \[f = \frac{N_{b}}{2}\left(\frac{R-r}{Rsin(\phi_{t})}\right) \tag{13}\]  
In practical applications, the performance of a propeller is commonly assessed using nondimensional coefficients such as thrust coefficients \(C_{T}\), torque coefficients \(C_{Q}\), power coefficients \(C_{P}\), as well as propeller efficiency \(\eta\) (Equations (14)-(17)). These parameters are well established in the literature and facilitate meaningful comparisons across different designs and operating conditions [46,76]. \[dT=N_{b}\frac{1}{2}\rho V_{e}^{2}(C_{l}\cos\phi-C_{d}\sin\phi)cdr \tag{6}\]0 \[dT=N_{b}\frac{1}{2}\rho V_{e}^{2}(C_{l}\cos\phi-C_{d}\sin\phi)cdr \tag{6}\]1 (15) \[dT=N_{b}\frac{1}{2}\rho V_{e}^{2}(C_{l}\cos\phi-C_{d}\sin\phi)cdr \tag{6}\]2 (16) \[dT=N_{b}\frac{1}{2}\rho V_{e}^{2}(C_{l}\cos\phi-C_{d}\sin\phi)cdr \tag{6}\]3  
However, classical BEMT exhibits several fundamental limitations when applied to propellers operating in the HAPS regime. A first and major limitation arises from the strong dependence of airfoil aerodynamic characteristics on Reynolds number in this regime. At low Reynolds numbers (\(10^{4}\)-\(10^{5}\)), phenomena such as laminar separation bubbles, delayed transition to turbulence, and early flow separation significantly alter the lift and drag behavior of airfoils. Consequently, the aerodynamic polars exhibit sharp variations with Reynolds number, unlike in higher-Reynolds conditions where they tend to be relatively insensitive. BEMT does not generally account for the substantial spanwise variation in the Reynolds number along the blade. Even if the same airfoil geometry is used throughout the span, the local Reynolds number, governed by the local relative velocity and chord length, can vary by more than an order of magnitude from root to tip. Because low-Reynolds airfoil characteristics are so sensitive to the Reynolds number, each section effectively operates with a different aerodynamic polar. Classical BEMT formulations typically ignore this, applying a single representative polar to the entire blade, which introduces significant errors in the estimation of aerodynamic forces. Consequently, applying BEMT without incorporating spanwise-resolved aerodynamic data may result in sub-optimal performance estimates and hinder the identification of a truly optimized blade geometry. As shown in [108], neglecting the influence of Reynolds number variation can lead to an overestimation of thrust and efficiency by up to 25% under typical stratospheric conditions. Moreover, BEMT inherently neglects viscous flow effects, relying on an inviscid formulation that assumes the forces on each blade section depend solely on local flow angles and circulation. At low Reynolds numbers, viscous phenomena dominate the flow behavior, affecting boundary layer development, delaying or preventing reattachment after separation, and increasing overall drag. Such effects fall entirely outside the scope of classical BEMT, compromising its ability to accurately predict thrust and power requirements in these regimes. The theory also employs simplified models to account for tip and root losses, most commonly through Prandtl's tip loss factor. These corrections were derived under assumptions valid primarily for high-Reynolds, inviscid, attached flow and do not reliably capture the diffused, viscous-dominated tip vortices or altered induction seen at low Reynolds numbers. Reynolds numbers.