## 4 Propeller Theory for HAPSs  
### Development on Theoretical Background  
Nowadays, as discussed in previous chapters, the propulsion system of HAPS platforms typically employs propellers. The low atmospheric density in the stratosphere, which at 20 km altitude is approximately 1/14 of the air density at sea level, coupled with low flight speeds (10-30 m/s), results in HAPS platforms operating at low Reynolds numbers and advance ratios (depending on rpm and propeller diameter) [66, 67, 68, 69]. Additionally, the lower temperature increases the Mach number, especially at the blade tip. Consequently, numerous studies have focused on optimizing propellers for HAPS platforms. The development of propeller theory dates back to the 19 th century with contributions from [PERSON] and [PERSON] [70; 71]. [PERSON] introduced Blade Element Theory (BET) in [72], although his approach did not consider the induced velocity of the propeller. The method aimed at minimizing induced losses was initially proposed by [PERSON] and [PERSON] [73] and further developed by [PERSON] [74]. [PERSON] and [PERSON] established criteria for minimizing energy loss, primarily focusing on propellers operating under light loads, while [PERSON] in [75] demonstrated that [PERSON]'s condition could also be applied to propellers with heavy disk loadings. [PERSON] formulated an estimation of the airflow around helicoidal vortex sheets, which provides more accurate predictions for propellers with lower advance ratios and becomes increasingly effective as the number of blades increases [76]. Integrating momentum theory, BET and Vortex Theory, [PERSON] in [49; 77] introduced a practical design theory for propellers with minimal induced loss. The aerodynamic design of propellers is still primarily conducted using two-dimensional methods, such as BEMT or Vortex Theory, even for propellers operating under high-altitude conditions [51; 67; 69; 78; 79; 80; 81; 82; 83; 84; 85]. BEMT enables rapid evaluation of a propeller when planform parameters and airfoil cross-section characteristics are known. Furthermore, given specific flow conditions, airfoil characteristics, and thrust requirements, BEMT facilitates rapid inverse design of a propeller's planform, including chord length and pitch angle distribution, based on Minimum Induced Loss (MIL) criteria. However, designs based on BEMT often lack high precision, leaving room for performance improvement. To enhance propeller aerodynamic efficiency, optimization techniques have been applied to blade shape design. [PERSON] et al. [67] refined chord length and pitch angle distribution using NSGA-II and a multi-island genetic algorithm (MIGA) based on BEMT, achieving a lightweight and efficient propeller. [PERSON] [80] improved BEMT accuracy by correcting its residual sequence and optimizing chord length and pitch angle through gradient-based methods. [PERSON] [68] performed multi-stage optimization for high-altitude pseudo-satellites, considering factors such as airfoil type, diameter, planform shape, and thickness, revealing important trends in propeller design. However, BEMT struggles to capture three-dimensional rotational effects like radial flow and root and tip influences, particularly at low Reynolds numbers. Its accuracy depends heavily on blade element aerodynamic coefficients, limiting computational precision. Thus, BEMT is primarily suitable for initial optimization and low-fidelity evaluations. More precise assessment methods are necessary to further enhance propeller efficiency. With the advancement of numerical simulation tools, the CFD method has become an essential aid in optimizing propeller design [86; 87; 88; 89; 90]. Compared to BEMT, CFD-based numerical simulations provide more accurate evaluations of propeller aerodynamic forces, thereby enhancing design precision. [PERSON] et al. [91] proposed a propeller design optimization approach using Bayesian optimization.