## 4 Propeller Theory for HAPSs
### Solidity  
The solidity, also known as Activity Factor (AF), is a critical parameter in the design of a propeller, particularly for applications involving high-altitude platforms. The AF is a dimensionless number representing the ratio of the blade area to the disk area of the propeller. Selecting an appropriate AF ensures that the propeller operates at optimal efficiency. If the AF is too low, the propeller blades might be too narrow, resulting in insufficient thrust. Conversely, a very high AF might cause excessive drag and power loss. For evaluating the performance of high-altitude pseudo-satellites, it is important to note that high-altitude platforms, unlike conventional aircraft, should work in a wide range of altitudes, and so in a wide range of densities. A propeller designed with an appropriate AF can maintain better performance under varying air densities, ensuring consistent thrust and efficiency at different altitudes. In [115], wind tunnel tests were conducted on five propellers with different blade tip widths to explore the effects of blade planform and advance ratio on both power coefficient and propulsive efficiency. The aim was to identify the optimal blade planform for achieving high thrust and a low advance ratio. Tests were conducted on five distinct propellers with varied blade planforms, assessing pitch angles ranging from \(20.5^{\circ}\) to \(40^{\circ}\) while maintaining specific geometric parameters and AFs as outlined in [115; 116]. In that study, a Reynolds number of \(1.5\times 10^{5}\) for the propeller, based on the chord length at \(r/R=0.7\), has been imposed. The AF increases with the widening of the propeller tip as shown in Figure 8. Analysis of the characteristic curve of this propeller with respect to the power coefficient (\(C_{P}\)) shows that the propeller with the highest AF yields a greater power coefficient, as illustrated in Figure 9.  
A comparison of the max efficiency value and power coefficients of blades with different AF has been illustrated in Figures 10 and 11. The main characteristics of the described propellers are shown in Table 2. One important result from this study, as shown the figures mentioned before, is that, for propellers operating at a low advance ratio (\(J<0.8\)), a blade with a narrower tip is advantageous for propulsive efficiency, although it limits power performance. Conversely, for propellers with an advance ratio exceeding 0.8 (\(J>0.8\)), those with wider tips exhibit superior power absorption capabilities while maintaining acceptable propulsive efficiency. The variation in the efficiency with blade planform and advance ratio is determined by the radial distributions of airflow angle  
Figure 8: Blades with increasing AF, from Prop-1 (AF = 43.4) to Prop-5 (AF = 149.8). Adapted from ref. [116].  
Figure 9: CP as a function of J at various AF ratios, at a constant pitch angle of 40\({}^{\circ}\)[115].  
and interference angle, which directly affects the efficiency of the blade elements. At low advance ratio, the low airflow angle and high interference angle of the blade element can result in reduced blade element efficiency, thereby decreasing the overall propulsive efficiency of the propeller. As the advance ratio increases, although the higher interference angle still causes the blade element efficiency to decrease, the increase in airflow angle could enhance the efficiency of the blade elements, leading to an overall improvement in propulsive efficiency.  
Figure 11: Characteristic of different propellers designed for different HAPS operations [115; 116; 117], in terms of maximum efficiency curves at different advance ratio values.  
Figure 10: Characteristics of different propellers designed for different HAPS operations [115; 116; 117], in terms of power coefficient \(C_{P}\) at maximum efficiency for different velocity values.