# Advanced Scale-Propeller Design Using a MATLAB Optimization Code
### Historical Perspective  
In the minutes of the (1890) Proceedings of the Institution of Civil Engineers [7], the use of propellers for marine applications was discussed, and it soon became clear that the theory they were developing was applicable to aeronautics as well. [PERSON] (1892) is credited with the development of Blade Element Theory (BET) [8]. BET is a method used to analyse propeller performance that, unlike actuator disk theory, considers the propeller geometry and aerodynamic properties. In blade element theory it is assumed that the flow over a blade section is two-dimensional. This means that the flow over a given blade section is unaffected by the flow over neighbouring sections--in this analysis, the effective velocity is simply the result of the axial velocity and the rotational velocity. The propeller blade is broken down into multiple sections, and the forces acting on each of these blade sections is determined. By integrating over the length of the blade, the overall blade forces can be developed. [PERSON] (1925) in his NASA technical note entitled: Simplified Propeller Design for Low-Powered Airplanes [9] detailed an empirical design method for propellers. The design process is accomplished through the use of charts and basic formulae and involves no detailed calculations. The data required for this design method is the aircraft airspeed, the brake horsepower and the revolutions per minute of the engine, and the density of air. From this data, the performance coefficient, advance ratio and efficiency of the propeller are found empirically. The article also educates on what materials are capable of withstanding the centripetal forces at different propeller diameters and rotational speeds. In his 1935 paper, entitled Airplane Propellers [10], [PERSON] formulated a method for analysing arbitrary propeller designs. The method is a combination of axial momentum theory and blade element theory, and includes corrections for momentum loss due to radial flow. The method is limited, as the contraction of the propeller wake is not considered. As a result, the theory is accurate for low disc loading (small thrust per unit disc area), but is only adequate for estimating performance at high disc loadings. Propeller Analysis from Experimental Data [11] is a 1940 report ([PERSON] & Crigler) in which propeller performance is analysed. The data used in this analysis was obtained in a 20-foot wind tunnel with a 4-foot diameter propeller, but the results apply to all propellers. Firstly, aerodynamic theory is used to create formulae which can be used to calculate both the axial and rotational energy losses in the wake of a propeller. These are then validated with wind tunnel data, and it is shown that, for an optimal propeller, the loss in efficiency due to rotational velocity is always very small. However, for a propeller with high-blade settings, this efficiency loss can become quite high. The load distribution along a propeller blade is also examined, and the maximum thrust generation is found to always be between 0.7 R-0.85 R (where R is the radius of the blade). Finally, the benefits of counter-rotating propellers over single propellers with an increased diameter are examined, and it was found that only small gains in propeller efficiency can be expected with counter-rotating blades. A book by [PERSON] (1948) entitled Theory of Propellers [12] was a development of the earlier propeller theory developed by [PERSON] (1892), and [PERSON] et al. (1890). [PERSON] focussed on the wake downstream of the propeller more than the propeller itself, as \"thrust, torque and efficiency are uniquely and completely established by the knowledge of the wake far behind the propeller and of the wake alone. To establish these quantities, in fact, it is absolutely unnecessary to know the design of the propeller\". [PERSON] introduces a mass coefficient, which represents the effective cross-section of the column of air behind the propeller divided by the projected propeller wake area.