Betz  described the wake flow conditions for a propeller of minimum induced loss. The condition is somewhat similar to the elliptic circulation distribution of a finite wing, which was found by Prandtl . Larrabee  reworked the theory in his 1979 paper Practical Design of Minimum Induced Loss Propellers, allowing for a step by step design of a propeller of specified RPM, advance velocity, diameter and thrust. In the present work the Larrabee model was programmed in VBA utilizing MS Excel and applied to a series of selected propellers. The minimum induced loss condition can be met for a chosen propeller duty point only. In order to evaluate off design points the blade element theory according to Glauert  was applied and implemented in the design tool. The tool itself offers complete freedom over the propeller parameters, including the choice of blade sections. Also a geometry export function into the ANSYS curve format was implemented to allow for a fast CFD computation of the current propeller design. For theory evaluation purposes a commercially available RC model propeller with 8″ in diameter was chosen as a reference. This propeller is listed in the wind tunnel measurements database of the University of Illinois . CFD computations of the reference propeller design showed very good agreement with the experimental data, therefore indicating an accurate CFD setup. The theory of minimum induced loss is based on the distribution of circulation along the propeller’s blades, which was also investigated using CFD. Satisfying the minimum induced loss condition, several design strategies are shown and evaluated, ultimately leading to a design which proved to be superior to the actual reference design. Hence the validity of the loss condition could be shown. Also deep insights into the circulation along propeller blades were gained which in turn allow for a further refinement of theory and design tool.
High Efficiency Propeller Design Based on the Betz Minimum Induced Loss Condition and CFD Validation on an APC 8″ Propeller
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Albert, S, Epple, P, Willinger, B, & Delgado, A. "High Efficiency Propeller Design Based on the Betz Minimum Induced Loss Condition and CFD Validation on an APC 8″ Propeller." Proceedings of the ASME 2013 International Mechanical Engineering Congress and Exposition. Volume 7B: Fluids Engineering Systems and Technologies. San Diego, California, USA. November 15–21, 2013. V07BT08A052. ASME. https://doi.org/10.1115/IMECE2013-64382
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