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Minimum-induced-loss propeller for a man-powered airplane: Larrabee's example from NASA CP-2085

Tool	Propeller Design
Source	Larrabee, E. E. (1979). "Design of Propellers for Motorsoarers." In Science and Technology of Low Speed and Motorless Flight, NASA Conference Publication 2085, Part 1, NASA Langley Research Center, Hampton, VA, pp. 285–303 (equations 16 to 20, Figure 6).
Level	Graduate

Problem

A two-blade propeller of 3.81 m diameter turning at 110 rpm must give 53.3 N of thrust at a flight speed of 5 m/s in air of density 1.178 kg/m³ (the climb condition of a man-powered airplane). Find the displacement velocity of the minimum-induced-loss loading, the power and efficiency, and the chord and blade-angle distribution for sections working at a lift coefficient of 0.7 with a lift-to-drag ratio of 50.

Inputs to Enter

Thrust 53.3 N, flight speed 5 m/s, 110 rpm, diameter 3.81 m, 2 blades, hub radius 0.1 m, density 1.178 kg/m³
Sections: design lift coefficient 0.7, L/D 50, lift slope 5.9 per radian, zero-lift angle −3°, C_l,max 1.3; Prandtl's tip-loss factor

Open the Propeller Design tool (loaded with this example)

Answers

Speed ratio λ = V/(ΩR) = 0.2279 (book 0.2278), J = 0.7158; thrust loading T_c = 2T/(ρV²πR²) = 0.3175 (book 0.3175)
Exact design (integrals iterated with ζ): I_1 1.2882, I_2 0.1019, J_1 1.3662, J_2 0.5622, ζ = 0.2515, P_c = 0.3791, η = 0.8375, shaft power 318.2 W
Larrabee's light-loading form with the same constant D/L of 0.02: ζ = 0.2640, η = 0.8269; the book, with a D/L rising from 0.02 to 0.033 at the tip, prints I_1 1.2125, I_2 0.0888, J_1 1.3151, J_2 0.5626, ζ = 0.2671, P_c = 0.3914, η = 0.8113, 328 W
With the D/L curve read from the book's Figure 6 the light-loading integrals come out 1.2245, 0.0891, 1.3278, 0.5688 and η = 0.8125: within about 1 % of the printed values (test script)
Induced efficiency 0.8893, profile efficiency 0.9418, actuator-disc limit 0.9312; C_T 0.06389, C_P 0.05460
Blade: 22.68° and 218.9 mm chord at 0.75 R, widest chord 374.9 mm at r/R = 0.333, activity factor 73.2 per blade, solidity 0.0797
Blade-element analysis of the designed blade at the design advance ratio returns 53.3000 N and 318.2 W; best fixed-pitch efficiency 0.856 at J = 0.823, zero thrust at J = 1.092

Notes

The tool's default inputs are this case; press Submit. The book gives the design conditions and the integrals of its light-loading method, not the section lift coefficient or a chord table, and its drag-to-lift ratio varies along the blade where the tool takes one value; the tool's exact design therefore differs from the printed efficiency by 3 %, most of it the lower drag. The 1 % left between the tool's light-loading integrals and the printed ones is within the reading of the D/L curve and the book's unstated integration rule.