Propeller Design

Propeller Design

Propeller Design

What the tool computes

The blade of least induced loss for one flight condition. You state what the propeller must do (a thrust to give, or a shaft power to absorb), the flight speed, the rotational speed, the diameter, the number of blades and the lift coefficient and lift-to-drag ratio at which the blade sections are to work. The tool returns the chord and the blade angle at every radius, the efficiency and the coefficients of the design point, and then checks the blade away from that point by a blade-element analysis over the advance ratio. The method is the vortex theory of Betz, Prandtl and Goldstein in the form given by Larrabee and completed by Adkins and Liebeck, who removed the light-loading and small-angle approximations.

The Betz condition and the displacement velocity

Each blade sheds a sheet of trailing vortices that winds back as a screw surface. Betz showed that the energy left in the slipstream is least, for a given thrust, when these sheets move back as if they were rigid: every point of a sheet then moves normal to itself, and the whole screw appears to slide aft at one speed v′, the displacement velocity. This is the propeller's counterpart of the uniform downwash behind an elliptically loaded wing. Its ratio to the flight speed, ζ = v′/V, is the one unknown of the design.

The fluid at the blade has received half of the motion it will have far behind, and that motion is normal to the sheet, so normal to the resultant velocity W at the section. With λ = V/(ΩR) the speed ratio (the advance ratio J = V/(nD) is πλ), ξ = r/R and x = Ωr/V = ξ/λ, the geometry of the velocity triangle gives the inflow angle directly:

tan φ = λ (1 + ζ/2)/ξ.

The tangent of φ times the radius is the same at every station: the blade elements lie on a screw of constant pitch, 2πRλ(1 + ζ/2).

Tip loss: Prandtl's factor and Goldstein's function

With a finite number of blades the fluid between two sheets does not move at the full sheet velocity; near the edge of the sheets it slips round them. The fraction F of the sheet velocity taken by the fluid at a radius sets the circulation the blades must carry there, B Γ = 2π r F v′ cos φ sin φ, or in the normalised form used on the page

G = B Γ Ω/(2π V² ζ) = F x cos φ sin φ.

Prandtl found F from the flow round the edges of a stack of parallel plates spaced as the sheets are at the tip: F = (2/π) arccos(e^−f), f = (B/2)(1 − ξ)/sin φ_t, where φ_t is the helix angle at the tip, tan φ_t = ξ tan φ = λ(1 + ζ/2). It is accurate when the sheets are close together compared with the radius: low advance ratio or many blades.

Goldstein solved the potential flow about the screw surfaces themselves. His circulation function K(ξ) = B Γ Ω/(2π(V + w)w) depends on the number of blades and on the pitch of the sheets, λ₂ = pitch/(2πR). In the notation above the Betz–Prandtl approximation is K = F μ²/(1 + μ²) with μ = ξ/λ₂, so Goldstein's result is used by putting F = K/cos² φ with λ₂ = ξ tan φ. The tool carries K for 2 to 12 blades and λ₂ from 0.05 to 4 in a table computed by a numerical solution of Goldstein's problem (finite volumes in the helical variable, with the mesh graded to the edge of the sheet); the values agree with those of Kramer and of Lock and Yeatman reprinted in NACA Report 775 within 0.007 and 2 %, and with Goldstein's own 1929 table within 0.008 except in the outer quarter of the radius, where his hand-computed values are up to 0.015 away (Kramer's later values there agree with the tool). For two blades at high advance ratio Goldstein's loading is noticeably lower toward the tip than Prandtl's and the efficiency a little lower; for three or more blades at ordinary advance ratios the two differ little. "None" sets F = 1, the limit of infinitely many blades, in which the chord does not close at the tip.

Thrust, power and the solution for ζ

The lift on an element is ρ W Γ per unit radius, normal to W; the drag is ε times the lift along W, ε = C_d/C_l. Resolving along the shaft and round it, and writing the axial and rotational interference factors that the momentum equations then require,

a = (ζ/2) cos² φ (1 − ε tan φ),   a′ = (ζ/2x) cos φ sin φ (1 + ε/tan φ),   W = V(1 + a)/sin φ,

the thrust and power coefficients T_c = 2T/(ρV²πR²) and P_c = 2P/(ρV³πR²) become

T_c = I₁ ζ − I₂ ζ²,   P_c = J₁ ζ + J₂ ζ²,

I₁ = ∫ 4ξG(1 − ε tan φ) dξ,   I₂ = ∫ λ(I₁′/2ξ)(1 + ε/tan φ) sin φ cos φ dξ,   J₁ = ∫ 4ξG(1 + ε/tan φ) dξ,   J₂ = ∫ (J₁′/2)(1 − ε tan φ) cos² φ dξ,

the integrals running from the hub to the tip. For a required thrust ζ is the smaller root of the first quadratic, for a given power the positive root of the second; because φ and F contain ζ the integrals are recomputed and the root taken again until ζ no longer changes. The efficiency is η = T_c/P_c. If the first quadratic has no real root the disc cannot carry the thrust at that speed with those sections. Larrabee's original procedure evaluates the integrals once, at ζ = 0 (light loading); the page prints its result beside the exact one.

The page separates the losses: the actuator-disc limit 2/(1 + √(1 + T_c)) is the efficiency of an ideal disc with the same thrust loading; the induced efficiency is that of the same design with frictionless sections (it adds the swirl and the tip loss); the ratio of the actual to the induced efficiency is the profile loss.

Chord and blade angle

The circulation fixes the product of chord and lift coefficient, Γ = ½ W c C_l:

W c C_l = 4π λ G V R ζ/B.

With every section at the design lift coefficient this gives the chord; the blade angle is β = α + φ, α being the angle of attack at which the section gives that lift coefficient. The blade angle is measured from the plane of rotation to the reference line of the section data (the chord line, or the flat face of a flat-bottomed section if its data are given from the face). A higher design lift coefficient narrows the blade in proportion and leaves less margin to the stall at low speed; the efficiency changes only through the lift-to-drag ratio. The station table lists, with chord and angle, the geometric pitch 2πr tan β and the two projections of the chord that a maker of a wooden blank needs. The chord closes to zero at the tip as the square root of the distance from it; in practice the last few percent are rounded.

The activity factor is the conventional measure of a blade's power absorption, AF = (100,000/16) ∫ (c/D)(r/R)³ d(r/R) from 0.15 to 1, per blade; the solidity is the blade area over the disc area.

Off-design analysis

A blade designed for one advance ratio is then held at fixed pitch and analysed at others. Each element of chord c and angle β at radius r must satisfy, with σ = Bc/(2πr), C_y = C_l cos φ − C_d sin φ and C_x = C_l sin φ + C_d cos φ at the angle of attack β − φ,

a/(1 + a) = σ C_y/(4 F sin² φ),   a′/(1 − a′) = σ C_x/(4 F sin φ cos φ),   tan φ = V(1 + a)/(Ω r (1 − a′)).

These are the momentum equations of the annulus with the same factor F as in the design (Larrabee's radially graded momentum theory), solved for φ by bisection in a form that remains regular at zero flight speed. The design relations above are exact solutions of them, so the analysis of the designed blade at the design advance ratio returns the design thrust and power; the page states both. Away from the design point the loading is no longer that of minimum induced loss, sections toward the root stall first as the advance ratio falls, and the thrust goes through zero somewhat above the design advance ratio. The static thrust is the J = 0 point of this analysis; the page also gives the speed at which the fixed-pitch blade absorbs the design power at rest (C_P constant, so n varies as the cube root of the power) and the thrust there, beside the momentum-theory limit (2ρAP²)^1/3. With most of the blade stalled these static figures rest on post-stall section data and are estimates.

The section data are either a simple model (a straight lift curve of the entered slope and zero-lift angle between the entered maximum and minimum lift coefficients, a drag that rises as the square of the departure from the design lift coefficient, and a flat-plate blend beyond the stall) or a pasted table of angle of attack, lift coefficient and drag coefficient, interpolated linearly and continued as a flat plate outside its range.

Checks

The design was checked against an independent numerical optimisation: with every section at the design lift coefficient, the chords of 24 annuli were varied to maximise the thrust at fixed power. For a lightly loaded frictionless propeller the optimiser gains nothing over the constant-displacement-velocity blade (less than one part in 10⁹); at a power loading P_c of 0.3 it gains 0.002 % without drag and 0.015 % with a section drag-to-lift ratio of 0.02. The Betz condition is thus the exact optimum for light loading and very nearly so otherwise.

The default case is the man-powered-airplane propeller of Larrabee's NASA paper (two blades, 3.81 m, 110 rpm, 53.3 N at 5 m/s). With the drag-to-lift curve read from his figure, his light-loading integrals and efficiency (0.8113) are reproduced within about 1 %. The four-blade example of NACA Report 924 (2,000 hp at 425 mph), designed there with Theodorsen's charts, gives an ideal efficiency of 0.930 against the report's 0.929 and blade widths within 4 % of its table.

Limits

One section polar is used at every radius and speed: there is no Reynolds-number or Mach-number correction, and the tip Mach number is only reported and flagged. The slipstream contraction and the flow field of a fuselage or nacelle are neglected. The theory is that of a lifting line: wide blades, swept blades and very low aspect ratios are outside it. The tool sizes the aerodynamic shape only; thickness, stresses and vibration of the blade are not treated.