Propeller Actuator-Disk Thrust

Ideal thrust, induced velocity, and Froude efficiency for a propeller or rotor from disk loading and flight speed.

InputT = 2·ρ·A·(V₀ + w)·w , A = πD²/4 , η = V₀/(V₀ + w) = 1/(1 + w/V₀)

Disk diameter (m)or enter ft — converts to mFreestream velocity (m/s)or enter mph — converts to m/sThrust (N)or enter lbf — converts to NAir density (kg/m³)

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The engineering

Actuator-disk (momentum) theory treats the propeller as an infinitely thin disk that adds a pressure jump to the flow, ignoring blades entirely. It gives the best-case thrust-to-power picture: for a given thrust and disk area, the induced velocity w is whatever the momentum balance demands, and the Froude efficiency tells you the ceiling no real prop can beat.

The lesson is disk loading. A big, lightly loaded disk (low T/A) needs only a small induced velocity, so η approaches 100% — that is why helicopters use huge rotors and why ducted fans lose out to open props in efficiency at low speed. At static thrust (V₀ = 0) η collapses to zero because useful power T·V₀ vanishes even though the disk still burns power moving air.

This is an ideal ceiling: real props lose to profile drag, tip losses, swirl, and non-uniform loading, so shaft power runs 15–30% above the disk power here. Use it to sanity-check a design, not to size a gearbox.

Where this math comes from

The momentum theory of propellers traces to W. J. M. Rankine (1865) and R. E. Froude (1889), who modeled a marine screw as a disk that accelerates a stream tube — the induced velocity being split half ahead of and half behind the disk falls straight out of their bookkeeping. It was ship-propeller work long before it was aeronautical.

Hermann Glauert consolidated and extended the theory for aircraft in his 1926 Elements of Aerofoil and Airscrew Theory and the 1935 Durand Aerodynamic Theory volume, joining simple momentum results to blade-element analysis. His treatment is still the standard entry point, and the efficiency ceiling this card computes is exactly Glauert's ideal.

1865W. J. M. RankineIntroduces momentum theory for the marine screw propeller.
1889R. E. FroudeRefines the actuator-disk model; induced velocity splits half fore and half aft.
1920Albert BetzDerives the ideal actuator-disk power limit for extracting/adding energy to a stream.
1926Hermann GlauertUnifies momentum and blade-element theory for aircraft airscrews.
1935Hermann GlauertDurand's Aerodynamic Theory publishes the definitive propeller treatment.

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