A tool for actuator-disk theory (Froude momentum theory), which models a propeller as an infinitely thin disk. Adjust the diameter, thrust and airspeed to see the induced velocity added to the air, the slipstream velocity, the ideal power required, and the Froude efficiency — the upper limit of propeller efficiency — update in real time.
Parameters
Propeller diameter D
m
Diameter of the actuator disk
Thrust T
N
Thrust produced by the propeller
Airspeed V
m/s
Forward speed. 0 = static thrust (hover)
Air density ρ
kg/m³
1.225 at sea-level standard atmosphere; falls with altitude
Results
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Disk area (m²)
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Induced velocity v_i (m/s)
—
Wake velocity (m/s)
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Ideal power required (kW)
—
Froude efficiency η (%)
—
Disk loading (N/m²)
—
Actuator disk and slipstream
Air approaching at the freestream speed V is accelerated at the disk by the induced velocity v_i. The slipstream contracts and speeds up to V+2v_i far downstream.
Froude efficiency vs disk loading
Induced velocity v_i vs thrust T
Theory & Key Formulas
$$A = \pi\left(\frac{D}{2}\right)^{2}, \qquad T = 2\rho A\,(V+v_i)\,v_i$$
Disk area A and the thrust T from conservation of momentum. ρ: air density, V: airspeed, v_i: induced velocity.
Froude (ideal) propulsive efficiency and ideal power required. A larger disk gives a smaller induced velocity and therefore a higher ideal efficiency.
What is Actuator-Disk Theory?
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A propeller has all those blades spinning in a complicated way. Does "actuator-disk theory" calculate those blades?
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That is the fun part — this theory never looks at the blades at all. It replaces the propeller with a single, infinitely thin disk. The disk does just one thing: it adds momentum to the air passing through it. Blade count, airfoil shape, twist — all ignored. It sounds crude, but it lets you write down the theoretical upper limit of propeller efficiency in a few lines.
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How can it produce thrust without looking at the blades? How is it calculated?
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It uses only conservation of momentum. A propeller accelerates air backwards and gets thrust from the reaction. The speed increment added to the air at the disk is called the "induced velocity v_i". The thrust can be written T = 2ρA(V+v_i)v_i, where A is the disk area, ρ the air density and V the airspeed. You just solve for v_i. Raise the thrust T on the left and you will see both the induced velocity v_i and the wake velocity climb steadily.
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The wake velocity comes out as V+2v_i. The disk only added v_i — why is it doubled downstream?
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Good question. The disk produces thrust through a pressure jump, but the acceleration of the air happens over a region both upstream and downstream of the disk. Exactly half the v_i is added before the air reaches the disk, the other half after it passes. So far downstream the slipstream has been accelerated to V+2v_i. At the same time the streamtube contracts, narrowing as the flow speeds up. Watch the animation above: the flow arrows lengthen behind the disk and the streamtube necks down.
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There is also a number called "Froude efficiency". What does that represent?
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The Froude efficiency is the ideal propulsive efficiency, η = V/(V+v_i). It is the useful work (thrust × speed) divided by the ideal power required, T(V+v_i). Because a propeller must accelerate air, it always leaves kinetic energy behind in the slipstream. That is a loss, so the efficiency can never reach 100%. The interesting case is hover, V=0. Put that into the formula and η=0 — at static thrust the ideal efficiency is zero and all the power goes into the wake. That is why hovering a helicopter is so power-hungry.
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So to make an efficient propeller, what should I make bigger?
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The diameter. That answer comes straight out of the theory. For the same thrust, a larger disk means "accelerate a lot of air a little", so the induced velocity v_i is small. The smaller v_i is, the closer η = V/(V+v_i) gets to one. Look at the "efficiency vs disk loading" chart below: the lower the disk loading (thrust ÷ area), the higher the efficiency. That is why helicopter rotors are so large, and why a glider-tug propeller has a big diameter. Conversely, a small, highly-loaded disk is compact but inherently inefficient.
Frequently Asked Questions
Actuator-disk theory (Froude momentum theory) is the simplest aerodynamic model of a propeller, a helicopter rotor or a wind turbine. It replaces the rotor with an infinitely thin disk that ignores the blades entirely and does just one thing: it adds momentum to the air passing through it. By applying conservation of mass, momentum and energy across the disk, the theory delivers the induced velocity (the extra speed the disk imparts to the air) and the ideal propulsive efficiency, the Froude efficiency. That efficiency is an absolute upper limit that no real propeller can beat.
With disk area A, air density ρ and airspeed V, conservation of momentum gives the thrust as T = 2ρA(V+v_i)v_i. Solving this quadratic for v_i yields v_i = (−V + √(V² + 2T/(ρA)))/2. The induced velocity is the speed increment added to the air at the disk. Far downstream, the slipstream has been accelerated to V+2v_i — twice the increment added at the disk eventually appears in the wake.
The Froude (ideal) propulsive efficiency is η = V/(V+v_i), the useful power T·V divided by the ideal power required T·(V+v_i). A propeller produces thrust by accelerating air, so it always leaves kinetic energy behind in the slipstream. That residual wake energy is a loss, so the efficiency is always below 100%. As the induced velocity v_i approaches zero the efficiency approaches one, but then the thrust also goes to zero. In hover (V=0) the Froude efficiency is 0 — all the power goes into the slipstream.
The design lesson falls straight out of the theory. For a given thrust, a larger disk accelerates a larger mass of air to a smaller induced velocity. The smaller v_i is, the higher the Froude efficiency η = V/(V+v_i). That is exactly why efficient propellers, and especially helicopter rotors, are made as large in diameter as is practical. Conversely, a small, highly-loaded disk (high disk loading) is compact but inherently inefficient.
Real-World Applications
Early-stage propeller design: Detailed propeller design needs blade-element momentum theory (BEMT) or CFD, but in the conceptual stage before that, actuator-disk theory gives an instant estimate of "what ideal efficiency this diameter, thrust and cruise speed allow". If the ideal efficiency is too low, the diameter can be revised before any blade detailing is done. Real propeller efficiency is the Froude efficiency multiplied by the airfoil drag, the swirl loss in the wake and the finite-blade tip loss.
Helicopter rotors and hover performance: A helicopter rotor is exactly a large actuator disk. Hover is V=0, so the Froude efficiency concept gives 0, but momentum theory still predicts the induced velocity and the induced power in hover — and that explains why larger rotors hover more efficiently. The lower the disk loading, the less power a helicopter needs to support a given weight.
Drone and eVTOL rotor sizing: Multicopters and eVTOL aircraft must choose rotor diameter and rotor count within a limited airframe size. Actuator-disk theory shows quantitatively that a few large rotors give a lower disk loading — and therefore better hover efficiency and longer flight time — than many small rotors. It also feeds endurance estimates from battery capacity.
Contrast with wind turbines: The same momentum theory applies to wind turbines, and reversing the direction yields the Betz limit (the 59.3% upper bound on power that can be extracted from the wind). A propeller adds energy to the air; a wind turbine extracts it — the fact that the same actuator-disk framework unifies both shows just how powerful this theory is.
Common Misconceptions and Pitfalls
The biggest misconception is equating the Froude efficiency with the efficiency of a real propeller. Actuator-disk theory gives only the ideal efficiency that accounts for the momentum loss (the kinetic energy left in the wake). A real propeller additionally suffers airfoil friction and pressure drag (profile losses), the swirl loss of the rotating wake, and the tip loss from a finite number of blades. Real propeller efficiency is always lower than the Froude efficiency — typically around 0.80 to 0.85 in cruise. Read this tool's output as "the upper limit it can never exceed".
Next, "the Froude efficiency is 0 at static thrust (hover), so hovering is pointless". Because the Froude efficiency is defined as η=V/(V+v_i), it is necessarily 0 at V=0 — but that is merely a consequence of the definition: with no forward motion there is no useful work in the forward direction. Hovering itself is clearly useful. To evaluate hover performance you use a different metric called the "figure of merit", not the Froude efficiency. Forward flight and hover must be judged on different scales.
Finally, the assumption that "raising the disk loading just makes things more compact with no penalty". Shrinking the disk at the same thrust raises the disk loading, increases the induced velocity v_i, and increases the ideal power required T(V+v_i). In other words, downsizing always comes with more power and lower efficiency. Furthermore, a high-disk-loading rotor has a fast wake and a fierce downwash that blasts the ground. That is why high-disk-loading aircraft such as the V-22 Osprey kick up dust violently and have to be choosy about landing sites. Never forget that compactness trades off against efficiency and operability.
How to Use
Enter propeller diameter (0.5–5 m) in diaNum; the simulator calculates disk area A = πD²/4
Set thrust requirement (100–50,000 N) and freestream velocity (0–50 m/s) in thrustNum and vNum
Input fluid density (0.8–1.225 kg/m³ for air/water) in rhoNum and run; outputs induced velocity v_i, wake velocity w, ideal power P_ideal = T(V + v_i), Froude efficiency η = 2/(1 + w/V), and disk loading T/A
Worked Example
Marine propeller: D=0.8 m (A=0.503 m²), thrust T=5,000 N, freestream V=8 m/s, seawater ρ=1,025 kg/m³. Actuator-disk theory yields v_i ≈ 2.14 m/s, wake velocity w ≈ 4.28 m/s, ideal power P_ideal ≈ 50.7 kW, Froude efficiency η ≈ 0.79 (79%), disk loading ≈ 9,940 N/m². Increasing thrust to 8,000 N raises v_i proportionally and reduces efficiency due to higher momentum losses.
Practical Notes
Disk loading above 2,000 N/m² indicates cavitation risk in marine propulsion; UAV propellers typically operate at 50–500 N/m² for efficiency
Froude efficiency degrades rapidly in static thrust (V=0); helicopter rotors mitigate this via collective pitch and blade design optimization
Actuator-disk theory neglects rotational wake and blade geometry; actual propeller performance requires CFD or blade-element momentum (BEM) coupling for detailed wake analysis