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Duct Thrust Share

Momentum theory of the ducted fan: how the thrust divides between the rotor and the duct lip, the induced velocity at the disk, the exit velocity, the ideal power and the power saving against an open rotor of the same diameter at the same thrust, all as functions of the duct's exit area ratio σd (exit area over disk area). A cylindrical duct (σd = 1) puts half the thrust on the lip and needs 1/√2 of the open rotor's ideal power; a diffuser (σd > 1) moves more thrust to the duct and saves more power until its walls separate. An optional hover figure of merit turns the ideal power into shaft power, and the chart sweeps σd so the diffuser's worth can be read against its length and separation risk.

  • Rotor and duct thrust shares, disk and exit velocities, ideal power and hover figure of merit from the exit area ratio

  • Comparison with the open rotor of the same diameter: induced velocity, ideal power and the power saving at equal thrust

  • Diffuser: half-angle and length for the exit area ratio, with the separation limits usually applied

  • Chart of thrust share, ideal power and induced velocity against σd, with the entered duct marked

Duct Thrust Share

Unit System: /

Fan and Thrust






Duct




Ducted-Fan Momentum Theory | Method page
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Duct Thrust Share

Ducted-Fan Momentum Theory

A duct changes the fan's performance in one dominant way: it fixes the area of the jet leaving the fan. An open rotor's wake contracts to half the disk area far downstream; a duct with exit area σd A holds the jet at that area, and the lip of the duct carries part of the thrust through the suction it develops in turning the inflow.

The one-dimensional model

Air arrives at the inlet with velocity Vc (zero in hover), passes the rotor disk of area A at velocity vd, and leaves the exit of area σd A at ambient pressure with velocity we. Continuity gives vd = σd we. The thrust is the momentum flux change, T = ṁ (we − Vc) with ṁ = ρ σd A we, and the ideal power is the kinetic energy added, P = ½ ṁ (we² − Vc²). For a given thrust the exit velocity follows from a quadratic.

Bernoulli upstream and downstream of the disk gives the pressure jump across the rotor, Δp = ½ ρ (we² − Vc²); the rotor thrust is Δp A and the duct carries the remainder. In hover the rotor share is 1/(2 σd): half the thrust for a cylindrical duct, less for a diffuser.

Comparison with the open rotor

The open rotor is the same model with σd = ½. In hover its ideal power is T3/2 / √(2 ρ A); the ducted fan's is T3/2 / √(4 σd ρ A), the ratio 1/√(2 σd). A cylindrical duct therefore needs 71 percent of the open rotor's ideal power at the same thrust, a diffuser with σd = 1.5 needs 58 percent. The saving is real only while the duct flow stays attached: conical diffusers separate beyond a wall half-angle of roughly 7°, and short ducts earlier, so the exit area ratios of practical fans stay near 1 to 1.3. Measured ducted fans also lose part of the saving to lip and wall friction, which momentum theory does not contain.

Figure of merit

The hover figure of merit entered on the form is the ideal power divided by the shaft power, referred to the ducted ideal. Good ducted fans reach 0.65 to 0.80. The Ducted Fan Performance tool computes the shaft power of a given blade and reports both the open-rotor and the ducted figures of merit.

What the blade sees

The blade works in the disk velocity vd = σd we, which for a diffuser is larger than the open rotor's disk velocity at the same thrust. The blade pitch must therefore be designed for the duct it will fly in; a blade optimized as an open rotor and then ducted runs at a lower angle of attack and gives less thrust than expected.



References

Leishman, J. G. (2006). Principles of Helicopter Aerodynamics, 2nd ed. Cambridge University Press, New York.

Stepniewski, W. Z., and Keys, C. N. (1984). Rotary-Wing Aerodynamics. Dover Publications, Mineola, NY.

Pereira, J. L. (2008). Hover and Wind-Tunnel Testing of Shrouded Rotors for Improved Micro Air Vehicle Design. PhD dissertation, University of Maryland, College Park, MD.

Mort, K. W., and Gamse, B. (1967). A Wind-Tunnel Investigation of a 7-Foot-Diameter Ducted Propeller. NASA Technical Note D-4142, Ames Research Center, Moffett Field, CA.

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