Duct Thrust Share: Method

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 σ_d 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 V_c (zero in hover), passes the rotor disk of area A at velocity v_d, and leaves the exit of area σ_d A at ambient pressure with velocity w_e. Continuity gives v_d = σ_d w_e. The thrust is the momentum flux change, T = ṁ (w_e − V_c) with ṁ = ρ σ_d A w_e, and the ideal power is the kinetic energy added, P = ½ ṁ (w_e² − V_c²). 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 (w_e² − V_c²); the rotor thrust is Δp A and the duct carries the remainder. In hover the rotor share is 1/(2 σ_d): half the thrust for a cylindrical duct, less for a diffuser.

Comparison with the open rotor

The open rotor is the same model with σ_d = ½. In hover its ideal power is T^3/2 / √(2 ρ A); the ducted fan's is T^3/2 / √(4 σ_d ρ A), the ratio 1/√(2 σ_d). A cylindrical duct therefore needs 71 percent of the open rotor's ideal power at the same thrust, a diffuser with σ_d = 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 v_d = σ_d w_e, 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.