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Ducted Fan Performance

Thrust, torque, power and figure of merit of a ducted fan whose blade geometry is known, from blade-element momentum theory: the chord and pitch at a few stations from hub to tip, the airfoil's lift and drag polynomials, the rotor speed and the climb or inlet velocity. The induced velocity of every annulus is solved exactly from the balance of the blade-element thrust with the momentum thrust (the Ducted Fan Optimization tool's small-angle inflow is a few percent higher at the hub). One operating point is written out in full and a sweep of rotor speed, inlet velocity or a pitch offset draws thrust, power and figure of merit against the swept variable: the inverse of the optimizer, which designs the blade for one power.

  • Blade geometry entered as stations (radius, chord, pitch) and interpolated linearly between them; paste the optimizer's table or a measured blade

  • Airfoil lift and drag as 6th-order polynomials in angle of attack with the optimizer's Reynolds and Mach corrections

  • Exact annulus momentum balance; no tip loss (the duct), no swirl recovery; standard atmosphere at the entered altitude

  • Thrust, torque, power, blade area, solidity, open-rotor and ducted figures of merit, element table and span distributions

  • Sweep of rpm, climb velocity or collective pitch offset with charts of thrust, power and figure of merit

Ducted Fan Performance

Unit System: /

Rotor and Operating Point







Blade Stations (hub to tip)
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Airfoil




















Sweep









Resolution and Duct



Blade Element Method and Conventions | Method page
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Ducted Fan Performance

Ducted Fan Performance: Blade Element Method and Conventions

The Ducted Fan Optimization tool answers the design question: which chord and pitch distribution gives the most thrust for a given shaft power. This tool answers the analysis question for a blade that already exists: what thrust, torque and power does it produce at a given rotor speed, climb or inlet velocity and altitude, and how do they change when one of those is varied.

Blade geometry

The blade is entered as stations from hub to tip: radius, chord and pitch angle θ measured from the plane of rotation to the chord line. Between stations the chord and pitch are interpolated linearly; outside the first and last station the end values are held. The optimizer's element table can be pasted in directly (every row, or a few of them), or a measured blade can be entered.

Blade element momentum theory

The blade is divided into elements between the hub radius and the tip radius. At each element the air arrives with the axial velocity Vc + vi (inlet velocity plus the induced velocity at the disk) and the tangential velocity Ω r. The inflow angle is φ = atan((Vc + vi) / (Ω r)), the angle of attack α = θ − φ, and the section lift and drag follow from the airfoil polynomials. The thrust of the element per unit radius is dT/dr = Nb (dL cos φ − dD sin φ) and its in-plane force dFt/dr = Nb (dL sin φ + dD cos φ), from which torque and power follow with the radius and the rotor speed.

The induced velocity is the one that makes this blade-element thrust equal to the momentum thrust of the same annulus, dT/dr = 4 π ρ r (Vc + vi) vi. The tool solves that balance exactly at every element by bisection. The optimizer uses instead the closed-form small-angle solution of the same balance with a local linear lift model; near the hub, where the inflow angle is large, that solution leaves the balance open by up to 20 percent and its thrust comes out a few percent higher. The difference for the optimizer's own 48-inch rotor is about 2 percent in thrust.

As in the optimizer, no tip-loss factor is applied (the duct suppresses the tip vortex), the swirl in the wake is not recovered, and the duct's own thrust is not included: the thrust reported is the blade thrust. The ducted ideal power and figure of merit are given alongside the open-rotor ones so the blade can be judged against either reference.

Airfoil data and corrections

Lift and drag are 6th-order polynomials in the angle of attack in degrees, the same form as the optimizer. The Reynolds-number correction scales the coefficients by the optimizer's fits referenced to Re = 106 at 7°, and the Mach correction by its fits referenced to M = 0. Elements whose angle of attack falls outside the fitted range are marked with an asterisk in the element table and counted in the notes: the polynomials are extrapolated there and the loads of those elements should not be trusted.

Figures of merit

The ideal power of an open rotor is T3/2 / √(2 ρ A), its wake contracting to half the disk area. A duct with exit area σd A has the ideal power T3/2 / √(4 σd ρ A): for a cylindrical duct (σd = 1) that is 1/√2 of the open rotor's, which is the classic halving of the induced velocity by the duct. The figure of merit is the ideal power divided by the shaft power; both references are reported. The optimizer prints √2 times the open-rotor value and calls 1.414 its maximum, which is the same quantity scaled.

Sweeps

With the blade fixed, the rotor speed, the inlet velocity or a collective pitch offset (added to every station's pitch) can be swept between two limits. Thrust, shaft power and figure of merit are charted against the swept variable with the operating point marked, and the sweep table lists the values with the number of stalled elements at each point. Thrust rises roughly with the square of the rotor speed and power with its cube until the sections stall; an inlet velocity unloads the blade and reduces both.

Atmosphere

Density, temperature and speed of sound come from the International Standard Atmosphere at the entered pressure altitude (troposphere lapse of 6.5 K/km to 11 km, isothermal above); viscosity from Sutherland's law. At sea level the density is 1.225 kg/m³ (0.002377 slug/ft³) and the speed of sound 340.3 m/s (1,116 ft/s).



References

Glauert, H. (1935). "Airplane Propellers." In Durand, W. F. (ed.), Aerodynamic Theory, Vol. IV, Division L, Julius Springer, Berlin, pp. 169–360.

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

Johnson, W. (1980). Helicopter Theory. Princeton University Press, Princeton, NJ.

Weick, F. E. (1930). Aircraft Propeller Design. McGraw-Hill, New York.

International Civil Aviation Organization (1993). Manual of the ICAO Standard Atmosphere, 3rd ed. Doc 7488, ICAO, Montreal.

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