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Multi-Fan Vehicle Sizing

First-order hover and climb power budget of a vehicle lifted by several ducted fans: from the takeoff weight, the number of fans, their diameter and exit area ratio, the disk loading, the ideal and actual hover power with a figure of merit and a drive efficiency, the thrust margin for control, the power to climb at a given rate, and the battery energy and mass for a required hover time from the pack's specific energy and usable fraction. Everything comes from momentum theory, so the result is a sizing estimate to decide how many fans of what size a weight can afford before any blade is drawn; the Ducted Fan Performance tool then checks a blade against the thrust each fan must deliver.

  • Disk loading, induced velocity, ideal and shaft power per fan and total from momentum theory with the duct's exit area ratio

  • Thrust margin for control and gusts (fan thrust at the margin, power at that thrust)

  • Climb power at a given rate of climb and the power to hover at altitude

  • Battery energy, mass and fraction of takeoff weight for the hover time, with the drive efficiency and usable depth of discharge

  • Chart of hover power and battery mass against fan diameter

Multi-Fan Vehicle Sizing

Unit System: /

Vehicle and Fans








Mission







Hover Power and Battery Budget | Method page
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Multi-Fan Vehicle Sizing

Hover Power and Battery Budget

Before any blade is drawn, a lift-fan vehicle is sized by momentum theory alone: the weight sets the thrust, the fans' disk area sets the induced velocity, and the two set the ideal power. Two efficiencies, the fan's figure of merit and the drive's electrical efficiency, turn that into battery power, and the hover time turns the power into a battery mass. The battery mass then changes the weight, and the loop is iterated by hand until it closes.

Hover

Each fan carries T = W / N. For a duct with exit area σd A the exit velocity in hover is we = √(T / (ρ σd A)), the induced velocity at the disk is σd we, and the ideal power is ½ ṁ we² = T3/2 / √(4 σd ρ A). The shaft power is the ideal power divided by the figure of merit, the battery power the shaft power divided by the drive efficiency. Disk loading T / A is the single number that governs the budget: hover power per unit thrust grows with its square root.

Thrust margin

A hovering vehicle controls itself by changing the thrust of its fans, and it must ride gusts and climb. The fans and drives are therefore sized for a maximum thrust kT times the hover thrust, 1.5 to 2.0 for multirotors, and the power at that thrust grows with kT3/2. That power sets the motor and controller rating, not the hover power.

Climb

In a steady vertical climb at Vc the thrust still equals the weight (the airframe's drag is neglected here) but the air arrives at the disk with the climb speed, so the momentum power rises: P = ½ ṁ (we² − Vc²) with ṁ = ρ σd A we and T = ṁ (we − Vc). The increase over hover is roughly T Vc/2 for slow climbs.

Battery

The energy for the hover time is the battery power times the time; dividing by the usable depth of discharge gives the pack energy, and dividing by the pack's specific energy gives its mass. Lithium-ion packs deliver 150 to 220 Wh/kg at the pack level; a usable fraction of 0.7 to 0.85 keeps a reserve and the cells' life. When the battery comes out above about a third of the takeoff weight the design does not close: larger fans, fewer minutes or a better figure of merit are the levers.

Fan diameter trade

The chart holds everything but the fan diameter fixed. Hover power falls with the diameter because the induced velocity falls with it; the airframe, the duct weight and the footprint push the other way. N fans of diameter D have the same disk area as one fan of diameter D √N, so the count trades against the diameter at constant power.



References

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.

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

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