Hovering Helicopter

A helicopter needs a minimum of a 100 hp engine to hover (1 hp = 746 W). Estimate the minimum power necessary to hover for the motor of a 10 times reduced model of this helicopter (assuming that it is made of the same materials).

SOLUTION

The simplest model for a helicopter rotor is a disk of area A inducing a uniform flow of air with velocity close to the rotor and ω far downstream. For an estimate, this model is adequate (see for instance Johnson, Helicopter Theory). We disregard any energy loss due to turbulence and consider air to be an incompressible fluid. The rate of mass flowing through the area A of the rotor will be

(1)

The thrust T is equal to the momentum change per unit time of the air from velocity 0 to ω:

(2)

The power P is equal to the energy change of the same amount of air per unit time (seconds):

(3)

From (2) and (3) we have ω = 2v, and substituting ṁ and ω from (1) into (3), we can get

(4)

and therefore

(5)

If the helicopter is hovering, it means that its weight W is equal to the thrust T in our ideal case. Then

(6)

The resulting power is

(7)

Since the weight W is proportional to the volume L³, of the helicopter, and the area A is, of course, proportional to L², we find

(8)

So, for a model of the helicopter

where p and l are the power and size of the model, respectively. For a 1/10th size model: