The Torque on a Blade Element

The determination of rotor thrust as a function of blade angle is of some importance, but more important is the answer to the question, "How much power is required to produce the thrust?” The first approximation of the power required is obtained using the Figure of Merit method described earlier. The second approximation may be made with the blade element theory.

The increment of power, A P, produced by a blade element, is:

AP = AQfl

where the incremental torque, AQ, is equal to the drag on the blade element times the radius of the element. Just as for airplanes, the drag is made up of two parts: induced and profile. From Figure 1.6, it may be seen that since the lift is perpendicular to the local velocity, it is tilted back by the induced angle, ф. Thus the induced drag due to lift, ДЦ, is:

AD, = ДТф

The profile drag, AD0, is the force parallel to the local flow; and, since ф is considered to be a small angle, the component of profile drag contributing to torque is taken as being equal to the entire profile drag. Thus the increment of torque, AQ, is:

AQ = <Діф + Ad0)

The drag can be treated in the same way that lift was:

AD0 = ^(£ir)2cdcAr

where the drag coefficient, cd, may be considered a constant at this stage of the analysis.

The equation for incremental torque can now be written as:

^(Пг)2С/сАгф {Cir)2cdcAr

Expressions for Ci and ф for an ideally twisted blade have already been derived. Using these, the torque per running foot is: