Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/318

§ 203 to say, the propeller blade should be designed as an aerofoil for minimum gliding; we shall therefore take $$\gamma$$ from this point to denote the minimum gliding angle as independently ascertained. If the conditions of least gliding angle are infringed, the present theory continues to apply, but the result will be the best efficiency possible under the adverse conditions imposed, and not the real maximum.

Now $$\gamma ,$$ and therefore $$\cot \gamma ,$$ is constant in our expression; consequently we have to solve—

for maximum value where $$\theta$$ is the only variable. Differentiating in respect of $$\theta$$ we get—

or

transforming we get—

hence

which we may express in another way and say: The angle of greatest efficiency is 45 degrees, minus half the least gliding angle.

Thus, if we take the gliding angle in the case of air to be 10 degrees for any particular aspect proportions, the angle of greatest efficiency will be 40 degrees; or, taking the probable equivalent for water as 6 degrees, the appropriate angle becomes 42 degrees. The figures cited are probable figures for blades of about 4 : 1 ratio, as founded on experiment; it is known that the tabulated theoretical figures of § 181 are too low, from causes already discussed.

§ 204. Efficiency of the Screw Propeller, General Solution.—From Equation (1), § 202, we obtain