Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/329

Now, energy per second for pterygoid aerofoil handling mass per second $=m_{1}$ is

${\frac {m_{1}\ V^{2}\ (\beta ^{2}-\alpha ^{2})}{2}},$

and energy per second on the simple Newtonian basis for the same mass handled per second would be

${\frac {m_{1}\ V^{2}\ (\beta ^{2}+\alpha ^{2})}{2}},$

or, by (1), if $m_{2}$ is the mass per second dealt with by our hypothetical device, we shall have

${\frac {m_{2}}{m_{1}}}={\frac {m_{1}\ V^{2}\ (\beta +\alpha )^{2}}{m_{1}\ V^{2}\ (\beta ^{2}-\alpha ^{2})}}$ $={\frac {(\beta +\alpha )\ (\beta +\alpha )}{(\beta +\alpha )\ (\beta -\alpha )}}$

That is to say, the sectional area of the fluid stratum which would be acted upon will be—

${\frac {1+\epsilon }{1-\epsilon }}\ \kappa \ A$

Now, we may evidently regard the aerofoil, with its accompanying peripteral system, as the equivalent of the hypothetical device which we have temporarily assumed. The peripteral system actually constitutes a kind of tool or appliance by which the aerofoil is able to deal in effect with more air than actually comes within its sweep. This extended “sphere of influence” of the aerofoil will be termed the peripteral zone, and its cross-section, ${\frac {1+\epsilon }{1-\epsilon }}\ \kappa \ A$ is the peripteral area.

§ 211. The Screw Propeller: Number of Blades.—The number of blades in a propeller must be determined by the conditions of their non-interference. It is evident that if the peripteral areas of adjacent blades overlap, the total amount of fluid operated upon will be insufficient and the efficiency must diminish. We must therefore secure that the hehces on which the different blades are based are separated in effect by an area, measured on

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