Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/311

any possible disturbances that may be set up in the fluid by the body in motion.

Let	F	be the sum of the propulsive forces.
„	m	„  mass of fluid handled per second.
„	$V$	„  velocity of vessel, the term vessel being used to denote the body propelled.
„	v	„  a uniform sternward velocity imparted to the fluid operated upon.

All in Absolute Units.

Then we have (§ 3) $\mathrm {F} =\mathrm {m} \ \mathrm {v} ,$ and the work done usefully per second is

and the energy left in the fluid per second, that is, lost power, is

or total energy per second

$=\mathrm {m} \ \mathrm {v} \ V+{\frac {\mathrm {m} \ \mathrm {v} ^{2}}{2}}$

or efficiency

This, according to Rankine,^[1] is the theoretical limit to the efficiency of a propeller. It will be shown subsequently that this assertion requires qualification.

If we depart from the simplicity of the assumption and suppose that the different portions of the fluid acted upon receive different velocities, the foregoing demonstration requires appropriate modification; the v of expression (1) and the v of expression (2) are not the same quantity, the v² in the latter expression becoming the mean square of the velocity v instead of the square of the mean. For a given value of m the efficiency must thus be less than if the velocity v were uniform over the

1. “Miscellaneous Scientific Papers,” Rankine, XXXIII.