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National Advisory Committee for Aeronautics, Technical Notes - Notes on Propeller Design - I - The Energy Losses of Propeller

The knowledge of the different kinds of energy losses of the
prepeller and of the magnitude of the losses in each single case
is of great value to the designer. There are three different
kinds of energy losses, and the most important has been the least
often discussed in the publications of recent years. This is the
friction between the air and the blade when whirled through it.
Suppose the propeller to be well shaped, so that each blade ele—
ment is working under a proper angle of attack.' Corresponding to
the induced drag of an ordinary wing, there are then coming into
action the slip stream loss and other similar losses to be dis—
cussed afterwards. Besides, there is the friction of the blade
surface moved through the air.

The drag coefficient which expresses the relative magnitude
of this friction depends, it is true, on the blade section and on
its angle of attack or, what amounts to the same thing, on its
momentary lift coefficient. But the variability of the drag co—
efficient for reasonable angles of attack is much smaller than
often supposed, the variation for different sections as well as
for different angles of attack being small. There is a certain
minimum of the drag coefficient existing, which it seems can ale
ways be obtained under reasonable conditions by the proper choice
of the section, whether the desired lift coefficient be smaller
or greater. Hence it is admissible to assume the drag coeffi—
cient CD to be constant for all propellers under those particu—
lar conditions for which it is chiefly designed.

The energy loss produced by the drag is the sum of all these
losses of each single blade element. Let i be the number of
blades, b their breadth at the point considered, v the veloc—
ity of the blade element relative to the air, r the distance
from axis, dr the length of the blade element, and D the pro—
peller diameter. The entire loss per unit time due to friction.