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National Advisory Committee for Aeronautics, Report - Mathematical Equations for Heat Conduction in the Fins of Air Cooled Engines

At the request and with the support of the Engineering Division of theAirService, United
States Army, McCook Field, the Bureau of Standards undertook some laboratory investiga-
tions dealing with air~cooled aviation engines, the results of which were submitted to the
Committee on Power Plants for Aircraft and by that committee recommended for publication
as a technical report of the National Advisory Committee for Aeronautics. In connection
with laboratory measurements of the heat-dissipating power of typical engine cylinders, a
mathematical analysis of fin behavior was made and is given in this communication.

The introduction contains a. description of the paper which will assist the general reader
who is not interested in mathematical detail in finding those parts of the paper most likely
to prove useful to him. A recapitulation of the‘mathematical developments is given in
Section IV and forms the statement of conclusions reached in so far as a mathematical paper
of this type may be said to have conclusions. Numerical examples illustrative of these con-
clusions are then given, followed by a very brief suggestion of possible application of the
equations.

The problem considered is that of reducing actual geometrical area of fin-cooling surface,
which is, of course, not uniform in temperature, to equivalent “cooling” area at one definite
temperature, namely, that prevailing on the cylinder wall at the point of attachment of the
fin. This makes it possible to treat all the cooling surface as if it were part of the cylinder
wall and 100 per cent efiective.

The quantities involved in the equations are the geometrical dimensions of the fin, thermal
conductivity of the material composing it, and the coefiicient of surface heat dissipation betWeen
the fin and the air stream. Several assumptions of physical nature are thus necessarily involved
in making the problem possible of solution. These are set forth in detail, and the limitations
which result from them in applying the equations to numerical calculations are carefully
pointed out.