§ 89. _Flow of Air in Long Pipes._--When air flows through a long
pipe, by far the greater part of the work expended is used in
overcoming frictional resistances due to the surface of the pipe. The
work expended in friction generates heat, which for the most part must
be developed in and given back to the air. Some heat may be
transmitted through the sides of the pipe to surrounding materials,
but in experiments hitherto made the amount so conducted away appears
to be very small, and if no heat is transmitted the air in the tube
must remain sensibly at the same temperature during expansion. In
other words, the expansion may be regarded as isothermal expansion,
the heat generated by friction exactly neutralizing the cooling due to
the work done. Experiments on the pneumatic tubes used for the
transmission of messages, by R. S. Culley and R. Sabine (_Proc. Inst.
Civ. Eng._ xliii.), show that the change of temperature of the air
flowing along the tube is much less than it would be in adiabatic
expansion.
§ 90. _Differential Equation of the Steady Motion of Air Flowing in a
Long Pipe of Uniform Section._--When air expands at a constant
absolute temperature [tau], the relation between the pressure p in
pounds per square foot and the density or weight per cubic foot G is
given by the equation
p/G = c[tau], (1)
where c = 53.15. Taking [tau] = 521, corresponding to a temperature of
60° Fahr.,
c[tau] = 27690 foot-pounds. (2)
The equation of continuity, which expresses the condition that in
steady motion the same weight of fluid, W, must pass through each
cross section of the stream in the unit of time, is