When the variation of pressure is very small, it is no longer safe to neglect the variation of level of the pipe. For that case we may neglect the work done by expansion, and then
z0 − z1 − p0/G0 − p1/G1 − ζ (v2/2g) (l/m) = 0,
(10)
precisely equivalent to the equation for the flow of water, z0 and z1 being the elevations of the two ends of the pipe above any datum, p0 and p1 the pressures, G0 and G1 the densities, and v the mean velocity in the pipe. This equation may be used for the flow of coal gas.
§ 92. Distribution of Pressure in a Pipe in which Air is Flowing.—From equation (7a) it results that the pressure p, at l ft. from that end of the pipe where the pressure is p0, is
p = p0 √ (1 − ζ lu02 / mgcτ);
(11)
which is of the form
p = √ (al + b)
for any given pipe with given end pressures. The curve of free surface level for the pipe is, therefore, a parabola with horizontal axis. Fig. 100 shows calculated curves of pressure for two of Sabine’s experiments, in one of which the pressure was greater than atmospheric pressure, and in the other less than atmospheric pressure. The observed pressures are given in brackets and the calculated pressures without brackets. The pipe was the pneumatic tube between Fenchurch Street and the Central Station, 2818 yds. in length. The pressures are given in inches of mercury.