Since the limits between which h can vary are in practical cases not very distant, it is easy to approximate to values sufficiently accurate.
§ 88. Water Hammer.—If in a pipe through which water is flowing a sluice is suddenly closed so as to arrest the forward movement of the water, there is a rise of pressure which in some cases is serious enough to burst the pipe. This action is termed water hammer or water ram. The fluctuation of pressure is an oscillating one and gradually dies out. Care is usually taken that sluices should only be closed gradually and then the effect is inappreciable. Very careful experiments on water hammer were made by N. J. Joukowsky at Moscow in 1898 (Stoss in Wasserleitungen, St Petersburg, 1900), and the results are generally confirmed by experiments made by E. B. Weston and R. C. Carpenter in America. Joukowsky used pipes, 2, 4 and 6 in. diameter, from 1000 to 2500 ft. in length. The sluice closed in 0.03 second, and the fluctuations of pressure were automatically registered. The maximum excess pressure due to water-hammer action was as follows:—
| Pipe 4-in. diameter. | Pipe 6-in. diameter. | ||
| Velocity ft. per sec. | Excess Pressure. ℔ per sq. in. | Velocity ft. per sec. | Excess Pressure. ℔ per sq. in. |
| 0.5 | 31 | 0.6 | 43 |
| 2.9 | 168 | 3.0 | 173 |
| 4.1 | 232 | 5.6 | 369 |
| 9.2 | 519 | 7.5 | 426 |
In some cases, in fixing the thickness of water mains, 100 ℔ per sq. in. excess pressure is allowed to cover the effect of water hammer. With the velocities usual in water mains, especially as no valves can be quite suddenly closed, this appears to be a reasonable allowance (see also Carpenter, Am. Soc. Mech. Eng., 1893).
IX. FLOW OF COMPRESSIBLE FLUIDS IN PIPES
§ 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 τ, 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 = cτ,
(1)
where c = 53.15. Taking τ = 521, corresponding to a temperature of 60° Fahr.,