Q1 < Q2 + Q3 in case I.,
or
Q1 + Q2 < Q3 in case III.,
the value of h is too great.
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 | Excess Pressure. | Velocity | Excess Pressure. |
| ft. per sec. | lb. per sq. in. | ft. per sec. | lb. 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 lb. 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).