§ 51. Divergent Conoidal Mouthpiece.—Suppose a mouthpiece so designed that there is no abrupt change in the section or velocity of the stream passing through it. It may have a form at the inner end approximately the same as that of a simple contracted vein, and may then enlarge gradually, as shown in fig. 60. Suppose that at EF it becomes cylindrical, so that the jet may be taken to be of the diameter EF. Let ω, v, p be the section, velocity and pressure at CD, and Ω, v1, p1 the same quantities at EF, pa being as usual the atmospheric pressure, or pressure on the free surface AB. Then, since there is no loss of energy, except the small frictional resistance of the surface of the mouthpiece,
h + pa/G = v2/2g + p/G = v12/2g + p1/G.
If the jet discharges into the air, p1 = pa; and
v12/2g = h;
v1 = √(2gh);
or, if a coefficient is introduced to allow for friction,
v1 = cv √(2gh);
where cv is about 0.97 if the mouthpiece is smooth and well formed.
Q = Ω v1 = cv Ω √(2gh).
| Fig. 61. |