Mouthpieces—Head Constant
§ 49. Cylindrical Mouthpieces.—When water issues from a short cylindrical pipe or mouthpiece of a length at least equal to l1⁄2 times its smallest transverse dimension, the stream, after contraction within the mouthpiece, expands to fill it and issues full bore, or without contraction, at the point of discharge. The discharge is found to be about one-third greater than that from a simple orifice of the same size. On the other hand, the energy of the fluid per unit of weight is less than that of the stream from a simple orifice with the same head, because part of the energy is wasted in eddies produced at the point where the stream expands to fill the mouthpiece, the action being something like that which occurs at an abrupt change of section.
Let fig. 58 represent a vessel discharging through a cylindrical mouthpiece at the depth h from the free surface, and let the axis of the jet XX be taken as the datum with reference to which the head is estimated. Let Ω be the area of the mouthpiece, ω the area of the stream at the contracted section EF. Let v, p be the velocity and pressure at EF, and v1, p1 the same quantities at GH. If the discharge is into the air, p1 is equal to the atmospheric pressure pa.
The total head of any filament which goes to form the jet, taken at a point where its velocity is sensibly zero, is h + pa/G; at EF the total head is v2/2g + p/G; at GH it is v12/2g + p1/G.
Between EF and GH there is a loss of head due to abrupt change of velocity, which from eq. (3), § 36, may have the value
(v − v1)2/2g.
Adding this head lost to the head at GH, before equating it to the heads at EF and at the point where the filaments start into motion,—
h + pa/G = v2/2g + p/G = v12/2g + p1/G + (v − v1)2/2g.
But ωv = Ωv1, and ω = ccΩ, if cc is the coefficient of contraction within the mouthpiece. Hence
v = Ωv1/ω = v1/cc.