where [Omega] is the area of the external end of the mouthpiece, and
c_c[Omega] the section of the contracted jet beyond the mouthpiece.
_Convergent Mouthpieces (Castel's Experiments).--Smallest diameter of
orifice = 0.05085 ft. Length of mouthpiece = 2.6 Diameters._
+------------+--------------+--------------+--------------+
| |Coefficient of|Coefficient of|Coefficient of|
| Angle of | Contraction, | Velocity, | Discharge, |
|Convergence.| c_c | c_v | c |
+------------+--------------+--------------+--------------+
| 0° 0´ | .999 | .830 | .829 |
| 1° 36´ | 1.000 | .866 | .866 |
| 3° 10´ | 1.001 | .894 | .895 |
| 4° 10´ | 1.002 | .910 | .912 |
| 5° 26´ | 1.004 | .920 | .924 |
| 7° 52´ | .998 | .931 | .929 |
| 8° 58´ | .992 | .942 | .934 |
| 10° 20´ | .987 | .950 | .938 |
| 12° 4´ | .986 | .955 | .942 |
| 13° 24´ | .983 | .962 | .946 |
| 14° 28´ | .979 | .966 | .941 |
| 16° 36´ | .969 | .971 | .938 |
| 19° 28´ | .953 | .970 | .924 |
| 21° 0´ | .945 | .971 | .918 |
| 23° 0´ | .937 | .974 | .913 |
| 29° 58´ | .919 | .975 | .896 |
| 40° 20´ | .887 | .980 | .869 |
| 48° 50´ | .861 | .984 | .847 |
+------------+--------------+--------------+--------------+
The maximum coefficient of discharge is that for a mouthpiece with a
convergence of 13°24´.
The values of c_v and c_c must here be determined by experiment. The
above table gives values sufficient for practical purposes. Since the
contraction beyond the mouthpiece increases with the convergence, or,
what is the same thing, c_c diminishes, and on the other hand the loss
of energy diminishes, so that c_v increases with the convergence,
there is an angle for which the product c_c c_v, and consequently the
discharge, is a maximum.
[Illustration: FIG. 59.]
§ 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 [omega], v, p be the section, velocity and pressure
at CD, and [Omega], v1, p1 the same quantities at EF, p_a 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 + p_a/G = v²/2g + p/G = v1²/2g + p1/G.