§ 19. Coefficients for Bellmouths and Bellmouthed Orifices.—If an orifice is furnished with a mouthpiece exactly of the form of the contracted vein, then the whole of the contraction occurs within the mouthpiece, and if the area of the orifice is measured at the smaller end, cc must be put = 1. It is often desirable to bellmouth the ends of pipes, to avoid the loss of head which occurs if this is not done; and such a bellmouth may also have the form of the contracted jet. Fig. 19 shows the proportions of such a bellmouth or bell-mouthed orifice, which approximates to the form of the contracted jet sufficiently for any practical purpose.

For such an orifice L. J. Weisbach found the following values of the coefficients with different heads.

Head over orifice, in ft. = h.661.6411.4855.77337.93
Coefficient of velocity = cv.959.967.975.994.994
Coefficient of resistance = cr.087.069.052.012.012

As there is no contraction after the jet issues from the orifice, cc = 1, c = cv; and therefore

Q = cvω √ (2gh) = ω √ { 2gh / (1 + cr }.

§ 20. Coefficients for Sharp-edged or virtually Sharp-edged Orifices.—There are a very large number of measurements of discharge from sharp-edged orifices under different conditions of head. An account of these and a very careful tabulation of the average values of the coefficients will be found in the Hydraulics of the late Hamilton Smith (Wiley & Sons, New York, 1886). The following short table abstracted from a larger one will give a fair notion of how the coefficient varies according to the most trustworthy of the experiments.

Coefficient of Discharge for Vertical Circular Orifices, Sharp-edged, with free Discharge into the Air. Q = cω √ (2gh).

Head
measured to
Centre of
Orifice.		 Diameters of Orifice.
.02 .04 .10 .20 .40 .60 1.0
Values of C.
0.3 .. .. .621 .. .. .. ..
0.4 .. .637 .618 .. .. .. ..
0.6 .655 .630 .613 .601 .596 .588 ..
0.8 .648 .626 .610 .601 .597 .594 .583
1.0 .644 .623 .608 .600 .598 .595 .591
2.0 .632 .614 .604 .599 .599 .597 .595
4.0 .623 .609 .602 .599 .598 .597 .596
8.0 .614 .605 .600 .598 .597 .596 .596
20.0 .601 .599 .596 .596 .596 .596 .594

At the same time it must be observed that differences of sharpness in the edge of the orifice and some other circumstances affect the results, so that the values found by different careful experimenters are not a little discrepant. When exact measurement of flow has to be made by a sharp-edged orifice it is desirable that the coefficient for the particular orifice should be directly determined.

The following results were obtained by Dr H. T. Bovey in the laboratory of McGill University.