G1 = p1/53.2τ1 ℔ per cubic foot.
Hence the weight of air discharged is
W = G1Q1 = cω √ [2gγp1G1ψ / (γ − 1)]
= 2.043cωp1 √ (ψ/τ1).
(5)
Weisbach found the following values of the coefficient of discharge c:—
| Conoidal mouthpieces of the form of the | |||
| contracted vein with effective pressures | c = | ||
| of .23 to 1.1 atmosphere | 0.97 | to | 0.99 |
| Circular sharp-edged orifices | 0.563 | ” | 0.788 |
| Short cylindrical mouthpieces | 0.81 | ” | 0.84 |
| The same rounded at the inner end | 0.92 | ” | 0.93 |
| Conical converging mouthpieces | 0.90 | ” | 0.99 |
§ 64. Limit to the Application of the above Formulae.—In the formulae above it is assumed that the fluid issuing from the orifice expands from the pressure p1 to the pressure p2, while passing from the vessel to the section of the jet considered in estimating the area ω. Hence p2 is strictly the pressure in the jet at the plane of the external orifice in the case of mouthpieces, or at the plane of the contracted section in the case of simple orifices. Till recently it was tacitly assumed that this pressure p2 was identical with the general pressure external to the orifice. R. D. Napier first discovered that, when the ratio p2/p1 exceeded a value which does not greatly differ from 0.5, this was no longer true. In that case the expansion of the fluid down to the external pressure is not completed at the time it reaches the plane of the contracted section, and the pressure there is greater than the general external pressure; or, what amounts to the same thing, the section of the jet where the expansion is completed is a section which is greater than the area ccω of the contracted section of the jet, and may be greater than the area ω of the orifice. Napier made experiments with steam which showed that, so long as p2/p1 > 0.5, the formulae above were trustworthy, when p2 was taken to be the general external pressure, but that, if p2/p1 < 0.5, then the pressure at the contracted section was independent of the external pressure and equal to 0.5p1. Hence in such cases the constant value 0.5 should be substituted in the formulae for the ratio of the internal and external pressures p2/p1.
It is easily deduced from Weisbach’s theory that, if the pressure external to an orifice is gradually diminished, the weight of air discharged per second increases to a maximum for a value of the ratio
| p2/p1 = {2/(γ + 1)}γ−1/γ = 0.527 for air = 0.58 for dry steam. |