W = G1Q1 = c[omega] [root][2g[gamma]p1G1[psi]/([gamma] - 1)]
= 2.043c[omega]p1 [root]([psi]/[tau]1). (5)
Weisbach found the following values of the coefficient of discharge
c:--
Conoidal mouthpieces of the form of the \
contracted vein with effective > c =
pressures 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
[omega]. 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 c_c[omega] of
the contracted section of the jet, and may be greater than the area
[omega] 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/([gamma] + 1)}^([gamma] - 1/[gamma])
= 0.527 for air
= 0.58 for dry steam.
For a further decrease of external pressure the discharge
diminishes,--a result no doubt improbable. The new view of Weisbach's
formula is that from the point where the maximum is reached, or not
greatly differing from it, the pressure at the contracted section
ceases to diminish.
A. F. Fliegner showed (_Civilingenieur_ xx., 1874) that for air
flowing from well-rounded mouthpieces there is no discontinuity of the
law of flow, as Napier's hypothesis implies, but the curve of flow
bends so sharply that Napier's rule may be taken to be a good
approximation to the true law. The limiting value of the ratio p2/p1,
for which Weisbach's formula, as originally understood, ceases to
apply, is for air 0.5767; and this is the number to be substituted for
p2/p1 in the formulae when p2/p1 falls below that value. For later
researches on the flow of air, reference may be made to G. A. Zeuner's
paper (_Civilingenieur_, 1871), and Fliegner's papers (_ibid._, 1877,
1878).