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I'm Helping The Pope With Public Relations


Actually if it's up to me...the Pope is on his own, because I spent most of the day worrying about Nerdy stuff like this (just glance at it, else your eyes will glaze over and you'll collapse into some sort of catatonic state:


Theory of Discharge in an Orifice


The Roman engineer Frontinus, who was in charge of the water supply under Augustus, used short pipes of graduated sizes to meter water delivered to different users.


This was purely empirical, since the effects of pressure, or "head," and orifice size were not known quantitatively until Torricelli, in 1643, showed that the velocity of efflux was given by Vi = √2gh. We still calculate the velocity from Bernoulli's principle, that h + p/ρg + V2/2, is a constant along a streamline in irrotational flow, which is equivalent to the conservation of energy.


We'll consider here the case of zero initial velocity, as at the surface of a liquid in a container with an orifice in the side. We assume that a streamline starts at the surface, a distance h above the orifice, and neglect the pressure on the surface of the liquid, since it would cancel out anyway. The streamline then leads somehow to the orifice, and out into the jet that issues from it. We choose the point at which the streamlines are parallel a short distance from the orifice, and find that the velocity there is Vi = √2gh, as given by Torricelli's theorem.


A jet surrounded only by air (or another fluid of small density) is called a free jet, and is acted upon by gravity. A jet surrounded by fluid is called a submerged jet. If the fluid is the same as that of the jet, then buoyancy eliminates the effect of gravity on it. A submerged jet is also subject to much greater friction at its boundary. We shall consider here only free jets of water, and neglect the viscosity of water, which is small, but finite.


A cross section of a circular orifice of diameter Do is shown. The thickness of the wall is assumed small compared to the diameter of the orifice. Because of the convergence of the streamlines approaching the orifice, the cross section of the jet decreases slightly until the pressure is equalized over the cross-section, and the velocity profile is nearly rectangular.

This point of minimum area is called the vena contracta. Beyond the vena contracta, friction with the fluid outside the jet (air) slows it down, and the cross section increases perforce. This divergence is usually quite small, and the jet is nearly cylindrical with a constant velocity. The jet is held together by surface tension, of course, which has a stronger effect the smaller the diameter of the jet.

The area A of the vena contracta is smaller than the area Ao of the orifice because the velocity is higher there (converging streamlines). For a sharp-edged, or "ideal" circular orifice, A/Ao = Cc = π/(π + 2) = 0.611. Cc is called the coefficient of contraction. For a sharp orifice, is usually estimated to be 0.62, a figure that can be used if the exact value is not known. For an orifice that resembles a short tube, Cc = 1, but then there are turbulence losses that affect the discharge.

The average velocity V is defined so that it gives the correct rate of discharge when it is assumed constant over the vena contracta, or Q = VA. Then, we can write V = CvVi, where Cv is the coefficient of velocity. The coefficient of velocity is usually quite high, between 0.95 and 0.99. Combining the results of this paragraph and the preceding one, the discharge Q = VA = CvViCcAo = CdAoVi. Cd, the coefficient of discharge, allows us to use the ideal velocity and the orifice area in calculating the discharge.

Aren't you happy that I'm worring about such things so that you don't have to?

Well...You're quite welcome...