How this instrument works
The coefficient of discharge, Cd, is the ratio of the flow a device actually delivers to the flow an ideal, frictionless calculation says it should deliver: Cd = Qactual ⁄ Qtheoretical. The theoretical figure comes from applying Bernoulli's equation to the device's geometric opening and the pressure or head driving fluid through it, as though the fluid were inviscid and the streamlines followed the hole's edges exactly. Real fluid never cooperates that fully, so Cd is always a fraction — never a unit conversion, never a fudge factor, but a direct, measured statement of how good a particular opening is at delivering the flow the mathematics promised.
Two separate losses hide inside that single number. As fluid rounds a sharp edge it cannot turn the corner instantly, so the jet keeps narrowing past the opening until it reaches a minimum section called the vena contracta — a pure geometry loss, captured in a contraction coefficient Cc. Viscosity then shaves a little more off the velocity itself, captured in a velocity coefficient Cv. Multiply the two and Cd = Cc·Cv: a thin sharp-edged orifice loses heavily to contraction and lands near 0.61, while a smoothly rounded nozzle barely contracts at all and reaches 0.97 or higher.
Cd is a property of geometry and Reynolds number, not of the fluid or the pressure alone, and treating it as a fixed constant is a common error in sizing work. Orifice-plate flow meters used for custody-transfer gas metering under the ISO 5167 standard do not assume a textbook 0.61; they compute Cd from the Reader-Harris/Gallagher correlation, which adjusts for bore-to-pipe ratio, tapping position, and Reynolds number, because a plate sized on the wrong assumed Cd under- or over-bills a pipeline by a real, costly margin.
- Enter the Measured actual flow rate in L/s — read from a flow meter, or timed by catching a known volume at the outlet.
- Enter the Theoretical (ideal) flow rate in L/s — the frictionless value Bernoulli's equation predicts for the same geometry and driving head.
- Read the Coefficient of discharge: the dimensionless fraction showing how much of that ideal flow the real device delivers.
- Compare the figure to standard values for the geometry — about 0.61 for a sharp orifice, 0.80 for a short tube, 0.97 or more for a rounded nozzle.
Worked example — an 8 L/s short-tube mouthpiece
A tank drains through a short cylindrical tube fitted flush to its wall — a mouthpiece a little longer than its own diameter — rather than through a thin sharp-edged hole. Applying Bernoulli's equation to the tube's bore and the head above it predicts a theoretical flow of 10 L/s. Catching and timing the actual discharge at the outlet measures 8 L/s.
Cd = actualFlow ⁄ theoreticalFlow = 8 ⁄ 10 = 0.8. That sits well above the roughly 0.61 typical of a sharp-edged plate of the same bore, because the jet's vena contracta forms just inside the tube mouth and then reattaches to the tube wall, refilling to nearly the full bore before it exits — area, and flow, that a thin plate would simply have thrown away. Fluid mechanics texts list 0.80 as the standard figure for exactly this short-tube geometry.
Questions
Why is the coefficient of discharge always less than one?
Because a real jet loses area to the vena contracta as it rounds the opening's edge, and loses a little speed to viscous friction, while the theoretical flow assumes neither loss. Cd = Cc·Cv bundles both effects into one measured fraction; a value of 1 would mean a perfectly frictionless jet filling the geometric opening exactly, which no real device achieves.
What is a typical Cd for a sharp-edged orifice versus a rounded nozzle?
A thin sharp-edged orifice plate typically runs about 0.61, since the flow cannot turn the sharp corner and contracts hard just past it. A short cylindrical tube reaches around 0.80, and a smoothly rounded flow nozzle or venturi throat, which barely contracts the jet at all, commonly reaches 0.95 to 0.98. Geometry, not the fluid, decides the number.
Is the coefficient of discharge the same thing as the flow coefficient Cv used for valves?
No, and confusing them is a common sizing mistake. Coefficient of discharge is a dimensionless ratio of two flow rates, always somewhere between roughly 0.5 and 1. A valve's flow coefficient Cv is a dimensioned capacity rating — gallons per minute of water at a 1 psi drop — used to select valve size, not to compare a real flow against an ideal one.
Does the coefficient of discharge change with flow rate?
Yes, through the Reynolds number, though the dependence flattens out at higher flow. At low Reynolds numbers viscous effects dominate and Cd drifts noticeably; above roughly Re = 10⁴ for most orifice and nozzle geometries it settles to a nearly constant value, which is why published Cd tables quote a number alongside a minimum Reynolds number for it to apply.
Where does the theoretical flow rate in this calculator come from?
From Bernoulli's equation applied to the device's geometric opening with no losses assumed: for a tank orifice, Qtheoretical = A times the square root of 2gh, using the hole's full area A and head h above it; for a pressure-driven nozzle, the equivalent Bernoulli expression using the pressure drop. Manufacturers sometimes list this as the device's nominal or rated capacity.
Can the coefficient of discharge exceed 1?
Not for a genuine discharge device — a real opening cannot deliver more flow than an ideal, lossless one predicts. A result above 1 signals a measurement error, a theoretical flow calculated for the wrong geometry, or a units mismatch between the two flow-rate fields, not a real device outperforming physics.