Orifice Flow Calculator

Background.

Punch a hole in the side of a water tank and the jet obeys a rule Evangelista Torricelli worked out in the 1640s: the water leaves at the speed it would reach falling freely from the surface down to the hole, √(2gh). Two and a half meters of water above an opening drives a jet at about 7 m/s. Multiply that speed by the area of the hole and you have the flow an ideal, frictionless liquid would deliver.

Real orifices deliver less, and the discharge coefficient C_d accounts for the shortfall. Water converging on a sharp-edged hole cannot turn the corner instantly, so the jet keeps narrowing for a short distance beyond the opening, to a waist called the vena contracta, and a little speed is lost to friction at the edge. For a sharp-edged orifice the combined effect cuts the flow to roughly 62% of the ideal, the default here. A smoothly rounded, bell-mouthed entrance lets the stream follow the wall and brings the coefficient close to 1. Choosing it well matters more than anything else on the page, because the flow is directly proportional to it.

The head difference is the driving height: from the free water surface to the center of the orifice when the jet discharges into air, or between the two water surfaces when the orifice is submerged. Flow grows only with its square root, so quadrupling the head merely doubles the discharge.

This is the steady-flow model. It assumes the head holds constant, which a draining tank does not, and that the fluid is an incompressible liquid with negligible viscous effects. Gases at large pressure ratios, cavitating jets and very small or slow orifices fall outside it. For an orifice plate inside a pipe, ISO 5167-2 adds an approach-velocity factor that this simple equation leaves out, so the coefficient you enter must already allow for it.

What is orifice flow calculator?

Orifice flow is the volume of liquid passing each second through an opening in a wall, plate or tank under a steady head, given here in cubic meters per second. The theoretical figure is the opening's area times Torricelli's velocity √(2gΔh). The actual flow is that figure scaled by a discharge coefficient, found by experiment for each orifice shape, which captures how much the jet contracts and slows as it leaves.

How to use this calculator.

1. 01Enter the discharge coefficient for your orifice. 0.62 suits a sharp-edged circular hole in a thin plate; for anything else take the value from the manufacturer, a calibration or a hydraulics handbook.
2. 02Enter the orifice area in square meters, not the diameter. For a circular hole A = πd²/4, so a 5 cm hole is 0.0019635 m².
3. 03Leave gravitational acceleration at standard gravity, 9.80665 m/s², unless you are working to a site value; the variation across Earth's surface changes the flow by less than 0.3%.
4. 04Enter the head difference in meters: water surface to orifice center for free discharge, or upstream minus downstream level for a submerged orifice.
5. 05Read the flow in m³/s. Multiply by 1,000 for liters per second or by 3,600 for cubic meters per hour.

The formula.

Q = C_d × A × √(2gΔh). Q is the volumetric flow rate in m³/s, C_d the dimensionless discharge coefficient, A the orifice area in m², g the gravitational acceleration in m/s² and Δh the head difference in meters. The square-root term is Torricelli's ideal jet velocity, which follows from Bernoulli's equation when the potential energy of a fall through Δh turns entirely into kinetic energy. Multiplying by A gives the ideal flow, and C_d corrects it for the contraction of the jet and the friction at the edge. Flow is directly proportional to C_d and to A, so a 10% error in either is a 10% error in Q. It grows only with the square root of g and Δh, so doubling the head adds about 41%. Every input must be positive, and the result describes one instant at a constant head: a falling level, gas compressibility, cavitation and viscous effects at small openings are not modeled.

A worked example.

Frequently asked questions.

References& sources.

1. [1]U.S. Bureau of Reclamation, Water Measurement Manual, 3rd ed.
2. [2]ISO 5167-2:2022, Measurement of fluid flow by pressure differential devices, Part 2 (PAYWALL/BIBLIOGRAPHIC).
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.