The orifice flow calculator above works out how much fluid passes through a hole. That sounds trivial and it is not, because the answer straight out of the energy equation is always too large. Real jets contract after they leave the bore, real edges shed energy, and a real orifice in a pipe sits in a stream that is already moving. This tool applies all three corrections and shows both the honest number and the idealised one.
Arb Digital publishes free engineering and physics calculators that state their assumptions instead of hiding them. This page assumes an incompressible fluid, steady flow and a discharge coefficient you supply, and it says so on the result whenever those assumptions are being stretched.
What This Orifice Flow Calculator Does
You give it a driving head or a pressure difference, the opening diameter, a discharge coefficient and the fluid density. It returns the volumetric flow rate as the headline number, in litres per second, with four supporting figures.
The mean velocity at the bore is the flow rate divided by the geometric area of the hole. It is deliberately not the velocity at the jet's narrowest point, which is higher. Quoting the bore-referenced velocity is the convention that makes continuity work when you multiply it back by the bore area.
The mass flow rate matters whenever you are filling by weight or checking a heat balance, and the hourly figure is the unit most pump curves use. The ideal frictionless velocity shows what the energy equation alone would give, so comparing it with the mean velocity shows how much the discharge coefficient is doing.
The optional upstream pipe bore switches on the velocity of approach correction. In a large tank the fluid is essentially still before it reaches the hole, so that correction is one and can be ignored. In a pipe the fluid already carries kinetic energy toward the plate, and ignoring it underestimates the flow.
How to Use It
- Choose head or pressure first. A tank draining to atmosphere is a head problem; an orifice plate between two gauges is a pressure problem. The tool converts internally, so either route reaches the same physics.
- Measure the head to the centre of the opening. Not to its top edge and not to the bottom of the tank. For a large hole in a shallow tank the head varies noticeably across the opening and the centreline value is the standard approximation.
- Enter the bore, not the jet. If you have measured the visible stream you have measured the vena contracta, which is smaller. Use the drilled or machined size.
- Pick a discharge coefficient that matches the edge. This is the single biggest source of error on the page. A sharp-edged plate and a rounded nozzle of identical bore differ by more than half again in flow.
- Add the pipe bore only when there is one. Leaving it at zero tells the tool to treat the upstream fluid as stationary, which is right for a reservoir and wrong for a pipeline.
The Formula: How Orifice Discharge Is Calculated
Start from the energy equation along a streamline from the still surface to the jet. It reduces to a single ideal velocity, videal = √(2ΔP ÷ ρ). When the driving force is a column of liquid rather than a gauge pressure, ΔP = ρgh, and the density cancels to leave videal = √(2gh) — Torricelli's result, which is the same expression as the impact speed of an object dropped through the same height. Physics LibreTexts section 52.4, Torricelli's Theorem, derives it and points out that the falling-object parallel is exact, not a coincidence.
The full discharge equation is Q = Cd A √(2ΔP ÷ ρ) ÷ √(1 − β4), where A is the bore area, β is the ratio of orifice diameter to pipe diameter, and the last factor is the velocity of approach correction. When β is zero, which is the large-tank case, that factor is one and drops out. The underlying energy balance is the one set out in OpenStax University Physics Volume 1, section 14.6 on Bernoulli's equation.
Work the defaults through. A 25 mm hole under 3 m of water at 998 kg/m³, sharp edged, in a large tank. Bore area = π ÷ 4 × 0.0252 = 4.909 × 10−4 m². Ideal velocity = √(2 × 9.80665 × 3) = 7.671 m/s. Multiply by the area and by Cd = 0.61: Q = 0.61 × 4.909 × 10−4 × 7.671 = 2.297 × 10−3 m³/s, or 2.30 litres per second. Mass flow = 2.297 × 10−3 × 998 = 2.29 kg/s, and the hourly figure is 8.27 m³/h. The mean velocity at the bore is 0.61 × 7.671 = 4.68 m/s, well below the ideal 7.67 m/s.
The Vena Contracta Is Where the Coefficient Comes From
Fluid does not turn corners instantly. As it converges on a sharp-edged hole it arrives from every direction, and the inward radial momentum keeps squeezing the stream after it has passed the plate. The jet reaches its narrowest cross-section about half a bore diameter downstream. That waist is the vena contracta.
For a sharp-edged circular hole with fully developed contraction the waist area is close to 0.62 of the bore area. Multiply that by a velocity coefficient near 0.98, which accounts for the modest energy loss on the way through, and you land at roughly 0.61 — the number that appears again and again in orifice work and the default on this page. It is not an empirical fudge; it is the product of two separately measurable factors.
Round the entrance and the contraction largely disappears, because the fluid is guided rather than forced to turn. A well-shaped bellmouth nozzle reaches 0.97 to 0.99, so it passes over half again as much fluid as a plate of the same bore. Changing the edge profile is usually cheaper than changing anything else.
Why the Flow Is Not Proportional to Pressure
Doubling the head does not double the flow. Flow scales with the square root of driving pressure, so doubling the head multiplies discharge by about 1.41 and quadrupling it doubles the discharge. This trips people up when sizing a drain or predicting how long a tank takes to empty. Diameter is far stronger: it is a square law, so a 50 mm hole passes four times what a 25 mm hole passes at the same head.
This is a different regime from a long narrow tube, where flow is linear in pressure and scales with the fourth power of the radius. Our Poiseuille's law calculator handles that laminar viscous case. The distinction is physical: an orifice loses energy to turbulence and inertia, a capillary loses it to viscous shear along the wall, and the two produce completely different exponents.
Where This Sits Beside Our Other Flow Tools
This page models a restriction where the loss is concentrated at one point. That is a different problem from friction along a length of pipe, which the pipe flow calculator handles with a Darcy friction factor over a full pressurised run. Use that page for the pipeline and this one for the plate in it.
For plain continuity between area, velocity and flow with no loss model, the flow rate calculator does that arithmetic. For the general streamline energy balance between two points, see the Bernoulli equation calculator. For a ditch or part-full culvert with a free surface, use the open channel flow calculator.
Two converters help when your inputs arrive in the wrong units: the pressure converter for psi, bar and inches of water column, and the flow rate converter for gallons per minute against litres per second. If you need to check whether the approach flow is laminar or turbulent, the Reynolds number calculator takes it from there.
Gases, Cavitation and the Limits of This Model
Everything here assumes the fluid density does not change as it passes through. For liquids that is safe. For gases it holds only while the pressure ratio stays mild. Beyond that the gas expands as it accelerates, an expansibility factor is needed, and once the downstream absolute pressure falls below about 53 per cent of the upstream value the flow chokes at the local speed of sound. A calculator built on incompressible assumptions keeps returning a rising number past that point, and it is wrong.
Liquids have their own ceiling. If the pressure at the vena contracta falls to the vapour pressure, bubbles form and collapse violently downstream. That is cavitation, and it caps the flow much as choking caps a gas, while eroding metal.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Leaving the discharge coefficient at 1 — that is the frictionless ideal and it overstates a sharp-edged hole by about 64 per cent. There is no orifice anywhere with a coefficient of one.
- Measuring the jet instead of the bore — the visible stream is the vena contracta and is narrower than the hole. Using it as the area double-counts the contraction that is already inside the coefficient.
- Assuming flow is proportional to pressure — it follows the square root. Doubling the head raises the discharge by about 41 per cent, not 100 per cent.
- Ignoring the approach velocity in a pipe — once the orifice bore exceeds about a third of the pipe bore the correction is no longer negligible and the flow is higher than the simple formula suggests.
- Using this for a gas at a large pressure ratio — compressibility and choking take over. The incompressible answer keeps rising when the real flow has already stopped rising.
Related Free Tools From Arb Digital
For friction along a pipe run rather than loss at a restriction, use the pipe flow calculator, and for the underlying streamline energy balance the Bernoulli equation calculator. Simple continuity between area, velocity and flow is handled by the flow rate calculator, while a free-surface channel needs the open channel flow calculator. Narrow-tube viscous flow belongs to the Poiseuille's law calculator. Convert awkward units with the pressure converter and the flow rate converter, and classify the flow regime with the Reynolds number calculator. Everything Arb Digital publishes is listed at the free online tools hub.
Frequently Asked Questions
Around 0.61 for a sharp-edged hole in a thin plate, which is the most common case. About 0.80 for a short square-edged tube where the jet reattaches inside the bore, and 0.97 to 0.99 for a smoothly rounded nozzle. If the component has a manufacturer's figure, use that instead of any generic value.
Two effects combine. The jet contracts to about 62 per cent of the bore area a short way downstream, and a small amount of energy is lost as the fluid turns into the hole. Multiplying the two gives roughly 0.61, so a sharp-edged orifice passes only about 61 per cent of the frictionless prediction.
No. Flow follows the square root of the driving pressure, so doubling it multiplies the discharge by about 1.41. You need four times the pressure to double the flow. Diameter is far more powerful: doubling the bore quadruples the discharge at the same head.
It is the narrowest point of the jet, roughly half a bore diameter downstream of the opening, where the inward momentum of the converging fluid has finished squeezing the stream. Its area is about 0.62 of the bore for a sharp-edged circular hole, and that ratio is the contraction part of the discharge coefficient.
Only at small pressure ratios, within a few per cent of the upstream absolute pressure, where the density barely changes. Beyond that a gas needs an expansibility factor, and once the downstream absolute pressure falls below about 53 per cent of the upstream value the flow chokes at sonic velocity and no longer responds to further downstream reduction.
Whenever the orifice sits in a pipe rather than a large tank. The fluid arrives already moving, so it carries kinetic energy into the restriction. The correction factor is one divided by the square root of one minus beta to the fourth power, and it becomes significant once the bore ratio passes about a third.
This page models a single concentrated restriction where the loss happens at one point and the driving term is the pressure difference across it. The pipe flow calculator models distributed wall friction along a length of full pipe using a Darcy friction factor. A real system usually needs both.
This tool is provided for educational and preliminary engineering use. It assumes steady incompressible flow with a discharge coefficient you supply, and it does not replace a metering standard, a manufacturer's data sheet or review by a qualified engineer for any installation where flow accuracy or safety matters.
