Orifice Flow Calculator
Torricelli with a tax: the jet's ideal velocity from the head above the hole, then the discharge coefficient's honest discount.
Orifice Flow Calculator
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What this result does not account for
- Incompressible, free jet; no submergence
- Constant head snapshot
In short: A 20 mm hole 2 m below the water line jets at 6.264184 m/s — v = √(2gH) = √39.24 — and with a sharp-edge Cd of 0.62 the tank discharges 1.220130 L/s. The coefficient is the whole lesson: the ideal jet would fill 1.967951 L/s, but the flow contracts through the hole and the vena contracta swallows 38% of the promise before the stream is a metre old.
Formula
Q = Cd·A·√(2gH) ··· v_ideal = √(2gH) ··· A = πd²/4
Torricelli prices the falling: a jet leaves a hole at the speed a drop would gain falling from the surface to the hole — the same square root as every gravity story. The discharge coefficient then discounts for the vena contracta, the jet's shrinkage as streamlines that approached from the sides fail to turn the corner: sharp edges pay the most, rounded entries the least.
Worked Example
- Pick the edge geometry's Cd.
- Enter the hole diameter.
- Enter the head above the centreline.
- Read ideal jet, taxed discharge, and the tax.
Defaults: Cd 0.62, 20 mm, 2 m → ideal 1.967951 L/s, taxed 1.220130 L/s. The head square-root check: 0.5 m head gives half the speed (√(1/4)) — the same root that prices falling objects.
Strengths & Limits Of This Model
Where this engine is strong
- Ideal jet and taxed discharge both shown
- The contraction tax named in watts of flow
Where it stops
- No drain-time integral
- No metering-plate pipe geometry
Practical Use Cases
Tank drainage
time to empty estimates
Weirs and sluices
small-aperture discharge
Teaching
Torricelli with receipts
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Orifice Flow Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why 0.62 for a sharp orifice?
Because the jet contracts: streamlines approaching from off-axis cannot turn a right-angle corner, so the jet pinches to about 0.62 of the hole's area just outside the plate — the vena contracta. Experiments across centuries land the sharp-edge coefficient between 0.60 and 0.65, and 0.62 is the convention's centre of mass. A rounded entry guides the streamlines around the corner and buys most of the loss back.
Why the square root in the velocity?
The same reason free fall has one: head H is potential energy per weight, and equating it to kinetic energy ½v²/g solves to v = √(2gH). A jet from 2 m below the surface leaves exactly as fast as a stone dropped 2 m lands. Torricelli found this a century before Bernoulli formalised it — the physics was always one root deep.
Does the discharge stay constant as the tank drains?
No — the head falls, and flow follows the square root of what remains: at a quarter of the head the jet runs at half speed. Draining is exponential-flavoured, fast at the top and miserly at the bottom, which is why the last fifth of any tank takes a fifth of the patience. This page prices one instant; the integral is calculus's share.
Why does a short tube reach Cd 0.80?
Because a stub of pipe bolted to the hole guides the streamlines through the corner instead of letting them shear: the vena contracta forms INSIDE the tube and re-expands before exit, recovering most of what a sharp plate loses. The classic ladder runs sharp plate 0.62, short tube 0.80, well-rounded entry toward 0.98. Geometry, not fluid, sets the tax rate.
What changes Cd?
Edge sharpness first (sharp 0.60–0.65, rounded toward 0.9+, short tube about 0.8), then Reynolds number at small scales, then whether the hole is in a thin plate or a thick wall. Standards like ISO 5167 tabulate Cd for metering plates with upstream piping included. The default 0.62 is the honest choice when the edge is sharp and the history is thin.
Can I use pressure instead of head?
Yes — H and Δp are the same information: v = √(2Δp/ρ) with Δp = ρgH. The orifice-plate flow meter in a pressurised line is exactly this page run on a measured Δp, with ISO tabulated Cd. The tank form uses head because gravity supplies it for free and a ruler can measure it.
Where does the jet land?
One root further: the jet leaves horizontally at √(2gH) and falls under gravity, so range = 2√(H·hₑ) where hₑ is the drop to the ground — geometry's bonus on top of Torricelli's root. Classic fountain problems are this page plus a projectile. The water only ever spends the energy the head bought.
How does this differ from full-bore pipe flow?
The pipe page trusts the geometry: full bore, no contraction, Cd = 1 — the ideal ceiling. The orifice admits the hole's shrinkage and pays Cd for it. Between them sits the truth of most systems: pipes lose to friction along the way, orifices lose to contraction at the door. Both charge the same square-root heart.