What this calculator does
An orifice calculator finds the flow rate of liquid escaping through a hole of known size under a given head of fluid above it, using the standard orifice equation Q = Cd × A × √(2gh). This shows up in tank drainage, weir and dam outlet design, irrigation emitters, and any pipe or vessel with a defined discharge opening.
The result from √(2gh) alone is the theoretical velocity a frictionless jet would reach falling from height h under gravity, the same expression as free-fall velocity. Real orifices lose some of that ideal flow to friction and jet contraction as fluid squeezes through the opening, which is exactly what the discharge coefficient, Cd, is there to correct for.
The formula
The orifice area is calculated from the diameter as a circle, π × (d/2)². The ideal velocity of the jet is √(2 × g × h), where h is the head of fluid above the orifice and g is standard gravity. Multiplying area by ideal velocity by the discharge coefficient gives the actual volumetric flow rate, Q = Cd × A × √(2gh).
| Term | Meaning |
|---|---|
| Q | Volumetric flow rate through the orifice, in cubic metres per second. |
| Cd | Discharge coefficient: the fraction of the theoretical flow actually achieved, accounting for friction and jet contraction. Around 0.6 to 0.62 is typical for a sharp-edged circular orifice, closer to 0.98 for a well-rounded nozzle. |
| A | Cross-sectional area of the orifice opening. |
| h | Head: the height of the fluid surface above the orifice, which drives the flow. |
| g | Standard gravity, 9.80665 m/s². |
The inputs explained
| Field | What to enter |
|---|---|
| Orifice diameter (mm) | The diameter of the orifice opening. |
| Discharge coefficient | The discharge coefficient for this orifice shape and edge condition. 0.6 is a common starting point for a sharp-edged orifice, but it is an editable estimate, not a fixed constant, since it depends on the exact geometry. |
| Head (height of fluid above orifice) (m) | The height of the fluid surface above the centre of the orifice, sometimes called the head or driving pressure head. |
When to use it
Sizing a tank drain hole
Given a target drain time or flow rate, adjusting the orifice diameter here shows how much larger a hole needs to be to move more water at a given head.
Estimating irrigation emitter output
Drip and micro-irrigation emitters are small orifices operating under line pressure expressed as an equivalent head, so this equation estimates the flow each emitter delivers.
Checking a discharge coefficient assumption
Running the same diameter and head at a couple of different Cd values shows how sensitive the estimated flow is to that coefficient, which matters most when the real edge condition of the orifice is uncertain.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does head affect flow through a 20 mm sharp-edged orifice?
The same orifice under a range of heads.
| Head | Flow rate (litres per second) | Theoretical (ideal) velocity |
|---|---|---|
| 0.5 m | 0.590 L/s | 3.132 m/s |
| 1 m | 0.835 L/s | 4.429 m/s |
| 2 m | 1.181 L/s | 6.263 m/s |
| 4 m | 1.670 L/s | 8.857 m/s |
How does orifice diameter affect flow at a 1 m head?
A range of orifice diameters at a fixed 1 m head.
| Orifice diameter | Flow rate (litres per second) | Orifice area |
|---|---|---|
| 10 mm | 0.209 L/s | 0.00007854 m² |
| 20 mm | 0.835 L/s | 0.000314 m² |
| 35 mm | 2.557 L/s | 0.000962 m² |
| 50 mm | 5.217 L/s | 0.001963 m² |
Questions
What discharge coefficient should I use?
For a sharp-edged circular orifice in a thin plate, around 0.6 to 0.62 is the commonly cited figure. A rounded or bell-mouthed entry reduces the losses and pushes Cd closer to 0.95 to 0.98. This value depends on the exact edge geometry, so treat it as an estimate to refine against measured flow where accuracy matters.
What counts as the head, h?
It is the vertical height of the fluid surface above the orifice opening, not the diameter of the tank or pipe. For a pressurised line rather than an open tank, an equivalent head can be found from pressure ÷ (fluid density × g).
Does this work for gases as well as liquids?
This equation is for incompressible flow, which describes liquids well. Gas flow through an orifice depends on compressibility and pressure ratio and needs a different equation once the pressure drop is significant.
Why is the actual flow always less than Cd × A × root(2gh) would suggest is the maximum?
It is not less than that, that expression already is the actual flow: the discharge coefficient Cd is what converts the larger theoretical, frictionless flow rate down to a realistic figure, accounting for the jet contracting as it leaves a sharp edge and for friction losses.
For force calculations involving pressurised gas rather than orifice flow, see the pneumatic cylinder force calculator.