Orifice Flow Calculator
Find discharge through a sharp-edged orifice from diameter, head, and discharge coefficient.
What it does
An orifice is a known opening in a plate, tank wall, or pipe end. Flow through it depends on the size of the opening and the height of water above it. The relationship is one of the oldest in hydraulics: discharge equals a coefficient times area times the square root of twice gravity times head. It is used for tank outlets, detention basin controls, flow measurement plates, and simple pond drains.
The coefficient Cd accounts for the contraction of the jet and the friction in the opening. A sharp-edged round hole with a free discharge typically has a Cd near 0.61. A rounded or bell-mouthed opening can be higher, while a thick plate or a submerged outlet can change the value. Using the wrong coefficient is one of the easiest ways to be off by 20 percent or more.
Head must be measured from the water surface to the center of the orifice, not to its top edge. The equation also assumes the jet discharges freely into air and that the approach velocity in the tank is small. If the outlet is submerged, the driving head is the difference between the two water surfaces, and the coefficient may change. A long tube behaves more like a pipe than a thin-plate orifice.
The square root relationship means flow rises quickly at first and then more slowly as head increases. Going from 1 ft to 4 ft of head doubles the flow, not quadruples it. Going from a 2 in hole to a 4 in hole quadruples the area and roughly quadruples the flow at the same head. Both effects are captured by the equation.
This page handles a round orifice with a free discharge. It returns flow in gpm and cfs for imperial units, or L/s and m3/h for metric, plus jet velocity and open area. Use it for a first estimate. For a real outlet structure, confirm the coefficient, submergence, and downstream conditions with the design reference or a hydraulic engineer.
How to use
- Choose imperial or metric units.
- Enter the open orifice diameter, not the nominal pipe size.
- Enter the head from the water surface to the center of the orifice.
- Enter the discharge coefficient. Start with 0.61 for a sharp-edged round hole and a free discharge.
- Read the flow, jet velocity, and area. Check that the flow units match the rest of your work.
- If the outlet is submerged or the plate is thick, adjust Cd or use a pipe flow method instead.
Formula
- Q
- Volumetric discharge, in cfs, gpm, m3/s, or L/s.
- Cd
- Discharge coefficient, dimensionless.
- A
- Open area of the orifice, in ft2 or m2.
- d
- Orifice diameter, in ft or m.
- h
- Head from the water surface to the orifice center, in ft or m.
- g
- Gravity, 32.174 ft/s2 or 9.80665 m/s2.
The equation assumes a thin plate, a free jet, and a steady head. The coefficient Cd bundles the vena contracta and the losses. A sharp-edged orifice is often near 0.61, a well-rounded opening can approach 0.98, and a submerged or thick-walled opening needs its own value. If the orifice discharges into a pipe, the pipe friction and the downstream head become part of the problem, and a pipe flow calculation is the better tool.
Worked example
Example: 2 in orifice under 5 ft of head
A tank has a sharp-edged 2 in round hole with its center 5 ft below the water surface. Use Cd 0.61.
- Convert diameter to feet: 2 / 12 = 0.1667 ft.
- Area = pi x 0.1667 squared / 4 = 0.02182 ft2.
- Jet velocity = sqrt(2 x 32.174 x 5) = 17.94 ft/s.
- Discharge = 0.61 x 0.02182 x 17.94 = 0.2387 cfs.
- Convert to gpm: 0.2387 x 448.831 = 107.1 gpm.
- Metric mirror: 50.8 mm gives 0.002027 m2, 5.47 m/s, and 6.76 L/s.
- Check: 6.76 L/s x 15.8503 = 107.1 gpm.
Reference table
| Diameter | Area | Discharge |
|---|---|---|
| 1 in (25.4 mm) | 0.00545 ft2 (5.07 cm2) | 26.8 gpm (1.69 L/s) |
| 2 in (50.8 mm) | 0.02182 ft2 (20.27 cm2) | 107.1 gpm (6.76 L/s) |
| 3 in (76.2 mm) | 0.04909 ft2 (45.60 cm2) | 241.1 gpm (15.21 L/s) |
| 4 in (101.6 mm) | 0.08727 ft2 (81.07 cm2) | 428.6 gpm (27.04 L/s) |
Limits
- Assumes a thin plate, a free discharge, and a steady head.
- Cd varies with edge shape, Reynolds number, and submergence.
- Ignores approach velocity in the tank or channel.
- Not suitable for long tubes, pipes, or partially full outlets.
Common mistakes
- Using the pipe diameter instead of the orifice diameter.
- Mixing inches with g = 32.174 ft/s2 without converting to feet.
- Measuring head to the top of the orifice.
- Using Cd 1.0 and ignoring jet contraction.
- Applying a free-flow coefficient to a submerged outlet.
Try it yourself
Quick checks. Answers sit tucked under each question.
A 1 in sharp-edged orifice sits under 5 ft of head with Cd 0.61. What is the flow?
Area = 0.005454 ft2, velocity = 17.94 ft/s, Q = 0.0597 cfs = 26.8 gpm.
A 4 in sharp-edged orifice sits under 5 ft of head with Cd 0.61. What is the flow?
Area = 0.08727 ft2, velocity = 17.94 ft/s, Q = 0.9548 cfs = 428.6 gpm.
A 50 mm orifice sits under 1 m of head with Cd 0.61. What is the flow?
Area = 0.001963 m2, velocity = 4.43 m/s, Q = 0.005304 m3/s = 5.3 L/s.
Questions
What is an orifice?
It is a known opening in a plate or wall. The flow through it is set by the opening size and the head above it.
What Cd should I use?
Start with 0.61 for a sharp-edged round hole with a free discharge. Use a tested or standard value for other shapes.
Does the pipe size matter?
The orifice diameter controls the area. The pipe size matters only if it changes the approach conditions or the downstream head.
What is the difference between free and submerged flow?
Free flow discharges into air. Submerged flow discharges into water, and the driving head is the difference between the two water levels.
Can I use this for a bucket with a hole?
Yes, as a first estimate. The head drops as the bucket drains, so the flow falls over time. Use the starting head for the initial rate.
Why does doubling the diameter quadruple the flow?
Area grows with the square of diameter. At the same head, velocity is unchanged, so flow grows by the same factor as area.
Sources
- US Bureau of Reclamation: Water Measurement Manual, orifice relationships
- USDA ARS: Orifice plates for furrow flow measurement
- Iowa DNR: Stormwater manual, detention basin outlet structures