Orifice Flow Calculator
Liquid flow through an orifice plate, nozzle or hole: Q = Cd·A·√(2ΔP/ρ) with the 1/√(1−β⁴) approach factor. Solve for flow, pressure drop or bore size.
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About this toolWhat is the Orifice Flow Calculator?
An orifice is a hole of known size that converts pressure into velocity, and the relationship is simple enough to run backwards: measure the pressure drop and you know the flow; choose a flow and pressure and you can size the hole. This calculator applies Q = Cd × E × A × √(2ΔP/ρ) for an orifice in a pipe or a free hole in a tank wall, and solves for flow, differential pressure or orifice bore. With the defaults, a 50 mm sharp-edged plate in a 100 mm water line at 25 kPa passes 8.75 L/s.
Discharge coefficient presets cover the common devices with their typical ranges: sharp-edged plates and holes around 0.61, short tubes about 0.82, bell-mouth inlets about 0.97, flow nozzles about 0.98 and classical Venturi tubes about 0.985. If you have a calibrated value, type it in.
In a pipe the beta ratio, β = d/D, matters because the fluid already moves toward the orifice. The velocity of approach factor E = 1/√(1 − β⁴) accounts for that, adding 3.3% at β = 0.5 and 15% at β = 0.7. For sharp orifice plates the tool also estimates the permanent pressure loss with the ISO 5167-2 expression, about 73% of the measured differential at β = 0.5.
- Solves for flow, differential pressure or orifice bore
- In-pipe (with β and approach factor) or free-discharge configurations
- Typical Cd presets for sharp orifices, short tubes, bell-mouths, nozzles and Venturi tubes
- Pressure input as Pa, kPa, bar, MPa, psi or metres and feet of liquid head
- ISO 5167-2 permanent pressure loss estimate and a beta-ratio range warning
How to use the Orifice Flow Calculator
- Choose what to solve for: flow rate, pressure drop or orifice bore.
- Choose whether the orifice sits in a pipe (enter the pipe bore) or is a free hole in a tank wall (no approach velocity).
- Pick a device type to fill in a typical discharge coefficient, or enter your own.
- Enter the known values. For a tank, give the head as metres or feet of liquid in the pressure unit picker.
- Read the flow, differential pressure, bore, beta ratio, jet and pipe velocities and, for plates, the permanent pressure loss.
Worked examples
Orifice plate in a water line
Input
50 mm sharp-edged plate in a 100 mm pipe, ΔP = 25 kPa, Cd = 0.61
Output
Q = 8.75 L/s (31.5 m³/h), β = 0.5, E = 1.0328, permanent loss about 18.3 kPa
Swapping to a flow nozzle (Cd 0.98) passes 14.1 L/s through the same bore at the same ΔP, with much less permanent loss.
Hole in a tank wall
Input
10 mm sharp hole, 2 m of water above it
Output
Q = 18.0 L/min
Torricelli alone would give 29.5 L/min; the 0.61 coefficient is mostly the jet contracting.
Sizing a restriction orifice
Input
20 m³/h of water, 50 kPa drop, 80 mm line, Cd 0.61
Output
Orifice bore d = 33.8 mm (β = 0.42)
Check the result against cavitation if the downstream pressure is low; a multi-stage orifice may be needed.
Background and insight
The ideal equation comes from Bernoulli between the upstream pipe and the vena contracta, the narrowest point of the jet just past the plate, plus continuity. The real jet contracts to about 61% of the hole area for a sharp edge and loses a little energy, so the ideal flow is multiplied by a discharge coefficient Cd. For a free orifice under a head h the pressure term becomes ρgh and the equation reduces to Torricelli's law times Cd.
ISO 5167 is the international standard for differential-pressure meters. Part 2 covers orifice plates and gives the Reader-Harris/Gallagher equation for C as a function of β, Reynolds number and tapping arrangement, valid for β from 0.1 to 0.75 with specified straight pipe lengths. Part 3 covers nozzles and Venturi nozzles and Part 4 Venturi tubes. Those details matter for metering accurate to 1%; a fixed Cd near 0.61 is fine for estimates and restriction orifices.
For gases and steam ISO 5167 adds an expansibility factor ε, which drops below 1 as the pressure ratio across the plate falls. This calculator is for liquids, where ε = 1.
Common use cases
Flow measurement estimates
Turn a differential pressure reading across an orifice plate into a flow rate.
Restriction orifices
Size a plate to limit flow or drop a set pressure in a bypass, purge or minimum-flow line.
Leaks and drain holes
Estimate how fast liquid escapes through a hole under a known head or pressure.
Sprinklers and nozzles
Relate nozzle size, pressure and flow when a K-factor or Cd is known.
Tips and limitations
- Flow goes with the square root of ΔP, so doubling the differential raises flow only 41%. Meters lose accuracy at the low end of their range for this reason.
- Keep β between about 0.2 and 0.7 for orifice plates. Outside the ISO 5167-2 range of 0.1 to 0.75 the typical Cd is unreliable.
- The measured ΔP is not the permanent loss. Much of it recovers downstream, more for nozzles and most for Venturi tubes.
- Sharp edges matter. A worn or rounded upstream edge raises Cd and makes the meter read low.
- If downstream pressure is near the liquid vapour pressure the jet can cavitate or flash and the simple equation no longer holds.
Frequently asked questions
What is the orifice flow equation?
Q = Cd × A × √(2ΔP/ρ) / √(1 − β⁴), where Cd is the discharge coefficient, A the orifice area, ΔP the differential pressure, ρ the liquid density and β the orifice-to-pipe diameter ratio. For a hole in a large tank β is zero.
What is the discharge coefficient of a sharp-edged orifice?
About 0.6 to 0.62 at high Reynolds number, with 0.61 a common working value. ISO 5167-2 gives the Reader-Harris/Gallagher equation for precise values that depend on β, Re and tapping position.
What is the beta ratio of an orifice plate?
The orifice bore divided by the pipe bore, β = d/D. It sets the velocity of approach factor 1/√(1 − β⁴) and affects Cd and permanent loss. Typical plates use β between about 0.2 and 0.7.
How do you calculate flow through a hole in a tank?
Q = Cd × A × √(2gh), with h the liquid height above the hole. A 10 mm sharp-edged hole under 2 m of water passes 0.61 × 78.5 mm² × 6.26 m/s = 0.30 L/s, or about 18 L/min.
How much pressure is lost permanently across an orifice plate?
Less than the measured differential, because some pressure recovers downstream. ISO 5167-2 gives Δϖ/ΔP = (√(1 − β⁴(1 − C²)) − Cβ²)/(√(1 − β⁴(1 − C²)) + Cβ²), about 73% at β = 0.5 and 51% at β = 0.7.
Does this orifice calculator work for gas or steam?
Only roughly, for very small pressure ratios. Gas flow needs the expansibility factor from ISO 5167 and the density at the upstream conditions. Use a compressible-flow method for gas and steam metering.
Is there an in-depth guide to the Orifice Flow Calculator?
Yes. “Orifice Flow Equation: Discharge Coefficient, Beta Ratio and ISO 5167” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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