Calculate the mass and volume flow through a standard orifice plate from the measured differential pressure to DIN EN ISO 5167-2:2023. The discharge coefficient follows the Reader-Harris/Gallagher equation and is found iteratively, because it depends on the Reynolds number it helps to produce.
The application limits of the standard for diameters, diameter ratio and Reynolds number are checked. Installation conditions, pipe roughness, edge sharpness and plate flatness are not – those are covered by clause 6 of the standard and by ISO 5167-1.
Enter orifice geometry and differential pressure.
Calculate mass and volume flow from the differential pressure across a standard orifice plate, including discharge coefficient, expansibility factor and permanent pressure loss.
Pipe inside diameter D and orifice bore d set the diameter ratio β, and the tapping arrangement sets the distance ratios L₁ and L₂′. Add the measured differential pressure Δp with density and dynamic viscosity of the fluid at operating conditions; for gases also the absolute pressure and the isentropic exponent.
Mass flow follows from qₘ = C·ε·A·√(2·Δp·ρ₁)/√(1−β⁴). The discharge coefficient C comes from the Reader-Harris/Gallagher equation and depends on β, the Reynolds number and the tapping arrangement; since the Reynolds number in turn depends on the flow being sought, C is determined iteratively.
Water at 20 °C in DN 100 with a 50 mm orifice and flange tappings: at 0.2 bar differential pressure this gives β = 0.5, Re_D ≈ 98,700, C = 0.6062 and a mass flow of 7.77 kg/s, or 467 l/min. About 0.146 bar of the differential pressure remains as a permanent loss in the plant.
Sources and limits: DIN EN ISO 5167-2:2023-08 defines geometry, discharge coefficient, expansibility factor and application limits for standard orifice plates; DIN EN ISO 5167-1:2023-08 provides the flow equation and the treatment of measurement uncertainty. Installation requirements, pipe roughness, edge sharpness and plate flatness have to be checked separately per clause 6 and ISO 5167-1.
Calculate mass and volume flow from the differential pressure across a standard orifice plate, including discharge coefficient, expansibility factor and permanent pressure loss.
A standard orifice plate is a thin disc with a sharp-edged bore, fitted across the pipe. The flow contracts and accelerates through it, and by the energy equation the static pressure drops. What is measured is not the flow itself but this pressure difference between a tapping upstream and one downstream – hence the name differential pressure method. The relation is quadratic: flow grows with the square root of the differential pressure, so doubling the differential pressure yields only about 41 % more flow. The method needs no moving parts and no in-service calibration, because the standard derives the discharge coefficient from a measurement database built up over decades.
The pressure downstream of an orifice is not uniform: the jet keeps contracting for a distance behind the plate and then gradually recovers. Where the measurement is taken therefore co-determines the measured pressure difference – and with it the discharge coefficient.
ISO 5167-2 recognises exactly three standardised arrangements: corner tappings right at the plate, D and D/2 tappings one pipe diameter upstream and half a diameter downstream, and flange tappings 25.4 mm either side of the plate. Each substitutes its own distance ratios L₁ and L₂′ into the equation for C.
What matters is what the standard explicitly forbids: substituting arbitrary L₁/L₂′ pairs that match none of the three arrangements. The calculator therefore derives them from the selection rather than offering a free field. For flange tappings they additionally depend on the pipe diameter, because the fixed 25.4 mm distance is put in ratio to D.
The discharge coefficient of the Reader-Harris/Gallagher equation depends on the Reynolds number. The Reynolds number in turn follows from the mass flow – the very quantity that C is meant to help calculate. The equation is therefore implicit and cannot be solved in one step.
In practice this is why an orifice calculation is tedious by hand and why tabulated values for C are only an approximation – the standard itself notes that its tables are not intended for accurate interpolation and that extrapolation is not permitted.
This calculator solves the equation numerically. Since mass flow is proportional to C, Re_D = k·C with a constant k, and what is sought is the root of C_RHG(Re_D(C)) − C. Because C falls monotonically with rising Reynolds number, this function is strictly monotonic and can be bracketed reliably – unlike a plain fixed-point iteration, which runs away at very low Reynolds numbers.
qₘ = C · ε · (π/4) d² · √(2 · Δp · ρ₁) / √(1 − β⁴)
β = d / DC = 0.5961 + 0.0261 β² − 0.216 β⁸ + 0.000521 (10⁶ β / Re_D)^0.7 + …ε = 1 − (0.351 + 0.256 β⁴ + 0.93 β⁸) · (1 − (p₂/p₁)^(1/κ))Re_D = 4 qₘ / (π · D · η)Δϖ = (√(1 − β⁴(1 − C²)) − C β²) / (√(1 − β⁴(1 − C²)) + C β²) · Δp| Symbol / input | Meaning |
|---|---|
| D, d | Pipe inside diameter and orifice bore at operating temperature. |
| β | Diameter ratio d/D; ISO 5167-2 permits 0.1 to 0.75. |
| Δp | Measured differential pressure between the upstream and downstream tapping. |
| ρ₁, η | Density and dynamic viscosity of the fluid at the upstream cross-section. |
| C | Discharge coefficient per Reader-Harris/Gallagher, depending on β, Re_D and tapping arrangement. |
| ε | Expansibility factor; 1 for liquids, below 1 for gases. |
| Δϖ | Permanent pressure loss that remains with the plant. |
Pipe inside diameter D and orifice bore d set the diameter ratio β, and the tapping arrangement sets the distance ratios L₁ and L₂′. Add the measured differential pressure Δp with density and dynamic viscosity of the fluid at operating conditions; for gases also the absolute pressure and the isentropic exponent.
Enter pipe inside diameter and orifice bore at operating temperature and select the tapping arrangement. Add the measured differential pressure with the density and dynamic viscosity of the fluid at operating conditions; for gases also the upstream absolute pressure and the isentropic exponent. The calculator determines the discharge coefficient iteratively and returns mass and volume flow, Reynolds number, expansibility factor, permanent pressure loss and the uncertainty of C.
Water at 20 °C in DN 100 with a 50 mm orifice and flange tappings: at 0.2 bar differential pressure this gives β = 0.5, Re_D ≈ 98,700, C = 0.6062 and a mass flow of 7.77 kg/s, or 467 l/min. About 0.146 bar of the differential pressure remains as a permanent loss in the plant.
The mass flow is the actual measurement result; the volume flow follows from it via the operating density you entered and, for gases, is explicitly not a flow at reference conditions. The reported uncertainty applies to the discharge coefficient only – the uncertainty of the overall result is larger, because differential pressure, density and both diameters contribute their own. The permanent pressure loss is the figure that costs money in service: it is markedly below the differential pressure because part of it is recovered downstream, but it stays with the plant for good. Not checked here are the installation conditions: the straight pipe lengths upstream and downstream, pipe roughness, roundness, edge sharpness and flatness of the plate. Those requirements decide whether the stated uncertainty is achievable at all.
Diameters in millimetres, differential and absolute pressure in bar, density in kg/m³, viscosity in mPa·s, mass flow in kg/s and volume flow in litres per minute. Diameter ratio, discharge coefficient, expansibility factor and Reynolds number are dimensionless.
β is the real design decision and a trade-off. A small β produces a large differential pressure at a given flow, which makes the measurement more robust against disturbances at the transmitter – but costs a lot of permanent pressure loss that the pump or compressor has to supply continuously.
A large β reverses both: little pressure loss, but a small differential pressure and hence a more sensitive measurement. On top of that the uncertainty of the discharge coefficient rises again above β = 0.6, from 0.5 % to as much as 0.75 % at β = 0.75.
The standard limits the usable range to 0.1 to 0.75. In practice the usual design range is considerably narrower, because that is where the uncertainty is at its minimum while the pressure loss is still acceptable. The calculator shows both figures at once, so the compromise can be read off directly.
The application limits for diameters, diameter ratio and Reynolds number from clause 5.3.1 are checked and flagged when violated. Nothing else is.
In particular the installation requirements of clause 6 are not covered: the straight pipe lengths required upstream and downstream, which vary considerably with the preceding disturbance – bend, valve, reducer – and the possible use of a flow conditioner. Nor are pipe roughness, roundness and cylindricity, the edge sharpness and flatness of the plate, or its alignment.
These are not side issues: the stated uncertainty of the discharge coefficient applies only when they are met. A numerically sound orifice on too short a pipe run does not measure to 0.5 %.
Sizing and checking differential-pressure measuring sections in pipework, sanity-checking an existing orifice measurement, estimating the permanent pressure loss of a measuring point, and choosing the diameter ratio between measurement accuracy and energy loss.
DIN EN ISO 5167-2:2023-08 defines geometry, discharge coefficient, expansibility factor and application limits for standard orifice plates; DIN EN ISO 5167-1:2023-08 provides the flow equation and the treatment of measurement uncertainty. Installation requirements, pipe roughness, edge sharpness and plate flatness have to be checked separately per clause 6 and ISO 5167-1.
Common mistake: Don't enter the density at reference conditions – what counts is the density at the upstream tapping cross-section at operating pressure and temperature, which for gases can differ by orders of magnitude. Nor may gauge pressure be used as p₁ for gases; the expansibility factor needs absolute pressure. And β cannot be raised freely to cut pressure loss: above 0.75 the measurement leaves the validity range of the standard.
It contracts and accelerates the flow, which lowers the static pressure. The pressure difference between a tapping upstream and one downstream is measured, and the flow follows from it. There are no moving parts.
Through qₘ = C·ε·A·√(2·Δp·ρ₁)/√(1−β⁴) to ISO 5167-1, where A is the area of the orifice bore, C the discharge coefficient and ε the expansibility factor. The relation is a square root, not a linear one.
Because it grows with its square root. Four times the differential pressure doubles the flow; twice the differential pressure yields only about 41 % more. That also limits the usable turndown of an orifice measuring point.
It links the ideal flow from the energy equation to the actual one and captures jet contraction and friction. For standard orifice plates the Reader-Harris/Gallagher equation gives it from β, the Reynolds number and the tapping arrangement; it is typically around 0.60.
Because it depends on the Reynolds number, which depends on the flow being sought. The equation is therefore implicit. This calculator solves it numerically rather than using a fixed table value.
It accounts for a gas expanding across the pressure drop, so its density in the orifice bore is lower than upstream. For liquids it is 1 by definition, for gases below 1. The equation in the standard applies only for p₂/p₁ ≥ 0.75.
It is a trade-off: a small β means a large differential pressure and a robust measurement but high pressure loss; a large β the other way round. In addition the uncertainty of C rises again above β = 0.6. The permitted range is 0.1 to 0.75.
Markedly smaller than the differential pressure, because part of it is recovered downstream – how much depends strongly on β. The calculator reports it; it is the figure that costs pump energy in continuous operation.
Orifice bore of at least 12.5 mm, pipe diameter from 50 mm to 1000 mm, β between 0.1 and 0.75, and a minimum Reynolds number that depends on β and the tapping arrangement. The calculator checks all four and reports violations.
The density at the upstream tapping cross-section at operating pressure and temperature, not at reference conditions. For gases the two easily differ by orders of magnitude.
Yes, that is what the expansibility factor is for. Besides the differential pressure it then needs the upstream absolute pressure and the isentropic exponent, and the pressure ratio p₂/p₁ must be at least 0.75.
For the discharge coefficient alone 0.5 % in the range 0.2 ≤ β ≤ 0.6, more outside it; pipes below 71.12 mm add a further allowance. The uncertainty of the overall result is larger, because differential pressure, density and the diameters contribute their own.
No. Installation conditions, straight pipe lengths, pipe roughness, roundness, and the edge sharpness and flatness of the plate all determine the achievable uncertainty. Those requirements are in clause 6 of the standard and in ISO 5167-1 and are not checked here.
No. It covers standard orifice plates to ISO 5167-2 only. Nozzles and Venturi nozzles are dealt with in ISO 5167-3 and classical Venturi tubes in ISO 5167-4 – each with its own equation for the discharge coefficient.