Orifice plate flow metering (ISO 5167-2)
The orifice plate is the most widely used differential-pressure primary element for flow metering in pipework: a concentric-bore disc creates a pressure differential Δp proportional to the square of the flow rate, computed rigorously per ISO 5167-2.
When to use
Use an orifice plate calculation when you need to measure (or impose) the flow rate of a liquid, gas or steam in a closed conduit with a simple, low-cost element that has no moving parts. It is the standard solution for inferential metering paired with a differential-pressure transmitter: you set the operating flow rate and the desired span Δp (the transmitter's full-scale value) and the method returns the geometry — beta ratio β and bore diameter d — that produces exactly that differential. It also serves to size a fixed restriction (restriction orifice) that imposes a known K factor or permanent pressure loss. The plate is only valid in fully developed turbulent flow, with sufficient straight runs upstream and downstream; outside those conditions, switch to a flow nozzle, Venturi tube or a mass flow meter.
What an orifice plate is
The orifice plate is the most widespread flow-metering element in industry: a thin disc with a concentric square-edged bore, mounted between pipe flanges. By forcing the flow through a reduced section, it creates a pressure drop that can be measured by a differential-pressure transmitter. Because that drop bears a well-defined relationship to the flow rate, the flow is metered inferentially — with no moving parts, at low cost, and on physics fully standardized by ISO 5167-2.
The principle is Bernoulli’s energy balance combined with conservation of mass: as the fluid accelerates through the bore, the static pressure drops; the flow rate is proportional to the square root of that differential. From this comes the plate’s signature characteristic — Δp grows with the square of the flow rate, which gives excellent resolution near full scale and poor resolution at low flows.
The metering equation and the discharge coefficient
The mass flow rate comes from the general ISO 5167-1 equation:
q_m = (C / √(1 − β⁴)) · ε · (π/4) · d² · √(2 · Δp · ρ₁)
Each term has a physical role:
- β = d/D is the diameter ratio. The factor E = 1/√(1 − β⁴) corrects for the approach velocity — the fluid already reaches the bore with a non-negligible velocity.
- C is the discharge coefficient. It captures the difference between ideal and real flow: the jet contracts beyond the bore (the vena contracta) and loses energy to friction. C is not constant — it depends on β, the Reynolds number and the tap type.
- ε is the expansibility factor: 1 for liquids, less than 1 for gases and steam, because the compressible fluid loses density as it accelerates.
C is computed with the empirical Reader-Harris/Gallagher equation (the 1998 version adopted by ISO 5167-2:2003), fitted to thousands of experimental points. It sums a base term (~0.5961), corrections in β, a Reynolds term and terms specific to the pressure taps, plus an addition for small pipes (D < 71.12 mm). In fully developed turbulence, C tends to stabilize near 0.60–0.61.
How sizing works, step by step
Sizing is an inverse, implicit problem: you know the operating flow rate and the desired span Δp (the transmitter’s full-scale value) and you look for the geometry that produces them.
- Define the design point. Flow rate Q (converted to q_m via density), span Δp, internal pipe diameter D and fluid properties (ρ, viscosity μ; for gas, also pressure and κ).
- Assume an initial β and compute C with Reader-Harris/Gallagher and ε (1 for a liquid).
- Solve for the flow rate using the general equation and compare it with the target flow. Because C depends on β and on Re_D — which in turn depends on the flow — the system is coupled.
- Iterate (a few fixed-point steps converge) until β closes the equation. The bore is d = β·D.
- Compute the permanent loss Δω and the bore velocity as a check.
Permanent loss: what really costs you
The most misunderstood point in practice is the difference between the Δp read across the taps and the permanent pressure loss Δω. Downstream of the plate the jet re-expands and recovers only part of the pressure — for a square-edged orifice, the smaller part; what remains, irreversibly, is Δω. ISO 5167-2 (§5.4.2) gives the ratio between the two:
Δω/Δp = (√(1 − β⁴(1 − C²)) − C·β²) / (√(1 − β⁴(1 − C²)) + C·β²)
This fraction depends strongly on β: for β = 0.75, about 45% of the differential stays as permanent loss; for β = 0.20, the loss reaches ~95%. Unlike a Venturi tube (which recovers most of the differential), the square-edged plate loses the majority of it — the recovered part is the minority. In pumped systems, Δω is energy the pump pays for continuously — which is why a high β is desirable when efficiency is the priority, even though it shrinks the span.
Practical design considerations
- Useful β range: 0.20 to 0.75. Small bores clog and amplify manufacturing error; a high β loses accuracy and validity.
- Check the Reynolds number. The standard requires Re_D ≥ 5000 and ≥ 170·β²·D. In viscous liquids or at low flow, C loses its certification.
- Straight runs. Respect the minimum straight lengths upstream and downstream (a function of β and the fittings) — without them, the velocity profile distorts the real C.
- Consistent taps. Declare and install the same type (flange, corner or D-D/2): each uses different terms in the C equation.
- Gas and steam. Include ε and respect Δp/p₁ ≤ 0.25; above that, compressibility falls outside the standard’s range.
Link to the standards
The geometry, the coefficient C (Reader-Harris/Gallagher), the expansibility factor and the permanent pressure loss follow ISO 5167-1/2 (and its Brazilian adoption, ABNT NBR ISO 5167). For custody-transfer metering of natural gas, AGA Report No. 3 / API MPMS 14.3 prevail; in the U.S. context, ASME MFC-3M is the equivalent. The result is a traceable, auditable and inexpensive measurement — provided the plate is well manufactured, correctly installed and operated within the standard’s validity range.
Formulas and fundamentals
q_m = (C / √(1 − β⁴)) · ε · (π/4) · d² · √(2 · Δp · ρ₁) Mass flow rate q_m (kg/s) as a function of the discharge coefficient C (dimensionless), the approach velocity factor E = 1/√(1 − β⁴), the expansibility factor ε, the bore diameter d (m), the differential Δp (Pa) and the upstream density ρ₁ (kg/m³). For a liquid, ε = 1.
β = d / D Ratio of the bore diameter d to the internal pipe diameter D (same unit). It is the geometric parameter that governs C, ε and the permanent pressure loss. ISO 5167-2 is valid for 0.10 ≤ β ≤ 0.75.
C = 0.5961 + 0.0261·β² − 0.216·β⁸ + 0.000521·(10⁶·β/Re_D)^0.7 + (0.0188 + 0.0063·A)·β^3.5·(10⁶/Re_D)^0.3 + (0.043 + 0.080·e^(−10·L₁) − 0.123·e^(−7·L₁))·(1 − 0.11·A)·β⁴/(1−β⁴) − 0.031·(M'₂ − 0.8·M'₂^1.1)·β^1.3 C depends on β, on the Reynolds number Re_D referred to D, and on the tap location via L₁ and L'₂ (dimensionless distances). A = (19000·β/Re_D)^0.8 and M'₂ = 2·L'₂/(1−β). For D < 71.12 mm, add the term +0.011·(0.75−β)·(2.8−D/25.4).
ε = 1 − (0.351 + 0.256·β⁴ + 0.93·β⁸)·[1 − (p₂/p₁)^(1/κ)] Corrects for the density drop of gas/steam between upstream and the bore. p₂/p₁ is the absolute pressure ratio (≈ 1 − Δp/p₁) and κ the isentropic exponent (air ≈ 1.40). For an incompressible liquid, ε = 1 exactly.
Δω = (√(1 − β⁴·(1 − C²)) − C·β²) / (√(1 − β⁴·(1 − C²)) + C·β²) · Δp Δω is the fraction of the differential that is NOT recovered downstream — the real pressure penalty imposed on the system, distinct from the Δp read between the taps. It grows as β decreases (smaller bore = more blockage = higher permanent loss).
Standards & methods
- ISO 5167-1 (flow measurement by differential-pressure devices — principles and general equation)
- ISO 5167-2 (orifice plates — geometry, Reader-Harris/Gallagher coefficient C, expansibility and permanent pressure loss §5.4.2)
- ASME MFC-3M (orifice metering — North American equivalent)
- AGA Report No. 3 / API MPMS 14.3 (custody-transfer metering of natural gas by orifice plate)
- ABNT NBR ISO 5167 (Brazilian adoption of the ISO 5167 series)
Typical reference values
| Quantity | Typical range | Note |
|---|---|---|
| Diameter ratio β = d/D | 0.20 to 0.75 | ISO 5167-2 is valid for 0.10 ≤ β ≤ 0.75; the practical range avoids clogging (low β) and loss of accuracy (high β). |
| Discharge coefficient C | 0.60 to 0.63 | Concentric square-edged plate in developed turbulence; tends to 0.5961 as Re_D → ∞. |
| Minimum Reynolds number Re_D | ≥ 5000 and ≥ 170·β²·D | D in mm. Below this, C falls outside the certified range and uncertainty rises sharply. |
| Expansibility factor ε (liquid) | = 1.000 | For gas/steam, ε < 1 and decreases with Δp/p₁; require Δp/p₁ ≤ 0.25 to stay within the standard. |
| Permanent loss Δω/Δp | 0.45 (β=0.75) to 0.95 (β=0.20) | The smaller β is, the larger the fraction of pressure permanently dissipated. Most of the differential is lost — a square-edged plate recovers far less than a Venturi. |
| Typical uncertainty of C | 0.5% to 0.75% | For a new, well-manufactured plate installed per the standard (flange taps). |
Worked example
Plate for water metering with a Δp transmitter
Inputs
- Fluid
- Water at 20 °C —
- Density ρ₁
- 998 kg/m³
- Internal pipe diameter D
- 100 mm
- Operating flow rate Q
- 80 m³/h
- Desired span Δp
- 250 mbar
- Tap type
- Flange —
Results
- Diameter ratio β
- 0.648 —
- Bore diameter d
- 64.8 mm
- Discharge coefficient C
- 0.604 —
- Reynolds number Re_D
- 2.8 × 10⁵ —
- Permanent loss Δω
- 144 mbar
With Q = 80 m³/h (0.0222 m³/s → q_m ≈ 22.2 kg/s) and a span of 250 mbar (25,000 Pa), the iteration solves the implicit flow equation and converges to β = 0.648 — that is, a 64.8 mm bore in a 100 mm pipe. The coefficient C settles at ≈ 0.604, typical of a square-edged plate with Re_D in the order of 10⁵ (full turbulence, within the standard's range). The permanent loss is ≈ 144 mbar — about 58% of the 250 mbar span — because a square-edged plate recovers only a modest fraction of the differential downstream (here ~42%); most of it is lost irreversibly. Even at this relatively high β the penalty is substantial. Had a smaller β been chosen (a narrower bore, greater span rangeability), the permanent-loss fraction would rise further still, raising the operating cost in pumping energy.
Common mistakes
- Confusing the Δp measured between the taps (transmitter span) with the permanent pressure loss Δω — the latter is always smaller and is what actually costs the system pressure.
- Using β outside 0.20–0.75 — too small a bore clogs and amplifies manufacturing error; too high a β falls outside the coefficient's validity and increases uncertainty.
- Omitting the expansibility factor ε for gas/steam (treating it as a liquid), which overestimates the flow rate; and ignoring the Δp/p₁ ≤ 0.25 limit.
- Not checking the Reynolds number: in a viscous liquid or at low flow, Re_D drops below the standard's minimum and C loses its certification.
- Mixing the tap type (flange, corner, D-D/2) between the C calculation and the physical installation — each tap uses different terms in the equation.
- Ignoring the minimum straight runs upstream/downstream (a function of β and the fittings), which distorts the velocity profile and the real C.
Frequently asked questions
What is the discharge coefficient C and why is it not constant?
C corrects the ideal flow rate for the real contraction of the jet (vena contracta) and for friction. It depends on β, the Reynolds number and the tap type, which is why ISO 5167-2 computes it with the empirical Reader-Harris/Gallagher equation rather than a fixed value. In full turbulence (high Re) it stabilizes near 0.60–0.61.
What is the difference between the Δp across the taps and the permanent loss Δω?
The Δp across the taps is the differential the transmitter reads — it includes pressure that will still recover downstream as the jet re-expands. The permanent loss Δω is the irreversible fraction that stays in the system forever (§5.4.2 of the standard). Δω is always smaller than Δp and is what matters for pumping-energy consumption.
How do I choose the beta ratio?
β is a trade-off: high values (0.6–0.75) give low permanent loss but a small span and higher uncertainty; low values (0.2–0.4) give a good span and good resolution but high permanent loss and a risk of clogging. Keep 0.20 ≤ β ≤ 0.75 and size from the transmitter span Δp you have selected.
Do I need to account for the expansibility factor ε?
Only for compressible fluids (gas and steam): the density drops between upstream and the bore, and ε < 1 corrects for it. For liquids, ε = 1 exactly. The standard requires the ratio Δp/p₁ not to exceed 0.25; above that the plate falls outside its validity range.
What pressure-tap type should I use?
The three standardized by ISO 5167-2 are corner taps, flange taps (25.4 mm from the face) and D-D/2 taps. Each enters the C calculation with different terms, so the type declared in the design must match the one installed. Flange taps are the most common in process; corner taps are typical of small-bore pipes.
Does the orifice plate work for any flow rate?
No. It requires a Reynolds number above the standard's minimum (fully developed turbulent flow) and has modest rangeability — because Δp varies with the square of the flow rate, below about 30% of full scale the reading loses resolution. For very low flows, viscous liquids or a wide range, consider a flow nozzle, a Venturi tube or a mass flow meter.
Glossary
- Orifice plate
- Thin disc with a concentric square-edged bore that creates a pressure differential used to infer the flow rate (ISO 5167-2).
- Differential-pressure primary element
- Device that constricts the flow, generating a pressure drop proportional to the square of the flow rate (plate, nozzle, Venturi).
- Diameter ratio (β)
- Ratio d/D of the bore to the internal pipe diameter; the primary geometric parameter of the plate.
- Discharge coefficient (C)
- Factor that corrects the ideal flow rate for the real contraction and friction of the jet; ~0.60–0.63 for a square-edged plate.
- Expansibility factor (ε)
- Factor (≤ 1) that corrects for the density change of gases/steam crossing the plate; equal to 1 for liquids.
- Permanent pressure loss (Δω)
- Irreversible fraction of the pressure differential that is not recovered downstream; the real energy cost of the plate.