How Orifice-Plate and Venturi Flow Measurement Works (ISO 5167)

The principle

A restriction in the pipe — an orifice plate, a nozzle or a Venturi tube — forces the fluid to speed up, and its static pressure falls. From conservation of mass and Bernoulli’s equation, the pressure drop between an upstream tap and the restriction grows with the square of the flow. Measuring that difference and inverting the relationship gives the flow.

The international standard ISO 5167 turns this into a practical, empirically calibrated equation. For mass flow:

q_m = C / √(1 − β⁴) · ε · (π/4) d² · √(2 Δp ρ₁)

Here d is the bore (or throat) diameter, D the pipe diameter, β = d/D, Δp the measured differential pressure and ρ₁ the fluid density at the upstream tap. C is the discharge coefficient and ε the expansibility factor.

The discharge coefficient C

C corrects the ideal equation for the real flow pattern: the jet contracts after a sharp-edged orifice, friction acts, and the pressure taps sit at specific positions. For a standard orifice plate C is about 0.60 and depends on β, on the pipe Reynolds number and on where the taps are (corner, flange or D and D/2). The Reader-Harris/Gallagher equation in ISO 5167-2 gives C to within about 0.5 % for typical geometries. A Venturi tube is far more efficient: C is about 0.984 to 0.995 depending on how the convergent section is finished.

Because C depends on Reynolds number, and Reynolds number depends on the flow you are trying to find, a correct calculation iterates: guess the flow, compute C, recompute the flow, and repeat until they agree. The ISO 5167 DP flow calculator does this automatically.

Expansibility ε for steam and gas

A liquid does not change density as it passes the restriction, so ε = 1. A gas or steam expands as its pressure falls, which lowers the density at the throat and reduces the flow for a given Δp. For an orifice plate ISO gives ε = 1 − (0.351 + 0.256β⁴ + 0.93β⁸)(1 − (p₂/p₁)^(1/κ)), where κ is the isentropic exponent. Nozzles and Venturi tubes use a different expression derived from isentropic expansion.

For superheated steam κ is not the textbook 1.3. At 30 bar and 400 °C the real isentropic exponent calculated from the IAPWS-IF97 equations is about 1.286, and at 100 bar and 540 °C about 1.276. The effect on ε is small but systematic, and it is simply free accuracy once the steam properties are computed properly.

Density is the biggest lever

Because density sits under a square root, a 1 % error in ρ₁ produces a 0.5 % error in flow. For steam that matters more than almost anything else, since density changes quickly with pressure and temperature. Treating steam as an ideal gas overstates its specific volume: at 30 bar and 400 °C the compressibility factor is about 0.96, so the ideal-gas density is roughly 4 % too low and the computed flow about 2 % too high. At 100 bar and 540 °C the compressibility factor is about 0.935 and the error is larger still. Use real steam tables (see Steam Properties) and always use absolute pressure.

Thermal expansion of the bore and pipe

Orifice dimensions are specified at 20 °C but the plate and pipe run hot. Austenitic stainless steel expands roughly 17 × 10⁻⁶ per kelvin and low-alloy steel roughly 13 × 10⁻⁶ per kelvin. At 540 °C the bore has grown by 1–2 %, which changes the flow by a comparable amount. The correction is d = d₂₀ [1 + α(T − 20)], applied to both d and D. These coefficients are material dependent, so a good calculator treats them as inputs you can override.

A worked example

Take a 250 mm pipe with a 150 mm flange-tapped orifice in 30 bar (absolute), 400 °C steam and a measured Δp of 250 mbar. The calculator finds a steam density of 10.06 kg/m³, a pipe Reynolds number of about 1.7 million, C = 0.6045, ε = 0.9974 and a mass flow of 29.55 t/h, with an expanded uncertainty of roughly 1–2 % once typical instrument uncertainties are included. The permanent pressure loss is about 157 mbar, or 63 % of the Δp, which is why orifice plates are not chosen where pressure recovery matters.

Where installations actually lose accuracy

Straight lengths and flow conditioning. ISO accuracy assumes the specified upstream and downstream straight pipe. Elbows, valves and reducers close to the plate distort the profile and shift C by more than the stated uncertainty.

Edge wear. The upstream edge of an orifice must be sharp. Erosion rounds it, which raises C and makes the meter under-read.

Impulse lines. Blocked, leaking or unequally filled impulse lines (condensate pots at different levels in steam service) give false Δp readings.

Wet steam. ISO 5167 is for single-phase flow. Near saturation, steam carries moisture and the meter over-reads.

Square-root scaling. Flow varies as √Δp, so at 10 % of flow the Δp is only 1 % of its full-scale value. Transmitter turn-down limits accuracy at low load; two ranges or a smaller bore are often used.

Choosing between element types

Orifice plates are cheap and standardised but have a high permanent loss. ISA 1932 and long-radius nozzles lose less pressure and tolerate high velocities and steam erosion better. A classical Venturi recovers most of the pressure and copes well with dirty fluids and large-diameter lines, at much higher cost. Which one suits you depends on pressure budget, accuracy requirement and maintenance access.

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