OpenSpecEngineering

Turns the differential pressure across an orifice plate into a mass and volumetric flow rate, using the ISO 5167-2 flow equation [1]. Instrument and process engineers use it to check a flowmeter reading, to set a transmitter range or to see what flow a given plate gives.

How the flow is calculated

q_m = C·E·ε·(π/4)·d²·√(2·ΔP·ρ₁)

where q_m is the mass flow, C the discharge coefficient, d the bore diameter of the plate, ΔP the differential pressure between the taps and ρ₁ the fluid density at the upstream tap. The velocity of approach factor E = 1/√(1 − β⁴) corrects for the flow already moving in the pipe, with β = d/D and D the pipe inside diameter. The expansibility factor ε corrects for a gas expanding through the bore; it is 1.0 for a liquid. The volumetric flow is q_v = q_m/ρ₁, at upstream conditions.

The page does not work out C or ε for you. Take C from the plate’s sizing sheet, or from the Reader–Harris/Gallagher equation in ISO 5167-2 [1, 2]; it depends on β, the Reynolds number and the tap positions, and is near 0.60 for most plates. For a gas, ISO 5167-2 gives ε = 1 − (0.351 + 0.256β⁴ + 0.93β⁸)·[1 − (p₂/p₁)^1/κ], with p₂/p₁ the downstream-to-upstream pressure ratio and κ the isentropic exponent. Picking a pipe size fills D from ASME B36.10M [3], and picking a fluid fills its density [4–7].

Limits and common mistakes

• ISO 5167-2 covers D from 50 to 1000 mm, d of at least 12.5 mm, β from 0.10 to 0.75 and a pipe Reynolds number of at least 5000. Outside that, C is not known to the standard’s uncertainty.
• Use the measured inside diameter of the pipe, not its nominal size.
• ΔP is the reading between the taps, not the permanent pressure loss, which is smaller.
• ρ₁ is the density at the upstream tap. For a gas, use the flowing pressure and temperature, not standard conditions.
• The plate needs enough straight pipe upstream. Bends and valves close to it shift C.

Worked example

A 4 in. Schedule 40 cooling-water line has an orifice plate with a 60 mm bore and flange taps. The transmitter reads 25 kPa. The water is at 20 °C, and the sizing sheet gives C = 0.609. What is the flow?

1. Choose Pipe Size NPS 4 and Schedule Sch 40.
2. Enter Orifice Bore Diameter 60 mm and Differential Pressure 25 kPa.
3. Choose Fluid Water, 20°C, and enter Discharge Coefficient 0.609 and Expansibility Factor 1.
4. Press Solve. The Solution shows: Diameter ratio β = d/D: 0.58674
   Velocity of approach factor E: 1.06511
   Mass flow qm: 12.9568 kg/s
   Mass flow qm: 46,644.4 kg/h
   Volumetric flow qv: 46.7285 m³/h

Check by hand. β = 60/102.26 = 0.5867 and the bore area is 2.827×10⁻³ m². √(2 × 25,000 × 998.2) = 7,065, so q_m = 0.609 × 1.0651 × 2.827×10⁻³ × 7,065 ≈ 12.96 kg/s. The Reynolds number is about 161,000, and the Reader–Harris/Gallagher equation for flange taps gives C ≈ 0.609 there, so the coefficient fits. Flow goes as √ΔP: a reading of 100 kPa would double it.