LOGOS Learning · 11
Orifice flow equation: how an orifice plate measures flow
The orifice flow equation turns the measured differential pressure into flow: Q = C / sqrt(1 - beta^4) * (pi/4) * d^2 * sqrt(2 * dP / rho), per ISO 5167-2. Because dP grows with the square of Q, the 4-20 mA signal needs square-root extraction to read flow linearly.
Change the flow and the bore diameter. The transmitter only sees a pressure difference; flow comes from its square root.
Quick test
Three questions about this lesson. Got one wrong? The explanation shows right away.
1. Square-root extraction on the signal serves to…
2. An orifice plate measures flow because…
3. Reducing β (smaller bore) at the same flow…
Why this happens
Arrows compare with 100 % flow, same β. ISO 5167-2, flange taps, Reader-Harris/Gallagher coefficient.
Orifice equation (ISO 5167)
The plate turns flow into a pressure difference: the liquid speeds up through the bore, pressure drops, and the transmitter reads that drop between the H and L taps. C barely changes with flow, so the Q–ΔP relation is practically fixed.
ΔP grows with the square of flow
At half flow only a quarter of the ΔP is left. At 10 % flow the ΔP is 1 % of span, close to the transmitter error itself. That is why an orifice with a single transmitter has a short turndown.
4–20 mA signal
Without square-root extraction, at 50 % flow the signal reads 8 mA, which an unaware system takes as 25 %. Extraction can be done in the transmitter or in the controller, never in both.
Permanent pressure loss
Part of the drop recovers after the vena contracta; the rest is permanent loss and adds to the pump head. A smaller β gives more signal, but also more loss.
Formulas in plain text
- Orifice flow equation (ISO 5167-2, liquid)
Q = C / sqrt(1 - beta^4) * (pi / 4) * d^2 * sqrt(2 * dP / rho)- Q = volumetric flow (m3/s) · C = discharge coefficient, Reader-Harris/Gallagher (about 0.6) · beta = d / D, diameter ratio (-) · d = orifice bore diameter (m) · D = pipe internal diameter (m) · dP = differential pressure between taps (Pa) · rho = fluid density (kg/m3)
- Square law
dP / dPmax = (Q / Qmax)^2- dPmax = differential pressure at full-scale flow (Pa) · Qmax = full-scale flow (m3/h)
- 4-20 mA transmitter signal
I = 4 + 16 * sqrt(dP / dPmax) | I = 4 + 16 * dP / dPmax- I = transmitter output (mA) · first form: with square-root extraction (signal proportional to flow) · second form: without extraction (signal proportional to dP)
- Permanent pressure loss (ISO 5167-2)
dw = (sqrt(1 - beta^4 * (1 - C^2)) - C * beta^2) / (sqrt(1 - beta^4 * (1 - C^2)) + C * beta^2) * dP- dw = unrecovered pressure loss (Pa)
Frequently asked questions
What is an orifice plate used for?
It measures flow in a pipe: a thin plate with a bore makes the liquid speed up, the pressure drops, and a differential pressure transmitter reads that drop. Flow is then calculated from the square root of dP with the ISO 5167 equation.
What beta ratio should an orifice plate have?
ISO 5167-2 covers beta from 0.1 to 0.75, and designs usually stay between about 0.2 and 0.7. A smaller beta gives a larger dP signal but also a larger permanent pressure loss.
Why does an orifice flowmeter need square-root extraction?
Because dP is proportional to Q^2: at 50 % flow the dP is 25 % of span and a linear 4-20 mA signal reads 8 mA. Extraction is done in the transmitter or in the controller, never in both.
What is the turndown of an orifice plate?
With a single transmitter it is short, typically around 3:1 to 4:1. At 10 % flow the dP is only 1 % of span, close to the error of the transmitter itself.