Orifice Plate Flow

An orifice plate is a disc with a hole, bolted between flanges, and it has survived a century of better ideas because it is cheap, has no moving parts, and is described by an international standard anyone can audit. Drop the flow through a restriction and the pressure falls by an amount proportional to the square of velocity, so the flow comes back out as a square root. A 50 mm bore in a 100 mm line with 20 kPa across it passes about 469 L/min of water. Double the flow and you need four times the differential, which is exactly why an orifice meter's usable turndown is only about 3 or 4 to 1: at a third of full flow the differential has fallen to a ninth and the transmitter is reading noise.

The discharge coefficient near 0.61 is not a fudge factor, it is the vena contracta. The jet keeps converging for a diameter or so downstream of the plate, so the true minimum flow area is smaller than the drilled hole. ISO 5167 gives the Reader-Harris/Gallagher equation for $C_d$ as a function of $\eta$ and Reynolds number, and it stays within a few tenths of a percent when the installation follows the standard. The velocity-of-approach factor $1/\sqrt{1-\eta^4}$ is a separate correction, accounting for the fact that the upstream fluid was already moving.

What surprises people is how much of the accuracy lives in the pipe rather than the plate. The standard demands long straight runs, typically 10 to 40 diameters upstream depending on what disturbance sits there, because a swirl left over from two out-of-plane elbows will bias the reading by several percent and no calibration will find it. The other quiet killer is a nicked or rounded upstream edge: an orifice plate installed backwards, or one that has been in erosive service for a few years, reads high by a margin that nothing on the transmitter reveals.