Both of this lesson's meters run on the same idea: narrow the pipe, and the stream must speed up to get through. Speeding up costs pressure, by Bernoulli, so the pressure drop across the restriction reports the flow. Measure a pressure, infer a flow — no moving parts anywhere.

$Q = \dfrac{C_d}{\sqrt{1-\beta^{4}}}\cdot\dfrac{\pi d^{2}}{4}\sqrt{\dfrac{2\Delta P}{\rho}}$, read as Q equals C-d over root one minus beta to the fourth, times the bore area, times root two delta-P over rho. The letters: $Q$ is volumetric flow in m³/s; $d$ is the bore of the restriction (the orifice hole, or the venturi throat) in metres and $D$ is the pipe inside diameter — the two are never interchangeable; $\beta$ — beta — is their ratio $d/D$, a bare number, typically 0.4 to 0.7. $\Delta P$ is the differential pressure in pascals and $\rho$ — rho — the fluid density in kg/m³.

Two of those terms are corrections and both earn their place. $\dfrac{1}{\sqrt{1-\beta^{4}}}$ is the velocity-of-approach factor: the fluid was already moving before it met the restriction, and this remembers it. $C_d$ is the discharge coefficient, and it is the honest one. A sharp-edged plate's jet keeps contracting AFTER it clears the hole — the vena contracta — so only about 0.61 of the ideal flow is really there. A machined venturi guides the stream instead of tearing it, and earns about 0.98.

That difference is why both exist. The plate is a disc of steel between two flanges, cheap forever, and it destroys perhaps ten times the permanent head the venturi does — a pumping bill for the life of the plant. The venturi costs real money once and gives most of its pressure back.

Carry the square root out of the room with you. Flow goes as $\sqrt{\Delta P}$: four times the differential is only twice the flow. Which means at a tenth of full flow the transmitter sees a hundredth of full scale, and that is exactly why DP meters are poor at the bottom of their range and every one of them has a low-flow cut-off.