Coefficient of Discharge Calculator

The formula

orifice Q = C_d A√(2gh); weir Q = C_d (2/3)^{3/2}√g b H^{3/2}

Why a weir makes a flow meter

Discharge through an orifice or over a weir depends only on the head and the geometry, so a single depth measurement gives the flow. That is why weirs are standard in open-channel gauging: no moving parts, and one staff-gauge reading is enough.

The head exponent sets the sensitivity. A rectangular weir goes as H^1.5, a V-notch as H^2.5. The V-notch is therefore far more sensitive, which makes it excellent at low flows — the depth changes appreciably when the discharge does — and correspondingly demanding of accurate head measurement, since a 1% depth error becomes a 2.5% flow error.

Torricelli's result is the ideal case: fluid leaves an orifice at exactly the speed a body would reach falling through the same head. Real jets fall short, mostly because the stream contracts just downstream of a sharp opening. That vena contracta accounts for most of the 0.62 discharge coefficient, and rounding the entry recovers nearly all of it.

These formulas assume constant head. A draining tank's head falls as it empties, so the actual emptying time is longer than a constant-rate estimate — for a tank of uniform section, by a factor of two.