Francis Formula: Rectangular Weir

James B. Francis measured flow over sharp-crested weirs at the Lowell canals in the 1850s and produced the coefficient that still bears his name: Q in cubic feet per second equals 3.33 times the crest length times the head to the three-halves power, both in feet. Converting the half power on length gives 1.838 for metres and cubic metres per second, the number SI texts round to 1.84. A 4 ft crest under 0.5 ft of head passes 3.33 × 4 × 0.354 = 4.71 cfs, about 2114 gpm. The solver applies the constant on the ft–second basis whatever units you type, converting first.

The three-halves power is what makes a weir a good meter and a demanding one: a 1% error in head becomes a 1.5% error in flow, so the head must be measured properly. Read it in a stilling well at least four times the maximum head upstream of the crest: measuring at the crest itself catches the drawdown and reads low every time. The formula as written is for a SUPPRESSED weir, whose crest runs the full channel width; if the weir is contracted, Francis's own correction shortens the effective length by 0.1H for each end contraction. And the nappe must be aerated and springing clear: if the sheet clings to the downstream face, the pressure underneath drops and the weir passes more flow than the equation says, which is a very common and entirely invisible metering error in wastewater plants.