Civil Engineering · Hydraulics
Weir Flow (Rectangular)
Discharge over a rectangular sharp-crested weir using the Francis formula. Does not apply to V-notch, Cipolletti, or broad-crested weirs, which use different formulas.
Standard method — stated scope limits apply
Weir
Enter crest length and head.
About This Calculation
A weir is a precisely-shaped obstruction across an open channel that creates a predictable relationship between upstream water depth (head) and flow rate over the crest — the rectangular (Francis formula) weir is one of the oldest and most widely used flow measurement methods for open channels, irrigation systems, and wastewater facilities, valued for having no moving parts and a well-established, simple empirical formula relating head to flow.
Frequently Asked Questions
Why does weir flow scale with head to the 3/2 power, not linearly?
The flow rate over a weir crest is essentially the integral of velocity (which itself depends on depth below the free surface, via Torricelli’s/Bernoulli’s relation) across the flowing cross-section — working through that integration for a rectangular weir produces the characteristic head^1.5 relationship the Francis formula captures empirically.
Why does the Francis formula include an end-contraction correction?
A rectangular weir narrower than the full channel width causes the flow to contract inward at each end as it approaches the crest, effectively reducing the useful flow width below the weir’s actual physical crest length — the end-contraction term corrects for this, and is omitted (suppressed) only when the weir spans the full channel width with no contraction possible.
How accurate is a well-maintained rectangular weir compared to other flow measurement methods?
A properly installed and maintained sharp-crested rectangular weir can achieve accuracy in the range of ±2–5% under the Francis formula’s valid conditions — comparable to many closed-pipe flow measurement methods, though it requires adequate approach channel length and a properly vented nappe (air space under the falling sheet of water) to maintain that accuracy.