# Free weir formula

The formula is derived from the original formula of Poleni (1717).

In a free flow, the flow rate depends only on the upstream water level $$h_{amont}$$:

$Q = Cd \sqrt{2g} L h_{amont}^{3/2}$

With:

• Q the flow in m3/s
• Cd the discharge coefficient
• $$g$$ the acceleration of gravity 9.81 m/s2
• L the width of the weir in m
• hamont the upstream water level above the crest of the weir in m

A flow coefficient value $$C_d = 0.4$$ is generally a good approximation for a rectangular weir. For more complex weir shapes (trapezoidal, circular...) or to take into account the characteristics of the longitudinal profile (thin-crested weir, thick-crested weir), one can refer to the CETMEF weir leaflet (CETMEF, 2005).

CETMEF (2005). Notice sur les déversoirs : synthèse des lois d’écoulement au droit des seuils et déversoirs. Compiègne : Centre d’Études Techniques Maritimes Et Fluviales. 89 p.