Chezy Equation

Antoine Chezy worked out this relation in 1769 while designing a water supply canal for Paris, which makes it the oldest equation on this site by more than a century. The derivation is a force balance, not a curve fit: the gravity component driving a slug of water down a slope is balanced by the boundary shear resisting it, and when you write both out, velocity comes out proportional to the square root of $RS$. A coefficient of 50, a hydraulic radius of 1.5 metres and a slope of 0.001 give $50\sqrt{0.0015} = 1.94$ m/s.

Manning's equation, which the trade uses far more, is Chezy with the coefficient written as $C = R^{1/6}/n$. The two are not rivals, they are the same equation with different bookkeeping, and the reason Manning won is that $n$ is very nearly constant for a given surface while $C$ is not. Chezy's C carries units, m^0.5/s in SI, and it drifts with the depth of flow, which means a value measured in a canal running two metres deep does not transfer to the same canal running half a metre deep.

Chezy's C is still worth knowing for two reasons. It is what appears in the theoretically grounded Colebrook and Darcy-Weisbach treatments of open-channel friction, where $C = \sqrt{8g/f}$, so it is the bridge between pipe-flow theory and channel practice. And it is the coefficient that Ganguillet, Kutter and Bazin all wrote their competing formulas for in the nineteenth century, which is why old canal records quote C rather than n. Typical values run 30 for rough natural channels to 90 or more for smooth concrete. If you have an n and want a C, remember the depth dependence: at $R = 1$ m the conversion is just $1/n$, and at other depths it is not.