Manning's Equation for Velocity

Robert Manning, an Irish engineer with no formal training in mathematics, offered this in 1889 as a tidy fit to the open-channel data of his day, and it has outlived every theoretically superior rival. Velocity goes as the two-thirds power of hydraulic radius — flow area divided by wetted perimeter — and the square root of slope, divided by a roughness coefficient. A concrete sewer with n = 0.013 flowing at a hydraulic radius of 0.5 m on a 0.1% grade carries water at 1.53 m/s.

The catch is that n is not dimensionless despite being written as a bare number: the SI form uses a hidden coefficient of 1.0 $\mathrm{m}^{1/3}/\mathrm{s}$, and the US customary form must carry 1.486 (which is $3.2808^{1/3}$) to use feet. The solver works in SI throughout and converts your entries, so an n from any handbook applies unchanged. Values worth remembering: 0.010–0.013 for smooth concrete, plastic or vitrified clay pipe, 0.014–0.017 for corrugated or old brick, 0.025–0.035 for a natural earth channel, and 0.05–0.15 for a weedy floodplain. The sanitary engineer's use of the equation is almost always to check the self-cleansing velocity — 0.6 m/s or 2 ft/s at minimum daily flow — because a sewer that runs too slow deposits solids that go septic, generate hydrogen sulfide and eat the crown of the pipe.