Hydraulic Radius Calculator

The hydraulic radius is the ratio of flow area A to wetted perimeter P, and it is best understood as a force balance rather than a geometric curiosity. For steady uniform flow down a slope S, the downslope component of the water weight on a reach of length L is rho x g x A x L x S, and it is resisted by boundary shear acting over the contact area P x L. Setting the two equal gives the mean boundary shear stress tau = rho x g x R x S, so R is exactly the quantity that converts channel shape into drag per unit area of bed. That is why it, and not depth or width alone, appears in resistance formulas. A wide shallow channel whose width exceeds about 20 times its depth has R approximately equal to the flow depth. A full circular pipe has R = D/4 regardless of size, so a 0.6 m sewer gives R = 0.15 m. Trapezoidal irrigation canals are commonly designed with R between 0.5 and 2 m, small upland streams sit near 0.15 to 0.4 m, and large lowland rivers reach 2 to 5 m. Two caveats matter. In a partly full circular pipe R does not rise monotonically with depth: it peaks near 81 percent of full depth, which is why a sewer running just below the crown carries a higher velocity than one running full. And R loses meaning in compound sections where a deep main channel adjoins a shallow rough floodplain; those must be subdivided and computed separately, or a single averaged R will overstate conveyance.

The ratio of area to wetted perimeter entered engineering as the hydraulic mean depth in the work of Antoine de Chezy, the French engineer commissioned in 1769 to study the proposed Yvette canal for supplying Paris with water. Chezy compared the Courpalet canal with the Seine and concluded that velocity varied with the square root of the product of slope and this ratio, giving V = C x sqrt(R x S). His manuscript circulated little and was largely unknown until Clemens Herschel translated and republished it in 1897. The nineteenth century was then spent trying to pin down the coefficient C. Henry Darcy and Henri Bazin ran extensive pipe and canal experiments in France, published in 1865, showing that C depended on both boundary roughness and on R itself. Ganguillet and Kutter in Switzerland produced a cumbersome empirical fit in 1869 that introduced the roughness coefficient n. Robert Manning, an engineer with Ireland's Office of Public Works, presented in 1889 a far simpler power-law alternative in which R appears raised to the two-thirds power; it was printed in the Transactions of the Institution of Civil Engineers of Ireland in 1891. Ven Te Chow's 1959 textbook Open Channel Hydraulics tabulated the geometric expressions for trapezoidal, circular, and irregular sections still used today, and the same A/P ratio was later generalised as the hydraulic diameter 4R used in pipe friction work.