Hydraulic radius and wetted perimeter explained
Understand hydraulic radius, wetted perimeter and hydraulic diameter, why pipes give d/4, and how the value drives Manning and pipe friction work.
Area, perimeter and the ratio between them
Flow through a channel or pipe is governed by two geometric quantities. The flow area is the cross-section the fluid actually occupies, and the wetted perimeter is the part of the boundary the fluid touches and drags against. Hydraulic radius is simply the first divided by the second, Rh = A / P. Because friction acts along the wetted boundary while flow is carried through the whole area, a shape that packs a large area behind a short perimeter moves water efficiently.
Why a full pipe collapses to d/4
The circular case is worth memorising because it appears everywhere. A pipe of diameter d flowing full has area pi times (d/2) squared and wetted perimeter pi times d, the full circumference. Dividing one by the other cancels pi and one factor of d, leaving Rh equal to d over 4. So a 200 mm pipe has a hydraulic radius of 50 mm, and doubling the diameter doubles the hydraulic radius. This clean result is why round pipes are the reference against which other shapes are compared.
Open channels versus closed ducts
A rectangular shape can be wetted in two different ways, and the distinction changes the perimeter. In an open channel the water has a free surface at the top, so only the bottom and the two sides are wetted, giving a perimeter of width plus two depths. A rectangular duct running completely full is sealed on all sides, so its perimeter is twice the sum of width and depth. Both share the same area of width times depth, which means the open channel, with its shorter wetted perimeter, has the larger hydraulic radius.
Feeding Manning and pipe friction
Hydraulic radius is rarely the final answer; it is an input to flow formulas. In open-channel design Manning's equation uses Rh raised to the two-thirds power together with the channel slope and a roughness coefficient to predict velocity and discharge. In pressurised pipe work the hydraulic diameter, defined as four times the hydraulic radius, lets non-circular ducts reuse the friction relationships developed for round pipes. Getting the hydraulic radius right is therefore the first step in sizing sewers, storm drains, irrigation canals and ventilation ducts.