Hydraulic Radius Calculator

Background.

Water moving along a channel loses energy to the bed and banks it rubs against, and its carrying capacity comes from the area it fills. Hydraulic radius is the ratio of the two: the cross-sectional area of the flow divided by the length of boundary the water actually touches. A large value means plenty of water per metre of friction surface, a compact section that flows fast for its slope. A small value means a thin layer of water spread over a lot of boundary, losing more of its energy to friction.

That is why the number sits inside the standard open-channel formulas. In Manning's equation, used for rivers, canals, culverts and storm drains, velocity rises with the hydraulic radius to the power of two-thirds, so two channels with the same area, slope and lining can carry water at quite different speeds purely because of their shape.

The name misleads people. Hydraulic radius is not the radius of anything: for a round pipe running full it works out to a quarter of the diameter, not half. For a wide, shallow river it comes close to the depth, because the banks add little wetted length compared with the bed.

Only the wetted boundary counts. In an open channel the top of the water touches air, not a wall, so the free surface is left out; in a pipe running full, the whole circumference is wetted. A culvert flowing part-full is an open channel for this purpose. Area and perimeter must also describe the same cross-section at the same water level, taken from one survey or one design depth.

The page performs the division and returns R in metres. Working out A and P from a width, depth or side slope, and the flow calculations that use R, are separate steps.

What is hydraulic radius calculator?

Hydraulic radius, written R or R_h, is the flow area A divided by the wetted perimeter P: the cross-section of moving water divided by the length of solid boundary in contact with it, both measured in the same plane across the flow. It has units of length, metres here. It sums up in one number how compact a flow section is, and it is the length scale in which friction formulas for open channels and part-full conduits are written.

How to use this calculator.

1. 01Enter the flow cross-sectional area in m²: the area of water only, at the water level you are analysing, not the full channel capacity unless it is running full.
2. 02Enter the wetted perimeter in m: the bed plus the wetted part of each side, measured along the boundary. For a rectangular channel of width b and water depth y, it is b + 2y.
3. 03Leave the free surface out of the perimeter. Count the top only when water fills a closed conduit right up to its crown.
4. 04Use the result as R in Manning's or Chézy's equation, or multiply it by 4 to get the hydraulic diameter that friction formulas for non-circular ducts expect.

The formula.

R = A ÷ P, with A in square metres and P in metres, so R comes out in metres. A few standard cases are worth knowing. A rectangular channel of width b and depth y has R = by ÷ (b + 2y); when the channel is very wide this approaches y, and for a fixed area the largest R comes from a width twice the depth. A circular pipe flowing full, or exactly half full, has R = D ÷ 4. The related hydraulic diameter is 4A ÷ P, four times R, defined so that it equals the real diameter of a full round pipe, and mixing the two up is a common source of factor-of-four errors. Because R enters Manning's equation as R^(2/3), a 10% error in R shifts the computed velocity by about 6.6%. Both inputs must be greater than zero, and they must belong to the same cross-section.

A worked example.

Frequently asked questions.

References& sources.

1. [1]FHWA HEC-22, Urban Drainage Design Manual, 4th ed., open-channel hydraulic definitions.
2. [2]USACE EM 1110-2-1601, Hydraulic Design of Flood Control Channels.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.