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The Hydraulic Radius Calculator computes the cross-sectional area, wetted perimeter, and hydraulic radius for three common open-channel shapes: rectangular, circular (partially filled pipe), and trapezoidal. The hydraulic radius is a critical parameter that appears in virtually every open-channel flow equation, including Manning's, Chezy's, and the Darcy-Weisbach formula adapted for open channels.
Defined as the ratio of flow area to wetted perimeter: $$R_h = \frac{A}{P}$$
the hydraulic radius represents the effective depth that determines the balance between gravitational driving force and frictional resistance. A larger R_h means a more hydraulically efficient cross-section — more area moving water relative to the boundary resisting it. This concept is central to channel design, sewer engineering, irrigation planning, and flood analysis.
For a rectangular channel: A = by, P = b + 2y. For a trapezoidal channel with side slope z:1: A = (b + zy)y, P = b + 2y√(1 + z²). For a partially filled circular pipe: the area and perimeter depend on the central angle θ = 2·arccos((r − y)/r).
The hydraulic radius depends on the channel geometry:
Rectangular channel (width b, depth y): $$A = b \cdot y, \quad P = b + 2y, \quad R_h = \frac{by}{b + 2y}$$
Trapezoidal channel (bottom width b, side slope z:1, depth y): $$A = (b + zy)y, \quad P = b + 2y\sqrt{1+z^2}, \quad R_h = \frac{(b+zy)y}{b + 2y\sqrt{1+z^2}}$$
Circular pipe (diameter d, flow depth y): The central angle subtended by the water surface is $$\theta = 2\arccos\!\left(\frac{r-y}{r}\right)$$ where r = d/2. Then: $$A = \frac{r^2}{2}(\theta - \sin\theta), \quad P = r\theta, \quad R_h = \frac{r}{2}\left(1 - \frac{\sin\theta}{\theta}\right)$$
A higher hydraulic radius indicates a more efficient channel cross-section. For a given area, the shape that minimizes the wetted perimeter (and thus maximizes R_h) is the semicircle. Among practical shapes, a half-hexagon (trapezoid with 60° sides) is optimal. Increasing depth relative to width generally improves efficiency until the channel becomes too narrow. For circular pipes, maximum R_h occurs at approximately 81% fill depth — not when full.
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A 3 m wide rectangular channel with 1.5 m water depth: A = 4.5 m², P = 6 m, Rh = 0.75 m.
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A trapezoidal channel (b = 2 m, z = 2, y = 1 m): A = 4 m², P ≈ 6.47 m, Rh ≈ 0.618 m. The sloped sides add to the wetted perimeter.
The hydraulic radius Rh = A/P is the ratio of the cross-sectional flow area to the wetted perimeter. It characterizes the efficiency of a channel cross-section: a larger Rh means less boundary friction per unit of flowing water, resulting in higher velocity for the same slope and roughness.
No, though they are related. The average (hydraulic mean) depth is A/T where T is the top width. The hydraulic radius is A/P where P is the wetted perimeter. For a very wide, shallow channel, Rh ≈ depth, but for deeper or narrower channels they differ significantly.
The semicircle maximizes Rh for a given area. Among practical shapes: the best rectangle has depth = width/2; the best trapezoid has a 60° side angle (z = 1/√3 ≈ 0.577). These shapes minimize wetted perimeter and construction cost for a given flow capacity.
As fill depth increases beyond about 81% of the diameter, the wetted perimeter grows faster than the area because the pipe walls curve inward. The incremental perimeter per unit of added area increases, reducing the ratio A/P. This is why sewers are sometimes designed to flow at less than full capacity.
The side slope z:1 means the channel wall runs z units horizontally for every 1 unit vertically. So z = 1.5 means 1.5 horizontal to 1 vertical (about 34° from horizontal). Steeper slopes have smaller z values. z = 0 would be a vertical wall (rectangular channel).
Yes. For a full circular pipe, Rh = D/4. This allows the Darcy-Weisbach equation to be written in terms of Rh, creating a unified framework for both open-channel and pressurized flow analysis.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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