Hydraulic Radius Equation Calculator

Hydraulic radius: water area A divided by wetted perimeter P
Hydraulic radius equals area divided by wetted perimeter

Solution

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Hydraulic Radius (Open Channel)

The hydraulic radius is the flow cross-sectional area divided by the wetted perimeter. It measures how efficiently a channel shape conveys water and appears in Manning’s and Chezy’s equations.

Rₕ = A / Pₘ

Pipe Hydraulic Radius (Circular Pipe)

For a partially filled circular pipe, the hydraulic radius is computed from the flow depth and pipe radius using geometry of circular segments. For a full pipe of diameter D, Rₕ = D/4.

Rₕ = A / Pₘ (circular pipe)

Froude Number

The Froude number classifies open-channel flow as subcritical (Fr < 1), critical (Fr = 1), or supercritical (Fr > 1). It is the ratio of flow velocity to the speed of a shallow-water gravity wave.

Fr = v / √(g × hₘ)

Mean Depth

Mean depth is the cross-sectional flow area divided by the top water surface width. It is used in the Froude number calculation and in energy equation analysis of open-channel flow.

hₘ = A / T

How It Works

This calculator handles two key open-channel parameters. The hydraulic radius (Rₕ = A/Pₘ) measures channel efficiency by dividing the flow area by the wetted perimeter. The Froude number (Fr = v/√(ghₘ)) classifies flow as subcritical (Fr < 1), critical, or supercritical (Fr > 1).

Example Problem

A rectangular channel is 3 m wide with a flow depth of 1 m. What is the hydraulic radius?

  1. Identify the knowns. Rectangular channel width b = 3 m and flow depth y = 1 m. The water surface forms a free boundary at the top and is not part of the wetted perimeter.
  2. Identify what we're solving for. We want the hydraulic radius Rₕ — the flow cross-sectional area divided by the wetted perimeter, in meters.
  3. Compute the cross-sectional flow area for a rectangle: A = b × y = 3 m × 1 m = **3 m²**.
  4. Compute the wetted perimeter — the length of channel boundary in contact with the fluid (one bottom plus two side walls, excluding the free surface): Pₘ = b + 2y = 3 + 2(1) = **5 m**.
  5. Write the hydraulic radius formula and substitute: Rₕ = A / Pₘ = 3 m² / 5 m.
  6. Simplify the arithmetic to get the result: Rₕ = **0.6 m** — note that a semicircular channel of the same area would have a higher Rₕ and thus less friction per unit of flow.

When to Use Each Variable

  • Solve for Hydraulic Radius — when you know the flow area and wetted perimeter, e.g., preparing inputs for Manning's equation to find flow velocity.
  • Solve for Cross-Sectional Area — when you know the hydraulic radius and wetted perimeter, e.g., determining the flow area from measured channel properties.
  • Solve for Wetted Perimeter — when you know the area and hydraulic radius, e.g., back-calculating the contact surface from flow measurements.
  • Solve for Pipe Hydraulic Radius — when working with a partially filled circular pipe, e.g., finding Rh for a sewer pipe at a given flow depth.
  • Solve for Froude Number — when you know velocity and mean depth, e.g., classifying open-channel flow as subcritical or supercritical.
  • Solve for Mean Depth — when you know the flow area and top width, e.g., preparing inputs for the Froude number calculation.

Key Concepts

The hydraulic radius (Rh = A/Pw) is the ratio of flow cross-sectional area to wetted perimeter. It measures how efficiently a channel shape conveys water — a higher Rh means less friction per unit of flow area. The Froude number (Fr = v/√(g·hm)) classifies open-channel flow: Fr < 1 is subcritical (slow, deep), Fr = 1 is critical, and Fr > 1 is supercritical (fast, shallow). These parameters are fundamental to Manning's equation and energy analysis in open channels.

Applications

  • Channel design: optimizing cross-section shape for maximum flow at minimum excavation cost
  • Storm sewer design: calculating flow capacity of partially filled circular pipes
  • Dam spillway analysis: using the Froude number to predict hydraulic jump location and energy dissipation
  • River engineering: classifying flow conditions for flood modeling and bridge scour analysis

Common Mistakes

  • Confusing hydraulic radius with the physical radius of a pipe — for a full pipe, Rh = D/4, not D/2
  • Including the free water surface in the wetted perimeter — only surfaces in contact with the channel boundary count
  • Using hydraulic depth instead of mean depth for the Froude number — mean depth is A/T (area over top width)
  • Assuming a rectangular approximation for irregular channels — measure the actual cross-section for accurate Rh and Fr

Frequently Asked Questions

What does hydraulic radius represent physically?

Rₕ is the flow cross-sectional area divided by the wetted perimeter — the length of channel boundary in contact with the fluid. It shows up in Manning's and Chezy's equations as the geometric driver of flow capacity: higher Rₕ means less drag per unit of cross-section.

Why is a semicircular channel hydraulically efficient?

For a given flow area, the semicircle minimizes the wetted perimeter, maximizing Rₕ. That's why drainage culverts, irrigation ditches with curved liners, and many designed channels approximate a half-pipe — same flow rate at a smaller energy cost or smaller channel.

What is the hydraulic radius of a circular pipe flowing full?

For a full pipe of diameter D, area is πD²/4 and the wetted perimeter is πD, so Rₕ = D/4. A 200 mm full pipe has Rₕ = 0.05 m. Partially full pipes have a different (and not always smaller) Rₕ that depends on flow depth.

How is the Froude number used in open-channel design?

Fr = v / √(g × hₘ) classifies flow regime. Fr < 1 is subcritical (slow, deep) — typical of rivers. Fr > 1 is supercritical (fast, shallow) — typical of steep flumes and downstream of spillways. Crossing Fr = 1 produces a hydraulic jump, useful for energy dissipation in stilling basins.

What is the difference between mean depth and flow depth?

Flow depth y is the vertical water height at a point. Mean depth hₘ = A / T is the cross-sectional area divided by the free-surface top width — an averaged depth used in Fr and energy calculations. For a rectangular channel they are equal; for trapezoidal or natural channels they differ.

Why isn't the free water surface part of the wetted perimeter?

Friction only develops where the fluid contacts a solid boundary. The free water surface is in contact with air, where shear stress is negligible compared to channel walls. Including it would overstate the wetted perimeter and underpredict the hydraulic radius.

When can I use Rₕ for a partially full circular pipe instead of Rₕ = D/4?

Only when the pipe runs completely full. For partial flow, area and wetted perimeter both depend on depth via circular-segment geometry — Rₕ peaks at about 0.30 m for a fully flowing 1.0 m pipe but actually reaches a slightly higher maximum near 81% depth. Use the partial-pipe formula or a pipe-flow chart.

Worked Examples

Municipal Stormwater Channel

What hydraulic radius does a 3 m wide concrete channel flowing 0.6 m deep have?

A municipal stormwater conveyance is a rectangular concrete channel 3 m wide flowing at a uniform depth of 0.6 m during a design storm. Compute the hydraulic radius R = A / P_w that feeds Manning's equation for the channel's design discharge.

  • Knowns: bottom width b = 3 m, flow depth y = 0.6 m
  • Cross-sectional area A = b × y = 3 × 0.6 = 1.8 m²
  • Wetted perimeter P_w = b + 2y = 3 + 2 × 0.6 = 4.2 m
  • R = A / P_w = 1.8 / 4.2

R ≈ 0.429 m

For rectangular channels, the deeper the flow relative to the width, the closer R approaches y/2 (a very wide flow's wetted perimeter is dominated by the floor). Hydraulic radius normalizes channel shape so Manning's n and the friction slope work across all cross-sections from triangular gutters to trapezoidal flood channels.

Sanitary Sewer Design

What is the hydraulic radius of a 1 m diameter sewer flowing half-full?

Municipal sanitary sewers are typically designed for half-full flow to leave headspace for surges and ventilation. For a circular pipe of radius r = 0.5 m (1 m inside diameter) flowing at depth d = 0.5 m, compute the hydraulic radius using the partially-filled pipe geometry.

  • Knowns: pipe radius r = 0.5 m, flow depth d = 0.5 m (half-full)
  • Wetted angle θ = 2 × arccos((r − d)/r) = 2 × arccos(0) = π radians
  • Wetted area A = (πr²) − r²(θ − sin θ)/2 = π(0.25) − 0.25(π − 0)/2 ≈ 0.393 m²
  • Wetted perimeter P_w = 2πr − rθ = π − 0.5π = 0.5π ≈ 1.571 m
  • R = A / P_w = 0.393 / 1.571

R ≈ 0.25 m (= D / 4)

It's a useful fact: a circular pipe flowing half-full has the same hydraulic radius as flowing full — both work out to D / 4. The full-flow capacity is twice the half-full capacity, but the friction-velocity term R^(2/3) in Manning is identical, so the velocity is the same in both cases.

Hydroelectric Spillway

What Froude number characterizes 4 m/s flow at 0.6 m mean depth on a spillway chute?

A hydroelectric dam spillway delivers flow down a steep chute. At one cross-section the velocity averages 4 m/s and the mean depth (area divided by top width) is 0.6 m. Compute the Froude number Fr = v / √(g × hm) to classify the flow regime — supercritical Fr > 1 means waves cannot travel upstream and the chute is governed by the upstream control.

  • Knowns: v = 4 m/s, h_m = 0.6 m, g = 9.80665 m/s²
  • Fr = v / √(g × h_m)
  • Fr = 4 / √(9.80665 × 0.6)
  • Fr = 4 / √5.884
  • Fr = 4 / 2.426

Fr ≈ 1.649 (supercritical flow)

Supercritical flow on a spillway is intentional — once Fr > 1, surges and ripples can't propagate back up the chute, so the upstream reservoir level and weir crest fully control the discharge. The transition from supercritical back to subcritical at the spillway toe is a hydraulic jump, where energy is dissipated in a turbulent roller (the apron / stilling basin is sized to contain it).

Hydraulic Radius & Open-Channel Flow Formulas

Four related equations describe the geometry and flow regime of an open channel:

Rh = A / PwHydraulic radius — flow area over wetted perimeter
Rh = D / 4Full circular pipe (special case)
hm = A / TMean depth — flow area over free-surface top width
Fr = v / √(g × hm)Froude number — classifies flow regime

Where:

  • Rh — hydraulic radius (m or ft)
  • A — flow cross-sectional area (m² or ft²)
  • Pw — wetted perimeter — length of channel boundary in contact with the fluid (m or ft); the free water surface does NOT count
  • D — pipe diameter (m), only for the full-pipe special case
  • hm — mean (hydraulic) depth (m)
  • T — top width of the free water surface (m)
  • Fr — Froude number (dimensionless); Fr < 1 subcritical, Fr = 1 critical, Fr > 1 supercritical
  • v — mean flow velocity (m/s)
  • g — gravitational acceleration (9.81 m/s² on Earth)

These geometric parameters feed directly into Manning's equation (v = (1/n) × Rh2/3 × S1/2) and into specific-energy and momentum analysis of open-channel flow. For partially filled circular pipes, Rh must be computed from circular-segment geometry rather than the D/4 shortcut.

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