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Hydraulic Radius Equation Calculator

Hydraulic radius equals area divided by wetted perimeter

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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. Area: A = 3 × 1 = 3 m²
  2. Wetted perimeter: Pₘ = 3 + 2(1) = 5 m
  3. Rₕ = 3 / 5 = 0.6 m

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 is hydraulic radius and why does it matter?

Hydraulic radius is the flow area divided by the wetted perimeter. It appears in Manning’s and Chezy’s equations and measures how efficiently a channel shape conveys water. A semicircular channel has the highest Rh for a given area.

What does the Froude number tell you?

It classifies open-channel flow regime. Fr < 1 is subcritical (slow, deep); Fr > 1 is supercritical (fast, shallow). At Fr = 1, a hydraulic jump can form, which is important for spillway and stilling basin design.

What is the hydraulic radius of a full circular pipe?

For a pipe of diameter D flowing full, Rh = D/4. A 200 mm pipe has Rh = 0.05 m.

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