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Hydraulic Radius Calculator

Hydraulic radius quantifies the relationship between a channel's flow cross-section and the walls in contact with the liquid. Engineers use it to predict flow velocity, energy losses, and sediment transport in open channels and partially filled pipes. This calculator handles five common channel geometries: full pipes, partially filled pipes, rectangles, trapezoids, and triangles—essential for drainage design, river modelling, and culvert analysis.

Last updated:

Creators Davide Borchia, PhD
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What Is Hydraulic Radius?

Hydraulic radius is a geometric property that describes how efficiently a channel conveys flow. It is the ratio of the cross-sectional area of flowing liquid to the wetted perimeter—the length of channel boundary actually touching the water.

Consider a rectangular channel carrying water. The cross-sectional area equals width multiplied by depth. The wetted perimeter includes the bottom and both sides in contact with water, but not the free surface. A larger hydraulic radius means a given flow volume encounters less friction with the channel walls, resulting in faster velocity and lower energy loss.

This concept applies equally to pipes (whether full or partially filled) and open channels of any shape. Hydraulic radius is fundamental to Manning's equation and other flow models used in civil and environmental engineering.

Hydraulic Radius Formula

All channel shapes follow the same core relationship:

Hydraulic Radius (R) = Cross-sectional Area (A) ÷ Wetted Perimeter (P)

For specific geometries:

Rectangle: R = (b × y) ÷ (b + 2y)

Full Pipe: R = r ÷ 2

Partial Pipe: R = [r² ÷ 2 × (θ − sin(θ))] ÷ (r × θ)

Trapezoid: R = [y × (B + b) ÷ 2] ÷ [b + 2y√(1 + z²)]

Triangle: R = [y × B ÷ 2] ÷ [2y√(1 + z²)]

  • A — Cross-sectional area of the flowing liquid
  • P — Wetted perimeter—length of channel boundary in contact with liquid
  • b — Bottom width (rectangular or trapezoidal channel)
  • y — Water depth
  • r — Pipe radius
  • θ — Central angle (in radians) subtended by the liquid surface in a partial pipe
  • z — Side slope ratio (horizontal distance per unit vertical rise)

Calculating Wetted Perimeter for Different Shapes

Wetted perimeter depends entirely on channel geometry and water depth. In a full circular pipe, the entire circumference contacts water: P = 2πr. In a partially filled pipe, only the arc touching liquid counts, calculated from the central angle subtended by the water surface.

For rectangular channels, the wetted perimeter is straightforward: sum the bottom width and both sides submerged. Trapezoidal and triangular channels require accounting for sloped side walls, which increases perimeter compared to their vertical height alone.

The key distinction is that wetted perimeter never includes the free surface (the top boundary where water meets air). This is why a river's wetted perimeter excludes the water-air interface, even though the total width might be much larger.

Channel Geometries and Their Applications

  • Rectangular channels: Found in engineered canals, concrete-lined ditches, and laboratory flumes. Simple geometry makes calculations straightforward and design intuitive.
  • Trapezoidal channels: Common in natural rivers and earthen canals. The sloped banks (defined by slope ratio z) increase stability and reduce erosion compared to vertical walls.
  • Triangular channels: Appear in small mountain streams, V-shaped erosion gullies, and some specialized hydraulic structures. Symmetric triangles are rare; asymmetric variants require separate left and right slope ratios.
  • Full pipes: Sewer systems, pressurized water mains, and closed conduits. The hydraulic radius simplifies to exactly half the pipe radius, independent of diameter.
  • Partially filled pipes: Storm drains and gravity sewers operating below capacity. Wetted perimeter increases nonlinearly as depth rises, causing hydraulic radius to peak near 93% full before decreasing slightly.

Common Pitfalls and Practical Considerations

Accurate hydraulic calculations require careful attention to geometry definition and unit consistency.

  1. Slope ratio vs. angle — The slope parameter z is the horizontal distance per unit vertical rise, not the angle in degrees. A 45° slope corresponds to z = 1; a gentler 30° slope is z ≈ 1.73. Confusing these will produce wrong wetted perimeter and hydraulic radius by 20–40%.
  2. Free surface exclusion — The most common mistake is including the water surface width in the wetted perimeter. For a 5 m wide rectangular channel, the free surface is not part of the perimeter calculation—only the bottom (5 m) and two sides (2y) count. Open-channel calculators must account for this; pressure pipe models do not.
  3. Unit consistency — Hydraulic radius inherits the units of your input dimensions. If width is in feet and depth in metres, results are meaningless. Always verify all inputs share the same unit system before proceeding with Manning's equation or Darcy-Weisbach calculations.
  4. Partial pipe central angle — For partially filled pipes, the central angle θ must be computed correctly from the fill depth. When h equals r (half-pipe), θ ≈ π radians (180°), giving wetted perimeter πr and hydraulic radius r/4. Errors in this angle compound into severely incorrect flow predictions.

Frequently Asked Questions

Why is hydraulic radius important in pipe flow design?

Hydraulic radius directly controls flow resistance. Larger hydraulic radius means lower friction per unit cross-section, enabling faster flow or shallower channel gradients to achieve the same velocity. In sewerage and drainage design, maximizing hydraulic radius within cost constraints is key to reducing excavation depth and pipe diameter, saving capital expense while maintaining adequate flow rates.

What is the relationship between hydraulic radius and pipe diameter?

For a full circular pipe, hydraulic radius equals exactly half the radius, or one-quarter the diameter (R = D/4). This simple relationship breaks down for partially filled pipes, where hydraulic radius depends on both the pipe radius and the fill depth. Interestingly, the hydraulic radius of a half-filled pipe is not simply half the full-pipe value; it is actually smaller, illustrating the nonlinear effect of water depth.

How do you account for non-rectangular channel cross-sections?

Each shape has a distinct area and perimeter formula that must be applied before dividing to obtain hydraulic radius. Trapezoidal channels include a slope term (z) that affects both the top width and the length of sloped banks. Triangular channels reduce to simple proportions of height and base width. For natural channels with irregular shapes, hydraulic radius can be estimated by surveying the actual cross-section, planimetering its area, and measuring the wetted boundary length directly.

When does hydraulic radius peak for a partially filled pipe?

As a pipe fills from empty to full, the hydraulic radius initially increases because area grows faster than wetted perimeter. However, near maximum capacity (approximately 93% full), the hydraulic radius reaches a peak and then decreases slightly as the pipe fills completely. This counterintuitive dip occurs because the wetted perimeter grows more rapidly in the final stage, reducing the area-to-perimeter ratio despite increasing volume.

How is slope ratio defined in trapezoidal and triangular channels?

Slope ratio z is the horizontal distance per unit vertical rise. A z of 1 means the side recedes 1 unit horizontally for every 1 unit vertically, equivalent to a 45° angle. A z of 0.5 represents a steeper bank (63° angle), while z of 2 is gentler (27° angle). This convention is standard in hydraulic engineering; it is not degrees or the tangent of an angle.

Can hydraulic radius be used for sediment transport calculations?

Yes. Hydraulic radius is an input to sediment transport models and scour prediction equations. Larger hydraulic radius typically indicates higher flow capacity and potential scour; smaller values in narrow, shallow channels may favour deposition. Combining hydraulic radius with flow velocity and water temperature allows engineers to assess whether a channel will erode, remain stable, or accumulate sediment over time.

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