Library / Hydraulics, open channels and fluid power / HYD-21
Hydraulics, open channels and fluid power
Hydraulic radius of a trapezoidal channel
Flow area divided by wetted perimeter for a trapezoidal channel with bed width b, depth y and side slope z (horizontal run per unit vertical rise on each bank).
- Ref
- HYD-21
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- Rh [m]
The hydraulic radius is the flow area divided by the wetted perimeter. For a trapezoid with bed width b, depth y and side slope z (horizontal per unit vertical), Rh = y(b + zy)/[b + 2y√(1 + z²)]. The wetted perimeter is the bed plus the two sloping banks and excludes the free surface. Manning’s and Chezy’s equations use Rh to account for the shape of the section.
Flow area divided by wetted perimeter for a trapezoidal channel with bed width b, depth y and side slope z (horizontal run per unit vertical rise on each bank). The wetted perimeter excludes the free surface.
Calculator
- Flow area, y(b + zy)4.75 m²
- Wetted perimeter, b + 2y√(1 + z²)6.5 m
- Top width, b + 2zy5.5 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A section 4 m wide with 1 m of water has an area of 4.75 m² and a wetted perimeter of 6.5 m, so Rh = 0.7308 m.
- Rh is always less than the depth, and approaches it only for a very wide channel.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| y Flow depth | 0.05 to 10 m | 0.04894 | 3.966 | 3.917 | +10% | +7.6 % |
| b Bed width | 0 to 100 m | 0.3 | 0.9829 | 0.6829 | +10% | +2.14 % |
| z Side slope (horizontal per unit vertical) | 0 to 5 | 0.6339 | 0.7322 | 0.09834 | +10% | +0.149 % |
Sweep of Flow depth
| Flow depth (m) | Hydraulic radius (m) |
|---|---|
| 0.05 | 0.04894 |
| 1.294 | 0.8889 |
| 2.537 | 1.448 |
| 3.781 | 1.921 |
| 5.025 | 2.357 |
| 6.269 | 2.773 |
| 7.512 | 3.177 |
| 8.756 | 3.574 |
| 10 | 3.966 |
Sweep one variable
Variables and how each one contributes
Hydraulic radius
What the equation gives you. Everything else feeds it.
Bed width
The bed width, in m. A wider bed raises the hydraulic radius, towards the depth. Raising it by 10 % (to 4.4) moves the result by +2.14 %, from 0.7308 to 0.7464 m.
Flow depth
The flow depth, in m. A greater depth raises the hydraulic radius. Raising it by 10 % (to 1.1) moves the result by +7.6 %, from 0.7308 to 0.7863 m.
Side slope (horizontal per unit vertical)
The side slope: horizontal run per unit vertical rise on each bank. Flatter banks add area at the top but also perimeter, so the radius changes slowly. Raising it by 10 % (to 0.825) moves the result by +0.149 %, from 0.7308 to 0.7319 m.
Worked example with the values above
- Area: y(b + zy) = 1 × (4 + 0.75 × 1) = 4.75 m².
- Wetted perimeter: 4 + 2 × 1 × √(1 + 0.75²) = 6.5 m.
- Rh = 4.75 / 6.5 = 0.73077 m.
Assumptions and limits
- Prismatic channels with a steady, nearly uniform flow, in SI units.
- A free surface at atmospheric pressure and a roughness that is the same all round the wetted perimeter.
Common mistakes
- Including the free surface in the wetted perimeter.
- Using the Manning formula with n in US units.
- Using the wide-channel result for a narrow channel, where the hydraulic radius is well below the depth.
Statistics and uncertainty
Manning’s n is the main source of uncertainty and is commonly known to within 10 % to 30 %. Depth and flow follow a power of 3/5 in the normal-depth formula, so a 10 % error in n gives about a 6 % error in depth.
Where it is used
- Designing canals, drains, culverts and flumes.
- Estimating flood depth and checking bed shear for scour.
Maths you need
- Wetted perimeter: the length of channel boundary in contact with the water
- Manning’s equation: V = R^(2/3) S^(1/2) / n
- Hydraulic radius: area divided by wetted perimeter