Library / Hydraulics, open channels and fluid power / HYD-05
Hydraulics, open channels and fluid power
Free-surface elevation change in a rotating liquid
Height difference between two radial positions on the free surface of a liquid rotating as a rigid body about a vertical axis at angular speed ω.
- Ref
- HYD-05
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- z2 - z1 [m]
Liquid rotating as a rigid body takes a parabolic free surface. The height at radius r above the axis level is ω² r²/(2g), so two radial positions differ by ω²(r₂² − r₁²)/(2g). It rises with the square of the speed and the square of the radius. Pressure above the surface must be uniform and the rotation must be steady.
Height difference between two radial positions on the free surface of a liquid rotating as a rigid body about a vertical axis at angular speed ω. The surface is a paraboloid and gas pressure above it is uniform.
Calculator
- Rim speed at the outer radius, ω r₂1.2 m/s
- Height of the outer position above the axis level, ω² r₂²/(2g)0.07339 m
- Speed in rpm, 60ω/(2π)38.2 rpm
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.
- At 4 rad/s (38.2 rpm) the surface at r = 0.3 m stands 0.06524 m above that at r = 0.1 m.
- Doubling the speed quadruples the rise, 0.261 m, so fast spin-up makes a deep depression at the centre.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ω Angular speed | 0.1 to 500 rad/s | 4.077e-5 | 1,019 | 1,019 | +10% | +21 % |
| r1 Inner radius | 0 to 5 m | −20.31 | 0.07339 | 20.39 | +10% | −2.62 % |
| r2 Outer radius | 0.01 to 5 m | −0.008073 | 20.38 | 20.39 | +10% | +23.6 % |
| g Gravitational acceleration | 1 to 30 m/s² | 0.02133 | 0.64 | 0.6187 | +10% | −9.09 % |
Sweep of Angular speed
| Angular speed (rad/s) | Surface elevation change (m) |
|---|---|
| 0.1 | 4.077e-5 |
| 0.29 | 3.429e-4 |
| 0.8409 | 0.002883 |
| 2.438 | 0.02424 |
| 7.071 | 0.2039 |
| 20.5 | 1.714 |
| 59.46 | 14.42 |
| 172.4 | 121.2 |
| 500 | 1,019 |
Sweep one variable
Variables and how each one contributes
Surface elevation change
What the equation gives you. Everything else feeds it.
Angular speed
The angular speed of the rotation, in rad/s. Raises the elevation change with the square of the speed. Raising it by 10 % (to 4.4) moves the result by +21 %, from 0.06524 to 0.07894 m.
Inner radius
The inner radial position on the free surface, in m. A larger inner radius reduces the difference between the two positions. Raising it by 10 % (to 0.11) moves the result by −2.62 %, from 0.06524 to 0.06353 m.
Outer radius
The outer radial position on the free surface, in m. A larger outer radius raises the elevation change with its square. Raising it by 10 % (to 0.33) moves the result by +23.6 %, from 0.06524 to 0.08065 m.
Gravitational acceleration
The gravitational acceleration, in m/s². Lowers the elevation change in inverse proportion. Raising it by 10 % (to 10.79) moves the result by −9.09 %, from 0.06524 to 0.05931 m.
Worked example with the values above
- Square terms: r₂² − r₁² = 0.09 − 0.01 = 0.08 m².
- ω² = 16 rad²/s².
- Δz = 16 × 0.08 / (2 × 9.81) = 0.06524 m.
Assumptions and limits
- A liquid at rest with constant density and atmospheric pressure at the free surface.
- Gauge pressure is used, with atmospheric pressure acting on the other side of the surface.
Common mistakes
- Measuring depth from the wrong datum, instead of from the free surface to the centroid.
- Using the plate length where the vertical depth is needed.
- Using absolute pressure, which adds a force that atmospheric pressure on the other side cancels.
Statistics and uncertainty
Hydrostatic results depend on the density and on a depth that is measured from the free surface, so a 1 % error in depth gives about a 1 % error in force and a larger one in moments, which go with higher powers of depth.
Where it is used
- Sizing gates, hatches, windows and tank walls.
- Reading pressure differences from manometers and finding the surface shape of rotating liquids.
Maths you need
- Hydrostatic pressure: p = ρ g h, increasing linearly with depth
- Centroid: the geometric centre of an area
- Second moment of area: I, in m⁴, about an axis through the centroid