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Hydraulics, open channels and fluid power
Surge-tank oscillation period
Period of small undamped oscillations of an open surge tank of plan area As connected to a reservoir by a conduit of length L and cross-section Ap, with a rigid water column and fixed downstream discharge.
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- Hydraulics, open channels and fluid power
- Result
- T [s]
A surge tank of plan area As connected to a reservoir by a conduit of length L and cross-section Ap forms a mass–spring system: the water column in the conduit is the mass, and the tank the spring. For small undamped oscillations about the steady level, the period is T = 2π√[L As/(g Ap)]. A larger tank or a longer conduit gives a slower oscillation. Friction damps the oscillation in practice.
Period of small undamped oscillations of an open surge tank of plan area As connected to a reservoir by a conduit of length L and cross-section Ap, with a rigid water column and fixed downstream discharge.
Calculator
- Frequency, 1/T0.02492 Hz
- Natural angular frequency, √(g Ap/(L As))0.1566 rad/s
- Period if the tank area doubles56.74 s
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.
- The water level oscillates with a period of 40.12 s (0.669 min), about 0.0249 cycles per second.
- Doubling the tank area lengthens the period to 56.74 s, which makes it slower to react but gives it more capacity.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| As Surge-tank plan area | 0.1 to 1,000 m² | 6.344 | 634.4 | 628 | +10% | +4.88 % |
| L Conduit length | 1 to 20,000 m | 4.012 | 567.4 | 563.4 | +10% | +4.88 % |
| Ap Conduit cross-section | 0.01 to 100 m² | 4.012 | 401.2 | 397.2 | +10% | −4.65 % |
| g Gravitational acceleration | 1 to 30 m/s² | 22.94 | 125.7 | 102.7 | +10% | −4.65 % |
Sweep of Surge-tank plan area
| Surge-tank plan area (m²) | Oscillation period (s) |
|---|---|
| 0.1 | 6.344 |
| 0.3162 | 11.28 |
| 1 | 20.06 |
| 3.162 | 35.67 |
| 10 | 63.44 |
| 31.62 | 112.8 |
| 100 | 200.6 |
| 316.2 | 356.7 |
| 1,000 | 634.4 |
Sweep one variable
Variables and how each one contributes
Oscillation period
What the equation gives you. Everything else feeds it.
Conduit length
The conduit length, in m. Raises the period with the square root of the length. Raising it by 10 % (to 110) moves the result by +4.88 %, from 40.12 to 42.08 s.
Surge-tank plan area
The plan area of the surge tank, in m². Raises the period with the square root of the area. Raising it by 10 % (to 4.4) moves the result by +4.88 %, from 40.12 to 42.08 s.
Gravitational acceleration
The gravitational acceleration, in m/s². Lowers the period with a square-root dependence. Raising it by 10 % (to 10.79) moves the result by −4.65 %, from 40.12 to 38.25 s.
Conduit cross-section
The conduit cross-sectional area, in m². A larger conduit lowers the period with a square-root dependence. Raising it by 10 % (to 1.1) moves the result by −4.65 %, from 40.12 to 38.25 s.
Worked example with the values above
- Group: L As/(g Ap) = 100 × 4 / (9.81 × 1) = 40.775 s².
- Square root: 6.3855 s.
- T = 2π × that = 40.121 s.
Assumptions and limits
- A single-phase liquid in a uniform line, with a linear elastic wall and the wave travelling without loss.
- Rapid closure, with the valve moving in less than the wave return time.
Common mistakes
- Using the closure time in place of the wave return time to decide whether a closure is rapid.
- Using the sound speed in free liquid, which is above the speed in a pipe.
- Using gauge pressure where absolute pressure is required for the gas law.
Statistics and uncertainty
Wave speed is sensitive to trapped air and pipe support, so surge predictions can be off by large factors. The first Joukowsky peak is usually the largest, but reflections and column separation can create bigger ones.
Where it is used
- Checking pipelines and pump stations for water hammer.
- Sizing surge tanks and gas-charged accumulators.
Maths you need
- Water hammer: a pressure wave from a sudden velocity change
- Bulk modulus: B, the stiffness of the liquid
- Polytropic process: p Vⁿ = constant