Library / Hydraulics, open channels and fluid power / HYD-48
Hydraulics, open channels and fluid power
Valve-to-reservoir wave return time
Time for a pressure wave to travel from a valve to the reservoir and back along a single uniform line, from the line length L and wave speed a.
- Ref
- HYD-48
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- tr [s]
A pressure wave generated at a valve travels the line length L to a constant-head reservoir, reflects and returns, taking tr = 2L/a. A closure that takes less than tr is rapid, and the full Joukowsky pressure acts. A slower closure lets reflected waves cancel part of the rise. It assumes a single uniform line with no branches, and a constant wave speed.
Time for a pressure wave to travel from a valve to the reservoir and back along a single uniform line, from the line length L and wave speed a. It separates rapid from slow closure.
Calculator
- One-way travel time, L/a0.75 s
- Fundamental period of the line, 4L/a3 s
- Round trips in 10 s6.667
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 wave takes 1.5 s to go to the reservoir and back along 750 m of pipe at 1,000 m/s.
- A valve closing in under 1.5 s is a rapid closure; a closure time of ten times this keeps the surge to about a tenth of the Joukowsky value.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Line length | 1 to 20,000 m | 0.002 | 40 | 40 | +10% | +10 % |
| a Wave speed | 100 to 1,500 m/s | 1 | 15 | 14 | +10% | −9.09 % |
Sweep of Line length
| Line length (m) | Round-trip time (s) |
|---|---|
| 1 | 0.002 |
| 3.448 | 0.006897 |
| 11.89 | 0.02378 |
| 41.01 | 0.08202 |
| 141.4 | 0.2828 |
| 487.7 | 0.9754 |
| 1,682 | 3.364 |
| 5,800 | 11.6 |
| 20,000 | 40 |
Sweep one variable
Variables and how each one contributes
Round-trip time
What the equation gives you. Everything else feeds it.
Line length
The length of line between the valve and the reservoir, in m. Raises the return time in direct proportion. Raising it by 10 % (to 825) moves the result by +10 %, from 1.5 to 1.65 s.
Wave speed
The pressure-wave speed, in m/s. A faster wave shortens the return time in inverse proportion. Raising it by 10 % (to 1,100) moves the result by −9.09 %, from 1.5 to 1.364 s.
Worked example with the values above
- Round-trip distance: 2L = 1,500 m.
- Divide by the wave speed 1,000 m/s.
- tr = 1.5 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