Library / Hydraulics, open channels and fluid power / HYD-46
Hydraulics, open channels and fluid power
Rapid-closure pressure surge
Pressure rise from the Joukowsky equation when the flow speed at a downstream valve drops quickly from U₀ to U₁ in a line with pressure-wave speed a.
- Ref
- HYD-46
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- Δp [Pa]
When the flow at a downstream valve is stopped or slowed quickly, a pressure wave travels back up the line. The first wave raises the pressure by the Joukowsky amount Δp = ρ a (U₀ − U₁), with a the pressure-wave speed and U the signed flow speeds before and after. It holds for closure faster than the wave return time. Reflections from the reservoir can later give higher or lower pressures.
Pressure rise from the Joukowsky equation when the flow speed at a downstream valve drops quickly from U₀ to U₁ in a line with pressure-wave speed a. Later reflections can produce larger peaks.
Calculator
- Surge in bar15 bar
- Equivalent head rise, a(U₀ − U₁)/9.81152.9 m
- Speed change, U₀ − U₁1.5 m/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.
- Slowing the flow from 2 to 0.5 m/s raises the pressure by 1,500,000 Pa (15 bar), equal to 152.9 m of water.
- Even a modest speed change of 1 m/s in a steel pipe gives about 10 bar, so valves on long pipelines must close slowly.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| U0 Flow speed before the change | −10 to 10 m/s | −1.05e7 | 9,500,000 | 2e7 | +0.4 m/s | +26.7 % |
| U1 Flow speed after the change | −10 to 10 m/s | −8,000,000 | 1.2e7 | 2e7 | +0.4 m/s | −26.7 % |
| ρ Liquid density | 500 to 2,000 kg/m³ | 750,000 | 3,000,000 | 2,250,000 | +10% | +10 % |
| a Pressure-wave speed | 100 to 1,500 m/s | 150,000 | 2,250,000 | 2,100,000 | +10% | +10 % |
Sweep of Flow speed after the change
| Flow speed after the change (m/s) | Pressure surge (Pa) |
|---|---|
| −10 | 1.2e7 |
| −7.5 | 9,500,000 |
| −5 | 7,000,000 |
| −2.5 | 4,500,000 |
| 0 | 2,000,000 |
| 2.5 | −500,000 |
| 5 | −3,000,000 |
| 7.5 | −5,500,000 |
| 10 | −8,000,000 |
Sweep one variable
Variables and how each one contributes
Pressure surge
What the equation gives you. Everything else feeds it.
Liquid density
The liquid density, in kg/m³. Raises the surge in direct proportion. Raising it by 10 % (to 1,100) moves the result by +10 %, from 1,500,000 to 1,650,000 Pa.
Pressure-wave speed
The pressure-wave speed in the line, in m/s. Raises the surge in direct proportion. Raising it by 10 % (to 1,100) moves the result by +10 %, from 1,500,000 to 1,650,000 Pa.
Flow speed before the change
The flow speed before the change, in m/s. Raises the surge in direct proportion. Raising it by 10 % (to 2.2) moves the result by +13.3 %, from 1,500,000 to 1,700,000 Pa.
Flow speed after the change
The flow speed after the change, in m/s. A higher remaining speed means less speed change, so the surge falls. Raising it by 10 % (to 0.55) moves the result by −3.33 %, from 1,500,000 to 1,450,000 Pa.
Worked example with the values above
- Speed change: U₀ − U₁ = 2 − 0.5 = 1.5 m/s.
- ρ a = 1,000 × 1,000 = 1,000,000 kg/(m²·s).
- Δp = 1,500,000 Pa.
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