Calculation
Δp = ϱ · a · Δv
The full Joukowsky surge occurs only if the closing time is shorter than the reflection time 2·l/a; slower closure reduces the surge proportionally.
Joukowsky-Gleichung
Water hammer pressure surge (Joukowsky equation)
The full Joukowsky surge occurs only if the closing time is shorter than the reflection time 2·l/a; slower closure reduces the surge proportionally.
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Result
Select a target and calculate.
Understand the inputs
- Pressure surge Δp — Maximum additional pressure from the sudden velocity change.
- Density ϱ — Density of the flowing medium, about 1,000 kg/m³ for water.
- Pressure wave speed a — Speed of sound in the filled pipe; about 1,000 m/s for water in thin-walled pipes, about 1,300 m/s in thick-walled hydraulic lines.
- Velocity change Δv — Sudden change in flow velocity, usually from v1 to v2 = 0 on closure.
Example
ϱ = 1,000 kg/m³, a = 1,000 m/s and Δv = 2 m/s give Δp = 20 bar.
Assumptions and limits
Applies to a closing time shorter than the reflection time 2·l/a (ideal, instantaneous surge); pipe friction and elastic pipe-wall expansion are not modeled separately.
Understand Water hammer pressure surge (Joukowsky equation)
This calculator determines the maximum pressure surge (Joukowsky surge) from very rapid valve closure, from fluid density, pressure wave speed and the change in flow velocity.
What does this quantity describe?
When a flowing liquid in a pipe is decelerated very quickly (e.g. by rapidly closing a valve), a pressure surge Δp = ϱ·a·Δv results. The full Joukowsky surge occurs only if the closing time tS is shorter than the reflection time tR = 2·l/a; slower closure reduces the surge proportionally.
Think of a long train stopping abruptly: the faster the braking and the heavier or faster the train, the larger the shock that propagates along its entire length. In a pipeline, the pressure wave plays this role.
Formula and variables
Δp = ϱ · a · Δv
Δp = ϱ · a · Δvϱ = Δp / (a · Δv)Δv = Δp / (ϱ · a)
| Symbol / input | Meaning |
|---|---|
| Pressure surge Δp | Maximum additional pressure from the sudden velocity change. |
| Density ϱ | Density of the flowing medium, about 1,000 kg/m³ for water. |
| Pressure wave speed a | Speed of sound in the filled pipe; about 1,000 m/s for water in thin-walled pipes, about 1,300 m/s in thick-walled hydraulic lines. |
| Velocity change Δv | Sudden change in flow velocity, usually from v1 to v2 = 0 on closure. |
Choose the inputs correctly
ϱ is the density of the flowing medium, a the pressure wave (sonic) speed in the filled pipe, and Δv the sudden change in flow velocity, usually from v1 to v2 = 0 on full closure.
How to use the calculator
Enter ϱ, a and Δv to get the maximum pressure surge Δp. Also check whether the actual closing time is shorter than the reflection time 2·l/a, so the full surge actually occurs.
Worked example
ϱ = 1,000 kg/m³, a = 1,000 m/s and Δv = 2 m/s give Δp = 1,000 kg/m³ · 1,000 m/s · 2 m/s = 2,000,000 Pa = 20 bar.
Understand the result and units
Δp = 20 bar shows how even a moderate velocity change of 2 m/s at high wave speed can produce a substantial additional pressure surge — far more than a simple Bernoulli analysis would suggest.
ϱ is given in kg/m³, a and Δv in m/s; Δp results in Pa and is often expressed in bar.
Why closing time governs the magnitude of the surge
The full Joukowsky surge Δp = ϱ·a·Δv occurs only if the valve closes faster than the reflection time tR = 2·l/a — the time a pressure wave needs to travel from the valve to the nearest reflection point (a vessel or pipe junction) and back. If the closing time tS is stretched over several reflection times, pressure can partially equalize before closure completes, and the resulting surge is correspondingly lower per Δp = ϱ·a·Δv·(tR/tS). For a safe design, the valve's actual closing characteristic must therefore always be compared against the line's reflection time.
Typical applications
The Joukowsky equation is used to estimate the maximum pressure surge from rapid valve closure, pump trip or check-valve slam in water, oil and hydraulic lines, to adequately pressure-rate piping and fittings.
Assumptions, limits and common mistakes
The formula applies to a closing time shorter than the reflection time 2·l/a (an ideal, instantaneous surge) and neglects pipe friction as well as elastic expansion of the pipe wall itself. For slower closure (tS ≫ tR), the actual pressure surge is noticeably lower than the maximum value calculated here.
Common mistake: A common mistake is applying the formula regardless of the valve's actual closing time — without comparing it to the reflection time 2·l/a, the surge is substantially overestimated for actually slow closure.
Frequently asked questions
What wave speed a do I use for water?
About 1,000 m/s in thin-walled pipes, about 1,300 m/s in comparatively thick-walled hydraulic lines.
What if the valve closes slowly rather than instantly?
Then the surge is lower, proportional to the ratio of reflection time to closing time, than the maximum value calculated here.
How do I determine the reflection time?
tR = 2·l/a, with l the pipe length between the valve and the nearest reflection point (a vessel or pipe junction).
Does the formula apply to gases too?
The classical Joukowsky equation is formulated for nearly incompressible liquids; gases require different approaches due to their higher compressibility.
How do I protect a system from excessive surge?
Through slower valve closure, surge vessels, relief valves, or a lower normal-operation flow velocity.