Engineering task and calculation objective
The STOS module calculates water hammer (pressure surge) in liquid-filled piping: the maximum pressure rise per Joukowsky, the actual pressure surge at the valve taking the closing characteristic into account, and the maximum and minimum overpressure at the valve. The basis is DIN 2413 Part 1 (section 4.5, dynamic pressure components — a German piping design standard) and the valve engineering literature (Wagner, "Regelarmaturen").
You need to calculate water hammer whenever a flow is decelerated rapidly: when shut-off and control valves close, when pumps trip, or when starting up against filled lines. The pressure rise superimposes on the operating pressure and can overload the pipe wall, expansion joints, supports and valves; on the low-pressure side, cavitation threatens, followed by cavitation shock when the liquid column rejoins.
The central quantities are the pressure wave speed in the pipe — depending on the bulk modulus and density of the medium and the elasticity of the pipe wall — and the ratio of closing time to the reflection time of the line. Only if the valve closes faster than the reflected wave returns does the full Joukowsky surge occur.
Standard and calculation basis: DIN 2413 Teil 1 / Abs. 4.5 Wagner "Regelarmaturen" 1.Auflage (Vogel Verlag)
Calculation workflow
- Record pipe and fluid data: Required are the pipe inside diameter, wall thickness, modulus of elasticity of the pipe material and pipe length, plus the density and bulk modulus of the liquid and the flow velocity in the initial state.
- Determine the pressure wave speed: The speed of sound of the free liquid follows from the bulk modulus and density; the compliance of the pipe wall reduces it per the Korteweg relation as a function of diameter, wall thickness and modulus of elasticity. The result is the governing wave speed of the line.
- Calculate the maximum Joukowsky surge: The maximum pressure rise follows from density, wave speed and the change in flow velocity. It represents the upper limit that occurs when the entire liquid column is decelerated instantaneously.
- Assess the valve closing process: The closing time is compared with the reflection time of the line (travel time of the wave to the reflection point and back). If the valve closes more slowly, the reflected wave partially relieves the pressure again; the pressure surge at the valve is then calculated correspondingly lower, taking the closing characteristic into account.
- Check the overpressure and underpressure limits: The module reports the maximum and minimum overpressure at the valve. The maximum pressure is used as the basis for the piping design (e.g. wall thickness verification per DIN 2413 including the dynamic pressure component); the minimum pressure is checked against the vapor pressure to exclude cavitation and separation of the liquid column.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Density of medium | ρ | kg/m³ |
| Acoustic velocity | a | m/s |
| Flow velocity | w | m/s |
| Max. pressure increase (Joukowsky) | dpm | bar |
| Pipe length | L | m |
| Closing time | Δt | s |
| Shock coefficient | z | - |
| Effective pressure increase | dpw | bar |
| Head of pump | Δh | m |
| Shock pressure coefficient | K | - |
| Modulus of elasticity of medium | EF | N/mm² |
| Modulus of elasticity of wall material | EW | N/mm² |
| Wall thickness | s | mm |
| Inside diameter | di | mm |
| Poisson number wall | μ | - |
| 1 | 1 | kg/m³ |
| 1 | 1 | bar |
| 2 | 2 | kg/m³ |
| 2 | 2 | bar |
| Force on support | F | N |
| Pump failure? | J/N | – |
| Inertial of all rotating parts | J | kg·m² |
| Rotational speed | n0 | 1/s |
| Efficiency | η | – |
Worked example
In a 500 m long cold water line of steel, DN 100 (114.3 × 3.6 mm, inside diameter 107.1 mm), water flows at 2.0 m/s. A gate valve at the end of the line closes instantaneously (closing time shorter than the reflection time). This worked example finds the pressure wave speed, the maximum water hammer per Joukowsky, and the reflection time of the line.
Given values
| Pipe length L | 500 m |
| Inside diameter di | 107.1 mm |
| Wall thickness s | 3.6 mm |
| Flow velocity v | 2.0 m/s |
| Density of water ρ | 1,000 kg/m³ |
| Bulk modulus of water K | 2.1 · 10⁹ Pa |
| Modulus of elasticity of steel E | 2.1 · 10¹¹ Pa |
Solution
Speed of sound of the free liquid
a0 = √(K/ρ) = √(2.1 · 10⁹ / 1,000) = 1,449 m/s
Wave speed in the pipe (Korteweg)
a = a0 / √(1 + K · di / (E · s))
K · di / (E · s) = (2.1 · 10⁹ · 0.1071) / (2.1 · 10¹¹ · 0.0036) = 0.2975
a = 1,449 / √1.2975 = 1,272 m/s
Maximum water hammer per Joukowsky
Δp = ρ · a · Δv = 1,000 · 1,272 · 2.0 = 2,544,000 Pa ≈ 25.4 bar
This pressure rise superimposes on the steady-state operating pressure at the valve.
Reflection time of the line
Tr = 2 · L / a = 2 · 500 / 1,272 = 0.79 s
Since the valve closes faster than Tr, the full Joukowsky surge occurs. Only closing times well above 0.79 s reduce the pressure surge.
Result
| Wave speed a | 1,272 m/s |
| Maximum water hammer Δp | 25.4 bar |
| Reflection time Tr | 0.79 s |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
When does the full Joukowsky surge occur and when may I reduce it?
The full surge occurs when the closing time is shorter than the reflection time Tr = 2L/a of the line — then the relief wave reflected at the end of the line can no longer reach the valve before it is fully closed. For longer closing times, the surge decreases roughly in the ratio of Tr to the closing time, but only if the valve throttles uniformly over its entire stroke. Valves that only become effective over the last millimeters of travel (e.g. ball valves) produce almost the full surge despite a nominally long stroking time.
Why is the wave speed in the pipe lower than the speed of sound of the liquid?
The elastic expansion of the pipe wall acts like an additional compressibility of the system. Per Korteweg, the wave speed drops the more, the larger the diameter-to-wall-thickness ratio and the softer the pipe material. In steel lines it is typically 900 to 1,300 m/s, in plastic lines (PE, PVC) only 200 to 500 m/s — which relieves plastic lines considerably with respect to water hammer.
Does the calculation also apply to gases and vapors?
No. The Joukowsky model assumes a nearly incompressible liquid column. In gas and steam lines, densities are low and compressibility is high, so fast stroking processes there produce different phenomena (pressure waves, condensation-induced shocks) that must be treated separately. Critical, by contrast, are liquid-filled lines with high flow velocity and great length.
Which measures reduce an inadmissibly high pressure surge?
Effective measures are: extending the closing time (especially of the last third of the stroke), limiting the flow velocity at the design stage, air vessels or suction tanks close to the source of the disturbance, flywheels on pumps, and low-surge closing laws in the valve control. Safety valves help only to a limited extent, since their response time can be of the same order as the wave travel time.