LOGOS Learning · 14
Joukowsky equation and water hammer: the danger of closing fast
The Joukowsky equation gives the largest water hammer surge for a sudden velocity change: dP = rho * a * dv, or in head, dH = a * dv / g. It holds when the valve closes faster than the critical time 2L/a; slower closures are estimated with Michaud.
Pick the material and the closing time and press "Close". The pressure wave runs along the main, hits the reservoir and comes back.
Quick test
Three questions about this lesson. Got one wrong? The explanation shows right away.
1. On a slow closure, doubling the closing time makes the surge…
2. A closure is rapid when the closing time is…
3. A simple way to reduce surge when closing a valve is to…
Why this happens
Arrows compare with a 15 s closure in the same material.
Wave speed
The pressure wave travels through the water, but the pipe deforms with it. A stiff pipe (steel) keeps the wave fast; a flexible pipe (HDPE) expands more under pressure, so the wave is slow and the surge smaller.
Instantaneous closure (Joukowsky)
The largest surge possible for this velocity change. With 0.8 m/s in steel, that is 10.5 bar on top of the 11.6 bar operating pressure.
Critical closing time
The wave travels to the reservoir and back in 2L/a. If the valve closes faster than that, the reflected relief arrives too late and the valve takes the full Joukowsky surge.
Slow closure (Michaud)
The formula does not apply here: the closure is rapid and the limit is Joukowsky. Michaud only applies when Tc > 2L/a.
Formulas in plain text
- Joukowsky equation
dH = a * dv / g (dP = rho * a * dv)- dH = surge head (m) · dP = surge pressure (Pa) · a = pressure wave speed (m/s) · dv = velocity change (m/s) · rho = fluid density (kg/m3) · g = 9.81 m/s2
- Pressure wave speed in a pipe
a = sqrt(K / rho) / sqrt(1 + psi * K * D / (E * e))- K = bulk modulus of the fluid (Pa) · D = internal diameter (m) · E = elastic modulus of the pipe wall (Pa) · e = wall thickness (m) · psi = anchoring factor (1 with expansion joints)
- Critical closing time
Tc <= 2 * L / a -> full Joukowsky surge- Tc = valve closing time (s) · L = pipe length to the reflection point (m)
- Michaud formula (slow closure)
dH ~= 2 * L * v0 / (g * Tc) (Tc > 2L/a)- v0 = initial velocity (m/s)
Frequently asked questions
What is the Joukowsky equation used for?
It gives the maximum pressure rise when the flow velocity changes suddenly, such as a fast valve closure or a pump trip: dP = rho * a * dv. It is the upper bound of the water hammer surge for that velocity change.
What is the wave speed of water in a pipe?
About 1,400 to 1,480 m/s in a perfectly rigid pipe. The pipe wall stretches with the pressure, so steel lines are typically around 1,000 to 1,300 m/s and plastic pipes (PVC, HDPE) much slower, which lowers the surge.
What is the critical closing time of a valve?
It is 2L/a, the time the pressure wave takes to travel to the reservoir and back. If the valve closes in less than that, the relief wave arrives too late and the valve sees the full Joukowsky surge.
When should the Michaud formula be used instead of Joukowsky?
Only for slow closures, when the closing time is longer than 2L/a. Michaud, dH ~= 2*L*v0/(g*Tc), is a simple estimate; for rapid closures the limit is Joukowsky.