Water Hammer Pressure Surge (Joukowsky Equation)
Water Hammer Pressure Surge (Joukowsky Equation)
Water hammer is the sharp pressure spike that travels through a pipeline when flow velocity changes abruptly — a valve slamming shut, a pump tripping off — and it is one of the most damaging transient loads a piping system can see, capable of bursting pipes, blowing out gaskets, or damaging pumps if not accounted for in design. The Joukowsky equation gives the theoretical maximum pressure surge for an instantaneous velocity change, built from the fluid density, the pressure wave (acoustic) speed in the pipe, and the size of the velocity change itself, and it is the starting point every pipeline surge analysis and surge-protection device (surge tanks, relief valves, slow-closing valves) is sized against.
The Joukowsky pressure surge is dP = rho·a·dv, the fluid density times the pressure wave speed times the velocity change. where rho is the fluid density, a is the pressure wave (acoustic) propagation speed in the pipe, and dv is the sudden change in flow velocity.
Inputs
rho_w := 1000 kg/m^3=1000 kg/m^3
a_wave := 1200 m/s=1200 m/s
dv_change := 1.5 m/s=1.5 m/s
Multiplying the fluid density, wave speed, and velocity change together gives the peak pressure surge an instantaneous velocity change would generate, the worst-case load a rapid valve closure or pump trip could impose on the pipe.
dP_surge := rho_w·a_wave·dv_change=1.8 MPa
Results
A surge of about 1.8 MPa (roughly 18 bar) from a modest 1.5 m/s velocity change shows just how severe water hammer can be — pressure spikes of this magnitude can easily exceed a pipe's rated working pressure even when the steady-state operating pressure is comfortably within limits. This is exactly why quick-closing valves and sudden pump trips are treated as design-basis transient events: the fix is almost always to slow the valve closure time or add surge-relief protection rather than to thicken the pipe wall, since the Joukowsky surge scales directly with how fast the velocity change happens, not just how large it is.
Assumptions & limitations
- Instantaneous (or very rapid) velocity change assumed — the theoretical maximum surge.
- Rigid pipe wall / incompressible-adjustment wave speed a assumed already correctly determined.
- No allowance for surge-relief devices or gradual valve closure.
- Single, straight pipeline segment — junctions and branches would need a full transient analysis.