Engineering
Joukowsky Water Hammer Pressure Surge & Piezometric Head Rise
Estimate the ideal initial pressure rise from a sudden velocity reduction, its equivalent fluid head and the pipe round-trip wave time. Compare an entered closure time without treating the result as a pipe-safety certification.
An ideal initial surge, not a guaranteed maximum
Joukowsky relates a sudden velocity change to an initial pressure change. Here B4 is a nonnegative velocity reduction, defined as initial velocity minus final velocity for the chosen downstream-closure scenario. A negative value is rejected instead of applying an unexplained sign convention.
The calculation assumes a liquid-filled pipe and an effective wave speed. Reflections, network geometry, frictional line packing, cavitation and vapor-pocket collapse can change the maximum pressure. This is not a conservative bound for every piping system or proof that a pipe will withstand closure.
Wave speed depends on the fluid and pipe dimensions, material and support. The example values 1200 and 400 m/s are illustrative, not universal steel or PVC properties.
Calculate pressure rise in MPa
B2 is density in kg/m³, B3 effective wave speed in m/s and B4 velocity reduction in m/s. Divide the pressure in pascals by one million to obtain MPa.
The baseline rise is 3 MPa, equivalent to 30 bar. This is a pressure increment; operating pressure is not included. Slower closure does not make this ideal instantaneous-change value an actual predicted peak.
=IF(COUNT(B2:B7)<>6,"Complete numeric inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<=0,B6<=0,B7<0),"Invalid physical input",B2*B3*B4/1000000))Calculate equivalent head rise
B5 is positive gravitational acceleration in m/s². Divide wave speed times velocity reduction by B5 to obtain head in meters of the modeled fluid.
=IF(COUNT(B2:B7)<>6,"Complete numeric inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<=0,B6<=0,B7<0),"Invalid physical input",B3*B4/B5))Calculate round-trip wave time
For the simple pipe model, B6 is pipe length in meters and 2*L/a is the time for a wave to travel to the reflecting boundary and back. E4 displays this previously missing output.
=IF(COUNT(B2:B7)<>6,"Complete numeric inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<=0,B6<=0,B7<0),"Invalid physical input",2*B6/B3))Compare the entered closure time
B7 is closure time in seconds. E5 reports whether it is at or below the simple round-trip time. It is a timing comparison, not a validation of the closure law or the network boundary conditions.
For closure above this interval, use a transient model rather than treating the displayed instantaneous surge as the actual peak. This worksheet does not automatically reduce the surge by a guessed factor.
=IF(COUNT(B2:B7)<>6,"Complete numeric inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<=0,B6<=0,B7<0),"Invalid physical input",IF(B7<=2*B6/B3,"At or below round-trip time","Above round-trip time: transient analysis needed")))Try the example
Use the original values or change them to test your understanding.