Simple analysis of line packing, attenuation, and rarefaction phenomena in water hammer

The classical Joukowsky formula for the head jump associated with a sudden flow stoppage is an engineering rule of thumb that helps estimate peak surge pressures in pipelines. However, the Joukowsky formula does not take account of friction. Although this is usually of little concern for short pipelines, the effects of pipe friction can cause significant deviations for long pipelines and for high frictional flows, where rarefaction, line packing, and attenuation phenomena can be significant. This paper describes some extensions of the classical Joukowsky formula that take frictional effects into account. By using suitable nondimensional variables based on the system parameters, a single universal formulation of the underlying water hammer equations is developed and some simple analytic approximations are derived. Both instantaneous and finite-duration stoppages of flow are considered. For an instantaneous stoppage, a first-order analysis shows that the head behind the surge front is uniform in space but continues to fall with time at a nondimensional rate of 0.5. A finite shutdown can usefully be thought of as being intermediate between two instantaneous shutdown cases, one occurring at the start of the finite shutdown period, and the other at the finish. The intuitive notion that a finite shutdown period has a moderating effect on the subsequent development of rarefaction is shown to be incorrect for long pipelines.