Energy Relations in Transient ClosedConduit Flow
by Bryan W. Karney, (Asst. Prof., Dept. of Civ. Engrg., Univ. of Toronto, Toronto, M5S 1A4, Canada)
Journal of Hydraulic Engineering, Vol. 116, No. 10, October 1990, pp. 11801196, (doi: http://dx.doi.org/10.1061/(ASCE)07339429(1990)116:10(1180))
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Journal Paper 
Abstract: 
When the rate of flow in a closed conduit is changed, largescale conversions of mechanical energy often occur, particularly if the pipeline is carrying water or some other slightly compressible liquid. Mathematical expressions describing these transient energy transformations are motivated from first principles and derived by mathematical manipulation of the governing continuity and momentum equations. The resulting expression accounts for the kinetic energy of the fluid, the internal energy associated with fluid compressibility and pipeline elasticity effects, the energy dissipated by friction, and the work done at the ends of the conduit. The energy approach provides an integrated view of transient conditions in the pipeline and is thus a simple, efficient, and logically consistent way of comparing the transient response of different systems and solution techniques. In particular, compressibility effects are shown to be negligible when the ratio of the change in internal energy to the change in kinetic energy is much less than one. This rule helps to distinguish the “rigid water column” model of unsteady flow from the more complex waterhammer theory. 
