In this paper, the effect of a partially closed in-line valve, viscoelasticity, and unsteady friction on the transient behavior of a pressurized pipe is examined. Such an analysis is executed by considering global energy quantities evaluated by means of a one-dimensional numerical model calibrated on the basis of a huge amount of laboratory tests. In the numerical experiments, the effect of the initial conditions and in-line valve characteristics has been analyzed by considering different values of the initial Reynolds number, N0, in-line valve head loss coefficient, χ, and location, δ. By introducing dimensionless quantities, exponential laws are shown to interpolate the time-history of maxima of both pressure and global energy quantities reliably with the related coefficients being a function of N0, χ, and δ. Thus, the links between the decay of pressure peaks at single sections and the dissipation of the global kinetic and internal energy are established. Moreover, it is shown that a given decay of pressure peaks may derive from very different transients. This result has crucial implications to inverse transient analysis based on the evaluation of the pressure decay at a given section with particular attention to the uniqueness of the solution.