A perturbation solution to the transient pipe flow problem

The analysis of hydraulic transients in pipes is traditionally carried out numerically. An approximate analytical solution of the nonlinear problem of unsteady flow in pipes is presented herein. The continuity and momentum equations are combined to give a nonlinear second-order hyperbolic partial differential equation whose approximate solution is derived using a perturbation technique called the delta expansion. It is shown that the zeroth-order perturbation solution gives ample accuracy for practical application. Useful series expansions for small values of the waterhammer parameter k are also obtained which compare extremely well with the results of the method of characteristics especially at small times. The explicit and simple form of the solution is suitable for further mathematical analysis and manipulation such as determining the maximal values and their time of occurrence. They are especially useful in determining the minimum time of closure possible while keeping the transient pressure in the pipe system within a specified limit.