A boundary layer approach for obtaining a fifth order polynomial equation for the unsteady velocity-time history of startup flow in a pipe

An incompressible fluid is initially at rest in a horizontal pipeline. A valve in the line is suddenly opened and the fluid accelerates due to an imposed, constant pressure difference. The fluid volume eventually reaches a steady state average velocity. The descriptive equation used to model the acceleration of the fluid volume is a first order, nonlinear ordinary differential equation containing average velocity, pressure difference, and friction factor. Friction factor depends on velocity which varies during the time that the fluid accelerates. For laminar flow, the equation is readily linearlized and integrated. In the traditional solution method for turbulent flow, the friction factor is assumed to be a constant, which makes the equation linear and leads to a solution. An alternative approach is to rewrite the differential equation in difference form and solve it using a stepwise numerical procedure, which must be performed for a specific fluid-pipe combination. In this study, we present a more general approach using techniques associated with classical boundary layer methods to generate a velocity-time equation. The method yields several polynomial equations, from first to fifth order. The results are used to model two specific fluid-pipe combinations, and the results are compared to those obtained numerically as well as to those obtained assuming constant friction factor. The results indicate that the technique can be used with little error to model the acceleration of a fluid volume within a pipe