Explicit Solution for Pipe Diameter Problem Using Lambert <i>W</i>-Function

Abstract

Determining the pipe diameter is one of the principal problems encountered in designing and analyzing pipe flow lines. However, the direct determination of a pipe’s diameter is not possible because of the implicit form of the Colebrook resistance flow formula through commercial pipes. Traditionally, the pipe diameter is determined using a trial procedure. In this paper, the pipe diameter problem was solved using explicit equations in terms of the Lambert W-function. The maximum relative errors of the developed solutions are less than 0.013% for the rough and smooth flow regimes and less than 0.8% and 0.6% for the transition flow region between them. In addition, a method for determining pipe diameter under uncertainty, including design graphs, is presented. It is hoped that the developed solution for predicting pipe diameter will be helpful in the analysis of pipe flow and the design of pipelines and water distribution networks.