A new approach for explicit approximation of the Colebrook–White formula for pipe flows

A novel approach is presented for explicit assessment of the friction factor of the Darcy-Weisbach formula for a pipe turbulent flow, a topic being especially useful in practical applications requiring a large number of pipes that have to be calculated a great many times in a short time. In such applications, in fact, an explicit formula shortening computation time, with respect to a trial-and-error solution of the Colebrook-White formula, is very advisable. To this aim, in the paper analytical simplicity is pursued besides accuracy, the latter being kept high enough for practical purposes. Unlike previous studies, the ratio between the actual friction factor and that relating to a fully turbulent flow is analysed as a function of the relative roughness and the relative Reynolds number (i.e., the ratio between the actual Reynolds number and the Reynolds number separating the transition regime from the fully turbulent regime). By processing a dataset obtained by systematically solving the C-W formula over suited ranges of the Reynolds number and the relative roughness, two expressions are obtained: a simpler first-step accuracy expression giving generally acceptable accuracy for most engineering practical purposes; and a second-step accuracy expression allowing adequately high accuracy for all situations.