Explicit approximation of the Colebrook-White formula based on the friction Reynolds number

Abstract

Several practical applications require a big number of pipes to be calculated a great many times in a short time. In such cases, an explicit formula for determination of the friction factor of the Darcy-Weisbach formula is advisable for noticeably shortening the computation time, with respect to a trial-and-error solution of the Colebrook-White (C-W) formula. In the present paper, unlike previous studies, an explicit formula is obtained based on the result by Colebrook himself that the deviation, here named δ, between the reciprocals of the square roots of the actual friction factor and that relating to a fully turbulent flow in the same pipe is a function of the friction Reynolds number only, Re*. To this aim, a criterion for better estimating the limit Re* value up to which a transitional regime can occur is also given, a limit value that can differ very much from the usual value 70. The explicit formula was achieved by processing a dataset, consisting of a big number of dyads (Re*, δ) generated over a wide range of relative roughnesses and, for each of the latter, over the Re* range where a transitional regime can occur, the latter range reaching the “new” limits for transitional flow as assumed here. The simple formula gives acceptable accuracy for practical engineering purposes.