A Haaland Based Explicit Solution for Colebrook-White Equation

Abstract

AbstractIn hydraulic flow modeling and experimentation, friction factor was usually determined using the Moody chart. Values of friction factor obtained from Moody chart can exhibit reading uncertainties. Since Moody chart itself is a result of the resolution of Colebrook-White equation, commonly performed iteratively, an alternative analytical solution of the implicit Colebrook-White equation would be of great usefulness towards more accurate determination of Darcy friction factor. In this paper, through differentiating then integrating Colebrook-White equation, a mathematical development was established. Then, as function of the Reynolds number and the pipe roughness, an explicit expression of the friction factor was given. As for the integration term, it was determined based on Haaland’s correlation. For validation purposes, the developed expression’ results were compared to exact values of friction factor and to those obtained from Haaland’s expression. For a set of couples Reynolds number-relative roughness, it was obtained a tiny mean relative error of obtained friction factors with reference to exact values. Additionally, outcomes of the developed formula compare very favorably with those of Haaland’s approximation. KeywordsFriction factorColebrook-White equationHaaland formulaAnalytical expression