Approximate Analytical Solutions for the Colebrook Equation

Abstract

Friction factor determination is important for modeling fluid flows in pipes. The Colebrook equation has been widely used for estimating the pipe friction factor in fully developed turbulent regime. Because of the implicit nature of Colebrook equation, various regression-based approximations, Lambert W-function-based solutions, series-based solutions, and analytical approximations are developed for explicitly determining the friction factor. This study focuses on approximate analytical solutions for the Colebrook equation with a minimum number of natural logarithms and noninteger powers (lower computational cost). For this, a new mathematically equivalent representation of the Colebrook equation is presented. This form consists of two nonlinear equations which are very well suited for developing the analytical solutions for the friction factor. The simple analytical solutions developed in this study with the maximum relative errors less than .85, .25, .54, and .28% (solutions with different degrees of accuracy) are among the most accurate analytical approximations to the Colebrook equation. Simple form and superior efficiency of the proposed solutions make them preferable to currently available approximate solutions to the Colebrook equation.