Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright-Omega function

Abstract: The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor f. To date, the captured flow friction factor f can be extracted from the logarithmic form analytically only in the term of the Lambert W-function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert W-function also known as the Wright Omega-function. The Wright Omega-function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term y=W(e^x) of the Lambert W-function to the series expansions that further can be easily evaluated in computers without causing overflow run-time errors. Although the Colebrook equation transformed through the Lambert W-function is identical to the original expression in term of accuracy, a further evaluation of the Lambert W-function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contains only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with the relative error of no more than 0.0096%. The presented approximations are in the form suitable for everyday engineering use, they are both accurate and computationally efficient.