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Comment published on 12 March 2019, see Mathematics 2019, 7(3), 253.
Correction published on 12 October 2019, see Mathematics 2019, 7(10), 951.
Open AccessArticle

Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function

by Dejan Brkić 1,2,* and Pavel Praks 1,3,*
1
European Commission, Joint Research Centre (JRC), 21027 Ispra, Italy
2
IRC Alfatec, 18000 Niš, Serbia
3
IT4Innovations, VŠB—Technical University of Ostrava, 708 00 Ostrava, Czech Republic
*
Authors to whom correspondence should be addressed.
Mathematics 2019, 7(1), 34; https://doi.org/10.3390/math7010034
Received: 23 October 2018 / Revised: 21 December 2018 / Accepted: 24 December 2018 / Published: 31 December 2018
(This article belongs to the Special Issue Special Functions and Applications)
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 ω-function. The Wright ω-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 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 terms of accuracy, a further evaluation of the Lambert W -function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contain only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with a relative error of no more than 0.0096%. The presented approximations are in a form suitable for everyday engineering use, and are both accurate and computationally efficient. View Full-Text
Keywords: Colebrook equation; hydraulic resistance; Lambert W-function; Wright ω-function; explicit approximations; computational burden; turbulent flow; friction factor Colebrook equation; hydraulic resistance; Lambert W-function; Wright ω-function; explicit approximations; computational burden; turbulent flow; friction factor
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Brkić, D.; Praks, P. Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function. Mathematics 2019, 7, 34.

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