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

Author

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  • Dejan Brkić

    (European Commission, Joint Research Centre (JRC), 21027 Ispra, Italy
    IRC Alfatec, 18000 Niš, Serbia)

  • Pavel Praks

    (European Commission, Joint Research Centre (JRC), 21027 Ispra, Italy
    IT4Innovations, VŠB—Technical University of Ostrava, 708 00 Ostrava, Czech Republic)

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 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.

Suggested Citation

  • Dejan Brkić & Pavel Praks, 2018. "Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function," Mathematics, MDPI, vol. 7(1), pages 1-15, December.
  • Handle: RePEc:gam:jmathe:v:7:y:2018:i:1:p:34-:d:194177
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    References listed on IDEAS

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    1. Dejan Brkić, 2011. "Iterative Methods for Looped Network Pipeline Calculation," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 25(12), pages 2951-2987, September.
    2. Praks, Pavel & Kopustinskas, Vytis & Masera, Marcelo, 2015. "Probabilistic modelling of security of supply in gas networks and evaluation of new infrastructure," Reliability Engineering and System Safety, Elsevier, vol. 144(C), pages 254-264.
    3. Brkic, Dejan, 2009. "An improvement of Hardy Cross method applied on looped spatial natural gas distribution networks," Applied Energy, Elsevier, vol. 86(7-8), pages 1290-1300, July.
    4. Barry, D.A & Parlange, J.-Y & Li, L & Prommer, H & Cunningham, C.J & Stagnitti, F, 2000. "Analytical approximations for real values of the Lambert W-function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 53(1), pages 95-103.
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    1. Su, Lingqi & Nie, Ting & On Ho, Chi & Yang, Zheng & Calvez, Philippe & Jain, Rishee K. & Schwegler, Ben, 2022. "Optimizing pipe network design and central plant positioning of district heating and cooling System: A Graph-Based Multi-Objective genetic algorithm approach," Applied Energy, Elsevier, vol. 325(C).

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