Abstract
In 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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Colebrooks, C.F.: Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws. J. Inst. Civ. Eng. 11, 133–156 (1939). https://doi.org/10.1680/ijoti.1939.13150
Praks, P., Brkić, D.: Review of new flow friction equations: constructing Colebrook’s explicit correlations accurately. Rev. Int. Métodos Numér. Cálc. Diseño Ing. 36(3) (2020). https://doi.org/10.23967/j.rimni.2020.09.001
Pimenta, B.D., Robaina, A.D., Peiter, M.X., Mezzomo, W., Kirchner, J.H., Ben, L.H.B.: Performance of explicit approximations of the coefficient of head loss for pressurized conduits. Rev. Brasil. Eng. Agríc. Ambient. 22(5), 301–307 (2018). https://doi.org/10.1590/1807-1929/agriambi.v22n5p301-307
Clamond, D.: Efficient resolution of the Colebrook equation. Ind. Eng. Chem. Res. 48, 3665–3671 (2009). https://doi.org/10.1021/ie801626g
Hafsi, Z.: Accurate explicit analytical solution for Colebrook-White equation. Mech. Res. Commun. 111 (2021). https://doi.org/10.1016/j.mechrescom.2020.103646https://doi.org/10.1016/j.mechrescom.2020.103646
Moody, L.: Friction factors for pipe flow. Trans. ASME 66(8), 671–678 (1944)
Wood, D.J.: An explicit friction factor relationship. Civ. Eng. 36(12), 60–61 (1966)
Swamee, P., Jain, A.: Explicit equations for pipe-flow problems. J. Hydraul. Div. (ASCE) 102(5), 657–664 (1976)
Shaikh, M.M., Massan, S., Wagan, A.I.: A new explicit approximation to Colebrook’s friction factor in rough pipes under highly turbulent cases. Int. J. Heat Mass Transf. 88, 538–543 (2015). https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.006
Brkić, D.: A note on explicit approximations to Colebrook’s friction factor in rough pipes under highly turbulent cases. Int. J. Heat Mass Transf. 93, 513–515 (2016). https://doi.org/10.1016/j.ijheatmasstransfer.2015.08.109
Brkić, D., Praks, P.: Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function. Mathematics 7(34) (2019). https://doi.org/10.3390/math7010034
Mikata, Y., Walczak, W.S.: Exact analytical solutions of the Colebrook-White equation. J. Hydraul. Eng. 142(2) (2016). https://doi.org/10.1061/(ASCE)HY.1943-7900.0001074
Haaland, S.E.: Simple and explicit formulas for the friction factor in turbulent flow. Trans. ASME. J. Fluids Eng. 105(1), 89–90 (1983). https://doi.org/10.1115/1.3240948
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Ayed, L., Choura, O., Hafsi, Z., Elaoud, S. (2023). A Haaland Based Explicit Solution for Colebrook-White Equation. In: Walha, L., et al. Design and Modeling of Mechanical Systems - V. CMSM 2021. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-031-14615-2_70
Download citation
DOI: https://doi.org/10.1007/978-3-031-14615-2_70
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-14614-5
Online ISBN: 978-3-031-14615-2
eBook Packages: EngineeringEngineering (R0)