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Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes

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Abstract

The implicit Colebrook–White equation is the accepted method for accurately estimating the friction factor for turbulent flow in pipes. This study reviews 28 explicit equations for approximating the friction factor to integrate both the accuracy to the implicit Colebrook–White equation and the relative computational efficiency of the explicit equations. A range of 901 Reynolds numbers were selected for the review between Re ≥ 4 ×103 and  ≤ 4 × 108 and 20 relative pipe roughness values were selected between \(\varepsilon \mathord{\left/ {\vphantom {\varepsilon D}} \right.}D\ge 10^{-6}\le 10^{-1}\), thus producing a matrix of 18,020 points for each explicit equation, covering all the values to be encountered in pipeline hydraulic analysis for turbulent flow. The accuracy of the estimation of the friction factor using the explicit equations to the value obtained using the implicit Colebrook–White equation were calculated and reported as absolute, relative percentage and mean square errors. To determine the relative computational efficiency, 300 million friction factor calculations were performed using randomly generated values for the Reynolds number and the relative pipe roughness values between the limits specified for each of the explicit equations and compared to the time taken by the Colebrook–White equation. Finally, 2D and 3D contour models were generated showing both the range and magnitude of the relative percentage accuracy across the complete range of 18,020 points for each explicit equation.

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References

  1. Farshad, F., Rieke, H., Garber, J.: New developments in surface roughness measurements, characterization, and modeling fluid flow in pipe. J. Petrol. Sci. Eng. 29, 139–150 (2001)

    Article  Google Scholar 

  2. Colebrook, 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)

    Google Scholar 

  3. Colebrook, C.F., White, C.M.: Experiments with fluid friction in roughened pipes. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 161, 367–381 (1937)

    Article  Google Scholar 

  4. Prandtl, L.: Essentials of Fluid Dynamics. Hafner, New York (1952)

    MATH  Google Scholar 

  5. Von Karman, T.: The fundamentals of the statistical theory of turbulence. J. Aeronaut. Sci. 4, 131–138 (1934)

    Google Scholar 

  6. Perry, R.H., Green, D.W.: Perry’s Chemical Engineers’ Handbook, 7th edn. McGraw-Hill, London (1997)

    Google Scholar 

  7. Moody, L.F.: Friction factors for pipe flow. Trans. Am. Soc. Mech. Eng. 66, 671–684 (1944)

    Google Scholar 

  8. Moody, L.F.: An approximate formula for pipe friction factors. Trans. Am. Soc. Mech. Eng. 69, 1005–1006 (1947)

    Google Scholar 

  9. Gregory, G.A., Fogarasi, M.: Alternative to standard friction factor equation. Oil Gas J. 83, 120–127 (1985)

    Google Scholar 

  10. Guo, B., Lyons, W.C., Ghalambor, A.: Petroleum Production Engineering: A Computer-Assisted Approach. Gulf Professional, Oxford (2007)

    Google Scholar 

  11. Takács, G.: Gas Lift Manual Tulsa. Pennwell Books, Oklahoma (2004)

    Google Scholar 

  12. Özger, M., Yıldırım, G.: Determining turbulent flow friction coefficient using adaptive neuro-fuzzy computing technique. Adv. Eng. Software 40, 281–287 (2009)

    Article  MATH  Google Scholar 

  13. Schroeder, D.W. Jr.: A tutorial on pipe flow equations. In: PSIG Annual Meeting Bonita Springs. Pipeline Simulation Interest Group, Florida (2010)

  14. Sonnad, J.R., Goudar, C.T.: Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook–White equation. J. Hydraul. Div. Am. Soc. Civ. Eng. 132, 863–867 (2006)

    Article  Google Scholar 

  15. Sonnad, J.R., Goudar, C.T.: Constraints for using Lambert W function-based explicit Colebrook–White equation. J. Hydraul. Eng. Am. Soc. Civ. Eng. 130, 929–931 (2004)

    Article  Google Scholar 

  16. Yıldırım, G.: Computer-based analysis of explicit approximations to the implicit Colebrook–White equation in turbulent flow friction factor calculation. Adv. Eng. Softw. 40, 1183–1190 (2009)

    Article  MATH  Google Scholar 

  17. Wood, D.J.: An Explicit Friction Factor Relationship, vol. 60. Civil Engineering American Society of Civil Engineers (1966)

  18. Serghides, T.K.: Estimate friction factor accurately. Chem. Eng. 91, 63–64 (1984)

    Google Scholar 

  19. Chen, N.H.: An explicit equation for friction factor in pipes. Ind. Eng. Chem. Fund. 18, 296 (1979)

    Article  Google Scholar 

  20. Zigrang, D.J., Sylvester, N.D.: A review of explicit friction factor equations. J. Energ. Resour. Tech. 107, 280–283 (1985)

    Article  Google Scholar 

  21. Chen, J.J.J.: A simple explicit formula for the estimation of pipe friction factor. Proced. Inst. Civ. Eng. 77, 49–55 (1984)

    Article  Google Scholar 

  22. Goudar, C.T., Sonnad, J.R.: Explicit friction factor in correlation ifor turbulent flow in rough pipe. Hydrocarb. Process 86, 103–105 (2007)

    Google Scholar 

  23. Goudar, C.T., Sonnad, J.R.: Comparision of the iterative approximations to the colebrook-white equation. Hydrocarb. Process 87, 79–83 (2008)

    Google Scholar 

  24. Manadilli, G.: Replace implicit equations with signomial functions. Chem. Eng. 104, 129 (1997)

    Google Scholar 

  25. Romeo, E., Royo, C., Monzón, A.: Improved explicit equations for estimation of the friction factor in rough and smooth pipes. Chem. Eng. J. 86, 369–374 (2002)

    Article  Google Scholar 

  26. Brkić, D.: Review of explicit approximations to the Colebrook relation for flow friction. J. Petrol. Sci. Eng. 77, 34–48 (2011)

    Article  Google Scholar 

  27. Eck, B.: Technische Stromungslehre. Springer, New York (1973)

    Google Scholar 

  28. Rao, A.R., Kumar, B.: Friction factor for turbulent pipe flow. Division of Mechanical Sciences, Civil Engineering Indian Institute of Science Bangalore, India ID Code 9587 (2007)

  29. Buzzelli, D.: Calculating friction in one step. Mach. Des. 80, 54–55 (2008)

    Google Scholar 

  30. Avci, A., Karagoz, I.: A novel explicit equation for friction factor in smooth and rough pipes. J. Fluids Eng. 131, 061203 (2009)

    Article  Google Scholar 

  31. Papaevangelou, G., Evangelides, C., Tzimopoulos, C.: A New Explicit Relation for the Friction Factor Coefficient in the Darcy–Weisbach Equation, pp. 166–172. Protection and Restoration of the Environment Corfu, Greece: University of Ioannina Greece and Stevens Institute of Technology New Jersey (2010)

  32. Round, G.F.: An explicit approximation for the friction factor—Reynolds number relation for rough and smooth pipes. Can. J. Chem. Eng. 58, 122 (1980)

    Article  Google Scholar 

  33. Genić, S., Arandjelović, I., Kolendić, P., Jarić, M., Budimir, N., Genić, V.: A review of explicit approximations of Colebrook’s equation. FME Transactions 39, 67–71 (2011)

    Google Scholar 

  34. Tsal, R.J.: Altshul–Tsal friction factor equation. Heat-Pi**-Air Cond. 8, 30–45 (1989)

    Google Scholar 

  35. Fang, X., Xu, Y., Zhou, Z.: New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations. Nucl. Eng. Des. 241, 897–902 (2011)

    Article  Google Scholar 

  36. Zigrang, D.J., Sylvester, N.D.: Explicit approximations to the solution of Colebrook’s friction factor equation. AIChE J. 28, 514–515 (1982)

    Article  Google Scholar 

  37. Barr, D.I.H.: Solutions of the Colebrook-white function for resistance to uniform turbulent flow. Proc. Inst. Civ. Eng. Part 2: Res. Theor. 71, 529–535 (1981)

    Google Scholar 

  38. Shacham, M.: Comments on: ‘An explicit equation for friction factor in pipe’. Ind. Eng. Chem. Fund. 19, 228–229 (1980)

    Article  Google Scholar 

  39. Haaland, S.E.: Simple and explicit formulas for the friction factor in turbulent pipe flow. J. Fluids Eng. 105, 89–90 (1983)

    Article  Google Scholar 

  40. Jain, A.K.: Accurate explicit equation for friction factor. J. Hydraul. Div. Am. Soc. Civ. Eng. 102, 674–677 (1976)

    Google Scholar 

  41. Churchill, S.W.: Friction factor equation spans all fluid flow regimes. Chem. Eng. J. 91, 91–92 (1977)

    Google Scholar 

  42. Swamee, D.K., Jain, A.K.: Explicit equations for pipe flow problems. J. Hydraul. Div. Am. Soc. Civ. Eng. 102, 657–664 (1976)

    Google Scholar 

  43. Churchill, S.W.: Empirical expressions for the shear stress in turbulent flow in commercial pipe. AIChE J. 19, 375–376 (1973)

    Article  Google Scholar 

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Authors and Affiliations

  1. CB&I, 40, Eastbourne Terrace, London, W2 6LG, UK

    Herbert Keith Winning

  2. Dr. Buckinghamshire New University, Queen Alexandra Road, High Wycombe, Buckinghamshire, HP11 2JZ, UK

    Tim Coole

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  1. Herbert Keith Winning
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Correspondence to Herbert Keith Winning.

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Winning, H.K., Coole, T. Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes. Flow Turbulence Combust 90, 1–27 (2013). https://doi.org/10.1007/s10494-012-9419-7

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  • DOI: https://doi.org/10.1007/s10494-012-9419-7

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