Correlation for the calculation of turbulent friction in pipes
Main Article Content
Abstract
One of the essential parameters in hydraulic systems of pipe networks is the friction factor lambda. The friction factor is determined using the implicit Colebrook-White equation through iterative methods, which makes its application challenging. In this work, a correlation based on the recursive method is developed to calculate the friction factor using the Colebrook-White equation. Two empirical relationships are proposed to finalise the correlation, with coefficients and exponents calibrated in Excel 2019. The results of the two proposed relationships were compared with the Swamee-Jain and Haaland relationships for recursive increments. For the lambda8 correlation, the maximum percentage error of the friction factor was 0,0000017%, for a relative roughness of 0,00001 and a Reynolds number of 4000. Additionally, the calculations yielded seven exact decimal digits for the friction factor. For Reynolds numbers greater than 4000, the percentage error decreases. As a result, it is concluded that the correlation based on the proposed explicit relationships satisfies the solution of the implicit Colebrook-White equation
Keywords
correlación, ecuación de Colebrook-White, error porcentual, factor de fricción, método recursivo Correlation, Colebrook-White equation, Percentage error, Friction factor, Recursive method
References
[1] F. M. White, Fluid Mechanics. McGraw Hill, 2011. [Online]. Available: https://bit.ly/3ICSyXO
[2] Y. A. Cengel and J. M. Cimbala, Fluid Mechanics: Fundamentals and Applications. McGraw- HillHigher Education, 2011. [Online]. Available: https://bit.ly/43sCUX3
[3] S. L. B. Tolentino Masgo, “Estudio experimental y numérico de la presión del flujo de agua en un tubo Venturi,” Ingenius, Revista de Ciencia y Tecnología, no. 23, pp. 9–22, 2020. [Online]. Available: https://doi.org/10.17163/ings.n23.2020.01
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[10] Y. Mikata and W. S. Walczak, “Exact analytical solutions of the colebrook-white equation,” Journal of Hydraulic Engineering, vol. 142, no. 2, p. 04015050, 2016. [Online]. Available: https://doi.org/10.1061/(ASCE)HY.1943-7900.0001074
[11] P. Rollmann and K. Spindler, “Explicit representation of the implicit colebrook–white equation,” Case Studies in Thermal Engineering, vol. 5, pp. 41–47, 2015. [Online]. Available: https://doi.org/10.1016/j.csite.2014.12.001
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[13] T. K. Seguides, “Estimate friction factor accurately,” Chemical Engineering Journal, vol. 91, pp. 63–64, 1984. [Online]. Available: https://bit.ly/3oqUTyd
[14] A. R. Vatankhah, “Approximate analytical solutions for the colebrook equation,” Journal of Hydraulic Engineering, vol. 144, no. 5, p. 06018007, 2018. [Online]. Available: https://doi.org/10.1061/(ASCE)HY.1943-7900.0001454
[15] N. Azizi, R. Homayoon, and M. R. Hojjati, “Predicting the Colebrook-white friction factor in the pipe flow by new explicit correlations,” Journal of Fluids Engineering, vol. 141, no. 5, Nov 2018, 051201. [Online]. Available: https://doi.org/10.1115/1.4041232
[16] N. H. Chen, “An explicit equation for friction factor in pipe,” Industrial & Engineering Chemistry Fundamentals, vol. 18, no. 3, pp. 296–297, 1979. [Online]. Available: https://doi.org/10.1021/i160071a019
[17] B. J. Schorle, S. W. Churchill, and M. Shacham, “Comments on: "an explicit equation for friction factor in pipe",” Industrial & Engineering Chemistry Fundamentals, vol. 19, no. 2, pp. 228–228, 1980. [Online]. Available: https://doi.org/10.1021/i160074a019
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[20] E. Romeo, C. Royo, and A. Monzon, “Improved explicit equations for estimation of the friction factor in rough and smooth pipes,” Chemical Engineering Journal, vol. 86, pp. 369–374, 04 2002. [Online]. Available:http://dx.doi.org/10.1016/S13858947(01)00254-6
[21] U. Offor and S. Alabi, “An accurate and computationally efficient explicit friction factor model,” Advances in Chemical Engineering and Science, no. 6, pp. 237–245, 2016. [Online]. Available: http://dx.doi.org/10.4236/aces.2016.63024
[22] I. Santos-Ruiz, J. R. Bermúdez, F. R. López-Estrada, V. Puig, and L. Torres, “Estimación experimental de la rugosidad y del factor de fricción en una tubería,” in Memorias del Congreso Nacional de Control Automático, San Luis Potosí, México,, 2018, pp. 489–494. [Online]. Available: https://bit.ly/3oubv8g
[23] A. P. Olivares-Gallardo, R. A. Guerra-Rojas, and M. A. Alfaro-Guerra, “Evaluación experimental de la solución analítica exacta de la ecuación de colebrook-white,” Ingeniería Investigación y Tecnología, vol. 2, pp. 1–11, 2019. [Online]. Available: https://doi.org/10.22201/fi.25940732e.2019.20n2.021
[24] J. R. Perez Pupo, J. N. Guerrero, and M. Batista Zaldivar, “On the explicit expressions for the determination of the friction factor in turbulent regime,” Revista mexicana de ingeniería química, vol. 19, pp. 313–334, 01 2020. [Online]. Available: https://bit.ly/3BUtqrD
[25] D. Brkic and Z. Stajic, “Excel VBA-based user defined functions for highly precise colebrook’s pipe flow friction approximations: a comparative overview,” FACTA UNIVERSITATIS Series: Mechanical Engineering, vol. 19, no. 2, pp. 253–269, 2021. [Online]. Available: https://doi.org/10.22190/FUME210111044B
[26] P. Praks and D. Brkic, “Advanced iterative procedures for solving the implicit Colebrook equation for fluid flow friction,” Advances in Civil Engineering, 2018. [Online]. Available: https://doi.org/10.1155/2018/5451034
[27] A. A. Lamri and S. M. Easa, “Computationally efficient and accurate solution for colebrook equation based on lagrange theorem,” Journal of Fluids Engineering, vol. 144, p. 014504, 2022. [Online]. Available: https://doi.org/10.1115/1.4051731
[2] Y. A. Cengel and J. M. Cimbala, Fluid Mechanics: Fundamentals and Applications. McGraw- HillHigher Education, 2011. [Online]. Available: https://bit.ly/43sCUX3
[3] S. L. B. Tolentino Masgo, “Estudio experimental y numérico de la presión del flujo de agua en un tubo Venturi,” Ingenius, Revista de Ciencia y Tecnología, no. 23, pp. 9–22, 2020. [Online]. Available: https://doi.org/10.17163/ings.n23.2020.01
[4] O. Reynolds, “Xxix. an experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels,” Philosophical Transactions of the Royal Society of London, vol. 174, pp. 935–982, 1883. [Online]. Available: https://doi.org/10.1098/rstl.1883.0029
[5] C. F. Colebrook, C. M. White, and G. I. Taylor, “Experiments with fluid friction in roughened pipes,” Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences, vol. 161, no. 906, pp. 367–381, 1937. [Online]. Available: https://doi.org/10.1098/rspa.1937.0150
[6] C. F. Colebrook, “Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws,” Journal of the Institution of Civil Engineers, vol. 11, no. 4, pp. 133–156, 1939. [Online]. Available: https://doi.org/10.1680/ijoti.1939.13150
[7] L. F. Moody and N. J. Princeton, “Friction factor for pipe flow,” Transaction of ASME, vol. 66, pp. 671–684, 1944. [Online]. Available: https://bit.ly/3BRgxyL
[8] P. K. Swamee and A. K. Jain, “Explicit equations for pipe-flow problems,” Journal of the Hydraulics Division, vol. 102, no. 5, pp. 657–664, 1976. [Online]. Available: https://doi.org/10.1061/JYCEAJ.0004542
[9] S. E. Haaland, “Simple and explicit formulas for the friction factor in turbulent pipe flow,” Journal of Fluids Engineering, vol. 105, no. 1, pp. 89–90, Mar 1983. [Online]. Available: https://doi.org/10.1115/1.3240948
[10] Y. Mikata and W. S. Walczak, “Exact analytical solutions of the colebrook-white equation,” Journal of Hydraulic Engineering, vol. 142, no. 2, p. 04015050, 2016. [Online]. Available: https://doi.org/10.1061/(ASCE)HY.1943-7900.0001074
[11] P. Rollmann and K. Spindler, “Explicit representation of the implicit colebrook–white equation,” Case Studies in Thermal Engineering, vol. 5, pp. 41–47, 2015. [Online]. Available: https://doi.org/10.1016/j.csite.2014.12.001
[12] D. Biberg, “Fast and accurate approximations for the colebrook equation,” Journal of Fluids Engineering, vol. 139, no. 3, Dec 2016, 031401. [Online]. Available: https://doi.org/10.1115/1.4034950
[13] T. K. Seguides, “Estimate friction factor accurately,” Chemical Engineering Journal, vol. 91, pp. 63–64, 1984. [Online]. Available: https://bit.ly/3oqUTyd
[14] A. R. Vatankhah, “Approximate analytical solutions for the colebrook equation,” Journal of Hydraulic Engineering, vol. 144, no. 5, p. 06018007, 2018. [Online]. Available: https://doi.org/10.1061/(ASCE)HY.1943-7900.0001454
[15] N. Azizi, R. Homayoon, and M. R. Hojjati, “Predicting the Colebrook-white friction factor in the pipe flow by new explicit correlations,” Journal of Fluids Engineering, vol. 141, no. 5, Nov 2018, 051201. [Online]. Available: https://doi.org/10.1115/1.4041232
[16] N. H. Chen, “An explicit equation for friction factor in pipe,” Industrial & Engineering Chemistry Fundamentals, vol. 18, no. 3, pp. 296–297, 1979. [Online]. Available: https://doi.org/10.1021/i160071a019
[17] B. J. Schorle, S. W. Churchill, and M. Shacham, “Comments on: "an explicit equation for friction factor in pipe",” Industrial & Engineering Chemistry Fundamentals, vol. 19, no. 2, pp. 228–228, 1980. [Online]. Available: https://doi.org/10.1021/i160074a019
[18] D. J. Zigrang and N. D. Sylvester, “A review of explicit friction factor equations,” Journal of Energy Resources Technology, vol. 107, no. 2, pp. 280–283, 1985. [Online]. Available: https://doi.org/10.1115/1.3231190
[19] J. Sousa, M. d. C. Cunha, and A. S. Marques, “An explicit solution of the colebrook-white equation through simulated annealing,” Water Industry Systems: Modelling and Optimization Applications, vol. 2, pp. 347–355, 1999.
[20] E. Romeo, C. Royo, and A. Monzon, “Improved explicit equations for estimation of the friction factor in rough and smooth pipes,” Chemical Engineering Journal, vol. 86, pp. 369–374, 04 2002. [Online]. Available:http://dx.doi.org/10.1016/S13858947(01)00254-6
[21] U. Offor and S. Alabi, “An accurate and computationally efficient explicit friction factor model,” Advances in Chemical Engineering and Science, no. 6, pp. 237–245, 2016. [Online]. Available: http://dx.doi.org/10.4236/aces.2016.63024
[22] I. Santos-Ruiz, J. R. Bermúdez, F. R. López-Estrada, V. Puig, and L. Torres, “Estimación experimental de la rugosidad y del factor de fricción en una tubería,” in Memorias del Congreso Nacional de Control Automático, San Luis Potosí, México,, 2018, pp. 489–494. [Online]. Available: https://bit.ly/3oubv8g
[23] A. P. Olivares-Gallardo, R. A. Guerra-Rojas, and M. A. Alfaro-Guerra, “Evaluación experimental de la solución analítica exacta de la ecuación de colebrook-white,” Ingeniería Investigación y Tecnología, vol. 2, pp. 1–11, 2019. [Online]. Available: https://doi.org/10.22201/fi.25940732e.2019.20n2.021
[24] J. R. Perez Pupo, J. N. Guerrero, and M. Batista Zaldivar, “On the explicit expressions for the determination of the friction factor in turbulent regime,” Revista mexicana de ingeniería química, vol. 19, pp. 313–334, 01 2020. [Online]. Available: https://bit.ly/3BUtqrD
[25] D. Brkic and Z. Stajic, “Excel VBA-based user defined functions for highly precise colebrook’s pipe flow friction approximations: a comparative overview,” FACTA UNIVERSITATIS Series: Mechanical Engineering, vol. 19, no. 2, pp. 253–269, 2021. [Online]. Available: https://doi.org/10.22190/FUME210111044B
[26] P. Praks and D. Brkic, “Advanced iterative procedures for solving the implicit Colebrook equation for fluid flow friction,” Advances in Civil Engineering, 2018. [Online]. Available: https://doi.org/10.1155/2018/5451034
[27] A. A. Lamri and S. M. Easa, “Computationally efficient and accurate solution for colebrook equation based on lagrange theorem,” Journal of Fluids Engineering, vol. 144, p. 014504, 2022. [Online]. Available: https://doi.org/10.1115/1.4051731
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Published
Jul 6, 2023
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Section
Mechanical Engineering - Fluids
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