Explicit pipe friction factor equations: evaluation, classification, and proposal
DOI:
https://doi.org/10.17533/udea.redin.20230928Keywords:
Colebrook equation, turbulent fluid, relative roughness, Reynolds numberAbstract
The Colebrook equation has been used to estimate the friction factor (f) in turbulent fluids. In this regard, many equations have been proposed to eliminate the iterative process of the Colebrook equation. The goal of this article was to perform an evaluation, classification, and proposal of the friction factor for better development of hydraulic projects. In this study, Gene Expression Programming (GEP), Newton-Raphson, and Python algorithms were applied. The accuracy and model selection were performed with the Maximum Relative Error (∆f/f), Percentage Standard Deviation (PSD), Model Selection Criterion (MSC), and Akaike Information Criterion (AIC). Of the 30 equations evaluated, the Vatankhah equation was the most accurate and simplest to obtain the friction factor with a classification of very high, reaching a value of ∆f/f<0.5% and 1.5<PSD<1.6. A new equation was formulated to obtain the explicit f with fast convergence and accuracy. It was concluded that the combination of GEP, error theory, and selection criteria provides a more reliable and strengthened model.
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References
L. F. Ospina-H., M. E. López-G., C. A. Palacio, and J. F. Jiménez-M.,“Dispositivo de reynolds para el estudio reológico de fluidos nonewtonianos independientes del tiempo: Diseño, construccióny realización de pruebas preliminares,”Revista Colombiana deMateriales, no. 3, Oct. 19, 2012. [Online]. Available:https://revistas.udea.edu.co/index.php/materiales/article/view/1322745
W. H. Alawee, Y. A. Almolhem, B. Yusuf, T. Mohammad, and H. A.Dhahad, “Variation of coefficient of friction and friction head lossesalong a pipe with multiple outlets,”Water, vol. 12, no. 3, Mar. 17,2020. [Online]. Available:https://doi.org/10.3390/w12030844
F. J. Mejía, “Relación de las curvas de energía específica y pendientede fricción con las zonas de flujo libre en canales,”Revista Escuelade Ingeniería de Antioquia, vol. 9, pp. 69–75, Jan-Jun. 2008.
J. A. Gómez-Camperos, P. J. García-Guarín, and C. Nolasco-Serna,“Modelo numérico de detección de fugas para sistema de tuberias,”AiBi Revista De Investigación, Administración E Ingeniería, vol. 8,no. 2, Apr. 22, 2020. [Online]. Available:https://doi.org/10.15649/2346030X.723
C. A. García-Ubaque and E. O. L.-M. nd M. C. García-Vaca,“Determination of the inside diameter of pressure pipes for drinkingwater systems using artificial neural networks,”Revista Facultadde Ingeniería, vol. 31, no. 59, Mar. 25, 2022. [Online]. Available:https://doi.org/10.19053/01211129.v31.n59.2022.14037
C. F. Colebrook, “Turbulent flow in pipe with particular referenceto the transition region between the smooth and rough pipe laws,”Journal of the Institution of Civil Engineers, vol. 11, no. 4, Jun. 05,2015. [Online]. Available:https://doi.org/10.1680/ijoti.1939.13150
R. T. de A. Minhoni, F. F. S. Pereira, T. B. G. da Silva, E. R.Castro, and J. C. C. Saad, “The performance of explicit formulasfor determining the darcy-weisbach friction factor,”EngenhariaAgrícola, vol. 40, no. 2, Mar-Apr. 2020. [Online]. Available:https://doi.org/10.1590/1809-4430-Eng.Agric.v40n2p258-265/2020
E. O. Ladino-Moreno, C. A. García-Ubaque, and M. C.García-Vaca, “Darcy-weisbach resistance coefficient determinationusing newton-raphson approach for android 4.0,”Tecnura,vol. 23, no. 60, Apr-Jun. 2019. [Online]. Available:https://doi.org/10.14483/22487638.14929
L. F. Moody, “An approximate formula for pipe friction factors,”Trans.ASME, vol. 69, no. 12, pp. 1005–1011, 1947.
P. K. Swamee and A. K. Jain, “Explicit equations for pipe-flowproblems,”Journal of the Hydraulics Division, vol. 102, no. 5, May.1976. [Online]. Available:https://doi.org/10.1061/JYCEAJ.0004542
D. J. Zigranga and N. D. Sylvester, “Explicit approximationsto the solution of colebrook’s friction factor equation,”AIChEJournal, vol. 28, no. 3, May. 1982. [Online]. Available:https://doi.org/10.1002/aic.690280323
E. Romeo, C. Royo, and A. Monzón, “Improved explicit equationsfor estimation of the friction factor in rough and smooth pipes,”Chemical Engineering Journal, vol. 86, no. 3, Dec. 06, 2001. [Online].Available:https://doi.org/10.1016/S1385-8947(01)00254-6
G. Papaevangelou, C. Evangelides, and C. D. Tzimopoulos, “A newexplicit equation for the friction coefficient in the darcy-weisbachequation,” inProceedings of the Tenth Conference on Protection andRestoration of the Environment: PRE10, Corfú, Gr., 2010. [Online].Available:https://tinyurl.com/mr25my46
D. Brkić, “An explicit approximation of colebrook’s equation forfluid flow friction factor,”Petroleum Science and Technology, vol. 29,no. 15, Jun. 09, 2001. [Online]. Available:https://doi.org/10.1080/10916461003620453
S. Genić and B. Jaćimović, “Reconsideration of the friction factordata and equations for smooth, rough and transition pipe flow,” inITM Web Conf. 1st International Conference on Computational Methodsand Applications in Engineering (ICCMAE 2018), Timisoara, RO., 2019.[Online]. Available:https://tinyurl.com/mr25my46
A. R. Vatankhah, “Approximate analytical solutions for the colebrookequation,”Journal of Hydraulic Engineering, vol. 144, no. 5, Mar.15, 2018. [Online]. Available:https://doi.org/10.1061/(ASCE)HY.1943-7900.0001454
D. Brkić and P. Praks, “Accurate and efficient explicit approximationsof the colebrook flow friction equation based on the wrightω-function,”Mathematics, vol. 7, no. 10, Dec. 31, 2018. [Online].Available:https://doi.org/10.3390/math7010034
D. Brkić and P. Praks”, “Review of new flow frictionequations: Constructing colebrook explicit correlations accurately,”International Journal of Numerical Methods for Calculation and Designin Engineering, vol. 36, no. 3, May. 13, 2020. [Online]. Available:https://doi.org/10.23967/j.rimni.2020.09.001
C. Yu, X. Yu, L. Zhang, B. Neupane, and J. Zhang, “Approximateapproach for improving pressure attenuation accuracy duringhydraulic transients,”International Journal of Numerical Methods forCalculation and Design in Engineering, vol. 22, no. 3, Mar. 01, 2022.[Online]. Available:https://doi.org/10.2166/ws.2021.394
G. S. Custódio-Assunção, D. Marcelin, J. C. V. Hohendorff-Filho, D. J.Schiozer, and M. S.-D. Castro, “Friction factor equations accuracyfor single and two-phase flows,” International Conference on Ocean,Offshore, and Arctic Engineering (OMAE), 2020.
B. Achour, A. Bedjaoiu, M. Khattaaoui, and M. Debabeche,“Contribution au calcul des ecoulements uniformes surface libreet en charge,”ILarhyss Journal, vol. 1, Mar. 01, 2002. [Online].Available:https://tinyurl.com/4fkfuw4r
L. Zeghadnia, J. Loup-Robert, and B. Achour, “Explicit solutions forturbulent flow friction factor: A review, assessment and approachesclassification,”International Journal of Numerical Methods forCalculation and Design in Engineering, vol. 10, no. 1, Mar. 2019.[Online]. Available:https://doi.org/10.1016/j.asej.2018.10.007
X. Fang, Y. Xu, and Z. Zhou, “New correlations of single-phasefriction factor for turbulent pipe flow and evaluation of existingsingle-phase friction factor correlations,”Nuclear Engineering andDesign, vol. 241, no. 3, Mar. 2011. [Online]. Available:https://doi.org/10.1016/j.nucengdes.2010.12.019
A. Bachir and A. Llyes, “New formulation of the darcy-weisbachfriction factor,”LARHYSS Journal, vol. 17, no. 3, Oct. 18 2020.[Online]. Available:https://www.asjp.cerist.dz/en/article/134661
W. Khan, “Numerical simulation of chun-hui he’s iteration methodwith applications in engineering,”International Journal of NumericalMethods for Heat & Fluid Flow, vol. 32, no. 3, Jan. 20, 2022. [Online].Available:https://doi.org/10.1108/HFF-04-2021-0245
E. O. Ladino-Moreno, C. A. G. Ubaque, and M. C. García-Vaca,“Modelado del factor de fricción en tuberías a presión utilizandoredes neuronales de aprendizaje bayesiano,”Ciencia en desarrollo,vol. 13, no. 1, Jan-Jun. 2022. [Online]. Available:https://doi.org/10.19053/01217488.v13.n1.2022.13241
M. Arif, M. Mohammed, U. Farooq, F. Bashir-Farooq, M. K.Elbashir, andet al., “Numerical and theoretical investigation toestimate darcy friction factor in water network problem based onmodified chun-hui he’s algorithm and applications,”MathematicalProblems in Engineering, Jan. 20, 2022. [Online]. Available:https://doi.org/10.1155/2022/8116282
M. Milošević, D. Brkić, P. Praks, D. Litričin, and Z. Stajić,“Hydraulic losses in systems of conduits with flow from laminarto fully turbulent: A new symbolic regression formulation,”Axioms, vol. 11, no. 5, Mar. 06, 2022. [Online]. Available:https://doi.org/10.3390/axioms11050198
S. Samadianfard, M. Taghi-Sattari, O. Kisi, and H. Kazemi,“Determining flow friction factor in irrigation pipes using datamining and artificial intelligence approaches,”Applied ArtificialIntelligence, vol. 28, no. 8, Oct. 14, 2014. [Online]. Available:https://doi.org/10.1080/08839514.2014.952923
M. Najafzadeh, J. Shiri, G. Sadeghi, and A. Ghaemi, “Predictionof the friction factor in pipes using model tree,”ISH Journalof Hydraulic Engineering, vol. 24, no. 1, May. 19, 2017. [Online].Available:https://doi.org/10.1080/09715010.2017.1333926
U. Herbert-Offor and S. Boladale-Alabi, “An accurate andcomputationally efficient explicit friction factor model,”Advancesin Chemical Engineering and Science, vol. 6, no. 3, Mar. 30, 2016.[Online]. Available:https://doi.org/10.4236/aces.2016.63024
E. Temizhan, H. Mirtagioglu, and M. Mendes, “Which correlationcoefficient should be used for investigating relations betweenquantitative variables?”American Scientific Research Journalfor Engineering, Dec. 2021. [Online]. Available:https://www.researchgate.net/publication/359579944
H. Akaike, “A new look at the statistical model identification,”IEEETransactions on Automatic Control, vol. 19, no. 6, Dec. 1974. [Online].Available:https://doi.org/10.1109/TAC.1974.1100705
G. Schwarz, “Estimating the dimension of a model,”IHydraulicFriction Losses in the Piping, vol. 6, no. 2, Mar. 1978. [Online]. Available:https://www.jstor.org/stable/2958889
C. L. Mallows, “Some comments oncp,”Technometrics, vol. 42,no. 1, Mar. 12, 2012. [Online]. Available:https://doi.org/10.1080/00401706.2000.10485984
G. Srbislav, A. Ivan, K. Petara, J. Marko, B. Nikola, andG. Vojislavc, “Some comments oncp,”Technometrics, vol. 39,2011. [Online]. Available:https://scindeks.ceon.rs/article.aspx?artid=1451-20921102067G
D. Altshul,Hydraulic Friction Losses in the Pipelines. Moscow:Gosenergoizdat, 1963.
A. Avci and I. Karagoz, “A novel explicit equation for friction factorin smooth and rough pipes,”J. Fluids Eng, vol. 131, no. 6, Jun. 2009.[Online]. Available:https://doi.org/10.1115/1.3129132
D. Barr and C. White, “Technical note. solutions of thecolebrook-white functions for resistance to uniform turbulentflows,”Proceedings of the Institution of Civil Engineers, vol. 71,no. 2, Jun. 17, 2015. [Online]. Available:https://doi.org/10.1680/iicep.1981.1895
N. H. Chen, “An explicit equation for friction factor in pipe,”Ind. Eng.Chem. Fundamen, vol. 18, no. 3, Aug. 01, 1979. [Online]. Available:https://doi.org/10.1021/i160071a019
S. W. Churchill, “Empirical expressions for the shear stress inturbulent flow in commercial pipe,”Aiche Journal, vol. 19, Mar. 1973.[Online]. Available:https://doi.org/10.1002/AIC.690190228
S. W. Churchill”, “Friction factor equation spans all fluid flowregimes,”Chemical Engineering, vol. 84, no. 24, 1977. [Online].Available:https://tinyurl.com/yeynd48k
B. J. Eck, “Use of a smoothed model for pipe friction loss,”Journalof Hydraulic Engineering, vol. 143, no. 1, Aug. 30, 2017. [Online].Available:https://doi.org/10.1061/(ASCE)HY.1943-7900.0001239
A. Ghanbari, F. Farshad, and H. H. Rieke, “Newly developed frictionfactor correlation for pipe flow and flow assurance,”Journal ofChemical Engineering and Materials Science, vol. 2, no. 6, Jun. 2011.[Online]. Available:http://www.academicjournals.org/jcems
S. E. Haaland, “Simple and explicit formulas for the friction factor inturbulent pipe flow,”Journal of Chemical Engineering and MaterialsScienceJ. Fluids Eng, vol. 105, no. 1, Mar. 01, 1983. [Online].Available:https://doi.org/10.1115/1.3240948
G. Manadilli, “Replace implicit equations with signomial functions,”Chemical Engineering, vol. 104, 1997. [Online]. Available:https://api.semanticscholar.org/CorpusID:126188836
B. J. McKeon, M. V. Zagarola, and A. J. Smits, “A new frictionfactor relationship for fully developed pipe flow,”Journal ofFluid Mechanics, vol. 538, Aug. 17, 2005. [Online]. Available:https://doi.org/10.1017/S0022112005005501
K. F. Pávlov, P. G. Romankov, and A. A. Noskov,Problemasyejemplospara el curso de operaciones básicas y aparatos en tecnología química.Moscú: Edición Mir, 1981.
G. F. Round, “An explicit approximation for the frictionfactor-reynolds number relation for rough and smooth pipes,”Chemical Engineering, vol. 58, no. 1, Feb. 1980. [Online]. Available:https://doi.org/10.1002/cjce.5450580119
N. H. Chen, “An explicit equation for friction factor in pipe,”Ind.Eng. Chem. Fundamen, vol. 18, no. 3, Aug. 1979. [Online]. Available:https://doi.org/10.1021/i160071a019
A. K. Jain, “Accurate explicit equation for friction factor,”Journal ofthe Hydraulics Division, vol. 102, no. 5, May. 1976. [Online]. Available:https://doi.org/10.1021/i160071a019
F. referencia, “Falta referencia,”Falta referencia, vol. Faltareferencia, no. 5, May. 1976. [Online]. Available:Faltareferencia
A. Olivares-Gallardo, R. Guerra-Rojas, and M. Alfaro-Guerra, “Newexplicit correlation to compute the friction factor under turbulentflow in pipes,”Revista Brasileira de Engenharia Agrícola e Ambiental,vol. 25, no. 7, Jul. 2021. [Online]. Available:10.1590/1807-1929/agriambi.v25n7p439-445-v247
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