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Journal articles on the topic 'Colebrook equation'

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Published: 5 June 2025

Last updated: 2 August 2025

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1

Zigrang, D. J., and N. D. Sylvester. "A Review of Explicit Friction Factor Equations." Journal of Energy Resources Technology 107, no. 2 (1985): 280–83. http://dx.doi.org/10.1115/1.3231190.

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A review of the explicit friction factor equations developed to replace the Colebrook equation is presented. Explicit friction factor equations are developed which yield a very high degree of precision compared to the Colebrook equation. A new explicit equation, which offers a reasonable compromise between complexity and accuracy, is presented and recommended for the calculation of all turbulent pipe flow friction factors for all roughness ratios and Reynold’s numbers.

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2

Praks, Pavel, and Dejan Brkić. "Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction." Advances in Civil Engineering 2018 (December 10, 2018): 1–18. http://dx.doi.org/10.1155/2018/5451034.

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The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re

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3

Zeghadnia, Lotfi, Bachir Achour та Jean Robert. "Discussion of “Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function” by DejanBrkić; and Pavel Praks, Mathematics 2019, 7, 34; doi:10.3390/math7010034". Mathematics 7, № 3 (2019): 253. http://dx.doi.org/10.3390/math7030253.

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The Colebrook-White equation is often used for calculation of the friction factor in turbulent regimes; it has succeeded in attracting a great deal of attention from researchers. The Colebrook–White equation is a complex equation where the computation of the friction factor is not direct, and there is a need for trial-error methods or graphical solutions; on the other hand, several researchers have attempted to alter the Colebrook-White equation by explicit formulas with the hope of achieving zero-percent (0%) maximum deviation, among them Dejan Brkić and Pavel Praks. The goal of this paper is

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4

Praks, Pavel, та Dejan Brkić. "Suitability for coding of the Colebrook’s flow friction relation expressed by symbolic regression approximations of the Wright-ω function". Reports in Mechanical Engineering 1, № 1 (2020): 174–79. http://dx.doi.org/10.31181/rme200101174p.

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This article analyses a form of the empirical Colebrook’s pipe flow friction equation given originally by the Lambert W-function and recently also by the Wright ω-function. These special functions are used to explicitly express the unknown flow friction factor of the Colebrook equation, which is in its classical formulation given implicitly. Explicit approximations of the Colebrook equation based on approximations of the Wright ω-function given by an asymptotic expansion and symbolic regression were analyzed in respect of speed and accuracy. Numerical experiments on 8 million Sobol’s quasi-Mon

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5

Pimenta, Bruna D., Adroaldo D. Robaina, Marcia X. Peiter, Wellington Mezzomo, Jardel H. Kirchner, and Luis H. B. Ben. "Performance of explicit approximations of the coefficient of head loss for pressurized conduits." Revista Brasileira de Engenharia Agrícola e Ambiental 22, no. 5 (2018): 301–7. http://dx.doi.org/10.1590/1807-1929/agriambi.v22n5p301-307.

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ABSTRACT One of the parameters involved in the design of pressurized hydraulic systems is the pressure drop in the pipes. The verification of the pressure drop can be performed through the Darcy-Weisbach formulation, which considers a coefficient of head loss (f) that can be estimated by the implicit Colebrook-White equation. However, for this determination, it is necessary to use numerical methods or the Moody diagram. Because of this, numerous explicit approaches have been proposed to overcome such limitation. In this sense, the objective of this study was to analyze the explicit approximati

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6

Cahyono, Muhammad. "Hybrid Models for Solving the Colebrook–White Equation Using Artificial Neural Networks." Fluids 7, no. 7 (2022): 211. http://dx.doi.org/10.3390/fluids7070211.

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This study proposes hybrid models to solve the Colebrook–White equation by combining explicit equations available in the literature to solve the Colebrook–White equation with an error function. The hybrid model is in the form of fH=fo−eA. fH is the friction factor value f predicted by the hybrid model, fo is the value of f calculated using several explicit formulas for the Colebrook–White equation, and eA is the error function determined using the neural network procedures. The hybrid equation consists of a series of hyperbolic tangent functions whose number corresponds to the number of neuron

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7

Brkić, Dejan, and Žarko Ćojbašić. "Intelligent Flow Friction Estimation." Computational Intelligence and Neuroscience 2016 (2016): 1–10. http://dx.doi.org/10.1155/2016/5242596.

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Nowadays, the Colebrook equation is used as a mostly accepted relation for the calculation of fluid flow friction factor. However, the Colebrook equation is implicit with respect to the friction factor (λ). In the present study, a noniterative approach using Artificial Neural Network (ANN) was developed to calculate the friction factor. To configure the ANN model, the input parameters of the Reynolds Number (Re) and the relative roughness of pipe (ε/D) were transformed to logarithmic scales. The 90,000 sets of data were fed to the ANN model involving three layers: input, hidden, and output lay

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8

Falade, A., A. Olaberinjo, M. Oyewola, F. Babalola, and S. Adaramola. "KPIM of Gas Transportation: Robust Modification of Gas Pipeline Equations." Latvian Journal of Physics and Technical Sciences 45, no. 5 (2008): 39–47. http://dx.doi.org/10.2478/v10047-008-0024-4.

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KPIM of Gas Transportation: Robust Modification of Gas Pipeline Equations Studies of the flow conditions of natural gases in pipelines have led to the development of complex equations for relating the volume transmitted through a gas pipeline to the various factors involved, thus deciding the optimum pressures and pipeline dimensions to be used. From equations of this type, various combinations of pipe diameter and wall thickness for a desired rate of gas throughput can be calculated. This research work presents modified forms of the basic gas flow equation for horizontal flow developed by Wey

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9

Clamond, Didier. "Efficient Resolution of the Colebrook Equation." Industrial & Engineering Chemistry Research 48, no. 7 (2009): 3665–71. http://dx.doi.org/10.1021/ie801626g.

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10

López-Silva, Maiquel, Hernández Dayma Sadami Carmenates, Nancy Delgado-Hernández, and Bereche Nataly Noelia Chunga. "Explicit pipe friction factor equations: evaluation, classification, and proposal." Revista Facultad de Ingeniería, Universidad de Antioquia, no. 111 (October 9, 2023): 38–47. https://doi.org/10.17533/udea.redin.20230928.

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The Colebrook equation has been used to estimate the friction factor (<em>f</em>) 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 (∆<em>f/f</em>), Percentage Standard Deviation (PS

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11

Brkić, Dejan, та Pavel Praks. "Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function". Mathematics 7, № 1 (2018): 34. http://dx.doi.org/10.3390/math7010034.

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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

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12

Brkić, Dejan, та Pavel Praks. "Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to Discussion". Mathematics 7, № 5 (2019): 410. http://dx.doi.org/10.3390/math7050410.

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This reply gives two corrections of typographical errors in respect to the commented article, and then provides few comments in respect to the discussion and one improved version of the approximation of the Colebrook equation for flow friction, based on the Wright ω-function. Finally, this reply gives an exact explicit version of the Colebrook equation expressed through the Wright ω-function, which does not introduce any additional errors in respect to the original equation. All mentioned approximations are computationally efficient and also very accurate. Results are verified using more than

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13

Vatankhah, Ali R. "Approximate Analytical Solutions for the Colebrook Equation." Journal of Hydraulic Engineering 144, no. 5 (2018): 06018007. http://dx.doi.org/10.1061/(asce)hy.1943-7900.0001454.

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14

Brkić, Dejan, and Pavel Praks. "Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression." Computation 7, no. 3 (2019): 48. http://dx.doi.org/10.3390/computation7030048.

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The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and Wright Ω functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Padé approximant of the first orde

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15

NIKITIN, Maxim Nikolaevich, Tatyana Sergeevna SOLOVYOVA, and Olga Vladimirovna SHLYAHTINA. "SOLUTIONS IN EXPLICIT FORM FOR DETERMINING THE HYDRAULIC RESISTANCE COEFFICIENT FOR TURBULENT FLOW." Urban construction and architecture 9, no. 4 (2019): 39–46. http://dx.doi.org/10.17673/vestnik.2019.04.7.

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A comparative analysis of explicit solutions of the Colebrook-White equation is carried out. The median values of relative deviations, coefficients of determination and computational complexities for each approximation were obtained. The results of the iterative solution of the Colebrook-White equation by successive substitution method were used as the intrinsic solution. Approximations by B. Eck and A.R. Vatankhah were identified as the most effective in terms of computational complexity. It was shown that widely used approximations by P.R.H. Blasius, A.D. Altshul and J. Nikuradze although si

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16

Mikata, Yozo, and Walter S. Walczak. "Exact Analytical Solutions of the Colebrook-White Equation." Journal of Hydraulic Engineering 142, no. 2 (2016): 04015050. http://dx.doi.org/10.1061/(asce)hy.1943-7900.0001074.

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17

Hafsi, Zahreddine. "Accurate explicit analytical solution for Colebrook-White equation." Mechanics Research Communications 111 (January 2021): 103646. http://dx.doi.org/10.1016/j.mechrescom.2020.103646.

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18

Rollmann, P., and K. Spindler. "Explicit representation of the implicit Colebrook–White equation." Case Studies in Thermal Engineering 5 (March 2015): 41–47. http://dx.doi.org/10.1016/j.csite.2014.12.001.

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19

Qiu, Mengning, and Avi Ostfeld. "A Head Formulation for the Steady-State Analysis of Water Distribution Systems Using an Explicit and Exact Expression of the Colebrook–White Equation." Water 13, no. 9 (2021): 1163. http://dx.doi.org/10.3390/w13091163.

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Steady-state demand-driven water distribution system (WDS) solution is the bedrock for much research conducted in the field related to WDSs. WDSs are modeled using the Darcy–Weisbach equation with the Swamee–Jain equation. However, the Swamee–Jain equation approximates the Colebrook–White equation, errors of which are within 1% for ϵ/D∈[10−6,10−2] and Re∈[5000,108]. A formulation is presented for the solution of WDSs using the Colebrook–White equation. The correctness and efficacy of the head formulation have been demonstrated by applying it to six WDSs with the number of pipes ranges from 454

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20

Praks, Pavel, та Dejan Brkić. "Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to the Discussion by Majid Niazkar". Mathematics 8, № 5 (2020): 796. http://dx.doi.org/10.3390/math8050796.

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In this reply, we present updated approximations to the Colebrook equation for flow friction. The equations are equally computational simple, but with increased accuracy thanks to the optimization procedure, which was proposed by the discusser, Dr. Majid Niazkar. Our large-scale quasi-Monte Carlo verifications confirm that the here presented novel optimized numerical parameters further significantly increase accuracy of the estimated flow friction factor.

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21

Praks, Pavel, and Dejan Brkić. "Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction." Water 10, no. 9 (2018): 1175. http://dx.doi.org/10.3390/w10091175.

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Widely used in hydraulics, the Colebrook equation for flow friction relates implicitly to the input parameters; the Reynolds number, Re and the relative roughness of an inner pipe surface, ε/D with an unknown output parameter; the flow friction factor, λ; λ = f (λ, Re, ε/D). In this paper, a few explicit approximations to the Colebrook equation; λ ≈ f (Re, ε/D), are generated using the ability of artificial intelligence to make inner patterns to connect input and output parameters in an explicit way not knowing their nature or the physical law that connects them, but only knowing raw numbers,

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22

OKYERE, Mavis Sika, Lucas Nana Wiredu DAMOAH, Emmanuel NYANKSON, and David Sasu KONADU. "SYNERGETIC EFFECT OF A DRAG REDUCER AND PIPELINE INTERNAL COATING ON CAPACITY ENHANCEMENT IN OIL AND GAS PIPELINES: A CASE STUDY." European Journal of Materials Science and Engineering 7, no. 3 (2022): 195–210. http://dx.doi.org/10.36868/ejmse.2022.07.03.195.

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A case study of a 20-inch diameter gas transmission pipeline assesses theoretically how the synergetic use of pipeline internal coating and drag reducing agent increases the flow rate of a pipeline and its impact on the pipe internal friction. The American Gas Association (AGA) equation and Modified Colebrook-White equations were both used to estimate the capacity of the pipeline in its existing state, after internally coating and after injecting a drag reducing agent in the internally coated pipeline. By means of both AGA equations and Modified Colebrook–White equations, it was observed that

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23

Benavides-Muñoz, Holger Manuel. "Modification and Improvement of the Churchill Equation for Friction Factor Calculation in Pipes." Water 16, no. 16 (2024): 2328. http://dx.doi.org/10.3390/w16162328.

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Accurate prediction of the friction factor is fundamental for designing and calibrating fluid transport systems. While the Colebrook–White equation is the benchmark for precision due to its physical basis, its implicit nature hinders practical applications. Explicit correlations like Churchill’s equation are commonly used but often sacrifice accuracy. This study introduces two novel modifications to Churchill’s equation to enhance predictive capabilities. Developed through a rigorous analysis of 240 test cases and validated against a dataset of 21,000 experiments, the proposed Churchill B(Re)

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24

Niazkar, Majid. "Discussion of “Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function” by Dejan Brkić and Pavel Praks, Mathematics 2019, 7, 34; doi:10.3390/math7010034". Mathematics 8, № 5 (2020): 793. http://dx.doi.org/10.3390/math8050793.

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Estimating the Darcy–Weisbach friction factor is crucial to various engineering applications. Although the literature has accepted the Colebrook–White formula as a standard approach for this prediction, its implicit structure brings about an active field of research seeking for alternatives more suitable in practice. This study mainly attempts to increase the precision of two explicit equations proposed by Brkić and Praks. The results obviously demonstrate that the modified relations outperformed the original ones from nine out of 10 accuracy evaluation criteria. Finally, one of the improved e

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25

Olivares Gallardo, Alan Paulo, Rodrigo Alejandro Guerra Rojas, and Marco Antonio Alfaro Guerra. "Evaluación experimental de la solución analítica exacta de la ecuación de Colebrook-White." Ingeniería Investigación y Tecnología 20, no. 2 (2019): 1–11. http://dx.doi.org/10.22201/fi.25940732e.2019.20n2.021.

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En el presente trabajo se diseñó y construyó un sistema hidráulico para la determinación experimental delfactor de fricción en una tubería de PVC bajo flujo turbulento. Este sistema permite conducir el agua por una tubería donde se midieron el caudal y la diferencia de presión para calcular la pérdida de carga entre dos puntos de control, considerando las propiedades asociadas a la tubería como es la rugosidad relativa, su diámetro y longitud. Utilizando la ecuación de Darcy-Weisbach se determinó el factor de fricción experimental. El objetivo del presente trabajo fue evaluar en forma experime

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26

Easa, Said M., Ahmed A. Lamri, and Dejan Brkić. "Reliability-Based Criterion for Evaluating Explicit Approximations of Colebrook Equation." Journal of Marine Science and Engineering 10, no. 6 (2022): 803. http://dx.doi.org/10.3390/jmse10060803.

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Numerous explicit approximations of the Colebrook equation have been developed and evaluated based on two criteria: prediction accuracy and computational efficiency. This paper introduces a new evaluation criterion based on the reliability of each equation. The reliability is defined by the coefficient of variation (CV) of the explicit friction factor that is a function of the variabilities of component random variables (roughness height of the internal pipe surface and kinematic viscosity of the fluid). The coefficient of variation of the friction factor depends on its first derivative for ro

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27

Augusto, Gerardo L., Alvin B. Culaba, and Renan Ma T. Tanhueco. "Pipe Sizing of District Cooling Distribution Network Using Implicit Colebrook-White Equation." Journal of Advanced Computational Intelligence and Intelligent Informatics 20, no. 1 (2016): 76–83. http://dx.doi.org/10.20965/jaciii.2016.p0076.

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An implicit solution of Colebrook-White equation was used in calculating the friction factor for commercial steel pipes using Newton-Raphson method with Reynolds number ranging from 4.0 × 103to 1.3 × 107. Initial value for iterative friction factor estimation was based on expanded form of Colebrook-White equation for larger values of Reynolds number with tolerance value of 1.0 × 10-8. Numerical results were compared with known explicit solutions and iterative procedure proposed by Lester in which, their mean difference, root-mean square deviation, mean relative error and correlation coefficien

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28

Praks, Pavel, and Dejan Brkić. "Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation." Mathematics 8, no. 1 (2019): 26. http://dx.doi.org/10.3390/math8010026.

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The empirical logarithmic Colebrook equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use of transcendental functions, such as logarithm or non-integer power, which require execution of the additional number of floating-point operations in computer processor units. Instead of these, we use only rational expressions that are executed directly in the processo

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29

MATTHEW, G. D. "THE COLEBROOK-WHITE EQUATION - AN OFT CITED RESULT BUT NEGLECTED DERIVATION ?" Proceedings of the Institution of Civil Engineers 89, no. 1 (1990): 39–45. http://dx.doi.org/10.1680/iicep.1990.5250.

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30

Sonnad, Jagadeesh R., and Chetan T. Goudar. "Constraints for Using Lambert W Function-Based Explicit Colebrook–White Equation." Journal of Hydraulic Engineering 130, no. 9 (2004): 929–31. http://dx.doi.org/10.1061/(asce)0733-9429(2004)130:9(929).

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31

Lira, Ignacio. "On the Uncertainties Stemming from Use of the Colebrook-White Equation." Industrial & Engineering Chemistry Research 52, no. 22 (2013): 7550–55. http://dx.doi.org/10.1021/ie4001053.

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32

., U. H. Offor. "PERFORMANCE EVALUATION OF THE EXPLICIT APPROXIMATIONS OF THE IMPLICT COLEBROOK EQUATION." International Journal of Research in Engineering and Technology 05, no. 08 (2016): 1–12. http://dx.doi.org/10.15623/ijret.2016.0508001.

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33

Praks, Pavel, and Dejan Brkić. "Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation." Mathematics 8, no. 1 (2019): 26. https://doi.org/10.3390/math8010026.

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The empirical logarithmic Colebrook equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use of transcendental functions, such as logarithm or non-integer power, which require execution of the additional number of floating-point operations in computer processor units. Instead of these, we use only rational expressions that are executed directly in the processo

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34

Brkić, Dejan, and Zoran Stajić. "EXCEL VBA-BASED USER DEFINED FUNCTIONS FOR HIGHLY PRECISE COLEBROOK’S PIPE FLOW FRICTION APPROXIMATIONS: A COMPARATIVE OVERVIEW." Facta Universitatis, Series: Mechanical Engineering 19, no. 2 (2021): 253. http://dx.doi.org/10.22190/fume210111044b.

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This review paper gives Excel functions for highly precise Colebrook’s pipe flow friction approximations developed by users. All shown codes are implemented as User Defined Functions – UDFs written in Visual Basic for Applications – VBA, a common programming language for MS Excel spreadsheet solver. Accuracy of the friction factor computed using nine to date the most accurate explicit approximations is compared with the sufficiently accurate solution obtained through an iterative scheme which gives satisfying results after sufficient number of iterations. The codes are given for the presented

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Praks, Pavel, and Dejan Brkić. "Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation." Processes 6, no. 8 (2018): 130. http://dx.doi.org/10.3390/pr6080130.

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The Colebrook equation is implicitly given in respect to the unknown flow friction factor λ; λ = ζ ( R e , ε * , λ ) which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. A common approach to solve it is through the Newton–Raphson iterative procedure or through the fixed-point iterative procedure. Both require in some cases, up to seven iterations. On the other hand, numerous more powerful iterative methods such as three- or two-point methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implici

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36

Ciprian, Bacoţiu. "The Quest for the Ideal Darcy-Weisbach Friction Factor Equation from the Perspective of a Building Services Engineer." Ovidius University Annals of Constanta - Series Civil Engineering 21, no. 1 (2019): 65–73. http://dx.doi.org/10.2478/ouacsce-2019-0008.

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Abstract The calculation of the friction factor involved in the Darcy-Weisbach equation has a key role in the accurate assessment of distributed head losses. For the turbulent flow regime, this friction factor was mathematically expressed in the form of the Colebrook-White (C-W) equation, widely accepted by engineers and scientists. Nevertheless, the C-W equation is an implicit one and must be solved using numerical methods. This is a major disadvantage for the average engineer, who always prefers an explicit equation which could be easily integrated into his familiar spreadsheet environment.

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37

Brkić, Dejan. "Comparison of the Lambert W‐function based solutions to the Colebrook equation." Engineering Computations 29, no. 6 (2012): 617–30. http://dx.doi.org/10.1108/02644401211246337.

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38

Santos-Ruiz, Ildeberto, Francisco-Ronay López-Estrada, Vicenç Puig, and Guillermo Valencia-Palomo. "Simultaneous Optimal Estimation of Roughness and Minor Loss Coefficients in a Pipeline." Mathematical and Computational Applications 25, no. 3 (2020): 56. http://dx.doi.org/10.3390/mca25030056.

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This paper presents a proposal to estimate simultaneously, through nonlinear optimization, the roughness and head loss coefficients in a non-straight pipeline. With the proposed technique, the calculation of friction is optimized by minimizing the fitting error in the Colebrook–White equation for an operating interval of the pipeline from the flow and pressure measurements at the pipe ends. The proposed method has been implemented in MATLAB and validated in a serpentine-shaped experimental pipeline by contrasting the theoretical friction for the estimated coefficients obtained from the Darcy–W

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39

Winning, Herbert Keith, and Tim Coole. "Improved method of determining friction factor in pipes." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 4 (2015): 941–49. http://dx.doi.org/10.1108/hff-06-2014-0173.

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Purpose – The purpose of this paper is to present an improved computational method for determining the friction factor for turbulent flow in pipes. Design/methodology/approach – Given that the absolute pipe roughness is generally constant in most systems, and that there are few changes to the pipe diameter, the proposed method uses a simplified equation for systems with a specific relative pipe roughness. The accuracy of the estimation of the friction factor using the proposed method is compared to the values obtained using the implicit Colebrook-White equation while the computational efficien

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40

Vatankhah, Ali R. "Closure to “Approximate Analytical Solutions for the Colebrook Equation” by Ali R. Vatankhah." Journal of Hydraulic Engineering 146, no. 2 (2020): 07019013. http://dx.doi.org/10.1061/(asce)hy.1943-7900.0001666.

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Brkić, Dejan, and Pavel Praks. "Discussion of “Approximate Analytical Solutions for the Colebrook Equation” by Ali R. Vatankhah." Journal of Hydraulic Engineering 146, no. 2 (2020): 07019011. http://dx.doi.org/10.1061/(asce)hy.1943-7900.0001667.

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42

Lamri, Ahmed Amine. "Discussion of “Approximate Analytical Solutions for the Colebrook Equation” by Ali R. Vatankhah." Journal of Hydraulic Engineering 146, no. 2 (2020): 07019012. http://dx.doi.org/10.1061/(asce)hy.1943-7900.0001668.

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43

Sonnad, Jagadeesh R., and Chetan T. Goudar. "Explicit Reformulation of the Colebrook−White Equation for Turbulent Flow Friction Factor Calculation." Industrial & Engineering Chemistry Research 46, no. 8 (2007): 2593–600. http://dx.doi.org/10.1021/ie0340241.

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44

Brkić, Dejan. "Revised Friction Groups for Evaluating Hydraulic Parameters: Pressure Drop, Flow, and Diameter Estimation." Journal of Marine Science and Engineering 12, no. 9 (2024): 1663. http://dx.doi.org/10.3390/jmse12091663.

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Suitable friction groups are provided for solving three typical hydraulic problems. While the friction group based on viscous forces is used for calculating the pressure drop or head loss in pipes and open channels, commonly referred to as the Type 1 problem in hydraulic engineering, additional friction groups with similar behaviors are introduced for calculating steady flow discharge as the Type 2 problem and, for estimating hydraulic diameter as the Type 3 problem. Contrary to the viscous friction group, the traditional Darcy–Weisbach friction factor demonstrates a negative correlation with

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Moukam, Tchawe Tchawe, Tientcheu Nsiewe Maxwell, Djiako Thomas, Nkontchou Ngongang François Legrand, Tcheukam-Toko Dénis, and Kenmeugne Bienvenu. "Graphical Application of The Colebrook-White Equation to the Songloulou and Lagdo Dam Penstocks." DS Journal of Digital Science and Technology 3, no. 2 (2024): 1–8. http://dx.doi.org/10.59232/dst-v3i2p101.

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Heydari, Amir, Elhameh Narimani, and Fatemeh Pakniya. "Explicit Determinations of the Colebrook Equation for the Flow Friction Factor by Statistical Analysis." Chemical Engineering & Technology 38, no. 8 (2015): 1387–96. http://dx.doi.org/10.1002/ceat.201400590.

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Sonnad, Jagadeesh R., and Chetan T. Goudar. "Turbulent Flow Friction Factor Calculation Using a Mathematically Exact Alternative to the Colebrook–White Equation." Journal of Hydraulic Engineering 132, no. 8 (2006): 863–67. http://dx.doi.org/10.1061/(asce)0733-9429(2006)132:8(863).

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Samadianfard, Saeed. "Gene expression programming analysis of implicit Colebrook–White equation in turbulent flow friction factor calculation." Journal of Petroleum Science and Engineering 92-93 (August 2012): 48–55. http://dx.doi.org/10.1016/j.petrol.2012.06.005.

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Aji, Anas Satria, and Rudi Siswanto. "RE-DESIGN SISTEM DISTRIBUSI AIR BERSIH DAN FIRE HYDRANT DI GEDUNG PLN UP3B KALSELTENG." JTAM ROTARY 3, no. 1 (2021): 29. http://dx.doi.org/10.20527/jtam_rotary.v3i1.3464.

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Sistem penyediaan dan distribusi air bersih di gedung-gedung bertingkat sering mengalami kendala. Tidak terkecuali PT. PLN (Persero) Gedung UP3B Kalselteng yang tidak luput dari permasalahan tersebut. Untuk itu perlu dilakukan perancangan ulang sistem distribusi dan penyediaan air bersih serta hidran kebakaran pada gedung. Tujuan dari penelitian ini antara lain menganalisis kebutuhan dan distribusi air bersih dan hidran kebakaran di dalam gedung. Kebutuhan air bersih dihitung berdasarkan jumlah penduduk yang mengkonsumsi air bersih, jenis dan jumlah unit beban alat perpipaan. Perhitungan head

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Praks, Pavel, and Dejan Brkić. "One-Log Call Iterative Solution of the Colebrook Equation for Flow Friction Based on Padé Polynomials." Energies 11, no. 7 (2018): 1825. http://dx.doi.org/10.3390/en11071825.

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The 80 year-old empirical Colebrook function ξ, widely used as an informal standard for hydraulic resistance, relates implicitly the unknown flow friction factor λ, with the known Reynolds number Re and the known relative roughness of a pipe inner surface ε*; λ=ξ(Re,ε*,λ). It is based on logarithmic law in the form that captures the unknown flow friction factor λ in a way that it cannot be extracted analytically. As an alternative to the explicit approximations or to the iterative procedures that require at least a few evaluations of computationally expensive logarithmic function or non-intege

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