Explicit Expressions for Calculating the Turbulent Friction Coefficient: A Review

Nkontchou Ngongang François; Tchawe Tchawe Moukam; Tientcheu Nsiewe Maxwell; Djiako Thomas; Kenmeugne Bienvenu

doi:doi:10.11648/j.ijepe.20261504.13

Research Article |

| Peer-Reviewed

Explicit Expressions for Calculating the Turbulent Friction Coefficient: A Review

Nkontchou Ngongang François

, Tchawe Tchawe Moukam^*

, Tientcheu Nsiewe Maxwell, Djiako Thomas, Kenmeugne Bienvenu

Published in International Journal of Energy and Power Engineering (Volume 15, Issue 4)

Received: 30 May 2026 Accepted: 22 June 2026 Published: 28 July 2026

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Abstract

Research on friction in pipes remains complex. This complexity is due to the numerous parameters that must be considered when calculating the coefficient of friction f. Due to this complexity, many researchers have proposed multiple formulas for determining the friction coefficient. Today, there are a large number of explicit expressions for calculating the friction coefficient for each flow regime, whether for smooth or rough pipes. In this review, we set out to study the evolution of work on the friction coefficient and to present it. For this purpose, a large number of reviews have been used. Once this has been done, we identified more than 120 expressions, most of which were explicit. We have thus noted that many of these expressions are approximations of the implicit Colebrook-White equation. Similarly, a good number are derived either from direct developments using computer tools, or from mathematical methods that are sometimes very long. However, in general, we have found that all these expressions only take into account the three parameters present in the Colebrook-White formula. These parameters are: Reynolds number Re, the absolute roughness ɛ and the diameter D. We would like to point out that these parameters were deemed insufficient to minimise the difference between the calculated values and the values observed on site with regard to the friction coefficient.

Published in	International Journal of Energy and Power Engineering (Volume 15, Issue 4)
DOI	10.11648/j.ijepe.20261504.13
Page(s)	115-123
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Friction Coefficient, Colebrook-White, Explicit Approximation, Explicit Expression, Implicit Expression

1. Introduction

To determine the flow rate of a fluid in a pipe or pipe network, it is important to first calculate the head losses caused by friction on the walls. To achieve this, it is first necessary to calculate the coefficient of friction (f). This coefficient is linked to both head losses and the viscous effects of the fluid

[1]

. Traditionally, the coefficient of friction is estimated using the Colebrook-White equation or the Moody diagram.

In general, the determination of the value of the coefficient of friction depends on the fluid’s flow regime in the pipe and the pipe’s physical characteristics. Consequently, there are several possible scenarios.

For smooth pipes, the expressions for calculating f are similar regardless of the flow regime. The literature suggests that there are no major difficulties in this case. Once the flow regime becomes turbulent, as it is most often the case in hydraulic structures, we observe a wide variety of expressions for calculating friction coefficient f. This variety is even greater when the pipe is rough. Some of these expressions have been obtained by processing experimental data

[2]

The Moody diagram which has been used by engineers since 1945 is semi-empirical and the conditions it describes (fully developed flow, isothermal, incompressible, dissipative, quasi-stationary) are rarely sufficiently achieved in a real-world scenari

[3]

The literature review conducted as part of this work shows that there are two main approaches to calculating friction coefficient. There is an implicit approach fully represented by the Colebrook-White formula, and an explicit approach with a wide variety of expressions in the literature. It should be noted that a large number of explicit approaches are based on the Colebrook-White equation.

The Colebrook and White formula is one of the most frequently used expressions for calculating the coefficient of friction in the turbulent regime

[4]

. It is given by Eq. (1) below.

f = - 2 \log (\frac{ε}{3.7 D} + \frac{2.51}{R_{e} \sqrt{f}})

(1)

With

\frac{ε}{D}

the relative pipe roughness and Re the Reynold's number.

This formula is not practical, as the expression is implicit. Iterations are required to perform the calculation. To find the value of f defined in this equation, you need to use numerical algorithms, which is not easy. This is why its use is not recommended in practical calculations by engineers or for students in courses related to fluid mechanics

[5]

. The difficulties in determining the coefficient of friction using Eq. (1) have led many authors to develop explicit equations that can be used directly.

In this review, we present a comprehensive set of formulas for calculating the coefficient of friction in pipes.

2. Current Situation

Table 1 below shows the platforms used for publishing new coefficient of friction formulas; scientific articles published by various journals with a high percentage for a high percentage (around 85%) of the total sources consulted.

Table 1. Bibliographical sources consulted.

Consulted	Number
Newspaper articles	118
Books	10
Reports	5
Lectures	2
Book chapters	1
Unpublished	3

Table 2 below shows the authors who published at least two expressions for calculating friction.

Table 2. Authors with the highest number of publications.

Author	Number of publications
Brkic D.	9
Vatankhah A.	4
Swamee P. K.	3
Achour B.	2
Avci A	2
Bedjaoui A.	2
Chen J.	2
Chernikin V.	2
Churchill S.	2
Genic S.	2
Giustolisi O.	2
Goudar C.	2
Jain A. K.	2
Karagoz I.	2
Kouchakzadeh S.	2
Mileikovskyi V.	2
Niazkar M.	2
Petresin E.	2
Praks P.	3
Sonnad J.	2
Tkachenko T.	2
Zigrang D.	2

Similarly, we aimed to place the development of research on this parameter over time. Figure 1 below illustrates the evolution of research into the coefficient of friction in pipes, showing a high level of interest after the year 2000.

Download: Download full-size image

Figure 1. Publication of friction coefficient formulas by year.

3. Approaches to Solving the Colebrook-White Equation

In general, there are six approaches available for solving the Colebrook-White equation

[6]

1) Graphical solutions, such as those using Moody or Rouse diagrams.

2) The i Iterative schemes, for example, the Newton-Raphson method

[7]

3) Explicit approximate relations

[8, 9]

4) Lambert's W function

[6, 10]

5) The trial-and-error method

[7, 11]

6) Artificial intelligence models

[12-14]

Of all these approaches, the second is technically considered not only the most accurate, but also the reference solution for calculating friction coefficient. However, it requires more computational effort, which is its only drawback for proper implementation.

Authors have used various techniques and methods to propose new explicit expressions for calculating the turbulent friction coefficient. Computerized methods are increasingly being used to determine friction in pipes. Below, we list some authors along with the methods used to calculate the coefficient of friction:

1) Artificial Intelligence (AI) models, including Artificial Neural Networks (ANNs)

[12, 15-19]

;

2) The adaptive neurofuzzy technique

[13]

;

3) Gene expression programming

[14, 20-22]

4) The adaptive neuro-fuzzy inference system

[23]

4. Classification of Explicit Relations

In this section, we listed almost all the explicit formulas for calculating the turbulent friction coefficient available in the literature. They are presented chronologically and separately for both rough and smooth pipes. We note, however, that many of them are approximations of the Colebrook and White equation.

4.1. Expressions for Rough Pipes

We examine the explicit formulations developed to calculate the turbulent friction coefficient in rough pipes. Over the decades, a vast number of approximations of the Colebrook-White implicit equation have been proposed. Given the large number of these equations, it is impossible to present them all in the main body of the text. Therefore, Table 3 presents a rigorous selection of these explicit relationships. This subset highlights pioneering historical formulas, the most widely used approximations, high-precision mathematical models, as well as recent advances based on artificial intelligence or Lambert’s W-function. For a comprehensive overview, an exhaustive compilation of all explicit relationships identified in this study for rough pipes is provided in the Supplementary Material.

Table 3. Selected explicit approximations for rough pipes.

Author and Year

Formula

Method / Main characteristic

Range of application

Moody

[24]

$f = 0.001375 (1 + {(2 . 10^{4} \frac{ε}{D} + \frac{10^{6}}{R_{e}})}^{\frac{1}{3}})$

The first explicit form proposed for the Colebrook-White equation.

4x10³ ≤ Re ≤ 10⁸ and 0 ≤ (ε/D) ≤ 10^-2

Wood

[25]

$f = 0.094 {(\frac{ε}{D})}^{0.225} + 0.53 (\frac{ε}{D}) + 88 {(\frac{ε}{D})}^{0.44} . {R_{e}}^{- ν}$

where $ν$ is given by: $ν = 1.62 {(\frac{ε}{D})}^{0.134}$

An attempt to provide a simplified formula.

Re > 4000; and 0 ≤ (ε/D) ≤ 5×10^-2

Churchill

[26]

$\frac{1}{\sqrt{f}} = - 2 \log (\frac{ε}{3.71 D} + {(\frac{7}{R_{e}})}^{0.9})$

An empirical expression that replaces the Colebrook-White equation.

4000 < Re < 10⁸ and 10-6 < ε/D < 5×10^-2

Swamee-Jain

[8]

$\frac{1}{\sqrt{f}} = - 2 \log (\frac{ε}{3.71 D} + \frac{5.74}{{R_{e}}^{0.9}})$

Widely used and considered the best explicit approximation.

5×103 < Re < 10⁸ and 10^-6 < ε/D < 10^-2

Churchill

[27]

$f = 8 {({(\frac{8}{R_{e}})}^{12} + \frac{1}{{(A + B)}^{3 / 2}})}^{1 / 12}$

Where:

$A = {(2.457 \ln (\frac{1}{{(\frac{7}{R_{e}})}^{0.9} + \frac{0.27 ε}{D}}))}^{16}$

$B = {(\frac{37530}{R_{e}})}^{16}$

This formula applies to all fluid flow regimes.

4×10³ < Re < 10⁸ and 10⁻⁶ < ε/D < 0.05

Chen

[28]

$\frac{1}{\sqrt{f}} = - 2 \log (\frac{ε}{3.7065 D} - \frac{5.0452}{R_{e}} \log (\frac{1}{2.8257} {(\frac{ε}{D})}^{1.1098} + \frac{5.8506}{{R_{e}}^{0.8981}}))$

Detailed solution.

4×10³ ≤ Re ≤ 4×10⁸and 10^-7 ≤ (ε/D) ≤ 5×10^-2

Round

[29]

$\frac{1}{\sqrt{f}} = - 1.8 \log (\frac{R_{e}}{0.135 R_{e} (\frac{ε}{D}) + 6.5})$

A relatively simple explicit approximation.

4×10³ ≤ Re ≤ 10⁸ and 0 ≤ (ε/D) ≤ 10^-2

Barr

[30]

$\frac{1}{\sqrt{f}} = - 2 \log (\frac{ε}{3.7 D} + \frac{4.518 \log (\frac{R_{e}}{7})}{R_{e} (1 + \frac{{R_{e}}^{0.52}}{29} {(\frac{ε}{D})}^{0.7})})$

Does not require any internal iterative calculations.

2300 ≤ Re ≤ 10⁸ and 0 ≤ (ε/D) ≤ 5×10^-2

Zigrang and Sylvester

[31]

$\frac{1}{\sqrt{f}} = - 2 \log (\frac{ε / D}{3.7} - \frac{5.02}{R_{e}} \log (\frac{ε / D}{3.7} - \frac{13}{R_{e}}))$

$\frac{1}{\sqrt{f}} = - 2 \log (\frac{ε / D}{3.7} - \frac{5.02}{R_{e}} \log (\frac{ε / D}{3.7} - \frac{5.02}{R_{e}} \log (\frac{ε / D}{3.7} + \frac{13}{R_{e}})))$

It does not use an internal iterative procedure to achieve high accuracy.

4×10³ ≤ Re ≤ 10⁸ and 10^-6 ≤ (ε/D) ≤ 5×10^-2

Haaland

[32]

$\frac{1}{\sqrt{f}} = - \frac{1.8}{n} \log ({(\frac{ε}{3.75 D})}^{1.1 n} + {(\frac{6.9}{R_{e}})}^{n})$

A simple and clear formula, a straightforward solution.

3×10³≤ Re ≤1.5×10⁸ and 0 ≤ (ε/D) ≤ 5×10^-2

Serghides

[33]

$\frac{1}{\sqrt{f}} = A - \frac{{(B - A)}^{2}}{C - 2 B + A}$

With:

$A = - 2 \log (\frac{ε}{3.7 D} + \frac{12}{R_{e}})$

$B = - 2 \log (\frac{ε}{3.7 D} + \frac{2.51 A}{R_{e}})$

$C = - 2 \log (\frac{ε}{3.7 D} + \frac{2.51 B}{R_{e}})$

An approximation of the implicit form that is valid for all ranges.

2300 ≤ Re ≤ 10⁸ and 10^-6 ≤ (ε/D) ≤ 5×10^-2

Romeo et al.,

[34]

$\frac{1}{\sqrt{f}} = - 2 \log (\frac{ε / D}{3.7065} - (\frac{5.0272}{R_{e}}) \log (\frac{ε / D}{3.827} - (\frac{4.567}{R_{e}}) \log ({(\frac{ε / D}{7.7918})}^{0.9924} + {(\frac{5.3326}{208.815 + R_{e}})}^{0.9345})))$

Obtained using nonlinear multivariate regression.

3×10³ ≤ Re ≤ 1.5×10⁸ and 0 ≤ (ε/D) ≤ 5×10^-2

Sonnad and Goudar

[35]

$\frac{1}{\sqrt{f}} = 0.8686 \ln (\frac{0.4587 R_{e}}{S^{S / (S + 1)}})$

With:

$S = 0.124 R_{e} \frac{ε}{D} + \ln (0.4587 R_{e})$

A mathematical approximation used as a basis by many authors.

4×10⁴ ≤ Re ≤ 10⁸ and 10^-6 ≤ (ε/D) ≤ 5×10^-2

[36]

$f = \frac{1}{c {(W_{0} (\frac{\exp (\frac{a}{bc})}{bc}) - \frac{a}{b})}^{2}}$

W₀ is the main branch of Lambert's W function (Chapeau-Blondeau and Monir, 2002).

With:

$a = \frac{1}{3.7} \frac{ε}{D}; b = \frac{1.257}{R_{e}}; c = 1.7372$

Use d'Alembert's method and the principal branch of Lambert's W-function.

Not specified.

Brkic

[37]

$\frac{1}{\sqrt{f}} = - 2 \log (\frac{2.18}{R_{e}} (\frac{R_{e}}{1.816 \ln (\frac{1.1 R_{e}}{1 + 1.1 R_{e}})}) + \frac{ε}{3.71 D})$

Use different solutions of Lambert's W-function.

2300 ≤ Re ≤ 10⁸ and 0 ≤ (ε/D) ≤ 5×10^-2

Samadianfard

[14]

$f = (\frac{{R_{e}}^{ε / D} - 0.6315093}{{R_{e}}^{1 / 3} + R_{e} (\frac{ε}{D})}) + 0.0275308 {(\frac{6.929841}{R_{e}} + \frac{ε}{D})}^{\frac{1}{9}} + (\frac{10^{ε / D}}{\frac{ε}{D} + 4.781616}) (\sqrt{\frac{ε}{D}} + \frac{9.997001}{R_{e}})$

Explicit calculation based on genetic programming.

4×10³≤ Re ≤ 10⁸ and 0 ≤ (ε/D) ≤ 5×10^-2

Vatankhah

[38]

$f = {(\frac{\frac{2.51}{R_{e}} + 1.1513 δ}{δ - \frac{ε / D}{3.71} - 2.3026 δ \log (δ)})}^{2}$

Where:

$δ = \frac{6.0173}{R_{e} {(0.07 (\frac{ε}{D}) + {R_{e}}^{- 0.885})}^{0.109}} + \frac{ε / D}{3.71}$

Optimization of relative error as an objective function.

4×10³ ≤ Re ≤ 10⁸ and 10^-6 ≤ (ε/D) ≤ 5×10^-2

Offor and Alabi

[16]

$\frac{1}{\sqrt{f}} = - 2 \log (\frac{ε}{3.71 D} - \frac{1.975}{R_{e}} \ln ({(\frac{ε}{3.93 D})}^{1.092} + \frac{7.627}{R_{e} + 395.9}))$

Use of artificial neural networks (ANNs).

4×10³ ≤ Re ≤ 10⁸ and 10^-6 ≤ (ε/D) ≤ 5×10^-2

Azizi et al.

[39]

$f = {(1.805 \log (\frac{{(\frac{ε}{D})}^{1.108}}{4.267} + \frac{5.164}{{R_{e}}^{0.966}}))}^{- 2}$

Explicit correlation.

2×10³ ≤ Re ≤ 10⁸ and 10^-6 ≤ (ε/D) ≤ 5×10^-2

López-Silva et al.

[40]

$f = {0.219 ((\frac{0.028 ε}{D}) + (\frac{0.896}{R_{e}}))}^{0.25}$

Model formulated using genetic expression programming (GEP).

4000 < Re < 10⁸ and 10^-6 <ε/D < 10^-2.

4.2. Expressions for Smooth Pipes

Similar to rough pipes, a wide variety of expressions have been developed to estimate the friction coefficient in smooth pipes (ε/D=0). To provide a concise and relevant overview, Table 4 highlights the most significant explicit relationships found in the literature. This selection covers both the classic fundamental models for laminar and turbulent flow regimes and more specific correlations, notably those adapted to Bingham plastic fluids. As in the previous section, the exhaustive list of all explicit formulas identified for smooth pipes is available in the Supplementary Material.

Table 4. Selected explicit approximations for rough pipes.

Author and Year

Formula

Method / Main characteristic

Range of application

Hazen and Poiseuille cited by

[41]

$f = \frac{64}{R_{e}}$

Standard basic formula for laminar flow.

Re ≤ 2100

Blasius

[42]

$f = \frac{0.3164}{R_{e}^{0.25}} for 2100 \leq R_{e} \leq 2 \times 10^{4}$

$f = \frac{0.184}{R_{e}^{0.2}} for 2 \times 10^{4} \leq R_{e}$

The simplest numerical equation for smooth turbulent flow.

3×10³ ≤ Re ≤ 10⁵

Colebrook-White cited by

[32]

$\frac{1}{\sqrt{f}} = 1.8 \log_{10} (\frac{R_{e}}{6.9})$

A specialized design developed specifically for smooth pipes.

4×10³ ≤ Re ≤ 10⁸

Moody

[41]

$f = \frac{0.184}{R_{e}^{0.2}}$

Classical approximation for smooth pipes.

3×10³ ≤ Re ≤ 10⁷

Filonienko

[43]

$f = \frac{1}{{(1.82 \log (R_{e}) - 1.64)}^{2}}$

An explicit correlation widely used in the literature.

3×10³ ≤ Re ≤ 10⁷

Darby and Melson

[44]

$f_{T} = 10^{a} R_{e}^{- 0.193}$

$a = - 1.47 (1 + 0.14 e^{(- 2.9 \times 10^{- 5})} H_{e})$

An approximation for Bingham-type plastic fluids covering all flow regimes.

Not specified.

McKeon et al.

[45]

$\frac{1}{\sqrt{f}} = 1.930 \log (R_{e} \sqrt{f}) - 0.537$

High-precision modern correlation.

Not specified.

Danish et al.

[46]

$f_{L} = \frac{K_{1} + \frac{4 K_{2}}{{(K_{1} + \frac{K_{1} K_{2}}{{K_{1}}^{4} + 3 K_{2}})}^{3}}}{1 + \frac{3 K_{2}}{{(K_{1} + \frac{K_{1} K_{2}}{{K_{1}}^{4} + 3 K_{2}})}^{4}}}$

Where:

$K_{1} = \frac{16}{R_{e}} + \frac{16 H_{e}}{6 {R_{e}}^{2}} et K_{2} = - \frac{16 {H_{e}}^{4}}{3 {R_{e}}^{8}}$

Explicit approximations using Adomian's decomposition method.

Not specified.

Swamee-Aggarwal

[47]

$f_{L} = \frac{64}{R_{e}} + \frac{10.67 + 0.1414 {(\frac{H_{e}}{R_{e}})}^{1.143}}{(1 + 0.0149 {(\frac{H_{e}}{R_{e}})}^{1.16}) R_{e}} (\frac{H_{e}}{R_{e}})$

Direct approximation of the implicit Buckingham-Reiner equation.

Not specified.

Fang et al.

[48]

$f = 0.25 {(\log (\frac{150.39}{R_{e}^{0.98865}} - \frac{152.66}{R_{e}}))}^{- 2}$

Correlation for smooth pipes.

Not specified

Brkic

[6]

New 1:

$\frac{1}{\sqrt{f}} = - 2 \log (- \frac{5.0272}{R_{e}} \log (- \frac{4.567}{R_{e}} \log ({(\frac{5.3326}{208.815 + R_{e}})}^{0.9345})))$

New 2:

$\frac{1}{\sqrt{f}} = - 2 \log (- \frac{5.02}{R_{e}} \log (- \frac{5.02}{R_{e}} \log (\frac{13}{R_{e}})))$

New 3:

$\frac{1}{\sqrt{f}} = - 2 \log (\frac{95}{R_{e}^{0.983}} - \frac{96.82}{R_{e}})$

New 4:

$f = {(S_{1} - \frac{{(S_{2} - S_{1})}^{2}}{S_{3} - 2 S_{2} + S_{1}})}^{- 2}$

Based on combinations of previous approximations (Romeo, Serghides, etc.).

Not specified

Morrison

[49]

$f = (\frac{0.0076 {(\frac{3170}{R_{e}})}^{0.165}}{1 + {(\frac{3170}{R_{e}})}^{7.0}}) + \frac{16}{R_{e}}$

A simple explicit correlation that matches Prandtl's equation at high Re numbers.

4000≤ Re ≤ 10⁶

Genic and Jacimovic

[50]

$f = \{\begin{matrix} \frac{0.3164}{{R_{e}}^{1 / 4}} \\ \frac{0.118}{{R_{e}}^{1 / 6}} \end{matrix}$

Statistical expansion based on the form of the Blasius equation.

Re < 150×10³

5. Conclusions

This document provides an exhaustive presentation of the expressions that have been developed for determining the coefficient of friction. Formulas have been presented according to two approaches: the implicit approach and the explicit approach. Some of these formulas for calculating the coefficient of friction can be up to 65% inaccurate, as noted by Cipra

[51]

. This occurs when comparing theoretical values obtained from the formulas with the practical data taken from the pipe under investigation and a large discrepancy is observed between both values. This discrepancy exists with all formulas, whether obtained using mathematical methods or artificial intelligence. Today, with the evolution of computer technology, we can easily solve the Colebrook-White equation. However, this universally used equation also produces significant errors. Yoo and Singh found that the Colebrook equation produced an average error of more than 11%, noting that, the roughness of commercial pipes varies considerably, depending on the size and type of pipe

[52, 53]

. Since roughness is a variable factor and should not be considered constant. We recommend that the friction coefficient value be calculated for a specific pipe and for well-defined fluids in a well-defined space-time. Haddad proposes in his work study that sediment deposits on the pipe walls be taken into account for a better assessment of friction

[54]

. In the same light, Tchawe recommends a graphical determination of this evolving parameter at every moment and at different points of the pipe including an improved analytical method to minimise errors in determining the coefficient of friction and head loss

[55]

Abbreviations

f	Friction Coefficient (or Friction Factor)
Re	Reynolds Number
D	Pipe Diameter
ε	Absolute Roughness
ε/D	Relative Roughness
ν	Kinematic Viscosity
ANN	Artificial Neural Networks
GEP	Genetic Expression Programming
He	Hedstrom Number
Pr	Prandtl Number

Acknowledgments

First and foremost, we would like to express our gratitude to Professor Djeumako Bonaventure (may he rest in peace), who supervised this work. We would also like to thank all the individuals and organizations that assisted us in any way with the completion of this project.

Author Contributions

Nkontchou Ngongang François: Conceptualization, Data curation, Formal Analysis, Investigation, Software, Writing – original draft

Tchawe Tchawe Moukam: Data curation, Formal Analysis, Investigation, Methodology, Project administration, Validation, Visualization, Writing – review & editing

Tientcheu Nsiewe Maxwell: Formal Analysis, Visualization

Djiako Thomas: Formal Analysis, Visualization

Kenmeugne Bienvenu: Formal Analysis, Supervision, Validation

Data Availability Statement

The data supporting the findings of this research have been reported in this manuscript.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Plascencia, G., Díaz-Damacillo, L., Robles-Agudo, R. Estimation of friction factor: Review of recent approaches. SN Applied Sciences. 2020, 2(163). https://doi.org/10.1007/s42452-020-1938-6
[2]	Perez Franco, D. Historical Evolution of Pressure Drop Formulas. Hydraulic Engineering and the Environment. 2002, 23, 3-8.
[3]	McGovern, J. Friction Factor Diagrams for Pipe Flow. Available from: https://arrow.dit.ie/engschmecart/28 (accessed 12 August 2018).
[4]	Colebrook, F., White, M. Experiments with fluid friction in roughened pipes. Proceedings of the Royal Society A. 1937, 161, 367-381. https://doi.org/10.1098/rspa.1937.0150
[5]	Diniz, V., Souza, P. A. Explicit formulae for friction factor calculations. WIT Transactions on Ecology and the Environment. 2009, 125, 369-380. https://doi.org/10.2495/WRM090331
[6]	Brkic, D. Review of explicit approximations to the Colebrook relation. Journal of Petroleum Science and Engineering. 2011, 77, 34-48. https://doi.org/10.1016/j.petrol.2011.02.006
[7]	Niazkar, M., Afzali, S. H. Analysis of water distribution networks using MATLAB and Excel: H-based methods. Computer Applications in Engineering Education. 2017, 25(1), 129-141. https://doi.org/10.1002/cae.21786
[8]	Swamee, P., Jain, A. Explicit equations for pipe-flow problems. Journal of the Hydraulics Division. 1976, 102(5), 657-664. https://doi.org/10.1061/JYCEAJ.0004542
[9]	Buzzelli, D. Calculating friction in one step. Machine Design. 2008, 80(12), 54-55.
[10]	Vatankhah, A. Analytical solutions for Colebrook equation. Journal of Hydraulic Engineering. 2018, 144(5), 06018007. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001454
[11]	Niazkar, M., Talebbeydokhti, N., Afzali, S. H. Relationship between Hazen-Williams and Colebrook factor. European Water. 2017, 58, 513-520.
[12]	Shayya, W. H., Sablani, S. S. Neural network for friction factor calculation. Computers and Electronics in Agriculture. 1998, 21(3), 219-228. https://doi.org/10.1016/S0168-1699(98)00044-0
[13]	Ozger, M., Yıldırım, G. Determining friction coefficient using neuro-fuzzy computing. Advances in Engineering Software. 2009, 40, 281-287. https://doi.org/10.1016/j.advengsoft.2008.05.003
[14]	Samadianfard, S. Gene expression programming analysis of Colebrook equation. Journal of Petroleum Science and Engineering. 2012, 92, 48-55. https://doi.org/10.1016/j.petrol.2012.06.005
[15]	Samadianfard, S., Sattari, M. T., Kisi, O., Kazemi, H. Flow friction factor using AI approaches. Applied Artificial Intelligence. 2014, 28, 793-813.
[16]	Offor, U. H., Alabi, S. Accurate explicit friction factor model. Advances in Chemical Engineering and Science. 2016, 6(3), 237-245. https://doi.org/10.4236/aces.2016.63024
[17]	Brkic, D., Cojbasic, Z. Intelligent flow friction estimation. Computational Intelligence and Neuroscience. 2016. https://doi.org/10.1155/2016/5242596
[18]	Nusrat, P., Sadaf, Z., Mohammad, D. AI-based friction factor models for piping networks. Chemical Engineering Communications. 2019.
[19]	Cahyono, M. Hybrid models for solving the Colebrook-White equation using artificial neural networks. Fluids. 2022, 7(211). https://doi.org/10.3390/fluids7070211
[20]	Salmasi, F., Khatibi, R., Ghorbani, M. A. Friction factor using AI and explicit equations. Turkish Journal of Engineering and Environmental Sciences. 2012, 36(2), 121-138.
[21]	Najafzadeh, M., Shiri, J., Sadeghi, G., Ghaemi, A. Prediction of friction factor using model tree. ISH Journal of Hydraulic Engineering. 2017. https://doi.org/10.1080/09715010.2017.1333926
[22]	Yousif, Q. A., Alomar, O. R., Mohamed, I. A., Najim, M. K. Optimal friction factor correlation using genetic algorithm. EAI Endorsed Transactions. 2020. https://doi.org/10.4108/eai.28-6-2020.2297927
[23]	Bardestani, S., Givehchi, M., Younesi, E., Sajjadi, S., Shamshirband, S., Petkovic, D. Predicting turbulent flow friction coefficient using ANFIS technique. Signal, Image and Video Processing. 2016. https://doi.org/10.1007/s11760-016-0948-8
[24]	Moody, L. F. Approximate formula for pipe friction factors. Transactions of the ASME. 1947, 69(12), 1005-1011.
[25]	Wood, D. J. Explicit friction factor relationship. Journal of the Hydraulics Division. 1966.
[26]	Churchill, S. W. Empirical expressions for shear stress in turbulent flow. AIChE Journal. 1973, 19(2), 375-376. https://doi.org/10.1002/aic.690190228
[27]	Churchill, S. W. Friction-factor equation spans all fluid flow regimes. Chemical Engineering. 1977, 84, 91-92.
[28]	Chen, J. Systematic explicit solutions of Prandtl and Colebrook-White equations for pipe flow. Proceedings of the Institution of Civil Engineers. 1985, 79, 383-389.
[29]	Round, G. Explicit approximation for friction factor-Reynolds relation. Canadian Journal of Chemical Engineering. 1980, 58(1), 122-123. https://doi.org/10.1002/cjce.5450580119
[30]	Barr, D. Solutions of the Colebrook-White function for resistance to uniform turbulent flow. Proceedings of the Institution of Civil Engineers. 1981, 71(2), 529-536. https://doi.org/10.1680/iicep.1981.1895
[31]	Zigrang, D., Sylvester, N. Explicit approximations to the Colebrook equation. AIChE Journal. 1982, 28(3), 514-515. https://doi.org/10.1002/aic.690280323
[32]	Haaland, S. Simple and explicit formulas for the friction factor in turbulent pipe flow. Journal of Fluids Engineering. 1983, 105(1), 89-90. https://doi.org/10.1115/1.3240948
[33]	Serghides, T. Estimate friction factor accurately. Chemical Engineering. 1984, 91(5), 63-64.
[34]	Romeo, E., Royo, C., Monzon, A. Improved explicit equation for friction factor. Chemical Engineering Journal. 2002, 86(3), 369-374. https://doi.org/10.1016/S0009-2509(01)00538-2
[35]	Sonnad, J. R., Goudar, C. T. Explicit reformulation of Colebrook equation. Industrial & Engineering Chemistry Research. 2007, 46, 2593-2600. https://doi.org/10.1021/ie0340241
[36]	More, A. A. Analytical solutions for Colebrook and White equation. Chemical Engineering Science. 2006, 61, 5515-5519. https://doi.org/10.1016/j.ces.2006.04.003
[37]	Brkic, D. Explicit approximations of Colebrook’s friction factor in rough pipes. International Journal of Heat and Mass Transfer. 2016, 93, 513-515.
[38]	Vatankhah, A. Comment on “Gene expression programming analysis of implicit Colebrook-White equation in turbulent flow friction factor calculation”. J Petroleum Sci Eng. 2014, 124, 402-405.
[39]	Azizi, N., Homayoon, R., Hojjati, M. R. Predicting the Colebrook-White friction factor in pipe flow by explicit correlations. Journal of Fluids Engineering. 2018, 141, 051201-051208. https://doi.org/10.1115/1.4041232
[40]	López-Silva, M., Carmenates-Hernández, D. S., Delgado-Hernández, N., Chunga-Bereche, N. Explicit pipe friction factor equations: Evaluation, classification, and proposal. Journal of the School of Engineering, University of Antioquia. 2023. https://doi.org/10.17533/udea.redin.20230928
[41]	Moody, L. F. Friction factors for pipe flow. Transactions of the American Society of Mechanical Engineers (ASME). 1944, 66, 671-684.
[42]	Blasius, H. The law of similarity in friction processes in liquids. VDI Research. 1913, 131, 1-41.
[43]	Filonienko, G. Friction factor for turbulent pipe flow. Thermal energy. 1954, 1, 4-40.
[44]	Darby, R., Melson, J. Prediction of friction factor for Bingham plastics. Chemical Engineering. 1981, 28, 59-61.
[45]	McKeon, B. J., Swanson, C. J., Zagarola, M. V., Donnelly, R. J., Smits, A. J. Friction factors for smooth pipe flow. Journal of Fluid Mechanics. 2004, 511, 41-44. https://doi.org/10.1017/S0022112004009796
[46]	Danish, M., Kumar, S., Kumar, S. Explicit analytical expressions of friction factor for Bingham fluids. Communications in Nonlinear Science and Numerical Simulation. 2011, 16(1), 239-251.
[47]	Swamee, P. K., Aggarwal, N. Explicit equations for laminar flow of Bingham fluids. Journal of Petroleum Science and Engineering. 2011, 76, 178-184.
[48]	Fang, X., Xu, Y., Zhou, Z. New correlations of friction factor for turbulent pipe flow. Nuclear Engineering and Design. 2011, 241, 897-902. https://doi.org/10.1016/j.nucengdes.2010.12.019
[49]	Morrison, F. A. Data Correlation for Friction Factor in Smooth Pipes. Department of Chemical Engineering, Michigan Technological University, Houghton, MI.
[50]	Genic, S., Jacimovic, B. Reconsideration of friction factor data. ITM Web of Conferences. 2019, 29, 02001. https://doi.org/10.1051/itmconf/20192902001
[51]	Cipra, B. A new theory of turbulence causes a stir among experts. Science. 1996, 272(5264), 951. https://doi.org/10.1126/science.272.5264.951
[52]	Yoo, D., Singh, V. Explicit design of commercial pipes with no secondary losses. Journal of Irrigation and Drainage Engineering ASCE. 2004, 130(5), 437-440. https://doi.org/10.1061/(ASCE)0733-9437(2004)130:5(437)
[53]	Yoo, D., Singh, V. Explicit design of commercial pipes with secondary losses. Journal of Hydro-environment Research. 2010, 4(1), 37-45.
[54]	Haddad, A. Evaluation and correlation of friction head losses in smooth and rough pipes. The Eurasia Proceedings of Science, Technology, Engineering and Mathematics. 2019, 7, 357-362. https://doi.org/10.5505/epstem.2019.733
[55]	Tchawe, T. Study and Characterization of Turbulent Friction in the Penstock and in the Intake Dam of a Hydroelectric Station: Case of Cameroon Dams. Ph.D. Thesis, University of Ngaoundere, Ngaoundere, 2019. https://doi.org/10.13140/RG.2.2.13744256

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François, N. N., Moukam, T. T., Maxwell, T. N., Thomas, D., Bienvenu, K. (2026). Explicit Expressions for Calculating the Turbulent Friction Coefficient: A Review. International Journal of Energy and Power Engineering, 15(4), 115-123. https://doi.org/10.11648/j.ijepe.20261504.13

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François, N. N.; Moukam, T. T.; Maxwell, T. N.; Thomas, D.; Bienvenu, K. Explicit Expressions for Calculating the Turbulent Friction Coefficient: A Review. Int. J. Energy Power Eng. 2026, 15(4), 115-123. doi: 10.11648/j.ijepe.20261504.13

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François NN, Moukam TT, Maxwell TN, Thomas D, Bienvenu K. Explicit Expressions for Calculating the Turbulent Friction Coefficient: A Review. Int J Energy Power Eng. 2026;15(4):115-123. doi: 10.11648/j.ijepe.20261504.13

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@article{10.11648/j.ijepe.20261504.13,
  author = {Nkontchou Ngongang François and Tchawe Tchawe Moukam and Tientcheu Nsiewe Maxwell and Djiako Thomas and Kenmeugne Bienvenu},
  title = {Explicit Expressions for Calculating the Turbulent Friction Coefficient: A Review},
  journal = {International Journal of Energy and Power Engineering},
  volume = {15},
  number = {4},
  pages = {115-123},
  doi = {10.11648/j.ijepe.20261504.13},
  url = {https://doi.org/10.11648/j.ijepe.20261504.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijepe.20261504.13},
  abstract = {Research on friction in pipes remains complex. This complexity is due to the numerous parameters that must be considered when calculating the coefficient of friction f. Due to this complexity, many researchers have proposed multiple formulas for determining the friction coefficient. Today, there are a large number of explicit expressions for calculating the friction coefficient for each flow regime, whether for smooth or rough pipes. In this review, we set out to study the evolution of work on the friction coefficient and to present it. For this purpose, a large number of reviews have been used. Once this has been done, we identified more than 120 expressions, most of which were explicit. We have thus noted that many of these expressions are approximations of the implicit Colebrook-White equation. Similarly, a good number are derived either from direct developments using computer tools, or from mathematical methods that are sometimes very long. However, in general, we have found that all these expressions only take into account the three parameters present in the Colebrook-White formula. These parameters are: Reynolds number Re, the absolute roughness ɛ and the diameter D. We would like to point out that these parameters were deemed insufficient to minimise the difference between the calculated values and the values observed on site with regard to the friction coefficient.},
 year = {2026}
}

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TY - JOUR
T1 - Explicit Expressions for Calculating the Turbulent Friction Coefficient: A Review
AU - Nkontchou Ngongang François
AU - Tchawe Tchawe Moukam
AU - Tientcheu Nsiewe Maxwell
AU - Djiako Thomas
AU - Kenmeugne Bienvenu
Y1 - 2026/07/28
PY - 2026
N1 - https://doi.org/10.11648/j.ijepe.20261504.13
DO - 10.11648/j.ijepe.20261504.13
T2 - International Journal of Energy and Power Engineering
JF - International Journal of Energy and Power Engineering
JO - International Journal of Energy and Power Engineering
SP - 115
EP - 123
PB - Science Publishing Group
SN - 2326-960X
UR - https://doi.org/10.11648/j.ijepe.20261504.13
AB - Research on friction in pipes remains complex. This complexity is due to the numerous parameters that must be considered when calculating the coefficient of friction f. Due to this complexity, many researchers have proposed multiple formulas for determining the friction coefficient. Today, there are a large number of explicit expressions for calculating the friction coefficient for each flow regime, whether for smooth or rough pipes. In this review, we set out to study the evolution of work on the friction coefficient and to present it. For this purpose, a large number of reviews have been used. Once this has been done, we identified more than 120 expressions, most of which were explicit. We have thus noted that many of these expressions are approximations of the implicit Colebrook-White equation. Similarly, a good number are derived either from direct developments using computer tools, or from mathematical methods that are sometimes very long. However, in general, we have found that all these expressions only take into account the three parameters present in the Colebrook-White formula. These parameters are: Reynolds number Re, the absolute roughness ɛ and the diameter D. We would like to point out that these parameters were deemed insufficient to minimise the difference between the calculated values and the values observed on site with regard to the friction coefficient.
VL - 15
IS - 4
ER -

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Nkontchou Ngongang François

Department of Mechanical Engineering, National School of Agro-Industrial Sciences of the University of Ngaoundere, Ngaoundere, Cameroon

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http://orcid.org/0009-0003-5181-3159
Tchawe Tchawe Moukam

Department of Mechanical Engineering, National School of Agro-Industrial Sciences of the University of Ngaoundere, Ngaoundere, Cameroon

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http://orcid.org/0000-0001-5449-7954
Tientcheu Nsiewe Maxwell

Department of Department of Fundamental Sciences, Chemical Engineering and Mineral Industries School of the University of Ngaoundere, Ngaoundere, Cameroon

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Djiako Thomas

Department of Energy and Mechanical Engineering, Higher Institute for Applied Technology of the University Institute of the Gulf of Guinea, Douala, Cameroon

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Kenmeugne Bienvenu

Department of Mechanical Engineering, National Advanced school of Engineering of the University of Yaounde 1, Yaounde, Cameroon

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Figure 1

Figure 1. Publication of friction coefficient formulas by year.

Plain Text BibTeX RIS

APA Style

François, N. N., Moukam, T. T., Maxwell, T. N., Thomas, D., Bienvenu, K. (2026). Explicit Expressions for Calculating the Turbulent Friction Coefficient: A Review. International Journal of Energy and Power Engineering, 15(4), 115-123. https://doi.org/10.11648/j.ijepe.20261504.13

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ACS Style

François, N. N.; Moukam, T. T.; Maxwell, T. N.; Thomas, D.; Bienvenu, K. Explicit Expressions for Calculating the Turbulent Friction Coefficient: A Review. Int. J. Energy Power Eng. 2026, 15(4), 115-123. doi: 10.11648/j.ijepe.20261504.13

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AMA Style

François NN, Moukam TT, Maxwell TN, Thomas D, Bienvenu K. Explicit Expressions for Calculating the Turbulent Friction Coefficient: A Review. Int J Energy Power Eng. 2026;15(4):115-123. doi: 10.11648/j.ijepe.20261504.13

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@article{10.11648/j.ijepe.20261504.13,
  author = {Nkontchou Ngongang François and Tchawe Tchawe Moukam and Tientcheu Nsiewe Maxwell and Djiako Thomas and Kenmeugne Bienvenu},
  title = {Explicit Expressions for Calculating the Turbulent Friction Coefficient: A Review},
  journal = {International Journal of Energy and Power Engineering},
  volume = {15},
  number = {4},
  pages = {115-123},
  doi = {10.11648/j.ijepe.20261504.13},
  url = {https://doi.org/10.11648/j.ijepe.20261504.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijepe.20261504.13},
  abstract = {Research on friction in pipes remains complex. This complexity is due to the numerous parameters that must be considered when calculating the coefficient of friction f. Due to this complexity, many researchers have proposed multiple formulas for determining the friction coefficient. Today, there are a large number of explicit expressions for calculating the friction coefficient for each flow regime, whether for smooth or rough pipes. In this review, we set out to study the evolution of work on the friction coefficient and to present it. For this purpose, a large number of reviews have been used. Once this has been done, we identified more than 120 expressions, most of which were explicit. We have thus noted that many of these expressions are approximations of the implicit Colebrook-White equation. Similarly, a good number are derived either from direct developments using computer tools, or from mathematical methods that are sometimes very long. However, in general, we have found that all these expressions only take into account the three parameters present in the Colebrook-White formula. These parameters are: Reynolds number Re, the absolute roughness ɛ and the diameter D. We would like to point out that these parameters were deemed insufficient to minimise the difference between the calculated values and the values observed on site with regard to the friction coefficient.},
 year = {2026}
}

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