Performance of explicit approximations of the coefficient of head loss for pressurized conduits

Pimenta, Bruna D.; Robaina, Adroaldo D.; Peiter, Marcia X.; Mezzomo, Wellington; Kirchner, Jardel H.; Ben, Luis H. B.

doi:10.1590/1807-1929/agriambi.v22n5p301-307

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 approximations of the f for pressurized conduits in comparison to the Colebrook-White formulation, determining the most precise ones so that they can be used as an alternative solution that is valid for the turbulent flow regime. Twenty nine explicit equations found in the literature were analysed, determining the f through the Reynolds number in the range of 4 × 10³ ≤ Re ≤ 10⁸ and a relative roughness (Ɛ/D) of 10^-6 ≤ Ɛ/D ≤ 5 × 10^-2, and obtaining 160 points for each equation. The performance index and relative error of the formulations were analyzed in relation to the Colebrook-White equation. Considering the equations analyzed, we found seven that presented excellent performance and high precision, highlighting the formulation of Offor & Alabi, which can be used as an alternative to the Colebrook-White standard equation.

Key words:
head loss; Darcy-Weisbach; turbulent flow regime; performance index

RESUMO

Um dos parâmetros envolvido no dimensionamento de sistemas hidráulicos pressurizados é a perda de carga das tubulações. Essa verificação pode ser realizada através da formulação de Darcy-Weisbach, que considera um coeficiente de perda de carga (f) que pode ser mensurado pela equação implícita de Colebrook-White. No entanto, para essa determinação é necessário utilizar métodos numéricos ou o diagrama de Moody. Devido a isso, numerosas aproximações explícitas são propostas para superar essa limitação. Nesse sentido, o objetivo desse trabalho é analisar as aproximações explícitas do f para condutos pressurizados em comparação a formulação de Colebrook-White, determinando as mais precisas para que possam ser uma solução alternativa, válidas para o regime de fluxo turbulento. Foram analisadas 29 equações explícitas encontradas na literatura, determinando o f através do número de Reynolds (Re) na faixa de 4 × 10³ ≤ Re ≤ 10⁸ e rugosidade relativa (Ɛ/D) de 10^-6 ≤ Ɛ/D ≤ 5 × 10^-2, obtendo 160 pontos para cada equação. O índice de desempenho e o erro relativo das formulações foram analisados em relação a equação de Colebrook-White. Considerando as equações analisadas, sete apresentaram excelente desempenho e alta precisão, destacando a formulação de Offor & Alabi, a qual pode ser utilizada como alternativa à equação padrão de Colebrook-White.

Palavras-chave:
perda de carga; Darcy-Weisbach; regime de fluxo turbulento; índice de desempenho

Introduction

The estimation of head loss in pressurized conduits is a significant problem in optimization studies, hydraulic analysis of ducts, and water distribution systems (Bardestani et al., 2017Bardestani, S.; Givehchi, M.; Younesi, E.; Sajjadi, S.; Shamshirband, S.; Petkovic, D. Predicting turbulent flow friction coefficient using ANFIS technique. Signal, Image and Video Processing, v.11, p.341-347, 2017. https://doi.org/10.1007/s11760-016-0948-8
https://doi.org/10.1007/s11760-016-0948-... ).

The Colebrook-White (1937)Colebrook, C. F.; White, C. M. Experiments with fluid friction in roughened pipes. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, v.161, p.367-381, 1937. https://doi.org/10.1098/rspa.1937.0150
https://doi.org/10.1098/rspa.1937.0150... (CW) equation has been considered as the most accurate approximation for the determination of the head loss coefficient (f) and has been used as a reference standard; it uses the Reynolds number (Re) and the relative roughness of the pipe (Ɛ/D) (Heydari et al., 2015Heydari, A.; Narimani, E.; Pakniya, F. Explicit determinations of the Colebrook equation for the flow friction factor by statistical analysis. Chemical Engineering & Technology, v.38, p.1387-1396, 2015. https://doi.org/10.1002/ceat.201400590
https://doi.org/10.1002/ceat.201400590... ; Brkić & Ćojbašić, 2016Brkić, D.; Ćojbašić, Ž. Intelligent flow friction estimation. Computational Intelligence and Neuroscience, v.2016, p.1-10, 2016. https://doi.org/10.1155/2016/5242596
https://doi.org/10.1155/2016/5242596... ) and is valid for a wide range of applicability: 2 × 10³ < Re < 10⁸ and 0 ≤ Ɛ/D ≥ 0.05. However, it is implicit in relation to f and requires an iterative process for the solution (Brkić, 2016Brkić, D. A note on explicit approximations to Colebrook’s friction factor in rough pipes under highly turbulent cases. International Journal of Heat and Mass Transfer, v.93, p.513-515, 2016. https://doi.org/10.1016/j.ijheatmasstransfer.2015.08.109
https://doi.org/10.1016/j.ijheatmasstran... ; Brkić & Ćojbašić, 2017Brkić, D.; Ćojbašić, Ž. Evolutionary optimization of Colebrook’s turbulent flow friction approximations. Fluids, v.2, p.1-27, 2017. https://doi.org/10.3390/fluids2020015
https://doi.org/10.3390/fluids2020015... ).

Several researchers have sought to find explicit equations that could be used as alternatives to the CW equation (Assefa & Kaushal, 2015Assefa, K. M.; Kaushal, D. R. A comparative study of friction factor correlations for high concentrate slurry flow in smooth pipes. Journal of Hydrology and Hydromechanics, v.63, p.13-20, 2015. https://doi.org/10.1515/johh-2015-0008
https://doi.org/10.1515/johh-2015-0008... ; Mikata & Walczak, 2015Mikata, Y.; Walczak, W. S. Exact analytical solutions of the Colebrook-White equation. Journal of Hydraulic Engineering, v.142, p.1-6, 2015.). According to Brkić & Ćojbašić (2017)Brkić, D.; Ćojbašić, Ž. Evolutionary optimization of Colebrook’s turbulent flow friction approximations. Fluids, v.2, p.1-27, 2017. https://doi.org/10.3390/fluids2020015
https://doi.org/10.3390/fluids2020015... , explicit approximations give a relatively good prediction of the f and can accurately reproduce the CW equation and the Moody (1944)Moody, L. F. Friction factors for pipe flow. Transactions ASME, v.66, p.671-678, 1944. diagram. In some of these explicit equations, their relative error is so small that they can be used directly instead of the CW equation (Çoban, 2012Çoban, M. T. Error analysis of non-iterative friction factor formulas relative to Colebrook-White equation for the calculation of pressure drop in pipes. Journal of Naval Science and Engineering, v.8, p.1-13, 2012.).

Therefore, the objective of this research was to analyze some explicit approximations of the pressure loss coefficient for pressurized conduits, determining the most accurate ones so that they can be used as an alternative to the CW formulation.

Material and Methods

The determination of the f of all equations were performed using a Microsoft Excel worksheet, with Re values in the range of 4 × 10³ ≤ Re ≤ 10⁸ and Ɛ/D of 10^-6 ≤ Ɛ/D ≤ 5 × 10^-2, and 160 points of data for each approximation analyzed were obtained. The CW formulation, Eq. 1, can be identified by:

\frac{1}{\sqrt{f}} = - 2 \log (\frac{ε}{3.7 D} + \frac{2.51}{R e \sqrt{f}})

(1)

where:

f - is the coefficient of head loss of the Darcy-Weisbach formulation (dimensionless);

Ɛ/D - is the relative roughness of the pipe (m); and

Re - is the Reynolds number (dimensionless).

In all twenty nine explicit equations of the f from different authors were analysed, years of publication, and range of applicability involving Re and Ɛ/D, as listed in Table 1. Their choice was determined to evaluate most of the equations available in the literature. In this study, any model devoid of iterations was considered explicit.

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Table 1
Explicit approximations for the determination of the head loss coefficient (f), with their respective authors, years of publication, and application ranges

The precision, related to the distance of the values of the explicit equations in relation to CW, was determined by the concordance index (d) proposed by Willmott (1981)Willmott, C. J. On the validation of models. Physical Geography, v.2, p.184-194, 1981.. The values ranged from 0 (without a match) to 1 (perfect match).

The Pearson correlation coefficient (r) allows quantifying the degree of association between the two variables involved in the analysis. The closer to 1, the greater the degree of linear statistical dependence between the variables, and the closer to zero, the lower the strength of that relationship.

The equations were evaluated using the performance index (Id) adapted from Camargo & Sentelhas (1997)Camargo, A. P.; Sentelhas, P. C. Avaliação do desempenho de diferentes métodos de estimativa da evapotranspiração potencial no estado de São Paulo, Brasil. Revista Brasileira de Agrometeorologia, v.5, p.89-97, 1997., whose value is the product of d and r. The criteria for interpreting d, r, Id, and their respective classifications are presented in Table 2.

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Table 2
Criteria for interpreting the concordance index, the precision index, the performance index, and their respective classifications

After sorting the equations that had a performance index rated as “Excellent,” the mean of the relative error (MRE) was calculated. According to Sadeghi et al. (2015)Sadeghi, S.; Peters, R. T.; Lamm, F. R. Design of zero slope microirrigation laterals: Effect of the friction factor variation. Journal of Irrigation and Drainage Engineering, v.141, p.1-9, 2015. https://doi.org/10.1061/(ASCE)IR.1943-4774.0000905
https://doi.org/10.1061/(ASCE)IR.1943-47... , it is a very useful parameter for evaluating practically the most precise model for the estimation of the f.

The values of the MRE were classified as follows: “Very good,” MRE ≤ 0.55; “Good,” 0.55 < MRE ≤ 1.00; “Average,” 1.00 < MRE ≤ 2.00; “Weak,” 2.00 < MRE ≤ 3.00; and “Poor,” MRE > 3.00.

Results and Discussion

All the explicit equations in relation to the CW standard presented d values very close to 1.00, being classified as “Excellent,” thus possessing a high degree of accuracy among the variables involved.

The r of most of the explicit equations also provided values very close to 1.00, demonstrating a good association of the variables involved. Eqs. 14, 25, 27, and 28 were classified as “Excellent,” but had correlation coefficient (r) of less than 0.99. Meanwhile, Eq. 20 had a lower value, with r = 0.93478, being classified as “Optimum,” and presented a lower correlation between the variables involved.

According to the Id, all the equations presented an “Excellent” classification, with values close to 1.00 except for Eq. 20, which obtained an Id of 0.93444.

Analyzing the performance coefficients, we can conclude that all explicit equations obtained a satisfactory performance for the estimation of the f when compared with the implicit CW formulation. Because of this, a statistical analysis was performed using the relative error (RE) to evaluate the most accurate model for estimating the f.

The approximations of Eqs. 8, 10, 12, 17, 19, 21, 22, 24, 26, and 30 presented an MRE lower than 0.55%, all being classified as “Very good.” The lowest value found was that of Eq. 30, with MRE = 0.30%.

Models 2, 3, 5, 9, 14, 23, 27, 29, 28, and 20 presented an MRE above 1.00%, with the last two standing out owing to their higher MRE of 10.24 and 16.15%, respectively. The other equations were classified as “Good.”

The mean values of the RE found in this study are in agreement with those of Brkić (2011b)Brkić, D. Review of explicit approximations to the Colebrook relation for flow friction. Journal of Petroleum Science and Engineering, v.77, p.34-48, 2011b. https://doi.org/10.1016/j.petrol.2011.02.006
https://doi.org/10.1016/j.petrol.2011.02... , who carried out a review of 26 explicit approximations based on the RE criterion and concluded that most of the explicit models available are very precise, with the exception of those of Moody (1947)Moody, L. F. An approximate formula for pipe friction factors. Transactions ASME, v.69, p.1005-1011, 1947., Wood (1966)Wood, D. J. An explicit friction factor relationship. Civil Engineering, v.36, p.60-61, 1966., Eck (1973)Eck, B. Technische Stromungslehre. New York: Springer, 1973. 324p., Round (1980)Round, G. F. An explicit approximation for the friction factor-Reynolds number relation for rough and smooth pipes. The Canadian Journal of Chemical Engineering, v.58, p.122-123, 1980. https://doi.org/10.1002/cjce.5450580119
https://doi.org/10.1002/cjce.5450580119... , and Rao & Kumar (2007)Rao, A. R.; Kumar, B. Friction factor for turbulent pipe flow. Bangalore: Division of Mechanical Science, Civil Engineering Indian Institute of Science Bangalore, 2007. 16p..

According to Winning & Coole (2013)Winning, H. K.; Coole, T. Explicit friction factor accuracy and computational efficiency for turbulent flow in pipes. Flow, Turbulence and Combustion, v.90, p.1-27, 2013. https://doi.org/10.1007/s10494-012-9419
https://doi.org/10.1007/s10494-012-9419... , when 28 explicit equations of the f were compared with CW, the most precise approximations were those obtained by the equations of Zigrang & Sylvester (1982)Zigrang, D. J.; Sylvester, N. D. Explicit approximations to the solution of Colebrook’s friction factor equation. AIChE Journal, v.28, p.514-515, 1982. https://doi.org/10.1002/aic.690280323
https://doi.org/10.1002/aic.690280323... , Romeo et al. (2002)Romeo, E.; Royo, C.; Monzón, A. Improved explicit equation for estimation of the friction factor in rough and smooth pipes. Chemical Engineering Journal, v.86, p.369-374, 2002. https://doi.org/10.1016/S1385-8947(01)00254-6
https://doi.org/10.1016/S1385-8947(01)00... , and Buzzelli (2008)Buzzelli, D. Calculating friction in one step. Machine Design, v.80, p.54-55, 2008.. This study found similar values of accuracy, with the exception of Romeo et al. (2002)Romeo, E.; Royo, C.; Monzón, A. Improved explicit equation for estimation of the friction factor in rough and smooth pipes. Chemical Engineering Journal, v.86, p.369-374, 2002. https://doi.org/10.1016/S1385-8947(01)00254-6
https://doi.org/10.1016/S1385-8947(01)00... , which presented higher values of RE.

Brkić (2011a)Brkić, D. New explicit correlations for turbulent flow friction factor. Nuclear Engineering and Design, v.241, p.4055-4059, 2011a. https://doi.org/10.1016/j.nucengdes.2011.07.042
https://doi.org/10.1016/j.nucengdes.2011... , Winning & Coole (2013)Winning, H. K.; Coole, T. Explicit friction factor accuracy and computational efficiency for turbulent flow in pipes. Flow, Turbulence and Combustion, v.90, p.1-27, 2013. https://doi.org/10.1007/s10494-012-9419
https://doi.org/10.1007/s10494-012-9419... and Offor & Alabi (2016)Offor, U. H.; Alabi, S. B. An accurate and computationally efficient friction factor model. Advances in Chemical Engineering and Science, v.6, p.237-245, 2016. https://doi.org/10.4236/aces.2016.63024
https://doi.org/10.4236/aces.2016.63024... analyzing explicit equations of the f, found that the RE values of Rao & Kumar’s (2007)Rao, A. R.; Kumar, B. Friction factor for turbulent pipe flow. Bangalore: Division of Mechanical Science, Civil Engineering Indian Institute of Science Bangalore, 2007. 16p. equation were the highest in relation to all the explicit equations analyzed in their research, being consistent with what was obtained in this study.

The discrepancy between the RE values found in this study and those obtained by Brkić (2016)Brkić, D. A note on explicit approximations to Colebrook’s friction factor in rough pipes under highly turbulent cases. International Journal of Heat and Mass Transfer, v.93, p.513-515, 2016. https://doi.org/10.1016/j.ijheatmasstransfer.2015.08.109
https://doi.org/10.1016/j.ijheatmasstran... proposed equation is possibly due to the fact that the approximation obtained by this study covers a limited range of applicability of Re and Ɛ/D, with values of 10⁶ ˂ Re ˂ 10⁸ and 10^-2 ˂ Ɛ/D ˂ 5 × 10^-2 only, respectively.

For an approximation of the range of applicability that the CW equation provides, only the explicit equations covering 4 × 10³ ≤ Re ≤ 10⁸ and 10^-6 ≤ Ɛ/D ≤ 5 × 10^-2 and MRE < 0.55% will be valid. This is applied because some highly accurate approximations are valid only at limited Re and Ɛ/D intervals and, thus, may incorrectly estimate the f.

Of the 29 explicit approximations of the f analyzed, only 7 satisfied these conditions, which were Eqs. 8, 10, 19, 21, 22, 26, and 30. These approximations are presented in Figure 1A-G, which shows the RE distribution for the entire Re range of 4 × 10³ ≤ Re ≤ 10⁸ and the Ɛ/D of 10^-6 ≤ Ɛ/D ≤ 5 × 10^-2.

Figure 1
Distribution of the relative error estimate (RE%), Reynolds number (Re), and relative roughness (Ɛ/D) produced by the equations of A) Chen (1979)Chen, N. H. An explicit equation for friction factor in pipes. Industrial & Engineering Chemistry Fundamentals, v.18, p.296-297, 1979. https://doi.org/10.1021/i160071a019
https://doi.org/10.1021/i160071a019... ; B) Shacham (1980)Shacham, M. Comment on "Explicit equation for friction factor in pipe". Industrial & Engineering Chemistry Fundamentals, v.19, p.228-229, 1980. https://doi.org/10.1021/i160074a019
https://doi.org/10.1021/i160074a019... ; C) Sonnad & Goudar (2006)Sonnad, J. R.; Goudar, C. T. Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation. Journal of Hydraulic Engineering, v.132, p.863-867, 2006. https://doi.org/10.1061/(ASCE)0733-9429(2006)132:8(863)
https://doi.org/10.1061/(ASCE)0733-9429(... ; D) Buzzelli (2008)Buzzelli, D. Calculating friction in one step. Machine Design, v.80, p.54-55, 2008.; E) Vantankhah & Kouchakzadeh (2008)Vatankhah, A. R.; Kouchakzadeh, S. Discussion of “Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation” by Jagadeesh R. Sonnad and Chetan T. Goudar. Journal Hydraulic Engineering, v.134, p.1-9, 2008. https://doi.org/10.1061/(ASCE)0733-9429(2008)134:8(1187)
https://doi.org/10.1061/(ASCE)0733-9429(... ; F) Fang et al. (2011)Fang, X.; Xu, Y.; Zhou, Z. New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing singlephase friction factor correlations. Nuclear Engineering and Design, v.241, p.897-902, 2011. https://doi.org/10.1016/j.nucengdes.2010.12.019
https://doi.org/10.1016/j.nucengdes.2010... , and G) Offor & Alabi (2016)Offor, U. H.; Alabi, S. B. An accurate and computationally efficient friction factor model. Advances in Chemical Engineering and Science, v.6, p.237-245, 2016. https://doi.org/10.4236/aces.2016.63024
https://doi.org/10.4236/aces.2016.63024... , when compared to the Colebrook-White (1937)Colebrook, C. F.; White, C. M. Experiments with fluid friction in roughened pipes. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, v.161, p.367-381, 1937. https://doi.org/10.1098/rspa.1937.0150
https://doi.org/10.1098/rspa.1937.0150... equation ()

A joint analysis of Figure 1A-G shows that the equation of Sonnad & Goudar (2006)Sonnad, J. R.; Goudar, C. T. Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation. Journal of Hydraulic Engineering, v.132, p.863-867, 2006. https://doi.org/10.1061/(ASCE)0733-9429(2006)132:8(863)
https://doi.org/10.1061/(ASCE)0733-9429(... presented the highest value of the maximum RE and the lowest value of the minimum RE in relation to the others, with values of 3.17 and 0.003%, respectively.

For Chen (1979)Chen, N. H. An explicit equation for friction factor in pipes. Industrial & Engineering Chemistry Fundamentals, v.18, p.296-297, 1979. https://doi.org/10.1021/i160071a019
https://doi.org/10.1021/i160071a019... , the minimum RE value was 0.019% for an Ɛ/D of 10^-5 and an Re of 5 × 10⁶, and the maximum RE value was 1.837% for an Ɛ/D of 10^-4 and an Re of 4 × 10³. Shacham (1980)Shacham, M. Comment on "Explicit equation for friction factor in pipe". Industrial & Engineering Chemistry Fundamentals, v.19, p.228-229, 1980. https://doi.org/10.1021/i160074a019
https://doi.org/10.1021/i160074a019... presented a minimum RE value of 0.069% for an Ɛ/D of 5 × 10^-6 and an Re of 5 × 10⁷, and a maximum RE value of 1.270% for an Ɛ/D of 10^-6 and an Re of 4 × 10³.

For Buzzelli (2008)Buzzelli, D. Calculating friction in one step. Machine Design, v.80, p.54-55, 2008., the minimum RE value was 0.007% for an Ɛ/D of 5 × 10^-6 and an Re of 5 × 10⁷, and the maximum RE value was 2.156% for an Ɛ/D of 10^-6 and an Re of 4 × 10³. Vantankhah & Kouchakzadeh (2008)Vatankhah, A. R.; Kouchakzadeh, S. Discussion of “Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation” by Jagadeesh R. Sonnad and Chetan T. Goudar. Journal Hydraulic Engineering, v.134, p.1-9, 2008. https://doi.org/10.1061/(ASCE)0733-9429(2008)134:8(1187)
https://doi.org/10.1061/(ASCE)0733-9429(... presented a minimum RE value of 0.01% for an Ɛ/D of 5 × 10^-6 and an Re of 5 × 10⁷, and a maximum RE value of 2.112% for an Ɛ/D of 10^-6 and an Re of 4 × 10³.

Fang et al. (2011)Fang, X.; Xu, Y.; Zhou, Z. New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing singlephase friction factor correlations. Nuclear Engineering and Design, v.241, p.897-902, 2011. https://doi.org/10.1016/j.nucengdes.2010.12.019
https://doi.org/10.1016/j.nucengdes.2010... presented a minimum RE value of 0.009% for an Ɛ/D of 2 × 10^-3 and an Re for 10⁵, and a maximum RE value of 2.375% for an Ɛ/D of 10^-6 and an Re of 4 × 10³. For Offor & Alabi (2016)Offor, U. H.; Alabi, S. B. An accurate and computationally efficient friction factor model. Advances in Chemical Engineering and Science, v.6, p.237-245, 2016. https://doi.org/10.4236/aces.2016.63024
https://doi.org/10.4236/aces.2016.63024... , the minimum value of RE was 0.005% for an Ɛ/D of 5 × 10^-6 and an Re of 10⁸, and the maximum RE value was 2.128% for an Ɛ/D of 5 × 10^-6 and an Re of 4 × 10³.

Conclusions

The equations of Chen (1979)Chen, N. H. An explicit equation for friction factor in pipes. Industrial & Engineering Chemistry Fundamentals, v.18, p.296-297, 1979. https://doi.org/10.1021/i160071a019
https://doi.org/10.1021/i160071a019... , Shacham (1980)Shacham, M. Comment on "Explicit equation for friction factor in pipe". Industrial & Engineering Chemistry Fundamentals, v.19, p.228-229, 1980. https://doi.org/10.1021/i160074a019
https://doi.org/10.1021/i160074a019... , Sonnad & Goudar (2006)Sonnad, J. R.; Goudar, C. T. Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation. Journal of Hydraulic Engineering, v.132, p.863-867, 2006. https://doi.org/10.1061/(ASCE)0733-9429(2006)132:8(863)
https://doi.org/10.1061/(ASCE)0733-9429(... , Buzzelli (2008)Buzzelli, D. Calculating friction in one step. Machine Design, v.80, p.54-55, 2008., Vantankhah & Kouchakzadeh (2008)Vatankhah, A. R.; Kouchakzadeh, S. Discussion of “Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation” by Jagadeesh R. Sonnad and Chetan T. Goudar. Journal Hydraulic Engineering, v.134, p.1-9, 2008. https://doi.org/10.1061/(ASCE)0733-9429(2008)134:8(1187)
https://doi.org/10.1061/(ASCE)0733-9429(... , Fang et al. (2011)Fang, X.; Xu, Y.; Zhou, Z. New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing singlephase friction factor correlations. Nuclear Engineering and Design, v.241, p.897-902, 2011. https://doi.org/10.1016/j.nucengdes.2010.12.019
https://doi.org/10.1016/j.nucengdes.2010... , and Offor & Alabi (2016)Offor, U. H.; Alabi, S. B. An accurate and computationally efficient friction factor model. Advances in Chemical Engineering and Science, v.6, p.237-245, 2016. https://doi.org/10.4236/aces.2016.63024
https://doi.org/10.4236/aces.2016.63024... showed higher performance indexes and precision when compared to the Colebrook-White approximation.
The equation of Offor & Alabi (2016)Offor, U. H.; Alabi, S. B. An accurate and computationally efficient friction factor model. Advances in Chemical Engineering and Science, v.6, p.237-245, 2016. https://doi.org/10.4236/aces.2016.63024
https://doi.org/10.4236/aces.2016.63024... , in relation to the explicit models analyzed, stood out from the others, presenting the highest performance index and precision, apart from covering the widest range of Reynolds number applicability and showing the highest relative roughness, and, therefore, can be used as an alternative to the implicit Colebrook-White equation.

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» https://doi.org/10.1515/johh-2015-0008
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» https://doi.org/10.1007/s11760-016-0948-8
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» https://doi.org/10.1016/j.nucengdes.2011.07.042
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Brkić, D. A note on explicit approximations to Colebrook’s friction factor in rough pipes under highly turbulent cases. International Journal of Heat and Mass Transfer, v.93, p.513-515, 2016. https://doi.org/10.1016/j.ijheatmasstransfer.2015.08.109
» https://doi.org/10.1016/j.ijheatmasstransfer.2015.08.109
Brkić, D.; Ćojbašić, Ž. Intelligent flow friction estimation. Computational Intelligence and Neuroscience, v.2016, p.1-10, 2016. https://doi.org/10.1155/2016/5242596
» https://doi.org/10.1155/2016/5242596
Brkić, D.; Ćojbašić, Ž. Evolutionary optimization of Colebrook’s turbulent flow friction approximations. Fluids, v.2, p.1-27, 2017. https://doi.org/10.3390/fluids2020015
» https://doi.org/10.3390/fluids2020015
Buzzelli, D. Calculating friction in one step. Machine Design, v.80, p.54-55, 2008.
Camargo, A. P.; Sentelhas, P. C. Avaliação do desempenho de diferentes métodos de estimativa da evapotranspiração potencial no estado de São Paulo, Brasil. Revista Brasileira de Agrometeorologia, v.5, p.89-97, 1997.
Chen, N. H. An explicit equation for friction factor in pipes. Industrial & Engineering Chemistry Fundamentals, v.18, p.296-297, 1979. https://doi.org/10.1021/i160071a019
» https://doi.org/10.1021/i160071a019
Churchill, S. W. Empirical expressions for the shear stress in turbulent flow in commercial pipe. AIChE Journal, v.19, p.375-376, 1973. https://doi.org/10.1002/aic.690190228
» https://doi.org/10.1002/aic.690190228
Çoban, M. T. Error analysis of non-iterative friction factor formulas relative to Colebrook-White equation for the calculation of pressure drop in pipes. Journal of Naval Science and Engineering, v.8, p.1-13, 2012.
Colebrook, C. F.; White, C. M. Experiments with fluid friction in roughened pipes. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, v.161, p.367-381, 1937. https://doi.org/10.1098/rspa.1937.0150
» https://doi.org/10.1098/rspa.1937.0150
Eck, B. Technische Stromungslehre. New York: Springer, 1973. 324p.
Fang, X.; Xu, Y.; Zhou, Z. New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing singlephase friction factor correlations. Nuclear Engineering and Design, v.241, p.897-902, 2011. https://doi.org/10.1016/j.nucengdes.2010.12.019
» https://doi.org/10.1016/j.nucengdes.2010.12.019
Ghanbari, A.; Farshad, F.; Rieke, H. Newly developed friction factor correlation for pipe flow and flow assurance. Journal of Chemical Engineering and Materials Science, v.2, p.83-86, 2011.
Haaland, S. E. Simple and explicit formulas for friction factor in turbulent pipe flow. Journal of Fluids Engineering, v.105, p.89-90, 1983. https://doi.org/10.1115/1.3240948
» https://doi.org/10.1115/1.3240948
Heydari, A.; Narimani, E.; Pakniya, F. Explicit determinations of the Colebrook equation for the flow friction factor by statistical analysis. Chemical Engineering & Technology, v.38, p.1387-1396, 2015. https://doi.org/10.1002/ceat.201400590
» https://doi.org/10.1002/ceat.201400590
Jain, A. K. Accurate explicit equation for friction factor. Journal of the Hydraulics Division, v.102, p.674-677, 1976.
Manadilli, G. Replace implicit equations with signomial functions. Chemical Engineering, v.104, p.129-129, 1997.
Mikata, Y.; Walczak, W. S. Exact analytical solutions of the Colebrook-White equation. Journal of Hydraulic Engineering, v.142, p.1-6, 2015.
Moody, L. F. Friction factors for pipe flow. Transactions ASME, v.66, p.671-678, 1944.
Moody, L. F. An approximate formula for pipe friction factors. Transactions ASME, v.69, p.1005-1011, 1947.
Offor, U. H.; Alabi, S. B. An accurate and computationally efficient friction factor model. Advances in Chemical Engineering and Science, v.6, p.237-245, 2016. https://doi.org/10.4236/aces.2016.63024
» https://doi.org/10.4236/aces.2016.63024
Papaevangelou, G.; Evangelides, C.; Tzimopoulos, C. A new explicit equation for the friction coefficient in the Darcy-Weisbach equation. In: Proceedings of the Tenth Conference on Protection and Restoration of the Environment, v.166, p.6-9, 2010.
Rao, A. R.; Kumar, B. Friction factor for turbulent pipe flow. Bangalore: Division of Mechanical Science, Civil Engineering Indian Institute of Science Bangalore, 2007. 16p.
Robaina, D. A. Análise de equações explícitas para o cálculo do coeficiente “f” da fórmula universal de perda de carga. Ciência Rural, v.22, p.157-159, 1992. https://doi.org/10.1590/S0103-84781992000200006
» https://doi.org/10.1590/S0103-84781992000200006
Romeo, E.; Royo, C.; Monzón, A. Improved explicit equation for estimation of the friction factor in rough and smooth pipes. Chemical Engineering Journal, v.86, p.369-374, 2002. https://doi.org/10.1016/S1385-8947(01)00254-6
» https://doi.org/10.1016/S1385-8947(01)00254-6
Round, G. F. An explicit approximation for the friction factor-Reynolds number relation for rough and smooth pipes. The Canadian Journal of Chemical Engineering, v.58, p.122-123, 1980. https://doi.org/10.1002/cjce.5450580119
» https://doi.org/10.1002/cjce.5450580119
Sadeghi, S.; Peters, R. T.; Lamm, F. R. Design of zero slope microirrigation laterals: Effect of the friction factor variation. Journal of Irrigation and Drainage Engineering, v.141, p.1-9, 2015. https://doi.org/10.1061/(ASCE)IR.1943-4774.0000905
» https://doi.org/10.1061/(ASCE)IR.1943-4774.0000905
Shacham, M. Comment on "Explicit equation for friction factor in pipe". Industrial & Engineering Chemistry Fundamentals, v.19, p.228-229, 1980. https://doi.org/10.1021/i160074a019
» https://doi.org/10.1021/i160074a019
Shaikh, M. M.; Massan S. ur R.; Wagan, A. I. A new explicit approximation to Colebrook’s friction factor in rough pipes under highly turbulent cases. International Journal of Heat and Mass Transfer, v.88, p.538-543, 2015. https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.006
» https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.006
Sonnad, J. R.; Goudar, C. T. Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation. Journal of Hydraulic Engineering, v.132, p.863-867, 2006. https://doi.org/10.1061/(ASCE)0733-9429(2006)132:8(863)
» https://doi.org/10.1061/(ASCE)0733-9429(2006)132:8(863)
Sousa, J.; Cunha, M. da C.; Marques, A. S. An explicit solution of the Colebrook-White equation through simulated annealing. Water Industry Systems: Modelling and Optimization Applications, v.2, p.347-355, 1999.
Swamee, P. K.; Jain, A. K. Explicit equations for pipe flow problems. Journal of the Hydraulics Division, v.102, p.657-664, 1976.
Tsal, R. J. Altshul-Tsal friction factor equation. Heating, Piping and Air Conditioning, v.8, p.30-45, 1989.
Vatankhah, A. R.; Kouchakzadeh, S. Discussion of “Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation” by Jagadeesh R. Sonnad and Chetan T. Goudar. Journal Hydraulic Engineering, v.134, p.1-9, 2008. https://doi.org/10.1061/(ASCE)0733-9429(2008)134:8(1187)
» https://doi.org/10.1061/(ASCE)0733-9429(2008)134:8(1187)
Willmott, C. J. On the validation of models. Physical Geography, v.2, p.184-194, 1981.
Winning, H. K.; Coole, T. Explicit friction factor accuracy and computational efficiency for turbulent flow in pipes. Flow, Turbulence and Combustion, v.90, p.1-27, 2013. https://doi.org/10.1007/s10494-012-9419
» https://doi.org/10.1007/s10494-012-9419
Wood, D. J. An explicit friction factor relationship. Civil Engineering, v.36, p.60-61, 1966.
Zigrang, D. J.; Sylvester, N. D. Explicit approximations to the solution of Colebrook’s friction factor equation. AIChE Journal, v.28, p.514-515, 1982. https://doi.org/10.1002/aic.690280323
» https://doi.org/10.1002/aic.690280323

Publication Dates

Publication in this collection
May 2018

History

Received
09 June 2017
Accepted
08 Dec 2017

This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

[1] Assefa, K. M.; Kaushal, D. R. A comparative study of friction factor correlations for high concentrate slurry flow in smooth pipes. Journal of Hydrology and Hydromechanics, v.63, p.13-20, 2015. https://doi.org/10.1515/johh-2015-0008
» https://doi.org/10.1515/johh-2015-0008

[2] Avci, A.; Karagoz, I. A novel explicit equation for friction factor in smooth and rough pipes. Journal of Fluids Engineering, v.131, p.1-2, 2009. https://doi.org/10.1115/1.3129132
» https://doi.org/10.1115/1.3129132

[3] 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, v.11, p.341-347, 2017. https://doi.org/10.1007/s11760-016-0948-8
» https://doi.org/10.1007/s11760-016-0948-8

[4] Barr, D. I. H. Solutions of the Colebrook-White function for resistance to uniform turbulent flow. Proceedings of the Institution of Civil Engineers, v.71, p.529-536, 1981.

[5] Brkić, D. New explicit correlations for turbulent flow friction factor. Nuclear Engineering and Design, v.241, p.4055-4059, 2011a. https://doi.org/10.1016/j.nucengdes.2011.07.042
» https://doi.org/10.1016/j.nucengdes.2011.07.042

[6] Brkić, D. Review of explicit approximations to the Colebrook relation for flow friction. Journal of Petroleum Science and Engineering, v.77, p.34-48, 2011b. https://doi.org/10.1016/j.petrol.2011.02.006
» https://doi.org/10.1016/j.petrol.2011.02.006

[7] Brkić, D. A note on explicit approximations to Colebrook’s friction factor in rough pipes under highly turbulent cases. International Journal of Heat and Mass Transfer, v.93, p.513-515, 2016. https://doi.org/10.1016/j.ijheatmasstransfer.2015.08.109
» https://doi.org/10.1016/j.ijheatmasstransfer.2015.08.109

[8] Brkić, D.; Ćojbašić, Ž. Intelligent flow friction estimation. Computational Intelligence and Neuroscience, v.2016, p.1-10, 2016. https://doi.org/10.1155/2016/5242596
» https://doi.org/10.1155/2016/5242596

[9] Brkić, D.; Ćojbašić, Ž. Evolutionary optimization of Colebrook’s turbulent flow friction approximations. Fluids, v.2, p.1-27, 2017. https://doi.org/10.3390/fluids2020015
» https://doi.org/10.3390/fluids2020015

[10] Buzzelli, D. Calculating friction in one step. Machine Design, v.80, p.54-55, 2008.

[11] Camargo, A. P.; Sentelhas, P. C. Avaliação do desempenho de diferentes métodos de estimativa da evapotranspiração potencial no estado de São Paulo, Brasil. Revista Brasileira de Agrometeorologia, v.5, p.89-97, 1997.

[12] Chen, N. H. An explicit equation for friction factor in pipes. Industrial & Engineering Chemistry Fundamentals, v.18, p.296-297, 1979. https://doi.org/10.1021/i160071a019
» https://doi.org/10.1021/i160071a019

[13] Churchill, S. W. Empirical expressions for the shear stress in turbulent flow in commercial pipe. AIChE Journal, v.19, p.375-376, 1973. https://doi.org/10.1002/aic.690190228
» https://doi.org/10.1002/aic.690190228

[14] Çoban, M. T. Error analysis of non-iterative friction factor formulas relative to Colebrook-White equation for the calculation of pressure drop in pipes. Journal of Naval Science and Engineering, v.8, p.1-13, 2012.

[15] Colebrook, C. F.; White, C. M. Experiments with fluid friction in roughened pipes. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, v.161, p.367-381, 1937. https://doi.org/10.1098/rspa.1937.0150
» https://doi.org/10.1098/rspa.1937.0150

[16] Eck, B. Technische Stromungslehre. New York: Springer, 1973. 324p.

[17] Fang, X.; Xu, Y.; Zhou, Z. New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing singlephase friction factor correlations. Nuclear Engineering and Design, v.241, p.897-902, 2011. https://doi.org/10.1016/j.nucengdes.2010.12.019
» https://doi.org/10.1016/j.nucengdes.2010.12.019

[18] Ghanbari, A.; Farshad, F.; Rieke, H. Newly developed friction factor correlation for pipe flow and flow assurance. Journal of Chemical Engineering and Materials Science, v.2, p.83-86, 2011.

[19] Haaland, S. E. Simple and explicit formulas for friction factor in turbulent pipe flow. Journal of Fluids Engineering, v.105, p.89-90, 1983. https://doi.org/10.1115/1.3240948
» https://doi.org/10.1115/1.3240948

[20] Heydari, A.; Narimani, E.; Pakniya, F. Explicit determinations of the Colebrook equation for the flow friction factor by statistical analysis. Chemical Engineering & Technology, v.38, p.1387-1396, 2015. https://doi.org/10.1002/ceat.201400590
» https://doi.org/10.1002/ceat.201400590

[21] Jain, A. K. Accurate explicit equation for friction factor. Journal of the Hydraulics Division, v.102, p.674-677, 1976.

[22] Manadilli, G. Replace implicit equations with signomial functions. Chemical Engineering, v.104, p.129-129, 1997.

[23] Mikata, Y.; Walczak, W. S. Exact analytical solutions of the Colebrook-White equation. Journal of Hydraulic Engineering, v.142, p.1-6, 2015.

[24] Moody, L. F. Friction factors for pipe flow. Transactions ASME, v.66, p.671-678, 1944.

[25] Moody, L. F. An approximate formula for pipe friction factors. Transactions ASME, v.69, p.1005-1011, 1947.

[26] Offor, U. H.; Alabi, S. B. An accurate and computationally efficient friction factor model. Advances in Chemical Engineering and Science, v.6, p.237-245, 2016. https://doi.org/10.4236/aces.2016.63024
» https://doi.org/10.4236/aces.2016.63024

[27] Papaevangelou, G.; Evangelides, C.; Tzimopoulos, C. A new explicit equation for the friction coefficient in the Darcy-Weisbach equation. In: Proceedings of the Tenth Conference on Protection and Restoration of the Environment, v.166, p.6-9, 2010.

[28] Rao, A. R.; Kumar, B. Friction factor for turbulent pipe flow. Bangalore: Division of Mechanical Science, Civil Engineering Indian Institute of Science Bangalore, 2007. 16p.

[29] Robaina, D. A. Análise de equações explícitas para o cálculo do coeficiente “f” da fórmula universal de perda de carga. Ciência Rural, v.22, p.157-159, 1992. https://doi.org/10.1590/S0103-84781992000200006
» https://doi.org/10.1590/S0103-84781992000200006

[30] Romeo, E.; Royo, C.; Monzón, A. Improved explicit equation for estimation of the friction factor in rough and smooth pipes. Chemical Engineering Journal, v.86, p.369-374, 2002. https://doi.org/10.1016/S1385-8947(01)00254-6
» https://doi.org/10.1016/S1385-8947(01)00254-6

[31] Round, G. F. An explicit approximation for the friction factor-Reynolds number relation for rough and smooth pipes. The Canadian Journal of Chemical Engineering, v.58, p.122-123, 1980. https://doi.org/10.1002/cjce.5450580119
» https://doi.org/10.1002/cjce.5450580119

[32] Sadeghi, S.; Peters, R. T.; Lamm, F. R. Design of zero slope microirrigation laterals: Effect of the friction factor variation. Journal of Irrigation and Drainage Engineering, v.141, p.1-9, 2015. https://doi.org/10.1061/(ASCE)IR.1943-4774.0000905
» https://doi.org/10.1061/(ASCE)IR.1943-4774.0000905

[33] Shacham, M. Comment on "Explicit equation for friction factor in pipe". Industrial & Engineering Chemistry Fundamentals, v.19, p.228-229, 1980. https://doi.org/10.1021/i160074a019
» https://doi.org/10.1021/i160074a019

[34] Shaikh, M. M.; Massan S. ur R.; Wagan, A. I. A new explicit approximation to Colebrook’s friction factor in rough pipes under highly turbulent cases. International Journal of Heat and Mass Transfer, v.88, p.538-543, 2015. https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.006
» https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.006

[35] Sonnad, J. R.; Goudar, C. T. Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation. Journal of Hydraulic Engineering, v.132, p.863-867, 2006. https://doi.org/10.1061/(ASCE)0733-9429(2006)132:8(863)
» https://doi.org/10.1061/(ASCE)0733-9429(2006)132:8(863)

[36] Sousa, J.; Cunha, M. da C.; Marques, A. S. An explicit solution of the Colebrook-White equation through simulated annealing. Water Industry Systems: Modelling and Optimization Applications, v.2, p.347-355, 1999.

[37] Swamee, P. K.; Jain, A. K. Explicit equations for pipe flow problems. Journal of the Hydraulics Division, v.102, p.657-664, 1976.

[38] Tsal, R. J. Altshul-Tsal friction factor equation. Heating, Piping and Air Conditioning, v.8, p.30-45, 1989.

[39] Vatankhah, A. R.; Kouchakzadeh, S. Discussion of “Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation” by Jagadeesh R. Sonnad and Chetan T. Goudar. Journal Hydraulic Engineering, v.134, p.1-9, 2008. https://doi.org/10.1061/(ASCE)0733-9429(2008)134:8(1187)
» https://doi.org/10.1061/(ASCE)0733-9429(2008)134:8(1187)

[40] Willmott, C. J. On the validation of models. Physical Geography, v.2, p.184-194, 1981.

[41] Winning, H. K.; Coole, T. Explicit friction factor accuracy and computational efficiency for turbulent flow in pipes. Flow, Turbulence and Combustion, v.90, p.1-27, 2013. https://doi.org/10.1007/s10494-012-9419
» https://doi.org/10.1007/s10494-012-9419

[42] Wood, D. J. An explicit friction factor relationship. Civil Engineering, v.36, p.60-61, 1966.

[43] Zigrang, D. J.; Sylvester, N. D. Explicit approximations to the solution of Colebrook’s friction factor equation. AIChE Journal, v.28, p.514-515, 1982. https://doi.org/10.1002/aic.690280323
» https://doi.org/10.1002/aic.690280323

Authors (Year)	Explicit equation of the f	Applicable ranges	Equation number
Moody (1947)Moody, L. F. An approximate formula for pipe friction factors. Transactions ASME, v.69, p.1005-1011, 1947.	$f = 0.0055 [1 + {(2 \times 10^{4} \frac{ε}{D} + \frac{10^{6}}{R e})}^{1 / 3}]$	4 × 10³ ≤ Re ≤ 10⁸ 0 ≤ Ɛ/D ≤ 10^-2	(2)
Wood (1966)Wood, D. J. An explicit friction factor relationship. Civil Engineering, v.36, p.60-61, 1966.	$f = 0.094 {(\frac{ε}{D})}^{0.225} + 0.53 (\frac{ε}{D}) + 88 {(\frac{ε}{D})}^{0.44} R e^{- [1.62 {(\frac{ε}{D})}^{0.134}]}$	4 × 10³ ≤ Re ≤ 5 × 10⁷ 10^-5 ≤ Ɛ/D ≤ 4 × 10^-2	(3)
Churchill (1973)Churchill, S. W. Empirical expressions for the shear stress in turbulent flow in commercial pipe. AIChE Journal, v.19, p.375-376, 1973. https://doi.org/10.1002/aic.690190228 https://doi.org/10.1002/aic.690190228...	$\frac{1}{\sqrt{f}} = - 2 \log [\frac{ε}{3.71 D} + {(\frac{7}{R e})}^{0.9}]$	Not specified	(4)
Eck (1973)Eck, B. Technische Stromungslehre. New York: Springer, 1973. 324p.	$\frac{1}{\sqrt{f}} = - 2 \log (\frac{ε}{3.71 D} + \frac{15}{R e})$	0 ≤ Ɛ/D ≤ 10^-2	(5)
Jain (1976)Jain, A. K. Accurate explicit equation for friction factor. Journal of the Hydraulics Division, v.102, p.674-677, 1976.	$\frac{1}{\sqrt{f}} = - 2 \log [\frac{ε}{3.715 D} + {(\frac{6.943}{R e})}^{0.9}]$	5 × 10³ ≤ Re ≤ 10⁷	(6)
Swamee & Jain (1976)Swamee, P. K.; Jain, A. K. Explicit equations for pipe flow problems. Journal of the Hydraulics Division, v.102, p.657-664, 1976.	$\frac{1}{\sqrt{f}} = - 2 \log (\frac{ε}{3.7 D} + \frac{5.74}{R e^{0.9}})$	5 × 10³ ≤ Re ≤ 10⁸ 10^-6 ≤ Ɛ/D ≤ 5 × 10^-2	(7)
Chen (1979)Chen, N. H. An explicit equation for friction factor in pipes. Industrial & Engineering Chemistry Fundamentals, v.18, p.296-297, 1979. https://doi.org/10.1021/i160071a019 https://doi.org/10.1021/i160071a019...	$\frac{1}{\sqrt{f}} = - 2 \log [\frac{ε}{3.7065 D} - \frac{5.0452}{R e} \log (\frac{1}{2.8257} {(\frac{ε}{D})}^{1.1098} + \frac{5.8506}{R e^{0.8981}})]$	4 × 10³ ≤ Re ≤ 4 × 10⁸ 10^-7 ≤ Ɛ/D ≤ 5 × 10^-2	(8)
Round (1980)Round, G. F. An explicit approximation for the friction factor-Reynolds number relation for rough and smooth pipes. The Canadian Journal of Chemical Engineering, v.58, p.122-123, 1980. https://doi.org/10.1002/cjce.5450580119 https://doi.org/10.1002/cjce.5450580119...	$\frac{1}{\sqrt{f}} = 1.8 \log [\frac{R e}{0.135 R e (\frac{ε}{D}) + 6.5}]$	4 × 10³ ≤ Re ≤ 10⁸ 0 ≤ Ɛ/D ≤ 5 × 10^-2	(9)
Shacham (1980)Shacham, M. Comment on "Explicit equation for friction factor in pipe". Industrial & Engineering Chemistry Fundamentals, v.19, p.228-229, 1980. https://doi.org/10.1021/i160074a019 https://doi.org/10.1021/i160074a019...	$\frac{1}{\sqrt{f}} = - 2 \log [\frac{ε}{3.7 D} - \frac{5.02}{R e} \log (\frac{ε}{3.7 D} + \frac{14.5}{R e})]$	4 × 10³ ≤ Re ≤ 4 × 10⁸	(10)
Barr (1981)Barr, D. I. H. Solutions of the Colebrook-White function for resistance to uniform turbulent flow. Proceedings of the Institution of Civil Engineers, v.71, p.529-536, 1981.	$\frac{1}{\sqrt{f}} = - 2 \log \{\frac{ε}{3.7 D} + \frac{4.518 \log (\frac{R e}{7})}{R e [1 + \frac{R e^{0.52}}{29} {(\frac{ε}{D})}^{0.7}]}\}$	Not specified	(11)
Zigrang & Sylvester (1982)Zigrang, D. J.; Sylvester, N. D. Explicit approximations to the solution of Colebrook’s friction factor equation. AIChE Journal, v.28, p.514-515, 1982. https://doi.org/10.1002/aic.690280323 https://doi.org/10.1002/aic.690280323...	$\frac{1}{\sqrt{f}} = - 2 \log \{\frac{ε}{3.7 D} - \frac{5.02}{R e} \log [\frac{ε}{3.7 D} - \frac{5.02}{R e} \log (\frac{ε}{3.7 D} + \frac{13}{R e})]\}$	4 × 10³ ≤ Re ≤ 10⁸ 4 × 10^-5 ≤ Ɛ/D ≤ 5 × 10^-2	(12)
Haaland (1983)Haaland, S. E. Simple and explicit formulas for friction factor in turbulent pipe flow. Journal of Fluids Engineering, v.105, p.89-90, 1983. https://doi.org/10.1115/1.3240948 https://doi.org/10.1115/1.3240948...	$\frac{1}{\sqrt{f}} = - 1.8 \log [{(\frac{ε}{3.7 D})}^{1.11} + \frac{6.9}{R e}]$	4 × 10³ ≤ Re ≤ 10⁸ 10^-6 ≤ Ɛ/D ≤ 5 × 10^-2	(13)
Tsal (1989)Tsal, R. J. Altshul-Tsal friction factor equation. Heating, Piping and Air Conditioning, v.8, p.30-45, 1989.	$A = 0.11 {(\frac{68}{R e} + \frac{ε}{D})}^{0.25}$ If A ≥ 0.018 than f = A If A < 0.018 than f = 0.0028 + 0.85A	4 × 10³ ≤ Re ≤ 10⁸ 0 ≤ Ɛ/D ≤ 5 × 10^-2	(14)
Robaina (1992)Robaina, D. A. Análise de equações explícitas para o cálculo do coeficiente “f” da fórmula universal de perda de carga. Ciência Rural, v.22, p.157-159, 1992. https://doi.org/10.1590/S0103-84781992000200006 https://doi.org/10.1590/S0103-8478199200...	$\frac{1}{\sqrt{f}} = - 2 \log (0.27 \frac{ε}{D} + \frac{5.62}{R e^{0.9}})$	4 × 10³ ≤ Re ≤ 4 × 10⁷ 10^-5 ≤ Ɛ/D ≤ 10^-2	(15)
Manadilli (1997)Manadilli, G. Replace implicit equations with signomial functions. Chemical Engineering, v.104, p.129-129, 1997.	$\frac{1}{\sqrt{f}} = - 2 \log (\frac{ε}{D} + \frac{95}{R e^{0.983}} + \frac{96.82}{R e})$	5.235 × 10³ ≤ Re ≤ 10⁸ 0 ≤ Ɛ/D ≤ 5 × 10^-2	(16)
Sousa et al. (1999)Sousa, J.; Cunha, M. da C.; Marques, A. S. An explicit solution of the Colebrook-White equation through simulated annealing. Water Industry Systems: Modelling and Optimization Applications, v.2, p.347-355, 1999.	$\frac{1}{\sqrt{f}} = - 2 \log [\frac{ε}{3.7 D} - \frac{5.16}{R e} \log (\frac{ε}{3.7 D} + \frac{5.09}{R e^{0.87}})]$	Not specified	(17)
Romeo et al. (2002)Romeo, E.; Royo, C.; Monzón, A. Improved explicit equation for estimation of the friction factor in rough and smooth pipes. Chemical Engineering Journal, v.86, p.369-374, 2002. https://doi.org/10.1016/S1385-8947(01)00254-6 https://doi.org/10.1016/S1385-8947(01)00...	$\frac{1}{\sqrt{f}} = - 2 \log \{\frac{ε}{3.7065 D} - \frac{5.0272}{R e} \log [\frac{ε}{3.827 D} - \frac{4.567}{R e} \log ({(\frac{ε}{7.7918 D})}^{0.9924} + {(\frac{5.3326}{208.815 + R e})}^{0.9345})]\}$	3 × 10³ ≤ Re ≤ 1.5 × 10⁸ 0 ≤ Ɛ/D ≤ 5 × 10^-2	(18)
Sonnad & Goudar (2006)Sonnad, J. R.; Goudar, C. T. Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation. Journal of Hydraulic Engineering, v.132, p.863-867, 2006. https://doi.org/10.1061/(ASCE)0733-9429(2006)132:8(863) https://doi.org/10.1061/(ASCE)0733-9429(...	$\frac{1}{\sqrt{f}} = 0.8686 \ln (\frac{0.4587 R e}{G^{\frac{G}{G + 1}}})$ $G = 0.124 R e \frac{ε}{D} + \ln (0.4587 R e)$	4 × 10³ ≤ Re ≤ 10⁸ 10^-6 ≤ Ɛ/D ≤ 5 × 10^-2	(19)
Rao & Kumar (2007)Rao, A. R.; Kumar, B. Friction factor for turbulent pipe flow. Bangalore: Division of Mechanical Science, Civil Engineering Indian Institute of Science Bangalore, 2007. 16p.	$\frac{1}{\sqrt{f}} = 2 \log [\frac{{(2 \frac{ε}{D})}^{- 1}}{(\frac{0.444 + 0.135 R e}{R e}) β}]$ $β = 1 - 0.55 e^{- 0.33 {[\ln (\frac{R e}{6.5})]}^{2}}$	Not specified	(20)
Buzzelli (2008)Buzzelli, D. Calculating friction in one step. Machine Design, v.80, p.54-55, 2008.	$\frac{1}{\sqrt{f}} = B_{1} - [\frac{B_{1} + 2 \log (\frac{B_{2}}{R e})}{1 + \frac{2.18}{B_{2}}}]$ $B_{1} = \frac{[0.744 \ln (R e) - 1.41]}{(1 + 1.32 \sqrt{\frac{ε}{D}})}$ $B_{2} = \frac{ε}{3.7 D} R e + 2.51 B_{1}$	3 × 10³ ≤ Re ≤ 1.5 × 10⁸ 0 ≤ Ɛ/D ≤ 5 × 10^-2	(21)
Vantankhah & Kouchakzadeh (2008)Vatankhah, A. R.; Kouchakzadeh, S. Discussion of “Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation” by Jagadeesh R. Sonnad and Chetan T. Goudar. Journal Hydraulic Engineering, v.134, p.1-9, 2008. https://doi.org/10.1061/(ASCE)0733-9429(2008)134:8(1187) https://doi.org/10.1061/(ASCE)0733-9429(...	$\frac{1}{\sqrt{f}} = 0.8686 \ln [\frac{0.4587 R e}{{(S - 0.31)}^{(\frac{S}{S + 0.9633})}}]$ $S = 0.124 R e (\frac{ε}{D}) + \ln (0.4587 R e)$	4 × 10³ ≤ Re ≤ 10⁸ 10^-6 ≤ Ɛ/D ≤ 5 × 10^-2	(22)
Avci & Karagoz (2009)Avci, A.; Karagoz, I. A novel explicit equation for friction factor in smooth and rough pipes. Journal of Fluids Engineering, v.131, p.1-2, 2009. https://doi.org/10.1115/1.3129132 https://doi.org/10.1115/1.3129132...	$f = \frac{6.4}{{\{\ln (R e) - \ln [1 + 0.01 R e \frac{ε}{D} (1 + 10 \sqrt{\frac{ε}{D}})]\}}^{2.4}}$	Not specified	(23)
Papaevangelou et al. (2010)Papaevangelou, G.; Evangelides, C.; Tzimopoulos, C. A new explicit equation for the friction coefficient in the Darcy-Weisbach equation. In: Proceedings of the Tenth Conference on Protection and Restoration of the Environment, v.166, p.6-9, 2010.	$f = \frac{0.2479 - 0.0000947 {(7 - \log R e)}^{4}}{{[\log (\frac{ε}{3.615 D} + \frac{7.366}{R e^{0.9142}})]}^{2}}$	10⁴ ≤ Re ≤ 10⁷ 10^-5 ≤ Ɛ/D ≤ 10^-3	(24)
Brkić (2011a)Brkić, D. New explicit correlations for turbulent flow friction factor. Nuclear Engineering and Design, v.241, p.4055-4059, 2011a. https://doi.org/10.1016/j.nucengdes.2011.07.042 https://doi.org/10.1016/j.nucengdes.2011...	$\frac{1}{\sqrt{f}} = - 2 \log (\frac{2.18 β}{R e} + \frac{ε}{3.71 D})$ $β = \ln \frac{R e}{1.816 \ln [\frac{1.1 R e}{\ln (1 + 1.1 R e)}]}$	Not specified	(25)
Fang et al. (2011)Fang, X.; Xu, Y.; Zhou, Z. New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing singlephase friction factor correlations. Nuclear Engineering and Design, v.241, p.897-902, 2011. https://doi.org/10.1016/j.nucengdes.2010.12.019 https://doi.org/10.1016/j.nucengdes.2010...	$f = 1.613 {\{\ln [0.234 {(\frac{ε}{D})}^{1.1007} - \frac{60.525}{R e^{1.1105}} + \frac{56.291}{R e^{1.0712}}]\}}^{- 2}$	3 × 10³ ≤ Re ≤ 10⁸ 0 ≤ Ɛ/D ≤ 5 × 10^-2	(26)
Ghanbari et al. (2011)Ghanbari, A.; Farshad, F.; Rieke, H. Newly developed friction factor correlation for pipe flow and flow assurance. Journal of Chemical Engineering and Materials Science, v.2, p.83-86, 2011.	$f = {\{- 1.52 \log [{(\frac{ε}{7.21 D})}^{1.042} + {(\frac{2.731}{R e})}^{0.9152}]\}}^{- 2.169}$	2.1 × 10³ ≤ Re ≤ 10⁸ 0 ≤ Ɛ/D ≤ 5 × 10^-2	(27)
Shaikh et al. (2015)Shaikh, M. M.; Massan S. ur R.; Wagan, A. I. A new explicit approximation to Colebrook’s friction factor in rough pipes under highly turbulent cases. International Journal of Heat and Mass Transfer, v.88, p.538-543, 2015. https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.006 https://doi.org/10.1016/j.ijheatmasstran...	$f = 0.25 {[\log (\frac{2.51}{α R e} + \frac{ε}{3.7 D})]}^{- 2}$ $α = {[1.14 - 2 \log (\frac{ε}{D})]}^{- 2}$	10⁴ ≤ Re ≤ 10⁸ 10^-4 ≤ Ɛ/D ≤ 5 × 10^-2	(28)
Brkić (2016)Brkić, D. A note on explicit approximations to Colebrook’s friction factor in rough pipes under highly turbulent cases. International Journal of Heat and Mass Transfer, v.93, p.513-515, 2016. https://doi.org/10.1016/j.ijheatmasstransfer.2015.08.109 https://doi.org/10.1016/j.ijheatmasstran...	$\frac{1}{\sqrt{f}} = - 2 \log [\frac{2.51 (1.14 - 2 \log (\frac{ε}{D}))}{R e} + \frac{ε}{3.71 D}]$	10⁶ < Re < 10⁸ 10^-2 < Ɛ/D < 5 × 10^-2	(29)
Offor & Alabi (2016)Offor, U. H.; Alabi, S. B. An accurate and computationally efficient friction factor model. Advances in Chemical Engineering and Science, v.6, p.237-245, 2016. https://doi.org/10.4236/aces.2016.63024 https://doi.org/10.4236/aces.2016.63024...	$\frac{1}{\sqrt{f}} = - 2 \log \{\frac{ε}{3.71 D} - \frac{1.975}{R e} [\ln ({(\frac{ε}{3.93 D})}^{1.092} + (\frac{7.627}{R e + 395.9}))]\}$	4 × 10³ ≤ Re ≤ 10⁸ 0 ≤ Ɛ/D ≤ 5 × 10^-2	(30)

Concordance index (d)	Correlation coefficient (r)	Performance index (Id)	Classification
0.95-1.00	0.95-1.00	0.90-1.00	Excellent
0.89-0.95	0.89-0.95	0.80-0.90	Optimum
0.84-0.89	0.84-0.89	0.70-0.80	Very Good
0.77-0.84	0.77-0.84	0.60-0.70	Good
0.71-0.77	0.71-0.77	0.50-0.60	Moderately Good
0.63-0.71	0.63-0.71	0.40-0.50	Moderate
0.55-0.63	0.55-0.63	0.30-0.40	Moderately Poor
0.45-0.55	0.45-0.55	0.20-0.30	Poor
0.32-0.45	0.32-0.45	0.10-0.20	Very Poor
0.00-0.32	0.00-0.32	0.00-0.10	Bad

Brasil