Desempenho de aproximações explícitas do coeficiente de perda de carga para condutos pressurizados

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

Acessibilidade / Reportar erro

Brasil

Revista Brasileira de Engenharia Agrícola e Ambiental

Español English

Brasil

Español English

sumário « anterior atual seguinte »

Sumário

ARTICLE • Rev. bras. eng. agríc. ambient. 22 (5) • Maio 2018 • https://doi.org/10.1590/1807-1929/agriambi.v22n5p301-307 copiar

Desempenho de aproximações explícitas do coeficiente de perda de carga para condutos pressurizados

Autoria SCIMAGO INSTITUTIONS RANKINGS

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

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

Unidade Acadêmica de Engenharia Agrícola Unidade Acadêmica de Engenharia Agrícola, UFCG, Av. Aprígio Veloso 882, Bodocongó, Bloco CM, 1º andar, CEP 58429-140, Campina Grande, PB, Brasil, Tel. +55 83 2101 1056 - Campina Grande - PB - Brazil
E-mail: revistagriambi@gmail.com

Acompanhe os números deste periódico no seu leitor de RSS

[1] brunadpimenta@gmail.com (Corresponding author)