THE PERFORMANCE OF EXPLICIT FORMULAS FOR DETERMINING THE DARCY-WEISBACH FRICTION FACTOR

Minhoni, Renata T. de A.; Pereira, Francisca F. S.; Silva, Tatiane B. G. da; Castro, Evanize R.; Saad, João C. C.

doi:10.1590/1809-4430-Eng.Agric.v40n2p258-265/2020

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Technical Paper, Soil and Water Engineering • Eng. agríc. (Online) 40 (2) • Mar-Apr 2020 • https://doi.org/10.1590/1809-4430-Eng.Agric.v40n2p258-265/2020 copy

THE PERFORMANCE OF EXPLICIT FORMULAS FOR DETERMINING THE DARCY-WEISBACH FRICTION FACTOR

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

The Darcy-Weisbach equation is the most recommended equation for determining the pressure loss in pressurized pipes because of its wide applicability. However, one of the largest obstacles to implementing this equation is the friction factor (f) calculation. This factor can be precisely determined using the Colebrook equation, which is implicit. Thus, the objective of this study was to compare six explicit equations for calculating the Darcy-Weisbach friction factor with the implicit Colebrook equation based on the relative error. Based on the results, the equations of Vatankhah and Offor & Alabi were the most highly recommended. These six explicit formulas showed a mean relative error of less than ± 1% compared to the Colebrook equation, except for the Swamee and Jain equation, for which the laminar regime generated a mean relative error of 1.83%.

KEYWORDS
absolute roughness; Colebrook equation; forced conduits; head loss; hydraulic load; Reynolds number

Equation	Reference
$f = \frac{0.25}{{[\log (\frac{ε}{3.7 \times D} + \frac{5.74}{R^{0.9}})]}^{2}}$	Swamee & Jain (1976)Swamee PK, Jain AK (1976) Explicit equations for pipe flow problems. Journal of Hydraulics Division 102:657-664.
$f = {{(\frac{64}{R})}^{8} + 9.5 \times {[\ln (\frac{ε}{3.7 \times D} + \frac{5.74}{R^{0.9}}) - {(\frac{2500}{R})}^{6}]}^{- 16}}^{0.125}$	Swamee & Swamee (2007)Swamee PK, Swamee N (2007) Full-range pipe-flow equations. Journal of Hydraulic Research 45:841-843.
$f = \frac{0.2479 - 0.0000947 {(7 - \log R)}^{4}}{{[\log (\frac{ε}{3.615 \times D} + \frac{7.366}{R^{0.9142}})]}^{2}}$	Papaevangelou et al., (2010)Papaevangelou G, Evangelides C, Tzimopoulos C (2010) A new explicit equation for the friction coeficiente in the Darcy-Weisbach equation. Proceedings of the Tenth Conference on Protection and Restoration of the Environment 166:6-9.
$f = 1.613 {{\ln [0.234 {(\frac{ε}{D})}^{1.1007} - \frac{60.525}{R^{1.1105}} + \frac{56.291}{R^{1.0712}}]}}^{- 2}$	Fang et al., (2011)Fang X, Xu Y, Zhou Z (2011) New correlations of single-phase friction fator for turbulent pipe flow and evaluation of existing single-phase friction fator correlations. Nuclear Engineering and Design 241:897-902.
$\frac{1}{f^{0.5}} = - 2 \log {\frac{ε}{3.71 D} - \frac{1.975}{R} [\ln ({(\frac{ε}{3.93 D})}^{1.092} + (\frac{7.627}{R + 395.9}))]}$	Offor & Alabi (2016)Offor UH, Alabi SB (2016) An Accurate and Computationally Efficient Explicit Friction Factor Model. Advances in Chemical Engineering and Science 6:237-245.
$\frac{1}{f^{0.5}} = 0.8686 l n [\frac{0.3984 R}{(0.8686 s) \frac{s - 0.645}{s + 0.39}}]$	Vatankhah (2018)Vatankhah AR (2018) Approximate analytical solutions for the colebrook equation. Journal of Hydraulic Engeneering 144(5). DOI: https://doi.org/10.1061/(ASCE)HY.1943-7900.0001454 https://doi.org/10.1061/(ASCE)HY.1943-79...

Material	DN (mm)	Di (mm)	ɛ (mm)	ɛ/D
PVC	35	35.71	0.003334	0.00009
	50	47.56		0.00007
	75	72.05		0.00005
LLDPE	10	9.55	0.008116	0.0008
	13	13.12		0.0006
	16	16.81		0.0005
	20	20.72		0.0004
	26	27.24		0.0003

ɛ/D	R
0.000001	4,000
0.000005	8,000
0.00001	10,000
0.00005	20,000
0.00007	40,000
0.00009	80,000
0.0001	100,000
0.0003	150,000
0.0004	200,000
0.0005	400,000
0.0006	800,000
0.0007	1,000,000
0.0008	2,000,000
0.001	4,000,000
0.005	8,000,000
0.05	10,000,000
	20,000,000
	40,000,000
	80,000,000
	100,000,000

Reference/Equation	Overall mean	Mean		Range	Standard deviation
Reference/Equation	Overall mean	Laminar regime	Turbulent regime	Range	Standard deviation
Swamee & Jain (1976)Swamee PK, Jain AK (1976) Explicit equations for pipe flow problems. Journal of Hydraulics Division 102:657-664.	0.00478	0.0183	0.0041	-0.0302 to 0.0071	0.0044
Swamee & Swamee (2007)Swamee PK, Swamee N (2007) Full-range pipe-flow equations. Journal of Hydraulic Research 45:841-843.	0.00385	0.0039	0.0038	-0.0190 to 0.0074	0.0030
Papaevangelou et al., (2010)Papaevangelou G, Evangelides C, Tzimopoulos C (2010) A new explicit equation for the friction coeficiente in the Darcy-Weisbach equation. Proceedings of the Tenth Conference on Protection and Restoration of the Environment 166:6-9.	0.00215	0.0026	0.0021	-0.0057 to 0.0079	0.0015
Fang et al., (2011)Fang X, Xu Y, Zhou Z (2011) New correlations of single-phase friction fator for turbulent pipe flow and evaluation of existing single-phase friction fator correlations. Nuclear Engineering and Design 241:897-902.	0.00184	0.0018	0.0018	-0.0049 to 0.0043	0.0011
Offor & Alabi (2016)Offor UH, Alabi SB (2016) An Accurate and Computationally Efficient Explicit Friction Factor Model. Advances in Chemical Engineering and Science 6:237-245.	0.00046	0.0039	0.0005	-0.0005 to 0.0013	0.0003
Vatankhah (2018)Vatankhah AR (2018) Approximate analytical solutions for the colebrook equation. Journal of Hydraulic Engeneering 144(5). DOI: https://doi.org/10.1061/(ASCE)HY.1943-7900.0001454 https://doi.org/10.1061/(ASCE)HY.1943-79...	0.00039	0.0001	0.0004	0.00001 to 0.0013	0.0003

Associação Brasileira de Engenharia Agrícola SBEA - Associação Brasileira de Engenharia Agrícola, Departamento de Engenharia e Ciências Exatas FCAV/UNESP, Prof. Paulo Donato Castellane, km 5, 14884.900 | Jaboticabal - SP, Tel./Fax: +55 16 3209 7619 - Jaboticabal - SP - Brazil
E-mail: revistasbea@sbea.org.br

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[1] *Corresponding author. São Paulo State University/ Botucatu - SP, Brazil. E-mail: renataminhoni@hotmail.com