Global Journal of Mathematics
Vol. 2, No. 2, April 17, 2015
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ISSN: 2395-4760
Survey in Colebrook equation approximations Nawfel Mohammed Baqer Engineering, Technical College in Najaf, Iraq.
E-mail:
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Abstract: In this review study, The Haaland equation was proposed by Norwegian Institute of Technology professor Haaland in 1984. It is used to solve directly for the Darcy–Weisbach friction factor f for a full-flowing circular pipe, and this review study involved free surface flow.
Keywords:
Colebrook, equation, friction factor.
Introduction: Compact forms: The Colebrook equation is an implicit equation that combines experimental results of studies of turbulent flow in smooth [2] [3] and rough pipes. It was developed in 1939 by C. F. Colebrook. The 1937 paper by C. F. Colebrook and C. M. White is often erroneously cited as the source of the equation. This is partly because Colebrook in a footnote (from his 1939 paper) acknowledges his debt to White for suggesting the mathematical method by which the smooth and rough pipe correlations could be combined. The equation is used to iteratively solve for the Darcy–Weisbach friction factor f. This equation is also known as the Colebrook–White equation. For conduits that are flowing completely full of fluid at Reynolds numbers greater than 4000, it is defined as:
or
where:
is the Darcy friction factor
Roughness height,
Hydraulic diameter,
Hydraulic radius,
(m, ft) (m, ft) — For fluid-filled, circular conduits, (m, ft) — For fluid-filled, circular conduits,
= D = inside diameter = D/4 = (inside diameter)/4
is the Reynolds number.
Solving : The Colebrook equation used to be solved numerically due to its apparent implicit nature. Recently, the Lambert W [4] function has been employed to obtain explicit reformulation of the Colebrook equation. Expanded forms: Additional, mathematically equivalent forms of the Colebrook equation are:
where: 1.7384... = 2 log (2 × 3.7) = 2 log (7.4)
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ISSN: 2395-4760
18.574 = 2.51 × 3.7 × 2 and
or
where: 1.1364... = 1.7384... − 2 log (2) = 2 log (7.4) − 2 log (2) = 2 log (3.7) 9.287 = 18.574 / 2 = 2.51 × 3.7. The additional equivalent forms above assume that the constants 3.7 and 2.51 in the formula at the top of this section are exact. The constants are probably values which were rounded by Colebrook during his curve fitting; but they are effectively treated as exact when comparing (to several decimal places) results from explicit formulae (such as those found elsewhere in this article) to the friction factor computed via Colebrook's implicit equation. Equations similar to the additional forms above (with the constants rounded to fewer decimal places—or perhaps shifted slightly to minimize overall rounding errors) may be found in various references. It may be helpful to note that they are essentially the same equation. Free surface flow: Another form of the Colebrook-White equation exists for free surfaces. Such a condition may exist in a pipe that is flowing partially full of fluid. For free surface flow:
Approximations of the Colebrook equation Haalandequation: The Haaland equation was proposed by Norwegian Institute of Technology professor Haaland in 1984. It is used to solve directly for the Darcy–Weisbach friction factor f for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation, but the discrepancy from experimental data is well within the accuracy of the data. It was developed by S. E. Haaland in 1983. The Haaland equation is defined as:
[5]
where:
is the Darcy friction factor
is the relative roughness is the Reynolds number.
Swamee–Jain equation: The Swamee–Jain equation is used to solve directly for the Darcy–Weisbach friction factor f for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation.
where f is a function of:
Roughness height, ε (m, ft)
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Pipe diameter, D (m, ft)
Reynolds number, Re (unitless).
Vol. 2, No. 2, April 17, 2015 ISSN: 2395-4760
Serghides'ssolution: Serghides's solution is used to solve directly for the Darcy–Weisbach friction factor f for a full-flowing circular pipe. It is an [6] approximation of the implicit Colebrook–White equation. It was derived using Steffensen's method. The solution involves calculating three intermediate values and then substituting those values into a final equation.
where f is a function of:
Roughness height, ε (m, ft)
Pipe diameter, D (m, ft)
Reynolds number, Re (unitless).
The equation was found to match the Colebrook–White equation within 0.0023% for a test set with a 70-point matrix 8 consisting of ten relative roughness values (in the range 0.00004 to 0.05) by seven Reynolds numbers (2500 to 10 ). Goudar–Sonnadequation: Goudar equation is the most accurate approximation to solve directly for the Darcy–Weisbach friction factor f for a full[7] flowing circular pipe. It is an approximation of the implicit Colebrook–White equation. Equation has the following form
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where f is a function of:
Roughness height, ε (m, ft)
Pipe diameter, D (m, ft)
Reynolds number, Re (unitless).
Brkićsolution: Brkić shows one approximation of the Colebrook equation based on the Lambert W-function
[8]
where Darcy friction factor f is a function of:
Roughness height, ε (m, ft)
Pipe diameter, D (m, ft)
Reynolds number, Re (unitless).
The equation was found to match the Colebrook–White equation within 3.15%. Blasius correlations: Early approximations by Blasius are given in terms of the Fanning friction factor in the Paul Richard Heinrich Blasius article. Table of Approximations: The following table lists historical approximations where:
[9]
Re, Reynolds number (unitless);
λ, Darcy friction factor (dimensionless);
ε, roughness of the inner surface of the pipe (dimension of length);
D, inner pipe diameter;
is the base-10 logarithm.
Note that the Churchill equation (1977) is the only one that returns a correct value for friction factor in the laminar flow region (Reynolds number < 2300). All of the others are for transitional and turbulent flow only. Table of Colebrook equation approximations Equation
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Author
Year
Moody
1947
Wood
1966
Ref
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where
Eck
1973
Jain and Swamee
1976
Churchill
1973
Jain
1976
Churchill
1977
Chen
1979
Round
1980
Barr
1981
Zigrang and Sylvester
1982
Haaland
1983
where
or
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or
where
where:
where:
Serghides
1984
Manadilli
1997
Monzon, Romeo, Royo
2002
Goudar, Sonnad 2006
Vatankhah, Kouchakzadeh
2008
Buzzelli
2008
where
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ISSN: 2395-4760
Avci, Kargoz
2009
Evangleids, Papaevangelou, 2010 Tzimopoulos
References: 1.
Manning, Francis S.; Thompson, Richard E. (1991). Oilfield Processing of Petroleum. Vol. 1:Natural Gas. PennWell Books. ISBN 0-87814-343-2, 420.
2.
Colebrook, C.F. (February 1939). "Turbulent flow in pipes, with particular reference to the transition region between smooth and rough pipe laws". Journal of the Institution of Civil Engineers (London).
3.
Colebrook, C. F. and White, C. M. (1937). "Experiments with Fluid Friction in Roughened Pipes". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 161 (906): 367–381.
4.
More, A. A. (2006). "Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes". Chemical Engineering Science 61 (16): 5515–5519. doi:10.1016/j.ces.2006.04.003.
5.
BS Massey Mechanics of Fluids 6th Ed ISBN 0-412-34280-4
6.
Serghides, T.K (1984). "Estimate friction factor accurately". Chemical Engineering Journal 91(5): 63–64.
7.
Goudar, C.T., Sonnad, J.R. (August 2008). "Comparison of the iterative approximations of the Colebrook–White equation". Hydrocarbon Processing Fluid Flow and Rotating Equipment Special Report(August 2008): 79–83.
8.
^Brkić, Dejan (2011). "An Explicit Approximation of Colebrook’s equation for fluid flow friction factor". Petroleum Science and Technology 29 (15): 1596–1602. doi:10.1080/10916461003620453.
9.
^Beograd, DejanBrkić (March 2012). "Determining Friction Factors in Turbulent Pipe Flow". Chemical Engineering: 34–39
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