Head loss due to friction for fluids traveling through pipes, tubes and ducts is a critical parameter for solving turbulent-flow problems in the chemical process industries. The Colebrook equation is used to assess hydraulic resistance for turbulent flow in both smooth- and rough-walled pipes. The equation contains a dimensionless fluid-flow friction coefficient that must be calculated for the properties of the pipe and the fluid flow.
Determining friction factors for the Colebrook equation requires either calculating iteratively or manipulating the equation to express friction factors explicitly. Iterative calculations can be carried out using a spreadsheet solver, but can require more computational time. Explicit expressions offer direct computation, but have a range of simplicity and corresponding error.
The Lambert W function may be a better method to express friction factors explicitly because it allows users to avoid iterative calculation and also reduce relative error. This article outlines methods for determining friction factors, and discusses how to use the Lambert W function. The Lambert W function is evaluated using real data in Part 2 of the feature.
Pipe-flow problems are challenging because they require determination of the fluid-flow friction factor (λ), a dimensionless term whose expression is a non-factorable polynomial. The friction factor is a complicated function of relative surface roughness and Reynolds number (Re), where, specifically, hydraulic resistance depends on flowrate. The situation is similar to that observed with electrical resistance when a diode is in circuit.
The hydraulics literature contains three forms of the Colebrook equation for which the friction factor is implicit, meaning that the term it has to be approximately solved using an iterative procedure because the term exists on both sides of the equation. Engineers have also developed a number of approximation formulas that express the friction factor explicitly, meaning that it is calculated directly rather than through an iterative process.
The equation proposed by Colebrook in 1939  describes a monotonic change in the friction factor as pipe surfaces transition from fully smooth to fully rough.
At the time it was developed, the implicit form of the Colebrook equation was too complex to be of great practical use. It may be difficult for many to recall the time, as recently as the 1970s, with no personal computers or even calculators that could do much more than add or subtract.
Many researchers, such as Coelho and Pinho , have adopted a modification of the implicit Colebrook equation, using 2.825 as the constant instead of 2.51. Alternatively, some engineers use the Fanning factor, which is different from the more commonly used Darcy friction factor. The Darcy friction factor is four times greater than the Fanning friction factor, but their physical meanings are equivalent.
In general, the following five approaches are available to solve the Colebrook equation:
• Graphical solutions using Moody or Rouse diagrams (useful only as an orientation)
• Iterative solutions using spreadsheet solvers (can be highly accurate to Colebrook standard, but require more computational resources)
• Using explicit Colebrook-equation approximations (less computation, but can introduce error)
• Lambert W function (avoids iterative calculations and allows reduction of relative error)
Trial-and-error method (obsolete)
Graphs based on the Colebrook equation represent the simplest, but most approximate approach to avoiding trial-and-error-based iterative solutions. In 1943, Rouse  developed a chart based upon the Colebrook equation that allowed it to be used more practically. In the Rouse diagram, Reynolds number is related to friction factor and the friction factor (λ) is implicit for both coordinates, (that is, ). In order to simplify this chart for more ordinary engineering use, Moody  adopted more convenient coordinates the following year, plotting Re versus f(λ). To be precise, as the primary axis, Rouse used and Re = f(λ) as the secondary axis, while Moody used only Re = f(λ).
Iterative spreadsheet solutions
Today, implicit equations such as the Colebrook can be solved easily and accurately using the Newton-Raphson iterative procedure and common software tools like Microsoft Excel 2007 (Figures 1 and 2). The maximum number of iterations in Excel 2007 is 32,767. To solve for the unknown friction factor λ, one must start by estimating the value of the friction factor on the right side of the equation, then solve for the new friction factor on the left. The new value would then be entered back on the right side, and the process continued until there is a balance on both sides of the equation within an arbitrary difference. This difference must be small, yet accommodate all values of relative roughness (ε/D) and values of Reynolds number without causing endless computations.
Note that the Colebrook equation consists of two parts: the first part is equal to zero in first iteration (meaning that 2.51 / (Re · √λ) = 0), but the second part has a value different than zero (ε / (D · 3.71) ≠ 0), so estimating the value in the first iteration is unnecessary. The initial value in the first iteration is ε / (D · 3.71).
In some cases, seemingly effective solutions are too simple to generate the required accuracy, and Excel is an ideal tool to solve these kinds of problems. Excel allows accuracies to within at least 0.01. The maximum accuracy can be set to 0.0000001.
To solve the implicit Colebrook equation, click the Excel “Office button” in the upper left of the screen (Figure 1). Then click “Excel options,” and “Formulas.” In the Formulas array, check the box for “Enable iterative calculation” and enter the maximum number of iterations desired. The maximum change allowed between two successive iterations also must be set in the program.
Many explicit approximations of the Colebrook equation are available (Table 1), including those from Moody , Wood , Eck , Swamee and Jain , Churchill [9–10], Jain , Chen , Round , Barr , Zigrang and Sylvester , Haaland , Serghides , Manadilli , Romeo and others. , Sonnad and Goudar , Buzzelli , Vatankhah and Kouchakzadeh , Avci and Karagoz , Papaevangelou et al. . These approximations vary in their degree of accuracy, depending upon the complexity of their functional forms. The more complex ones generally estimate friction factors with higher accuracy (Table 1). Only approximations proposed by Moody , Wood , Eck  and Round  have maximum errors greater than 5%. Accuracy of approximations to the Colebrook equation is examined by Zigrang and Sylvester , Gregory and Fogarasi  and Yildirim .
|Table 1. Explicit approximations to The Colebrook relation|
|-||Swamee and Jain-1976|
||Zigrang and Sylvester-1982|
|Romeo, Royo and Monzón, 2002|
|Sonnad and Goudar, 2006|
|Vatankhah and Kouchakzadeh, 2008|
|Avci and Karagoz, 2009|
|-||Papaevangelou, Evangelides and Tzimopoulos, 2010|
|Using above S, new S (noted as S1) can be calculated to reduce error: Or S can be calculated as:
|*Churchill relation from 1977 also covers laminar regime|
Using the Lambert W function
The approximation proposed by the author is developed using the Lambert W function. Functions involving exponents, logarithms and square roots are indispensible tools in solving broad classes of mathematical problems. With just four basic operations of arithmetic, any linear equation can be solved. Quadratic equations can be solved as well if square roots are added. For some classes of problems, trigonometric functions are useful. These functions can be classified as elementary. But for the solution of implicit equations such as the Colebrook, the best function is Lambert W [28–29]. The Lambert W function (Equation 2; Figure 3) and the Colebrook equations are transcendental.
The exponential function can be defined, in a variety of equivalent ways, as an infinite series. In particular, it may be defined by a power series in the form of a Taylor series expansion (Equation 3):
The Taylor series is a mathematical representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. Using a Taylor series, trigonometric functions can be written as Equations (4) and (5):
Similarly, the principal branch of the Lambert W function can be noted as in Equation (6):
A logical question that arises is why the Lambert W function is not an elementary function, while trigonometric, logarithmic, exponential and others are. Whether Lambert W ultimately attains such canonical status will depend on whether the wider mathematics community finds it sufficiently useful. Note that the Taylor series appears on most pocket calculators, so it is readily usable.
For real-number values of the argument x, the W function has two branches: W–1 and W0, where the latter is the principal branch. The evolution of the W function began with ideas proposed by J.H. Lambert in 1758 and the function was refined by L. Euler over the subsequent two decades. Only part of the principal branch of the Lambert W function will be used for solving the Colebrook equation. The equation can be written in explicit form in an exact mathematical way without any approximation involved (Equation 7):
Where x = Re·ln(10)/5.02. Also, procedures to arrive at the solution of the reformulated Lambert W function could find application in commercial software packages.
The Lambert W function is implemented in many mathematical systems, such as Mathematica by Wolfram Research, under the name ProductLog, or Matlab by MathWorks, under the name Lambert .
Regarding the name of the Colebrook equation, it is sometimes alternately known as the Colebrook-White equation, or the CW equation .Cedric White was not actually a coauthor of the paper where the equation was presented, but Cyril Frank Colebrook made a special point of acknowledging the important contribution of White for the development of the equation. So the letter W has additional symbolic value in the reformulated Colebrook equation.
Summary of uses
In solving the Colebrook equation approximately, the trial-and-error method is obsolete, and the graphical solution approach is useful only as an orientation. A spreadsheet solver, such as Excel, can generate accurate iterative solutions to the implicit Colebrook equation. The numerous explicit approximations available are also very accurate for solution to the equation. Finally, the new approach using the Lambert W function can be useful [32–34].
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13. Round, G.F. An explicit approximation for the friction factor-Reynolds number relation for rough and smooth pipes. Canadian J. Chem. Eng. 58(1), pp. 122–123. 1980.
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15. Zigrang, D.J. and Sylvester, N.D. Explicit approximations to the solution of Colebrook’s friction factor equation. AIChE Journal 28(3), 514–515. 1982.
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19. Romeo, E., Royo, C. and Monzon, A. Improved explicit equation for estimation of the friction factor in rough and smooth pipes. Chem. Eng. Journal 86(3), pp. 369–374. 2002.
20. Sonnad, J.R. and Goudar, C.T. Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation. J. Hydraul. Eng. ASCE 132 (8), pp. 863–867. 2006.
21. Buzzelli, D. Calculating friction in one step. Machine Design 80(12), 54–55. 2008.
22. Vatankhah, A.R. and Kouchakzadeh, S.K. Discussion of turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook–White equation. J. Hydraul Eng. ASCE 134(8), p. 1187. 2008.
23. Avci, A. and Karagoz, I. A novel explicit equation for friction factor in smooth and rough pipes. J. Fluids Eng. ASME 131(6), 061203, pp. 1–4. 2009.
24. Papaevangelou, G., Evangelides, C. and Tzimopoulos C. A new explicit equation for the friction coefficient in the Darcy-Weisbach equation. “Proceedings of the Tenth Conference on Protection and Restoration of the Environment: PRE10,” July 6–9, 2010, Greece, Corfu, 166, pp. 1–7. 2010.
25. Zigrang, D.J. and Sylvester, N.D. A Review of explicit friction factor equations. J. Energy Resources Tech. ASME 107(2), pp. 280–283. 1985.
26. Gregory, G.A. and Fogarasi, M. Alternate to standard friction factor equation. Oil & Gas Journal 83(13), pp. 120 and 125–127. 1985.
27. Yildirim, G. Computer-based analysis of explicit approximations to the implicit Colebrook–White equation in turbulent flow friction factor calculation. Adv. in Eng. Software 40(11), pp. 1183–1190. 2009.
28. Barry D.A., Parlange, J.-Y., Li, L., Prommer, H., Cunningham, C.J. and Stagnitti F. Analytical approximations for real values of the Lambert W function. Math. Computers in Simul. 53(1–2), pp. 95–103. 2000.
29. Boyd, J.P. Global approximations to the principal real-valued branch of the Lambert W function. Applied Math. Lett. 11(6), pp. 27–31. 1998.
30. Hayes, B. Why W? American Scientist 93(2), pp. 104–108. 2005.
31. Colebrook, C.F. and White C.M. Experiments with fluid friction in roughened pipes. Proc. Royal Society London Series A 161(906), pp. 367–381. 1937.
32. Keady, G. Colebrook-White formulas for pipe flow. J. Hydraul. Eng. ASCE 124(1), pp. 96–97. 1998.
33. More A.A. Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes. Chem. Eng. Science 61(16), pp. 5515–5519. 2006.
34. Sonnad, J.R. and Goudar, C.T. Constraints for using Lambert-W function-based explicit Colebrook-White equation. J. Hydraul. Eng. ASCE 130(9), pp. 929–931. 2004.
Dejan BrkiÄ‡ (StrumiÄka 88, 11050 Beograd, Serbia; Phone: +38 16425 43668; Email: email@example.com) received his doctoral degree in petroleum and natural engineering from University of Belgrade (Serbia) in 2010. He also holds M.S. degrees in petroleum engineering (2002) and in the treatment and transport of fluids (2005), both from the University of Belgrade. BrkiÄ‡ has published 15 research papers in international journals. His research interests include hydraulics and natural gas. BrkiÄ‡ is currently searching for a post-doctoral position abroad.
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