Friction Factor-Turbulent Pipe Flow

8/13/2019 Friction Factor-Turbulent Pipe Flow

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Head loss due to friction for fluidstraveling through pipes, tubesand ducts is a critical param-eter for solving turbulent-flow

problems in the chemical process in-dustries. The Colebrook equation isused to assess hydraulic resistancefor turbulent flow in both smooth-and rough-walled pipes. The equation

contains a dimensionless fluid-flowfriction coefficient that must be calcu-lated for the properties of the pipe andthe fluid flow.

Determining friction factors for theColebrook equation requires eithercalculating iteratively or manipulatingthe equation to express friction factorsexplicitly. Iterative calculations can becarried out using a spreadsheet solver,but can require more computationaltime. Explicit expressions offer directcomputation, but have a range of sim-

plicity and corresponding error.The Lambert W function may be a

better method to express friction fac-tors explicitly because it allows usersto avoid iterative calculation and alsoreduce relative error. This article out-lines methods for determining frictionfactors, and discusses how to use theLambert W function. The Lambert Wfunction is evaluated using real datain Part 2 of the feature (p. 40).

Colebrook equationPipe-flow problems are challengingbecause they require determination ofthe fluid-flow friction factor (), a di-

mensionless term whose expression isa non-factorable polynomial. The fric-tion factor is a complicated function ofrelative surface roughness and Reyn-olds number (Re), where, specifically,hydraulic resistance depends on flow-rate. The situation is similar to thatobserved with electrical resistancewhen a diode is in circuit.

The hydraulics literature containsthree forms of the Colebrook equationfor which the friction factor is implicit,meaning that the term it has to be ap-proximately solved using an iterativeprocedure because the term exists onboth sides of the equation. Engineershave also developed a number of ap-proximation formulas that express thefriction factor explicitly, meaning thatit is calculated directly rather thanthrough an iterative process.

The equation proposed by Colebrook

in 1939 [1] describes a monotonicchange in the friction factor as pipesurfaces transition from fully smoothto fully rough.

(1)

At the time it was developed, the im-plicit form of the Colebrook equationwas too complex to be of great prac-tical use. It may be difficult for manyto recall the time, as recently as the1970s, with no personal computers oreven calculators that could do muchmore than add or subtract.

Many researchers, such as Coelhoand Pinho [2], have adopted a modifica-tion of the implicit Colebrook equation,using 2.825 as the constant instead of2.51. Alternatively, some engineers usethe Fanning factor, which is differentfrom the more commonly used Darcyfriction factor. The Darcy friction fac-tor is four times greater than the Fan-

ning friction factor, but their physicalmeanings are equivalent.

Calculation approachesIn general, the following five ap-proaches are available to solve theColebrook equation: Graphical solutions using Moody or

Rouse diagrams (useful only as anorientation)

Iterative solutions using spread-sheet solvers (can be highly accurateto Colebrook standard, but require

more computational resources) Using explicit Colebrook-equation

approximations (less computation,but can introduce error)

Lambert W function (avoids itera-tive calculations and allows reduc-tion of relative error)Trial-and-error method (obsolete)

Graphical solutionsGraphs based on the Colebrook equa-tion represent the simplest, but mostapproximate approach to avoidingtrial-and-error-based iterative solu-tions. In 1943, Rouse [3] developed achart based upon the Colebrook equa-

Feature Report

34 CHEMICAL ENGINEERING WWW.CHE.COM MARCH 2012

Cover Story

Dejan BrkiBeograd, Serbia

Several approaches are reviewed

for calculating fluid-flow friction factors

in fluid mechanics problems

using the Colebrook equation

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Part 1

FIGURE 1. Microsoft Excel can carry out it-erative calculations as needed in solving theimplicit Colebrook equation

Determining Friction Factors

in Turbulent Pipe Flow


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Cover Story


TABLE 1. EXPLICIT APPROXIMATIONS TO THE COLEBROOK RELATION

Relation Auxiliary terms Name-year

- Moody-1947

Wood-1966

- Eck-1973

- Swamee and

Jain-1976

- Churchill-1973

- Jain-1976

1

0 9

2 457 7

0 27=

+

. ln

Re.

.

D

16

2

16

37530=

Re

Churchill*-1977

Chen-1979

- Round-1980

- Barr-1981

or

-

Zigrang and Syl-vester-1982

- Haaland-1983

Continues on next page


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CHEMICAL ENGINEERING WWW.CHE.COM MARCH 2012 37

TABLE 1. EXPLICIT APPROXIMATIONS TO THE COLEBROOK RELATION (Continued)

Relation Auxiliary terms Name, year

Serghides, 1984

- Manadilli, 1997

Romeo, Royo andMonzn, 2002

Sonnad and Gou-

dar, 2006

Vatankhah andKouchakzadeh,2008

Buzzelli, 2008

Avci and Karagoz,2009

-Papaevangelou,Evangelides andTzimopoulos, 2010

Using above S, new S(noted as S1) can be calculated to re-duce error:

Or Scan be calculated as:

*Churchill relation from 1977 also covers laminar regime


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Cover Story


the best function is Lambert W [2829]. The Lambert W function (Equa-tion 2; Figure 3) and the Colebrookequations are transcendental.

(2)

The exponential function can be de-fined, in a variety of equivalent ways,as an infinite series. In particular, itmay be defined by a power series inthe form of a Taylor series expansion(Equation 3):

(3)

The Taylor series is a mathemati-cal representation of a function asan infinite sum of terms calculatedfrom the values of its derivatives ata single point. Using a Taylor series,trigonometric functions can be writ-

ten as Equations (4) and (5):

(4)

(5)Similarly, the principal branch of

the Lambert W function can be notedas in Equation (6):

(6)

A logical question that arises iswhy the Lambert W function is notan elementary function, while trigo-nometric, logarithmic, exponentialand others are. Whether Lambert W

ultimately attains such canonical sta-tus will depend on whether the wider

mathematics community finds it suf-ficiently useful. Note that the Taylorseries appears on most pocket calcula-tors, so it is readily usable.

For real-number values of the ar-gument x, the W function has two

-e-1

W0+

W0-

W-1

0 1-1 2

-e-1 0 1-1 2x

Used for Colebrookequation

1

0

-1

-2

-3

-4

1

0

LambertW

W(x)

-1

-2

-3

-4

References

1. Colebrook, C.F. Turbulent flow in pipes withparticular reference to the transition regionbetween the smooth and rough pipe laws. J.Inst. Civil Eng.11(4), pp. 133156. 1939.

2. Coelho, P.M. and Pinho, C. Considerationsabout equations for steady state flow in nat-ural gas pipelines. J. Brazil. Soc. Mech. Sci.Eng.29(3), pp. 262273. 2007.

3. Rouse, H. Evaluation of boundary roughness.Proceedings of the 2nd Hydraulics Confer-ence, New York, 27, pp. 105116. 1943.

4. Moody, L.F. Friction factors for pipe flow.Trans. ASME66(8) pp. 671684. 1944.

5. Moody, L.F. An approximate formula for pipefriction factors. Trans. ASME 69(12), pp.10051011. 1947.

6. Wood, D.J. An explicit friction factor relation-ship. Civil Eng.36(12), pp. 6061. 1966.

7. Eck, B. Technische Stromungslehre. 1st ed.Springer, New York. 1973.

8. Swamee, P.K. and Jain, A.K. Explicit equa-tions for pipe flow problems.J. Hydraul. Div.ASCE102 (HY5), pp. 657664. 1976.

9. Churchill, S.W. Empirical expressions for theshear stressing turbulent flow in commer-cial pipe.AIChE Journal19(2), pp. 375376.1973.

10. Churchill, S.W. Friction-factor equation spansall fluid flow regimes. Chem. Eng.(New York)84 (24), pp. 9192. 1977.

11. Jain, A.K. Accurate explicit equation for fric-tion factor.J. Hydraul. Div. ASCE102 (HY5),pp. 674677. 1976.

12. Chen, N.H. An explicit equation for frictionfactor in pipes. Indust. Eng. Chem. Funda-

ment.18(3), pp. 296297. 1979.

13. Round, G.F. An explicit approximation for thefriction factor-Reynolds number relation forrough and smooth pipes. Canadian J. Chem.

Eng.58(1), pp. 122123. 1980.

14. Barr, D.I.H. Solutions of the Colebrook-Whitefunction for resistance to uniform turbulentflow.Proc. Inst. Civil Eng. 71(2), pp. 529536.1981.

15. Zigrang, D.J. and Sylvester, N.D. Explicit ap-

proximations to the solution of Colebrooksfriction factor equation. AIChE Journal28(3), 514515. 1982.

16. Haaland, S.E. Simple and explicit formulasfor friction factor in turbulent pipe flow. J.Fluids Eng. ASME105(1), pp. 8990. 1983.

17. Serghides, T.K. Estimate friction factor ac-curately. Chem. Eng. (New York) 91(5), pp.6364. 1984.

18. Manadilli, G. Replace implicit equations withsignomial functions. Chem. Eng. (New York)104 (8), pp. 129130. 1997.

19. Romeo, E., Royo, C. and Monzon, A. Improvedexplicit equation for estimation of the fric-tion factor in rough and smooth pipes. Chem.Eng. Journal86(3), pp. 369374. 2002.

20. Sonnad, J.R. and Goudar, C.T. Turbulent flowfriction factor calculation using a mathemat-ically exact alternative to the Colebrook-White equation.J. Hydraul. Eng. ASCE132(8), pp. 863867. 2006.

21. Buzzelli, D. Calculating friction in one step.Machine Design 80(12), 5455. 2008.

22. Vatankhah, A.R. and Kouchakzadeh, S.K.Discussion of turbulent flow friction factorcalculation using a mathematically exact al-ternative to the ColebrookWhite equation.J. Hydraul Eng. ASCE 134(8), p. 1187. 2008.

23. Avci, A. and Karagoz, I. A novel explicit equa-tion for friction factor in smooth and roughpipes. J. Fluids Eng. ASME131(6), 061203,pp. 14. 2009.

24. Papaevangelou, G., Evangelides, C. and Tzi-mopoulos C. A new explicit equation for thefriction coefficient in the Darcy-Weisbachequation. Proceedings of the Tenth Con-

ference on Protection and Restoration ofthe Environment: PRE10, July 69, 2010,Greece, Corfu, 166, pp. 17. 2010.

25. Zigrang, D.J. and Sylvester, N.D. A Review ofexplicit friction factor equations. J. EnergyResources Tech. ASME 107(2), pp. 280283.1985.

26.Gregory, G.A. and Fogarasi, M. Alternate tostandard friction factor equation. Oil & GasJournal83(13), pp. 120 and 125127. 1985.

27. Yildirim, G. Computer-based analysis of ex-plicit approximations to the implicit Cole-brookWhite equation in turbulent flow fric-tion factor calculation.Adv. in Eng. Software40(11), pp. 11831190. 2009.

28. Barry D.A., Parlange, J.-Y., Li, L., Prommer,H., Cunningham, C.J. and Stagnitti F. Ana-lytical approximations for real values of theLambert W function. Math. Computers inSimul.53(12), pp. 95103. 2000.

29. Boyd, J.P. Global approximations to the prin-cipal real-valued branch of the Lambert Wfunction. Applied Math. Lett. 11(6), pp. 27

31. 1998.

30. Hayes, B. Why W?American Scientist93(2),pp. 104108. 2005.

31. Colebrook, C.F. and White C.M. Experimentswith fluid friction in roughened pipes. Proc.Royal Society London Series A161(906), pp.367381. 1937.

32. Keady, G. Colebrook-White formulas for pipeflow.J. Hydraul. Eng. ASCE124(1), pp. 9697. 1998.

33. More A.A. Analytical solutions for the Cole-brook and White equation and for pressuredrop in ideal gas flow in pipes. Chem. Eng.Science 61(16), pp. 55155519. 2006.

34. Sonnad, J.R. and Goudar, C.T. Constraints

for using Lambert-W function-based explicitColebrook-White equation.J. Hydraul. Eng.ASCE130(9), pp. 929931. 2004.

FIGURE 3.The Lambert W functioncan be defined as an infinite series


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CHEMICAL ENGINEERING WWW.CHE.COM MARCH 2012 39

branches: W1and W0, where the lat-ter is the principal branch. The evo-lution of the W function began withideas proposed by J.H. Lambert in

1758 and the function was refinedby L. Euler over the subsequent twodecades. Only part of the principalbranch of the Lambert W functionwill be used for solving the Colebrookequation. The equation can be writ-ten in explicit form in an exact math-ematical way without any approxi-mation involved (Equation 7):

(7)

Where x =Reln(10)/5.02. Also, proce-dures to arrive at the solution of thereformulated Lambert W function

could find application in commercialsoftware packages.

The Lambert W function is im-plemented in many mathematical

systems, such as Mathematica byWolfram Research, under the nameProductLog, or Matlab by MathWorks,under the name Lambert [30].

Regarding the name of the Cole-brook equation, it is sometimes alter-nately known as the Colebrook-Whiteequation, or the CW equation [31].Ce-dric White was not actually a coauthorof the paper where the equation waspresented, but Cyril Frank Colebrookmade a special point of acknowledgingthe important contribution of White

for the development of the equation.So the letter W has additional sym-bolic value in the reformulated Cole-brook equation.

Summary of usesIn solving the Colebrook equation ap-proximately, the trial-and-error method

is obsolete, and the graphical solutionapproach is useful only as an orien-tation. A spreadsheet solver, such asExcel, can generate accurate iterative

solutions to the implicit Colebrook equa-tion. The numerous explicit approxima-tions available are also very accuratefor solution to the equation. Finally, thenew approach using the Lambert Wfunction can be useful [3234].

Edited by Scott Jenkins

AuthorDejan Brki (Strumika88, 11050 Beograd, Serbia;Phone: +38 16425 43668;Email: [email protected]) received his doc-toral degree in petroleum

and natural engineering fromUniversity of Belgrade (Ser-bia) in 2010. He also holdsM.S. degrees in petroleumengineering (2002) and in thetreatment and transport of

fluids (2005), both from the University of Bel-grade. Brki has published 15 research papersin international journals. His research interestsinclude hydraulics and natural gas. Brki iscurrently searching for a post-doctoral positionabroad.

Circle 10 on p. 70 or go to adlinks.che.com/40265-10


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The Nikuradse-Prandtl-von Kar-

man (NPK) equation is the mostwidely used expression to deter-mine friction factors for fluid flow

in smooth pipes [13]. It implicitlyrelates the friction factor (expressedhere as f; sometimes called ) to theReynolds number, Re. It is shown inEquation (1).

(1)

Because the friction factor appears onboth sides of the equation, an iterative

approach is required to arrive at accu-rate values forf. To simplify the prac-tical use of the NPK equation whileproviding highly accurate f valuesand eliminating the need for iterativeestimations off, several empirical ap-proximations have been proposed [4](see Part 1 of this feature, p. 34).

The authors employed the LambertW function to derive an explicit repre-sentation of the NPK equation for tur-bulent flow in smooth pipes and usedthe expression to determine the accu-

racy of numerous empirical approxi-mations of the NPK equation [4, 5].More recently, other studies have usednewer experimental data sets to derivean alternate relationship between fandRe[6, 7], and it is shown here:

(2)

While functionally identical to theNPK equation, Equation (2) has dif-ferent constants. Using two sources of

experimental data, a comparison wasmade between the new expression andthe widely used NPK equation and is

presented here. Specifically, explicit

representations of Equations (1) and(2) were derived using the Lambert Wfunction that allowed estimation offtomachine precision. Subsequently, ex-perimentalfversusRedata were com-pared with estimates from Equations(1) and (2) to determine their respec-tive deviations from observed values.

Equation (2) was shown to effec-tively describe experimental data inthe range of 31 103


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typically encountered in practice. Frictionfactor estimates from Equations (1) and(2), along with the percentage difference inestimates, are shown in Figure 4 for 1,000

logarithmically spacedRevalues in the 4 103

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Friction Factor-Turbulent Pipe Flow