Darcy Friction Factor Formulae

Darcy friction factor formulae 1

Darcy friction factor formulaeIn fluid dynamics, the Darcy friction factor formulae are equations based on experimental data and theory forthe Darcy friction factor. The Darcy friction factor is a dimensionless quantity used in the DarcyWeisbachequation, for the description of friction losses in pipe flow as well as open channel flow. It is also known as theDarcyWeisbach friction factor or Moody friction factor and is four times larger than the Fanning friction factor.[1]

Flow regimeWhich friction factor formula may be applicable depends upon the type of flow that exists: Laminar flow Transition between laminar and turbulent flow Fully turbulent flow in smooth conduits Fully turbulent flow in rough conduits Free surface flow.

Laminar flowThe Darcy friction factor for laminar flow (Reynolds number less than 2100) is given by the following formula:

where:

is the Darcy friction factor is the Reynolds number.

Transition flowTransition (neither fully laminar nor fully turbulent) flow occurs in the range of Reynolds numbers between 2300and 4000. The value of the Darcy friction factor may be subject to large uncertainties in this flow regime.

Turbulent flow in smooth conduitsThe Blasius correlation is the most simple equation for computing the Darcy friction factor. Because the Blasiuscorrelation has no term for pipe roughness, it is valid only to smooth pipes. However, the Blasius correlation issometimes used in rough pipes because of its simplicity. The Blasius correlation is valid up to the Reynolds number100000.


Turbulent flow in rough conduitsThe Darcy friction factor for fully turbulent flow (Reynolds number greater than 4000) in rough conduits is given bythe Colebrook equation.

Free surface flowThe last formula in the Colebrook equation section of this article is for free surface flow. The approximationselsewhere in this article are not applicable for this type of flow.

Choosing a formulaBefore choosing a formula it is worth knowing that in the paper on the Moody chart, Moody stated the accuracy isabout 5% for smooth pipes and 10% for rough pipes. If more than one formula is applicable in the flow regimeunder consideration, the choice of formula may be influenced by one or more of the following: Required precision Speed of computation required Available computational technology:

calculator (minimize keystrokes) spreadsheet (single-cell formula) programming/scripting language (subroutine).

Compact formsThe Colebrook equation is an implicit equation that combines experimental results of studies of turbulent flow insmooth 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 smoothand rough pipe correlations could be combined. The equation is used to iteratively solve for the DarcyWeisbachfriction factor f. This equation is also known as the ColebrookWhite 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, (m, ft) Hydraulic diameter, (m, ft) For fluid-filled, circular conduits, = D = inside diameter Hydraulic radius, (m, ft) For fluid-filled, circular conduits, = D/4 = (inside diameter)/4 is the Reynolds number.Note: Some sources use a constant of 3.71 in the denominator for the roughness term in the first equation above.[2]


SolvingThe Colebrook equation is usually solved numerically due to its implicit nature. Recently, the Lambert W functionhas been employed to obtain explicit reformulation of the Colebrook equation.You can solve the Colebrook equation by iteration using Newton raphson method. An example is provided in C#here [3]: http:/ / processimulation. blogspot. com/

Expanded formsAdditional, mathematically equivalent forms of the Colebrook equation are:

where:1.7384... = 2 log (2 3.7) = 2 log (7.4)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 sectionare exact. The constants are probably values which were rounded by Colebrook during his curve fitting; but they areeffectively treated as exact when comparing (to several decimal places) results from explicit formulae (such as thosefound 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 perhapsshifted slightly to minimize overall rounding errors) may be found in various references. It may be helpful to notethat they are essentially the same equation.

Free surface flowAnother form of the Colebrook-White equation exists for free surfaces. Such a condition may exist in a pipe that isflowing partially full of fluid. For free surface flow:

Approximations of the Colebrook equation

Haaland equationThe Haaland equation was proposed by Norwegian Institute of Technology professor Haaland in 1984. It is used tosolve directly for the DarcyWeisbach friction factor f for a full-flowing circular pipe. It is an approximation of theimplicit ColebrookWhite equation, but the discrepancy from experimental data is well within the accuracy of thedata. It was developed by S. E. Haaland in 1983.


The Haaland equation is defined as:

[4]

where:

is the Darcy friction factor is the relative roughness is the Reynolds number.

SwameeJain equationThe SwameeJain equation is used to solve directly for the DarcyWeisbach friction factor f for a full-flowingcircular pipe. It is an approximation of the implicit ColebrookWhite equation.

where f is a function of: Roughness height, (m, ft) Pipe diameter, D (m, ft) Reynolds number, Re (unitless).

Serghides's solutionSerghides's solution is used to solve directly for the DarcyWeisbach friction factor f for a full-flowing circular pipe.It is an approximation of the implicit ColebrookWhite equation. It was derived using Steffensen's method.[5]

The solution involves calculating three intermediate values and then substituting those values into a final equation.

where f is a function of: Roughness height, (m, ft) Pipe diameter, D (m, ft) Reynolds number, Re (unitless).The equation was found to match the ColebrookWhite equation within 0.0023% for a test set with a 70-point matrixconsisting of ten relative roughness values (in the range 0.00004 to 0.05) by seven Reynolds numbers (2500 to 108).


GoudarSonnad equationGoudar equation is the most accurate approximation to solve directly for the DarcyWeisbach friction factor f for afull-flowing circular pipe. It is an approximation of the implicit ColebrookWhite equation. Equation has thefollowing form[6]

where f is a function of: Roughness height, (m, ft) Pipe diameter, D (m, ft) Reynolds number, Re (unitless).

Brki solutionBrki shows one approximation of the Colebrook equation based on the Lambert W-function

where Darcy friction factor f is a function of: Roughness height, (m, ft) Pipe diameter, D (m, ft) Reynolds number, Re (unitless).The equation was found to match the ColebrookWhite equation within 3.15%.


Blasius correlationsEarly approximations by Paul Richard Heinrich Blasius in terms of the Fanning friction factor are given in onearticle of 1913:[7]

.Johann Nikuradse in 1932 proposed that this corresponds to a power law correlation for the fluid velocity profile.Mishra and Gupta in 1979 proposed a correction for curved or helically coiled tubes, taking into account theequivalent curve radius, Rc:

[8]

,

with,

where f is a function of: Pipe diameter, D (m, ft) Curve radius, R (m, ft) Helicoidal pitch, H (m, ft) Reynolds number, Re (unitless)valid for: Retr < Re < 10

5

6.7 < 2Rc/D < 346.0 0 < H/D < 25.4

Table of ApproximationsThe following table lists historical approximations where: Re, Reynolds number (unitless); , Darcy friction factor (dimensionless); , 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 laminarflow region (Reynolds number < 2300). All of the others are for transitional and turbulent flow only.

Table of Colebrook equation approximations

Equation Author Year Ref

Moody 1947

where

Wood 1966

Eck 1973

Jain andSwamee

1976


Churchill 1973

Jain 1976

where

Churchill 1977

Chen 1979

Round 1980

Barr 1981

or

Zigrang andSylvester

1982

Haaland[9][10] 1983

or

where

Serghides 1984

Manadilli 1997

Monzon,Romeo, Royo

2002

where:

Goudar,Sonnad

2006

where:

Vatankhah,Kouchakzadeh

2008


where

Buzzelli 2008

Avci, Kargoz 2009

Evangleids,Papaevangelou,Tzimopoulos

2010

[1][1] , 420 pages. See page 293.[2][2] VDI Heat Atlas second edition page 1058 (ISBN 978-3-540-77876-9)[3] http:/ / processimulation. blogspot. com/[4][4] BS Massey Mechanics of Fluids 6th Ed ISBN 0-412-34280-4[5] Serghides, T.K (1984). "Estimate friction factor accurately". Chemical Engineering Journal 91(5): 6364.[6] Goudar, C.T., Sonnad, J.R. (August 2008). "Comparison of the iterative approximations of the ColebrookWhite equation". Hydrocarbon

Processing Fluid Flow and Rotating Equipment Special Report(August 2008): 7983.[7] Trinh, On the Blasius correlation for friction factors, p. 1 (http:/ / arxiv. org/ ftp/ arxiv/ papers/ 1007/ 1007. 2466. pdf)[8] Adrian Bejan, Allan D. Kraus, Heat transfer handbook, John Wiley & Sons, 2003[9] The Haaland equation was proposed by Norwegian Institute of Technology professor Haaland in 1984. It is used to solve directly for the

DarcyWeisbach friction factor f for a full-flowing circular pipe. It is an approximation of the implicit ColebrookWhite equation, but thediscrepancy from experimental data is well within the accuracy of the data. It was developed by S. E. Haaland in 1983.

[10][10] BS Massey Mechanics of Fluids 6th Ed ISBN 0-412-34280-4

References

Further reading 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). doi:10.1680/ijoti.1939.13150 (http:/ / dx. doi. org/ 10. 1680/ ijoti. 1939. 13150).For the section which includes the free-surface form of the equation Computer Applications in HydraulicEngineering (5th ed.). Haestad Press. 2002., p.16.

Haaland, SE (1983). "Simple and Explicit Formulas for the Friction Factor in Turbulent Flow". Journal of FluidsEngineering (ASME) 105 (1): 8990. doi: 10.1115/1.3240948 (http:/ / dx. doi. org/ 10. 1115/ 1. 3240948).

Swamee, P.K.; Jain, A.K. (1976). "Explicit equations for pipe-flow problems". Journal of the Hydraulics Division(ASCE) 102 (5): 657664.

Serghides, T.K (1984). "Estimate friction factor accurately". Chemical Engineering 91 (5): 6364. Serghides'solution is also mentioned here (http:/ / www. cheresources. com/ colebrook2. shtml).

Moody, L.F. (1944). "Friction Factors for Pipe Flow". Transactions of the ASME 66 (8): 671684. Brki, Dejan (2011). "Review of explicit approximations to the Colebrook relation for flow friction". Journal of

Petroleum Science and Engineering 77 (1): 3448. doi: 10.1016/j.petrol.2011.02.006 (http:/ / dx. doi. org/ 10.1016/ j. petrol. 2011. 02. 006).

Brki, Dejan (2011). "W solutions of the CW equation for flow friction". Applied Mathematics Letters 24 (8):13791383. doi: 10.1016/j.aml.2011.03.014 (http:/ / dx. doi. org/ 10. 1016/ j. aml. 2011. 03. 014).


External links Web-based calculator of Darcy friction factors by Serghides' solution. (http:/ / www. calctool. org/ CALC/ eng/

civil/ friction_factor) Open source pipe friction calculator. (http:/ / pfcalc. sourceforge. net)

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Darcy friction factor formulaeFlow regimeLaminar flowTransition flowTurbulent flow in smooth conduitsTurbulent flow in rough conduitsFree surface flow

Choosing a formulaCompact formsSolvingExpanded formsFree surface flow

Approximations of the Colebrook equationHaaland equationSwameeJain equationSerghides's solutionGoudarSonnad equationBrki solutionBlasius correlationsTable of Approximations

ReferencesFurther readingExternal links

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