Darcy friction factor formulaeFrom Wikipedia, the free encyclopedia (Redirected from Swamee–Jain equation)

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

Contents

1 Flow regime1.1 Laminar flow1.2 Transition flow1.3 Turbulent flow in smooth conduits1.4 Turbulent flow in rough conduits1.5 Free surface flow

2 Choosing a formula3 Colebrook equation

3.1 Compact forms3.2 Solving3.3 Expanded forms3.4 Free surface flow

4 Approximations of the Colebrook equation4.1 Haaland equation4.2 Swamee–Jain equation4.3 Serghides's solution4.4 Goudar–Sonnad equation4.5 Brkić solution4.6 Blasius correlations4.7 Table of Approximations

5 References6 Further reading7 External links

Flow regime

Which friction factor formula may be applicable depends upon the type of flow that exists:

Laminar flowTransition between laminar and turbulent flowFully turbulent flow in smooth conduitsFully turbulent flow in rough conduitsFree surface flow.

Laminar flow

The 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 flow

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

Turbulent flow in smooth conduits

The Blasius equation is the most simple equation for solving the Darcy friction factor. Because the Blasius equation has no term for pipe roughness, it is validonly to smooth pipes. However, the Blasius equation is sometimes used in rough pipes because of its simplicity. The Blasius equation is valid up to the Reynoldsnumber 100000.

Turbulent flow in rough conduits

The Darcy friction factor for fully turbulent flow (Reynolds number greater than 4000) in rough conduits is given by the Colebrook equation.

Free surface flow

The last formula in the Colebrook equation section of this article is for free surface flow. The approximations elsewhere in this article are not applicable for thistype of flow.

Choosing a formula

Before choosing a formula it is worth knowing that in the paper on the Moody chart, Moody stated the accuracy is about ±5% for smooth pipes and ±10% forrough pipes. If more than one formula is applicable in the flow regime under consideration, the choice of formula may be influenced by one or more of thefollowing:

Required precisionSpeed of computation requiredAvailable computational technology:

calculator (minimize keystrokes)spreadsheet (single­cell formula)programming/scripting language (subroutine).

Colebrook equation

Compact forms

The Colebrook equation is an implicit equation that combines experimental results of studies of turbulent flow in smooth and rough pipes. It was developed in1939 by C. F. Colebrook.[2] The 1937 paper by C. F. Colebrook and C. M. White[3] is often erroneously cited as the source of the equation. This is partlybecause Colebrook in a footnote (from his 1939 paper) acknowledges his debt to White for suggesting the mathematical method by which the smooth and roughpipe correlations could be combined. The equation is used to iteratively solve for the Darcy–Weisbach friction factor f. This equation is also known as theColebrook–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 factorRoughness height, (m, ft)Hydraulic diameter, (m, ft) — For fluid­filled, circular conduits, = D = inside diameterHydraulic radius, (m, ft) — For fluid­filled, circular conduits, = D/4 = (inside diameter)/4

is the Reynolds number.

Solving

The Colebrook equation is used to be solved numerically due to its apparent implicit nature. Recently, the Lambert W function has been employed to obtainexplicit reformulation of the Colebrook equation.[4]

Expanded forms

Additional, 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 section are exact. The constants are probablyvalues which were rounded by Colebrook during his curve fitting; but they are effectively treated as exact when comparing (to several decimal places) resultsfrom 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 roundingerrors) 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 Colebrook­White equation exists for free surfaces. Such a condition may exist in a pipe that is flowing partially full of fluid. For free surfaceflow:

Approximations of the Colebrook equation

Haaland equation

The Haaland equation was proposed by Norwegian Institute of Technology professor Haaland in 1984. It is used to solve directly for the Darcy–Weisbachfriction factor f for a full­flowing circular pipe. It is an approximation of the implicit Colebrook–White equation, but the discrepancy from experimental data iswell 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 full­flowing circular pipe. It is an approximation of the implicitColebrook–White equation.

where f is a function of:

Roughness height, ε (m, ft)Pipe diameter, D (m, ft)Reynolds number, Re (unitless).

Serghides's solution

Serghides's solution is used to solve directly for the Darcy–Weisbach friction factor f for a full­flowing circular pipe. It is an approximation of the implicitColebrook–White equation. It was derived using Steffensen's method.[6]

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 70­point matrix consisting of ten relative roughness values(in the range 0.00004 to 0.05) by seven Reynolds numbers (2500 to 108).

Goudar–Sonnad equation

Goudar equation is the most accurate approximation to solve directly for the Darcy–Weisbach friction factor f for a full­flowing circular pipe. It is anapproximation of the implicit Colebrook–White equation. Equation has the following form[7]

where f is a function of:

Roughness height, ε (m, ft)Pipe diameter, D (m, ft)Reynolds number, Re (unitless).

Brkić solution

Brkić shows one approximation of the Colebrook equation based on the Lambert W­function[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 base­10 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). Allof 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

whereChurchill 1977

Chen 1979

Round 1980

Barr 1981

or Zigrang andSylvester 1982

Haaland 1983

or

where Serghides 1984

Manadilli 1997

Monzon,Romeo, Royo 2002

where:

Goudar,Sonnad 2006

where:

Vatankhah,Kouchakzadeh 2008

whereBuzzelli 2008

Avci, Kargoz 2009

Evangleids,Papaevangelou,Tzimopoulos

2010

References

1. ^ Manning, Francis S.; Thompson, Richard E. (1991). Oilfield Processing of Petroleum. Vol. 1: Natural Gas. PennWell Books. ISBN 0­87814­343­2, 420 pages. Seepage 293.

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 theInstitution 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. Bibcode:1937RSPSA.161..367C (http://adsabs.harvard.edu/abs/1937RSPSA.161..367C).doi:10.1098/rspa.1937.0150 (http://dx.doi.org/10.1098%2Frspa.1937.0150).

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 (http://dx.doi.org/10.1016%2Fj.ces.2006.04.003).

5. ^ BS Massey Mechanics of Fluids 6th Ed ISBN 0­412­34280­46. ^ 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 (http://dx.doi.org/10.1080%2F10916461003620453).

9. ^ Beograd, Dejan Brkić (March 2012). "Determining Friction Factors in Turbulent Pipe Flow"(http://www.che.com/processing_and_handling/liquid_gas_and_air_handling/9059.html). Chemical Engineering: 34–39.(subscription required)

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%2Fijoti.1939.13150).For the section which includes the free­surface form of the equation — Computer Applications in Hydraulic Engineering (5th ed.). Haestad Press. 2002,p. 16.Haaland, SE (1983). "Simple and Explicit Formulas for the Friction Factor in Turbulent Flow". Journal of Fluids Engineering (ASME) 105 (1): 89–90.doi:10.1115/1.3240948 (http://dx.doi.org/10.1115%2F1.3240948).Swamee, P.K.; Jain, A.K. (1976). "Explicit equations for pipe­flow problems". Journal of the Hydraulics Division (ASCE) 102 (5): 657–664.Serghides, T.K (1984). "Estimate friction factor accurately". Chemical Engineering 91 (5): 63–64. — 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): 671–684.Brkić, Dejan (2011). "Review of explicit approximations to the Colebrook relation for flow friction". Journal of Petroleum Science and Engineering 77(1): 34–48. doi:10.1016/j.petrol.2011.02.006 (http://dx.doi.org/10.1016%2Fj.petrol.2011.02.006).Brkić, Dejan (2011). "W solutions of the CW equation for flow friction". Applied Mathematics Letters 24 (8): 1379–1383. doi:10.1016/j.aml.2011.03.014(http://dx.doi.org/10.1016%2Fj.aml.2011.03.014).

External links

Web­based 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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