Darcy friction factor formulae

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The Darcy friction factor is also known as the Darcy–Weisbach friction factor and is four times larger than the Fanning friction factor.

Contents

[edit] Flow Regimes

Which 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


[edit] Laminar Flow

The Darcy friction factor for laminar flow (Reynolds number less than 2300) is given by the following formula:

f = \frac{64}{\mathrm{Re}}

where:


[edit] 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 friction factor may be subject to large uncertainties in this flow regime.


[edit] Turbulent Flow in Smooth Conduits

Empirical correlations exist for this flow regime.

Such correlations are included in the ASHRAE Handbook of Fundamentals.


[edit] 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.


[edit] 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 this type of flow.


[edit] Choosing a Formula

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 the following:

  • Required accuracy
  • Speed of computation required
  • Available computational technology:
  • calculator (minimize keystrokes)
  • spreadsheet (single-cell formula)
  • programming/scripting language (subroutine)


[edit] Colebrook Equation

[edit] Compact forms

The Colebrook Equation is an implicit equation which combines experimental results of studies of laminar and turbulent flow in pipes. It was developed in 1939 by C. F. Colebrook. It 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:

\frac{1}{\sqrt{f}}= -2 \log_{10} \left( \frac { \varepsilon/D_\mathrm{h}} {3.7} + \frac {2.51} {\mathrm{Re} \sqrt{f}} \right)
or
\frac{1}{\sqrt{f}}= -2 \log_{10} \left( \frac{\varepsilon}{14.8 R_\mathrm{h}} + \frac{2.51}{\mathrm{Re}\sqrt{f}} \right)

where:


Due to the implicit nature of the Colebrook equation, determination of a friction factor requires some iteration or a numerical solving method.


[edit] Expanded forms

Additional, mathematically-equivalent forms of the Colebrook equation are:

\frac{1}{\sqrt{f}}= 1.7384... -2 \log_{10} \left( \frac { 2 \varepsilon} {D_\mathrm{h}} + \frac {18.574} {\mathrm{Re} \sqrt{f}} \right)
where:
1.7384... = 2 log (2 * 3.7) = 2 log (7.4)
18.574 = 2.51 * 3.7 * 2

and

\frac{1}{\sqrt{f}}= 1.1364... + 2 \log_{10} (D_\mathrm{h} / \varepsilon) -2 \log_{10} \left( 1 + \frac { 9.287} {\mathrm{Re} (\varepsilon/D_\mathrm{h}) \sqrt{f}} \right)
or
\frac{1}{\sqrt{f}}= 1.1364... -2 \log_{10} \left( \frac {\varepsilon} {D_\mathrm{h}} + \frac {9.287} {\mathrm{Re} \sqrt{f}} \right)
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 helpful to note that they are essentially the same equation.


Though no explicit forms are provided for it elsewhere in this article, 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.

[edit] Free Surface Flow

For Free Surface Flow:
\frac{1}{\sqrt{f}} = -2 \log_{10} (\frac{\varepsilon}{12R_\mathrm{h}} + \frac{2.51}{\mathrm{Re}\sqrt{f}})

[edit] Approximations of the Colebrook Equation

[edit] Haaland equation

The Haaland 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. It was developed by S. E. Haaland in 1983.

The Haaland equation is defined as:

\frac{1}{\sqrt {f}} = -1.8 \log_{10} \left[ \left( \frac{\varepsilon/D}{3.7} \right)^{1.11} + \frac{6.9}{\mathrm{Re}} \right]

where:

[edit] 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.

f = \frac{0.25}{[log_{10} (\frac{\epsilon}{3.7D} + \frac{5.74}{Re^{0.9}})]^2}

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 1.0% for 10-6 < ε/D < 10-2 and 5000 < Re < 108.

[edit] Serghide's solution

Serghide's solution 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, and is derived by taken an iterative approach to solving the equation, then applying a Shanks transformation[citation needed] to the first three terms of the series.

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

A = -2\log_{10}\left( {\varepsilon/D\over 3.7} + {12\over \mbox{Re}}\right)
B = -2\log_{10} \left({\varepsilon/D\over 3.7} + {2.51 A \over \mbox{Re}}\right)
C = -2\log_{10} \left({\varepsilon/D\over 3.7} + {2.51 B \over \mbox{Re}}\right)
f = \left(A - \frac{(B - A)^2}{C - 2B + A}\right)^{-2}

where f is a function of:

The equation was found to match the Colebrook-White equation within 0.0031% for all practical applications.

A mathematical equivalent for the last equation is:

f = \left(\frac {C - 2B + A} {AC - B^2}\right)^2

This form may not be computationally equivalent for some values of (AC - B2) due to rounding errors in computerized calculations.


[edit] Blasius correlations

Early approximations by Blasius are given in terms of the Fanning friction factor in the Paul Richard Heinrich Blasius article.


[edit] References

  • C. F. Colebrook, "Turbulent flow in pipes, with particular reference to the transition region between smooth and rough pipe laws," Jour. Ist. Civil Engrs., London (Feb. 1939).
  • For the section which includes the free-surface form of the equation -- 2002, Computer Applications In Hydraulic Engineering, 5th Ed., Haestad Press, pg. 16
  • Haaland, SE (1983). "Simple and Explicit Formulas for the Friction Factor in Turbulent Flow". Trans. ASIVIE, J. of Fluids Engineering 103: 89–90. 
  • Swamee, P.K.; Jain, A.K. (1976). "Explicit equations for pipe-flow problems". Journal of the Hydraulics Division, ASCE 102 (5): 657–664. 
  • Serghide's solution is mentioned here -- http://www.cheresources.com/colebrook2.shtml
  • L.F. Moody, FRICTION FACTORS FOR PIPE FLOW, ASME Transations Vol.66, N.Y., N.Y. 1944

[edit] External Links

Web-based calculator of Darcy friction factors by Serghide's solution.