Darcy friction factor formulae

In fluid dynamics, the Darcy friction factor formulae are equations — based on experimental data and theory — for the Darcy friction factor. The Darcy friction 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]

Flow regime

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.

Laminar flow

The Darcy friction factor for laminar flow (Reynolds number less than 2000) is given by the following formula:

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

where:

• f is the Darcy friction factor
• Re 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 friction factor may be subject to large uncertainties in this flow regime.

Turbulent flow in smooth conduits

Empirical correlations exist for this flow regime. Such correlations are included in the ASHRAE Handbook of Fundamentals.

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 this type 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% for rough 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 the following:

• Required precision
• Speed of computation required
• Available computational technology:

• calculator (minimize keystrokes)
• spreadsheet (single-cell 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 in 1939 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 partly because Colebrook in a footnote (from his 1939 paper) acknowledges his debt to White for suggesting the mathematical method by which the smooth and rough pipe correlations could be combined. The equation 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:

• f is the Darcy friction factor
• Roughness height, ε (m, ft)
• Hydraulic diameter, D_h (m, ft) — For fluid-filled, circular conduits, D_h = D = inside diameter
• Hydraulic radius, R_h (m, ft) — For fluid-filled, circular conduits, R_h = D/4 = (inside diameter)/4
• Re is the Reynolds number.

Solving

The Colebrook equation used to be solved numerically due to its apparent implicit nature. Recently, the Lambert W function has been employed to obtained explicit reformulation of the Colebrook equation.

^[4]

Expanded forms

Additional, mathematically equivalent forms of the Colebrook equation are:

$\frac{1}{\sqrt{f}}= 1.7384\ldots -2 \log_{10} \left( \frac { 2 \varepsilon} {D_\mathrm{h}} + \frac {18.574} {\mathrm{Re} \sqrt{f}} \right)$

where:

1.7384... = 2 log (2 × 3.7) = 2 log (7.4)

18.574 = 2.51 × 3.7 × 2

and

$\frac{1}{\sqrt{f}}= 1.1364\ldots + 2 \log_{10} (D_\mathrm{h} / \varepsilon) -2 \log_{10} \left( 1 + \frac { 9.287} {\mathrm{Re} (\varepsilon/D_\mathrm{h}) \sqrt{f}} \right)$

or

$\frac{1}{\sqrt{f}}= 1.1364\ldots -2 \log_{10} \left( \frac {\varepsilon} {D_\mathrm{h}} + \frac {9.287} {\mathrm{Re} \sqrt{f}} \right)$

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 be helpful to note that they are essentially the same equation.

Free surface flow

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. For free surface flow:

$\frac{1}{\sqrt{f}} = -2 \log_{10} \left(\frac{\varepsilon}{12R_\mathrm{h}} + \frac{2.51}{\mathrm{Re}\sqrt{f}}\right).$

Approximations of the Colebrook equation

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, but the discrepancy from experimental data is well within the accuracy of the data. 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:

• f is the Darcy friction factor
• ε / D is the relative roughness
• Re 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 full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation.

$f = \frac{0.25}{\left[\log_{10} \left(\frac{\varepsilon}{3.7D} + \frac{5.74}{\mathrm{Re}^{0.9}}\right)\right]^2}$

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 full-flowing circular pipe. It is an approximation of the implicit Colebrook–White 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.

$A = -2\log_{10}\left( {\varepsilon/D\over 3.7} + {12\over \mbox{Re}}\right)$

$B = -2\log_{10} \left({\varepsilon/D\over 3.7} + {2.51 A \over \mbox{Re}}\right)$

$C = -2\log_{10} \left({\varepsilon/D\over 3.7} + {2.51 B \over \mbox{Re}}\right)$

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

where f is a function of:

• 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 70-point 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 full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation. Equation has the following form ^[6]

$a = {2 \over \ln(10)}$

$b = {\varepsilon/D\over 3.7}$

$d = {\ln(10)Re\over 5.02}$

s = bd + ln(d)

q = s^{s / (s + 1)}

$g = {bd + \ln{d \over q}}$

$z = {\ln{q \over g}}$

$D_{LA} = z{{g\over {g+1}}}$

$D_{CFA} = D_{LA} \left(1 + \frac{z/2}{(g+1)^2+(z/3)(2g-1)}\right)$

$\frac{1}{\sqrt {f}} = {a\left[ \ln\left( \frac{d}{q} \right) + D_{CFA} \right] }$

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 W-function^[7]

$S = ln\frac{Re}{\mathrm{1.816ln\frac{1.1Re}{\mathrm{ln(1+1.1Re)}}}}$

$\frac{1}{\sqrt {f}} = -2\log_{10} \left({\varepsilon/D\over 3.71} + {2.18 S \over \mbox{Re}}\right)$

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.

References

1. ^ Manning, Francis S.; Thompson, Richard E. (1991). Oilfield Processing of Petroleum. Vol. 1: Natural Gas. PennWell Books. ISBN 0878143432 , 420 pages. See page 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 the Institution 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. doi:10.1098/rspa.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.
5. ^ Serghides, T.K (1984). "Estimate friction factor accurately". Chemical Engineering Journal 91(5): 63–64.
6. ^ 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.
7. ^ 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. 

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. 
  For the section which includes the free-surface 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) 103 (5): 89–90. doi: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): 657–664. 
• Serghides, T.K (1984). "Estimate friction factor accurately". Chemical Engineering 91 (5): 63–64.  — Serghides' solution is also mentioned here.
• 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. 
• 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. 

External links

• Web-based calculator of Darcy friction factors by Serghides' solution.
• Applications of Colebrook Equation
• Open source pipe friction calculator.