Darcy friction factor formulae


In fluid dynamics, the Darcy friction factor formulae are equations that allow the calculation of the Darcy friction factor, a dimensionless quantity used in the Darcy–Weisbach equation, for the description of friction losses in pipe flow as well as open-channel flow.
The Darcy friction factor is also known as the Darcy–Weisbach friction factor, resistance coefficient or simply friction factor; by definition it is four times larger than the Fanning friction factor.

Notation

In this article, the following conventions and definitions are to be understood:
Which friction factor formula may be applicable depends upon the type of flow that exists:
Transition flow occurs in the range of Reynolds numbers between 2300 and 4000. The value of the Darcy friction factor is subject to large uncertainties in this flow regime.

Turbulent flow in smooth conduits

The Blasius correlation is the simplest equation for computing the Darcy friction
factor. Because the Blasius correlation has no term for pipe roughness, it
is valid only to smooth pipes. However, the Blasius correlation is sometimes
used in rough pipes because of its simplicity. The Blasius correlation is valid
up to the Reynolds number 100000.

Turbulent flow in rough conduits

The Darcy friction factor for fully turbulent flow in rough conduits can be modeled by the Colebrook–White 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:
The phenomenological Colebrook–White equation expresses the Darcy friction factor f as a function of Reynolds number Re and pipe relative roughness ε / Dh, fitting the data of experimental studies of turbulent flow in smooth and rough pipes.
The equation can be used to solve for the Darcy–Weisbach friction factor f.
For a conduit flowing completely full of fluid at Reynolds numbers greater than 4000, it is expressed as:
or
where:
Note: Some sources use a constant of 3.71 in the denominator for the roughness term in the first equation above.

Solving

The Colebrook equation is usually solved numerically due to its implicit nature. Recently, the Lambert W function has been employed to obtain explicit reformulation of the Colebrook equation.
or
will get:
then:

Expanded forms

Additional, mathematically equivalent forms of the Colebrook equation are:
and
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 results from explicit formulae to the friction factor computed via Colebrook's implicit equation.
Equations similar to the additional forms above 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:
The above equation is valid only for turbulent flow. Another approach for estimating f in free surface flows, which is valid under all the flow regimes is the following:
where a is:
and b is:
where Reh is Reynolds number where h is the characteristic hydraulic length and Rh is the hydraulic radius or the water depth. The Lambert W function can be calculated as follows:

Approximations of the Colebrook equation

Haaland equation

The Haaland equation was proposed in 1983 by Professor S.E. Haaland of the Norwegian Institute of Technology. It 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.
The Haaland equation is expressed:

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.

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

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

Brkić solution

Brkić shows one approximation of the Colebrook equation based on the Lambert W-function
The equation was found to match the Colebrook–White equation within 3.15%.

Blasius correlations

Early approximations for smooth pipes by Paul Richard Heinrich Blasius in terms of the Moody friction factor are given in one article of 1913:
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 the equivalent curve radius, Rc:
with,
where f is a function of:
valid for:
The following table lists historical approximations to the Colebrook–White relation for pressure-driven flow. Churchill equation, Cheng and Bellos et al. equations return an approximately correct value for friction factor in the laminar flow region. All of the others are for transitional and turbulent flow only.
EquationAuthorYearRangeRef
Moody1947

if then and if then		Tsal1989
Manadilli1997
Romeo, Royo, Monzon2002

where

Cheng2008all flow regimes
Avci, Kargoz2009
Evangelides, Papaevangelou, Tzimopoulos2010
Fang2011

,		Brkić2011

where		Bellos, Nalbantis, Tsakiris2018all flow regimes