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 DarcyâÂÂWeisbach 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 DarcyâÂÂWeisbach friction factor, resistance coefficient or simply friction factor; by definition it is four times larger than the Fanning friction factor.
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 (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 is subject to large uncertainties in this flow regime.
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.
The Darcy friction factor for fully turbulent flow (Reynolds number greater than 4000) in rough conduits can be modeled by the ColebrookâÂÂWhite equation.
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.
Before choosing a formula it is worth knowing that in the paper on the Moody chart, Moody stated the accuracy is about ñ5% for smooth pipes and ñ10% 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 empirical ColebrookâÂÂWhite equation (or Colebrook equation) expresses the Darcy friction factor f as a function of Reynolds number Re and pipe relative roughness õ / D<sub>h</sub>, fitting the data of experimental studies of turbulent flow in smooth and rough pipes. The equation can be used to (iteratively) solve for the DarcyâÂÂWeisbach 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.
The Colebrook equation is usually solved numerically due to its implicit nature. Recently, the Lambert W function has been employed to obtain an exact solution in an explicit reformulation of the Colebrook equation.
or
will get:
then:
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 (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.
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 (laminar, transition and turbulent) is the following:
where a is:
and b is:
where Re<sub>h</sub> is Reynolds number where h is the characteristic hydraulic length (hydraulic radius for 1D flows or water depth for 2D flows) and R<sub>h</sub> is the hydraulic radius (for 1D flows) or the water depth (for 2D flows). The Lambert W function can be calculated as follows:
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 DarcyâÂÂWeisbach friction factor f for a full-flowing circular pipe. It is an approximation of the implicit ColebrookâÂÂWhite equation, but the discrepancy from experimental data is well within the accuracy of the data.
The Haaland equation is expressed:
The SwameeâÂÂJain equation is used to solve directly for the DarcyâÂÂWeisbach friction factor f for a full-flowing circular pipe. It is an approximation of the implicit ColebrookâÂÂWhite equation.
Serghides's solution is used to solve directly for the DarcyâÂÂWeisbach friction factor f for a full-flowing circular pipe. It is an approximation of the implicit ColebrookâÂÂWhite 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 ColebrookâÂÂWhite 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 10<sup>8</sup>).
Goudar equation is the most accurate approximation to solve directly for the DarcyâÂÂWeisbach friction factor f for a full-flowing circular pipe. It is an approximation of the implicit ColebrookâÂÂWhite equation. Equation has the following form
BrkiÃÂ shows one approximation of the Colebrook equation based on the Lambert W-function
The equation was found to match the ColebrookâÂÂWhite equation within 3.15%.
BrkiÃÂ and Praks show one approximation of the Colebrook equation based on the Wright -function, a cognate of the Lambert W-function
The equation was found to match the ColebrookâÂÂWhite equation within 0.0497%.
Praks and BrkiÃÂ show one approximation of the Colebrook equation based on the Wright -function, a cognate of the Lambert W-function
The equation was found to match the ColebrookâÂÂWhite equation within 0.0012%.
Since Serghides's solution was found to be one of the most accurate approximation of the implicit ColebrookâÂÂWhite equation, Niazkar modified the Serghides's solution to solve directly for the DarcyâÂÂWeisbach friction factor f for a full-flowing circular pipe.
Niazkar's solution is shown in the following:
Niazkar's solution was found to be the most accurate correlation based on a comparative analysis conducted in the literature among 42 different explicit equations for estimating Colebrook friction factor.
Early approximations for smooth pipes by Paul Richard Heinrich Blasius in terms of the DarcyâÂÂWeisbach 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, R<sub>c</sub>:
with,
where f is a function of:
valid for:
The Swamee equation is used to solve directly for the DarcyâÂÂWeisbach friction factor (f) for a full-flowing circular pipe for all flow regimes (laminar, transitional, turbulent). It is an exact solution for the HagenâÂÂPoiseuille equation in the laminar flow regime and an approximation of the implicit ColebrookâÂÂWhite equation in the turbulent regime with a maximum deviation of less than 2.38% over the specified range. Additionally, it provides a smooth transition between the laminar and turbulent regimes to be valid as a full-range equation, 0 < Re < 10<sup>8</sup>.
The following table lists historical approximations to the ColebrookâÂÂWhite relation for pressure-driven flow. Churchill equation (1977) is the only equation that can be evaluated for very slow flow (Reynolds number < 1), but the Cheng (2008), and Bellos et al. (2018) equations also return an approximately correct value for friction factor in the laminar flow region (Reynolds number < 2300). All of the others are for transitional and turbulent flow only.