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.
[1] 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.
The Darcy friction factor for fully turbulent flow (Reynolds number greater than 4000) in rough conduits can be modeled by the Colebrook–White equation.
The last formula in the Colebrook equation section of this article is for free surface flow.
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 (or Colebrook 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.
[2][3] The equation can be used to (iteratively) 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.
[4] 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.
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.
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.
[9] It is used to solve directly for the Darcy–Weisbach friction factor f for a full-flowing circular pipe.
[11] Serghides's solution is used to solve directly for the Darcy–Weisbach friction factor f for a full-flowing circular pipe.
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 equation is the most accurate approximation to solve directly for the Darcy–Weisbach friction factor f for a full-flowing circular pipe.
Brkić and Praks show one approximation of the Colebrook equation based on the Wright
Praks and Brkić show one approximation of the Colebrook equation based on the Wright
Since Serghides's solution was found to be one of the most accurate approximation of the implicit Colebrook–White equation, Niazkar modified the Serghides's solution to solve directly for the Darcy–Weisbach friction factor f for a full-flowing circular pipe.
[17] 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.
[17] Early approximations for smooth pipes[18] by Paul Richard Heinrich Blasius in terms of the Darcy–Weisbach friction factor are given in one article of 1913:[19] Johann Nikuradse in 1932 proposed that this corresponds to a power law correlation for the fluid velocity profile.
[20] Mishra and Gupta in 1979 proposed a correction for curved or helically coiled tubes, taking into account the equivalent curve radius, Rc:[21] with, where f is a function of: valid for: The Swamee equation is used to solve directly for the Darcy–Weisbach friction factor (f) for a full-flowing circular pipe for all flow regimes (laminar, transitional, turbulent).
It is an exact solution for the Hagen–Poiseuille equation in the laminar flow regime and an approximation of the implicit Colebrook–White 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 < 108.
The following table lists historical approximations to the Colebrook–White relation[23] for pressure-driven flow.
Churchill equation[24] (1977) is the only equation that can be evaluated for very slow flow (Reynolds number < 1), but the Cheng (2008),[25] and Bellos et al. (2018)[8] equations also return an approximately correct value for friction factor in the laminar flow region (Reynolds number < 2300).