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: Flow regime. Which friction factor formula may be applicable  depends upon the type of flow that exists: 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 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 (Reynolds number greater than 4000) 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: Colebrook–White equation. 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 / "D", fitting the data of experimental studies of turbulent flow in smooth and rough pipes. 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. 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. formula_7 formula_8 or formula_9 formula_10 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 (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: 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: formula_18 where "a" is: formula_19 and "b" is: formula_20 where "Re" is Reynolds number where "h" is the characteristic hydraulic length (hydraulic radius for 1D flows or water depth for 2D flows) and "R" is the hydraulic radius (for 1D flows) or the  water depth (for 2D flows). The Lambert W function can be calculated as follows: formula_21 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 (in the range 0.00004 to 0.05) by seven Reynolds numbers (2500 to 10). 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%. Brkić-Praks solution. Brkić and Praks show one approximation of the Colebrook  equation based on the Wright formula_40-function, a cognate of the Lambert W-function The equation was found to match the Colebrook–White equation within 0.0497%. Praks-Brkić solution. Praks and Brkić show one approximation of the Colebrook equation based on the Wright formula_40-function, a cognate of the Lambert W-function The equation was found to match the Colebrook–White equation within 0.0012%. Niazkar's solution. 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. 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. Blasius correlations. Early approximations for smooth pipes by Paul Richard Heinrich Blasius in terms of the Darcy–Weisbach 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: with, where "f" is a function of: valid for: Table of Approximations. The following table lists historical approximations to the Colebrook–White 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. Darcy