Discussion: “A Novel Explicit Equation for Friction Factor in Smooth and Rough Pipes” (, and , 2009, ASME J. Fluids Eng., 131, p. 061203) PUBLIC ACCESS

[+] Author and Article Information
Rafael Ballesteros-Tajadura

Área de Mecánica de Fluidos Campus de Viesques, Universidad de Oviedo, 33271 Gijón, Spainrballest@uniovi.es

José González

Área de Mecánica de Fluidos Campus de Viesques, Universidad de Oviedo, 33271 Gijón, Spainaviados@uniovi.es

J. Fluids Eng 133(1), 015501 (Feb 01, 2011) (2 pages) doi:10.1115/1.4003157 History: Received July 09, 2009; Revised June 15, 2010; Published February 01, 2011; Online February 01, 2011

This paper shows a comparison of the most classic and historical formulations to obtain the friction factor for cylindrical pipes with novel approaches. Although there has been some evolution of the expressions, basically, the ideas and results remain very close to the historical formulations.


It was back in the late 1930s when Colebrook and White developed their well-known and widespread relation to obtain the Darcy–Weisbach friction factor, very often named “f.” They derived their expression from the combination of previous research by Prandtl, back in 1935, and von Karman, in 1938. Classical Refs. 1-5 explain all these concepts in detail.

In a mathematical arrangement, the results by Colebrook and White are equivalent to say that the proposed value for “f” can be derived fromDisplay Formula

to be used in the Darcy–Weisbach equation and obtain the pressure loss in a cylindrical pipe. The referred pressure loss can be obtained fromDisplay Formula
More recently, in 1983, Haaland proposed an explicit formula to obtain “f” as follows:Display Formula
Finally, in the June 2009 issue of the ASME Journal of Fluids Engineering, Avci and Karagoz published a paper (6) that introduces a “novel formulation.” They proposed a new expression for the friction factor, following the formulaDisplay Formula
The whole set of previously mentioned formulations, with their different expressions, has been considered and comparison between them is accomplished in the present paper.

When considering a low pipe relative roughness, for example, ε=0.0003 (that is, kS/D=0.0003), a plot as shown in Fig. 1 is obtained. In this plot, for high Reynolds numbers (higher than 106), a 1.2% difference between the classic formulations and the new equation is found.

A second and higher pipe relative roughness, for example, ε=0.03 (that is, kS/D=0.03), has been considered. The results of the comparison for a wide range of Reynolds numbers are presented in Fig. 2. For high Reynolds numbers (higher than 1.5×105), again, a 1.1% difference between the classic formulations and the new equation is found. The two classical formulations give almost the same value, while the proposed new approach differs from them, giving a slightly higher friction factor.

From the observed trends in the comparison between the classical approaches and the new proposed Eq. 4, one can conclude that
  • For a wide range of the pipe relative roughness, the three considered equations give almost the same friction factor, “f,” with only a 1% difference for the new approach.
  • The relevance of the proposed equation is limited if the classical formulations are considered.
  • The new approach presented in Ref. 6 only solves the explicitness of some classical equations, as for instance, Eq. 1. However, the explicitness was already overcome with formulations as the one by Haaland (4).
  • Therefore, the only relevance of Ref. 6 could be its Fig. 3. In that figure, the “Princeton superpipe” results are shown. Nevertheless, there is a lack of explanation about the way these results were obtained, and they are not explicitly referred to. A detailed comment on the results of the Princeton superpipe would be of great interest.

Although a deeper analysis on the discrepancies of the new formulation (Eq. 4) when compared with the classical approaches would be interesting, no improvement in using this new formulation is found.

The authors gratefully acknowledge the financial support from the “Ministerio de Ciencia e Innovación” (MICINN, Spain) under Project No. TRA2007-62708.


pipe diameter, m


friction factor


gravitational constant, m/s2


pipe roughness, m


pipe length, m


pressure drop in a pipe


flow rate, m3/s


Reynolds number


mean velocity, m/s


pipe relative roughness


fluid kinematic viscosity m2/s


fluid density, kg/m3

Copyright © 2011 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 2

Comparison of Eqs. 1,3,4 for high pipe relative roughness

Grahic Jump Location
Figure 1

Comparison of Eqs. 1,3,4 for low pipe relative roughness




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