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.
Brief Historical Review and Discussion
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. 1,2,3,4,5 explain all these concepts in detail.
When considering a low pipe relative roughness, for example, (that is, ), a plot as shown in Fig.1 is obtained. In this plot, for high Reynolds numbers (higher than ), a 1.2% difference between the classic formulations and the new equation is found.
A second and higher pipe relative roughness, for example, (that is, ), 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 ), 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
pipe roughness, m
pipe length, m
pressure drop in a pipe
mean velocity, m/s
pipe relative roughness
fluid kinematic viscosity