Discussion of “Turbulent Flow Friction Factor Calculation Using a Mathematically Exact Alternative to the Colebrook–White Equation” by Jagadeesh R. Sonnad and Chetan T. Goudar August 2006, Vol. 132, No. 8, pp. 863–867.
DOI: 10.1061/共ASCE兲0733-9429共2006兲132:8共863兲
Ali R. Vatankhah1 and Salah Kouchakzadeh2 1
Ph.D. Candidate, Irrigation and Reclamation Eng. Dept., Univ. of Tehran, P.O. Box 4111, Karaj 31587-77871, Iran. E-mail:
[email protected] 2 Professor, Irrigation and Reclamation Eng. Dept., Univ. of Tehran, P.O. Box 4111, Karaj 31587-77871, Iran. E-mail:
[email protected]
The authors are appreciated for presenting a novel equation with high accuracy against presently available equations for friction factor estimation. The discussers, however, would like to add a few points that might be of interest. Using a quadratic approximation of the Taylor series, Eq. 共19兲 of the original paper could be presented in the following form: h+
h h2 − = − ln共s兲 s 2s2
共1兲
Solving Eq. 共1兲 here for h yields:
冉 冑
h = s共s + 1兲 1 −
ln共s兲 共s + 1兲2
1+2
冊
共2兲
Eq. 共26兲 of the original paper could be rearranged as follows: 1
共3兲
冑 f = a共ln共d兲 + h兲
Fig. 1. Comparison of maximum percentage error in f computed by Eq. 共27兲 of the original paper and Eq. 共4兲 and Eq. 共6兲 of this discussion
The error associated with Eq. 共6兲 here for the entire practical range of R − / D is also presented in Fig. 1 here. It seems that the term “superior accuracy” should be attributed to the results of Eq. 共6兲 here since it provides solutions with maximum error of 0.05% compared to that of 1% and 0.15% associated with answers of Eq. 共27兲 of the original paper and Eq. 共4兲 here, respectively. The error of Eq. 共6兲 here for the whole practical domain of R and / D is presented in Fig. 2. The advantage of Eq. 共6兲 of this discussion could be better highlighted by indicating that the maximum error is restricted to a relatively small area characterized by low R and small / D. Also, it is worth mentioning that for the entire practical range of R- / D, the value of s 0.31.
Substituting Eq. 共2兲 here into Eq. 共3兲 of this discussion yields: 1
冑f = a
冉
冉 冑
ln共d兲 + s共s + 1兲 1 −
1+2
ln共s兲 共s + 1兲2
冊冊
共4兲
The errors associated with Eq. 共27兲 of the original paper and Eq. 共4兲 of this discussion over the entire range of / D and R encountered in practice are presented in Fig. 1 of this discussion. The figure indicates that the maximum error of Eq. 共4兲 of this discussion is about 0.15%, which is much smaller than that provided by Eq. 共27兲 of the original paper. However, the number of floating point operations for estimating f from Eq. 共4兲 here is higher than that required by Eq. 共27兲 of the original paper. Far more accuracy than that provided by Eq. 共27兲 of the original paper and Eq. 共4兲 here with a low number of floating point operations could also be reached. That is, by using a curve fitting based on the general form of Eq. 共20兲 of the original paper, Eq. 共18兲 of the original paper could be presented in the following form: h=
−s ln共s − 0.31兲 s + 0.9633
共5兲
Substituting Eq. 共5兲 of this discussion into Eq. 共3兲 here yields: 1
冑 f = a ln
冉
d 共s − 0.31兲s/共s+0.9633兲
冊
共6兲
Fig. 2. Percentage error associated with f for the entire practical domain of R- / D computed by Eq. 共6兲 of this discussion JOURNAL OF HYDRAULIC ENGINEERING © ASCE / AUGUST 2008 / 1187
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