Introduction
The estimation of head loss in pressurized conduits is a significant problem in optimization studies, hydraulic analysis of ducts, and water distribution systems (^{Bardestani et al., 2017}).
The ^{ColebrookWhite (1937)} (CW) equation has been considered as the most accurate approximation for the determination of the head loss coefficient (f) and has been used as a reference standard; it uses the Reynolds number (Re) and the relative roughness of the pipe (Ɛ/D) (^{Heydari et al., 2015}; ^{Brkić & Ćojbašić, 2016}) and is valid for a wide range of applicability: 2 × 10^{3} < Re < 10^{8} and 0 ≤ Ɛ/D ≥ 0.05. However, it is implicit in relation to f and requires an iterative process for the solution (^{Brkić, 2016}; ^{Brkić & Ćojbašić, 2017}).
Several researchers have sought to find explicit equations that could be used as alternatives to the CW equation (^{Assefa & Kaushal, 2015}; ^{Mikata & Walczak, 2015}). According to ^{Brkić & Ćojbašić (2017)}, explicit approximations give a relatively good prediction of the f and can accurately reproduce the CW equation and the ^{Moody (1944)} diagram. In some of these explicit equations, their relative error is so small that they can be used directly instead of the CW equation (^{Çoban, 2012}).
Therefore, the objective of this research was to analyze some explicit approximations of the pressure loss coefficient for pressurized conduits, determining the most accurate ones so that they can be used as an alternative to the CW formulation.
Material and Methods
The determination of the f of all equations were performed using a Microsoft Excel worksheet, with Re values in the range of 4 × 10^{3} ≤ Re ≤ 10^{8} and Ɛ/D of 10^{6} ≤ Ɛ/D ≤ 5 × 10^{2}, and 160 points of data for each approximation analyzed were obtained. The CW formulation, Eq. 1, can be identified by:
where:
f  is the coefficient of head loss of the DarcyWeisbach formulation (dimensionless);
Ɛ/D  is the relative roughness of the pipe (m); and
Re  is the Reynolds number (dimensionless).
In all twenty nine explicit equations of the f from different authors were analysed, years of publication, and range of applicability involving Re and Ɛ/D, as listed in Table 1. Their choice was determined to evaluate most of the equations available in the literature. In this study, any model devoid of iterations was considered explicit.
Authors (Year)  Explicit equation of the f  Applicable ranges  Equation number 

Moody (1947) 

4 × 10^{3} ≤ Re ≤ 10^{8} 0 ≤ Ɛ/D ≤ 10^{2} 
(2) 
Wood (1966) 

4 × 10^{3} ≤ Re ≤ 5 × 10^{7} 10^{5} ≤ Ɛ/D ≤ 4 × 10^{2} 
(3) 
Churchill (1973) 

Not specified  (4) 
Eck (1973) 

0 ≤ Ɛ/D ≤ 10^{2}  (5) 
Jain (1976) 

5 × 10^{3} ≤ Re ≤ 10^{7}  (6) 
Swamee & Jain (1976) 

5 × 10^{3} ≤ Re ≤ 10^{8} 10^{6} ≤ Ɛ/D ≤ 5 × 10^{2} 
(7) 
Chen (1979) 

4 × 10^{3} ≤ Re ≤ 4 × 10^{8} 10^{7} ≤ Ɛ/D ≤ 5 × 10^{2} 
(8) 
Round (1980) 

4 × 10^{3} ≤ Re ≤ 10^{8} 0 ≤ Ɛ/D ≤ 5 × 10^{2} 
(9) 
Shacham (1980) 

4 × 10^{3} ≤ Re ≤ 4 × 10^{8}  (10) 
Barr (1981) 

Not specified  (11) 
Zigrang & Sylvester (1982) 

4 × 10^{3} ≤ Re ≤ 10^{8} 4 × 10^{5} ≤ Ɛ/D ≤ 5 × 10^{2} 
(12) 
Haaland (1983) 

4 × 10^{3} ≤ Re ≤ 10^{8} 10^{6} ≤ Ɛ/D ≤ 5 × 10^{2} 
(13) 
Tsal (1989) 
If A ≥ 0.018 than f = A If A < 0.018 than f = 0.0028 + 0.85A 
4 × 10^{3} ≤ Re ≤ 10^{8} 0 ≤ Ɛ/D ≤ 5 × 10^{2} 
(14) 
Robaina (1992) 

4 × 10^{3} ≤ Re ≤ 4 × 10^{7} 10^{5} ≤ Ɛ/D ≤ 10^{2} 
(15) 
Manadilli (1997) 

5.235 × 10^{3} ≤ Re ≤ 10^{8} 0 ≤ Ɛ/D ≤ 5 × 10^{2} 
(16) 
Sousa et al. (1999) 

Not specified  (17) 
Romeo et al. (2002) 

3 × 10^{3} ≤ Re ≤ 1.5 × 10^{8} 0 ≤ Ɛ/D ≤ 5 × 10^{2} 
(18) 
Sonnad & Goudar (2006) 

4 × 10^{3} ≤ Re ≤ 10^{8} 10^{6} ≤ Ɛ/D ≤ 5 × 10^{2} 
(19) 
Rao & Kumar (2007) 

Not specified  (20) 
Buzzelli (2008) 

3 × 10^{3} ≤ Re ≤ 1.5 × 10^{8} 0 ≤ Ɛ/D ≤ 5 × 10^{2} 
(21) 
Vantankhah & Kouchakzadeh (2008) 

4 × 10^{3} ≤ Re ≤ 10^{8} 10^{6} ≤ Ɛ/D ≤ 5 × 10^{2} 
(22) 
Avci & Karagoz (2009) 

Not specified  (23) 
Papaevangelou et al. (2010) 

10^{4} ≤ Re ≤ 10^{7} 10^{5} ≤ Ɛ/D ≤ 10^{3} 
(24) 
Brkić (2011a) 

Not specified  (25) 
Fang et al. (2011) 

3 × 10^{3} ≤ Re ≤ 10^{8} 0 ≤ Ɛ/D ≤ 5 × 10^{2} 
(26) 
Ghanbari et al. (2011) 

2.1 × 10^{3} ≤ Re ≤ 10^{8} 0 ≤ Ɛ/D ≤ 5 × 10^{2} 
(27) 
Shaikh et al. (2015) 

10^{4} ≤ Re ≤ 10^{8} 10^{4} ≤ Ɛ/D ≤ 5 × 10^{2} 
(28) 
Brkić (2016) 

10^{6} < Re < 10^{8} 10^{2} < Ɛ/D < 5 × 10^{2} 
(29) 
Offor & Alabi (2016) 

4 × 10^{3} ≤ Re ≤ 10^{8} 0 ≤ Ɛ/D ≤ 5 × 10^{2} 
(30) 
The precision, related to the distance of the values of the explicit equations in relation to CW, was determined by the concordance index (d) proposed by ^{Willmott (1981)}. The values ranged from 0 (without a match) to 1 (perfect match).
The Pearson correlation coefficient (r) allows quantifying the degree of association between the two variables involved in the analysis. The closer to 1, the greater the degree of linear statistical dependence between the variables, and the closer to zero, the lower the strength of that relationship.
The equations were evaluated using the performance index (Id) adapted from ^{Camargo & Sentelhas (1997)}, whose value is the product of d and r. The criteria for interpreting d, r, Id, and their respective classifications are presented in Table 2.
Concordance index (d)  Correlation coefficient (r)  Performance index (Id)  Classification 

0.951.00  0.951.00  0.901.00  Excellent 
0.890.95  0.890.95  0.800.90  Optimum 
0.840.89  0.840.89  0.700.80  Very Good 
0.770.84  0.770.84  0.600.70  Good 
0.710.77  0.710.77  0.500.60  Moderately Good 
0.630.71  0.630.71  0.400.50  Moderate 
0.550.63  0.550.63  0.300.40  Moderately Poor 
0.450.55  0.450.55  0.200.30  Poor 
0.320.45  0.320.45  0.100.20  Very Poor 
0.000.32  0.000.32  0.000.10  Bad 
After sorting the equations that had a performance index rated as “Excellent,” the mean of the relative error (MRE) was calculated. According to ^{Sadeghi et al. (2015)}, it is a very useful parameter for evaluating practically the most precise model for the estimation of the f.
The values of the MRE were classified as follows: “Very good,” MRE ≤ 0.55; “Good,” 0.55 < MRE ≤ 1.00; “Average,” 1.00 < MRE ≤ 2.00; “Weak,” 2.00 < MRE ≤ 3.00; and “Poor,” MRE > 3.00.
Results and Discussion
All the explicit equations in relation to the CW standard presented d values very close to 1.00, being classified as “Excellent,” thus possessing a high degree of accuracy among the variables involved.
The r of most of the explicit equations also provided values very close to 1.00, demonstrating a good association of the variables involved. Eqs. 14, 25, 27, and 28 were classified as “Excellent,” but had correlation coefficient (r) of less than 0.99. Meanwhile, Eq. 20 had a lower value, with r = 0.93478, being classified as “Optimum,” and presented a lower correlation between the variables involved.
According to the Id, all the equations presented an “Excellent” classification, with values close to 1.00 except for Eq. 20, which obtained an Id of 0.93444.
Analyzing the performance coefficients, we can conclude that all explicit equations obtained a satisfactory performance for the estimation of the f when compared with the implicit CW formulation. Because of this, a statistical analysis was performed using the relative error (RE) to evaluate the most accurate model for estimating the f.
The approximations of Eqs. 8, 10, 12, 17, 19, 21, 22, 24, 26, and 30 presented an MRE lower than 0.55%, all being classified as “Very good.” The lowest value found was that of Eq. 30, with MRE = 0.30%.
Models 2, 3, 5, 9, 14, 23, 27, 29, 28, and 20 presented an MRE above 1.00%, with the last two standing out owing to their higher MRE of 10.24 and 16.15%, respectively. The other equations were classified as “Good.”
The mean values of the RE found in this study are in agreement with those of ^{Brkić (2011b)}, who carried out a review of 26 explicit approximations based on the RE criterion and concluded that most of the explicit models available are very precise, with the exception of those of ^{Moody (1947)}, ^{Wood (1966)}, ^{Eck (1973)}, ^{Round (1980)}, and ^{Rao & Kumar (2007)}.
According to ^{Winning & Coole (2013)}, when 28 explicit equations of the f were compared with CW, the most precise approximations were those obtained by the equations of ^{Zigrang & Sylvester (1982)}, ^{Romeo et al. (2002)}, and ^{Buzzelli (2008)}. This study found similar values of accuracy, with the exception of ^{Romeo et al. (2002)}, which presented higher values of RE.
^{Brkić (2011a)}, ^{Winning & Coole (2013)} and ^{Offor & Alabi (2016)} analyzing explicit equations of the f, found that the RE values of ^{Rao & Kumar’s (2007)} equation were the highest in relation to all the explicit equations analyzed in their research, being consistent with what was obtained in this study.
The discrepancy between the RE values found in this study and those obtained by ^{Brkić (2016)} proposed equation is possibly due to the fact that the approximation obtained by this study covers a limited range of applicability of Re and Ɛ/D, with values of 10^{6} ˂ Re ˂ 10^{8} and 10^{2} ˂ Ɛ/D ˂ 5 × 10^{2} only, respectively.
For an approximation of the range of applicability that the CW equation provides, only the explicit equations covering 4 × 10^{3} ≤ Re ≤ 10^{8} and 10^{6} ≤ Ɛ/D ≤ 5 × 10^{2} and MRE < 0.55% will be valid. This is applied because some highly accurate approximations are valid only at limited Re and Ɛ/D intervals and, thus, may incorrectly estimate the f.
Of the 29 explicit approximations of the f analyzed, only 7 satisfied these conditions, which were Eqs. 8, 10, 19, 21, 22, 26, and 30. These approximations are presented in Figure 1AG, which shows the RE distribution for the entire Re range of 4 × 10^{3} ≤ Re ≤ 10^{8} and the Ɛ/D of 10^{6} ≤ Ɛ/D ≤ 5 × 10^{2}.
A joint analysis of Figure 1AG shows that the equation of ^{Sonnad & Goudar (2006)} presented the highest value of the maximum RE and the lowest value of the minimum RE in relation to the others, with values of 3.17 and 0.003%, respectively.
For ^{Chen (1979)}, the minimum RE value was 0.019% for an Ɛ/D of 10^{5} and an Re of 5 × 10^{6}, and the maximum RE value was 1.837% for an Ɛ/D of 10^{4} and an Re of 4 × 10^{3}. ^{Shacham (1980)} presented a minimum RE value of 0.069% for an Ɛ/D of 5 × 10^{6} and an Re of 5 × 10^{7}, and a maximum RE value of 1.270% for an Ɛ/D of 10^{6} and an Re of 4 × 10^{3}.
For ^{Buzzelli (2008)}, the minimum RE value was 0.007% for an Ɛ/D of 5 × 10^{6} and an Re of 5 × 10^{7}, and the maximum RE value was 2.156% for an Ɛ/D of 10^{6} and an Re of 4 × 10^{3}. ^{Vantankhah & Kouchakzadeh (2008)} presented a minimum RE value of 0.01% for an Ɛ/D of 5 × 10^{6} and an Re of 5 × 10^{7}, and a maximum RE value of 2.112% for an Ɛ/D of 10^{6} and an Re of 4 × 10^{3}.
^{Fang et al. (2011)} presented a minimum RE value of 0.009% for an Ɛ/D of 2 × 10^{3} and an Re for 10^{5}, and a maximum RE value of 2.375% for an Ɛ/D of 10^{6} and an Re of 4 × 10^{3}. For ^{Offor & Alabi (2016)}, the minimum value of RE was 0.005% for an Ɛ/D of 5 × 10^{6} and an Re of 10^{8}, and the maximum RE value was 2.128% for an Ɛ/D of 5 × 10^{6} and an Re of 4 × 10^{3}.
Conclusions
The equations of ^{Chen (1979)}, ^{Shacham (1980)}, ^{Sonnad & Goudar (2006)}, ^{Buzzelli (2008)}, ^{Vantankhah & Kouchakzadeh (2008)}, ^{Fang et al. (2011)}, and ^{Offor & Alabi (2016)} showed higher performance indexes and precision when compared to the ColebrookWhite approximation.
The equation of ^{Offor & Alabi (2016)}, in relation to the explicit models analyzed, stood out from the others, presenting the highest performance index and precision, apart from covering the widest range of Reynolds number applicability and showing the highest relative roughness, and, therefore, can be used as an alternative to the implicit ColebrookWhite equation.