Performance of explicit approximations of the coefficient of head loss for pressurized conduits

Desempenho de aproximações explícitas do coeficiente de perda de carga para condutos pressurizados R E S U M O Um dos parâmetros envolvido no dimensionamento de sistemas hidráulicos pressurizados é a perda de carga das tubulações. Essa verificação pode ser realizada através da formulação de Darcy-Weisbach, que considera um coeficiente de perda de carga (f) que pode ser mensurado pela equação implícita de Colebrook-White. No entanto, para essa determinação é necessário utilizar métodos numéricos ou o diagrama de Moody. Devido a isso, numerosas aproximações explícitas são propostas para superar essa limitação. Nesse sentido, o objetivo desse trabalho é analisar as aproximações explícitas do f para condutos pressurizados em comparação a formulação de Colebrook-White, determinando as mais precisas para que possam ser uma solução alternativa, válidas para o regime de fluxo turbulento. Foram analisadas 29 equações explícitas encontradas na literatura, determinando o f através do número de Reynolds (Re) na faixa de 4 × 103 ≤ Re ≤ 108 e rugosidade relativa (Ɛ/D) de 10-6 ≤ Ɛ/D ≤ 5 × 10-2, obtendo 160 pontos para cada equação. O índice de desempenho e o erro relativo das formulações foram analisados em relação a equação de ColebrookWhite. Considerando as equações analisadas, sete apresentaram excelente desempenho e alta precisão, destacando a formulação de Offor & Alabi, a qual pode ser utilizada como alternativa à equação padrão de Colebrook-White.


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 Colebrook-White (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 × 10³ < 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 × 10³ ≤ 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: Table 1.Explicit approximations for the determination of the head loss coefficient (f), with their respective authors, years of publication, and application ranges Continues on the next page Continued from Table 1 where: f -is the coefficient of head loss of the Darcy-Weisbach 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.
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.
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.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), andBuzzelli (2008).This study found similar values of accuracy, with the exception of Romeo et al. (2002) The discrepancy between the RE values found in this study and those obtained by Brkić's (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 × 10³ ≤ 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.
A joint analysis of Figure 1A-G 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 × 10³.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 × 10³.
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 × 10³.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 × 10³.
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 Table 2. Criteria for interpreting the concordance index, the precision index, the performance index, and their respective classifications value of 2.375% for an Ɛ/D of 10 -6 and an Re of 4 × 10³.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 × 10³.
2. 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 Colebrook-White equation.