Reliability-based criterion for evaluating explicit approximations of Colebrook equation

Citation: Easa, S.M.; Lamri, A.A.; Brki´c, D. Reliability-Based Criterion for Evaluating Explicit Approximations of Colebrook Equation. J. Mar. Sci. Eng. 2022,10, 803. https://doi.org/10.3390/ jmse10060803 Academic Editor: Michael Hartnett Received: 18 May 2022 Accepted: 9 June 2022 Published: 11 June 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Journal of Marine Science and Engineering Article Reliability-Based Criterion for Evaluating Explicit Approximations of Colebrook Equation Said M. Easa 1, Ahmed A. Lamri 2,* and Dejan Brki´c 3,4 1Department of Civil Engineering, Toronto Metropolitan University (formerly Ryerson University), Toronto, ON M5B 2K3, Canada; [email protected] 2Department of Civil Engineering and Hydraulics, University of Mohamed Khider, P.O. Box 145 RP, Biskra 07000, Algeria 3Department of Electronic Engineering, University of Niš, 18000 Niš, Serbia; [email protected] 4IT4Innovations, VSB-Technical University of Ostrava, 708 00 Ostrava, Czech Republic *Correspondence: [email protected] Abstract: Numerous explicit approximations of the Colebrook equation have been developed and evaluated based on two criteria: prediction accuracy and computational efficiency. This paper introduces a new evaluation criterion based on the reliability of each equation. The reliability is defined by the coefficient of variation (CV) of the explicit friction factor that is a function of the variabilities of component random variables (roughness height of the internal pipe surface and kinematic viscosity of the fluid). The coefficient of variation of the friction factor depends on its first derivative for roughness height of the inner pipe surface and kinematic viscosity of the fluid and their correlation. Seven explicit approximations were evaluated using the new reliability-based criterion. The results show that all explicit approximations are very reliable, but variations exist regarding the reliability level. The reliabilities of the seven approximations is very close for the rough-flow regime and when the CV of the viscosity is minimal. However, for the smooth-flow regime, and when the CV of the roughness is minimal, various approximations showed substantially different reliabilities. The novelty of the proposed criterion is that it addresses an evaluation dimension that complements the accuracy and efficiency criteria. Keywords: Colebrook equation; explicit approximations; evaluation criterion; reliability 1. Introduction The pipe friction factor ffrom the Darcy–Weisbach formula is given by h=f·L·ρ·V2 2·D(1) where his the head loss in meters, which can be calculated using the Colebrook equation given by [1–4] 1 pf=−2·log10 ε 3.7·D+2.51 R·pf!(2) where: fis the dimensionless Darcy’s friction factor, Lis the length of pipe (m), ρis the density of the fluid (kg/m3), Vis the velocity of the fluid (m/sec), Dis the pipe diameter (m), εis the roughness of the inner pipe surface (m), and Ris the dimensionless Reynolds number, defined as R=V·D/ν, where νis the kinematic viscosity of the fluid (m2/sec). J. Mar. Sci. Eng. 2022,10, 803. https://doi.org/10.3390/jmse10060803 https://www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2022,10, 803 2 of 9 Both the Darcy–Weisbach and the Colebrook equations are empirical. Equation (2) is based on experimentation with the flow of air through pipes with a different inner pipe roughness ranging from smooth to fully rough. Additionally, it contains unknown friction factor fon both sides of the equal sign in a form from which it cannot be extracted analytically. To overcome this inconvenience, many explicit approximations have been developed for efficient use in everyday engineering practice. Genic et al. [ 5 ] and Pracks and Brkic [ 6 ] have presented excellent reviews and evaluations of the available explicit approximations of the Colebrook equation. The evaluation criteria used for evaluating the explicit approximations were prediction accuracy and computational efficiency. The reliability of the explicit approximations has not been previously addressed in the literature. This article introduces a new evaluation criterion based on the reliability of the explicit approximations that aims to complement the existing criteria. The new criterion was used to evaluate seven approximations of the Colebrook equation. The reliability is defined by the coefficient of variation of the explicit friction factor that is a function of the variabilities of component random variables (roughness height of the internal pipe surface and kinematic viscosity of the fluid). The coefficient of variation of the friction factor depends on its first derivative for roughness height of the inner pipe surface and kinematic viscosity of the fluid, and their correlation. 2. Materials and Methods 2.1. Explicit Approximations of Colebrook Equation Seven explicit approximations that are evaluated through the new reliability-based criterion are by Swamee and Jain [ 7 ], Haaland [ 8 ], Mikata and Walczak [ 9 ], Biberg [ 10 ], Vatankhah [11], Praks and Brki´c [6], and Lamri and Easa [12]. 2.1.1. Approximation by Swamee and Jain (1976) The approximation by Swamee and Jain [7] is given by 1 pf=−2 logε 3.7D+5.74 R0.9 (3) 2.1.2. Haaland’s Approximation (1983) The approximation by Haaland [8] is given by 1 pf≈ −1.8·log10ε 3.7·D1.11 +6.9 R=−1.8·log10(ζ1)(4) 2.1.3. Approximation by Mikata and Walczak (2015) The approximation by Mikata and Walczak [9] is given by 1 pf≈0.8686·ln0.458·R A1−ln(A1)=ζ3(5) where A1is defined as A1=0.124·R·ε D+ln(0.4587·R)(6) 2.1.4. Biberg’s Approximation (2017) The approximation by Biberg [10] is given by 1 pf≈A2·lnR 2.51·A2+1 A3−A3·ln(A3)=ζ4(7) where A2and A3are defined as A2=2 ln(10)(8) J. Mar. Sci. Eng. 2022,10, 803 3 of 9 A3=lnR 2.51·A2+R 9.287·A2·ε D(9) 2.1.5. Vatankhah’s Approximation (2018) The approximation by Vatankhah [11] is given by 1 pf≈0.8686·ln  0.3984·R (0.8686·A4) A4 A4+A5 =ζ5(10) where A4and A5are defined as A4=0.12363·R·ε D+ln(0.3984·R)(11) A5=1+1 1+A4 0.5·ln(0.8686·A4)−1+4·A4 3·(1+A4) (12) 2.1.6. Approximation by Praks and Brki´c (2020) The approximation by Praks and Brki´c [6] is given by 1 pf≈0.8686·A8−A9+A9 A6−0.5564·A9+1.207=0.8686·A8−A9+A9 ζ6(13) where A6,A7,A8, and A9are defined as A6=A7+A8(14) A7=R 8.0884·ε D(15) A8=ln(R)−0.7794 (16) A9=ln(A6)(17) 2.1.7. Approximation by Lamri and Easa (2022) The approximation by Lamri [13] and Lamri and Easa [12] is given by 1 pf≈A10 +2·log10(A11)·−1+0.862 A11 ·1+1 A11 ·log10A11 e2 (18) where A10 and A11 are defined as A10 =2·log10R 2.51(19) A11 =R 9.287·ε D+A10 (20) Brkic and Stajic [ 14 ] affirmed that Equation (18) is the most efficient available approximation of the Colebrook equation. 2.2. Proposed Reliability Criterion 2.2.1. Reliability Definition The friction factor of the Colebrook approximations is a nonlinear function of the random variables ν and ε , and therefore fis also a random variable. To derive the reliability of the friction factor f, it is necessary to linearize its solution. In general, let Ybe a nonlinear function of several random variables, X i ,I= 1, 2, . . . ,n, where nis the number of random J. Mar. Sci. Eng. 2022,10, 803 4 of 9 variables. That is, Y= F(X 1 ,X 2 , . . . ,X n ). Then, using the Taylor series, the expected value and variance of Y,E[Y] and var[Y], are given, respectively, by [15,16]. E[Y]∼ =F(µx1,µx1, . . . , µxn)(21) var[Y]∼ = n ∑ i=1∂F ∂Xi2 σ2 Xi + n ∑ i n ∑ j∂F ∂Xi ∂F ∂Xj!covXi,Xj(22) where µXi is the mean value of the random variable X i , σXi is the standard deviation (SD) of the random variable X i , and cov[X i , X j ] is the covariance of the random variables X i and X j . The covariance between two random variables is given by cov[X i ,X j ] = ρXi,Xj σXi σXj , where ρXi,Xj is the correlation coefficient between X i and X j . The coefficient of variation of Y, which is a dimensionless measure of reliability, is given by CVY=pvar[Y] E[Y](23) where CV Y is the coefficient of variation of the random variable Yand µY is the mean of the random variable Y. Similarly, CVxi =σXi/µXi. The coefficient of variation has been used in engineering applications to determine the reliability of a random variable. The smaller the CV value, the more reliable the random variable is. The definition of reliability is graphically presented in Figure 1. J. Mar. Sci. Eng. 2022, 10, x FOR PEER REVIEW 4 of 10 random variables. That is, Y = F(X 1 , X 2 , …, X n ). Then, using the Taylor series, the expected value and variance of Y, E[Y] and var[Y], are given, respectively, by [15,16]. E[Y] ≅ F(µ x1 , µ x1 , …, µ xn ) (21) 𝑣𝑎𝑟󰇟𝑌󰇠≅∑󰇡   󰇢𝜎   +∑∑󰇡   󰇢   𝑐𝑜𝑣𝑋,𝑋     , (22) where μ Xi is the mean value of the random variable X i , σ Xi is the standard deviation (SD) of the random variable X i , and cov[X i , X j ] is the covariance of the random variables X i and X j . The covariance between two random variables is given by cov[X i , X j ] = ρ Xi,Xj σ Xi σ Xj , where ρ Xi,Xj is the correlation coefficient between X i and X j . The coefficient of variation of Y, which is a dimensionless measure of reliability, is given by 𝐶𝑉=󰇟󰇠 󰇟󰇠 , (23) where CV Y is the coefficient of variation of the random variable Y and µ Y is the mean of the random variable Y. Similarly, CV xi = σ Xi / µ Xi . The coefficient of variation has been used in engineering applications to determine the reliability of a random variable. The smaller the CV value, the more reliable the random variable is. The definition of reliability is graphically presented in Figure 1. Figure 1. Definition of reliability. 2.2.2. First Derivatives of Friction Factor f Since the friction factor f of the Colebrook equation is a nonlinear function of the random variables v and ε, the mean and standard deviation of f are given by 𝐸󰇟 𝑓 󰇠≅𝐹󰇛𝜇,𝜇󰇜 𝑣𝑎𝑟󰇟𝑓󰇠≅󰇡 󰇢𝜎+󰇡 󰇢𝜎+󰇡 󰇢󰇡 󰇢𝑐𝑜𝑣󰇟𝑣,𝜀󰇠 𝑐𝑜𝑣󰇟𝑣,𝜀󰇠=𝜌,𝜎𝜎 𝐶𝑉=󰇟󰇠 󰇟󰇠⎭ ⎪ ⎬ ⎪ ⎫ , (24) The first derivatives of f for v and ε are needed for calculating the variance and expected value of f, and the coefficient of variation of f. These derivatives are presented next for the seven explicit approximations of the Colebrook equation, respectively, presented in Section 2.1. - Approximation by Swamee and Jain: Figure 1. Definition of reliability. 2.2.2. First Derivatives of Friction Factor f Since the friction factor fof the Colebrook equation is a nonlinear function of the random variables vand ε, the mean and standard deviation of fare given by E[f]∼ =F(µv,µε) var[f]∼ =∂f ∂v2σ2 v+∂f ∂ε 2σ2 ε+∂f ∂v∂f ∂ε cov[v,ε] cov[v,ε]=ρv,εσvσε CVf=√var[f] E[f]                (24) The first derivatives of ffor vand ε are needed for calculating the variance and expected value of f, and the coefficient of variation of f. These derivatives are presented next for the seven explicit approximations of the Colebrook equation, respectively, presented in Section 2.1. J. Mar. Sci. Eng. 2022,10, 803 5 of 9 - Approximation by Swamee and Jain: d f dε=4 3.7·D·ln(10)·[−2·log10(ζ2)]−3·[ln(ζ2)]−1 d f dν=5.934 ν0.1·ln(10)·π·D Q0.9·[−2·log10(ζ2)]−3·[ln(ζ2)]−1 ζ2=ε 3.7·D+5.74 R0.9          (25) - Haaland’s approximation: d f dε=3.6 ln(10)·[−1.8·log10(ζ1)]−3·1.11·ε0.11 (3.7·D)1.11 ·[ζ1]−1 d f dν=3.6 ln(10)·[−1.8·log10(ζ1)]−3·6.9·π·D 4·Q·[ζ1]−1 ζ1=ε 3.7·D1.11 +6.9 R            (26) - Approximation by Mikata and Walczak: d f dε=−2·0.8686·[ζ3]−3·−0.124·R D·(A1−ln(A1)) ·1−1 A1 d f dν=−2·0.8686·[ζ3]−3·h −4·Q R·π·D·ν2−(A1−ln(A1))−1·dA1 dν1−1 A1i ζ3=0.8686·ln0.458·R A1−ln(A1) dA1 dν=−0.124·4·Q·ε π·D2·ν2−1 ν                  (27) - Biberg’s approximation: d f dε=−2·A2·[ζ5]−3·h1 A3−1·R 9.287·z·A3·D−ln(A3)·R 9.287·z·A32·Di d f dν=−2·A2·[ζ5]−3·h−1 ν+1 A3·dA3 dν1 A3−1−ln(A3) A32·dA3 dνi ζ4=A2·lnR 2.51·A2+1 A3−1·ln(A3)            (28) - Vatankhah’s approximation: d f dε=2·(0.8686)2·[ζ5]−3·0.1236·A4·R D·(A4+A5)·(0.8686·A4) d f dν=2·0.8686·[ζ5]−3·h1 ν−1−1 ν·(A4+A5)i ζ5=0.8686·ln 0.3984·R (0.8686·A4) A4 A4+A5                   (29) - Approximation by Praks and Brki´c: d f dε=−2·0.8686·h0.8686·hA8−A9+A9 ζ6ii−3·"−dA6 dε·1 A6+ dA6 dε·1 A6·ζ6−A9dA6 dε−0.5564·dA6 dε·1 A6 A62# d f dν=−2·0.8686·h0.8686·hA8−A9+A9 ζ6ii−3·"−1 ν−dA6 dν·1 A6+ dA6 dε·1 A6·ζ6−A9dA6 dε−0.5564·dA6 dε·1 A6 A62# ζ6=A6−0.5564·A9+1.207 dA6 dν=−4·Q·ε 8.0884·πD2·ν2−1 ν                        (30) J. Mar. Sci. Eng. 2022,10, 803 6 of 9 - Approximation by Lamri and Easa: d f dε=−2·[ζ7]−3·h0.8645 A11 −1·2·R 9.287·D·A11ln(10)+2·log10(A11)·−0.8645·R 9.287·D·A112i d f dν=−2·[ζ7]−3·h −2 ν·ln(10)+0.8645 A11 −1·dA11 dν+log10(A11)·dA11 dνi ζ7=A10 +2·log10(A11)·0.8645 A11 −1 dA11 dν=−4·Q·ε 8.0884·πD2·ν2−2 ν·ln(10)                  (31) 2.2.3. Verification To verify the developed derivatives, the mean and standard deviation of fof the analytical formulas were compared with those of the Monte Carlo simulation. Given the means of ν and ε and the assumed values of CV, the expected value and standard deviation of fwere obtained. For simplicity, ν and ε were assumed to be uncorrelated ( ρv,ε = 0). The simulation involved generating 20,000 random values of the four random variables ν and ε , assuming their probability distributions to be normal. The generated random values of the two variables were then substituted in the respective equation of fof the corresponding approximation, resulting in 20,000 values of fthat were used to establish a frequency histogram. The analytical and MC simulation results of all explicit approximations were very close. This verifies the developed first derivatives. For example, a comparison of the simulation and mathematical results of the approximation by Lamri and Easa [ 12 ] is shown in Figure 2. The hollow columns represent a normal distribution with the respective mean and SD of the mathematical solution. The solid columns represent the frequency distribution based on simulation. It is noted that the means and standard deviations of the mathematical formula and simulation are very close, and the distribution of fis also close to normal. J. Mar. Sci. Eng. 2022, 10, x FOR PEER REVIEW 6 of 10  =−2·󰇟𝜁󰇠·󰇣󰇡.   −1󰇢·󰇡· .··  ·󰇛󰇜󰇢+2·𝑙𝑜𝑔󰇛 𝐴 󰇜·󰇡.· .··  󰇢󰇤  =−2·󰇟𝜁󰇠·󰇣󰇡 ·󰇛󰇜󰇢+󰇡.   −1󰇢·󰇡   󰇢+2·𝑙𝑜𝑔󰇛𝐴󰇜·󰇡   󰇢󰇤 𝜁=𝐴+2·𝑙𝑜𝑔󰇛𝐴󰇜·󰇡.   −1󰇢    =·· .··  ·  − ·󰇛󰇜⎭ ⎪ ⎪ ⎬ ⎪ ⎪ ⎫ , (31) 2.2.3. Verification To verify the developed derivatives, the mean and standard deviation of f of the analytical formulas were compared with those of the Monte Carlo simulation. Given the means of ν and ε and the assumed values of CV, the expected value and standard deviation of f were obtained. For simplicity, ν and ε were assumed to be uncorrelated (𝜌,= 0). The simulation involved generating 20,000 random values of the four random variables ν and ε, assuming their probability distributions to be normal. The generated random values of the two variables were then substituted in the respective equation of f of the corresponding approximation, resulting in 20,000 values of f that were used to establish a frequency histogram. The analytical and MC simulation results of all explicit approximations were very close. This verifies the developed first derivatives. For example, a comparison of the simulation and mathematical results of the approximation by Lamri and Easa [12] is shown in Figure 2. The hollow columns represent a normal distribution with the respective mean and SD of the mathematical solution. The solid columns represent the frequency distribution based on simulation. It is noted that the means and standard deviations of the mathematical formula and simulation are very close, and the distribution of f is also close to normal. Figure 2. Comparison of the frequency distributions of analytical and simulation for the approximation by Lamri and Easa [12]. 3. Results 3.1. Reliability-Based Ranking of Various Approximations The reliability of various approximations was evaluated for the following four cases: Case 1: Coefficient of variation of ε is zero (CV ε = 0, CV ν = 30%). Case 2: Coefficient of variation of v is zero (CV ε = 30%, CV ν = 0). Case 3: Smooth-flow regime (ε = 0, ν = 1x10 −6 ). Case 4: Rough-flow regime (ε = 0.001, ν = 2x10 −9 ). The reliabilities of some methods were very close in the preceding cases. To fairly reflect the ranking of various approximations, the minimum coefficient of variation of f, Figure 2. Comparison of the frequency distributions of analytical and simulation for the approximation by Lamri and Easa [12]. 3. Results 3.1. Reliability-Based Ranking of Various Approximations The reliability of various approximations was evaluated for the following four cases: Case 1: Coefficient of variation of εis zero (CVε= 0, CVν= 30%). Case 2: Coefficient of variation of vis zero (CVε= 30%, CVν= 0). Case 3: Smooth-flow regime (ε= 0, ν= 1 ×10−6). Case 4: Rough-flow regime (ε= 0.001, ν= 2 ×10−9). The reliabilities of some methods were very close in the preceding cases. To fairly reflect the ranking of various approximations, the minimum coefficient of variation of f, min J. Mar. Sci. Eng. 2022,10, 803 7 of 9 CV f , was calculated for each case, and the rankings of the approximations were determined as follows: Rank 1: min CVf≤CVf< 1.1 min CVf. Rank 2: 1.1 min CVf≤CVf< 1.2 min CVf. Rank 3: 1.2 min CVf≤CVf. Thus, an approximation is assigned Rank 1 if its CV f lies within 110% of min CV f and Rank 2 if its CV f lies within 110% and 120% of min CV f . If CV f is equal to or greater than 120% of min CVf, the approximation is assigned Rank 3. The ranking results are shown in Table 1. For each case, the minimum and maximum CV f are shown. For example, for Case 3, min CV f = 4.438 and max CV f = 5.449. The 110% and 120% thresholds are 4.438 and 4.881, respectively. The approximation of Swamee and Jain is assigned Rank 1 because their CV f is the minimum. The CV f of all other approximations are greater than 4.881 and therefore were assigned Rank 3. Table 1. Rankings of various approximations based on the reliability criterion. Approximation CV Level Flow regime Case 1: CVε= 0 Case 2: CVv= 0 Case 3: Smooth 1Case 4: Rough 2 CVν= 30% (CVf= 0.210–0.415) CVε= 30% (CVf= 8.172–8.354) ε= 0, ν= 1 ×10−6 (CVf= 4.438–5.449) ε= 0.001, ν= 2 ×10−9 (CVf= 5.701–5.706) Swamee and Jain 3 1 1 1 Haaland 1 1 3 1 Mikata and Walczak 2 1 3 1 Biberg 2 1 3 1 Vatankhah 3 1 3 1 Praks and Brkic 2 1 3 1 Lamri and Easa 2 1 3 1 1CVε= 0, CVν= 30%. 2CVε= 20%, CVν= 30%. Thus, for Case 1, Haaland’s approximation is the most reliable (Rank 1), while the approximations by Swamee and Jain and Vatankhah are the least reliable (Rank 3). The other approximations have Rank 2). For Case 3, the approximation by Swamee and Jain substantially outperforms the other approximations. For Cases 2 and 4, all approximations are assigned Rank 1 because their reliabilities are within 110% of the min CV f (8.990 and 6.627, respectively). 3.2. Sensitivity Analysis The variation of the reliability of various approximations was examined as vor ε varies. The variation CVfof various approximations as the roughness varies is shown in Figure 3 for CV ε = 0, CV ν = 30%, and ν = 1 × 10 −6 . The log( ε ) varied from − 1.5 to − 5. As noted, as the roughness decreases to about − 3, CV f of all approximations is almost the same. As the roughness decreases further, CV f of Swamee and Jain’s approximation becomes the best, and all other approximations exhibit substantially larger values, consistent with Case 3 of Table 1. The variation CV f of various approximations as the viscosity varies is shown in Figure 4 for the smooth-flow regime ( ε = 0). This is similar to Case 3 of Table 1, except that vvaries. As noted, Swamee and Jain’s approximation exhibits the best reliability for the entire range of v. The reliability of the other approximations is almost the same. However, for larger viscosity, Haaland’s approximation shows the least reliability. J. Mar. Sci. Eng. 2022,10, 803 8 of 9 J. Mar. Sci. Eng. 2022, 10, x FOR PEER REVIEW 8 of 10 Figure 3. Variation of the reliability of various approximations as roughness varies (ν = 1x10−6, CVε = 0, CVν = 30%). Figure 4. Variation of the reliability of various approximations as viscosity varies for the smoothflow regime (ε = 0). 4. Conclusions This article has introduced a new reliability-based criterion for evaluating the performance of the explicit approximations of the Colebrook equation. The proposed criterion addresses a performance dimension that complements the existing prediction accuracy and computational efficiency criteria. The criterion was used to evaluate the performance of seven explicit approximations. The new criterion is defined by the coefficient of variation of the explicit friction factor. The CVf of the friction factor is calculated using the variabilities of the two random parameters: roughness height of the internal pipe surface and kinematic viscosity of the fluid. The coefficient of variation of the friction factor depends on its first derivatives for these two parameters and their correlation. The results show that all explicit approximations are very reliable, but variations exist regarding the reliability level. The reliabilities of the seven approximations were close for the rough-flow regime and when the CV of viscosity is minimal (all approximations have Rank 1). For the smooth-flow regime, Swamee and Jain’s approximation showed the largest reliability (Rank 1), while the other approximations were close (Rank 3). When the Figure 3. Variation of the reliability of various approximations as roughness varies ( ν = 1 × 10 −6 , CVε= 0, CVν= 30%). J. Mar. Sci. Eng. 2022, 10, x FOR PEER REVIEW 8 of 10 Figure 3. Variation of the reliability of various approximations as roughness varies (ν = 1x10−6, CVε = 0, CVν = 30%). Figure 4. Variation of the reliability of various approximations as viscosity varies for the smoothflow regime (ε = 0). 4. Conclusions This article has introduced a new reliability-based criterion for evaluating the performance of the explicit approximations of the Colebrook equation. The proposed criterion addresses a performance dimension that complements the existing prediction accuracy and computational efficiency criteria. The criterion was used to evaluate the performance of seven explicit approximations. The new criterion is defined by the coefficient of variation of the explicit friction factor. The CVf of the friction factor is calculated using the variabilities of the two random parameters: roughness height of the internal pipe surface and kinematic viscosity of the fluid. The coefficient of variation of the friction factor depends on its first derivatives for these two parameters and their correlation. The results show that all explicit approximations are very reliable, but variations exist regarding the reliability level. The reliabilities of the seven approximations were close for the rough-flow regime and when the CV of viscosity is minimal (all approximations have Rank 1). For the smooth-flow regime, Swamee and Jain’s approximation showed the largest reliability (Rank 1), while the other approximations were close (Rank 3). When the Figure 4. Variation of the reliability of various approximations as viscosity varies for the smooth-flow regime (ε= 0). 4. Conclusions This article has introduced a new reliability-based criterion for evaluating the performance of the explicit approximations of the Colebrook equation. The proposed criterion addresses a performance dimension that complements the existing prediction accuracy and computational efficiency criteria. The criterion was used to evaluate the performance of seven explicit approximations. The new criterion is defined by the coefficient of variation of the explicit friction factor. The CV f of the friction factor is calculated using the variabilities of the two random parameters: roughness height of the internal pipe surface and kinematic viscosity of the fluid. The coefficient of variation of the friction factor depends on its first derivatives for these two parameters and their correlation. The results show that all explicit approximations are very reliable, but variations exist regarding the reliability level. The reliabilities of the seven approximations were close for the rough-flow regime and when the CV of viscosity is minimal (all approximations have Rank 1). For the smooth-flow regime, Swamee and Jain’s approximation showed the largest reliability (Rank 1), while the other approximations were close (Rank 3). When the CV of the roughness is minimal, Haaland’s approximation showed the largest reliability J. Mar. Sci. Eng. 2022,10, 803 9 of 9 (Rank 1), followed by the approximations by Mikata and Walczak, Biberg, Praks and Brkic, and Lamri and Easa (Rank 2). In contrast, Vatankhah’s approximation showed the least reliability (Rank 3). The sensitivity analysis shows that, for CV ε = 0, the performance of various explicit approximations is the same for large ε , and the performance varies as ε decreases. In addition, for the smooth-flow regime ( ε = 0), the relative performance remains almost the same for the entire range of v. Author Contributions: Conceptualization, S.M.E. and A.A.L.; methodology, S.M.E. and A.A.L.; software, A.A.L.; validation, S.M.E. and A.A.L.; formal analysis, S.M.E. and A.A.L.; investigation, S.M.E. and A.A.L.; resources, S.M.E.; data curation, A.A.L.; writing—original draft preparation, S.M.E., A.A.L. and D.B.; writing—review and editing, S.M.E., A.A.L. and D.B.; visualization, A.A.L.; supervision, S.M.E.; project administration, S.M.E. and D.B.; funding acquisition, S.M.E. and D.B. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Dean’s Scholarly, Research, and Creative fund for the first author. A. Lamri is grateful to Toronto Metropolitan University for the great support provided during his conduct of this research. D. Brki´c was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia and by the Technology Agency of the Czech Republic through the project “Center of Energy and Environmental Technologies” TK03020027. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: All data to repeat computations are given in the text. Conflicts of Interest: The authors declare no conflict of interest. References 1. Colebrook, C.F.; White, C.M. Experiments with Fluid Friction in Roughened Pipes. Proc. R. Soc. London. Ser. A—Math. Phys. Sci. 1937,161, 367–381. [CrossRef] 2. Colebrook, C.F. Turbulent Flow in Pipe with Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws. J. Inst. Civ. Eng. 1939,11, 133–156. [CrossRef] 3. Darcy, H. Recherches Expérimentales Relative au Mouvement de l’eau dans les Tuyaux; Mallet-Bachelier: Paris, France, 1857. (In French) 4. 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