Review of new flow friction equations: Constructing Colebrook’s explicit correlations accurately

Latest revision as of 14:47, 19 May 2021

Abstract

Using only a limited number of computationally expensive functions, we show a way how to construct accurate and computationally efficient approximations of the Colebrook equation for flow friction. The presented approximations are based on the asymptotic series expansion of the Wright ω-function and symbolic regression. The results are verified with 8 million of Quasi-Monte Carlo points covering the domain of interest for engineers. In comparison with the built-in “wrightOmega” feature of Matlab R2016a, the herein introduced related approximations of the Wright ω-function significantly accelerate the computation. With only two logarithms and several basic arithmetic operations used, the presented approximations are not only computationally efficient but also extremely accurate. The maximal relative error of the most promising approximation which is given in the form suitable for engineers’ use is limited to 0.0012%, while for a little bit more complex variant is limited to 0.000024%.

Keywords: Colebrook equation, hydraulic flow friction, wright $\omega$ -function, explicit approximations, symbolic regression, computational intelligence

1. Introduction

The empirical Colebrook equation [1] (Eq. (1)), widely used in hydraulics, implicitly relates in an entangled logarithmic form the known Reynolds number ${\textstyle Re}$ and the relative roughness of inner pipe surface ${\textstyle \epsilon }$ with the unknown Darcy flow friction factor ${\textstyle f}$ . The equation is based on the experiment by Colebrook and White from 1937 [2]

(1)

To date, the Colebrook equation can be explicitly expressed only in terms of the Lambert $W$ -function [3-5], the function introduced by Corless et al. [6]. Based on series expansion of its cognate, the Wright $\omega$ -function, introduced by Wright [7], explicit approximations of the Colebrook equation of any degree of accuracy can be constructed. With only two computationally expensive logarithmic functions and without exponential nor non-integer power functions, the herein presented approximations which are based on rational polynomial expressions that contain only simple arithmetic operations are also computationally efficient.

Praks and Brkić [8] already developed approximations of the Colebrook equation based on symbolic regression analyses. Symbolic regression is a classic interpretable machine learning method by bridging input data using mathematical expressions composed of some basic functions [9-16]. The study by Praks and Brkić [8] deals with symbolic regression with raw data based on the Colebrook equation solved iteratively and not through the Wright $\omega$ -function. Belkić [17] provided a detailed review of applications of transcendental Lambert and Euler functions, which includes mathematics, physics, chemistry, biomedicine, ecology and sociology (hydraulics in [18-20]). Recently, approximations of the Colebrook equation by Brkić and Praks [21,22] based on symbolic regression which goes through the Wright $\omega$ -function received a high ranking in large accuracy/speed benchmarks [23,24]. Further comparisons of the available approximations of the Colebrook equation can be seen in [25-35].

Here we will show explicit approximations of the Colebrook equation for flow friction based on the series expansion of ${\textstyle \omega (x)-}$ $x$ and symbolic regression. This paper has the following structure. After this Introduction, Section 2 brings transformation of the Colebrook equation through the Wright ω-function. As the Wright $\omega$ -function is computationally demanding. Section 3.1 introduces series expansion of ${\textstyle \omega (x)-}$ $x$ using asymptotic expansion and symbolic regression, and then Section 3.2 presents approximations suitable for engineering use, based on series expansions in Section 3.2.1, while on symbolic regression in Section 3.2.2. The novelty of this paper includes also introducing optimal constants, which significantly reduce the relative error of the asymptotic expansion for the Colebrook equation by carefully selected constants, which were estimated by global optimization. Moreover, the novel symbolic regression approximation of the Colebrook equation based on minimization of the relative error is introduced (Section 3.2.2). In comparison with the asymptotic expansion of the same complexity, the novel symbolic-based rational approximation reduces the maximal relative error significantly. Summary related to the presented approximations concerning the accuracy and speed of execution is given in Section 3.3. Finally, Section 4 includes conclusions and future work.

2. Transformation through the Wright ω-function

In 1998, Keady [3] was the first to found that the Colebrook equation can be rearranged in the explicit form in terms of the Lambert $W$ -function (however Keady’s solution has some undesired computational side effects related to a fast-growing term which can cause overflow error [4,5]). As explained by Hayes [36], the Lambert $W$ -function, named after the mathematician Johann Heinrich Lambert, who lived in XVIII century, is defined as the inverse of ${\textstyle F(x)=}$ $x\cdot {e}^{x}$ where ${\textstyle e}$ is an exponential function. The Lambert $W$ -function has provided to date the unique available closed-form explicit solution of the Colebrook equation in respect to the unknown Darcy’s flow friction factor ${\textstyle f}$ . According to Brkić and Praks [21,22], and based on work by Biberg [37] and by Rollmann and Spindler [38] among others, the Colebrook equation can be exactly explicitly solved by Eq. (2)

{\textstyle \left.{\begin{matrix}\displaystyle {\frac {1}{\sqrt {f}}}=\displaystyle {\frac {z}{2.51}}\cdot \left(\mathrm {B} \,+y\right)\\z=\displaystyle {\frac {2\cdot 2.51}{\mathrm {ln} \,\left(10\right)}}\\A=\displaystyle {\frac {Re}{z}}\cdot \displaystyle {\frac {\epsilon }{3.71}}\\B=\mathrm {ln} \,(Re)-ln(z)\\x=A+B\\y=W\left({e}^{x}\right)-x=\omega (x)-x\end{matrix}}\right\}}

(2)

In Eq. (2), the symbol $W$ denotes the Lambert $W$ -function and ${\textstyle \omega }$ represents the Wright $\omega$ -function, where ${\textstyle \omega (x)=}$ $W\left({e}^{x}\right)$ . Further, for the real engineering implementation ${\textstyle y}$ denotes an approximation of ${\textstyle \omega (x)-}$ $x$ , where ${\frac {2}{\mathrm {ln} \,\left.10\right.}}\approx 0.8686$ ; ${\frac {2\cdot 2.51}{\mathrm {ln} \,\left.10\right.}}\approx 2.18$ ; ${\textstyle 2.18\cdot 3.71\approx 8.0878;}$ $A\approx {\frac {Re\cdot \epsilon }{8.0878}}$ ; and ${\textstyle B\approx \mathrm {ln} \,\left(\displaystyle {\frac {Re}{2.18}}\right)\approx \mathrm {ln} \,\left(Re\right)-0.7794}$ .

In this review, we will also test approximations given by the asymptotic expansion of ${\textstyle y\approx \omega (x)-}$ $x$ , as we confirmed that the accuracy and especially the speed of the algorithm very depends on the used implementation of the Wright $\omega$ -function. We will start with approximations based on the asymptotic expansion of the Wright $\omega$ -function, and then we will introduce novel approximations of ${\textstyle y}$ based on the symbolic regression tool Eureqa [39,40], further noted as ${\textstyle {y}_{sr}}$ .

3. Methods and discussion

In addition to the already shown Eq. (3) [21] which represents only the first term of series expansion from Eqs. (3)-(7), we show additional terms in Section 3.1.1 and based on them, related approximations in the form suitable for engineering use in Section 3.2.1. Also, additional strategy on using symbolic regression to construct approximations of ${\textstyle \omega (x)-}$ $x$ is given in Section 3.1.2., where their form suitable for engineering use is given in Section 3.2.2. Besides at the end of Section 3.1.1., we show an efficient strategy on how to reduce the relative error of series expansion-based approximations using carefully selected constants or functions. Only one example of this promising strategy how to construct accurate explicit approximations has been already given in [21], while here we give additional expressions which are to date not only the most accurate but also very simple. These extreme accurate approximations are offered in the form suitable for human eyes (as the number of digits in constants is minimized), but also in forms suitable for coding. Summary of the accuracy and speed of the presented approximations is given in Section 3.3.

As can be seen in Figure 1 , the function ${\textstyle y=\omega (x)-}$ $x$ needs to be evaluated only in the domain 7.51< ${\textstyle x}$ <619, which is the one relevant for the Colebrook equation; Eq. (1). It is because the corresponding Reynolds number ${\textstyle Re}$ is between 4000 and 10⁸, while the roughness of inner pipe surface ${\textstyle \epsilon }$ is from 0 to 0.05, as defined in the Moody diagram [42].

Figure 1. The strictly monotonic decreasing function

for the Colebrook equation

3.1. Series expansion of ${\textstyle \omega (x)-x}$ 3.1. Series expansion of ω ( x ) − x {\textstyle \omega (x)-x}

3.1.1. Asymptotic expansion of ${\textstyle \omega (x)-x}$ about infinity3.1.1. Asymptotic expansion of ω ( x ) − x {\textstyle \omega (x)-x} about infinity

Eqs. (3)-(7) with ${\textstyle {s}_{i}\left(x\right)}$ terms ( ${\textstyle i=1,2,\cdots ,5}$ ) used for approximating ${\textstyle \omega \left(x\right)-}$ $x$ by a truncated series about infinity [41], where generally ${\textstyle {y}_{i}(x)=}$ $\sum _{i=1}^{+\infty }{s}_{i}(x)$ . The terms ${\textstyle {s}_{1}\left(x\right),\,{s}_{2}\left(x\right),\ldots ,\,{s}_{5}\left(x\right)}$ are given in Eqs. (3)-(7), see [45]. The corresponding first five approximations of ${\textstyle \omega \left(x\right)-}$ $x$ are summarized in Eqs. (8)-(12)

(3)

(4)

(5)

(6)

(7)

Winning and Coole [43,44] found that logarithmic, exponential, non-integer power terms and the other transcendental functions require extensive additional execution of floating-point operations in the Central Processor Units (CPUs) of modern computers compared with the simple arithmetic operations (+,-,*,/). The logarithmic function is needed only once in Eq. (2) to evaluate ${\textstyle \mathrm {ln} \,\left(Re\right)}$ and as can be seen from Eqs. (3)-(7), one additional logarithmic function is needed for evaluation of ${\textstyle \mathrm {ln} \,\left(x\right)}$ to approximate the ${\textstyle \omega (x)}$ .

The first five approximations of ${\textstyle \omega \left(x\right)-}$ $x$ (Eqs. (8)-(12)) are based on the series expansion of about infinity by Corless et al. [6,45]. These approximations are expressed in terms of ${\textstyle {s}_{i}(x)}$ functions from Eqs. (3-7). The coefficients of ${\textstyle \omega \left(x\right)}$ series expansion about infinity are unsigned Stirling numbers of the first kind [46] as pointed out by Rollmann and Spindler [38].

The symbol ${\textstyle y=wrightOmega(x)-x}$ represents the “exact” explicit accurate solution of the Colebrook equation given by Eq. (2), in which ${\textstyle y}$ is calculated by the Matlab’s built-in command wrightOmega (e.g. similar function does not yet exist in MS Excel [47-49] and user-defined Visual Basic for Applications - VBA script should be provided instead [50-55]). However, this wrightOmega solution is extremely slow; it requires more than 11 394 seconds, i.e. 189.9 minutes(!) to evaluate our benchmark, even the here presented approximations need, depends on the required accuracy, only seconds or even fragments of seconds. In our benchmark study, we measured time in seconds to evaluate the proposed explicit approximations of the Colebrook equation using 8 million of Quasi Monte-Carlo samples from the domain interesting for engineering use. The Monte Carlo method [55-57] for random sampling is based on a broad class of various computational algorithms while for this sampling, the LPTAU51 algorithm was used [58]. We will see from the accuracy analysis and speed tests of Eqs. (8)-(12) that it makes sense to introduce fast, but accurate approximations of the wrightOmega function

(8)

(9)

(10)

(11)

(12)

Eq. (8) is with a maximal relative error of no more than ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.153% and it needs 0.4 seconds to execute the benchmark sample; Eq. (9) is with ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.118% and 0.4 seconds; Eq. (10) is with ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.008% and 1.1 seconds; Eq. (11) with ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.00249% and 2.4 seconds, while Eq. (12) with ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.00247% and 4.4 seconds. The maximal error is defined with ${\textstyle {\delta }_{\%}=}$ $\displaystyle {\frac {f-{f}_{i}}{f}}\cdot 100\%$ where the maximal relative error of the approximated Darcy flow friction factor ${\textstyle {f}_{i}}$ is calculated with the reference to the accurate ${\textstyle f}$ calculated by the Matlab’s built-in command wrightOmega and Eq. (2). Alternatively, an accurate iterative solution of Eq. (1) is available, as explained in [26,54,59,60].

As can be seen in Eqs. (8)-(12), the function ${\textstyle {y}_{i}}$ approximates the function ${\textstyle \omega \left(x\right)-}$ $x$ by various quality and complexity. The larger ${\textstyle i}$ of the function ${\textstyle {y}_{i}}$ implies a more precise approximation of ${\textstyle \omega \left(x\right)-}$ $x$ , but the computation time increases.

Further, to increase the accuracy of expansions given with Eqs. (8)-(12), approximations can be written in a form ${\textstyle y+}$ $\alpha \approx W\left({e}^{x}\right)-x$ , where ${\textstyle \alpha }$ represents a real constant, which minimizes the relative error of the flow friction factor. The constants were obtained in Matlab by the global optimization solver Globalsearch. This optimal constant ${\textstyle \alpha }$ is 0.00056 for Eq. (8) gives Eq. (13), -0.0014 for Eq. (9) gives Eq. (14), and -0.000093 for Eq. (10) gives Eq. (15). This strategy also makes the distribution of the error more uniform, and the methodology for the relative error and time measurement is in correlation with Eqs. (8)-(12)

(13)

(14)

(15)

Eq. (13) is with the maximal relative error of no more than ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.129% and it needs 0.4 seconds to execute the benchmark sample, Eq. (14) with ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.0691% and 0.4 seconds, while Eq. (15) with ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.00527% and 1.1 seconds. From Eq. (13)-(15) we can observe that the optimal constants tend to zero.

By extending the approach for error minimization by adding a constant as in Eqs. (13)-(14), a more accurate approximation can be constructed using a function ${\textstyle \xi }$ for error minimization (instead of the constant ${\textstyle \alpha }$ ) which is found through symbolic regression [61]. Eq. (16) is based on Eq. (8) where function ${\textstyle \xi }$ reduce the relative error significantly (the maximal error is limited to 0.000391% with 1.1 seconds to execute the benchmark)

\left.{\begin{matrix}y\approx -\mathrm {ln} \,\left(x\right)+\displaystyle {\frac {\mathrm {ln} \,\left(x\right)}{x}}+\xi \\\xi =\displaystyle {\frac {0.3896\cdot \mathrm {ln} \,\left(x\right)\cdot \left(\mathrm {ln} \,\left(x\right)-1\right)-0.9873}{0.8421\cdot {x}^{2}+0.01274\cdot x\cdot {\left(\mathrm {ln} \,\left(x\right)\right)}^{4}+x+5.882}}\end{matrix}}\right\}

(16)

The error function ${\textstyle \xi }$ of Eq. (16) is developed using symbolic regression, while ${\textstyle \omega (x)-}$ $x$ will be approximated using the same approach in Section 3.1.2.

3.1.2. Symbolic regression-based expansions of ${\textstyle y=\omega \left(x\right)-x}$ for Colebrook equation3.1.2. Symbolic regression-based expansions of y = ω ( x ) − x {\textstyle y=\omega \left(x\right)-x} for Colebrook equation

An additional strategy for increasing the accuracy of ${\textstyle y=\omega \left(x\right)-x}$ for the Colebrook equation is given in Eqs. (17)-(20). These approximations were found by symbolic regression as already explained in [21] and software Eureqa is used to perform analyses [40] (very recently, also novel artificial intelligence - AI software AI Feynman: A physics-inspired method for symbolic regression is available [62]). These approximations are optimized for the domain 7.51< ${\textstyle x}$ <619, which is of the interest for the Colebrook equation.

With the strategy used for Eq. (16), the accuracy of Eq. (19) is significantly improved using a newly constructed error function ${\textstyle {\xi }_{1}}$ as shown in Eq. (20)

(17)

(18)

(19)

(20)

Eq. (17) is with the maximal relative error of no more than ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.0497% and it needs 0.42 seconds to execute the benchmark sample, Eq. (18) with ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.0105% and 0.42 seconds, Eq. (19) with ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.00229% and 0.42 seconds while Eq. (20) with ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.000024% and 1.1 seconds.

3.2. Approximations suitable for engineering use

3.2.1. Approximations and procedures based on series expansions of ${\textstyle \omega (x)-x}$ about infinity3.2.1. Approximations and procedures based on series expansions of ω ( x ) − x {\textstyle \omega (x)-x} about infinity

To use the procedure for solving Eq. (2) based on series expansions given with Eqs. (3)-(7) and Eqs. (8)-(12), one should follow the algorithm illustrated in Figure 2. It can be modified so ${\textstyle s}$ can be given as ${\textstyle s\approx {s}_{1}+}$ ${s}_{2}$ , ${\textstyle s\approx {s}_{1}+{s}_{2}+{s}_{3}}$ , etc. Also, Eqs. (8)-(12) can be extended to construct ${\textstyle {s}_{6},\,{s}_{7},\,{s}_{8},}$ etc. [45].

Following the algorithm from Figure 2, Eq. (12) can be executed (it introduces a relative error of no more than 0.00247% and that it needs around 4.4 seconds to execute 8 million of Quasi Monte-Carlo tested samples carefully selected from the domain of the Colebrook equation which is of interest for engineering practice).

Figure 2. Algorithm for the procedure for solving Eq. (2) based on series expansion by Eq. (12)

Besides, if ${\textstyle y\approx {s}_{1}}$ , Eq. (8), the relative error is up to 0.153% with around 0.2 seconds for the execution of 8 million of Quasi Monte-Carlo tested samples. Based on the algorithm from Figure 2 and using only ${\textstyle y\approx {s}_{1}}$ , Eq. (21) can be constructed (this equation is already presented in [21])

\left.{\begin{matrix}\displaystyle {\frac {1}{\sqrt {f}}}\approx 0.8686\cdot \left(B-C+\displaystyle {\frac {C}{x}}\right)\\A\approx \displaystyle {\frac {Re\cdot \epsilon }{8.0878}}\\B\approx \mathrm {ln} \,\left(Re\right)-0.7794\\C=\mathrm {ln} \,\left(x\right)\\x=A+B\,\end{matrix}}\right\}

(21)

Further accurate approximations based on expansions from Eqs. (13)-(14) are given in Eqs. (22)-(25). Based on the expansions from Eqs. (13)-(14), the appropriate explicit approximations to the Colebrook equation can be constructed: Eq. (13) → Eq. (22), Eq. (14) → Eq. (23), and Eq. (15) → Eq. (24).

To construct the approximations from Eqs. (22)-(24) based on Eqs. (13)-(14), the notation ${\textstyle \mathrm {C=ln} \,\left(x\right)}$ should be used to avoid multiple repetitions of the logarithmic function, as ${\textstyle C}$ is computed only once.

Comparing Eq. (22) with the maximal relative error of 0.129% and Eq. (21) with 0.153%, it is confirmed that adding the carefully chosen constant ${\textstyle \alpha }$ , gives more accurate results for the Colebrook equation

(22)

(23)

(24)

In Eqs. (22)-(24), ${\textstyle A\approx \displaystyle {\frac {Re\cdot \epsilon }{8.0878}}}$ , ${\textstyle B\approx \mathrm {ln} \,\left(Re\right)-0.7794}$ , and ${\textstyle C=}$ $\mathrm {ln} \,\left(x\right)$ , ${\textstyle x=A+B}$ . They are with the relative error of up to ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.1266%, 0.0691% and 0.00527%, respectively.

Eq. (25), which is based on Eq. (22), brings the relative error of up to 0.000391% thanks to the novel function ${\textstyle \xi }$ which reduce the error. Let us compare Eq. (22) and Eq. (25) by terms of accuracy and speed. The relative error of Eq. (25) is reduced significantly by the factor of 0.129%/0.000391% ~ 330. Of course, the computational time of novel Eq. (25) is 1.1 seconds, as it is slower than Eq. (22) by the factor 1.1/0.42~ 2.616, as Eq. (25) is more complex. However, Eq. (25) can be implemented easily in software codes, as ${\textstyle \xi }$ is a simple rational function, in which both the numerator and the denominator are polynomials. Here, only the error function ${\textstyle \xi }$ is developed using symbolic regression, whereas this technique is more thoroughly evaluated also in Section 3.2.2. The error function ${\textstyle \xi }$ is defined in Eq. (16)

{\textstyle \left.{\begin{matrix}\displaystyle {\frac {1}{\sqrt {f}}}\approx 0.86858896\cdot \left[B-C+\displaystyle {\frac {C}{x}}+\xi \,\right]\,\\\xi =\displaystyle {\frac {0.3896\cdot C\cdot \left(C-1\right)-0.9873}{0.8421\cdot {x}^{2}+0.01274\cdot x\cdot {C}^{4}+x+5.882}}\\A\approx \displaystyle {\frac {Re\cdot \epsilon }{8.0884}}\\B\approx \mathrm {ln} \,\left(Re\right)-0.779397\\C=\mathrm {ln} \,\left(x\right)\\x=A+B\end{matrix}}\right\}}

(25)

3.2.2. Approximations based on symbolic regression

Four simple but very accurate approximations based on symbolic regression are presented in Eqs. (26)-(29) where Eqs. (26) and (27) are already given in [21], whereas Eqs. (28) and (29) are novel (they are simple, but very accurate). Although Eureqa allows the only minimization of the absolute error [40], we can use it also for minimizing the relative error as Gholamy and Kreinovich [63] suggested (they showed how to use existing absolute-error-minimizing software to minimize relative errors). Consequently, Eureqa was used for the minimization of the maximal error (worst-case) of ${\textstyle \displaystyle {\frac {\left|{y}_{sr}-y\right|}{y}}}$ , where ${\textstyle {y}_{sr}}$ denotes a symbolic regression-based approximation of ${\textstyle y}$ . Eqs. (26)-(29) show that this formulation significantly reduces the approximation error

(26)

(27)

(28)

In Eqs. (26)-(28), ${\textstyle A\approx \displaystyle {\frac {Re\cdot \epsilon }{8.0884}}}$ , ${\textstyle B\approx \mathrm {ln} \,\left(Re\right)-0.7794}$ , ${\textstyle C=}$ $\mathrm {ln} \,\left(x\right)$ , and ${\textstyle x=A+B}$ . They are with the relative error of up to ${\textstyle {\left\{{\delta }_{\%}\right\}}_{max}}$ =0.0497%, 0.0105%, and 0.00337%, respectively (they require only 0.42 seconds to execute the benchmark study).

Comparing Eq. (26) which is already available in [21] with the here introduced Eq. (28), it can be seen that both approximations have similar complexity, but thanks to the minimizing of the relative error [63] through symbolic regression, the maximal relative error of the friction factor was reduced from 0.0497% of Eq. (26) to 0.00337% of Eq. (28), i.e. more than 14 times. Moreover, Eqs. (26)-(28) are based on Eqs. (17-20), in order to minimize the relative error of the friction factor of the Colebrook equation.

To date, the most accurate approximation of the Colebrook equation is by Vatankhah [64-66] with the relative error of no more than 0.0028% with three logarithmic and one non-integer power functions used (Clamond [67] reported that in a modern computer, evaluation of non-integer power goes also through logarithmic function).

Besides, the accuracy of Eq. (28), can be improved by optimization of coefficients as given in Eq. (29); details of optimization is given in [68,69]. For Eq. (29), the following vector is used [0.8685972, 0.779626, 8.0897, 0.5588, 1.2079] instead of [0.8686, 0.7794, 8.0884, 0.5564, 1.207] for Eq. (28), to decrease the maximal relative error from 0.00337% for Eq. (28) to 0.0012% for Eq. (29). Computation of the benchmark sample of 8 million of Quasi-Monte Carlo points for Eq. (29) takes only 0.388 seconds. However, our novel Eq. (29) with the relative error of no more than 0.0012% is not only two times more accurate, but with two logarithmic functions, it is also simpler (approaches with the only one logarithmic function [70], and even without [71], exist, but they are less accurate; the approach with one logarithm [70] is based on symbolic regression where the second logarithm is replaced using Padé approximants [72], while the algorithm without logarithms [70] is based entirely on rational functions, where both the numerator and the denominator are polynomials). Although the Eqs. (28) and (29) have the same complexity, the optimization of coefficients of Eq. (29) reduced the maximal relative significantly by the factor 0.00337/0.0012~2.8

(29)

In Eq. (29), ${\textstyle A\approx \displaystyle {\frac {Re\cdot \epsilon }{8.0897}}}$ , ${\textstyle B\approx \mathrm {ln} \,\left(Re\right)-0.779626}$ , ${\textstyle C=}$ $\mathrm {ln} \,\left(x\right)$ , and ${\textstyle x=A+B}$ .

Algorithm for solving Eq. (29) is given in Figure 3 .

Figure 3. Algorithm for solving Eq. (29)

Further, based on Eq. (20), thanks to the function for error minimization ${\textstyle {\xi }_{1}}$ , a novel very accurate approximation given with Eq. (30) can be constructed

{\textstyle \left.{\begin{matrix}\displaystyle {\frac {1}{\sqrt {f}}}\approx 0.868589\cdot \left[B+Y-{\xi }_{1}\right]\,\\{\xi }_{1}=\displaystyle {\frac {x\cdot {Y}^{2}+3.0636\cdot x\cdot Y+18.58}{19.5\cdot \left({Y}^{2}\cdot {x}^{2}+{x}^{3}\right)+169.9\cdot {Y}^{2}+1260\cdot x+18178}}\\Y=-C+\displaystyle {\frac {C}{x-0.5564\cdot C+1.207}}\\A\approx \displaystyle {\frac {Re\cdot \epsilon }{8.088387}}\\B\approx \mathrm {ln} \,\left(Re\right)-0.7793975\\C=\mathrm {ln} \,\left(x\right)\\x=A+B\end{matrix}}\right\}}

(30)

The computation speed of Eq. (30) is the same as Eq. (15), i.e. 1.1 seconds. The maximal relative error of Eq. (15) is 0.00527%, while the maximal relative error of Eq. 30 is only 0.000024%. Thus, the novel symbolic regression approximation of Eq. (30) reduces the relative error by factor 0.00527%/0.000024%≈219, which confirms the advantages of using symbolic regression for flow friction estimations.

3.3. Summary of accuracy and speed tests

The presented approximations of the Colebrook equation which are presented in this article have been compared in Table 1 with respect of accuracy and speed of execution. The column ‘time in sec’ is given for execution of the Colebrook equation for 8 million samples generated using the Sobol’s Quasi Monte Carlo algorithm, as already given through our paper.

Eqs. (8)-(12) and Eq. (21) are based on asymptotic expansions following algorithm from Figure 2. Eqs. (13)-(15) and Eqs. (22)-(24) are based on asymptotic expansions corrected with a carefully selected constant which minimizes the relative error, while Eq. (16) and Eq. (25) are with a carefully selected corrective function. Eqs. (17)-(19) and Eqs. (26)-(28) are based on symbolic regression which approximate accurately ${\textstyle \omega (x)-}$ $x$ ; while Eq. (29) is Eq. (28), but with optimised coefficients. Eq. (20) are also based on symbolic regression, but with corrective function. Finally, Eq. (30) is based on accurate approximation of ${\textstyle \omega (x)-}$ $x$ found by symbolic regression and also corrected with the carefully selected function for minimization of the maximal relative error (this corrective function is also found using symbolic regression). We can see from Table 1 that it is really worth to construct approximations, as the exact solution of the Colebrook equation; Eq. 2, given by the built-in wrightOmega function of Matlab is extremely slow (11 394 seconds, i.e. 189.9 minutes(!)).

Table 1. Comparison of the presented approximations with related error and speed of execution

${\textstyle y\approx \omega (x)-x}$	Approximation	Maximal relative error	Time
Exact solution by built-in wrightOmega function of Matlab
Eq. (2)		0% i.e. exact	11394 seconds ~ 189.9 minutes
Asymptotic expansions
Eq. (8)	Eq. (21) – Figure 2	0.153%	0.4 seconds
Eq. (9)	Figure 2	0.118%	0.4 seconds
Eq. (10)	Figure 2	0.008%	1.1 seconds
Eq. (11)	Figure 2	0.00249%	2.4 seconds
Eq. (12)	Figure 2	0.00247%	4.4 seconds
Asymptotic expansions with corrective constant
Eq. (13)	Eq. (22)	0.129%	0.4 seconds
Eq. (14)	Eq. (23)	0.0691%	0.4 seconds
Eq. (15)	Eq. (24)	0.00527%	1.1 seconds
Asymptotic expansions with corrective function obtained using symbolic regression
Eq. (16)	Eq. (25)	0.000391%	1.1 seconds
Symbolic regression
Eq. (17)	Eq. (26)	0.0497%	0.42 seconds
Eq. (18)	Eq. (27)	0.0105%	0.42 seconds
Eq. (19)	Eq. (28)	0.00337%	0.42 seconds
Symbolic regression with the optimized coefficients
Eq. (20); optimized Eq. (19)	Eq. (29) – Figure 3	0.0012%	0.388 seconds
Symbolic regression with the optimized coefficients and corrective function
Eq. (20)	Eq. (30)	0.000024%	1.1 seconds

4. Conclusions and future work

In this review, we showed very accurate and fast explicit approximations for the implicitly given Colebrook equation for flow friction. They are based on the Lambert W-function and its cognate the Wright ω-function. These explicit approximations are constructed on series expansion and symbolic regression analysis of the Wright ω-function which are much faster compared with the related Matlab built-in function. Moreover, numerical experiments on 8 million of quasi Monte-Carlo pairs show that the approximations based on symbolic regression provide for the fixed complexity more accurate flow friction approximations than the general approximations based on the series expansion. The reason is that the here presented symbolic regression approximations are optimized for the Colebrook equation, whereas the series expansion is more general.

The proposed approximations should be compared with very recent comparative studies of the available approximations of the Colebrook equation [23-26]. Such comparisons will confirm that the approximations proposed in this review are both accurate and computationally efficient.

In terms of the balance among simplicity for human eyes, accuracy and computational simplicity, the novel Eq. (29) with a relative error of up to 0.0012% is highly recommended (a simple algorithm shown in Figure 3 should be followed for engineering use). For a very accurate software implementation, its more complex variant given by Eq. (30) should be followed, as the maximal relative error is only 0.000024%.

Similar approach has been recently developed for flow friction caused by air flow through fuel cells [73].

Notations

The following symbols are used in this paper:

: variable that depends on

and

(dimensionless);

: variable that depends on

and

(dimensionless);

: variable that depends on

(dimensionless);

: Darcy (Moody) flow friction factor (dimensionless) – main output parameter

: approximation of

by a truncated series about infinity;

: Reynolds number (dimensionless) – main input parameter

: variable that depends on

and

(dimensionless);

: approximation of

, i.e. of

: approximation of

, i.e. of

found by symbolic regression

: approximation of

from Eq. (19) and used in Eq. (20)

: constant (dimensionless);

: constant (dimensionless); half of the smallest term in the asymptotic series

: relative roughness of inner pipe surface (dimensionless) – main input parameter

: relative error (%);

and

: error-functions developed using symbolic regression

: exponential function

: logarithm with base 10

: natural logarithm

: Lambert

-function

: wright

-function

: counter

wrightOmega: Matlab’s built-in function

Globalsearch: Matlab’s built-in solver

Funding

The work of Pavel Praks and Dejan Brkić has been supported by the Technology Agency of the Czech Republic through the project CEET - "Center of Energy and Environmental Technologies" TK03020027. Additionaly, the work of Dejan Brkić has been supported by the Ministry of Education, Youth and Sports of the Czech Republic through the National Programme of Sustainability (NPS II) project “IT4Innovations excellence in science-LQ1602”.

Conflicts of interest

The authors declare no conflict of interest. Neither the European Commission, the VŠB - Technical University of Ostrava, nor any person acting on their behalf is responsible for any use of this publication. Both authors have approved the final version of this paper.

Acknowledgments

We especially thank Marcelo Masera for his careful scientific supervision and kind approvals. We acknowledge support from the European Commission, Joint Research Centre (JRC), Directorate C: Energy, Transport, and Climate, Unit C3: Energy Security, Distribution, and Markets. This review paper has been registered in the internal system for publications of the European Commission, Joint Research Centre (JRC) under PubSy JRC120798.

Supplementary materials

The readers who want to transpose the presented methods into software codes would like to have a list of all required formulas in one place and in a condensed manner. For that purpose, all used equations are listed in this section, here: List of Equations: https://www.scipedia.com/public/Draft_Brkic_569749396

References

[1] Colebrook C.F. Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws. J. Inst. Civ. Eng., 11:133–156, 1939. doi:10.1680/ijoti.1939.13150

[2] Colebrook C., White C. Experiments with fluid friction in roughened pipes. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci., 161:367–381, 1937. doi:10.1098/rspa.1937.0150

[3] Keady G. Colebrook-White formula for pipe flows. J. Hydraul. Eng., 124:96–97, 1998. doi:10.1061/(ASCE)0733-9429(1998)124:1(96)

[4] Sonnad J.R., Goudar C.T. Constraints for using Lambert W function-based explicit Colebrook–White equation. J. Hydraul. Eng., 130:929-931, 2004. doi:10.1061/(ASCE)0733-9429(2004)130:9(929)

[5] Brkić D. Comparison of the Lambert W‐function based solutions to the Colebrook equation. Eng. Comput., 29:617-630, 2012. doi:10.1108/02644401211246337

[6] Corless R.M., Gonnet G.H., Hare D.E., Jeffrey D.J., Knuth D.E. On the Lambert W function. Adv. Comput. Math., 5:329–359, 1996. doi:10.1007/BF02124750

[7] Wright E.M. Solution of the equation ${\textstyle z\cdot {e}^{z}=}$ $a$ . Bull. Am. Math Soc., 65:89–93, 1959. doi:10.1090/S0002-9904-1959-10290-1

[8] Praks P., Brkić D. Symbolic regression-based genetic approximations of the Colebrook equation for flow friction. Water, 10:1175, 2018. doi:10.3390/w10091175

[9] Niazkar M., Talebbeydokhti N., Afzali S.-H. Bridge backwater estimation: A comparison between artificial intelligence models and explicit equations. Sci. Iran., 2020. doi:10.24200/SCI.2020.51432.2175

[10] Filippov A.E., Gorb S.N. Methods of the pattern formation in numerical modeling of biological problems. Facta Universitatis, Series: Mech. Eng., 17:217-242, 2019. doi:10.22190/FUME190227027F

[11] Gorkemli B., Karaboga D. A quick semantic artificial bee colony programming (qsABCP) for symbolic regression. Inf. Sci. (N.Y.), 502:346–362, 2019. doi:10.1016/j.ins.2019.06.052

[12] Niazkar M. Revisiting the estimation of Colebrook friction factor: A comparison between artificial intelligence models and CW based explicit equations. KSCE J. Civ. Eng., 23:4311–4326, 2019. doi:10.1007/s12205-019-2217-1

[13] Diyaley S., Chakraborty S. Optimization of multi-pass face milling parameters using metaheuristic algorithms. Facta Universitatis, Series: Mech. Eng., 17:365-383, 2019. doi:10.22190/FUME190605043D

[14] Brkić D., Ćojbašić Ž. Evolutionary optimization of Colebrook’s turbulent flow friction approximations. Fluids, 2:15, 2017. doi:10.3390/fluids2020015

[15] Brkić D., Ćojbašić Ž. Intelligent flow friction estimation. Comput. Intel. Neurosc., 2016:5242596, 2016. doi:10.1155/2016/5242596

[16] Ćojbašić Ž., Brkić D. Very accurate explicit approximations for calculation of the Colebrook friction factor. Int. J. Mech. Sci., 67:10-13, 2013. doi:10.1016/j.ijmecsci.2012.11.017

[17] Belkić Dž. All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells. J. Math. Chem., 57:59–106, 2019. doi:10.1007/s10910-018-0985-3

[18] Brkić, D. Lambert W function in hydraulic problems. Math. Balk. New Ser., 26:285–292, 2012. http://www.math.bas.bg/infres/MathBalk/MB-26/MB-26-285-292.pdf

[19] Brkić D. New explicit correlations for turbulent flow friction factor. Nucl. Eng. Des., 241:4055-4059, 2011. doi:10.1016/j.nucengdes.2011.07.042

[20] Brkić, D. W solutions of the CW equation for flow friction. Appl. Math. Lett., 24:1379-1383, 2011. doi:10.1016/j.aml.2011.03.014

[21] Brkić D., Praks P. Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright $\omega$ -function. Mathematics, 7:34, 2019. doi:10.3390/math7010034

[22] Brkić D., Praks P. Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function: Reply to Discussion. Mathematics, 7:410, 2019. doi:10.3390/math7050410

[23] Zeyu Z., Junrui C., Zhanbin L., Zengguang X., Peng L. Approximations of the Darcy–Weisbach friction factor in a vertical pipe with full flow regime. Water Sci. Technol.: Water Supply, 20:1321–1333, 2020. doi:10.2166/ws.2020.048

[24] Muzzo L.E., Pinho D., Lima L.E., Ribeiro L.F. Accuracy/Speed analysis of pipe friction factor correlations. In International Congress on Engineering and Sustainability in the XXI Century—INCREaSE 2019, Section: Water for Ecosystem and Society, Faro, Portugal, 9–11 October 2019; Monteiro, J., Silva, A.J., Mortal, A., Aníbal, J., da Silva, M.M., Oliveira, M. (Eds.); Springer Nature Switzerland AG 2020: Cham, Switzerland, 2019, pp. 664–679, doi:10.1007/978-3-030-30938-1_51

[25] Brkić D. Review of explicit approximations to the Colebrook relation for flow friction. J. Pet. Sci. Eng., 77:34–48, 2011. doi:10.1016/j.petrol.2011.02.006

[26] Brkić D. Determining friction factors in turbulent pipe flow. Chem. Eng. N.Y. 119:34-39, 2012.

[27] Anaya-Durand A.I., Cauich-Segovia G.I., Funabazama-Bárcenas O., Gracia-Medrano-Bravo V.A., Evaluación de ecuaciones de factor de fricción explícito para tuberías (Evaluation of explicit friction factor equations for pipes) /in Spanish/. Educ. Quím., 25:128-134, 2014. doi:10.1016/S0187-893X(14)70535-X

[28] Silva A.L.M.F., Mesquita A.L.A., Felipe A.M.P.F., Souza, J.A.D.S. Simulação do fator de atrito para o escoamento confinado de caulim com diferentes teores de sólidos pelo modelo de SWAMEE–JAIN (Simulation of the friction factor for the confined flow of kaolin with different solid contents using the SWAMEE – JAIN model) /in Portuguese/. Rev. Mater., 22, 2017. doi:10.1590/s1517-707620170002.0146

[29] Pimenta B.D., Robaina A.D., Peiter M.X., Mezzomo W., Kirchner J.H., Ben L.H. Performance of explicit approximations of the coefficient of head loss for pressurized conduits. Rev. Bras. Eng. Agríc. Ambient., 22:301-307, 2018. doi:10.1590/1807-1929/agriambi.v22n5p301-307

[30] Santos-Ruiz I., Bermúdez J.R., López-Estrada F.R., Puig V., Torres, L. Estimación experimental de la rugosidad y del factor de fricción en una tubería /in Spanish/. In Proceedings of the Memorias del Congreso Nacional de Control Automático, San Luis Potosí, Mexico, 10–12 October 2018. Available online: http://amca.mx/RevistaDigital/cnca2018/articulos/VieAT3-03.pdf

[31] Olivares A., Guerra R., Alfaro M., Notte-Cuello E., Puentes L. Evaluación experimental de correlaciones para el cálculo del factor de fricción para flujo turbulento en tuberías cilíndricas. Rev. Int. Métodos Numér. Cálc. Diseño Ing., 35:15, 2019. doi:10.23967/j.rimni.2019.01.001

[32] Alfaro-Guerra M., Guerra-Rojas R., Olivares-Gallardo A. Evaluación experimental de la solución analítica exacta de la ecuación de Colebrook-White (Experimental evaluation of exact analytical solution of the Colebrook-White Equation) (in Spanish). Ingeniería Investigación y Tecnología, 20:1–11. 2019. doi:10.22201/fi.25940732e.2019.20n2.021

[33] Minhoni R.T.D.A., Pereira, F.F., da Silva, T.B., Castro, E.R., Saad, J.C. The performance of explicit formulas for determining the Darcy-Weisbach friction factor. Eng. Agric., 40:258-265, 2020. doi:10.1590/1809-4430-eng.agric.v40n2p258-265/2020

[34] Pérez-Pupo J.R., Navarro-Ojeda M.N., Pérez-Guerrero J.N., Batista-Zaldívar M.A. On the explicit expressions for the determination of the friction factor in turbulent regime. Revista Mexicana de Ing. Quim., 19:313–334, 2020. doi:10.24275/rmiq/Fen497

[35] Niazkar M., Talebbeydokhti, N. Comparison of explicit relations for calculating Colebrook friction factor in pipe network analysis using h-based methods. Iran. J. Sci. Technol., Trans. Civ. Eng., 44:231–249, 2020. doi:10.1007/s40996-019-00343-2

[36] Hayes B. Why W? Am. Sci., 93:104–108, 2005.

[37] Biberg, D. Fast and accurate approximations for the Colebrook equation. J. Fluids Eng., 139:031401, 2017. doi:10.1115/1.4034950

[38] Rollmann P., Spindler K. Explicit representation of the implicit Colebrook–White equation. Case Stud. Therm. Eng., 5:41–47, 2015. doi:10.1016/j.csite.2014.12.001

[39] Schmidt M., Lipson H. Distilling free-form natural laws from experimental data. Science, 324:81–85, 2009. doi:10.1126/science.1165893

[40] Dubčáková R. Eureqa: Software review. Genet. Program. Evolvable Mach., 12:173–178, 2011. doi:10.1007/s10710-010-9124-z

[41]. Horchler A.D. Complex double-precision evaluation of the Wright Omega function. 2017. https://github.com/horchler/wrightOmegaq

[42] Moody L.F. Friction factors for pipe flow. T. ASME, 66:671–684, 1944.

[43] Winning H.K., Coole T. Explicit friction factor accuracy and computational efficiency for turbulent flow in pipes. Flow Turbul. Combust., 90:1–27, 2013. doi:10.1007/s10494-012-9419-7

[44] Winning H.K., Coole T. Improved method of determining friction factor in pipes. Int. J. Numer. Methods Heat Fluid Flow, 25:941–949, 2015. doi:10.1108/HFF-06-2014-0173

[45] Corless R.M., Jeffrey D.J. The Wright ω Function. In: Calmet J., Benhamou B., Caprotti O., Henocque L., Sorge V. (eds) Artificial Intelligence, Automated Reasoning, and Symbolic Computation. AISC 2002, Calculemus 2002. Lecture Notes in Computer Science, vol 2385. Springer, Berlin, Heidelberg, 2002. doi:10.1007/3-540-45470-5_10

[46] On-Line Encyclopedia of Integer Sequences; Sequence A008275 for Stirling numbers of first kind. http://oeis.org/A008275

[47] Brkić D. Spreadsheet-based pipe networks analysis for teaching and learning purpose. Spreadsheets Educ. (EJSiE), 9:4646, 2016. https://sie.scholasticahq.com/article/4646

[48] Niazkar M., Afzali S.H. Analysis of water distribution networks using MATLAB and Excel spreadsheet: h‐based methods. Comput. Appl. Eng. Educ., 25:129-141, 2017. doi:10.1002/cae.21786

[49] Niazkar M., Afzali S.H. Analysis of water distribution networks using MATLAB and Excel spreadsheet: Q‐based methods. Comput. Appl. Eng. Educ., 25:277-289, 2017. doi:10.1002/cae.21796

[50] Demir S., Karadeniz A., Demir N.M., Duman S. Excel VBA-based solution to pipe flow measurement problem. Spreadsheets Educ. (EJSiE), 10:4671, 2018. https://sie.scholasticahq.com/article/4671.pdf

[51] Niazkar M., Afzali S.H. Streamline performance of Excel in stepwise implementation of numerical solutions. Comput. Appl. Eng. Educ., 24:555–5664, 2016. doi:10.1002/cae.21731

[52] Niazkar M., Talebbeydokhti N., Afzali, S.H. Relationship between Hazen-William coefficient and Colebrook-White friction factor: Application in water network analysis. Eur. Water, 58:513-520, 2017. https://www.ewra.net/ew/pdf/EW_2017_58_74.pdf

[53] Niazkar M., Talebbeydokhti N., Afzali S.H. Novel grain and form roughness estimator scheme incorporating artificial intelligence models. Water Resour. Manag., 33:757-773, 2019. doi:10.1007/s11269-018-2141-z

[54] Brkić D. Solution of the implicit Colebrook equation for flow friction using Excel. Spreadsheets Educ. (EJSiE), 10:4663, 2017. https://sie.scholasticahq.com/article/4663

[55] Botchkarev A. Assessing Excel VBA suitability for Monte Carlo simulation. Spreadsheets Educ. (EJSiE), 8:4629, 2015. https://sie.scholasticahq.com/article/4629

[56] Christie D.S. Build your own Monte Carlo spreadsheet. Spreadsheets Educ. (EJSiE), 11:4672, 2018. https://sie.scholasticahq.com/article/4672

[57] Cordero A., Martí P., Victoria M. Optimización de topología robusta de estructuras continuas usando el método de Monte Carlo y modelos Kriging (Robust topology optimization of continuum structures using Monte Carlo method and Kriging models) /in Spanish/. Rev. Int. Métodos Numér. Cálc. Diseño Ing., 34:5, 2018. doi:10.23967/j.rimni.2017.5.005

[58] Sobol I.M., Turchaninov V.I., Levitan Y.L., Shukhman B.V. Quasi-random sequence generators; Distributed by OECD/NEA data bank; Keldysh Institute of Applied Mathematics; Russian Academy of Sciences: Moscow, Russia. 1992. https://ec.europa.eu/jrc/sites/jrcsh/files/LPTAU51.rar

[59] Praks P., Brkić D. Choosing the optimal multi-point iterative method for the Colebrook flow friction equation. Processes, 6:130, 2018. doi:10.3390/pr6080130

[60] Praks P., Brkić D. Advanced iterative procedures for solving the implicit Colebrook equation for fluid flow friction. Adv. Civ. Eng., 2018:5451034, 2018. doi:10.1155/2018/5451034

[61] Sun S., Ouyang R., Zhang B., Zhang T.Y. Data-driven discovery of formulas by symbolic regression. MRS Bull., 44:559-564, 2019. doi:10.1557/mrs.2019.156

[62] Udrescu S.-M., Tegmark M. AI Feynman: A physics-inspired method for symbolic regression. Sci. Adv. 6:eaay2631, 2020. doi:10.1126/sciadv.aay2631

[63] Gholamy A., Kreinovich V. How to use absolute-error-minimizing software to minimize relative error: Practitioner’s guide. Int. Math. Forum, 12:763-770, 2017. doi:10.12988/imf.2017.7761

[64] Vatankhah A.R. Approximate analytical solutions for the Colebrook equation. J. Hydraul. Eng., 144:06018007, 2018. doi:10.1061/(ASCE)HY.1943-7900.0001454

[65] Brkić D., Praks P. Discussion of Approximate analytical solutions for the Colebrook equation. By Ali R. Vatankhah. J. Hydraul. Eng., 146:07019011, 2020. doi:10.1061/(ASCE)HY.1943-7900.0001667

[66] Lamri A.A. Discussion of Approximate analytical solutions for the Colebrook equation. By Ali R. Vatankhah. J. Hydraul. Eng., 146:07019012, 2020. doi:10.1061/(ASCE)HY.1943-7900.0001668

[67] Clamond D. Efficient resolution of the Colebrook equation. Ind. Eng. Chem. Res., 48:3665-3671, 2009. doi:10.1021/ie801626g

[68] Niazkar M. Discussion of Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright $\omega$ -function by Dejan Brkić and Pavel Praks. Mathematics, 8:793, 2020. doi:10.3390/math8050793

[69] Praks P., Brkić D. Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright $\omega$ -function: Reply to the Discussion by Majid Niazkar. Mathematics, 8:796, 2020. doi:10.3390/math8050796

[70] Brkić D., Praks P. Colebrook’s flow friction explicit approximations based on fixed-point iterative cycles and symbolic regression. Computation, 7:48, 2019. doi:10.3390/computation7030048

[71] Praks P., Brkić D. Rational Approximation for solving an implicitly given Colebrook flow friction equation. Mathematics, 8:26, 2020. doi:10.3390/math8010026

[72] Praks P., Brkić D. One-log call iterative solution of the Colebrook equation for flow friction based on Padé polynomials. Energies, 11:1825, 2018. doi:10.3390/en11071825

[73] Brkić D., Praks P. Air-forced flow in Proton Exchange Membrane Fuel Cells: Calculation of fan-induced friction in open-cathode conduits with virtual roughness. Processes, 8:686, 2020. doi:10.3390/pr8060686

Latest revision as of 14:47, 19 May 2021

Abstract

1. Introduction

2. Transformation through the Wright ω-function

3. Methods and discussion

3.1. Series expansion of ${\textstyle \omega (x)-x}$ 3.1. Series expansion of ω ( x ) − x {\textstyle \omega (x)-x}

3.1.1. Asymptotic expansion of ${\textstyle \omega (x)-x}$ about infinity3.1.1. Asymptotic expansion of ω ( x ) − x {\textstyle \omega (x)-x} about infinity

3.1.2. Symbolic regression-based expansions of ${\textstyle y=\omega \left(x\right)-x}$ for Colebrook equation3.1.2. Symbolic regression-based expansions of y = ω ( x ) − x {\textstyle y=\omega \left(x\right)-x} for Colebrook equation

3.2. Approximations suitable for engineering use

3.2.1. Approximations and procedures based on series expansions of ${\textstyle \omega (x)-x}$ about infinity3.2.1. Approximations and procedures based on series expansions of ω ( x ) − x {\textstyle \omega (x)-x} about infinity

3.2.2. Approximations based on symbolic regression

3.3. Summary of accuracy and speed tests

4. Conclusions and future work

Notations

Funding

Conflicts of interest

Acknowledgments

Supplementary materials

References

Document information

Document Score

Share this document

Keywords

claim authorship

@@ Line 18: / Line 18: @@
 Using only a limited number of computationally expensive functions, we show a way how to construct accurate and computationally efficient approximations of the Colebrook equation for flow friction. The presented approximations are based on the asymptotic series expansion of the Wright ω-function and symbolic regression. The results are verified with 8 million of Quasi-Monte Carlo points covering the domain of interest for engineers. In comparison with the built-in “wrightOmega” feature of Matlab R2016a, the herein introduced related approximations of the Wright ω-function significantly accelerate the computation. With only two logarithms and several basic arithmetic operations used, the presented approximations are not only computationally efficient but also extremely accurate. The maximal relative error of the most promising approximation which is given in the form suitable for engineers’ use is limited to 0.0012%, while for a little bit more complex variant is limited to 0.000024%.
-'''Keywords''': Colebrook equation, Hydraulic flow friction, Wright <math>\omega</math>-function, explicit approximations, symbolic regression, computational intelligence
+'''Keywords''': Colebrook equation, hydraulic flow friction, wright <math>\omega</math>-function, explicit approximations, symbolic regression, computational intelligence
 ==1. Introduction==
@@ Line 42: / Line 42: @@
 ==2. Transformation through the Wright ω-function==
-In 1998, Keady [3] was the first to found that the Colebrook equation can be rearranged in the explicit form in terms of the Lambert <math>W</math>-function (however Keady’s solution has some undesired computational side effects related to a fast-growing term which can cause overflow error [4,5]). As explained by Hayes [36], the Lambert <math>W</math>-function, named after the mathematician Johann Heinrich Lambert, who lived in XVIII century, is defined as the inverse of <math display="inline">F(x)=</math><math>x\cdot {e}^{x}</math> where <math display="inline">e</math> is an exponential function. The Lambert <math>W</math>-function has provided to date the unique available closed-form explicit solution of the Colebrook equation in respect to the unknown Darcy’s flow friction factor <math display="inline">f</math>. According to Brkić and Praks [21,22], and based on work by Biberg [37] and by Rollmann and Spindler [38] among others, the Colebrook equation can be exactly explicitly solved by Eq. (2).
+In 1998, Keady [3] was the first to found that the Colebrook equation can be rearranged in the explicit form in terms of the Lambert <math>W</math>-function (however Keady’s solution has some undesired computational side effects related to a fast-growing term which can cause overflow error [4,5]). As explained by Hayes [36], the Lambert <math>W</math>-function, named after the mathematician Johann Heinrich Lambert, who lived in XVIII century, is defined as the inverse of <math display="inline">F(x)=</math><math>x\cdot {e}^{x}</math> where <math display="inline">e</math> is an exponential function. The Lambert <math>W</math>-function has provided to date the unique available closed-form explicit solution of the Colebrook equation in respect to the unknown Darcy’s flow friction factor <math display="inline">f</math>. According to Brkić and Praks [21,22], and based on work by Biberg [37] and by Rollmann and Spindler [38] among others, the Colebrook equation can be exactly explicitly solved by Eq. (2)
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 49: / Line 49: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\left. \begin{matrix}\displaystyle\frac{1}{\sqrt{f}}=\displaystyle\frac{z}{2.51}\cdot \left( \mathrm{B}\,+y\right) \\z=\displaystyle\frac{2\cdot 2.51}{\mathrm{ln}\,\left( 10\right) }\\\begin{matrix}A=\displaystyle\frac{Re}{z}\cdot \displaystyle\frac{\epsilon }{3.71}\\B=\mathrm{ln}\,(Re)-ln(z)\\\begin{matrix}x=A+B\\y=W\left( {e}^{x}\right) -x=\omega (x)-x\end{matrix}\end{matrix}\end{matrix}\right\}</math>
+| <math display="inline">\left. \begin{matrix}\displaystyle\frac{1}{\sqrt{f}}=\displaystyle\frac{z}{2.51}\cdot \left( \mathrm{B}\,+y\right) \\z=\displaystyle\frac{2\cdot 2.51}{\mathrm{ln}\,\left( 10\right) }\\A=\displaystyle\frac{Re}{z}\cdot \displaystyle\frac{\epsilon }{3.71}\\B=\mathrm{ln}\,(Re)-ln(z)\\x=A+B\\y=W\left( {e}^{x}\right) -x=\omega (x)-x\end{matrix}\right\}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (2)
@@ Line 55: / Line 55: @@
-In Eq. (2), the symbol <math>W</math> denotes the Lambert <math>W</math>-function and <math display="inline">\omega</math>  represents the Wright <math>\omega</math>-function, where <math display="inline">\omega (x)=</math><math>W\left( {e}^{x}\right)</math> . Further, for the real engineering implementation <math display="inline">y</math> denotes an approximation of <math display="inline">\omega (x)-</math><math>x</math>, where <math display="inline">\, \frac{2}{\mathrm{ln}\,\left. 10\right. }\approx 0.8686</math>; <math display="inline">\frac{2\cdot 2.51}{\mathrm{ln}\,\left. 10\right. }\approx 2.18</math>; <math display="inline">2.18\cdot 3.71\approx 8.0878;</math> <math display="inline">A\approx \frac{Re\cdot \epsilon }{8.0878}</math>; and <math display="inline">B\approx \mathrm{ln}\,\left( \frac{Re}{2.18}\right) \approx \mathrm{ln}\,\left( Re\right) -0.7794</math>.
+In Eq. (2), the symbol <math>W</math> denotes the Lambert <math>W</math>-function and <math display="inline">\omega</math>  represents the Wright <math>\omega</math>-function, where <math display="inline">\omega (x)=</math><math>W\left( {e}^{x}\right)</math>. Further, for the real engineering implementation <math display="inline">y</math> denotes an approximation of <math display="inline">\omega (x)-</math><math>x</math>, where <math>\frac{2}{\mathrm{ln}\,\left. 10\right. }\approx 0.8686</math>; <math>\frac{2\cdot 2.51}{\mathrm{ln}\,\left. 10\right. }\approx 2.18</math>; <math display="inline">2.18\cdot 3.71\approx 8.0878;</math> <math>A\approx \frac{Re\cdot \epsilon }{8.0878}</math>; and <math display="inline">B\approx \mathrm{ln}\,\left(\displaystyle \frac{Re}{2.18}\right) \approx \mathrm{ln}\,\left( Re\right) -0.7794</math>.
 In this review, we will also test approximations given by the asymptotic expansion of <math display="inline">y\approx \omega (x)-</math><math>x</math>, as we confirmed that the accuracy and especially the speed of the algorithm very depends on the used implementation of the Wright <math>\omega</math>-function. We will start with approximations based on the asymptotic expansion of the Wright <math>\omega</math>-function, and then we will introduce novel approximations of <math display="inline">y</math> based on the symbolic regression tool Eureqa [39,40], further noted as <math display="inline">{y}_{sr}</math>.
@@ Line 62: / Line 62: @@
 In addition to the already shown Eq. (3) [21] which represents only the first term of series expansion from Eqs. (3)-(7), we show additional terms in Section 3.1.1 and based on them, related approximations in the form suitable for engineering use in Section 3.2.1. Also, additional strategy on using symbolic regression to construct approximations of <math display="inline">\omega (x)-</math><math>x</math> is given in Section 3.1.2., where their form suitable for engineering use is given in Section 3.2.2. Besides at the end of Section 3.1.1., we show an efficient strategy on how to reduce the relative error of series expansion-based approximations using carefully selected constants or functions. Only one example of this promising strategy how to construct accurate explicit approximations has been already given in [21], while here we give additional expressions which are to date not only the most accurate but also very simple. These extreme accurate approximations are offered in the form suitable for human eyes (as the number of digits in constants is minimized), but also in forms suitable for coding. Summary of the accuracy and speed of the presented approximations is given in Section 3.3.
+As can be seen in [[#img-1|Figure 1]] , the function <math display="inline">y=\omega (x)-</math><math>x</math> needs to be evaluated only in the domain 7.51< <math display="inline">x</math><619, which is the one relevant for the Colebrook equation; Eq. (1). It is because the corresponding Reynolds number <math display="inline">Re</math> is between 4000 and 10<sup>8</sup>, while the roughness of inner pipe surface <math display="inline">\epsilon</math>  is from 0 to 0.05, as defined in the Moody diagram [42].
+<div id='img-1'></div>
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
+|-
+|style="padding:10px;"|  [[Image:Review_419592774240-image1.png|288px]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 1.''' The strictly monotonic decreasing function <math display="inline">y=\omega (x)-</math><math>x</math> for the Colebrook equation
+|}
 ===3.1. Series expansion of <math display="inline">\omega (x)-x</math>===
-====3.1.1. Assymptotic expansion of <math display="inline">\omega (x)-x</math> about infinity====
+====3.1.1. Asymptotic expansion of <math display="inline">\omega (x)-x</math> about infinity====
 Eqs. (3)-(7) with <math display="inline">{s}_{i}\left( x\right)</math>  terms (<math display="inline">i=1, 2,\cdots, 5</math>) used for approximating <math display="inline">\omega \left( x\right) -</math><math>x</math> by a truncated series about infinity [41], where generally <math display="inline">{y}_{i}(x)=</math><math>\sum _{i=1}^{+\infty }{s}_{i}(x)</math>. The terms <math display="inline">{s}_{1}\left( x\right) ,\, {s}_{2}\left( x\right) ,\ldots ,\, {s}_{5}\left( x\right)</math>  are given in Eqs. (3)-(7), see [45]. The corresponding first five approximations of <math display="inline">\omega \left( x\right) -</math><math>x</math> are summarized in Eqs. (8)-(12)
@@ Line 118: / Line 129: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (7)
 |}
-As can be seen in Figure 1, the function <math display="inline">y</math> needs to be evaluated only in the domain 7.51< <math display="inline">x</math><619, which is the one relevant for the Colebrook equation; Eq. (1). It is because the corresponding Reynolds number <math display="inline">Re</math> is between 4000 and 10<sup>8</sup>, while the roughness of inner pipe surface <math display="inline">\epsilon</math>  is from 0 to 0.05, as defined in the Moody diagram [42].
-{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
-|-
-|style="padding:10px;"|  [[Image:Review_419592774240-image1.png|288px]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding:10px;"| '''Figure 1.''' The strictly monotonic decreasing function <math display="inline">y=\omega (x)-</math><math>x</math> for the Colebrook equation
-|}
 Winning and Coole [43,44] found that logarithmic, exponential, non-integer power terms and the other transcendental functions require extensive additional execution of floating-point operations in the Central Processor Units (CPUs) of modern computers compared with the simple arithmetic operations (+,-,*,/). The logarithmic function is needed only once in Eq. (2) to evaluate <math display="inline">\mathrm{ln}\,\left( Re\right)</math>  and as can be seen from Eqs. (3)-(7), one additional logarithmic function is needed for evaluation of <math display="inline">\mathrm{ln}\,\left( x\right)</math>  to approximate the <math display="inline">\omega (x)</math>.
@@ Line 185: / Line 186: @@
 |}
-Eq. (8) is with a maximal relative error of no more than <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.153% and it needs 0.4 seconds to execute the benchmark sample; Eq. (9) is with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.118% and 0.4 seconds; Eq. (10) is with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.008% and 1.1 seconds; Eq. (11) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.00249% and 2.4 seconds, while Eq. (12) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.00247% and 4.4 seconds. The maximal error is defined with <math display="inline">{\delta }_{\%}=</math><math>\frac{f-{f}_{i}}{f}\cdot 100\%</math> where the maximal relative error of the approximated Darcy flow friction factor <math display="inline">{f}_{i}</math> is calculated with the reference to the accurate <math display="inline">f</math> calculated by the Matlab’s built-in command ''wrightOmega'' and Eq. (2). Alternatively, an accurate iterative solution of Eq. (1) is available, as explained in [26,54,59,60].
+Eq. (8) is with a maximal relative error of no more than <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.153% and it needs 0.4 seconds to execute the benchmark sample; Eq. (9) is with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.118% and 0.4 seconds; Eq. (10) is with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.008% and 1.1 seconds; Eq. (11) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.00249% and 2.4 seconds, while Eq. (12) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.00247% and 4.4 seconds. The maximal error is defined with <math display="inline">{\delta }_{\%}=</math><math>\displaystyle \frac{f-{f}_{i}}{f}\cdot 100\%</math> where the maximal relative error of the approximated Darcy flow friction factor <math display="inline">{f}_{i}</math> is calculated with the reference to the accurate <math display="inline">f</math> calculated by the Matlab’s built-in command ''wrightOmega'' and Eq. (2). Alternatively, an accurate iterative solution of Eq. (1) is available, as explained in [26,54,59,60].
 As can be seen in Eqs. (8)-(12), the function  <math display="inline">{y}_{i}</math> approximates the function  <math display="inline">\omega \left( x\right) -</math><math>x</math> by various quality and complexity. The larger <math display="inline">i</math> of the function <math display="inline">{y}_{i}</math> implies a more precise approximation of  <math display="inline">\omega \left( x\right) -</math><math>x</math>, but the computation time increases.
@@ Line 237: / Line 238: @@
 The error function <math display="inline">\xi</math>  of Eq. (16) is developed using symbolic regression, while <math display="inline">\omega (x)-</math><math>x</math> will be approximated using the same approach in Section 3.1.2.
-====3.1.2. Symbolic regression-based expansions of <math display="inline">y=\omega \left( x\right) -</math><math>x</math> for Colebrook equation====
+====3.1.2. Symbolic regression-based expansions of <math display="inline">y=\omega \left( x\right) -x</math> for Colebrook equation====
-An additional strategy for increasing the accuracy of <math display="inline">y=\omega \left( x\right) -</math><math>x</math> for the Colebrook equation is given in Eqs. (17)-(20). These approximations were found by symbolic regression as already explained in [21] and software Eureqa is used to perform analyses [40] (very recently, also novel artificial intelligence - AI software AI Feynman: A physics-inspired method for symbolic regression is available [62]). These approximations are optimized for the domain 7.51< <math display="inline">x</math><619, which is of the interest for the Colebrook equation.
+An additional strategy for increasing the accuracy of <math display="inline">y=\omega \left( x\right) -x</math> for the Colebrook equation is given in Eqs. (17)-(20). These approximations were found by symbolic regression as already explained in [21] and software Eureqa is used to perform analyses [40] (very recently, also novel artificial intelligence - AI software AI Feynman: A physics-inspired method for symbolic regression is available [62]). These approximations are optimized for the domain 7.51< <math display="inline">x</math><619, which is of the interest for the Colebrook equation.
 With the strategy used for Eq. (16), the accuracy of Eq. (19) is significantly improved using a newly constructed error function <math display="inline">{\xi }_{1}</math>'' ''as shown in Eq. (20)
@@ Line 288: / Line 289: @@
 ===3.2. Approximations suitable for engineering use===
-====3.2.1. Approximations and procedures based on series expansions of <math display="inline">\omega (x)-</math><math>x</math> about infinity====
+====3.2.1. Approximations and procedures based on series expansions of <math display="inline">\omega (x)-x</math> about infinity====
 To use the procedure for solving Eq. (2) based on series expansions given with Eqs. (3)-(7) and Eqs. (8)-(12), one should follow the algorithm illustrated in Figure 2. It can be modified so <math display="inline">s</math> can be given as <math display="inline">s\approx {s}_{1}+</math><math>{s}_{2}</math>, <math display="inline">s\approx {s}_{1}+{s}_{2}+{s}_{3}</math>, etc. Also, Eqs. (8)-(12) can be extended to construct <math display="inline">{s}_{6},\, {s}_{7},\, {s}_{8},</math> etc. [45].
-Following the algorithm from Figure 2, Eq. (12) can be executed (it introduces a relative error of no more than 0.00247% and that it needs around 4.4 seconds to execute 8 million of Quasi Monte-Carlo tested samples carefully selected from the domain of the Colebrook equation which is of interest for engineering practice).
+Following the algorithm from [[#img-2|Figure 2]], Eq. (12) can be executed (it introduces a relative error of no more than 0.00247% and that it needs around 4.4 seconds to execute 8 million of Quasi Monte-Carlo tested samples carefully selected from the domain of the Colebrook equation which is of interest for engineering practice).
+<div id='img-2'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
-|style="padding:10px;"| [[Image:Review_419592774240-image2.png|294px]]
+|style="padding:10px;"| [[Image:Review_419592774240-image2.png|360px]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" style="padding:10px;"| '''Figure 2'''. Algorithm for the procedure for solving Eq. (2) based on series expansion by Eq. (12)
 |}
 Besides, if <math display="inline">y\approx {s}_{1}</math>, Eq. (8), the relative error is up to 0.153% with around 0.2 seconds for the execution of 8 million of Quasi Monte-Carlo tested samples. Based on the algorithm from Figure 2 and using only <math display="inline">y\approx {s}_{1}</math>, Eq. (21) can be constructed (this equation is already presented in [21])
@@ Line 309: / Line 310: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\left. \begin{matrix}\displaystyle\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left( B-C+\displaystyle\frac{C}{x}\right) \\\begin{matrix}A\approx \displaystyle\frac{Re\cdot \epsilon }{8.0878}\\B\approx \mathrm{ln}\,\left( Re\right) -0.7794\\\begin{matrix}C=\mathrm{ln}\,\left( x\right) \\x=A+B\, \end{matrix}\end{matrix}\end{matrix}\right\}</math>
+| <math>\left. \begin{matrix}\displaystyle\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left( B-C+\displaystyle\frac{C}{x}\right) \\A\approx \displaystyle\frac{Re\cdot \epsilon }{8.0878}\\B\approx \mathrm{ln}\,\left( Re\right) -0.7794\\ C=\mathrm{ln}\,\left( x\right) \\x=A+B\, \end{matrix}\right\}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (21)
 |}
 Further accurate approximations based on expansions from Eqs. (13)-(14) are given in Eqs. (22)-(25). Based on the expansions from Eqs. (13)-(14), the appropriate explicit approximations to the Colebrook equation can be constructed: Eq. (13) → Eq. (22), Eq. (14) → Eq. (23), and Eq. (15) → Eq. (24).
@@ Line 319: / Line 319: @@
 To construct the approximations from Eqs. (22)-(24) based on Eqs. (13)-(14), the notation <math display="inline">\mathrm{C=ln}\,\left( x\right)</math>  should be used to avoid multiple repetitions of the logarithmic function, as <math display="inline">C</math> is computed only once.
-Comparing Eq. (22) with the maximal relative error of 0.129% and Eq. (21) with 0.153%, it is confirmed that adding the carefully chosen constant <math display="inline">\alpha</math> , gives more accurate results for the Colebrook equation
+Comparing Eq. (22) with the maximal relative error of 0.129% and Eq. (21) with 0.153%, it is confirmed that adding the carefully chosen constant <math display="inline">\alpha</math>, gives more accurate results for the Colebrook equation
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 352: / Line 352: @@
-In Eqs. (22)-(24), <math display="inline">A\approx \frac{Re\cdot \epsilon }{8.0878}</math>, <math display="inline">B\approx \mathrm{ln}\,\left( Re\right) -0.7794</math>, and <math display="inline">C=</math><math>\mathrm{ln}\,\left( x\right)</math> , <math display="inline">x=A+B</math>. They are with the relative error of up to <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.1266%, 0.0691% and 0.00527%, respectively.
+In Eqs. (22)-(24), <math display="inline">A\approx \displaystyle \frac{Re\cdot \epsilon }{8.0878}</math>, <math display="inline">B\approx \mathrm{ln}\,\left( Re\right) -0.7794</math>, and <math display="inline">C=</math><math>\mathrm{ln}\,\left( x\right)</math> , <math display="inline">x=A+B</math>. They are with the relative error of up to <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.1266%, 0.0691% and 0.00527%, respectively.
 Eq. (25), which is based on Eq. (22), brings the relative error of up to 0.000391% thanks to the novel function <math display="inline">\xi</math>  which reduce the error. Let us compare Eq. (22) and Eq. (25) by terms of accuracy and speed. The relative error of Eq. (25) is reduced significantly by the factor of 0.129%/0.000391% ~ 330. Of course, the computational time of novel Eq. (25) is 1.1 seconds, as it is slower than Eq. (22) by the factor 1.1/0.42~ 2.616, as Eq. (25) is more complex. However, Eq. (25) can be implemented easily in software codes, as <math display="inline">\xi</math>  is a simple rational function, in which both the numerator and the denominator are polynomials. Here, only the error function <math display="inline">\xi</math>  is developed using symbolic regression, whereas this technique is more thoroughly evaluated also in Section 3.2.2. The error function <math display="inline">\xi</math>  is defined in Eq. (16)
@@ Line 361: / Line 361: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\left. \begin{matrix}\frac{1}{\sqrt{f}}\approx 0.86858896\cdot \left[ B-C+\frac{C}{x}+\varsigma \, \right] \, \\\begin{matrix}\xi =\frac{0.3896\cdot C\cdot \left( C-1\right) -0.9873}{0.8421\cdot {x}^{2}+0.01274\cdot x\cdot {C}^{4}+x+5.882}\\A\approx \frac{Re\cdot \epsilon }{8.0884}\\\begin{matrix}B\approx \mathrm{ln}\,\left( Re\right) -0.779397\\\begin{matrix}C=\mathrm{ln}\,\left( x\right) \\x=A+B\end{matrix}\end{matrix}\end{matrix}\end{matrix}\right\}</math>
+| <math display="inline">\left. \begin{matrix}\displaystyle\frac{1}{\sqrt{f}}\approx 0.86858896\cdot \left[ B-C+\displaystyle\frac{C}{x}+\xi \, \right] \, \\\xi =\displaystyle\frac{0.3896\cdot C\cdot \left( C-1\right) -0.9873}{0.8421\cdot {x}^{2}+0.01274\cdot x\cdot {C}^{4}+x+5.882}\\A\approx \displaystyle\frac{Re\cdot \epsilon }{8.0884}\\B\approx \mathrm{ln}\,\left( Re\right) -0.779397\\C=\mathrm{ln}\,\left( x\right) \\x=A+B\end{matrix}\right\}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (25)
 |}
 ====3.2.2. Approximations based on symbolic regression====
-Four simple but very accurate approximations based on symbolic regression are presented in Eqs. (26)-(29) where Eqs. (26) and (27) are already given in [21], whereas Eqs. (28) and (29) are novel (they are simple, but very accurate). Although Eureqa allows the only minimization of the absolute error [40], we can use it also for minimizing the relative error as Gholamy and Kreinovich [63] suggested (they showed how to use existing absolute-error-minimizing software to minimize relative errors). Consequently, Eureqa was used for the minimization of the maximal error (worst-case) of <math display="inline">\frac{\left| {y}_{sr}-y\right| }{y}</math>, where <math display="inline">{y}_{sr}</math> denotes a symbolic regression-based approximation of <math display="inline">y</math>. Eqs. (26)-(29) show that this formulation significantly reduces the approximation error
+Four simple but very accurate approximations based on symbolic regression are presented in Eqs. (26)-(29) where Eqs. (26) and (27) are already given in [21], whereas Eqs. (28) and (29) are novel (they are simple, but very accurate). Although Eureqa allows the only minimization of the absolute error [40], we can use it also for minimizing the relative error as Gholamy and Kreinovich [63] suggested (they showed how to use existing absolute-error-minimizing software to minimize relative errors). Consequently, Eureqa was used for the minimization of the maximal error (worst-case) of <math display="inline">\displaystyle \frac{\left| {y}_{sr}-y\right| }{y}</math>, where <math display="inline">{y}_{sr}</math> denotes a symbolic regression-based approximation of <math display="inline">y</math>. Eqs. (26)-(29) show that this formulation significantly reduces the approximation error
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 401: / Line 400: @@
 |}
+In Eqs. (26)-(28), <math display="inline">A\approx \displaystyle \frac{Re\cdot \epsilon }{8.0884}</math>, <math display="inline">B\approx \mathrm{ln}\,\left( Re\right) -0.7794</math>, <math display="inline">C=</math><math>\mathrm{ln}\,\left( x\right)</math>, and <math display="inline">x=A+B</math>. They are with the relative error of up to <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.0497%, 0.0105%, and 0.00337%, respectively (they require only 0.42 seconds to execute the benchmark study).
-In Eqs. (26)-(28), <math display="inline">A\approx \frac{Re\cdot \epsilon }{8.0884}</math>, <math display="inline">B\approx \mathrm{ln}\,\left( Re\right) -0.7794</math>, <math display="inline">C=</math><math>\mathrm{ln}\,\left( x\right)</math> , and <math display="inline">x=A+B</math>. They are with the relative error of up to <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.0497%, 0.0105%, and 0.00337%, respectively (they require only 0.42 seconds to execute the benchmark study).
 Comparing Eq. (26) which is already available in [21] with the here introduced Eq. (28), it can be seen that both approximations have similar complexity, but thanks to the minimizing of the relative error [63] through symbolic regression, the maximal relative error of the friction factor was reduced from 0.0497% of Eq. (26) to 0.00337% of Eq. (28), i.e. more than 14 times. Moreover, Eqs. (26)-(28) are based on Eqs. (17-20), in order to minimize the relative error of the friction factor of the Colebrook equation.
@@ Line 421: / Line 419: @@
-In Eq. (29), <math display="inline">A\approx \frac{Re\cdot \epsilon }{8.0897}</math>, <math display="inline">B\approx \mathrm{ln}\,\left( Re\right) -0.779626</math>, <math display="inline">C=</math><math>\mathrm{ln}\,\left( x\right)</math> , and <math display="inline">x=A+B</math>.
+In Eq. (29), <math display="inline">A\approx \displaystyle \frac{Re\cdot \epsilon }{8.0897}</math>, <math display="inline">B\approx \mathrm{ln}\,\left( Re\right) -0.779626</math>, <math display="inline">C=</math><math>\mathrm{ln}\,\left( x\right)</math>, and <math display="inline">x=A+B</math>.
-Algorithm for solving Eq. (29) is given in Figure 3.
+Algorithm for solving Eq. (29) is given in [[#img-3|Figure 3]] .
+<div id='img-3'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 440: / Line 438: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\left. \begin{matrix}\displaystyle\frac{1}{\sqrt{f}}\approx 0.868589\cdot \left[ B+Y-{\xi }_{1}\right] \, \\\begin{matrix}\begin{matrix}{\xi }_{1}=\displaystyle\frac{x\cdot {Y}^{2}+3.0636\cdot x\cdot Y+18.58}{19.5\cdot \left( {Y}^{2}\cdot {x}^{2}+{x}^{3}\right) +169.9\cdot {Y}^{2}+1260\cdot x+18178}\\Y=-C+\displaystyle\frac{C}{x-0.5564\cdot C+1.207}\end{matrix}\\A\approx \displaystyle\frac{Re\cdot \epsilon }{8.088387}\\\begin{matrix}B\approx \mathrm{ln}\,\left( Re\right) -0.7793975\\\begin{matrix}C=\mathrm{ln}\,\left( x\right) \\x=A+B\end{matrix}\end{matrix}\end{matrix}\end{matrix}\right\}</math>
+| <math display="inline">\left. \begin{matrix}\displaystyle\frac{1}{\sqrt{f}}\approx 0.868589\cdot \left[ B+Y-{\xi }_{1}\right] \, \\{\xi }_{1}=\displaystyle\frac{x\cdot {Y}^{2}+3.0636\cdot x\cdot Y+18.58}{19.5\cdot \left( {Y}^{2}\cdot {x}^{2}+{x}^{3}\right) +169.9\cdot {Y}^{2}+1260\cdot x+18178}\\Y=-C+\displaystyle\frac{C}{x-0.5564\cdot C+1.207}\\
+A\approx \displaystyle\frac{Re\cdot \epsilon }{8.088387}\\B\approx \mathrm{ln}\,\left( Re\right) -0.7793975\\C=\mathrm{ln}\,\left( x\right) \\x=A+B\end{matrix}\right\}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (30)
@@ Line 448: / Line 447: @@
 The computation speed of Eq. (30) is the same as Eq. (15), i.e. 1.1 seconds. The maximal relative error of Eq. (15) is 0.00527%, while the maximal relative error of Eq. 30 is only 0.000024%. Thus, the novel symbolic regression approximation of Eq. (30) reduces the relative error by factor 0.00527%/0.000024%≈219, which confirms the advantages of using symbolic regression for flow friction estimations.
-====3.3. Summary of accuracy and speed tests====
+===3.3. Summary of accuracy and speed tests===
-The presented approximations of the Colebrook equation which are presented in this article have been compared in Table 1 with respect of accuracy and speed of execution. The column ‘time in sec’ is given for execution of the Colebrook equation for 8 million samples generated using the Sobol’s Quasi Monte Carlo algorithm, as already given through our paper.
+The presented approximations of the Colebrook equation which are presented in this article have been compared in [[#tab-1|Table 1]] with respect of accuracy and speed of execution. The column ‘time in sec’ is given for execution of the Colebrook equation for 8 million samples generated using the Sobol’s Quasi Monte Carlo algorithm, as already given through our paper.
+Eqs. (8)-(12) and Eq. (21) are based on asymptotic expansions following algorithm from Figure 2. Eqs. (13)-(15) and Eqs. (22)-(24) are based on asymptotic expansions corrected with a carefully selected constant which minimizes the relative error, while Eq. (16) and Eq. (25) are with a carefully selected corrective function. Eqs. (17)-(19) and Eqs. (26)-(28) are based on symbolic regression which approximate accurately <math display="inline">\omega (x)-</math><math>x</math>; while Eq. (29) is Eq. (28), but with optimised coefficients. Eq. (20) are also based on symbolic regression, but with corrective function. Finally, Eq. (30) is based on accurate approximation of <math display="inline">\omega (x)-</math><math>x</math> found by symbolic regression and also corrected with the carefully selected function for minimization of the maximal relative error (this corrective function is also found using symbolic regression). We can see from Table 1 that it is really worth to construct approximations, as the exact solution of the Colebrook equation; Eq. 2, given by the built-in ''wrightOmega'' function of Matlab is extremely slow (11 394 seconds, i.e. 189.9 minutes(!)).
+<div id='tab-1'></div>
 <div class="center" style="font-size: 75%;">'''Table 1'''. Comparison of the presented approximations with related error and speed of execution</div>
-{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:100%;"
+{| {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:60%;"
 |-
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|<math display="inline">y\approx \omega (x)-</math><math>x</math>
+!  <math display="inline">y\approx \omega (x)-x</math> !!  Approximation !! Maximal relative error !! Time
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|Approximation
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Maximal relative error
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Time
 |-
-|  colspan='4'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Exact solution by built-in ''wrightOmega'' function of Matlab
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Exact solution by built-in ''wrightOmega'' function of Matlab
 |-
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (2)
+|  style="text-align: center;vertical-align: top;"|Eq. (2)
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
+|  style="vertical-align: top;"|
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0% i.e. exact
+|  style="text-align: center;vertical-align: top;"|0% i.e. exact
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|11394 seconds ~
+|  style="text-align: center;"|11394 seconds ~<br/>189.9 minutes
-.9 minutes
 |-
-|  colspan='4'  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|Asymptotic expansions
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Asymptotic expansions
 |-
 |  style="text-align: center;vertical-align: top;"|Eq. (8)
@@ Line 492: / Line 489: @@
 |  style="text-align: center;vertical-align: top;"|2.4 seconds
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (12)
+|  style="text-align: center;vertical-align: top;"|Eq. (12)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Figure 2
+|  style="text-align: center;vertical-align: top;"|Figure 2
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.00247%
+|  style="text-align: center;vertical-align: top;"|0.00247%
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|4.4 seconds
+|  style="text-align: center;vertical-align: top;"|4.4 seconds
 |-
-|  colspan='4'  style="text-align: center;vertical-align: top;"|Asymptotic expansions with corrective constant
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Asymptotic expansions with corrective constant
 |-
 |  style="text-align: center;vertical-align: top;"|Eq. (13)
@@ Line 509: / Line 506: @@
 |  style="text-align: center;vertical-align: top;"|0.4 seconds
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (15)
+|  style="text-align: center;vertical-align: top;"|Eq. (15)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (24)
+|  style="text-align: center;vertical-align: top;"|Eq. (24)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.00527%
+|  style="text-align: center;vertical-align: top;"|0.00527%
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1.1 seconds
+|  style="text-align: center;vertical-align: top;"|1.1 seconds
 |-
-|  colspan='4'  style="text-align: center;vertical-align: top;"|Asymptotic expansions with corrective function obtained using symbolic regression
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Asymptotic expansions with corrective function obtained using symbolic regression
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (16)
+|  style="text-align: center;vertical-align: top;"|Eq. (16)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (25)
+|  style="text-align: center;vertical-align: top;"|Eq. (25)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.000391%
+|  style="text-align: center;vertical-align: top;"|0.000391%
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1.1 seconds
+|  style="text-align: center;vertical-align: top;"|1.1 seconds
 |-
-|  colspan='4'  style="text-align: center;vertical-align: top;"|Symbolic regression
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Symbolic regression
 |-
 |  style="text-align: center;vertical-align: top;"|Eq. (17)
@@ Line 533: / Line 530: @@
 |  style="text-align: center;vertical-align: top;"|0.42 seconds
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (19)
+|  style="text-align: center;vertical-align: top;"|Eq. (19)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (28)
+|  style="text-align: center;vertical-align: top;"|Eq. (28)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.00337%
+|  style="text-align: center;vertical-align: top;"|0.00337%
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.42 seconds
+|  style="text-align: center;vertical-align: top;"|0.42 seconds
 |-
-|  colspan='4'  style="text-align: center;vertical-align: top;"|Symbolic regression with the optimized coefficients
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Symbolic regression with the optimized coefficients
 |-
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|Eq. (20); optimized Eq. (19)
+|  style="vertical-align: top;text-align: center;"|Eq. (20);
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (29) – Figure 3
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.0012%
+optimized Eq. (19)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.388 seconds
+|  style="text-align: center;vertical-align: top;"|Eq. (29) – Figure 3
+|  style="text-align: center;vertical-align: top;"|0.0012%
+|  style="text-align: center;vertical-align: top;"|0.388 seconds
 |-
-|  colspan='4'  style="text-align: center;vertical-align: top;"|Symbolic regression with the optimized coefficients and corrective function
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Symbolic regression with the optimized coefficients and corrective function
 |-
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|Eq. (20)
+|  style="vertical-align: top;text-align: center;"|Eq. (20)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (30)
+|  style="text-align: center;vertical-align: top;"|Eq. (30)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.000024%
+|  style="text-align: center;vertical-align: top;"|0.000024%
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1.1 seconds
+|  style="text-align: center;vertical-align: top;"|1.1 seconds
 |}
-Eqs. (8-12) and Eq. (21) are based on asymptotic expansions following algorithm from Figure 2. Eqs. (13-15) and Eqs. (22-24) are based on asymptotic expansions corrected with a carefully selected constant which minimizes the relative error, while Eq. (16) and Eq. (25) are with a carefully selected corrective function. Eqs. (17-19) and Eqs. (26-28) are based on symbolic regression which approximate accurately <math display="inline">\omega (x)-</math><math>x</math>; while Eq. (29) is Eq. (28), but with optimised coefficients. Eq. (20) are also based on symbolic regression, but with corrective function. Finally, Eq. (30) is based on accurate approximation of <math display="inline">\omega (x)-</math><math>x</math> found by symbolic regression and also corrected with the carefully selected function for minimization of the maximal relative error (this corrective function is also found using symbolic regression). We can see from Table 1 that it is really worth to construct approximations, as the exact solution of the Colebrook equation; Eq. 2, given by the built-in ''wrightOmega'' function of Matlab is extremely slow (11 394 seconds, i.e.189.9 minutes(!),)
 ==4. Conclusions and future work==
@@ Line 562: / Line 558: @@
 The proposed approximations should be compared with very recent comparative studies of the available approximations of the Colebrook equation [23-26]. Such comparisons will confirm that the approximations proposed in this review are both accurate and computationally efficient.
-<span id='_Hlk39321495'></span>In terms of the balance among simplicity for human eyes, accuracy and computational simplicity, the novel Eq. (29) with a relative error of up to 0.0012% is highly recommended (a simple algorithm shown in Figure 3 should be followed for engineering use). For a very accurate software implementation, its more complex variant given by Eq. (30) should be followed, as the maximal relative error is only 0.000024%.
+In terms of the balance among simplicity for human eyes, accuracy and computational simplicity, the novel Eq. (29) with a relative error of up to 0.0012% is highly recommended (a simple algorithm shown in Figure 3 should be followed for engineering use). For a very accurate software implementation, its more complex variant given by Eq. (30) should be followed, as the maximal relative error is only 0.000024%.
-'''Contributors:''' Both authors contributed equally to this study. Pavel Praks worked mostly on the mathematical background of the proposed methods and procedures, while Dejan Brkić put them in a form suitable for engineering use. Pavel Praks was responsible for experiments with symbolic regression tools and global optimization. Dejan Brkić made the draft of this paper.
+Similar approach has been recently developed for flow friction caused by air flow through fuel cells [73].
-'''Funding:''' This work has been partially supported by the Technology Agency of the Czech Republic under the project “National Centre for Energy” TN01000007, “Energy System for Grids” TK02030039 and by the Ministry of Education, Youth and Sports of the Czech Republic through the National Programme of Sustainability (NPS II) project “IT4Innovations excellence in science-LQ1602”.
-'''Conflicts of Interest:''' The authors declare no conflict of interest. Neither the European Commission, the VŠB - Technical University of Ostrava, nor any person acting on their behalf is responsible for any use of this publication. Both authors have approved the final version of this paper.
-==Acknowledgments==
-We especially thank Marcelo Masera for his careful scientific supervision and kind approvals. We acknowledge support from the European Commission, Joint Research Centre (JRC), Directorate C: Energy, Transport, and Climate, Unit C3: Energy Security, Distribution, and Markets.
 ==Notations==
@@ Line 580: / Line 568: @@
 {| style="width: 100%;border-collapse: collapse;"
 |-
-| <math>A</math>
+| <math>A</math>: variable that depends on <math display="inline">Re</math> and <math display="inline">\epsilon</math>  (dimensionless); <math>A=\frac{Re}{z}\cdot \frac{\epsilon }{3.71}</math>
-| <span style="text-align: center; font-size: 75%;">variable that depends on </span> <math display="inline">Re</math><span style="text-align: center; font-size: 75%;"> and </span> <math display="inline">\epsilon</math> <span style="text-align: center; font-size: 75%;"> (dimensionless); </span> <math display="inline">A=</math><math>\frac{Re}{z}\cdot \frac{\epsilon }{3.71}</math>
 |-
-| <math>B</math>
+| <math>B</math>: variable that depends on <math display="inline">Re</math> and <math display="inline">z</math> (dimensionless); <math display="inline">B=</math><math>\mathrm{ln}\,(Re)-ln(z)\approx \mathrm{ln}\,\left( Re\right) -0.7794</math>
-| <span style="text-align: center; font-size: 75%;">variable that depends on </span> <math display="inline">Re</math><span style="text-align: center; font-size: 75%;"> and </span> <math display="inline">z</math><span style="text-align: center; font-size: 75%;"> (dimensionless); </span> <math display="inline">B=</math><math>\mathrm{ln}\,(Re)-ln(z)\approx \mathrm{ln}\,\left( Re\right) -0.7794</math>
 |-
-| <math>C</math>
+| <math>C</math>: variable that depends on <math display="inline">x</math> (dimensionless); <math display="inline">C=</math><math>\mathrm{ln}\,\left( x\right)</math>
-| <span style="text-align: center; font-size: 75%;">variable that depends on </span> <math display="inline">x</math><span style="text-align: center; font-size: 75%;"> (dimensionless); </span> <math display="inline">C=</math><math>\mathrm{ln}\,\left( x\right)</math>
 |-
-| <math>f</math>
+| <math>f</math>: Darcy (Moody) flow friction factor (dimensionless) – main output parameter
-| <span style="text-align: center; font-size: 75%;">Darcy (Moody) flow friction factor (dimensionless) – main output parameter</span>
 |-
-| <math>s</math>
+| <math>s</math>: approximation of <math display="inline">y</math> by a truncated series about infinity; <math display="inline">{y}_{i}=</math><math>\sum _{i=1}^{+\infty }{s}_{i}</math>
-| <span style="text-align: center; font-size: 75%;">approximation of </span> <math display="inline">y</math><span style="text-align: center; font-size: 75%;"> by a truncated series about infinity; </span> <math display="inline">{y}_{i}=</math><math>\sum _{i=1}^{+\infty }{s}_{i}</math>
 |-
-| <math>Re</math>
+| <math>Re</math>: Reynolds number (dimensionless) – main input parameter
-| <span style="text-align: center; font-size: 75%;">Reynolds number (dimensionless) – main input parameter</span>
 |-
-| <math>x</math>
+| <math>x</math>: variable that depends on <math display="inline">A</math> and <math display="inline">B</math> (dimensionless); <math display="inline">x=A+</math><math>B</math>
-| <span style="text-align: center; font-size: 75%;">variable that depends on </span> <math display="inline">A</math><span style="text-align: center; font-size: 75%;"> and </span> <math display="inline">B</math><span style="text-align: center; font-size: 75%;"> (dimensionless); </span> <math display="inline">x=A+</math><math>B</math>
 |-
-| <math>y</math>
+| <math>y</math>: approximation of <math display="inline">\omega (x)-</math><math>x</math>, i.e. of <math display="inline">W\left( {e}^{x}\right) -</math><math>x</math>
-| <span style="text-align: center; font-size: 75%;">approximation of </span> <math display="inline">\omega (x)-</math><math>x</math><span style="text-align: center; font-size: 75%;">, i.e. of </span> <math display="inline">W\left( {e}^{x}\right) -</math><math>x</math>
 |-
-| <math>{y}_{sr}</math>
+| <math>{y}_{sr}</math>: approximation of <math display="inline">\omega (x)-</math><math>x</math>, i.e. of <math display="inline">W\left( {e}^{x}\right) -</math><math>x</math> found by symbolic regression
-| <span style="text-align: center; font-size: 75%;">approximation of </span> <math display="inline">\omega (x)-</math><math>x</math><span style="text-align: center; font-size: 75%;">, i.e. of </span> <math display="inline">W\left( {e}^{x}\right) -</math><math>x</math><span style="text-align: center; font-size: 75%;"> found by symbolic regression</span>
 |-
-| <math>Y</math>
+| <math>Y</math>: approximation of <math display="inline">\omega (x)-</math><math>x</math> from Eq. (19) and used in Eq. (20)
-| <span style="text-align: center; font-size: 75%;">approximation of </span> <math display="inline">\omega (x)-</math><math>x</math><span style="text-align: center; font-size: 75%;"> from Eq. (19) and used in Eq. (20)</span>
 |-
-| <math>z</math>
+| <math>z</math>: constant (dimensionless); <math>z=\frac{2\cdot 2.51}{\mathrm{ln}\,\left( 10\right) }</math>
-| <span style="text-align: center; font-size: 75%;">constant (dimensionless); </span> <math display="inline">z=</math><math>\frac{2\cdot 2.51}{\mathrm{ln}\,\left( 10\right) }</math>
 |-
-| <math>\alpha</math>
+| <math>\alpha</math>: constant (dimensionless); half of the smallest term in the asymptotic series
-| <span style="text-align: center; font-size: 75%;">constant (dimensionless); half of the smallest term in the asymptotic series</span>
 |-
-| <math>\epsilon</math>
+| <math>\epsilon</math>: relative roughness of inner pipe surface (dimensionless) – main input parameter
-| <span style="text-align: center; font-size: 75%;">Relative roughness of inner pipe surface (dimensionless) – main input parameter</span>
 |-
-| <math>{\delta }_{\%}</math>
+| <math>{\delta }_{\%}</math>: relative error (%); <math display="inline">{\delta }_{\%}=\displaystyle \frac{f-{f}_{i}}{f}\cdot 100\%</math>
-| <span style="text-align: center; font-size: 75%;">Relative error (%); </span> <math display="inline">{\delta }_{\%}=</math><math>\frac{f-{f}_{i}}{f}\cdot 100\%</math>
 |-
-| <math display="inline">\xi</math> <span style="text-align: center; font-size: 75%;"> and </span> <math display="inline">{\xi }_{1}</math>
+| <math display="inline">\xi</math>  and <math display="inline">{\xi }_{1}</math>: error-functions developed using symbolic regression
-| <span style="text-align: center; font-size: 75%;">error-functions developed using symbolic regression</span>
 |-
-| <math>e</math>
+| <math>e</math>: exponential function
-| <span style="text-align: center; font-size: 75%;">exponential function</span>
 |-
-| <math>{log}_{10}</math>
+| <math>{log}_{10}</math>: logarithm with base 10
-| <span style="text-align: center; font-size: 75%;">logarithm with base 10</span>
 |-
-| <math>ln</math>
+| <math>ln</math>: natural logarithm
-| <span style="text-align: center; font-size: 75%;">natural logarithm</span>
 |-
-| <math>W</math>
+| <math>W</math>: Lambert <math display="inline">W</math>-function
-| <span style="text-align: center; font-size: 75%;">Lambert </span> <math display="inline">W</math><span style="text-align: center; font-size: 75%;">-function</span>
 |-
-| <math>\omega</math>
+| <math>\omega</math>: wright <math display="inline">\omega</math> -function
-| <span style="text-align: center; font-size: 75%;">Wright </span> <math display="inline">\omega</math> <span style="text-align: center; font-size: 75%;">-function</span>
 |-
-| <math>i</math>
+| <math>i</math>: counter
-| <span style="text-align: center; font-size: 75%;">Counter</span>
 |-
-| <span style="text-align: center; font-size: 75%;">''wrightOmega''</span>
+| ''wrightOmega'': Matlab’s built-in function
-| <span style="text-align: center; font-size: 75%;">Matlab’s built-in function</span>
 |-
-| <span style="text-align: center; font-size: 75%;">''Globalsearch''</span>
+| ''Globalsearch'': Matlab’s built-in solver
-| <span style="text-align: center; font-size: 75%;">Matlab’s built-in solver</span>
 |}
+<!--'''Contributors:''' Both authors contributed equally to this study. Pavel Praks worked mostly on the mathematical background of the proposed methods and procedures, while Dejan Brkić put them in a form suitable for engineering use. Pavel Praks was responsible for experiments with symbolic regression tools and global optimization. Dejan Brkić made the draft of this paper.-->
+==Funding==
+The work of Pavel Praks and Dejan Brkić has been supported by the Technology Agency of the Czech Republic through the project CEET - "Center of Energy and Environmental Technologies" TK03020027. Additionaly, the work of Dejan Brkić has been supported by the Ministry of Education, Youth and Sports of the Czech Republic through the National Programme of Sustainability (NPS II) project “IT4Innovations excellence in science-LQ1602”.
+==Conflicts of interest==
+The authors declare no conflict of interest. Neither the European Commission, the VŠB - Technical University of Ostrava, nor any person acting on their behalf is responsible for any use of this publication. Both authors have approved the final version of this paper.
+==Acknowledgments==
+We especially thank Marcelo Masera for his careful scientific supervision and kind approvals. We acknowledge support from the European Commission, Joint Research Centre (JRC), Directorate C: Energy, Transport, and Climate, Unit C3: Energy Security, Distribution, and Markets. This review paper has been registered in the internal system for publications of the European Commission, Joint Research Centre (JRC) under PubSy JRC120798.
+==Supplementary materials==
+The readers who want to transpose the presented methods into software codes would like to have a list of all required formulas in one place and in a condensed manner. For that purpose, all used equations are listed in this section, here: [[Draft_Brkic_569749396|List of Equations: https://www.scipedia.com/public/Draft_Brkic_569749396]]
 ==References==
-<span id='_Hlk32574718'></span>
+<div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
-:<span style="text-align: center; font-size: 75%;">[[#cite-_Hlk32574718|[1].]] Colebrook C.F. Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws. J. Inst. Civ. Eng., 11:133–156, 1939. [https://doi.org/10.1680/ijoti.1939.13150 doi:10.1680/ijoti.1939.13150]</span>
+[1] Colebrook C.F. Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws. J. Inst. Civ. Eng., 11:133–156, 1939. [https://doi.org/10.1680/ijoti.1939.13150 doi:10.1680/ijoti.1939.13150]
+[2] Colebrook C., White C. Experiments with fluid friction in roughened pipes. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci., 161:367–381, 1937. [http://dx.doi.org/10.1098/rspa.1937.0150 doi:10.1098/rspa.1937.0150]
-:<span style="text-align: center; font-size: 75%;">[2]. Colebrook C., White C. Experiments with fluid friction in roughened pipes. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci., 161:367–381, 1937. [http://dx.doi.org/10.1098/rspa.1937.0150 doi:10.1098/rspa.1937.0150]</span>
+[3] Keady G. Colebrook-White formula for pipe flows. J. Hydraul. Eng., 124:96–97, 1998. [https://doi.org/10.1061/(ASCE)0733-9429(1998)124:1(96) doi:10.1061/(ASCE)0733-9429(1998)124:1(96)]
-:<span style="text-align: center; font-size: 75%;">[3]. Keady G. Colebrook-White formula for pipe flows. J. Hydraul. Eng., 124:96–97, 1998. [https://doi.org/10.1061/(ASCE)0733-9429(1998)124:1(96) doi:10.1061/(ASCE)0733-9429(1998)124:1(96)]</span>
+[4] Sonnad J.R., Goudar C.T. Constraints for using Lambert W function-based explicit Colebrook–White equation. J. Hydraul. Eng., 130:929-931, 2004. [https://doi.org/10.1061/(ASCE)0733-9429(2004)130:9(929) doi:10.1061/(ASCE)0733-9429(2004)130:9(929)]
-:<span style="text-align: center; font-size: 75%;">[4]. Sonnad J.R., Goudar C.T. Constraints for using Lambert W function-based explicit Colebrook–White equation. J. Hydraul. Eng., 130:929-931, 2004. [https://doi.org/10.1061/(ASCE)0733-9429(2004)130:9(929) doi:10.1061/(ASCE)0733-9429(2004)130:9(929)]</span>
+[5] Brkić D. Comparison of the Lambert W‐function based solutions to the Colebrook equation. Eng. Comput., 29:617-630, 2012. [https://doi.org/10.1108/02644401211246337 doi:10.1108/02644401211246337]
-:<span style="text-align: center; font-size: 75%;">[5]. Brkić D. Comparison of the Lambert W‐function based solutions to the Colebrook equation. Eng. Comput., 29:617-630, 2012. [https://doi.org/10.1108/02644401211246337 doi:10.1108/02644401211246337]</span>
+[6] Corless R.M., Gonnet G.H., Hare D.E., Jeffrey D.J., Knuth D.E. On the Lambert W function. Adv. Comput. Math., 5:329–359, 1996. [https://doi.org/10.1007/BF02124750 doi:10.1007/BF02124750]
-:<span style="text-align: center; font-size: 75%;">[6]. Corless R.M., Gonnet G.H., Hare D.E., Jeffrey D.J., Knuth D.E. On the Lambert W function. Adv. Comput. Math., 5:329–359, 1996. [https://doi.org/10.1007/BF02124750 doi:10.1007/BF02124750]</span>
+[7] Wright E.M. Solution of the equation  <math display="inline">z\cdot {e}^{z}=</math><math>a</math>. Bull. Am. Math Soc., 65:89–93, 1959. [https://doi.org/10.1090/S0002-9904-1959-10290-1 doi:10.1090/S0002-9904-1959-10290-1]
-:<span style="text-align: center; font-size: 75%;">[7]. Wright E.M. Solution of the equation </span> <math display="inline">z\cdot {e}^{z}=</math><math>a</math><span style="text-align: center; font-size: 75%;">. Bull. Am. Math Soc., 65:89–93, 1959. [https://doi.org/10.1090/S0002-9904-1959-10290-1 doi:10.1090/S0002-9904-1959-10290-1]</span>
+[8] Praks P., Brkić D. Symbolic regression-based genetic approximations of the Colebrook equation for flow friction. Water, 10:1175, 2018. [https://doi.org/10.3390/w10091175 doi:10.3390/w10091175]
-:<span style="text-align: center; font-size: 75%;">[8]. Praks P., Brkić D. Symbolic regression-based genetic approximations of the Colebrook equation for flow friction. Water, 10:1175, 2018. [https://doi.org/10.3390/w10091175 doi:10.3390/w10091175]</span>
+[9] Niazkar M., Talebbeydokhti N., Afzali S.-H. Bridge backwater estimation: A comparison between artificial intelligence models and explicit equations. Sci. Iran., 2020. [https://doi.org/10.24200/SCI.2020.51432.2175 doi:10.24200/SCI.2020.51432.2175]
-:<span style="text-align: center; font-size: 75%;">[9]. Niazkar M., Talebbeydokhti N., Afzali S.-H. Bridge backwater estimation: A comparison between artificial intelligence models and explicit equations. Sci. Iran., [https://doi.org/10.24200/SCI.2020.51432.2175 doi:10.24200/SCI.2020.51432.2175]</span>
+[10] Filippov A.E., Gorb S.N. Methods of the pattern formation in numerical modeling of biological problems. Facta Universitatis, Series: Mech. Eng., 17:217-242, 2019. [https://doi.org/10.22190/FUME190227027F doi:10.22190/FUME190227027F]
-<span id='_Hlk32857029'></span>
+[11] Gorkemli B., Karaboga D. A quick semantic artificial bee colony programming (qsABCP) for symbolic regression. Inf. Sci. (N.Y.), 502:346–362, 2019. [https://doi.org/10.1016/j.ins.2019.06.052 doi:10.1016/j.ins.2019.06.052]
-:<span style="text-align: center; font-size: 75%;">[[#cite-_Hlk32857029|[10].]] Filippov A.E., Gorb S.N. Methods of the pattern formation in numerical modeling of biological problems. Facta Universitatis, Series: Mech. Eng., 17:217-242, 2019. [https://doi.org/10.22190/FUME190227027F doi:10.22190/FUME190227027F]</span>
-:<span style="text-align: center; font-size: 75%;">[11]. Gorkemli B., Karaboga D. A quick semantic artificial bee colony programming (qsABCP) for symbolic regression. Inf. Sci. (N.Y.), 502:346–362, 2019. [https://doi.org/10.1016/j.ins.2019.06.052 doi:10.1016/j.ins.2019.06.052]</span>
+[12] Niazkar M. Revisiting the estimation of Colebrook friction factor: A comparison between artificial intelligence models and CW based explicit equations. KSCE J. Civ. Eng., 23:4311–4326, 2019. [https://doi.org/10.1007/s12205-019-2217-1 doi:10.1007/s12205-019-2217-1]
-:<span style="text-align: center; font-size: 75%;">[12]. Niazkar M. Revisiting the estimation of Colebrook friction factor: A comparison between artificial intelligence models and CW based explicit equations. KSCE J. Civ. Eng., 23:4311–4326, 2019. [https://doi.org/10.1007/s12205-019-2217-1 doi:10.1007/s12205-019-2217-1]</span>
+[13] Diyaley S., Chakraborty S. Optimization of multi-pass face milling parameters using metaheuristic algorithms. Facta Universitatis, Series: Mech. Eng., 17:365-383, 2019. [https://doi.org/10.22190/FUME190605043D doi:10.22190/FUME190605043D]
-:<span style="text-align: center; font-size: 75%;">[13]. Diyaley S., Chakraborty S. Optimization of multi-pass face milling parameters using metaheuristic algorithms. Facta Universitatis, Series: Mech. Eng., 17:365-383, 2019. [https://doi.org/10.22190/FUME190605043D doi:10.22190/FUME190605043D]</span>
+[14] Brkić D., Ćojbašić Ž. Evolutionary optimization of Colebrook’s turbulent flow friction approximations. Fluids, 2:15, 2017. [https://doi.org/10.3390/fluids2020015 doi:10.3390/fluids2020015]
-:<span style="text-align: center; font-size: 75%;">[14]. Brkić D., Ćojbašić Ž. Evolutionary optimization of Colebrook’s turbulent flow friction approximations. Fluids, 2:15, 2017. [https://doi.org/10.3390/fluids2020015 doi:10.3390/fluids2020015]</span>
+[15] Brkić D., Ćojbašić Ž. Intelligent flow friction estimation. Comput. Intel. Neurosc., 2016:5242596, 2016. [https://doi.org/10.1155/2016/5242596 doi:10.1155/2016/5242596]
-:<span style="text-align: center; font-size: 75%;">[15]. Brkić D., Ćojbašić Ž. Intelligent flow friction estimation. Comput. Intel. Neurosc., 2016:5242596, 2016. [https://doi.org/10.1155/2016/5242596 doi:10.1155/2016/5242596]</span>
+[16] Ćojbašić Ž., Brkić D. Very accurate explicit approximations for calculation of the Colebrook friction factor. Int. J. Mech. Sci., 67:10-13, 2013. [https://doi.org/10.1016/j.ijmecsci.2012.11.017 doi:10.1016/j.ijmecsci.2012.11.017]
-:<span style="text-align: center; font-size: 75%;">[16]. Ćojbašić Ž., Brkić D. Very accurate explicit approximations for calculation of the Colebrook friction factor. Int. J. Mech. Sci., 67:10-13, 2013. [https://doi.org/10.1016/j.ijmecsci.2012.11.017 doi:10.1016/j.ijmecsci.2012.11.017] </span>
+[17] Belkić Dž. All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells. J. Math. Chem., 57:59–106, 2019. [https://doi.org/10.1007/s10910-018-0985-3 doi:10.1007/s10910-018-0985-3]
-:<span style="text-align: center; font-size: 75%;">[17]. Belkić Dž. All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells. J. Math. Chem., 57:59–106, 2019. [https://doi.org/10.1007/s10910-018-0985-3 doi:10.1007/s10910-018-0985-3]</span>
+[18] Brkić, D. Lambert ''W'' function in hydraulic problems. Math. Balk. New Ser., 26:285–292, 2012. [http://www.math.bas.bg/infres/MathBalk/MB-26/MB-26-285-292.pdf http://www.math.bas.bg/infres/MathBalk/MB-26/MB-26-285-292.pdf]
-:<span style="text-align: center; font-size: 75%;">[18]. Brkić, D. Lambert W function in hydraulic problems. Math. Balk. New Ser., 26:285–292, 2012, [http://www.math.bas.bg/infres/MathBalk/MB-26/MB-26-285-292.pdf http://www.math.bas.bg/infres/MathBalk/MB-26/MB-26-285-292.pdf]</span>
+[19] Brkić D. New explicit correlations for turbulent flow friction factor. Nucl. Eng. Des., 241:4055-4059, 2011. [https://doi.org/10.1016/j.nucengdes.2011.07.042 doi:10.1016/j.nucengdes.2011.07.042]
-:<span style="text-align: center; font-size: 75%;">[19]. Brkić D. New explicit correlations for turbulent flow friction factor. Nucl. Eng. Des., 241:4055-4059, 2011, [https://doi.org/10.1016/j.nucengdes.2011.07.042 doi:10.1016/j.nucengdes.2011.07.042]</span>
+[20] Brkić, D. W solutions of the CW equation for flow friction. Appl. Math. Lett., 24:1379-1383, 2011. [https://doi.org/10.1016/j.aml.2011.03.014 doi:10.1016/j.aml.2011.03.014]
-:<span style="text-align: center; font-size: 75%;">[20]. Brkić, D. W solutions of the CW equation for flow friction. Appl. Math. Lett., 24:1379-1383, 2011, [https://doi.org/10.1016/j.aml.2011.03.014 doi:10.1016/j.aml.2011.03.014]</span>
+[21] Brkić D., Praks P. Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright <math>\omega</math>-function. Mathematics, 7:34, 2019. [https://doi.org/10.3390/math7010034 doi:10.3390/math7010034]
-:<span style="text-align: center; font-size: 75%;">[21]. Brkić D., Praks P. Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function. Mathematics, 7:34, 2019. [https://doi.org/10.3390/math7010034 doi:10.3390/math7010034]</span>
+[22] Brkić D., Praks P. Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function: Reply to Discussion. Mathematics, 7:410, 2019. [https://doi.org/10.3390/math7050410 doi:10.3390/math7050410]
-:<span style="text-align: center; font-size: 75%;">[22]. Brkić D., Praks P. Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function: Reply to Discussion. Mathematics, 7:410, 2019. [https://doi.org/10.3390/math7050410 doi:10.3390/math7050410]</span>
+[23] Zeyu Z., Junrui C., Zhanbin L., Zengguang X., Peng L. Approximations of the Darcy–Weisbach friction factor in a vertical pipe with full flow regime. Water Sci. Technol.: Water Supply, 20:1321–1333, 2020. [https://doi.org/10.2166/ws.2020.048 doi:10.2166/ws.2020.048]
-:<span style="text-align: center; font-size: 75%;">[23]. Zeyu Z., Junrui C., Zhanbin L., Zengguang X., Peng L. Approximations of the Darcy–Weisbach friction factor in a vertical pipe with full flow regime. Water Sci. Technol.: Water Supply, 20, 2020. [https://doi.org/10.2166/ws.2020.048 doi:10.2166/ws.2020.048]</span>
+[24] Muzzo L.E., Pinho D., Lima L.E., Ribeiro L.F. Accuracy/Speed analysis of pipe friction factor correlations. In International Congress on Engineering and Sustainability in the XXI Century—INCREaSE 2019, Section: Water for Ecosystem and Society, Faro, Portugal, 9–11 October 2019; Monteiro, J., Silva, A.J., Mortal, A., Aníbal, J., da Silva, M.M., Oliveira, M. (Eds.); Springer Nature Switzerland AG 2020: Cham, Switzerland, 2019, pp. 664–679, [https://doi.org/10.1007/978-3-030-30938-1_51 doi:10.1007/978-3-030-30938-1_51]
-:<span style="text-align: center; font-size: 75%;">[24]. Muzzo L.E., Pinho D., Lima L.E., Ribeiro L.F. Accuracy/Speed analysis of pipe friction factor correlations. In International Congress on Engineering and Sustainability in the XXI Century—INCREaSE 2019, Section: Water for Ecosystem and Society, Faro, Portugal, 9–11 October 2019; Monteiro, J., Silva, A.J., Mortal, A., Aníbal, J., da Silva, M.M., Oliveira, M., Eds.; Springer Nature Switzerland AG 2020: Cham, Switzerland, 2019; pp. 664–679, [https://doi.org/10.1007/978-3-030-30938-1_51 doi:10.1007/978-3-030-30938-1_51]</span>
+[25] Brkić D. Review of explicit approximations to the Colebrook relation for flow friction. J. Pet. Sci. Eng., 77:34–48, 2011. [https://doi.org/10.1016/j.petrol.2011.02.006 doi:10.1016/j.petrol.2011.02.006]
-:<span style="text-align: center; font-size: 75%;">[25]. Brkić D. Review of explicit approximations to the Colebrook relation for flow friction. J. Pet. Sci. Eng., 77:34–48, 2011. [https://doi.org/10.1016/j.petrol.2011.02.006 doi:10.1016/j.petrol.2011.02.006]</span>
+[26] Brkić D. Determining friction factors in turbulent pipe flow. Chem. Eng. N.Y. 119:34-39, 2012.
-:<span style="text-align: center; font-size: 75%;">[26]. Brkić D. Determining friction factors in turbulent pipe flow. Chem. Eng. N.Y. 119:34-39, 2012.</span>
+[27] Anaya-Durand A.I., Cauich-Segovia G.I., Funabazama-Bárcenas O., Gracia-Medrano-Bravo V.A., Evaluación de ecuaciones de factor de fricción explícito para tuberías (Evaluation of explicit friction factor equations for pipes) /in Spanish/. Educ. Quím., 25:128-134, 2014. [https://doi.org/10.1016/S0187-893X(14)70535-X doi:10.1016/S0187-893X(14)70535-X]
-:<span style="text-align: center; font-size: 75%;">[27]. Anaya-Durand A.I., Cauich-Segovia G.I., Funabazama-Bárcenas O., Gracia-Medrano-Bravo V.A., Evaluación de ecuaciones de factor de fricción explícito para tuberías (Evaluation of explicit friction factor equations for pipes) /in Spanish/. Educ. Quím., 25:128-134, 2014. [https://doi.org/10.1016/S0187-893X(14)70535-X doi:10.1016/S0187-893X(14)70535-X]</span>
+[28] Silva A.L.M.F., Mesquita A.L.A., Felipe A.M.P.F., Souza, J.A.D.S. Simulação do fator de atrito para o escoamento confinado de caulim com diferentes teores de sólidos pelo modelo de SWAMEE–JAIN (Simulation of the friction factor for the confined flow of kaolin with different solid contents using the SWAMEE – JAIN model) /in Portuguese/. Rev. Mater., 22, 2017. [https://doi.org/10.1590/s1517-707620170002.0146 doi:10.1590/s1517-707620170002.0146]
-:<span style="text-align: center; font-size: 75%;">[28]. Silva A.L.M.F., Mesquita A.L.A., Felipe A.M.P.F., Souza, J.A.D.S. Simulação do fator de atrito para o escoamento confinado de caulim com diferentes teores de sólidos pelo modelo de SWAMEE–JAIN (Simulation of the friction factor for the confined flow of kaolin with different solid contents using the SWAMEE – JAIN model) /in Portuguese/. Rev. Mater., 22, 2017. [https://doi.org/10.1590/s1517-707620170002.0146 doi:10.1590/s1517-707620170002.0146]</span>
+[29] Pimenta B.D., Robaina A.D., Peiter M.X., Mezzomo W., Kirchner J.H., Ben L.H. Performance of explicit approximations of the coefficient of head loss for pressurized conduits. Rev. Bras. Eng. Agríc. Ambient., 22:301-307, 2018. [https://doi.org/10.1590/1807-1929/agriambi.v22n5p301-307 doi:10.1590/1807-1929/agriambi.v22n5p301-307]
-:<span style="text-align: center; font-size: 75%;">[29]. Pimenta B.D., Robaina A.D., Peiter M.X., Mezzomo W., Kirchner J.H., Ben L.H. Performance of explicit approximations of the coefficient of head loss for pressurized conduits. Rev. Bras. Eng. Agríc. Ambient., 22:301-307, 2018. [https://doi.org/10.1590/1807-1929/agriambi.v22n5p301-307 doi:10.1590/1807-1929/agriambi.v22n5p301-307]</span>
+[30] Santos-Ruiz I., Bermúdez J.R., López-Estrada F.R., Puig V., Torres, L. Estimación experimental de la rugosidad y del factor de fricción en una tubería /in Spanish/. In Proceedings of the Memorias del Congreso Nacional de Control Automático, San Luis Potosí, Mexico, 10–12 October 2018. Available online: [http://amca.mx/RevistaDigital/cnca2018/articulos/VieAT3-03.pdf http://amca.mx/RevistaDigital/cnca2018/articulos/VieAT3-03.pdf]
-:<span style="text-align: center; font-size: 75%;">[30]. Santos-Ruiz I., Bermúdez J.R., López-Estrada F.R., Puig V., Torres, L. Estimación experimental de la rugosidad y del factor de fricción en una tubería /in Spanish/. In Proceedings of the Memorias del Congreso Nacional de Control Automático, San Luis Potosí, Mexico, 10–12 October 2018; Available online: [http://amca.mx/RevistaDigital/cnca2018/articulos/VieAT3-03.pdf http://amca.mx/RevistaDigital/cnca2018/articulos/VieAT3-03.pdf]</span>
+[31] Olivares A., Guerra R., Alfaro M., Notte-Cuello E., Puentes L. Evaluación experimental de correlaciones para el cálculo del factor de fricción para flujo turbulento en tuberías cilíndricas. Rev. Int. Métodos Numér. Cálc. Diseño Ing., 35:15, 2019. [https://doi.org/10.23967/j.rimni.2019.01.001 doi:10.23967/j.rimni.2019.01.001]
-:<span style="text-align: center; font-size: 75%;">[31]. Olivares A., Guerra R., Alfaro M., Notte-Cuello E., Puentes L. Evaluación experimental de correlaciones para el cálculo del factor de fricción para flujo turbulento en tuberías cilíndricas (Experimental evaluation of correlations used to calculate friction factor for turbulent flow in cylindrical pipes) /in Spanish/. Rev. Int. Métodos Numér. Cálc. Diseño Ing., 35:15, 2019. [https://doi.org/10.23967/j.rimni.2019.01.001 doi:10.23967/j.rimni.2019.01.001]</span>
+[32] Alfaro-Guerra M., Guerra-Rojas R., Olivares-Gallardo A. Evaluación experimental de la solución analítica exacta de la ecuación de Colebrook-White (Experimental evaluation of exact analytical solution of the Colebrook-White Equation) (in Spanish). Ingeniería Investigación y Tecnología, 20:1–11. 2019. [http://dx.doi.org/10.22201/fi.25940732e.2019.20n2.021 doi:10.22201/fi.25940732e.2019.20n2.021]
-:<span style="text-align: center; font-size: 75%;">[32]. Alfaro-Guerra M., Guerra-Rojas R., Olivares-Gallardo A. Evaluación experimental de la solución analítica exacta de la ecuación de Colebrook-White (Experimental evaluation of exact analytical solution of the Colebrook-White Equation) /in Spanish/. Ingeniería Investigación y Tecnología, 20:1–11. 2019. [http://dx.doi.org/10.22201/fi.25940732e.2019.20n2.021 doi:10.22201/fi.25940732e.2019.20n2.021]</span>
+[33] Minhoni R.T.D.A., Pereira, F.F., da Silva, T.B., Castro, E.R., Saad, J.C. The performance of explicit formulas for determining the Darcy-Weisbach friction factor. Eng. Agric., 40:258-265, 2020. [https://doi.org/10.1590/1809-4430-eng.agric.v40n2p258-265/2020 doi:10.1590/1809-4430-eng.agric.v40n2p258-265/2020]
-:<span style="text-align: center; font-size: 75%;">[33]. Minhoni R.T.D.A., Pereira, F.F., da Silva, T.B., Castro, E.R., Saad, J.C. The performance of explicit formulas for determining the Darcy-Weisbach friction factor. Eng. Agric., 40:258-265, 2020. [https://doi.org/10.1590/1809-4430-eng.agric.v40n2p258-265/2020 doi:10.1590/1809-4430-eng.agric.v40n2p258-265/2020]</span>
+[34] Pérez-Pupo J.R., Navarro-Ojeda M.N., Pérez-Guerrero J.N., Batista-Zaldívar M.A. On the explicit expressions for the determination of the friction factor in turbulent regime. Revista Mexicana de Ing. Quim., 19:313–334, 2020. [https://doi.org/10.24275/rmiq/Fen497 doi:10.24275/rmiq/Fen497]
-:<span style="text-align: center; font-size: 75%;">[34]. Pérez-Pupo J.R., Navarro-Ojeda M.N., Pérez-Guerrero J.N., Batista-Zaldívar M.A. On the explicit expressions for the determination of the friction factor in turbulent regime. Revista Mexicana de Ing. Quim., 19:313–334, 2020. [https://doi.org/10.24275/rmiq/Fen497 doi:10.24275/rmiq/Fen497]</span>
+[35] Niazkar M., Talebbeydokhti, N. Comparison of explicit relations for calculating Colebrook friction factor in pipe network analysis using h-based methods. Iran. J. Sci. Technol., Trans. Civ. Eng., 44:231–249, 2020. [https://doi.org/10.1007/s40996-019-00343-2 doi:10.1007/s40996-019-00343-2]
-:<span style="text-align: center; font-size: 75%;">[35]. Niazkar M., Talebbeydokhti, N. Comparison of explicit relations for calculating Colebrook friction factor in pipe network analysis using h-based methods. Iran. J. Sci. Technol., Trans. Civ. Eng., 44:231–249, 2020. [https://doi.org/10.1007/s40996-019-00343-2 doi:10.1007/s40996-019-00343-2]</span>
+[36] Hayes B. Why ''W''? Am. Sci., 93:104–108, 2005.
-:<span style="text-align: center; font-size: 75%;">[36]. Hayes B. Why W? Am. Sci., 93:104–108, 2005.</span>
+[37] Biberg, D. Fast and accurate approximations for the Colebrook equation. J. Fluids Eng., 139:031401, 2017. [https://doi.org/10.1115/1.4034950 doi:10.1115/1.4034950]
-:<span style="text-align: center; font-size: 75%;">[37]. Biberg, D. Fast and accurate approximations for the Colebrook equation. J. Fluids Eng., 139:031401, 2017. [https://doi.org/10.1115/1.4034950 doi:10.1115/1.4034950]</span>
+[38] Rollmann P., Spindler K. Explicit representation of the implicit Colebrook–White equation. Case Stud. Therm. Eng., 5:41–47, 2015. [https://doi.org/10.1016/j.csite.2014.12.001 doi:10.1016/j.csite.2014.12.001]
-:<span style="text-align: center; font-size: 75%;">[38]. Rollmann P., Spindler K. Explicit representation of the implicit Colebrook–White equation. Case Stud. Therm. Eng., 5:41–47, 2015. [https://doi.org/10.1016/j.csite.2014.12.001 doi:10.1016/j.csite.2014.12.001]</span>
+[39] Schmidt M., Lipson H. Distilling free-form natural laws from experimental data. Science, 324:81–85, 2009. [https://doi.org/10.1126/science.1165893 doi:10.1126/science.1165893]
-:<span style="text-align: center; font-size: 75%;">[39]. Schmidt M., Lipson H. Distilling free-form natural laws from experimental data. Science, 324:81–85, 2009. [https://doi.org/10.1126/science.1165893 doi:10.1126/science.1165893]</span>
+[40] Dubčáková R. Eureqa: Software review. Genet. Program. Evolvable Mach., 12:173–178, 2011. [https://doi.org/10.1007/s10710-010-9124-z doi:10.1007/s10710-010-9124-z]
-:<span style="text-align: center; font-size: 75%;">[40]. Dubčáková R. Eureqa: Software review. Genet. Program. Evolvable Mach., 12:173–178, 2011. [https://doi.org/10.1007/s10710-010-9124-z doi:10.1007/s10710-010-9124-z]</span>
+[41]. Horchler A.D. Complex double-precision evaluation of the Wright Omega function. 2017. [https://github.com/horchler/wrightOmegaq https://github.com/horchler/wrightOmegaq]
-:<span style="text-align: center; font-size: 75%;">[41]. Horchler A.D. Complex double-precision evaluation of the Wright Omega function. 2017. [https://github.com/horchler/wrightOmegaq https://github.com/horchler/wrightOmegaq]</span>
+[42] Moody L.F. Friction factors for pipe flow. T. ASME, 66:671–684, 1944.
-:<span style="text-align: center; font-size: 75%;">[42]. Moody L.F. Friction factors for pipe flow. T. ASME, 66:671–684.</span>
+[43] Winning H.K., Coole T. Explicit friction factor accuracy and computational efficiency for turbulent flow in pipes. Flow Turbul. Combust., 90:1–27, 2013. [https://doi.org/10.1007/s10494-012-9419-7 doi:10.1007/s10494-012-9419-7]
-:<span style="text-align: center; font-size: 75%;">[43]. Winning H.K., Coole T. Explicit friction factor accuracy and computational efficiency for turbulent flow in pipes. Flow Turbul. Combust., 90:1–27, 2013. [https://doi.org/10.1007/s10494-012-9419-7 doi:10.1007/s10494-012-9419-7]</span>
+[44] Winning H.K., Coole T. Improved method of determining friction factor in pipes. Int. J. Numer. Methods Heat Fluid Flow, 25:941–949, 2015. [https://doi.org/10.1108/HFF-06-2014-0173 doi:10.1108/HFF-06-2014-0173]
-:<span style="text-align: center; font-size: 75%;">[44]. Winning H.K., Coole T. Improved method of determining friction factor in pipes. Int. J. Numer. Methods Heat Fluid Flow, 25:941–949, 2015. [https://doi.org/10.1108/HFF-06-2014-0173 doi:10.1108/HFF-06-2014-0173]</span>
+[45] Corless R.M., Jeffrey D.J. The Wright ω Function. In: Calmet J., Benhamou B., Caprotti O., Henocque L., Sorge V. (eds) Artificial Intelligence, Automated Reasoning, and Symbolic Computation. AISC 2002, Calculemus 2002. Lecture Notes in Computer Science, vol 2385. Springer, Berlin, Heidelberg, 2002. [https://doi.org/10.1007/3-540-45470-5_10 doi:10.1007/3-540-45470-5_10]
-:<span style="text-align: center; font-size: 75%;">[45]. Corless R.M., Jeffrey D.J. The Wright ω Function. In: Calmet J., Benhamou B., Caprotti O., Henocque L., Sorge V. (eds) Artificial Intelligence, Automated Reasoning, and Symbolic Computation. AISC 2002, Calculemus 2002. Lecture Notes in Computer Science, vol 2385. Springer, Berlin, Heidelberg, 2002. [https://doi.org/10.1007/3-540-45470-5_10 doi:10.1007/3-540-45470-5_10]</span>
+[46] On-Line Encyclopedia of Integer Sequences; Sequence A008275 for Stirling numbers of first kind. [http://oeis.org/A008275 http://oeis.org/A008275]
-:<span style="text-align: center; font-size: 75%;">[46]. On-Line Encyclopedia of Integer Sequences; Sequence A008275 for Stirling numbers of first kind. [http://oeis.org/A008275 http://oeis.org/A008275]</span>
+[47] Brkić D. Spreadsheet-based pipe networks analysis for teaching and learning purpose. Spreadsheets Educ. (EJSiE), 9:4646, 2016. [https://sie.scholasticahq.com/article/4646 https://sie.scholasticahq.com/article/4646]
-:<span style="text-align: center; font-size: 75%;">[47]. Brkić D. Spreadsheet-based pipe networks analysis for teaching and learning purpose. Spreadsheets Educ. (EJSiE), 9:4646, 2016. [https://sie.scholasticahq.com/article/4646 https://sie.scholasticahq.com/article/4646]</span>
+[48] Niazkar M., Afzali S.H. Analysis of water distribution networks using MATLAB and Excel spreadsheet: h‐based methods. Comput. Appl. Eng. Educ., 25:129-141, 2017. [https://doi.org/10.1002/cae.21786 doi:10.1002/cae.21786]
-:<span style="text-align: center; font-size: 75%;">[48]. Niazkar M., Afzali S.H. Analysis of water distribution networks using MATLAB and Excel spreadsheet: h‐based methods. Comput. Appl. Eng. Educ., 25:129-141, 2017. [https://doi.org/10.1002/cae.21786 doi:10.1002/cae.21786]</span>
+[49] Niazkar M., Afzali S.H. Analysis of water distribution networks using MATLAB and Excel spreadsheet: Q‐based methods. Comput. Appl. Eng. Educ., 25:277-289, 2017. [https://doi.org/10.1002/cae.21796 doi:10.1002/cae.21796]
-:<span style="text-align: center; font-size: 75%;">[49]. Niazkar M., Afzali S.H. Analysis of water distribution networks using MATLAB and Excel spreadsheet: Q‐based methods. Comput. Appl. Eng. Educ., 25:277-289, 2017. [https://doi.org/10.1002/cae.21796 doi:10.1002/cae.21796]</span>
+[50] Demir S., Karadeniz A., Demir N.M., Duman S. Excel VBA-based solution to pipe flow measurement problem. Spreadsheets Educ. (EJSiE), 10:4671, 2018. [https://sie.scholasticahq.com/article/4671.pdf https://sie.scholasticahq.com/article/4671.pdf]
-:<span style="text-align: center; font-size: 75%;">[50]. Demir S., Karadeniz A., Demir N.M., Duman S. Excel VBA-based solution to pipe flow measurement problem. Spreadsheets Educ. (EJSiE), 10:4671, 2018. [https://sie.scholasticahq.com/article/4671.pdf https://sie.scholasticahq.com/article/4671.pdf]</span>
+[51] Niazkar M., Afzali S.H. Streamline performance of Excel in stepwise implementation of numerical solutions. Comput. Appl. Eng. Educ., 24:555–5664, 2016. [https://doi.org/10.1002/cae.21731 doi:10.1002/cae.21731]
-:<span style="text-align: center; font-size: 75%;">[51]. Niazkar M., Afzali S.H. Streamline performance of Excel in stepwise implementation of numerical solutions. Comput. Appl. Eng. Educ., 24:555–5664, 2016. [https://doi.org/10.1002/cae.21731 doi:10.1002/cae.21731]</span>
+[52] Niazkar M., Talebbeydokhti N., Afzali, S.H. Relationship between Hazen-William coefficient and Colebrook-White friction factor: Application in water network analysis. Eur. Water, 58:513-520, 2017. [https://www.ewra.net/ew/pdf/EW_2017_58_74.pdf https://www.ewra.net/ew/pdf/EW_2017_58_74.pdf]
-:<span style="text-align: center; font-size: 75%;">[52]. Niazkar M., Talebbeydokhti N., Afzali, S.H. Relationship between Hazen-William coefficient and Colebrook-White friction factor: Application in water network analysis. Eur. Water, 58:513-520, 2017. [https://www.ewra.net/ew/pdf/EW_2017_58_74.pdf https://www.ewra.net/ew/pdf/EW_2017_58_74.pdf]</span>
+[53] Niazkar M., Talebbeydokhti N., Afzali S.H. Novel grain and form roughness estimator scheme incorporating artificial intelligence models. Water Resour. Manag., 33:757-773, 2019. [https://doi.org/10.1007/s11269-018-2141-z doi:10.1007/s11269-018-2141-z]
-:<span style="text-align: center; font-size: 75%;">[53]. Niazkar M., Talebbeydokhti N., Afzali S.H. Novel grain and form roughness estimator scheme incorporating artificial intelligence models. Water Resour. Manag., 33:757-773, 2019. [https://doi.org/10.1007/s11269-018-2141-z doi:10.1007/s11269-018-2141-z]</span>
+[54] Brkić D. Solution of the implicit Colebrook equation for flow friction using Excel. Spreadsheets Educ. (EJSiE), 10:4663, 2017. [https://sie.scholasticahq.com/article/4663 https://sie.scholasticahq.com/article/4663]
-:<span style="text-align: center; font-size: 75%;">[54]. Brkić D. Solution of the implicit Colebrook equation for flow friction using Excel. Spreadsheets Educ. (EJSiE), 10:4663, 2017. [https://sie.scholasticahq.com/article/4663 https://sie.scholasticahq.com/article/4663]</span>
+[55] Botchkarev A. Assessing Excel VBA suitability for Monte Carlo simulation. Spreadsheets Educ. (EJSiE), 8:4629, 2015. [https://sie.scholasticahq.com/article/4629 https://sie.scholasticahq.com/article/4629]
-:<span style="text-align: center; font-size: 75%;">[55]. Botchkarev A. Assessing Excel VBA suitability for Monte Carlo simulation. Spreadsheets Educ. (EJSiE), 8:4629, 2015. [https://sie.scholasticahq.com/article/4629 https://sie.scholasticahq.com/article/4629]</span>
+[56] Christie D.S. Build your own Monte Carlo spreadsheet. Spreadsheets Educ. (EJSiE), 11:4672, 2018. [https://sie.scholasticahq.com/article/4672 https://sie.scholasticahq.com/article/4672]
-:<span style="text-align: center; font-size: 75%;">[56]. Christie D.S. Build your own Monte Carlo spreadsheet. Spreadsheets Educ. (EJSiE), 11:4672, 2018. [https://sie.scholasticahq.com/article/4672 https://sie.scholasticahq.com/article/4672]</span>
+[57] Cordero A., Martí P., Victoria M. Optimización de topología robusta de estructuras continuas usando el método de Monte Carlo y modelos Kriging (Robust topology optimization of continuum structures using Monte Carlo method and Kriging models) /in Spanish/. Rev. Int. Métodos Numér. Cálc. Diseño Ing., 34:5, 2018. [https://doi.org/10.23967/j.rimni.2017.5.005 doi:10.23967/j.rimni.2017.5.005]
-:<span style="text-align: center; font-size: 75%;">[57]. Cordero A., Martí P., Victoria M. Optimización de topología robusta de estructuras continuas usando el método de Monte Carlo y modelos Kriging (Robust topology optimization of continuum structures using Monte Carlo method and Kriging models) /in Spanish/. Rev. Int. Métodos Numér. Cálc. Diseño Ing., 34:5, 2018. [https://doi.org/10.23967/j.rimni.2017.5.005 doi:10.23967/j.rimni.2017.5.005]</span>
+[58] Sobol I.M., Turchaninov V.I., Levitan Y.L., Shukhman B.V. Quasi-random sequence generators; Distributed by OECD/NEA data bank; Keldysh Institute of Applied Mathematics; Russian Academy of Sciences: Moscow, Russia. 1992. [https://ec.europa.eu/jrc/sites/jrcsh/files/LPTAU51.rar https://ec.europa.eu/jrc/sites/jrcsh/files/LPTAU51.rar]
-:<span style="text-align: center; font-size: 75%;">[58]. Sobol I.M., Turchaninov V.I., Levitan Y.L., Shukhman B.V. Quasi-random sequence generators; Distributed by OECD/NEA data bank; Keldysh Institute of Applied Mathematics; Russian Academy of Sciences: Moscow, Russia. 1992. [https://ec.europa.eu/jrc/sites/jrcsh/files/LPTAU51.rar https://ec.europa.eu/jrc/sites/jrcsh/files/LPTAU51.rar]</span>
+[59] Praks P., Brkić D. Choosing the optimal multi-point iterative method for the Colebrook flow friction equation. Processes, 6:130, 2018. [https://doi.org/10.3390/pr6080130 doi:10.3390/pr6080130]
-:<span style="text-align: center; font-size: 75%;">[59]. Praks P., Brkić D. Choosing the optimal multi-point iterative method for the Colebrook flow friction equation. Processes, 6:130, 2018. [https://doi.org/10.3390/pr6080130 doi:10.3390/pr6080130]</span>
+[60] Praks P., Brkić D. Advanced iterative procedures for solving the implicit Colebrook equation for fluid flow friction. Adv. Civ. Eng., 2018:5451034, 2018. [https://doi.org/10.1155/2018/5451034 doi:10.1155/2018/5451034]
-:<span style="text-align: center; font-size: 75%;">[60]. Praks P., Brkić D. Advanced iterative procedures for solving the implicit Colebrook equation for fluid flow friction. Adv. Civ. Eng., 2018:5451034, 2018. [https://doi.org/10.1155/2018/5451034 doi:10.1155/2018/5451034]</span>
+[61] Sun S., Ouyang R., Zhang B., Zhang T.Y. Data-driven discovery of formulas by symbolic regression. MRS Bull., 44:559-564, 2019. [https://doi.org/10.1557/mrs.2019.156 doi:10.1557/mrs.2019.156]
-:<span style="text-align: center; font-size: 75%;">[61]. Sun S., Ouyang R., Zhang B., Zhang T.Y. Data-driven discovery of formulas by symbolic regression. MRS Bull., 44:559-564, 2019. [https://doi.org/10.1557/mrs.2019.156 doi:10.1557/mrs.2019.156]</span>
+[62] Udrescu S.-M., Tegmark M. AI Feynman: A physics-inspired method for symbolic regression. Sci. Adv. 6:eaay2631, 2020. [https://doi.org/10.1126/sciadv.aay2631 doi:10.1126/sciadv.aay2631]
-:<span style="text-align: center; font-size: 75%;">[62]. Udrescu S.-M., Tegmark M. AI Feynman: A physics-inspired method for symbolic regression. Sci. Adv. 6:eaay2631, 2020. [https://doi.org/10.1126/sciadv.aay2631 doi:10.1126/sciadv.aay2631]</span>
+[63] Gholamy A., Kreinovich V. How to use absolute-error-minimizing software to minimize relative error: Practitioner’s guide. Int. Math. Forum, 12:763-770, 2017. [https://doi.org/10.12988/imf.2017.7761 doi:10.12988/imf.2017.7761]
-:<span style="text-align: center; font-size: 75%;">[63]. Gholamy A., Kreinovich V. How to use absolute-error-minimizing software to minimize relative error: Practitioner’s guide. Int. Math. Forum, 12:763-770, 2017. [https://doi.org/10.12988/imf.2017.7761 doi:10.12988/imf.2017.7761]</span>
+[64] Vatankhah A.R. Approximate analytical solutions for the Colebrook equation. J. Hydraul. Eng., 144:06018007, 2018. [https://doi.org/10.1061/(ASCE)HY.1943-7900.0001454 doi:10.1061/(ASCE)HY.1943-7900.0001454]
-:<span style="text-align: center; font-size: 75%;">[64]. Vatankhah A.R. Approximate analytical solutions for the Colebrook equation. J. Hydraul. Eng., 144:06018007, 2018. [https://doi.org/10.1061/(ASCE)HY.1943-7900.0001454 doi:10.1061/(ASCE)HY.1943-7900.0001454]</span>
+[65] Brkić D., Praks P. Discussion of Approximate analytical solutions for the Colebrook equation. By Ali R. Vatankhah. J. Hydraul. Eng., 146:07019011, 2020. [https://doi.org/10.1061/(ASCE)HY.1943-7900.0001667 doi:10.1061/(ASCE)HY.1943-7900.0001667]
-:<span style="text-align: center; font-size: 75%;">[65]. Brkić D., Praks P. Discussion of “Approximate analytical solutions for the Colebrook equation.”, by Ali R. Vatankhah. J. Hydraul. Eng., 146:07019011, 2020. [https://doi.org/10.1061/(ASCE)HY.1943-7900.0001667 doi:10.1061/(ASCE)HY.1943-7900.0001667]</span>
+[66] Lamri A.A. Discussion of Approximate analytical solutions for the Colebrook equation. By Ali R. Vatankhah. J. Hydraul. Eng., 146:07019012, 2020. [https://doi.org/10.1061/(ASCE)HY.1943-7900.0001668 doi:10.1061/(ASCE)HY.1943-7900.0001668]
-:<span style="text-align: center; font-size: 75%;">[66]. Lamri A.A. Discussion of “Approximate analytical solutions for the Colebrook equation.”, by Ali R. Vatankhah. J. Hydraul. Eng., 146:07019012, 2020. [https://doi.org/10.1061/(ASCE)HY.1943-7900.0001668 doi:10.1061/(ASCE)HY.1943-7900.0001668]</span>
+[67] Clamond D. Efficient resolution of the Colebrook equation. Ind. Eng. Chem. Res., 48:3665-3671, 2009. [https://doi.org/10.1021/ie801626g doi:10.1021/ie801626g]
-:<span style="text-align: center; font-size: 75%;">[67]. Clamond D. Efficient resolution of the Colebrook equation. Ind. Eng. Chem. Res., 48:3665-3671, 2009. [https://doi.org/10.1021/ie801626g doi:10.1021/ie801626g]</span>
+[68] Niazkar M. Discussion of Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright <math>\omega</math>-function by Dejan Brkić and Pavel Praks. Mathematics, 8:793, 2020. [https://doi.org/10.3390/math8050793 doi:10.3390/math8050793]
-:<span style="text-align: center; font-size: 75%;">[68]. Niazkar M. Discussion of “Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function” by Dejan Brkić and Pavel Praks. Mathematics, 8, 2020.</span>
+[69] Praks P., Brkić D. Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright <math>\omega</math>-function: Reply to the Discussion by Majid Niazkar. Mathematics, 8:796, 2020. [https://doi.org/10.3390/math8050796 doi:10.3390/math8050796]
-:<span style="text-align: center; font-size: 75%;">[69]. Praks P., Brkić D. Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function: Reply to the Discussion by Majid Niazkar. Mathematics, 8, 2020.</span>
+[70] Brkić D., Praks P. Colebrook’s flow friction explicit approximations based on fixed-point iterative cycles and symbolic regression. Computation, 7:48, 2019. [https://doi.org/10.3390/computation7030048 doi:10.3390/computation7030048]
-:<span style="text-align: center; font-size: 75%;">[70]. Brkić D., Praks P. Colebrook’s flow friction explicit approximations based on fixed-point iterative cycles and symbolic regression. Computation, 7:48, 2019: [https://doi.org/10.3390/computation7030048 doi:10.3390/computation7030048]</span>
+[71] Praks P., Brkić D. Rational Approximation for solving an implicitly given Colebrook flow friction equation. Mathematics, 8:26, 2020. [https://doi.org/10.3390/math8010026 doi:10.3390/math8010026]
-:<span style="text-align: center; font-size: 75%;">[71]. Praks P., Brkić D. Rational Approximation for solving an implicitly given Colebrook flow friction equation. Mathematics, 8:26, 2020. [https://doi.org/10.3390/math8010026 doi:10.3390/math8010026]</span>
+[72] Praks P., Brkić D. One-log call iterative solution of the Colebrook equation for flow friction based on Padé polynomials. Energies, 11:1825, 2018. [https://doi.org/10.3390/en11071825 doi:10.3390/en11071825]
-:<span style="text-align: center; font-size: 75%;">[72]. Praks P., Brkić D. One-log call iterative solution of the Colebrook equation for flow friction based on Padé polynomials. Energies, 11:1825, 2018. [https://doi.org/10.3390/en11071825 doi:10.3390/en11071825]</span>
+[73] Brkić D., Praks P. Air-forced flow in Proton Exchange Membrane Fuel Cells: Calculation of fan-induced friction in open-cathode conduits with virtual roughness. Processes, 8:686, 2020. [https://doi.org/10.3390/pr8060686 doi:10.3390/pr8060686]

Latest revision as of 14:47, 19 May 2021

Abstract

1. Introduction

2. Transformation through the Wright ω-function

3. Methods and discussion

3.1. Series expansion of ω ( x ) − x {\textstyle \omega (x)-x} 3.1. Series expansion of ω ( x ) − x {\textstyle \omega (x)-x}

3.1.1. Asymptotic expansion of ω ( x ) − x {\textstyle \omega (x)-x} about infinity3.1.1. Asymptotic expansion of ω ( x ) − x {\textstyle \omega (x)-x} about infinity

3.1.2. Symbolic regression-based expansions of y = ω ( x ) − x {\textstyle y=\omega \left(x\right)-x} for Colebrook equation3.1.2. Symbolic regression-based expansions of y = ω ( x ) − x {\textstyle y=\omega \left(x\right)-x} for Colebrook equation

3.2. Approximations suitable for engineering use

3.2.1. Approximations and procedures based on series expansions of ω ( x ) − x {\textstyle \omega (x)-x} about infinity3.2.1. Approximations and procedures based on series expansions of ω ( x ) − x {\textstyle \omega (x)-x} about infinity

3.2.2. Approximations based on symbolic regression

3.3. Summary of accuracy and speed tests

4. Conclusions and future work

Notations

Funding

Conflicts of interest

Acknowledgments

Supplementary materials

References

Document information

Document Score

Share this document

Keywords

claim authorship

3.1. Series expansion of ${\textstyle \omega (x)-x}$ 3.1. Series expansion of ω ( x ) − x {\textstyle \omega (x)-x}

3.1.1. Asymptotic expansion of ${\textstyle \omega (x)-x}$ about infinity3.1.1. Asymptotic expansion of ω ( x ) − x {\textstyle \omega (x)-x} about infinity

3.1.2. Symbolic regression-based expansions of ${\textstyle y=\omega \left(x\right)-x}$ for Colebrook equation3.1.2. Symbolic regression-based expansions of y = ω ( x ) − x {\textstyle y=\omega \left(x\right)-x} for Colebrook equation

3.2.1. Approximations and procedures based on series expansions of ${\textstyle \omega (x)-x}$ about infinity3.2.1. Approximations and procedures based on series expansions of ω ( x ) − x {\textstyle \omega (x)-x} about infinity