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

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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

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@@ Line 4: / Line 4: @@
 '''Review of new flow friction equations: Constructing Colebrook’s explicit correlations accurately'''
-==Pavel Praks <sup>1,2,</sup>* and Dejan Brkić <sup>1,2,</sup>*==
+==Pavel Praks and Dejan Brkić==
-<sup>1</sup> Joint Research Centre (JRC), European Commission, 21027 Ispra, Italy
+<span id='_Hlk15384487'></span><span style="text-align: center; font-size: 75%;">Joint Research Centre (JRC), European Commission, 21027 Ispra, Italy and IT4Innovations, VŠB—Technical University of Ostrava, 708 00 Ostrava, Czechia</span>
-<span id='_Hlk15384487'><sup>2</sup>IT4Innovations, VŠB—Technical University of Ostrava, 708 00 Ostrava, Czechia</span>
+Correspondence: [mailto:pavel.praks@vsb.cz pavel.praks@vsb.cz] (P.P.) - [https://orcid.org/0000-0002-3913-7800 https://orcid.org/0000-0002-3913-7800]; [mailto:dejanbrkic0611@gmail.com dejanbrkic0611@gmail.com] or [mailto:dejan.brkic@vsb.cz dejan.brkic@vsb.cz] (D.B.) - [https://orcid.org/0000-0002-2502-0601 https://orcid.org/0000-0002-2502-0601]
-* Correspondence: [mailto:pavel.praks@vsb.cz pavel.praks@vsb.cz] (P.P.); [mailto:dejanbrkic0611@gmail.com dejanbrkic0611@gmail.com] or [mailto:dejan.brkic@vsb.cz dejan.brkic@vsb.cz] (D.B.)
+-both authors contributed equally to this study
-both authors contributed equally to this study
+<span id='_Hlk39321727'></span>
 -->
+==Abstract==
-'''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 based on the asymptotic series expansion of the Wright ω-function and on 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 explicit solution of the Colebrook equation. Such balance between speed and accuracy could be achieved only using symbolic regression, a computational intelligence approach that can find optimal coefficients and the best structure of the equation. The presented numerical experiments show that the novel symbolic regression approximation reduced the maximal relative error from 0.045% to 0.00337%, i.e. more than 13 times, even the complexity remains almost unchanged. Moreover, we also provide a novel highly precise symbolic regression approximation (max. relative error 0.000024%), which, for the same speed as asymptotic expansion, reduces the relative error by factor 219. This research is motivated by estimation of flow rate using electrical parameters of pumps where direct measurement is not always possible such as in offshore underwater pipelines.
+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 ω-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==
-<span id='OLE_LINK1'></span><span id='OLE_LINK2'></span>The empirical Colebrook equation [1]; Eq. (1)., widely used in hydraulics, implicitly relates in an entangled logarithmic form the known Reynolds number <math display="inline">Re</math> and the relative roughness of inner pipe surface <math display="inline">\epsilon</math>  with the unknown Darcy flow friction factor <math display="inline">f</math>. The equation is based on the experiment by Colebrook and White from 1937 [2].
+The empirical Colebrook equation [1] (Eq. (1)), widely used in hydraulics, implicitly relates in an entangled logarithmic form the known Reynolds number <math display="inline">Re</math> and the relative roughness of inner pipe surface <math display="inline">\epsilon</math>  with the unknown Darcy flow friction factor <math display="inline">f</math>. The equation is based on the experiment by Colebrook and White from 1937 [2]
-<span id='_GoBack'></span> <math display="inline">\frac{1\, }{\sqrt{f}}=-2\cdot \mathrm{{log}_{10}}\,\left( \frac{2.51}{Re}\cdot \frac{1\, }{\sqrt{f}}+\frac{\epsilon }{3.71}\right)</math>  (1)
+{| class="formulaSCP" style="width: 100%; text-align: center;"
+|-
+|
+{| style="text-align: center; margin:auto;"
+|-
+| <math>\frac{1\, }{\sqrt{f}}=-2\cdot \mathrm{{log}_{10}}\,\left( \frac{2.51}{Re}\cdot \frac{1\, }{\sqrt{f}}+\frac{\epsilon }{3.71}\right)</math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
+|}
-To date, the Colebrook equation can be explicitly expressed only in terms of the Lambert ''W''-function, the function introduced by Corless et al. in 1996 [3]. Based on series expansion of its cognate, the Wright ''ω''-function, introduced by Wright [4], 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 are also computationally efficient.
+To date, the Colebrook equation can be explicitly expressed only in terms of the Lambert <math>W</math>-function [3-5], the function introduced by Corless et al. [6]. Based on series expansion of its cognate, the Wright <math>\omega</math>-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.
-Such balance between speed and accuracy could be achieved only using symbolic regression, a computational intelligence approach that can find optimal coefficients and the best structure of equation. This analysis is needed as an introduction for our method for estimation of flow rate using electrical parameters of pumps where direct measurement is not always possible such as in offshore underwater pipelines.
+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 <math>\omega</math>-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 <math>\omega</math>-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].
-The paper has the following structure. After this Introducion, Section 2 brings transformation of the Colebrook equation through the Wright ω-function. As the the Wright ω-function is computationally demanding, Section 3 introdudes series expansion of ω(x)-x using assymptotic expansion and symbolic regression. Then, Section 4 presents approximations suitable for engineering use. The novelty of this paper includes introducing of optimal constants and simple rational functions, which reduce the relative error of the assymptotic expansion for the Colebrook equation by global optimization and symbolic regression (Sections 3.1 and Section 4.1). Moreover, the novel symbolic regression approximation of the Colebrook equation based on minimization of the relative error is introduces in Section 4.2. In comparison with asymptotic expansion of the same complexity, the novel symbolic-based rational approximation reduces the maximal relative error from 0.045% to 0.00337%, i.e. more than 13 times. Finally, Section 5 includes conclusions and future work.
+Here we will show explicit approximations of the Colebrook equation for flow friction based on the series expansion of <math display="inline">\omega (x)-</math><math>x</math> 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 <math>\omega</math>-function is computationally demanding. Section 3.1 introduces series expansion of <math display="inline">\omega (x)-</math><math>x</math> 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 [5] was the first to found that the Colebrook equation can be rearranged in 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 [6,7]). As explained by Hayes [8], the Lambert ''W''-function, named after the mathematician Johann Heinrich Lambert, who lived in XVIII century, is defined as the inverse of ''F(x)=x·e<sup>x</sup>'' where <math display="inline">e</math> 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 <math display="inline">f</math>. According to Brkić and Praks [9,10], Biberg [11], Rollmann and Spindler [12], 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 40: / Line 49: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\left. \begin{matrix}\frac{1}{\sqrt{f}}=\frac{z}{2.51}\cdot \left( \mathrm{B}\,+y\right) \\z=\frac{2\cdot 2.51}{\mathrm{ln}\,\left( 10\right) }\\\begin{matrix}A=\frac{Re}{z}\cdot \frac{\epsilon }{3.71}\approx \frac{Re\cdot \epsilon }{8.0878}\\B=\mathrm{ln}\,(Re)-ln(z)\approx \mathrm{ln}\,\left( Re\right) -0.7794\\\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 46: / Line 55: @@
-In Eq. (2), the symbol ''W'' denotes the Lambert ''W''-function and <math display="inline">\omega</math>  represents the Wright ''ω''-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 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 ''ω''-function. We will start with approximations based on the asymptotic expansion of the Wright ''ω''-function, and then we will introduce novel approximations of <math display="inline">y</math> based on the symbolic regression tool Eureqa [13,14].
+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>.
 ==3. Methods and discussion==
-In addition to the already showed Eq. (3) [9] which represents only the first term of series expansion from Eqs. (3-7), in Section 3.1.1., we show additional terms. Using them, the accuracy of explicit approximations which is put in the form suitable for engineering use in Section 3.2.1. is increased significantly (In addition at the end of Section 3.1.1., we show an efficient strategy to reduce error using symbolic regression). Also additional strategy how to use symbolic regression to construct approximations of <math display="inline">\mathit{\boldsymbol{\omega (x)-x}}</math> is given in Section 3.1.2., where their form suitable for engineering use is given in 3.2.2.. Only one example of this really promising strategy how to construct accurate explicit approximations based on symbolic regression is given in [9], while here we give additional expressions which are to date the most accurate, but also very simple. These extreme accurate approximations are offered in the form suitable for human eyes, but also in forms suitable for coding.
+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">\mathit{\boldsymbol{\omega (x)-x}}</math>==
+===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 (''i''=1, 2, …, 5); used for approximating <math display="inline">\omega \left( x\right) -</math><math>x</math> by a truncated series about infinity [15], where generally <math display="inline">{y}_{i}=</math><math>\sum _{i=1}^{+\infty }{s}_{i}</math> as given in Eqs. (8-12).
+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)
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@@ Line 75: / Line 95: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">{s}_{2}(x)\, =\frac{\mathrm{ln}\,\left( x\right) }{2\cdot {x}^{2}}\cdot \left( \mathrm{ln}\,\left( x\right) -\right. </math><math>\left. 2\right)</math>
+| <math display="inline">{s}_{2}(x)\, =\displaystyle\frac{\mathrm{ln}\,\left( x\right) }{2\cdot {x}^{2}}\cdot \left( \mathrm{ln}\,\left( x\right) -\right. </math><math>\left. 2\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (4)
@@ Line 85: / Line 105: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">{s}_{3}(x)\, =\mathrm{\frac{\mathrm{ln}\,\left( x\right) }{6\cdot {x}^{3}}\cdot \, }\,\left( 2\cdot \, \mathrm{{ln}^{2}}\,\left( x\right) -9\cdot \mathrm{ln}\,\left( x\right) +6\right)</math>
+| <math display="inline">{s}_{3}(x)\, =\mathrm{\displaystyle\frac{\mathrm{ln}\,\left( x\right) }{6\cdot {x}^{3}}\cdot \, }\,\left( 2\cdot \, \mathrm{{ln}^{2}}\,\left( x\right) -9\cdot \mathrm{ln}\,\left( x\right) +6\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5)
@@ Line 95: / Line 115: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">{s}_{4}(x)\, =\mathrm{\frac{\mathrm{ln}\,\left( x\right) }{12\cdot {x}^{4}}\cdot \, }\,\left( 3\cdot \, \mathrm{{ln}^{3}}\,\left( x\right) -22\cdot \, \mathrm{{ln}^{2}}\,\left( x\right) +36\cdot \mathrm{ln}\,\left( x\right) -12\right)</math>
+| <math display="inline">{s}_{4}(x)\, =\mathrm{\displaystyle\frac{\mathrm{ln}\,\left( x\right) }{12\cdot {x}^{4}}\cdot \, }\,\left( 3\cdot \, \mathrm{{ln}^{3}}\,\left( x\right) -22\cdot \, \mathrm{{ln}^{2}}\,\left( x\right) +36\cdot \mathrm{ln}\,\left( x\right) -12\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (6)
@@ Line 105: / Line 125: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">{s}_{5}(x)\, =\mathrm{\frac{\mathrm{ln}\,\left( x\right) }{60\cdot {x}^{5}}\cdot \, }\,\left( 12\cdot \, \mathrm{{ln}^{4}}\,\left( x\right) -125\cdot \, \mathrm{{ln}^{3}}\,\left( x\right) +350\cdot \, \mathrm{{ln}^{2}}\,\left( x\right) -300\cdot \mathrm{ln}\,\left( x\right) +60\right)</math>
+| <math display="inline">{s}_{5}(x)\, =\mathrm{\displaystyle\frac{\mathrm{ln}\,\left( x\right) }{60\cdot {x}^{5}}\cdot \, }\,\left( 12\cdot \, \mathrm{{ln}^{4}}\,\left( x\right) -125\cdot \, \mathrm{{ln}^{3}}\,\left( x\right) +350\cdot \, \mathrm{{ln}^{2}}\,\left( x\right) -300\cdot \mathrm{ln}\,\left( x\right) +60\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (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 <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>.
-As can be seen in Figure 1, the function <math display="inline">y</math> needs to be accurate only in the domain ''7.51<x<619,'' which is the one relevant for the Colebrook equation; Eq. (1). On the other hand, the Reynolds number <math display="inline">Re</math> should be between 4000 and 10<sup>8</sup>, while roughness of inner pipe surface <math display="inline">\epsilon</math>  should be from 0 to 0.05.
+The first five approximations of <math display="inline">\omega \left( x\right) -</math><math>x</math> (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 <math display="inline">{s}_{i}(x)</math> functions from Eqs. (3-7). The coefficients of <math display="inline">\omega \left( x\right)</math>  series expansion about infinity are unsigned Stirling numbers of the first kind [46] as pointed out by Rollmann and Spindler [38].
-{| style="width: 100%;"
+The symbol <math display="inline">y=wrightOmega(x)-x</math> represents the “exact” explicit accurate solution of the Colebrook equation given by Eq. (2), in which <math display="inline">y</math> 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
-|-
-|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:Draft_Brkic_171642252-image1.png|396px]] </span>
-|}
-<span style="text-align: center; font-size: 75%;">'''Figure 1.''' Strictly monotonic decreasing of'' ''</span> <math display="inline">\omega (x)-</math><math>x</math><span style="text-align: center; font-size: 75%;"> for the Colebrook equation.</span>
-Winning and Coole [16,17] found that logarithmic, exponential, non-integer power terms and similar 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>  in order to approximate the <math display="inline">\omega (x)</math>.
-The first five approximations of <math display="inline">\omega \left( x\right) -</math><math>x</math>; Eqs. (8-12) based on the series expansion of about infinity by Corless et al. [3, 15]. These approximations are expressed in terms of <math display="inline">{s}_{i}(x)</math> functions from Eqs. (3-7). The coefficients of <math display="inline">\omega \left( x\right)</math>  series expansion about infinity are unsigned Stirling numbers of the first kind [18] as pointed out by Rollmann and Spindler [12].
-<math display="inline">y=wrightOmega(x)-x</math> represents the “exact” explicit accurate solution of the Colebrook equation given by Eq. (2), in which <math display="inline">y</math> is calculated by the Matlab’s built-in command ''wrightOmega'' (e.g. similar function does not yet exist in MS Excel [19,20] and user-defined Visual Basic for Applications - VBA script should be provided instead [21]). However, this ''wrightOmega'' solution is extremely slow; it requires more than 11394=189.9 minutes(!) to evaluate the benchmark, even the here presented approximations requires only seconds. The column “Time” represents time in seconds to evaluate the proposed explicit approximations of the Colebrook equation using 8 million of Quasi Monte-Carlo samples (a benchmark study). For this sampling, the LPTAU51 algorithm was used [22]. We can clearly see from accuracy analysis and speed test of Eqs. (8-12) that it makes sense to introduce fast, but accurate approximations of the ''wrightOmega'' function.
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 177: / 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>\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].
-Eq. (8) is with 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, while Eq. (12) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.00247% and 4.4 seconds. <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 [23-26].
+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.
-<span style="text-align: center; font-size: 75%;">As can be seen in Eqs. (8-12), the function </span> <math display="inline">{y}_{i}</math><span style="text-align: center; font-size: 75%;"> approximates the function </span> <math display="inline">\omega \left( x\right) -</math><math>x</math><span style="text-align: center; font-size: 75%;"> by various quality and complexity. The larger ''i'' of the function </span> <math display="inline">{y}_{i}</math><span style="text-align: center; font-size: 75%;"> implies a more precise approximation of </span> <math display="inline">\omega \left( x\right) -</math><math>x</math><span style="text-align: center; font-size: 75%;">, but the computation time increases.</span>
+Further, to increase the accuracy of expansions given with Eqs. (8)-(12), approximations can be written in a form <math display="inline">y+</math><math>\alpha \approx W\left( {e}^{x}\right) -x</math>, where <math display="inline">\alpha</math>  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 <math display="inline">\alpha</math>  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)
-Further, to increase the accuracy of expansions given with Eqs. (3-7), approximations can be written in a form <math display="inline">y-</math><math>\alpha \approx W\left( {e}^{x}\right) -x</math>, where the constant <math display="inline">\alpha</math>  represents a constant, which minimizes the relative error of the flow friction factor. The constant was obtained in Matlab by global optimization solver ''Globalsearch''. This optimal constant <math display="inline">\alpha</math>  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); as given in Eqs. (13-14). This strategy makes also the distribution of the error more uniform, and the methodology for the relative error and time mesurment is in correlation with Eqs. (8-12).
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@@ Line 189: / Line 197: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">y\approx {s}_{1}(x)+0.00056</math>
+| <math>y\approx {s}_{1}(x)+0.00056</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (13)
@@ Line 199: / Line 207: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">y\approx {s}_{1}\left( x\right) +{s}_{2}\left( x\right) -</math><math>0.0014</math>
+| <math>y\approx {s}_{1}\left( x\right) +{s}_{2}\left( x\right) -</math><math>0.0014</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (14)
@@ Line 209: / Line 217: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">y\approx {s}_{1}\left( x\right) +{s}_{2}\left( x\right) +</math><math>{s}_{3}\left( x\right) -0.000093</math>
+| <math>y\approx {s}_{1}\left( x\right) +{s}_{2}\left( x\right) +</math><math>{s}_{3}\left( x\right) -0.000093</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (15)
 |}
+Eq. (13) is with the maximal relative error of no more than <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.129% and it needs 0.4 seconds to execute the benchmark sample, Eq. (14) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.0691% and 0.4 seconds, while Eq. (15) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.00527% and 1.1 seconds. From Eq. (13)-(15) we can observe that the optimal constants tend to zero.
-Eq. (13) is with maximal relative error of no more than <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.129% and it needs 0.4 seconds to execute the benchmark sample, Eq. (14) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.0691% and 0.4 seconds, while Eq. (15) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.00527% and 1.1 seconds.
+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 <math display="inline">\xi</math>  for error minimization (instead of the constant <math display="inline">\alpha</math> ) which is found through symbolic regression [61]. Eq. (16) is based on Eq. (8) where function <math display="inline">\xi</math>  reduce the relative error significantly (the maximal error is limited to 0.000391% with 1.1 seconds to execute the benchmark)
-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 <math display="inline">\xi</math>  for error minimization (instead of the constant <math display="inline">\alpha</math> ) which is found through symbolic regression [27]. Eq. (16) is based on Eq. (8) where function <math display="inline">\xi</math>  reduce the relative error significantly.
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 224: / Line 231: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\left. \begin{matrix}y\approx -\mathrm{ln}\,\left( x\right) +\frac{\mathrm{ln}\,\left( x\right) }{x}+\xi \\\xi =\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\}</math>
+| <math>\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\}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (16)
 |}
+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.
-The error function <math display="inline">\xi</math>  of Eq. (16) is developed using symbolic regression, while in Section 3.2. <math display="inline">\omega (x)-</math><math>x</math> will be approximated using the same approach.
+====3.1.2. Symbolic regression-based expansions of <math display="inline">y=\omega \left( x\right) -x</math> for Colebrook equation====
-===3.1.2. Symbolic regression-based expansions of <math display="inline">y=\omega \left( x\right) -</math><math>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 Collebrook equation is given in Eqs. (17-20). These approximations were found by symbolic regression [9]. Software Eureqa is used to perform regression analyses [13,14]. These approximations are optimized for the domain 7.51<x<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).
+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)
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 243: / Line 249: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">{y}_{sr}\approx -\mathrm{ln}\,\left( x\right) +\frac{1.038\cdot \mathrm{ln}\,\left( x\right) }{x+0.332}</math>
+| <math>{y}_{sr}\approx -\mathrm{ln}\,\left( x\right) +\displaystyle\frac{1.038\cdot \mathrm{ln}\,\left( x\right) }{x+0.332}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (17)
@@ Line 253: / Line 259: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">{y}_{sr}\approx -\mathrm{ln}\,\left( x\right) +\mathrm{\frac{1.0119\cdot \mathrm{ln}\,\left( x\right) }{x}}\,+\frac{\mathrm{ln}\,\left( x\right) -2.3849}{{x}^{2}}</math>
+| <math>{y}_{sr}\approx -\mathrm{ln}\,\left( x\right) +\mathrm{\displaystyle\frac{1.0119\cdot \mathrm{ln}\,\left( x\right) }{x}}\,+\displaystyle\frac{\mathrm{ln}\,\left( x\right) -2.3849}{{x}^{2}}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (18)
@@ Line 263: / Line 269: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">{y}_{sr}\approx -\mathrm{ln}\,\left( x\right) +\frac{\mathrm{ln}\,\left( x\right) }{x-0.5564\cdot \mathrm{ln}\,\left( x\right) +1.207}=</math><math>Y</math>
+| <math>{y}_{sr}\approx -\mathrm{ln}\,\left( x\right) +\displaystyle\frac{\mathrm{ln}\,\left( x\right) }{x-0.5564\cdot \mathrm{ln}\,\left( x\right) +1.207}=</math><math>Y</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (19)
@@ Line 273: / Line 279: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">{y}_{sr}\approx Y-{\xi }_{1}\, =Y-\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}</math>
+| <math>{y}_{sr}\approx Y-{\xi }_{1}\, =Y-\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}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (20)
@@ Line 279: / Line 285: @@
-Eq. (17) is with maximal relative error of no more than <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.0497% and it needs 0.42 seconds to execute the benchmark sample, Eq. (18) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.0105% and 0.42 seconds, Eq. (19) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.00229% and 0.42 seconds while Eq. (20) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.000024% and 1.1 seconds.
+Eq. (17) is with the maximal relative error of no more than <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.0497% and it needs 0.42 seconds to execute the benchmark sample, Eq. (18) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.0105% and 0.42 seconds, Eq. (19) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.00229% and 0.42 seconds while Eq. (20) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.000024% and 1.1 seconds.
-==3.2. Approximations suitable for engineering use==
+===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. (more details in [28]).
+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].
-From Eq. (12), it can be seen that the algorithm from Figure 2 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).
-{| style="width: 100%;"
+<div id='img-2'></div>
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
-|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:Draft_Brkic_171642252-image2.png|342px]] </span>
+|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])
-<span style="text-align: center; font-size: 75%;">'''Figure 2.''' Algorithm for the procedure for solving Eq. (2) based on series expansions by Eqs. (3-7) and Eqs. (8-12).</span>
-In addition, 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, Eq. (21) can be constructed (this equation is already presented in [9]).
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 304: / Line 310: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\left. \begin{matrix}\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left( B-C+\frac{C}{x}\right) \\\begin{matrix}A\approx \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).
-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 <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.
-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 in order to avoid multiple repetitions of the logarithmic function, as C is computed only once.
-Comparing Eq. (22) with the maximal relative error of 0.1266% 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 321: / Line 326: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left[ B-C+\frac{C}{x}+\right. </math><math>\left. 0.00056\, \right]</math>
+| <math>\displaystyle\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left[ B-C+\displaystyle\frac{C}{x}+\right. </math><math>\left. 0.00056\, \right]</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (22)
@@ Line 331: / Line 336: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left[ B-C+\frac{C}{x}+\right. </math><math>\left. \frac{C}{2\cdot {x}^{2}}\cdot \left( C-2\right) -0.0014\right]</math>
+| <math>\displaystyle\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left[ B-C+\displaystyle\frac{C}{x}+\right. </math><math>\left. \displaystyle\frac{C}{2\cdot {x}^{2}}\cdot \left( C-2\right) -0.0014\right]</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (23)
@@ Line 347: / 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 significanty by the factor 0.129%/0.000391% ~ 330. Of course, the computational time of novel Eq. (25) is 1.1 seconds, as it is slower than Eq. (17) 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. Here, only the error function <math display="inline">\xi</math>  is developed using symbolic regression, whereas this technique is used in a more thorough way also in Section 4.2.
+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)
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 356: / 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}+\xi \, \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====
-===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">\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
-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 [9], whereas Eqs. (28) and (29) are novel simple, but very accurate approximations found by symbolic regression. Although Eureqa allows only minimization of the absolute error [13], we can use it also for minimizing the relative error; Gholamy and Kreinovich [29] 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) shows that this formulation significantly reduces the approximation error. Comparing Eq. (26) which is already available in [9] 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 [29] through symbolic regression, the maximal relative error of the friction factor was reduced from 0.0497% of Eq. (26) to 0.00229% of Eq. (28), i.e. more than 21 times. Moreover, Eqs. (26-28) are based on Eqs. (17-20) while Eq. (29) is improved Eq. (28) by adding the carefully chosen constant, 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 [30] with the relative error of no more than 0.0028% with three logarithmic and one noninteger power functions used (Clamond [31] reported that in a modern computer, evaluation of noniteger power goes also through logarithmic function). However, our novel Eq. (29) with the relative error of no more than 0.00224% is not only more accurate, but with two logarithmic function, it is also simpler (approaches with the only one logarithmic function [32], and even without [33], exist, but they are less accurate).
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 373: / Line 375: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left[ B-C+\frac{1.038\cdot C}{\mathrm{0.332+}\,x}\right] \,</math>
+| <math display="inline">\displaystyle\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left[ B-C+\displaystyle\frac{1.038\cdot C}{\mathrm{0.332+}\,x}\right] \,</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (26)
@@ Line 383: / Line 385: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left[ B-C+\mathrm{\frac{1.0119\cdot C}{x}}\,+\frac{\mathrm{C}\,-2.3849}{{x}^{2}}\right]</math>
+| <math display="inline">\displaystyle\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left[ B-C+\mathrm{\displaystyle\frac{1.0119\cdot C}{x}}\,+\frac{\mathrm{C}\,-2.3849}{{x}^{2}}\right]</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (27)
@@ Line 393: / Line 395: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left[ B-C+\frac{C}{x-0.5564\cdot C+1.207}\, \right]</math>
+| <math display="inline">\displaystyle\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left[ B-C+\displaystyle\frac{C}{x-0.5564\cdot C+1.207}\, \right]</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (28)
 |}
+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).
+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
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 403: / Line 413: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left[ B-C+\frac{C}{x-0.5564\cdot C+1.207}\, -\right. </math><math>\left. 0.00003\right]</math>
+| <math display="inline">\displaystyle\frac{1}{\sqrt{f}}\approx 0.8685972\cdot \left[ B-C+\displaystyle\frac{C}{x-0.5588\cdot C+1.2079}\, \right]</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (29)
@@ Line 409: / Line 419: @@
-In Eqs. (26-29), <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%, 0.00229% and 0.00224%, respectively.
+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>.
-<span style="text-align: center; font-size: 75%;">Coefficients of Eq. (28) can be further optimized, where the following vector should be used [0.8685972, 0.779626, 8.0897, 0.5588, 1.2079] instead of [0.8686, 0.7794, 8.0884, 0.5564, 1.207] in order to decrease the relative error to 0.0012%. The worst case for the updated coefficients is for </span> <math display="inline">Re</math><span style="text-align: center; font-size: 75%;">= 92957 and for </span> <math display="inline">\epsilon</math> <span style="text-align: center; font-size: 75%;">=3.1769·10<sup>-</sup><sup>7</sup> while computation of the benchmark sample tooks 0.388 seconds. </span>
+Algorithm for solving Eq. (29) is given in [[#img-3|Figure 3]] .
+<div id='img-3'></div>
-Eq.(26) - Eq.(29) require only 0.42 seconds. Algorithm for solving Eq. (29) is given in Figure 3.
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
-{| style="width: 100%;"
 |-
-|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:Draft_Brkic_171642252-image3.png|522px]] </span>
+|style="padding:10px;"|  [[Image:Review_419592774240-image3.png|426px]]
+|- style="text-align: center; font-size: 75%;"
+| colspan="1" style="padding:10px;"| '''Figure 3.''' Algorithm for solving Eq. (29)
 |}
-<span style="text-align: center; font-size: 75%;">'''Figure 3.''' Algorithm for solving Eq. (29).</span>
+Further, based on Eq. (20), thanks to the function for error minimization <math display="inline">{\xi }_{1}</math>, a novel very accurate approximation given with Eq. (30) can be constructed
-Based on Eq. (20), thanks to the function for error minimization <math display="inline">{\xi }_{1}</math>, a novel very accurate approximation given with Eq. (30) can be constructed.
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 430: / Line 438: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\left. \begin{matrix}\frac{1}{\sqrt{f}}\approx 0.868589\cdot \left[ B+Y-{\xi }_{1}\right] \, \\\begin{matrix}\begin{matrix}{\xi }_{1}=\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+\frac{C}{x-0.5564\cdot C+1.207}\end{matrix}\\A\approx \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 436: / Line 445: @@
-The computation speed of Eq.(30) is the same as Eq. (15), i.e. 1.1 seconds. However, let us compare the precision of the approximation given by the improved series expansion (see Eq. 15) with the precision of the novel symbolic regression-based approximation given by Eq. (30). The maximal relative error of Eq. (15) is <math display="inline">0.00527\%</math>, 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 <math display="inline">0.00527\%/0.000024\%\approx</math>  219, which confirms advantages of using symbolic regression for flow friction estimations.
+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.
-Before, Praks and Brkić [34] already developed approximations of the Colebrook equation based on symbolic regression analyses, which was recognized as a benchmark for feeding the automatic programming methods [35-38]. The study of Praks and Brkić [34] deals with symbolic regression with raw data based on the Colebrook equation by iterations and not through <math display="inline">\omega (x)-</math><math>x</math>. Belkić [39] provided a detailed review of applications of transcendental Lambert and Euler functions, which includes mathematics, physics, chemistry, biomedicine, ecologogy and sociology. Recently, approximations of the Colebrook equation based on symbolic regression of <math display="inline">\omega (x)-</math><math>x</math> [9] received a high ranking in large accuracy/speed benchmarks [40-43]. Further comparisons can be seen in [44-46].
+===3.3. Summary of accuracy and speed tests===
-==4. Conclusions and future work==
+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.
-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. Using methods presented in this paper, an explicit approximation of any desired accuracy can be constructed.
+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(!)).
-The proposed approximations should be compared with very recent comparative studies of the available approximations of the Colebrook equation [40-46]. Such comparisons will confirm that the approximations proposed in this review are both accurate and computationally efficient.
+<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>
-In terms of the balance among simplicity for human eyes, accuracy and computational simplicity, the novel Eq. (29) with an relative error of up to 0.00234% is highly recommended  (a simple algorithm shown in Figure 3 should be followed for engineering use). For very accurate software implementation, its more complex version given in Eq. (30) should be followed, as the maximal relative error is only 0.000024%. These formulas can be also used for indirect estimation of the flow friction rate using electrical parameters of pumps, when direct measurement is not possible, for example in offshore installations. For remote sensing control, the needed indirect electrical parameters of pump such as electrical pressure drop, must be measured accurately. Consequently, flow friction can be estimated indirectly from electrical parameters of the pump using the here proposed simple, but accurate flow friction equation.
+{| {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:60%;"
+|-
+!  <math display="inline">y\approx \omega (x)-x</math> !!  Approximation !! Maximal relative error !! Time
+|-
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Exact solution by built-in ''wrightOmega'' function of Matlab
+|-
+|  style="text-align: center;vertical-align: top;"|Eq. (2)
+|  style="vertical-align: top;"|
+|  style="text-align: center;vertical-align: top;"|0% i.e. exact
+|  style="text-align: center;"|11394 seconds ~<br/>189.9 minutes
+|-
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Asymptotic expansions
+|-
+|  style="text-align: center;vertical-align: top;"|Eq. (8)
+|  style="text-align: center;vertical-align: top;"|Eq. (21) – Figure 2
+|  style="text-align: center;vertical-align: top;"|0.153%
+|  style="text-align: center;vertical-align: top;"|0.4 seconds
+|-
+|  style="text-align: center;vertical-align: top;"|Eq. (9)
+|  style="text-align: center;vertical-align: top;"|Figure 2
+|  style="text-align: center;vertical-align: top;"|0.118%
+|  style="text-align: center;vertical-align: top;"|0.4 seconds
+|-
+|  style="text-align: center;vertical-align: top;"|Eq. (10)
+|  style="text-align: center;vertical-align: top;"|Figure 2
+|  style="text-align: center;vertical-align: top;"|0.008%
+|  style="text-align: center;vertical-align: top;"|1.1 seconds
+|-
+|  style="text-align: center;vertical-align: top;"|Eq. (11)
+|  style="text-align: center;vertical-align: top;"|Figure 2
+|  style="text-align: center;vertical-align: top;"|0.00249%
+|  style="text-align: center;vertical-align: top;"|2.4 seconds
+|-
+|  style="text-align: center;vertical-align: top;"|Eq. (12)
+|  style="text-align: center;vertical-align: top;"|Figure 2
+|  style="text-align: center;vertical-align: top;"|0.00247%
+|  style="text-align: center;vertical-align: top;"|4.4 seconds
+|-
+|  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)
+|  style="text-align: center;vertical-align: top;"|Eq. (22)
+|  style="text-align: center;vertical-align: top;"|0.129%
+|  style="text-align: center;vertical-align: top;"|0.4 seconds
+|-
+|  style="text-align: center;vertical-align: top;"|Eq. (14)
+|  style="text-align: center;vertical-align: top;"|Eq. (23)
+|  style="text-align: center;vertical-align: top;"|0.0691%
+|  style="text-align: center;vertical-align: top;"|0.4 seconds
+|-
+|  style="text-align: center;vertical-align: top;"|Eq. (15)
+|  style="text-align: center;vertical-align: top;"|Eq. (24)
+|  style="text-align: center;vertical-align: top;"|0.00527%
+|  style="text-align: center;vertical-align: top;"|1.1 seconds
+|-
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Asymptotic expansions with corrective function obtained using symbolic regression
+|-
+|  style="text-align: center;vertical-align: top;"|Eq. (16)
+|  style="text-align: center;vertical-align: top;"|Eq. (25)
+|  style="text-align: center;vertical-align: top;"|0.000391%
+|  style="text-align: center;vertical-align: top;"|1.1 seconds
+|-
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Symbolic regression
+|-
+|  style="text-align: center;vertical-align: top;"|Eq. (17)
+|  style="text-align: center;vertical-align: top;"|Eq. (26)
+|  style="text-align: center;vertical-align: top;"|0.0497%
+|  style="text-align: center;vertical-align: top;"|0.42 seconds
+|-
+|  style="text-align: center;vertical-align: top;"|Eq. (18)
+|  style="text-align: center;vertical-align: top;"|Eq. (27)
+|  style="text-align: center;vertical-align: top;"|0.0105%
+|  style="text-align: center;vertical-align: top;"|0.42 seconds
+|-
+|  style="text-align: center;vertical-align: top;"|Eq. (19)
+|  style="text-align: center;vertical-align: top;"|Eq. (28)
+|  style="text-align: center;vertical-align: top;"|0.00337%
+|  style="text-align: center;vertical-align: top;"|0.42 seconds
+|-
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Symbolic regression with the optimized coefficients
+|-
+|  style="vertical-align: top;text-align: center;"|Eq. (20);
-'''Author Contributions: '''P.P. worked mostly on the mathematical background of the proposed methods and procedures, while D.B. put it in a form suitable for engineering use. P.P. made also tests with symbolic regression tool and global optimization. D.B. made the draft of this paper.
+optimized Eq. (19)
+|  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;padding:10px;"|Symbolic regression with the optimized coefficients and corrective function
+|-
+|  style="vertical-align: top;text-align: center;"|Eq. (20)
+|  style="text-align: center;vertical-align: top;"|Eq. (30)
+|  style="text-align: center;vertical-align: top;"|0.000024%
+|  style="text-align: center;vertical-align: top;"|1.1 seconds
+|}
-'''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”.
+==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.
-'''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.
+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.
-'''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.
+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%.
-<span id='_Hlk32420670'></span>
+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 review:
+The following symbols are used in this paper:
 {| 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)
-| <math display="inline">y</math><span style="text-align: center; font-size: 75%;"> suitable for error minimization using function </span> <math display="inline">{\xi }_{1}</math>
 |-
-| <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]]
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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

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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