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

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

==Pavel Praks and Dejan Brkić==

<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>

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]

-both authors contributed equally to this study

<span id='_Hlk39321727'></span>
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==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 <math>\omega</math>-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 <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]

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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 <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.

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].

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 [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;" 
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{| style="text-align: center; margin:auto;" 
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| <math display="inline">\left. \begin{matrix}\displaystyle\frac{1}{\sqrt{f}}=\displaystyle\frac{z}{2.51}\cdot \left( \mathrm{B}\,+y\right) \\z=\displaystyle\frac{2\cdot 2.51}{\mathrm{ln}\,\left( 10\right) }\\\begin{matrix}A=\displaystyle\frac{Re}{z}\cdot \displaystyle\frac{\epsilon }{3.71}\\B=\mathrm{ln}\,(Re)-ln(z)\\\begin{matrix}x=A+B\\y=W\left( {e}^{x}\right) -x=\omega (x)-x\end{matrix}\end{matrix}\end{matrix}\right\}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
|}


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

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

==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 <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.

===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====

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)

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math display="inline">{s}_{1}(x)\, =\mathrm{ln}\,\left( x\right) \cdot \left( \frac{1}{x}-1\right)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
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{| style="text-align: center; margin:auto;" 
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| <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)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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)
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{| style="text-align: center; margin:auto;" 
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| <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)
|}

As can be seen in [[#img-1|Figure 1]] , the function <math display="inline">y</math> needs to be evaluated only in the domain 7.51< <math display="inline">x</math><619, which is the one relevant for the Colebrook equation; Eq. (1). It is because the corresponding Reynolds number <math display="inline">Re</math> is between 4000 and 10<sup>8</sup>, while the roughness of inner pipe surface <math display="inline">\epsilon</math>  is from 0 to 0.05, as defined in the Moody diagram [42].

<div id='img-1'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
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|style="padding:10px;"|  [[Image:Review_419592774240-image1.png|288px]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figure 1.''' The strictly monotonic decreasing function <math display="inline">y=\omega (x)-</math><math>x</math> for the Colebrook equation
|}  


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

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].

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

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math display="inline">{y}_{1}\approx {s}_{1}(x)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
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{| style="text-align: center; margin:auto;" 
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| <math display="inline">{y}_{2}\approx {s}_{1}\left( x\right) +{s}_{2}\left( x\right)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
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{| style="text-align: center; margin:auto;" 
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| <math display="inline">{y}_{3}\approx {s}_{1}\left( x\right) +{s}_{2}\left( x\right) +</math><math>{s}_{3}\left( x\right)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
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{| style="text-align: center; margin:auto;" 
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| <math display="inline">{y}_{4}\approx {s}_{1}\left( x\right) +{s}_{2}\left( x\right) +</math><math>{s}_{3}\left( x\right) +{s}_{4}\left( x\right)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
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{| style="text-align: center; margin:auto;" 
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| <math display="inline">{y}_{5}\approx {s}_{1}\left( x\right) +{s}_{2}\left( x\right) +</math><math>{s}_{3}\left( x\right) +{s}_{4}\left( x\right) +{s}_{5}\left( x\right)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
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Eq. (8) is with a maximal relative error of no more than <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.153% and it needs 0.4 seconds to execute the benchmark sample; Eq. (9) is with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.118% and 0.4 seconds; Eq. (10) is with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.008% and 1.1 seconds; Eq. (11) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.00249% and 2.4 seconds, while Eq. (12) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.00247% and 4.4 seconds. The maximal error is defined with <math display="inline">{\delta }_{\%}=</math><math>\displaystyle \frac{f-{f}_{i}}{f}\cdot 100\%</math> where the maximal relative error of the approximated Darcy flow friction factor <math display="inline">{f}_{i}</math> is calculated with the reference to the accurate <math display="inline">f</math> calculated by the Matlab’s built-in command ''wrightOmega'' and Eq. (2). Alternatively, an accurate iterative solution of Eq. (1) is available, as explained in [26,54,59,60].

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

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)

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>y\approx {s}_{1}(x)+0.00056</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
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{| style="text-align: center; margin:auto;" 
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| <math>y\approx {s}_{1}\left( x\right) +{s}_{2}\left( x\right) -</math><math>0.0014</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
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{| style="text-align: center; margin:auto;" 
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| <math>y\approx {s}_{1}\left( x\right) +{s}_{2}\left( x\right) +</math><math>{s}_{3}\left( x\right) -0.000093</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
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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.

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)

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (16)
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The error function <math display="inline">\xi</math>  of Eq. (16) is developed using symbolic regression, while <math display="inline">\omega (x)-</math><math>x</math> will be approximated using the same approach in Section 3.1.2.

====3.1.2. Symbolic regression-based expansions of <math display="inline">y=\omega \left( x\right) -x</math> for 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)

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>{y}_{sr}\approx -\mathrm{ln}\,\left( x\right) +\displaystyle\frac{1.038\cdot \mathrm{ln}\,\left( x\right) }{x+0.332}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (17)
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (18)
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (19)
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{| style="text-align: center; margin:auto;" 
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| <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)
|}


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===

====3.2.1. Approximations and procedures based on series expansions of <math display="inline">\omega (x)-x</math> about infinity====

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

Following the algorithm from [[#img-2|Figure 2]], Eq. (12) can be executed (it introduces a relative error of no more than 0.00247% and that it needs around 4.4 seconds to execute 8 million of Quasi Monte-Carlo tested samples carefully selected from the domain of the Colebrook equation which is of interest for engineering practice).
<div id='img-2'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"| [[Image:Review_419592774240-image2.png|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])

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\left. \begin{matrix}\displaystyle\frac{1}{\sqrt{f}}\approx 0.8686\cdot \left( B-C+\displaystyle\frac{C}{x}\right) \\\begin{matrix}A\approx \displaystyle\frac{Re\cdot \epsilon }{8.0878}\\B\approx \mathrm{ln}\,\left( Re\right) -0.7794\\\begin{matrix}C=\mathrm{ln}\,\left( x\right) \\x=A+B\, \end{matrix}\end{matrix}\end{matrix}\right\}</math>
|}
| 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).

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

Comparing Eq. (22) with the maximal relative error of 0.129% and Eq. (21) with 0.153%, it is confirmed that adding the carefully chosen constant <math display="inline">\alpha</math>, gives more accurate results for the Colebrook equation

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <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)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <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)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| 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) +\mathrm{\frac{C}{6\cdot {x}^{3}}\cdot \, }\,\left( 2\cdot \, \mathrm{{C}^{2}}\,-9\cdot C+6\right) -\right. </math><math>\left. 0.000093\right]</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (24)
|}


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

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

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math display="inline">\left. \begin{matrix}\displaystyle\frac{1}{\sqrt{f}}\approx 0.86858896\cdot \left[ B-C+\displaystyle\frac{C}{x}+\xi \, \right] \, \\\begin{matrix}\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}\\\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>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (25)
|}

====3.2.2. Approximations based on symbolic regression====

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

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <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)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <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)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <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;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <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)
|}


In Eq. (29), <math display="inline">A\approx \displaystyle \frac{Re\cdot \epsilon }{8.0897}</math>, <math display="inline">B\approx \mathrm{ln}\,\left( Re\right) -0.779626</math>, <math display="inline">C=</math><math>\mathrm{ln}\,\left( x\right)</math>, and <math display="inline">x=A+B</math>.

Algorithm for solving Eq. (29) is given in [[#img-3|Figure 3]] .
<div id='img-3'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|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) 
|}


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

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math display="inline">\left. \begin{matrix}\displaystyle\frac{1}{\sqrt{f}}\approx 0.868589\cdot \left[ B+Y-{\xi }_{1}\right] \, \\\begin{matrix}\begin{matrix}{\xi }_{1}=\displaystyle\frac{x\cdot {Y}^{2}+3.0636\cdot x\cdot Y+18.58}{19.5\cdot \left( {Y}^{2}\cdot {x}^{2}+{x}^{3}\right) +169.9\cdot {Y}^{2}+1260\cdot x+18178}\\Y=-C+\displaystyle\frac{C}{x-0.5564\cdot C+1.207}\end{matrix}\\ \\ \\
A\approx \displaystyle\frac{Re\cdot \epsilon }{8.088387}\\\begin{matrix}B\approx \mathrm{ln}\,\left( Re\right) -0.7793975\\\begin{matrix}C=\mathrm{ln}\,\left( x\right) \\x=A+B\end{matrix}\end{matrix}\end{matrix}\end{matrix}\right\}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (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 [[#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.
<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>

{| {| 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="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;padding:10px;"|Exact solution by built-in ''wrightOmega'' function of Matlab
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (2)
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0% i.e. exact
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;text-align: center;"|11394 seconds ~

189.9 minutes
|-
|  colspan='4'  style="border-top: 1pt solid black;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="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (12)
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Figure 2
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.00247%
|  style="border-bottom: 1pt solid black;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="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (15)
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (24)
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.00527%
|  style="border-bottom: 1pt solid black;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="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (16)
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (25)
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.000391%
|  style="border-bottom: 1pt solid black;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="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (19)
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (28)
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.00337%
|  style="border-bottom: 1pt solid black;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="border-bottom: 1pt solid black;vertical-align: top;text-align: center;"|Eq. (20); 

optimized Eq. (19)
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (29) – Figure 3 
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.0012%
|  style="border-bottom: 1pt solid black;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="border-bottom: 1pt solid black;vertical-align: top;text-align: center;"|Eq. (20)
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (30)
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.000024%
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1.1 seconds
|}


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

==4. Conclusions and future work==

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:

{| style="width: 100%;border-collapse: collapse;" 
|-
| <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>
|-
| <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>
|-
| <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> 
|-
| <math>f</math>: Darcy (Moody) flow friction factor (dimensionless) – main output parameter
|-
| <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>
|-
| <math>Re</math>: Reynolds number (dimensionless) – main input parameter
|-
| <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>
|-
| <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>
|-
| <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
|-
| <math>Y</math>: approximation of <math display="inline">\omega (x)-</math><math>x</math> from Eq. (19) and used in Eq. (20)
|-
| <math>z</math>: constant (dimensionless); <math>z=\frac{2\cdot 2.51}{\mathrm{ln}\,\left( 10\right) }</math>
|-
| <math>\alpha</math>: constant (dimensionless); half of the smallest term in the asymptotic series
|-
| <math>\epsilon</math>: relative roughness of inner pipe surface (dimensionless) – main input parameter
|-
| <math>{\delta }_{\%}</math>: relative error (%); <math display="inline">{\delta }_{\%}=>\displaystyle \frac{f-{f}_{i}}{f}\cdot 100\%</math>
|-
| <math display="inline">\xi</math>  and <math display="inline">{\xi }_{1}</math>: error-functions developed using symbolic regression
|-
| <math>e</math>: exponential function
|-
| <math>{log}_{10}</math>: logarithm with base 10
|-
| <math>ln</math>: natural logarithm
|-
| <math>W</math>: Lambert <math display="inline">W</math>-function
|-
| <math>\omega</math>: wright <math display="inline">\omega</math> -function
|-
| <math>i</math>: counter
|-
| ''wrightOmega'': Matlab’s built-in function
|-
| ''Globalsearch'': Matlab’s built-in solver
|}
<!--'''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.

<pdf>Media:Praks_Brkic_2020a_2895_Praks_Brkic_2020 List of equations.pdf</pdf>

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<div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">

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