# Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{8}and for 0 < ${\epsilon}^{*}$ < 0.05. The Colebrook equation is transcendental (cannot be expressed in terms of elementary functions); the implicitly given function in respect to the unknown flow friction factor, $f$:

## 2. Proposed Explicit Approximations and Comparative Analysis

#### 2.1. Transformation and Formulation

#### 2.2. Accuracy

#### 2.3. Complexity and Computational Burden

#### 2.4. Simplifications

## 3. Software Description

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Constants: | |

$a$ | any $>{10}^{5}$ |

Variables: | |

$A$ | variable that depends on $R$ and ${\epsilon}^{*}$ (dimensionless) |

$B$ | variable that depends on $R$ (dimensionless) |

$C$ | variable that depends on variables $A$ and $B$ (dimensionless) |

$f$ | Darcy (Moody) flow friction factor (dimensionless) |

$R$ | Reynolds number (dimensionless) |

$r$ | variable that depends on R (dimensionless) |

$x$ | variable in function on R and ${\epsilon}^{*}$ (dimensionless) |

${\epsilon}^{*}$ | Relative roughness of inner pipe surface (dimensionless) |

$\alpha $ | variables defined in Appendix A of this paper |

Functions: | |

$e$ | exponential function |

log10 | logarithm with base 10 |

$\mathrm{ln}$ | natural logarithm |

$s$ | Padé approximant |

$W$ | Lambert $W$-function |

ω | Wright ω-function |

## Appendix A

- -
- Here, developed Equations (3), (5), and (6); Equations (A1)–(A3):$$\frac{1}{\sqrt{f}}\approx 0.8686\xb7\left[B+C\xb7\left(\frac{1}{B+A}-1\right)\right]$$$$\frac{1}{\sqrt{f}}\approx 0.8686\xb7\left[B+\frac{1.038\xb7C}{0.332+B+A}-C\right]$$$$\frac{1}{\sqrt{f}}\approx 0.8686\xb7\left[B+\frac{1.0119\xb7C}{B+A}-C+\frac{C-2.3849}{{\left(B+A\right)}^{2}}\right]$$
- -
- Here, developed Equation (4); Equations (A4)–(A6):$$\frac{1}{\sqrt{f}}\approx 0.8686\xb7\left[B+\left(a\xb7{\left(B+A\right)}^{{a}^{-1}}-a\right)\xb7\left(\frac{1}{B+A}-1\right)\right]$$$$\frac{1}{\sqrt{f}}\approx 0.8686\xb7\left[B+\frac{1.038\xb7\left(a\xb7{\left(B+A\right)}^{{a}^{-1}}-a\right)}{0.332+B+A}-\left(a\xb7{\left(B+A\right)}^{{a}^{-1}}-a\right)\right]$$$$\frac{1}{\sqrt{f}}\approx 0.8686\xb7\left[B+\frac{1.0119\xb7\left(a\xb7{\left(B+A\right)}^{{a}^{-1}}-a\right)}{B+A}-\left(a\xb7{\left(B+A\right)}^{{a}^{-1}}-a\right)+\frac{\left(a\xb7{\left(B+A\right)}^{{a}^{-1}}-a\right)-2.3849}{{\left(B+A\right)}^{2}}\right]$$As parameter $a$ is larger, the approximation is more accurate. The value, $a>{10}^{5}$, gives the sufficiently accurate approximation for gas hydraulic modelling, as the corresponding maximal relative error is less than 0.007% for the analysed Colebrook model.
- -
- Here, developed Equation (11); Equation (A7):Parameter $B$ from the Equations (A1)–(A3) and Equations (A4)–(A6) should be calculated using Equation (A7).$$\begin{array}{c}B\approx s\xb7\left(0.0001086\xb7{s}^{6}+0.9824\right)-\frac{0.006206}{r}-r\xb7\left(0.000007237\xb7r-0.006656\right)+11.881\\ \begin{array}{c}r=\frac{R}{315,012.6}\\ s\approx s\left(r\right)=\frac{r\xb7\left(r\xb7\left(11\xb7r+27\right)-27\right)-11}{r\xb7\left(r\xb7\left(3\xb7r+27\right)+27\right)+3}\end{array}\end{array}\}$$
- -
- Buzzelli [39]; (A8):$$\begin{array}{c}\frac{1}{\sqrt{f}}\approx {\alpha}_{1}-\left(\frac{{\alpha}_{1}+2\xb7{\mathrm{log}}_{10}\left(\frac{{\alpha}_{2}}{R}\right)}{1+\frac{2.18}{{\alpha}_{2}}}\right)\\ {\alpha}_{1}\approx \frac{\left(0.774\xb7\mathrm{ln}\left(R\right)\right)-1.41}{1+1.32\xb7\sqrt{{\epsilon}^{*}}}\\ {\alpha}_{2}\approx \frac{{\epsilon}^{*}}{3.7}\xb7R+2.51\xb7{\alpha}_{1}\end{array}\}$$
- -
- Zigrang and Sylvester [42]; (A9):$$\begin{array}{c}\frac{1}{\sqrt{f}}\approx -2\xb7{\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.7}-\frac{5.02}{R}\xb7{\alpha}_{3}\right)\\ {\alpha}_{3}\approx {\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.7}-\frac{5.02}{R}\xb7{\alpha}_{4}\right)\\ {\alpha}_{4}\approx {\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.7}-\frac{13}{R}\right)\end{array}\}$$
- -
- Serghides [43]; (A10):$$\begin{array}{c}\begin{array}{c}\frac{1}{\sqrt{f}}\approx {\alpha}_{5}-\frac{{\left({\alpha}_{6}-{\alpha}_{5}\right)}^{2}}{{\alpha}_{7}-2\xb7{\alpha}_{6}+{\alpha}_{5}}\\ {\alpha}_{5}\approx -2\xb7{\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.7}-\frac{12}{R}\right)\end{array}\\ {\alpha}_{6}\approx -2\xb7{\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.7}-\frac{2.51}{R}\xb7{\alpha}_{5}\right)\\ {\alpha}_{7}\approx -2\xb7{\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.7}-\frac{2.51}{R}\xb7{\alpha}_{6}\right)\end{array}\}$$
- -
- Romeo et al. [41]; (A11):$$\begin{array}{c}\frac{1}{\sqrt{f}}\approx -2\xb7{\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.7065}-\frac{5.0272}{R}\xb7{\alpha}_{8}\right)\\ {\alpha}_{8}\approx {\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.827}-\frac{4.567}{R}\xb7{\alpha}_{9}\right)\\ {\alpha}_{9}\approx {\mathrm{log}}_{10}\left({\left(\frac{{\epsilon}^{*}}{7.7918}\right)}^{0.9924}+{\left(\frac{5.3326}{208.815+R}\right)}^{0.9345}\right)\end{array}\}$$
- -
- Vatankhah and Kouchakzadeh [40]; (A12):$$\begin{array}{c}\frac{1}{\sqrt{f}}\approx 0.8686\xb7\mathrm{ln}\left(\frac{0.4587\xb7R}{{\left({\alpha}_{10}-0.31\right)}^{{\alpha}_{11}}}\right)\\ {\alpha}_{10}\approx 0.124\xb7R\xb7{\epsilon}^{*}+\mathrm{ln}\left(0.1587\xb7R\right)\\ {\alpha}_{11}\approx \frac{{\alpha}_{10}}{{\alpha}_{10}+0.9633}\end{array}\}$$
- -
- Barr [44]; (A13):$$\begin{array}{c}\frac{1}{\sqrt{f}}\approx -2\xb7{\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.7}+\frac{4.518\xb7{\mathrm{log}}_{10}\left(\frac{R}{7}\right)}{{\alpha}_{12}}\right)\\ {\alpha}_{12}\approx R\xb7\left(1+\frac{{R}^{0.52}}{29}\xb7{\left({\epsilon}^{*}\right)}^{0.7}\right)\end{array}\}$$
- -
- Serghides-simple [43]; (A14):$$\begin{array}{c}\frac{1}{\sqrt{f}}\approx 4.781-\frac{{\left({\alpha}_{13}-4.781\right)}^{2}}{{\alpha}_{14}-2\xb7{\alpha}_{13}+4.781}\\ {\alpha}_{13}\approx -2\xb7{\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.7}-\frac{12}{R}\right)\\ {\alpha}_{14}\approx -2\xb7{\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.7}-\frac{2.51}{R}\xb7{\alpha}_{13}\right)\end{array}\}$$
- -
- Chen [45]; (A15):$$\begin{array}{c}\frac{1}{\sqrt{f}}\approx -2\xb7{\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.7065}-\frac{5.0452}{R}\xb7{\alpha}_{15}\right)\\ {\alpha}_{15}\approx {\mathrm{log}}_{10}\left(\frac{{\left({\epsilon}^{*}\right)}^{1.1098}}{2.8257}+\frac{5.8506}{{R}^{0.8981}}\right)\end{array}\}$$
- -
- Fang et al. [46]; (A16):$$\begin{array}{c}\frac{1}{\sqrt{f}}\approx {(1.613\xb7{(\mathrm{ln}(0.234\xb7{({\epsilon}^{*})}^{1.1007}-{\alpha}_{16}))}^{-2})}^{-2}\\ {\alpha}_{16}\approx \frac{60.525}{{R}^{1.1105}}+\frac{56.291}{{R}^{1.0712}}\end{array}\}$$
- -
- Papaevangelou et al. [47]; (A17):$$\frac{1}{\sqrt{f}}\approx {\left(\frac{0.2479-0.0000947\xb7{\left(7-{\mathrm{log}}_{10}\left(R\right)\right)}^{4}}{{\left({\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.615}+\frac{7.366}{{R}^{0.9142}}\right)\right)}^{2}}\right)}^{-2}$$
- -
- Vatankhah [14]; (A18):$$\begin{array}{c}\frac{1}{\sqrt{f}}\approx 0.8686\xb7\mathrm{ln}\left(\frac{0.3984\xb7R}{{\left(0.8686\xb7{\alpha}_{17}\right)}^{\frac{{\alpha}_{17}}{{\alpha}_{17}+{\alpha}_{18}}}}\right)\\ \begin{array}{c}{\alpha}_{17}\approx 0.12363\xb7R\xb7{\epsilon}^{*}+\mathrm{ln}\left(0.3984\xb7R\right)\\ {\alpha}_{18}\approx 1+\frac{1}{\frac{1+{\alpha}_{17}}{0.5\xb7\mathrm{ln}\left(0.8686\xb7{\alpha}_{17}\right)}-\frac{1+4\xb7{\alpha}_{17}}{3\xb7\left(1+{\alpha}_{17}\right)}}\end{array}\end{array}\}$$
- -
- Offor and Alabi [38]; (A19):$$\begin{array}{c}\frac{1}{\sqrt{f}}\approx -2\xb7{\mathrm{log}}_{10}\left(\frac{{\epsilon}^{*}}{3.71}-\frac{1.975}{R}\xb7{\alpha}_{19}\right)\\ {\alpha}_{19}\approx \mathrm{ln}({\left(\frac{{\epsilon}^{*}}{3.93}\right)}^{1.092}+\frac{7.627}{R+395.9})\end{array}\}$$

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**Figure 1.**Distribution of the relative error by the proposed explicit approximation of Colebrook’s equation; (

**a**) Equation (3), (

**b**) Equation (5) and (

**c**) Equation (6); comparison.

**Table 1.**Values of $W\left({e}^{x}\right)$ compared with its approximate replacement, $y\approx x-\mathrm{ln}x+\frac{\mathrm{ln}x}{x}$.

$\mathit{W}\mathbf{\left(}{\mathit{e}}^{\mathit{x}}\mathbf{\right)}$ | R = 4000 | R = 10^{4} | R = 10^{5} | R = 10^{6} | R = 10^{7} | R = 10^{8} |

${\epsilon}^{*}=$10^{−6} | 5.763586714 | 6.552354737 | 8.594740889 | 10.78188015 | 13.94025768 | 26.71930109 |

${\epsilon}^{*}=$10^{−5} | 5.767379666 | 6.562009418 | 8.694474328 | 11.80401384 | 24.50329461 | 125.7849498 |

${\epsilon}^{*}=$10^{−3} | 5.805329409 | 6.658658836 | 9.697953496 | 22.29514802 | 124.0554132 | #VALUE! |

${\epsilon}^{*}=$10^{−2} | 6.186774452 | 7.63459358 | 20.09639172 | 122.325789 | #VALUE! | #VALUE! |

${\epsilon}^{*}=$0.05 | 10.14320931 | 17.90904123 | 120.5960672 | #VALUE! | #VALUE! | #VALUE! |

$\mathit{y}$ | R = 4000 | R = 10^{4} | R = 10^{5} | R = 10^{6} | R = 10^{7} | R = 10^{8} |

${\epsilon}^{*}=$10^{−6} | 5.766606874 | 6.552971455 | 8.592338256 | 10.7784212 | 13.93654591 | 26.71669441 |

${\epsilon}^{*}=$10^{−5} | 5.770385511 | 6.562602762 | 8.691991603 | 11.80037821 | 24.50049484 | 136.3596559 |

${\epsilon}^{*}=$10^{−3} | 5.808193728 | 6.659024862 | 9.694862641 | 22.29214094 | 134.073966 | 1246.853296 |

${\epsilon}^{*}=$10^{−2} | 6.188374207 | 7.633218988 | 20.093168 | 131.7885643 | 1244.552558 | 12,371.62215 |

${\epsilon}^{*}=$0.05 | 10.13993873 | 17.90560354 | 129.5034606 | 1242.251823 | 12,369.31975 | 123,639.9564 |

**Table 2.**Number of computationally expensive functions in the available approximations of the Colebrook equations that introduce a relative error of no more than 1%.

^{1} Approximation | Maximal Relative Error % | Function | ||
---|---|---|---|---|

Logarithms | Non-Integer Powers | ^{2} TOTAL | ||

Vatankhah [14] | 0.0028% | 1 | 2 | 3(5) |

Here developed; Equation (6) | 0.0096% | 2 | 0 | 2 |

Here developed; Equation (5) | 0.045%, | 2 | 0 | 2 |

Offor and Alabi [38] | 0.0602% | 2 | 1 | 3(4) |

Here developed; Equation (3) | 0.13% | 2 | 0 | 2 |

Here developed; Equation (4) | 0.13% | 0 | 2 | 2(4) |

^{3} Buzzelli [39] | 0.14% | 2 | 0 | 2 |

Zigrang and Sylvester [42] | 0.14% | 3 | 0 | 3 |

Serghides [43] | 0.14% | 3 | 0 | 3 |

Romeo et al. [41] | 0.14% | 3 | 2 | 5(7) |

Vatankhah and Kouchakzadeh [40] | 0.15% | 2 | 1 | 3(4) |

Barr [44] | 0.27% | 2 | 2 | 4(6) |

Serghides-simple [43] | 0.35% | 2 | 0 | 2 |

Chen [45] | 0.36% | 2 | 2 | 4(6) |

Here developed; Equation (11) | Up to 0.4% | 1 | 0 | 1 |

Fang et al. [46] | 0.62% | 1 | 3 | 4(7) |

Papaevangelou et al. [47] | 0.82% | 2 | 1 | 3(4) |

^{1}All approximations are listed in the Appendix A of this paper,

^{2}in brackets: according to Clamond [12], non-integer powers require the evaluation of two computationally expensive functions–logarithm and exponential function,

^{3}in addition also contains one square root function.

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## Share and Cite

**MDPI and ACS Style**

Brkić, D.; Praks, P. Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function. *Mathematics* **2019**, *7*, 34.
https://doi.org/10.3390/math7010034

**AMA Style**

Brkić D, Praks P. Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function. *Mathematics*. 2019; 7(1):34.
https://doi.org/10.3390/math7010034

**Chicago/Turabian Style**

Brkić, Dejan, and Pavel Praks. 2019. "Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function" *Mathematics* 7, no. 1: 34.
https://doi.org/10.3390/math7010034