Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function
Abstract
:1. Introduction
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: | |
any | |
Variables: | |
variable that depends on and (dimensionless) | |
variable that depends on (dimensionless) | |
variable that depends on variables and (dimensionless) | |
Darcy (Moody) flow friction factor (dimensionless) | |
Reynolds number (dimensionless) | |
variable that depends on R (dimensionless) | |
variable in function on R and (dimensionless) | |
Relative roughness of inner pipe surface (dimensionless) | |
variables defined in Appendix A of this paper | |
Functions: | |
exponential function | |
log10 | logarithm with base 10 |
natural logarithm | |
Padé approximant | |
Lambert -function | |
ω | Wright ω-function |
Appendix A
- -
- Here, developed Equations (3), (5), and (6); Equations (A1)–(A3):
- -
- Here, developed Equation (4); Equations (A4)–(A6):As parameter is larger, the approximation is more accurate. The value, , 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 from the Equations (A1)–(A3) and Equations (A4)–(A6) should be calculated using Equation (A7).
- -
- Buzzelli [39]; (A8):
- -
- Zigrang and Sylvester [42]; (A9):
- -
- Serghides [43]; (A10):
- -
- Romeo et al. [41]; (A11):
- -
- Vatankhah and Kouchakzadeh [40]; (A12):
- -
- Barr [44]; (A13):
- -
- Serghides-simple [43]; (A14):
- -
- Chen [45]; (A15):
- -
- Fang et al. [46]; (A16):
- -
- Papaevangelou et al. [47]; (A17):
- -
- Vatankhah [14]; (A18):
- -
- Offor and Alabi [38]; (A19):
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R = 4000 | R = 104 | R = 105 | R = 106 | R = 107 | R = 108 | |
10−6 | 5.763586714 | 6.552354737 | 8.594740889 | 10.78188015 | 13.94025768 | 26.71930109 |
10−5 | 5.767379666 | 6.562009418 | 8.694474328 | 11.80401384 | 24.50329461 | 125.7849498 |
10−3 | 5.805329409 | 6.658658836 | 9.697953496 | 22.29514802 | 124.0554132 | #VALUE! |
10−2 | 6.186774452 | 7.63459358 | 20.09639172 | 122.325789 | #VALUE! | #VALUE! |
0.05 | 10.14320931 | 17.90904123 | 120.5960672 | #VALUE! | #VALUE! | #VALUE! |
R = 4000 | R = 104 | R = 105 | R = 106 | R = 107 | R = 108 | |
10−6 | 5.766606874 | 6.552971455 | 8.592338256 | 10.7784212 | 13.93654591 | 26.71669441 |
10−5 | 5.770385511 | 6.562602762 | 8.691991603 | 11.80037821 | 24.50049484 | 136.3596559 |
10−3 | 5.808193728 | 6.659024862 | 9.694862641 | 22.29214094 | 134.073966 | 1246.853296 |
10−2 | 6.188374207 | 7.633218988 | 20.093168 | 131.7885643 | 1244.552558 | 12,371.62215 |
0.05 | 10.13993873 | 17.90560354 | 129.5034606 | 1242.251823 | 12,369.31975 | 123,639.9564 |
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) |
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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
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 StyleBrkić, 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
APA StyleBrkić, D., & Praks, P. (2019). Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function. Mathematics, 7(1), 34. https://doi.org/10.3390/math7010034