Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction
The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation i...
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doaj-577965594f3f4451aa7cbadf011e390b2020-11-24T20:42:45ZengHindawi LimitedAdvances in Civil Engineering1687-80861687-80942018-01-01201810.1155/2018/54510345451034Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow FrictionPavel Praks0Dejan Brkić1European Commission, Joint Research Centre (JRC), Directorate C: Energy, Transport and Climate, Unit C3: Energy Security, Distribution and Markets, Via Enrico Fermi 2749, 21027 Ispra (VA), ItalyEuropean Commission, Joint Research Centre (JRC), Directorate C: Energy, Transport and Climate, Unit C3: Energy Security, Distribution and Markets, Via Enrico Fermi 2749, 21027 Ispra (VA), ItalyThe empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.http://dx.doi.org/10.1155/2018/5451034 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Pavel Praks Dejan Brkić |
spellingShingle |
Pavel Praks Dejan Brkić Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction Advances in Civil Engineering |
author_facet |
Pavel Praks Dejan Brkić |
author_sort |
Pavel Praks |
title |
Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction |
title_short |
Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction |
title_full |
Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction |
title_fullStr |
Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction |
title_full_unstemmed |
Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction |
title_sort |
advanced iterative procedures for solving the implicit colebrook equation for fluid flow friction |
publisher |
Hindawi Limited |
series |
Advances in Civil Engineering |
issn |
1687-8086 1687-8094 |
publishDate |
2018-01-01 |
description |
The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%. |
url |
http://dx.doi.org/10.1155/2018/5451034 |
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