Efficient Resolution of the Colebrook Equation

Clamond, Didier

doi:10.1021/ie801626g

Cited by 99 publications

(95 citation statements)

References 9 publications

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“…number of logarithmic and power expression stays unchanged [9,18]. Only change of integer power to non-integer power in some approximation can increase computational burden, but even than not significantly.…”

Section: Explicit Approximations Of Colebrook's Equationmentioning

confidence: 99%

“…The Colebrook equation (1) relates hydraulic flow friction (λ) through Reynolds number (R) and relative roughness (ε/D) of inner pipe surface but in implicit way; λ=f(λ, R, ε/D) [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. On the other hand, to express flow friction (λ) in implicit way a number of approximations can be used .…”

Section: Introductionmentioning

confidence: 99%

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Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations

Brkić¹,

Ćojbašić²

2017

Preprint

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Today, Colebrook's equation is mostly accepted as an informal standard for modeling of turbulent flow in hydraulically smooth and rough pipes including transient zone in between. The empirical Colebrook's equation relates the unknown flow friction factor (λ) with the known Reynolds number (R) and the known relative roughness of inner pipe surface (ε/D). It is implicit in unknown friction factor (λ). Implicit Colebrook's equation cannot be rearranged to derive friction factor (λ) directly and therefore it can be solved only iteratively [λ=f(λ, R, ε/D)] or using its explicit approximations [λ≈f(R, ε/D)]. Of course, approximations carry in certain error compared with the iterative solution where the highest level of accuracy can be reached after enough number of iterations. The explicit approximations give a relatively good prediction of the friction factor (λ) and can reproduce accurately Colebrook's equation and its Moody's plot. Usually, more complex models of approximations are more accurate and vice versa. In this paper, numerical values of parameters in various existing approximations are changed (optimized) using genetic algorithms to reduce maximal relative error. After this improvement computational burden stays unchanged while accuracy of approximations increases in some of the cases very significantly.

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Section: Explicit Approximations Of Colebrook's Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations

Brkić¹,

Ćojbašić²

2017

Preprint

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“…Numerous evaluations of flow friction factor such as in the case of complex networks of pipes pose extensive burden for computers, so not only an accurate but also a simplified solution is required. Calculation of complex water or gas distribution networks (Brkić 2009, Brkić 2011ab, Praks et al 2015, Praks et al 2017 which requires few evaluations of logarithmic function for each pipe, presents a significant and extensive burden which available computer resources hardly can easily manage (Clamond 2009, Giustolisi et al 2011, Danish et al 2011, Winning and Coole 2013, Vatankhah 2018. The Colebrook equation is based on logarithmic law where the unknown flow friction factor is given implicitly, i.e., it appears on both sides of Equation (1) in form , from which it cannot be extracted analytically; an exception is through the Lambert -function (Keady 1998, Sonnad and Goudar 2004, Brkić 2011cd, Brkić 2012ab, Biberg 2017, Brkić 2017a.…”

Section: Introductionmentioning

confidence: 99%

One-Log Call Iterative Solution of the Colebrook Equation for Flow Friction Based on Padé Polynomials

Praks¹,

Brkić²

2018

Preprint

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Abstract:The eighty years old empirical Colebrook function widely used as an informal standard for hydraulic resistance relates implicitly the unknown flow friction factor , with the known Reynolds number and the known relative roughness of a pipe inner surface ;. It is based on logarithmic law in the form that captures the unknown flow friction factor in a way from which it cannot be extracted analytically. As an alternative to the explicit approximations or to the iterative procedures that require at least a few evaluations of computationally expensive logarithmic function or non-integer powers, this paper offers an accurate and computationally cheap iterative algorithm based on Padé polynomials with only one -call in total for the whole procedure (expensive -calls are substituted with Padé polynomials in each iteration with the exception of the first). The proposed modification is computationally less demanding compared with the standard approaches of engineering practice, but does not influence the accuracy or the number of iterations required to reach the final balanced solution.

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“…It can be used to determine pressure drop or flow rate in such pipes. Although the accuracy of empirical equation of Colebrook can be disputable, it is sometimes essential to produce a fast, accurate, and robust resolution of this equation, which is particularly necessary for the scientific intensive computations and very often for comparisons [5]. Unfortunately, the Colebrook equation suffers from being implicit with respect to the friction factor ( ).…”

Section: Introductionmentioning

confidence: 99%

Intelligent Flow Friction Estimation

Brkić¹,

Ćojbašić²

2017

Preprint

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Nowadays, the Colebrook equation is used as a mostly accepted relation for the calculation of fluid flow friction factor. However, the Colebrook equation is implicit with respect to the friction factor ( ). In the present study, a noniterative approach using Artificial Neural Network (ANN) was developed to calculate the friction factor. To configure the ANN model, the input parameters of the Reynolds Number (Re) and the relative roughness of pipe ( / ) were transformed to logarithmic scales. The 90,000 sets of data were fed to the ANN model involving three layers: input, hidden, and output layers with, 2, 50, and 1 neurons, respectively. This configuration was capable of predicting the values of friction factor in the Colebrook equation for any given values of the Reynolds number (Re) and the relative roughness ( / ) ranging between 5000 and 10 8 and between 10 −7 and 0.1, respectively. The proposed ANN demonstrates the relative error up to 0.07% which had the high accuracy compared with the vast majority of the precise explicit approximations of the Colebrook equation.

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Efficient Resolution of the Colebrook Equation

Cited by 99 publications

References 9 publications

Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations

Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations

One-Log Call Iterative Solution of the Colebrook Equation for Flow Friction Based on Padé Polynomials

Intelligent Flow Friction Estimation

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Efficient Resolution of the Colebrook Equation

Cited by 99 publications

References 9 publications

Evolutionary Optimization of Colebrook&rsquo;s Turbulent Flow Friction Approximations

Evolutionary Optimization of Colebrook&rsquo;s Turbulent Flow Friction Approximations

One-Log Call Iterative Solution of the Colebrook Equation for Flow Friction Based on Pad&eacute; Polynomials

Intelligent Flow Friction Estimation

Contact Info

Product

Resources

About

Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations

Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations

One-Log Call Iterative Solution of the Colebrook Equation for Flow Friction Based on Padé Polynomials