Constraints for Using Lambert W Function-Based Explicit Colebrook–White Equation

Sonnad, Jagadeesh R.; Goudar, Chetan T.

doi:10.1061/(asce)0733-9429(2004)130:9(929)

Cited by 65 publications

(83 citation statements)

References 14 publications

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“…In this way the same accuracy is reached through the proposed less demanding procedure, after the same number of iterations as in the standard algorithm which uses -call in each iterative step. This is a good (Clamond 2009, Giustolisi et al 2011, Danish et al 2011, Winning and Coole 2013, Vatankhah 2018, Sonnad and Goudar 2004, Brkić 2012a, Winning and Coole 2015. The here presented iterative approach only introduces a computationally cheaper alternative to the standard iterative procedure.…”

Section: Discussionmentioning

confidence: 99%

“…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. The common way to solve it is to guess an initial value for friction factor and then to try to balance it using the iterative algorithm (Brkić 2017b) which needs to be terminated after the certain number of iterations when the final balanced value is reached.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Discussionmentioning

confidence: 99%

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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“…The Colebrook equation can be rearranged in explicit form only approximately [λ≈f(R, ε/D)] where approach with the Lambert W-function can be treated as partial exemption from this rule [6][7][8][60][61][62], but also, further evaluation of the Lambert W-function function is approximate.…”

Section: Introductionmentioning

confidence: 99%

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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“…Currently, the Colebrook White equation can be solved by personal computers and even an exact although complex solution was proposed [5,6]. Even with great accuracy [7,8], the evolution of computers reduces the usefulness of approximations with high computational costs.…”

Section: Introductionmentioning

confidence: 99%

An Improved Expression for a Classical Type of Explicit Approximation of the Colebrook White Equation with Only One Internal Iteration

Beluco¹,

Schettini²

2016

ijhe

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The solution of the Colebrook White equation continues to attract the attention of engineers and mathematicians. Some solutions proposed in recent years have two or more internal iterations and present relative complexity. At the same time, micro-computers and calculators have shown increased calculation capacity, virtually eliminating approximations that have a certain accuracy but that are too complex. A direct approximation of the equation allowing for quick and accurate solution is still useful. This technical note brings together some explicit approximations with only one internal iteration presented in the 1970s and 1980s into a single category and proposes a single model with five parameters to be determined. Moreover, this technical note also presents the values for these five parameters that minimize the mean square error related to the outcome of the Colebrook White equation, considered as a reference. The proposed model allows to obtain results with mean square error equal to 1.14 x 10 -8 , with maximum relative error of 4.44% and 96.78% of results below 1.00% of difference relative to Colebrook White equation, which can be considered quite satisfactory.

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Constraints for Using Lambert W Function-Based Explicit Colebrook–White Equation

Cited by 65 publications

References 14 publications

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

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

Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations

An Improved Expression for a Classical Type of Explicit Approximation of the Colebrook White Equation with Only One Internal Iteration

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Constraints for Using Lambert W Function-Based Explicit Colebrook–White Equation

Cited by 65 publications

References 14 publications

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

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

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

An Improved Expression for a Classical Type of Explicit Approximation of the Colebrook White Equation with Only One Internal Iteration

Contact Info

Product

Resources

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

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

Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations