Intelligent Flow Friction Estimation

Brkić, Dejan; Ćojbašić, Žarko

doi:10.31219/osf.io/79ktd

Cited by 6 publications

(8 citation statements)

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“…Also, based on our findings, we provide an approximation, Equation 28, with the error of no more than 0.69% and 0.0617%. e Colebrook equation can also be approximately simulated using Artificial Neural Networks [37][38][39]. (4) Lambert W-function: Until now, the only one known way to express the Colebrook equation exactly in explicit way is through the Lambert W-function, λ � W(Re, ε/D) [3,8,[40][41][42][43], where further evaluation of the Lambert W-function can be only approximated [44][45][46][47][48].…”

Section: Introductionmentioning

confidence: 99%

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Praks

Brkić

2018

Advances in Civil Engineering

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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%.

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Section: Introductionmentioning

confidence: 99%

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Praks

Brkić

2018

Advances in Civil Engineering

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“…A equação de Colebrook-White (1937) tem sido considerada a mais precisa aproximação para determinação do coeficiente de perda de carga, sendo utilizada como padrão referencial (YOO;SINGH, 2005;HEYDARI et al, 2015;SHAIKH et al, 2015;BRKIĆ;ĆOJBAŠIĆ 2016), sendo válida para um amplo intervalo de aplicabilidade: 2x10³ < Re ≤ 10 8 e 0 ≤ Ɛ/D ≥ 0,05. Trata-se de uma formulação implícita, cuja resolução é obtida através de processos iterativos (YILDIRIM, 2009; OFFOR; ALABI, 2016b; BRKIĆ, 2016; BRKIĆ; ĆOJBAŠIĆ, 2017) e é representada pela equação 2:…”

Section: Introductionunclassified

Desempenho E Precisão De Equações Explícitas Do Coeficiente De Perda De Carga Em Regime De Fluxo Turbulento

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Peiter

et al. 2018

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A equação de Colebrook-White é amplamente utilizada para dimensionamento do coeficiente de perda de carga (f) de Darcy-Weisbach, sendo estipulada como padrão. Por ser uma formulação implícita, com soluções iterativas, diversas equações explícitas foram desenvolvidas para calcular o "f". O objetivo desse estudo foi avaliar o desempenho dessas aproximações, através do índice de desempenho (Id) e a precisão, através da média do erro relativo (MER), das equações explícitas em relação a implícita padrão, variando-se o intervalo do número de Reynolds de 4x10³ ≤ Re ≤ 10 8 e da rugosidade relativa de 10-6 ≤ Ɛ/D ≤ 5x10-2. As equações de Chen (3), Sonnad e Goudar (8), Buzzelli (9), Vantankhah e Kouchakzadeh (10), Fang et al. (11) e Offor e Alabi (13) apresentaram elevado Id, classificando-as como "Excelente" e elevada precisão quando comparadas com a aproximação de Colebrook

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“…Samadianfard [47] uses genetic programming, a sort of genetic algorithms, to develop his own explicit approximations to the Colebrook equation. Also genetic algorithms can be used together with some other techniques of artificial intelligence such as neural networks [50][51][52].…”

Section: Genetic Algorithm Optimization Techniquementioning

confidence: 99%

“…Genetic optimization is an alternative to the traditional optimal search approaches which make difficult finding the global optimum for nonlinear and multimodal optimization problems. Thus, genetic algorithms have been successful for example in solving combinatorial problems, control applications of parameter identification and control structure design, as well as in many other areas [47][48][49][50][51][52].…”

Section: Genetic Algorithm Optimization Techniquementioning

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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Intelligent Flow Friction Estimation

Cited by 6 publications

References 41 publications

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Desempenho E Precisão De Equações Explícitas Do Coeficiente De Perda De Carga Em Regime De Fluxo Turbulento

Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations

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Intelligent Flow Friction Estimation

Cited by 6 publications

References 41 publications

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Desempenho E Precisão De Equações Explícitas Do Coeficiente De Perda De Carga Em Regime De Fluxo Turbulento

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

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Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations