scholarly journals Comparison of the Lambert W‐function based solutions to the Colebrook equation

2017 ◽  
Author(s):  
Dejan Brkić
Keyword(s):  
Iterative Procedure ◽  
Computer Code ◽  
Computer Hardware ◽  
Lambert W Function ◽  
Accuracy Analysis ◽  
Pipe Surface ◽  
Flow Friction ◽  
Relative Roughness ◽  
Modern Computer

Purpose– The Colebrook equation for determination of hydraulic resistances is implicit in fluid flow friction factor and hence it has to be approximately solved using iterative procedure or using some of the approximate explicit formulas developed by many authors. The purpose of this paper is to compare different Lambert W based solutions of the Colebrook equation and to make comparisons among them and identify some constraints in applicability of certain solutions.Design/methodology/approach– Alternate mathematically equivalents to the implicit Colebrook equation in explicit form with no approximation involved actually exist.Findings– These alternate equations were developed using Lambert W‐function. The paper compares various implementations of the Lambert W methodology and shows that some of these are less able than others to yield solutions using modern computer hardware. This is because the functions require the evaluation of terms with numerical values outside the ranges that can be expressed on most computers.Research limitations/implications– Some of existed transformations cannot be applied for high values of relative roughness of inner pipe surface and the Reynolds number. This limitation applied only for computer computations. Other presented transformations do not sufferer of this limitation.Practical implications– Presented procedures can be easily implemented in a computer code. Recommended solution can be used in all cases that can occur in engineering practice.Originality/value– The paper shows some possible practical procedures for solution of the transformed Colebrook equation. Accuracy analysis and comparisons of presented formulas are also performed and recommendation for use is shown.

scholarly journals solutions of the CW equation for flow friction

2017 ◽  
Author(s):  
Dejan Brkić
Keyword(s):  
Fluid Flow ◽  
Friction Factor ◽  
Explicit Form ◽  
Iterative Procedure ◽  
Lambert W Function ◽  
Explicit Formulas ◽  
Flow Friction ◽  
Transcendental Functions ◽  
Flow Through

The empirical Colebrook–White (CW) equation belongs to the group of transcendental functions. The CW function is used for the determination of hydraulic resistances associated with fluid flow through pipes, flow of rivers, etc. Since the CW equation is implicit in fluid flow friction factor, it has to be approximately solved using iterative procedure or using some of the approximate explicit formulas developed by many authors. Alternate mathematical equivalents to the original expression of the CW equation, but now in the explicit form developed using the Lambert W-function, are shown (with related solutions). The W-function is also transcendental, but it is used more general compared with the CW function. Hence, the solution to the W-function developed by mathematicians can be used effectively for the CW function which is of interest only for hydraulics.

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

2018 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Friction Factor ◽  
Iterative Methods ◽  
Lambert W Function ◽  
Symbolic Form ◽  
Pipe Surface ◽  
Flow Friction ◽  
Relative Roughness ◽  
Starting Point ◽  
High Level ◽  
Derivatives Of

Empirical Colebrook equation from 1939 is still accepted as an informal standard to calculate friction factor during the turbulent flow (4000 < Re < 108) through pipes from smooth with almost negligible relative roughness (ε/D→0) to the very rough (up to ε/D = 0.05) inner surface. The Colebrook equation contains flow friction factor λ in implicit logarithmic form where it is, aside of itself; λ, 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 many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate value of the friction factor λ from the Colebrook equation as accurate as possible. The most accurate way to achieve that is using some kind of iterative methods. Usually classical approach also known as simple fixed point method requires up to 8 iterations to achieve the high level of accuracy, but does not require derivatives of the Colebrook function as here presented accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method and 1st order: Newton-Raphson) which needs only 3 to 7 iteration and three-point iterative methods which needs only 1 to 4 iteration to achieve the same high level of accuracy. Strategies how to find derivatives of the Colebrook function in symbolic form, how to avoid use of the derivatives (Secant method) and how to choose optimal starting point for the iterative procedure are shown. Householder’s approach to the Colebrook’s equations expressed through the Lambert W-function is also analyzed. One approximation to the Colebrook equation based on the analysis from the paper with the error of no more than 0.0617% is shown.

scholarly journals An Explicit Approximation of Colebrook's Equation for Fluid Flow Friction Factor

2017 ◽  
Author(s):  
D. Brkić
Keyword(s):  
Fluid Flow ◽  
Friction Factor ◽  
Iterative Procedure ◽  
Lambert W Function ◽  
Explicit Formulas ◽  
Flow Friction

The Colebrook equation for determination of hydraulic resistances is implicit in fluid flow friction factor and hence it has to be approximately solved using iterative procedure or using some of the approximate explicit formulas which were developed by many authors. Here will be shown one approximation of the Colebrook equation based on Lambert W-function.

scholarly journals Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction

Water ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 1175 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Friction Factor ◽  
Computational Cost ◽  
Pipe Surface ◽  
Flow Friction ◽  
Relative Roughness ◽  
Programming Tool ◽  
Input And Output ◽  
Input Parameters ◽  
Logarithmic Law ◽  
Low Computational Cost

Widely used in hydraulics, the Colebrook equation for flow friction relates implicitly to the input parameters; the Reynolds number, Re and the relative roughness of an inner pipe surface, ε/D with an unknown output parameter; the flow friction factor, λ; λ = f (λ, Re, ε/D). In this paper, a few explicit approximations to the Colebrook equation; λ ≈ f (Re, ε/D), are generated using the ability of artificial intelligence to make inner patterns to connect input and output parameters in an explicit way not knowing their nature or the physical law that connects them, but only knowing raw numbers, {Re, ε/D}→{λ}. The fact that the used genetic programming tool does not know the structure of the Colebrook equation, which is based on computationally expensive logarithmic law, is used to obtain a better structure of the approximations, which is less demanding for calculation but also enough accurate. All generated approximations have low computational cost because they contain a limited number of logarithmic forms used for normalization of input parameters or for acceleration, but they are also sufficiently accurate. The relative error regarding the friction factor λ, in in the best case is up to 0.13% with only two logarithmic forms used. As the second logarithm can be accurately approximated by the Padé approximation, practically the same error is obtained also using only one logarithm.

scholarly journals Solution of the Implicit Colebrook Equation for Flow Friction Using Excel

2017 ◽  
Author(s):  
Dejan Brkić
Keyword(s):  
Reynolds Number ◽  
Friction Factor ◽  
Hydraulic Resistance ◽  
Iterative Solution ◽  
Pipe Surface ◽  
Implicit Equation ◽  
Flow Friction ◽  
Relative Roughness ◽  
Additional Task ◽  
Accepted Standard

Empirical Colebrook equation implicit in unknown ow friction factor (λ) is an accepted standard for calculation of hydraulic resistance in hydraulically smooth and rough pipes. e Colebrook equation gives friction factor (λ) implicitly as a function of the Reynolds number (Re) and relative roughness (ε/D) of inner pipe surface; i.e. λ0=f(λ0, Re, ε/D). e paper presents a problem that requires iterative methods for the solution. In particular, the implicit method used for calculating the friction factor λ0 is an application of xed- point iterations. e type of problem discussed in this "in the classroom paper" is commonly encountered in uid dynamics, and this paper provides readers with the tools necessary to solve similar problems. Students’ task is to solve the equation using Excel where the procedure for that is explained in this “in the classroom” paper. Also, up to date numerous explicit approximations of the Colebrook equation are available where as an additional task for students can be evaluation of the error introduced by these explicit approximations λ≈f(Re, ε/D) compared with the iterative solution of implicit equation which can be treated as accurate.

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

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Fixed Point ◽  
Friction Factor ◽  
Fixed Point Method ◽  
Point Method ◽  
Symbolic Form ◽  
Pipe Surface ◽  
Flow Friction ◽  
Relative Roughness ◽  
Starting Point ◽  
Derivatives Of

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%.

scholarly journals Symbolic Regression Based Genetic Approximations of the Colebrook Equation for Flow Friction

2018 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Friction Factor ◽  
Computational Cost ◽  
Pipe Surface ◽  
Flow Friction ◽  
Relative Roughness ◽  
Programming Tool ◽  
Input And Output ◽  
Input Parameters ◽  
Logarithmic Law ◽  
Low Computational Cost

Widely used in hydraulics, the Colebrook equation for flow friction relates implicitly to the input parameters; the Reynolds number, Re and the relative roughness of inner pipe surface, &epsilon;/D with the output unknown parameter; the flow friction factor, &lambda;; &lambda;=f(&lambda;, Re, &epsilon;/D). In this paper, a few explicit approximations to the Colebrook equation; &lambda;&asymp;f(Re, &epsilon;/D), are generated using the ability of artificial intelligence to make inner patterns to connect input and output parameters in explicit way not knowing their nature or the physical law that connects them, but only knowing raw numbers, {Re, &epsilon;/D}&rarr;{&lambda;}. The fact that the used genetic programming tool does not know the structure of the Colebrook equation which is based on computationally expensive logarithmic law, is used to obtain better structure of the approximations which is less demanding for calculation but also enough accurate. All generated approximations are with low computational cost because they contain a limited number of logarithmic forms used although for normalization of input parameters or for acceleration, but they are also sufficiently accurate. The relative error regarding the friction factor &lambda;, in best case is up to 0.13% with only two logarithmic forms used. As the second logarithm can be accurately approximated by the Pad&eacute; approximation, practically the same error is obtained also using only one logarithm.

scholarly journals Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations

2017 ◽  
Author(s):  
Dejan Brkić ◽  
Žarko Ćojbašić
Keyword(s):  
Turbulent Flow ◽  
Friction Factor ◽  
Iterative Solution ◽  
Good Prediction ◽  
Pipe Surface ◽  
Flow Friction ◽  
Relative Roughness ◽  
Complex Models ◽  
Transient Zone ◽  
Maximal Relative Error

Today, Colebrook&rsquo;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&rsquo;s equation relates the unknown flow friction factor (&lambda;) with the known Reynolds number (R) and the known relative roughness of inner pipe surface (&epsilon;/D). It is implicit in unknown friction factor (&lambda;). Implicit Colebrook&rsquo;s equation cannot be rearranged to derive friction factor (&lambda;) directly and therefore it can be solved only iteratively [&lambda;=f(&lambda;, R, &epsilon;/D)] or using its explicit approximations [&lambda;&asymp;f(R, &epsilon;/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 (&lambda;) and can reproduce accurately Colebrook&rsquo;s equation and its Moody&rsquo;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.

scholarly journals Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations

2017 ◽  
Author(s):  
Dejan Brkić ◽  
Žarko Ćojbašić
Keyword(s):  
Friction Factor ◽  
Iterative Solution ◽  
Evolutionary Optimization ◽  
Optimization Strategy ◽  
Pipe Surface ◽  
Turbulent Friction ◽  
Flow Friction ◽  
Relative Roughness ◽  
Transient Zone ◽  
Maximal Relative Error

This paper presents evolutionary optimization of explicit approximations of the empirical Colebrook’s equation that is used for the calculation of the turbulent friction factor (λ), i.e., for the calculation of turbulent hydraulic resistance in hydraulically smooth and rough pipes including the transient zone between them. The empirical Colebrook’s equation relates the unknown flow friction factor (λ) with the known Reynolds number (R) and the known relative roughness of the inner pipe surface (ε/D). It is implicit in the unknown friction factor (λ). The implicit Colebrook’s equation cannot be rearranged to derive the 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)], which introduce certain error compared with the iterative solution. The optimization of explicit approximations of Colebrook’s equation is performed with the aim to improve their accuracy, and the proposed optimization strategy is demonstrated on a large number of explicit approximations published up to date where numerical values of the parameters in various existing approximations are changed (optimized) using genetic algorithms to reduce maximal relative error. After that improvement, the computational burden stays unchanged while the accuracy of approximations increases in some of the cases very significantly.

Accuracy Analysis of Fringe Projection Systems Based on Blue Light Technology

2014 ◽  
Vol 615 ◽  
pp. 9-14 ◽  
Author(s):  
Claudio Bernal ◽  
Beatriz de Agustina ◽  
Marta María Marín ◽  
Ana Maria Camacho
Keyword(s):  
Blue Light ◽  
Point Clouds ◽  
Fringe Projection ◽  
Accuracy Analysis ◽  
Scanning Procedure ◽  
Structured Light Scanner ◽  
Cloud Of Points ◽  
3D Digitizing

Some manufacturers of 3D digitizing systems are developing and market more accurate, fastest and affordable systems of fringe projection based on blue light technology. The aim of the present work is the determination of the quality and accuracy of the data provided by the LED structured light scanner Comet L3D (Steinbichler). The quality and accuracy of the cloud of points produced by the scanner is determined by measuring a number of gauge blocks of different sizes. The accuracy range of the scanner has been established through multiple digitizations showing the dependence on different factors such as the characteristics of the object and scanning procedure. Although many factors influence, accuracies announced by manufacturer have been achieved under optimal conditions and it has been noted that the quality of the point clouds (density, noise, dispersion of points) provided by this system is higher than that obtained with laser technology devices.

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