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 Review of explicit approximations to the Colebrook relation for flow friction

2017 ◽  
Author(s):  
Dejan Brkić
Keyword(s):  
Friction Factor ◽  
Iterative Solution ◽  
Pipe System ◽  
Flow Friction ◽  
Relative Roughness ◽  
Wide Range ◽  
Accepted Standard ◽  
One Step ◽  
Hydraulic Friction Factor ◽  
Moody Diagram

Because of Moody's chart has demonstrated applicability of the Colebrook equation over a very wide range of Reynolds number and relative roughness values, this equation becomes the accepted standard of accuracy for calculated hydraulic friction factor. Colebrook equation suffers from being implicit in unknown friction factor and thus requires an iterative solution where convergence to 0.01% typically requires less than 7 iterations. Implicit Colebrook equation cannot be rearranged to derive friction factor directly in one step. Iterative calculus can cause a problem in simulation of flow in a pipe system in which it may be necessary to evaluate friction factor hundreds or thousands of times. This is the main reason for attempting to develop a relationship that is a reasonable approximation for the Colebrook equation but which is explicit in friction factor. A review of existing explicit approximation of the implicit Colebrook equation with estimated accuracy is shown in this paper. Estimated accuracy compared with iterative solution of implicit Colebrook equation is shown for the entire range of turbulence where Moody diagram should be used as the reference. Finally, it can be concluded that most of the available approximations of the Colebrook equation, with a few exceptions, are very accurate with deviations of no more than few percentages.

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’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.

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.

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

Energies ◽  
2018 ◽  
Vol 11 (7) ◽  
pp. 1825 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Friction Factor ◽  
Iterative Solution ◽  
Engineering Practice ◽  
Logarithmic Function ◽  
Flow Friction ◽  
Relative Roughness ◽  
Logarithmic Law ◽  
Proposed Modification ◽  
Iterative Procedures ◽  
Balanced Solution

The 80 year-old empirical Colebrook function ξ, widely used as an informal standard for hydraulic resistance, relates implicitly the unknown flow friction factor λ, with the known Reynolds number Re and the known relative roughness of a pipe inner surface ε*; λ=ξ(Re,ε*,λ). It is based on logarithmic law in the form that captures the unknown flow friction factor λ in a way that 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 log-call in total for the whole procedure (expensive log-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.

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 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 &lt; Re &lt; 108) through pipes from smooth with almost negligible relative roughness (&epsilon;/D&rarr;0) to the very rough (up to &epsilon;/D = 0.05) inner surface. The Colebrook equation contains flow friction factor &lambda; in implicit logarithmic form where it is, aside of itself; &lambda;, a function of the Reynolds number Re and the relative roughness of inner pipe surface &epsilon;/D; &lambda; = f (&lambda;, Re, &epsilon;/D). To evaluate the error introduced by many available explicit approximations to the Colebrook equation, &lambda; &asymp; f(Re, &epsilon;/D), it is necessary to determinate value of the friction factor &lambda; 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&rsquo;s approach (3rd order, 2nd order: Halley&rsquo;s and Schr&ouml;der&rsquo;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&rsquo;s approach to the Colebrook&rsquo;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 Intelligent Flow Friction Estimation

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Dejan Brkić ◽  
Žarko Ćojbašić
Keyword(s):  
Neural Network ◽  
Artificial Neural Network ◽  
Reynolds Number ◽  
Friction Factor ◽  
High Accuracy ◽  
Ann Model ◽  
Flow Friction ◽  
Relative Roughness ◽  
Friction Estimation ◽  
Artificial Neural Network Ann

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 (λ). 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 (ε/D) 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 (ε/D) ranging between 5000 and 108and between 10−7and 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.

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

2018 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Friction Factor ◽  
Iterative Solution ◽  
Engineering Practice ◽  
Logarithmic Function ◽  
Flow Friction ◽  
Relative Roughness ◽  
Logarithmic Law ◽  
Proposed Modification ◽  
Iterative Procedures ◽  
Balanced Solution

The eighty years old empirical Colebrook function &nbsp;widely used as an informal standard for hydraulic resistance relates implicitly the unknown flow friction factor , with the known Reynolds number &nbsp;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 &nbsp;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 Pad&eacute; polynomials with only one -call in total for the whole procedure (expensive -calls are substituted with Pad&eacute; 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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