The Experts below are selected from a list of 2649 Experts worldwide ranked by ideXlab platform
Dejan Brkic - One of the best experts on this subject based on the ideXlab platform.
-
Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression
Computation, 2019Co-Authors: Dejan Brkic, Pavel PraksAbstract:The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and Wright Ω functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Pade approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence.
-
accurate and efficient explicit approximations of the Colebrook flow friction equation based on the wright ω function reply to discussion
Mathematics, 2019Co-Authors: Dejan Brkic, Pavel PraksAbstract:This reply gives two corrections of typographical errors in respect to the commented article, and then provides few comments in respect to the discussion and one improved version of the approximation of the Colebrook equation for flow friction, based on the Wright ω-function. Finally, this reply gives an exact explicit version of the Colebrook equation expressed through the Wright ω-function, which does not introduce any additional errors in respect to the original equation. All mentioned approximations are computationally efficient and also very accurate. Results are verified using more than 2 million of Quasi Monte-Carlo samples.
-
Colebrook s flow friction explicit approximations based on fixed point iterative cycles and symbolic regression
Computation, 2019Co-Authors: Dejan Brkic, Pavel PraksAbstract:The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and Wright Ω functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Pade approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence.
-
accurate and efficient explicit approximations of the Colebrook flow friction equation based on the wright ω function
arXiv: Numerical Analysis, 2018Co-Authors: Dejan Brkic, Pavel PraksAbstract:The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor, f . To date, the captured flow friction factor, f , can be extracted from the logarithmic form analytically only in the term of the Lambert W -function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert W -function also known as the Wright ω-function. The Wright ω-function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term, y = W ( e x ) , of the Lambert W -function to series expansions that further can be easily evaluated in computers without causing overflow run-time errors. Although the Colebrook equation transformed through the Lambert W -function is identical to the original expression in terms of accuracy, a further evaluation of the Lambert W -function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contain only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with a relative error of no more than 0.0096%. The presented approximations are in a form suitable for everyday engineering use, and are both accurate and computationally efficient.
-
advanced iterative procedures for solving the implicit Colebrook equation for fluid flow friction
Advances in Civil Engineering, 2018Co-Authors: Pavel Praks, Dejan BrkicAbstract: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 (e/D ⟶ 0) to very rough (up to e/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 e/D: λ = f(λ, Re, e/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, e/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 Schroder’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%.
Pavel Praks - One of the best experts on this subject based on the ideXlab platform.
-
Discussion of “Approximate Analytical Solutions for the Colebrook Equation” by Ali R. Vatankhah
Journal of Hydraulic Engineering, 2020Co-Authors: Dejan Brkić, Pavel PraksAbstract:This contribution discuss computational effective approximations of the Colebrook equation, which is widely used for hydraulic gas modeling. The aim is to accelerate the hydraulic gas modeling procedure, which is used for risk assessment of gas networks.In our contribution, accurate and computational accurate explicit approximations were found using artificial intelligent techniques. Finally, fast iterative methods were successfully used for finding the approximate solution of the Colebrook equation.JRC.C.3-Energy Security, Distribution and Market
-
Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression
Computation, 2019Co-Authors: Dejan Brkic, Pavel PraksAbstract:The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and Wright Ω functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Pade approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence.
-
accurate and efficient explicit approximations of the Colebrook flow friction equation based on the wright ω function reply to discussion
Mathematics, 2019Co-Authors: Dejan Brkic, Pavel PraksAbstract:This reply gives two corrections of typographical errors in respect to the commented article, and then provides few comments in respect to the discussion and one improved version of the approximation of the Colebrook equation for flow friction, based on the Wright ω-function. Finally, this reply gives an exact explicit version of the Colebrook equation expressed through the Wright ω-function, which does not introduce any additional errors in respect to the original equation. All mentioned approximations are computationally efficient and also very accurate. Results are verified using more than 2 million of Quasi Monte-Carlo samples.
-
Colebrook s flow friction explicit approximations based on fixed point iterative cycles and symbolic regression
Computation, 2019Co-Authors: Dejan Brkic, Pavel PraksAbstract:The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and Wright Ω functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Pade approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence.
-
accurate and efficient explicit approximations of the Colebrook flow friction equation based on the wright ω function
arXiv: Numerical Analysis, 2018Co-Authors: Dejan Brkic, Pavel PraksAbstract:The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor, f . To date, the captured flow friction factor, f , can be extracted from the logarithmic form analytically only in the term of the Lambert W -function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert W -function also known as the Wright ω-function. The Wright ω-function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term, y = W ( e x ) , of the Lambert W -function to series expansions that further can be easily evaluated in computers without causing overflow run-time errors. Although the Colebrook equation transformed through the Lambert W -function is identical to the original expression in terms of accuracy, a further evaluation of the Lambert W -function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contain only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with a relative error of no more than 0.0096%. The presented approximations are in a form suitable for everyday engineering use, and are both accurate and computationally efficient.
Praks Pavel - One of the best experts on this subject based on the ideXlab platform.
-
Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function
2020Co-Authors: Brkić Dejan, Praks PavelAbstract:The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor, f . To date, the captured flow friction factor, f , can be extracted from the logarithmic form analytically only in the term of the Lambert W-function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert W-function also known as the Wright ω-function. The Wright ω-function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term, y=W(ex), of the Lambert W-function to series expansions that further can be easily evaluated in computers without causing overflow run-time errors. Although the Colebrook equation transformed through the Lambert W-function is identical to the original expression in terms of accuracy, a further evaluation of the Lambert W-function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contain only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with a relative error of no more than 0.0096%. The presented approximations are in a form suitable for everyday engineering use, and are both accurate and computationally efficient
-
Colebrook’s flow friction explicit approximations based on fixed-point iterative cycles and symbolic regression
2020Co-Authors: Brkić Dejan, Praks PavelAbstract:The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and WrightΩ functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Padé approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence
-
Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function: Reply to Discussion
2020Co-Authors: Brkić Dejan, Praks PavelAbstract:This reply gives two corrections of typographical errors in respect to the commented article, and then provides few comments in respect to the discussion and one improved version of the approximation of the Colebrook equation for flow friction, based on the Wright ω-function. Finally, this reply gives an exact explicit version of the Colebrook equation expressed through the Wright ω-function, which does not introduce any additional errors in respect to the original equation. All mentioned approximations are computationally efficient and also very accurate. Results are verified using more than 2 million of Quasi Monte-Carlo samples
-
What can students learn while solving Colebrook’s flow friction equation?
2020Co-Authors: Kić Deja, Praks PavelAbstract:Even a relatively simple equation such as Colebrook’s offers a lot of possibilities to students to increase their computational skills. The Colebrook’s equation is implicit in the flow friction factor and, therefore, it needs to be solved iteratively or using explicit approximations, which need to be developed using different approaches. Various procedures can be used for iterative methods, such as single the fixed-point iterative method, Newton–Raphson, and other types of multi-point iterative methods, iterative methods in a combination with Padé polynomials, special functions such as Lambert W, artificial intelligence such as neural networks, etc. In addition, to develop explicit approximations or to improve their accuracy, regression analysis, genetic algorithms, and curve fitting techniques can be used too. In this learning numerical exercise, a few numerical examples will be shown along with the explanation of the estimated pedagogical impact for university students. Students can see what the difference is between the classical vs. floating-point algebra used in computers
-
Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright omega-function
'MDPI AG', 2019Co-Authors: Brkić Dejan, Praks PavelAbstract:The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor, f. To date, the captured flow friction factor, f, can be extracted from the logarithmic form analytically only in the term of the Lambert W-function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert W-function also known as the Wright omega-function. The Wright omega-function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term, y = W (e(x)), of the Lambert W-function to series expansions that further can be easily evaluated in computers without causing overflow run-time errors. Although the Colebrook equation transformed through the Lambert W-function is identical to the original expression in terms of accuracy, a further evaluation of the Lambert W-function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contain only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with a relative error of no more than 0.0096%. The presented approximations are in a form suitable for everyday engineering use, and are both accurate and computationally efficient.Web of Science71art. no. 3
Brkić Dejan - One of the best experts on this subject based on the ideXlab platform.
-
Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function
2020Co-Authors: Brkić Dejan, Praks PavelAbstract:The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor, f . To date, the captured flow friction factor, f , can be extracted from the logarithmic form analytically only in the term of the Lambert W-function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert W-function also known as the Wright ω-function. The Wright ω-function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term, y=W(ex), of the Lambert W-function to series expansions that further can be easily evaluated in computers without causing overflow run-time errors. Although the Colebrook equation transformed through the Lambert W-function is identical to the original expression in terms of accuracy, a further evaluation of the Lambert W-function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contain only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with a relative error of no more than 0.0096%. The presented approximations are in a form suitable for everyday engineering use, and are both accurate and computationally efficient
-
Colebrook’s flow friction explicit approximations based on fixed-point iterative cycles and symbolic regression
2020Co-Authors: Brkić Dejan, Praks PavelAbstract:The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and WrightΩ functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Padé approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence
-
Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function: Reply to Discussion
2020Co-Authors: Brkić Dejan, Praks PavelAbstract:This reply gives two corrections of typographical errors in respect to the commented article, and then provides few comments in respect to the discussion and one improved version of the approximation of the Colebrook equation for flow friction, based on the Wright ω-function. Finally, this reply gives an exact explicit version of the Colebrook equation expressed through the Wright ω-function, which does not introduce any additional errors in respect to the original equation. All mentioned approximations are computationally efficient and also very accurate. Results are verified using more than 2 million of Quasi Monte-Carlo samples
-
Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright omega-function
'MDPI AG', 2019Co-Authors: Brkić Dejan, Praks PavelAbstract:The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor, f. To date, the captured flow friction factor, f, can be extracted from the logarithmic form analytically only in the term of the Lambert W-function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert W-function also known as the Wright omega-function. The Wright omega-function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term, y = W (e(x)), of the Lambert W-function to series expansions that further can be easily evaluated in computers without causing overflow run-time errors. Although the Colebrook equation transformed through the Lambert W-function is identical to the original expression in terms of accuracy, a further evaluation of the Lambert W-function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contain only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with a relative error of no more than 0.0096%. The presented approximations are in a form suitable for everyday engineering use, and are both accurate and computationally efficient.Web of Science71art. no. 3
-
Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright-Omega function
'MDPI AG', 2019Co-Authors: Brkić Dejan, Praks PavelAbstract:The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor f. To date, the captured flow friction factor f can be extracted from the logarithmic form analytically only in the term of the Lambert W-function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert W-function also known as the Wright Omega-function. The Wright Omega-function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term y=W(e^x) of the Lambert W-function to the series expansions that further can be easily evaluated in computers without causing overflow run-time errors. Although the Colebrook equation transformed through the Lambert W-function is identical to the original expression in term of accuracy, a further evaluation of the Lambert W-function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contains only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with the relative error of no more than 0.0096%. The presented approximations are in the form suitable for everyday engineering use, they are both accurate and computationally efficient.Comment: 15 pages, 58 references, 1 figure, 2 table
Ignacio Lira - One of the best experts on this subject based on the ideXlab platform.
-
on the uncertainties stemming from use of the Colebrook white equation
Industrial & Engineering Chemistry Research, 2013Co-Authors: Ignacio LiraAbstract:The Darcy-Weisbach and Colebrook-White equations are considered to be the most accurate formulas available for calculating head losses in pipes. Because of the implicit nature of the latter equation, various graphical methods and analytic approximations for explicitly determining the friction factor have been developed. Detailed comparisons among these approximations have revealed that they exhibit varying levels of accuracy depending upon the complexity of their functional forms. However, none of these comparisons has taken into account the crucial fact that the input quantities needed to compute the friction factor are obtained from measurements and, consequently, those quantities are subject to experimental uncertainties. The present paper proposes a procedure for evaluating the impact these uncertainties have on the dispersion of the values that can be attributed to the friction factor and other relevant quantities computed from the Colebrook-White equation. The procedure, which accords with current i...