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Dejan Brkić - One of the best experts on this subject based on the ideXlab platform.
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Review of new flow friction Equations: Constructing Colebrook explicit correlations accurately
arXiv: Numerical Analysis, 2020Co-Authors: Pavel Praks, Dejan BrkićAbstract:Using only a limited number of computationally expensive functions, we show a way how to construct accurate and computationally efficient approximations of the Colebrook Equation for flow friction. The presented approximations are based on the asymptotic series expansion of the Wright Omega-function and symbolic regression. The results are verified with 8 million of Quasi-Monte Carlo points covering the domain of interest for engineers. In comparison with the built-in wrightOmega feature of Matlab R2016a, the herein introduced related approximations of the Wright Omega-function significantly accelerate the computation. With only two logarithms and several basic arithmetic operations used, the presented approximations are not only computationally efficient but also extremely accurate. The maximal relative error of the most promising approximation which is given in the form suitable for engineers use is limited to 0.0012%, while for a little bit more complex variant is limited to 0.000024%.
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Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression
Computation, 2019Co-Authors: Dejan Brkić, 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.
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Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction
Advances in Civil Engineering, 2018Co-Authors: Pavel Praks, Dejan BrkićAbstract:The empirical Colebrook Equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000
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Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on Wright-Omega Function
2018Co-Authors: Dejan Brkić, 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 . To date, the captured flow friction factor can be extracted from the logarithmic form analytically only in the term of the Lambert -function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert -function also known as the Wright -function. The Wright -function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term of the Lambert -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 -function is identical to the original expression in term of accuracy, a further evaluation of the Lambert -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.
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Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction
Water, 2018Co-Authors: Pavel Praks, Dejan BrkićAbstract: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.
Dejan Brkic - One of the best experts on this subject based on the ideXlab platform.
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accurate and efficient explicit approximations of the Colebrook flow friction Equation based on the wright ω function reply to the discussion by majid niazkar
Mathematics, 2020Co-Authors: Pavel Praks, Dejan BrkicAbstract:In this reply, we present updated approximations to the Colebrook Equation for flow friction. The Equations are equally computational simple, but with increased accuracy thanks to the optimization procedure, which was proposed by the discusser, Dr. Majid Niazkar. Our large-scale quasi-Monte Carlo verifications confirm that the here presented novel optimized numerical parameters further significantly increase accuracy of the estimated flow friction factor.
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rational approximation for solving an implicitly given Colebrook flow friction Equation
Mathematics, 2019Co-Authors: Pavel Praks, Dejan BrkicAbstract:The empirical logarithmic Colebrook Equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use of transcendental functions, such as logarithm or non-integer power, which require execution of the additional number of floating-point operations in computer processor units. Instead of these, we use only rational expressions that are executed directly in the processor unit. The rational approximation was found using a combination of a Pade approximant and artificial intelligence (symbolic regression). Numerical experiments in Matlab using 2 million quasi-Monte Carlo samples indicate that the relative error of this new rational approximation does not exceed 0.866%. Moreover, these numerical experiments show that the novel rational approximation is approximately two times faster than the exact solution given by the Wright omega function.
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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.
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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.
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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.
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accurate and efficient explicit approximations of the Colebrook flow friction Equation based on the wright ω function reply to the discussion by majid niazkar
Mathematics, 2020Co-Authors: Pavel Praks, Dejan BrkicAbstract:In this reply, we present updated approximations to the Colebrook Equation for flow friction. The Equations are equally computational simple, but with increased accuracy thanks to the optimization procedure, which was proposed by the discusser, Dr. Majid Niazkar. Our large-scale quasi-Monte Carlo verifications confirm that the here presented novel optimized numerical parameters further significantly increase accuracy of the estimated flow friction factor.
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Review of new flow friction Equations: Constructing Colebrook explicit correlations accurately
arXiv: Numerical Analysis, 2020Co-Authors: Pavel Praks, Dejan BrkićAbstract:Using only a limited number of computationally expensive functions, we show a way how to construct accurate and computationally efficient approximations of the Colebrook Equation for flow friction. The presented approximations are based on the asymptotic series expansion of the Wright Omega-function and symbolic regression. The results are verified with 8 million of Quasi-Monte Carlo points covering the domain of interest for engineers. In comparison with the built-in wrightOmega feature of Matlab R2016a, the herein introduced related approximations of the Wright Omega-function significantly accelerate the computation. With only two logarithms and several basic arithmetic operations used, the presented approximations are not only computationally efficient but also extremely accurate. The maximal relative error of the most promising approximation which is given in the form suitable for engineers use is limited to 0.0012%, while for a little bit more complex variant is limited to 0.000024%.
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rational approximation for solving an implicitly given Colebrook flow friction Equation
Mathematics, 2019Co-Authors: Pavel Praks, Dejan BrkicAbstract:The empirical logarithmic Colebrook Equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use of transcendental functions, such as logarithm or non-integer power, which require execution of the additional number of floating-point operations in computer processor units. Instead of these, we use only rational expressions that are executed directly in the processor unit. The rational approximation was found using a combination of a Pade approximant and artificial intelligence (symbolic regression). Numerical experiments in Matlab using 2 million quasi-Monte Carlo samples indicate that the relative error of this new rational approximation does not exceed 0.866%. Moreover, these numerical experiments show that the novel rational approximation is approximately two times faster than the exact solution given by the Wright omega function.
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Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression
Computation, 2019Co-Authors: Dejan Brkić, 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.
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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.
Brkic Dejan - One of the best experts on this subject based on the ideXlab platform.
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Discussion of “Gene expression programming analysis of implicit Colebrook–White Equation in turbulent flow friction factor calculation” by Saeed Samadianfard [J. Pet. Sci. Eng. 92-93 (2012), 48-55]
2017Co-Authors: Dejan Brkić, Brkic DejanAbstract:Maximal relative error of the explicit approximation to the Colebrook Equation for flow friction presented in the discussed paper by Saeed Samadianfard [J. Pet. Sci. Eng. 92-93 (2012), 48-55; doi. 10.1016/j.petrol.2012.06.005] is investigated. Samadianfard claims that his approximation is very accurate with the maximal relative error of no more than 0.08152%. Here is shown that this error is about 7%. Related comments about the paper are also enclosed. ; JRC.F.3-Energy Security, Systems and Market
Zhanru Zhou - One of the best experts on this subject based on the ideXlab platform.
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New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations
Nuclear Engineering and Design, 2011Co-Authors: Xiande Fang, Yu Xu, Zhanru ZhouAbstract:Abstract The determination of single-phase friction factor of pipe flow is essential to a variety of industrial applications, such as single-phase flow systems, two-phase flow systems and supercritical flow systems. There are a number of correlations for the single-phase friction factor. It still remains an issue to examine similarities and differences between them to avoid misusing. This paper evaluates the correlations for the single-phase friction factor against the Nikuradse Equation and the Colebrook Equation, respectively. These two Equations are the base for the turbulent portion of the Moody diagram, and are deemed as the standard to test the explicit counterparts. The widely used correlations for smooth pipes, the Blasius correlation and the Filonenko correlation, have big errors in some Re ranges. Simpler forms of the single-phase friction factor covering large ranges are needed. For this reason, two new correlations of single-phase friction factor for turbulent flow are proposed, one for smooth pipes and the other for both smooth and rough pipes. Compared with the Nikuradse Equation, the new correlation for smooth pipes has the mean absolute relative error of 0.022%, with the maximum relative error of −0.045% in the Reynolds number ( Re ) range from 3000 through 10 8 . It is an idea replacement of the correlations of Blasius and Filonenko. The new correlation for both smooth and rough pipes has the mean absolute relative error of 0.16% and the maximum relative error of 0.50% compared with the Colebrook Equation in the range of Re = 3000–10 8 and Rr = 0.0–0.05, which is the most simplest correlation in that error band.