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Pal Efsing - One of the best experts on this subject based on the ideXlab platform.
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a duplex oxide cohesive zone model to simulate intergranular stress corrosion cracking
International Journal of Mechanical Sciences, 2021Co-Authors: Sedlak M Mosesson, Bo Alfredsson, Pal EfsingAbstract:Abstract A finite element model with slip-oxidation is proposed for solving intergranular stress corrosion cracking (IGSCC) with duplex oxides replicating the cyclic physics of the slip oxidation. The purpose is to investigate the crack growth effect due to different rate, compositions and kinetics of the duplex oxide. The finite element model is based on a coupling between cohesive zone Formulation, slip-oxidation model and a diffusion model. The cohesive zone Formulation includes a degradation Formulation which is linked to the slip-oxidation Formulation. The environmental properties in the slip-oxidation were obtained from the diffusion modeled with Fick's second law in one-dimension. This was then coupled to the structural model by a segregated solution scheme. The mesh of the cohesive zone adapts to the oxide thickness of the duplex oxide during the crack growth. The duplex oxide has the mathematical Form of a power law or a Logarithmic Form. The model showed matching results for all duplex oxide combinations in varying stress, but the inner Logarithmical oxide gave higher crack growth rates than the power law. The power law with the thicker inner oxide showed good results for the change of stress intensity factor and gave the best results when the yield stress was varied. Grain misorientation effect was higher for the duplex oxides with thicker outer oxides.
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
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%.
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accurate and efficient explicit approximations of the colebrook flow friction equation based on the wright omega 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 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.
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advanced iterative procedures for solving the implicit colebrook equation for fluid flow friction
arXiv: Computational Engineering Finance and Science, 2018Co-Authors: Pavel Praks, Dejan BrkicAbstract:Empirical Colebrook equation from 1939 is still accepted as an inFormal standard to calculate friction factor during the turbulent flow through pipes from smooth with almost negligible relative roughness to the very rough inner surface. The Colebrook equation contains flow friction factor in implicit Logarithmic Form where it is, aside of itself, a function of the Reynolds number Re and the relative roughness of inner pipe surface. To evaluate the error introduced by many available explicit approximations to the Colebrook equation, it is necessary to determinate value of the friction factor 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 approach (3rd order, 2nd order: Halley and Schroder 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 approach to the Colebrook 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.
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
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%.
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accurate and efficient explicit approximations of the colebrook flow friction equation based on the wright omega 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 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.
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advanced iterative procedures for solving the implicit colebrook equation for fluid flow friction
arXiv: Computational Engineering Finance and Science, 2018Co-Authors: Pavel Praks, Dejan BrkicAbstract:Empirical Colebrook equation from 1939 is still accepted as an inFormal standard to calculate friction factor during the turbulent flow through pipes from smooth with almost negligible relative roughness to the very rough inner surface. The Colebrook equation contains flow friction factor in implicit Logarithmic Form where it is, aside of itself, a function of the Reynolds number Re and the relative roughness of inner pipe surface. To evaluate the error introduced by many available explicit approximations to the Colebrook equation, it is necessary to determinate value of the friction factor 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 approach (3rd order, 2nd order: Halley and Schroder 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 approach to the Colebrook 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.
Praks Pavel - 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
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
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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
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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
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Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function
HAL CCSD, 2018Co-Authors: Brkić Dejan, Praks PavelAbstract:International audienceThe 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
Sedlak M Mosesson - One of the best experts on this subject based on the ideXlab platform.
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a duplex oxide cohesive zone model to simulate intergranular stress corrosion cracking
International Journal of Mechanical Sciences, 2021Co-Authors: Sedlak M Mosesson, Bo Alfredsson, Pal EfsingAbstract:Abstract A finite element model with slip-oxidation is proposed for solving intergranular stress corrosion cracking (IGSCC) with duplex oxides replicating the cyclic physics of the slip oxidation. The purpose is to investigate the crack growth effect due to different rate, compositions and kinetics of the duplex oxide. The finite element model is based on a coupling between cohesive zone Formulation, slip-oxidation model and a diffusion model. The cohesive zone Formulation includes a degradation Formulation which is linked to the slip-oxidation Formulation. The environmental properties in the slip-oxidation were obtained from the diffusion modeled with Fick's second law in one-dimension. This was then coupled to the structural model by a segregated solution scheme. The mesh of the cohesive zone adapts to the oxide thickness of the duplex oxide during the crack growth. The duplex oxide has the mathematical Form of a power law or a Logarithmic Form. The model showed matching results for all duplex oxide combinations in varying stress, but the inner Logarithmical oxide gave higher crack growth rates than the power law. The power law with the thicker inner oxide showed good results for the change of stress intensity factor and gave the best results when the yield stress was varied. Grain misorientation effect was higher for the duplex oxides with thicker outer oxides.
