Abstract
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.
Highlights
The Colebrook equation; Equation (1), is an empirical relation which, in its native form, relates implicitly the unknown Darcy’s flow friction factor, f, with the known Reynolds number, R, and the known relative roughness of inner pipe surface, ε∗ [1,2]
The best version of the presented explicit approximation gives the value of the flow friction factor, f, for which the Colebrook equation is in balance with the relative error of no more than 0.0096%
This is the first highly accurate explicit approximation of the Colebrook equation that contains only two computationally expensive functions, or even less if a combination of Padé approximations [15,21,22], and symbolic regression is used for a further reduction of the computational burden
Summary
The Colebrook equation; Equation (1), is an empirical relation which, in its native form, relates implicitly the unknown Darcy’s flow friction factor, f , with the known Reynolds number, R, and the known relative roughness of inner pipe surface, ε∗ [1,2] Engineers use it at defined domains of the input parameters: 4000 < R < 108 and for 0 < ε∗ < 0.05. The best version of the presented explicit approximation gives the value of the flow friction factor, f, for which the Colebrook equation is in balance with the relative error of no more than 0.0096% Such accuracy achieved without using a large number of computationally expensive logarithmic functions (or non-integer powers) is highly computationally efficient. This is the first highly accurate explicit approximation of the Colebrook equation that contains only two computationally expensive functions (two logarithms, or as an alternative, two functions with non-integer powers), or even less if a combination of Padé approximations [15,21,22], and symbolic regression is used for a further reduction of the computational burden (where one of the logarithms is approximated by simple rational functions with a moderate increase of the maximal relative error)
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