Colebrook Equation For Flow Friction Research Articles

Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to the Discussion by Majid Niazkar

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.

Discussion of “Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function” by Dejan Brkić and Pavel Praks, Mathematics 2019, 7, 34; doi:10.3390/math7010034

Estimating the Darcy–Weisbach friction factor is crucial to various engineering applications. Although the literature has accepted the Colebrook–White formula as a standard approach for this prediction, its implicit structure brings about an active field of research seeking for alternatives more suitable in practice. This study mainly attempts to increase the precision of two explicit equations proposed by Brkić and Praks. The results obviously demonstrate that the modified relations outperformed the original ones from nine out of 10 accuracy evaluation criteria. Finally, one of the improved equations estimates closer friction factors to those obtained by the Colebrook–White formula among 18 one-step explicit equations available in the literature based on three of the considered criteria.

Review of new flow friction equations: Constructing Colebrook’s explicit correlations accurately

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%.

Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation

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 Padé 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.

Correction: Brkić, D., and Praks, P. Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function. Mathematics 2019, 7, 34

Having in mind that in the title of the article contains term “Wright ω-function” and not its cognate “Lambert W-function”, the authors would like to change Equation (2) of [1] in order to contain both expressions, as follows [...]

Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression

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 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.

A quick semantic artificial bee colony programming (qsABCP) for symbolic regression

Artificial bee colony programming (ABCP) is a novel evolutionary computation based automatic programming method, which uses the basic structure of artificial bee colony (ABC) algorithm. In this paper, some studies were conducted to improve the performance of ABCP and three new versions of ABCP are introduced. One of these improvements is related to the convergence performance of ABCP. In order to increase the local search ability and achieve higher quality solutions in early cycles, quick ABCP algorithm was developed. Experimental studies validate the enhancement of the convergence performance when the quick ABC approach is used in ABCP. The second improvement introduced in this paper is about providing high locality. Using semantic similarity based operators in the information sharing mechanism of ABCP, semantic ABCP was developed and experiment results show that semantic based information sharing improves solution quality. Finally, combining these two methods, quick semantic ABCP is introduced. Performance of these novel methods was compared with some well known automatic programming algorithms on literature test problems. Additionally, ABCP based methods were used to find approximations of the Colebrook equation for flow friction. Simulation results show that, the proposed methods can be used to solve symbolic regression problems effectively.

Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to Discussion

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.

Discussion of “Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function” by DejanBrkić; and Pavel Praks, Mathematics 2019, 7, 34; doi:10.3390/math7010034

The Colebrook-White equation is often used for calculation of the friction factor in turbulent regimes; it has succeeded in attracting a great deal of attention from researchers. The Colebrook–White equation is a complex equation where the computation of the friction factor is not direct, and there is a need for trial-error methods or graphical solutions; on the other hand, several researchers have attempted to alter the Colebrook-White equation by explicit formulas with the hope of achieving zero-percent (0%) maximum deviation, among them Dejan Brkić and Pavel Praks. The goal of this paper is to discuss the results proposed by the authors in their paper:” Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function” and to propose more accurate formulas.

Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function

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.

Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction

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.

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation

The Colebrook equation is implicitly given in respect to the unknown flow friction factor λ; λ = ζ ( R e , ε * , λ ) which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. A common approach to solve it is through the Newton–Raphson iterative procedure or through the fixed-point iterative procedure. Both require in some cases, up to seven iterations. On the other hand, numerous more powerful iterative methods such as three- or two-point methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implicit Colebrook equation for flow friction accurately using the least possible number of iterations. The methods are thoroughly tested and those which require the least possible number of iterations to reach the accurate solution are identified. The most powerful three-point methods require, in the worst case, only two iterations to reach the final solution. The recommended representatives are Sharma–Guha–Gupta, Sharma–Sharma, Sharma–Arora, Džunić–Petković–Petković; Bi–Ren–Wu, Chun–Neta based on Kung–Traub, Neta, and the Jain method based on the Steffensen scheme. The recommended iterative methods can reach the final accurate solution with the least possible number of iterations. The approach is hybrid between the iterative procedure and one-step explicit approximations and can be used in engineering design for initial rough, but also for final fine calculations.

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

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.

Solution of the Implicit Colebrook Equation for Flow Friction Using Excel

Solution of the Implicit Colebrook Equation for Flow Friction Using Excel

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

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.