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

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 34 ◽  
Author(s):  
Dejan Brkić ◽  
Pavel Praks
Keyword(s):  
Friction Factor ◽  
Relative Error ◽  
Efficient Solution ◽  
Friction Loss ◽  
Lambert W Function ◽  
Series Expansions ◽  
Computationally Efficient ◽  
Flow Friction ◽  
Fast Growing ◽  
Loss Coefficients

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.

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

2018 ◽  
Author(s):  
Dejan Brkić ◽  
Pavel Praks
Keyword(s):  
Friction Factor ◽  
Relative Error ◽  
Efficient Solution ◽  
Friction Loss ◽  
Series Expansions ◽  
Wright Function ◽  
Lambert Function ◽  
Computationally Efficient ◽  
Flow Friction ◽  
Loss Coefficients

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.

scholarly journals solutions of the CW equation for flow friction

2017 ◽  
Author(s):  
Dejan Brkić
Keyword(s):  
Fluid Flow ◽  
Friction Factor ◽  
Explicit Form ◽  
Iterative Procedure ◽  
Lambert W Function ◽  
Explicit Formulas ◽  
Flow Friction ◽  
Transcendental Functions ◽  
Flow Through

The empirical Colebrook–White (CW) equation belongs to the group of transcendental functions. The CW function is used for the determination of hydraulic resistances associated with fluid flow through pipes, flow of rivers, etc. Since the CW equation is implicit in fluid flow friction factor, it has to be approximately solved using iterative procedure or using some of the approximate explicit formulas developed by many authors. Alternate mathematical equivalents to the original expression of the CW equation, but now in the explicit form developed using the Lambert W-function, are shown (with related solutions). The W-function is also transcendental, but it is used more general compared with the CW function. Hence, the solution to the W-function developed by mathematicians can be used effectively for the CW function which is of interest only for hydraulics.

scholarly journals Lambert W function in hydraulic problems

2017 ◽  
Author(s):  
Dejan Brkić
Keyword(s):  
Friction Factor ◽  
Explicit Form ◽  
Elementary Function ◽  
Approximate Solutions ◽  
Iteration Procedure ◽  
Lambert W Function ◽  
Implicit Functions ◽  
Flow Friction ◽  
Implicit Form

Darcy's flow friction factor is expressed in implicit form in some of the relations suchas Colebrook's and have to be solved by iteration procedure because the unknown frictionfactor appears on both sides of the equation. Lambert W function is implicitly elementarybut is not, itself, an elementary function. Implicit form of the Lambert W function allows usto transform other implicit functions in explicit form without any kind of approximations orsimplications involved. But unfortunately, the Lambert W function itself cannot be solvedeasily without approximation. Two original transformations in explicit form of Colebrook'srelation using Lambert W function will also be shown. Here will be shown ecient procedurefor approximate solutions of the transformed relations.

scholarly journals 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]

2017 ◽  
Author(s):  
Dejan Brkić ◽  
Brkic Dejan
Keyword(s):  
Gene Expression ◽  
Turbulent Flow ◽  
Friction Factor ◽  
Relative Error ◽  
Energy Security ◽  
Gene Expression Programming ◽  
Security Systems ◽  
Flow Friction ◽  
Programming Analysis ◽  
Maximal Relative Error

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

scholarly journals Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

2018 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Friction Factor ◽  
Iterative Methods ◽  
Lambert W Function ◽  
Symbolic Form ◽  
Pipe Surface ◽  
Flow Friction ◽  
Relative Roughness ◽  
Starting Point ◽  
High Level ◽  
Derivatives Of

Empirical Colebrook equation from 1939 is still accepted as an informal standard to calculate friction factor during the turbulent flow (4000 < Re < 108) through pipes from smooth with almost negligible relative roughness (ε/D→0) to the very rough (up to ε/D = 0.05) inner surface. The Colebrook equation contains flow friction factor λ in implicit logarithmic form where it is, aside of itself; λ, a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D; λ = f (λ, Re, ε/D). To evaluate the error introduced by many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate value of the friction factor λ 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’s approach (3rd order, 2nd order: Halley’s and Schröder’s 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’s approach to the Colebrook’s 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.

scholarly journals An Explicit Approximation of Colebrook's Equation for Fluid Flow Friction Factor

2017 ◽  
Author(s):  
D. Brkić
Keyword(s):  
Fluid Flow ◽  
Friction Factor ◽  
Iterative Procedure ◽  
Lambert W Function ◽  
Explicit Formulas ◽  
Flow Friction

The Colebrook equation for determination of hydraulic resistances is implicit in fluid flow friction factor and hence it has to be approximately solved using iterative procedure or using some of the approximate explicit formulas which were developed by many authors. Here will be shown one approximation of the Colebrook equation based on Lambert W-function.

scholarly journals Review of New Flow Friction Equations: Constructing Colebrook’s Explicit Correlations Accurately

2020 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Monte Carlo ◽  
Series Expansion ◽  
Relative Error ◽  
Asymptotic Series ◽  
Computationally Efficient ◽  
Arithmetic Operations ◽  
Flow Friction ◽  
Quasi Monte Carlo ◽  
Computationally Expensive ◽  
Maximal Relative Error

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 ω-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 ω-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%.

A Review of Explicit Friction Factor Equations

1985 ◽  
Vol 107 (2) ◽  
pp. 280-283 ◽  
Author(s):  
D. J. Zigrang ◽  
N. D. Sylvester
Keyword(s):  
Friction Factor ◽  
Pipe Flow ◽  
Turbulent Pipe Flow ◽  
Explicit Equation ◽  
Flow Friction ◽  
Friction Factors ◽  
High Degree ◽  
Very High

A review of the explicit friction factor equations developed to replace the Colebrook equation is presented. Explicit friction factor equations are developed which yield a very high degree of precision compared to the Colebrook equation. A new explicit equation, which offers a reasonable compromise between complexity and accuracy, is presented and recommended for the calculation of all turbulent pipe flow friction factors for all roughness ratios and Reynold’s numbers.

Sign in / Sign up

Export Citation Format

Share Document