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 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 Comparison of the Lambert W‐function based solutions to the Colebrook equation

2017 ◽  
Author(s):  
Dejan Brkić
Keyword(s):  
Iterative Procedure ◽  
Computer Code ◽  
Computer Hardware ◽  
Lambert W Function ◽  
Accuracy Analysis ◽  
Pipe Surface ◽  
Flow Friction ◽  
Relative Roughness ◽  
Modern Computer

Purpose– 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 developed by many authors. The purpose of this paper is to compare different Lambert W based solutions of the Colebrook equation and to make comparisons among them and identify some constraints in applicability of certain solutions.Design/methodology/approach– Alternate mathematically equivalents to the implicit Colebrook equation in explicit form with no approximation involved actually exist.Findings– These alternate equations were developed using Lambert W‐function. The paper compares various implementations of the Lambert W methodology and shows that some of these are less able than others to yield solutions using modern computer hardware. This is because the functions require the evaluation of terms with numerical values outside the ranges that can be expressed on most computers.Research limitations/implications– Some of existed transformations cannot be applied for high values of relative roughness of inner pipe surface and the Reynolds number. This limitation applied only for computer computations. Other presented transformations do not sufferer of this limitation.Practical implications– Presented procedures can be easily implemented in a computer code. Recommended solution can be used in all cases that can occur in engineering practice.Originality/value– The paper shows some possible practical procedures for solution of the transformed Colebrook equation. Accuracy analysis and comparisons of presented formulas are also performed and recommendation for use is shown.

scholarly journals 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

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 253 ◽  
Author(s):  
Lotfi Zeghadnia ◽  
Bachir Achour ◽  
Jean Robert
Keyword(s):  
Friction Factor ◽  
The Other ◽  
Maximum Deviation ◽  
Explicit Formulas ◽  
Flow Friction ◽  
Complex Equation ◽  
Other Hand

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.

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 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.

Determination of the Darcy Pipe Flow Friction Factor as a Routine in Visual Basic for MS Excel

2006 ◽  
Author(s):  
Ignacio R. Martín-Domínguez ◽  
Ma. Teresa Alarcón-Herrera ◽  
Jorge A. Escobedo-Bretado
Keyword(s):  
Friction Factor ◽  
Pipe Flow ◽  
Visual Basic ◽  
Flow Friction

Newtonian and Non-Newtonian Fluids: Velocity Profiles, Viscosity Data, and Laminar Flow Friction Factor Equations for Flow in a Circular Duct

2008 ◽  
Author(s):  
Melissa M. Simpson ◽  
William S. Janna
Keyword(s):  
Fluid Flow ◽  
Pressure Drop ◽  
Laminar Flow ◽  
Friction Factor ◽  
Newtonian Fluid ◽  
Velocity Profiles ◽  
Newtonian Fluids ◽  
Viscosity Data ◽  
Circular Duct ◽  
Flow Friction

Newtonian fluid flow in a duct has been studied extensively, and velocity profiles for both laminar and turbulent flows can be found in countless references. Non-Newtonian fluids have also been studied extensively, however, but are not given the same attention in the Mechanical Engineering curriculum. Because of a perceived need for the study of such fluids, data were collected and analyzed for various common non-Newtonian fluids in order to make the topic more compelling for study. The viscosity and apparent viscosity of non-Newtonian fluids are both defined in this paper. A comparison is made between these fluids and Newtonian fluids. Velocity profiles for Newtonian and non-Newtonian fluid flow in a circular duct are described and sketched. Included are profiles for dilatant, pseudoplastic and Bingham fluids. Only laminar flow is considered, because the differences for turbulent flow are less distinct. Also included is a procedure for determining the laminar flow friction factor which allows for calculating pressure drop. The laminar flow friction factor in classical non-Newtonian fluid studies is the Fanning friction factor. The equations developed in this study involve the Darcy-Weisbach friction factor which is preferred for Newtonian fluids. Also presented in this paper are viscosity data of Heinz Ketchup, Kroger Honey, Jif Creamy Peanut Butter, and Kraft Mayonnaise. These data were obtained with a TA viscometer. The results of this study will thus provide the student with the following for non-Newtonian fluids: • Viscosity data and how it is measured for several common non-Newtonian fluids; • A knowledge of velocity profiles for laminar flow in a circular duct for both Newtonian and non-Newtonian fluids; • A procedure for determining friction factor and calculating pressure drop for non-Newtonian flow in a duct.

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.

Application of generalized Fourier heat conduction law on MHD viscoinelastic fluid flow over stretching surface

2019 ◽  
Vol 30 (6) ◽  
pp. 3481-3496 ◽  
Author(s):  
Arif Hussain ◽  
Muhammad Yousaf Malik ◽  
Mair Khan ◽  
Taimoor Salahuddin
Keyword(s):  
Heat Flux ◽  
Fluid Flow ◽  
Friction Factor ◽  
Numerical Technique ◽  
Wall Heat Flux ◽  
Constitutive Law ◽  
Wall Friction ◽  
Theoretical Physics ◽  
Stretching Surface ◽  
Content Type

Purpose The purpose of current flow configuration is to spotlights the thermophysical aspects of magnetohydrodynamics (MHD) viscoinelastic fluid flow over a stretching surface. Design/methodology/approach The fluid momentum problem is mathematically formulated by using the Prandtl–Eyring constitutive law. Also, the non-Fourier heat flux model is considered to disclose the heat transfer characteristics. The governing problem contains the nonlinear partial differential equations with appropriate boundary conditions. To facilitate the computation process, the governing problem is transmuted into dimensionless form via appropriate group of scaling transforms. The numerical technique shooting method is used to solve dimensionless boundary value problem. Findings The expressions for dimensionless velocity and temperature are found and investigated under different parametric conditions. The important features of fluid flow near the wall, i.e. wall friction factor and wall heat flux, are deliberated by altering the pertinent parameters. The impacts of governing parameters are highlighted in graphical as well as tabular manner against focused physical quantities (velocity, temperature, wall friction factor and wall heat flux). A comparison is presented to justify the computed results, it can be noticed that present results have quite resemblance with previous literature which led to confidence on the present computations. Originality/value The computed results are quite useful for researchers working in theoretical physics. Additionally, computed results are very useful in industry and daily-use processes.

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