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 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 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 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 Very accurate explicit approximations for calculation of the Colebrook friction factor

2017 ◽  
Author(s):  
Žarko Ćojbašić ◽  
Dejan Brkić
Keyword(s):  
Genetic Algorithms ◽  
Turbulent Flow ◽  
Friction Factor ◽  
Relative Error ◽  
Iterative Procedure ◽  
Iterative Solution ◽  
Flow Through ◽  
Maximal Relative Error ◽  
Looped Network

To date, the Colebrook equation is mostly accepted as an unofficial standard for calculation of the friction factor in turbulent flow through pipes. Unfortunately, the unknown friction factor in the Colebrook equation is given implicitly. Therefore, the implicit Colebrook equation has to be solved in an iterative procedure or using some of the appropriate explicit correlations proposed by many authors. Although the iterative solution is simple and very accurate, it can cause some problems during the calculation of looped network of pipes or similar systems of pipes. Therefore, explicit approximations are favorable in these cases. Up to date, the most accurate approximations have maximal relative error of no more than 0.14% compared to the very accurate iterative solution. Here two explicit approximations are presented, based on already existing models which are improved using genetic algorithms optimization. They are with the maximal relative error of no more than 0.0083% and 0.0026%.

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 Determination of Transmission Coefficient for the Filtration Deposit Made of Coal Grains

2015 ◽  
Vol 31 (4) ◽  
pp. 151-160
Author(s):  
Tadeusz Piecuch ◽  
Jacek Piekarski ◽  
Łukasz Gajewski
Keyword(s):  
Fluid Flow ◽  
Transmission Coefficient ◽  
Constant Coefficient ◽  
Mineral Resources ◽  
Filtration Process ◽  
Filtration Equation ◽  
Resources Management ◽  
Flow Through ◽  
General Filtration

Abstract The study presents the method of determining constant coefficient b2, occurring in general filtration equation (1), in the second part of the denominator, that is in the expression for deposit resistance In the considerations, lack of the deposit’s compressibility was assumed, which means that the deposit porosity is constant. With such an assumption, constant coefficient b2 is equivalent with transmission coefficient, which occurs in commonly known and accepted equation – as the baseline equation – for fluid flow through a porous layer according to Darcy (Piecuch 2009, 2010). This study is another publication in the cycle of basic tests of the filtration process which constitute next publications of the authors, published in Rocznik Ochrona Środowiska [Annual Set the Environmental Protection] as well as in the magazine Gospodarka Surowcami Mineralnymi [Mineral Resources Management], to which the reader interested in these problems can refer. Another study will be the publication discussing the filtration process with the creation of sediment on the filtration deposit, hence in the general filtration equation (1) the value of sediment resistance RO will appear in the denominator. Relevant cycle of publications will study the possibilities of use of gravitational deposit filters in which the porous deposit will be the set of coal grains, while the fed mixture will also be the post-production suspension of coal grains.

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