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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61989100%3A27740%2F19%3A10242741" target="_blank" >RIV/61989100:27740/19:10242741 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.hindawi.com/journals/ace/2018/5451034/" target="_blank" >https://www.hindawi.com/journals/ace/2018/5451034/</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1155/2018/5451034" target="_blank" >10.1155/2018/5451034</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 &lt; Re &lt; 10(8)) through pipes with roughness between negligible relative roughness (epsilon/D -&gt; 0) to very rough (up to epsilon/D = 0.05). The Colebrook equation includes the flow friction factor lambda in an implicit logarithmic form, lambda being a function of the Reynolds number Re and the relative roughness of inner pipe surface epsilon/D: lambda = f(lambda, Re, epsilon/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, lambda approximate to f(Re, epsilon/D), it is necessary to determinate the value of the friction factor lambda from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder&apos;s approach (3rd order, 2nd order: Halley&apos;s and Schroder&apos;s method, and 1st order: Newton-Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook&apos; equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.

Název v anglickém jazyce

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

Popis výsledku anglicky

The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 &lt; Re &lt; 10(8)) through pipes with roughness between negligible relative roughness (epsilon/D -&gt; 0) to very rough (up to epsilon/D = 0.05). The Colebrook equation includes the flow friction factor lambda in an implicit logarithmic form, lambda being a function of the Reynolds number Re and the relative roughness of inner pipe surface epsilon/D: lambda = f(lambda, Re, epsilon/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, lambda approximate to f(Re, epsilon/D), it is necessary to determinate the value of the friction factor lambda from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder&apos;s approach (3rd order, 2nd order: Halley&apos;s and Schroder&apos;s method, and 1st order: Newton-Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook&apos; equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

20104 - Transport engineering

Projekt

—

Návaznosti

V - Vyzkumna aktivita podporovana z jinych verejnych zdroju

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Advances in Civil Engineering

ISSN

1687-8086

e-ISSN

—

Svazek periodika

2018

Číslo periodika v rámci svazku

Article number 5451034

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

18

Strana od-do

—

Kód UT WoS článku

000460259400001

EID výsledku v databázi Scopus

2-s2.0-85051786909