Abstract
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.
Highlights
The Colebrook function is to date the most used relation in engineering practice for evaluation of flow friction in pipes [1]
In Equation (1), x = √1 is introduced because of linearization of the unknown flow friction factor λ λ, Re is the Reynolds number, and ε∗ is the relative roughness of inner pipe surface
Because the Colebrook function in the examined iterative methods sometimes needs to be evaluated in two or three points, y and z are used in the same meaning as x, where they are dimensionless parameters that depend on the friction factor
Summary
The Colebrook function is to date the most used relation in engineering practice for evaluation of flow friction in pipes [1]. In Equation (1), x = √1 is introduced because of linearization of the unknown flow friction factor λ λ, Re is the Reynolds number, and ε∗ is the relative roughness of inner pipe surface (all quantities are dimensionless). The Colebrook equation is empirical [2] and its accuracy can be disputed (experiment of Nikuradse or recent experiments from Oregon or Princeton research groups [3]), but anyway, it is widely accepted in engineering practice. It is based on an experiment conducted by Colebrook and
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
