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Empirical Colebrook equation from 1939 is still accepted as an informalstandard to calculate friction factor during the turbulent flow through pipesfrom smooth with almost negligible relative roughness to the very rough innersurface. The Colebrook equation contains flow friction factor in implicitlogarithmic form where it is, aside of itself, a function of the Reynoldsnumber Re and the relative roughness of inner pipe surface. To evaluate theerror introduced by many available explicit approximations to the Colebrookequation, it is necessary to determinate value of the friction factor from theColebrook equation as accurate as possible. The most accurate way to achievethat is using some kind of iterative methods. Usually classical approach alsoknown as simple fixed point method requires up to 8 iterations to achieve thehigh level of accuracy, but does not require derivatives of the Colebrookfunction as here presented accelerated Householder approach (3rd order, 2ndorder: Halley and Schroder method and 1st order: Newton-Raphson) which needsonly 3 to 7 iteration and three-point iterative methods which needs only 1 to 4iteration to achieve the same high level of accuracy. Strategies how to findderivatives of the Colebrook function in symbolic form, how to avoid use of thederivatives (Secant method) and how to choose optimal starting point for theiterative procedure are shown. Householder approach to the Colebrook equationsexpressed through the Lambert W-function is also analyzed. One approximation tothe Colebrook equation based on the analysis from the paper with the error ofno more than 0.0617% is shown.

Advanced iterative procedures for solving the implicit Colebrook equation for fluid flow friction