  Empirical Colebrook equation from 1939 is still accepted as an informal
standard to calculate friction factor during the turbulent flow through pipes
from smooth with almost negligible relative roughness to the very rough inner
surface. The Colebrook equation contains flow friction factor in implicit
logarithmic form where it is, aside of itself, a function of the Reynolds
number Re and the relative roughness of inner pipe surface. To evaluate the
error introduced by many available explicit approximations to the Colebrook
equation, it is necessary to determinate value of the friction factor 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 approach (3rd order, 2nd
order: Halley and Schroder 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 approach to the Colebrook 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.
