An Improved Expression for a Classical Type of Explicit Approximation of the Colebrook White Equation with Only One Internal Iteration

The solution of the Colebrook White equation continues to attract the attention of engineers and mathematicians. Some solutions proposed in recent years have two or more internal iterations and present relative complexity. At the same time, micro-computers and calculators have shown increased calculation capacity, virtually eliminating approximations that have a certain accuracy but that are too complex. A direct approximation of the equation allowing for quick and accurate solution is still useful. This technical note brings together some explicit approximations with only one internal iteration presented in the 1970s and 1980s into a single category and proposes a single model with five parameters to be determined. Moreover, this technical note also presents the values for these five parameters that minimize the mean square error related to the outcome of the Colebrook White equation, considered as a reference. The proposed model allows to obtain results with mean square error equal to 1.14 x 10-8, with maximum relative error of 4.44% and 96.78% of results below 1.00% of difference relative to Colebrook White equation, which can be considered quite satisfactory.