Review of explicit approximations to the Colebrook relation for flow friction

Because of Moody's chart has demonstrated applicability of the Colebrook equation over a very wide range of Reynolds number and relative roughness values, this equation becomes the accepted standard of accuracy for calculated hydraulic friction factor. Colebrook equation suffers from being implicit in unknown friction factor and thus requires an iterative solution where convergence to 0.01% typically requires less than 7 iterations. Implicit Colebrook equation cannot be rearranged to derive friction factor directly in one step. Iterative calculus can cause a problem in simulation of flow in a pipe system in which it may be necessary to evaluate friction factor hundreds or thousands of times. This is the main reason for attempting to develop a relationship that is a reasonable approximation for the Colebrook equation but which is explicit in friction factor. A review of existing explicit approximation of the implicit Colebrook equation with estimated accuracy is shown in this paper. Estimated accuracy compared with iterative solution of implicit Colebrook equation is shown for the entire range of turbulence where Moody diagram should be used as the reference. Finally, it can be concluded that most of the available approximations of the Colebrook equation, with a few exceptions, are very accurate with deviations of no more than few percentages.

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► Empirical Colebrook equation is an accepted standard for calculation of flow friction factor.
► Colebrook equation is a transcendental function implicit in unknown friction coefficient and therefore it cannot be rearranged to derive friction factor directly.
► Colebrook equation has to be solved iteratively or using approximations where iterative calculus can cause a problem in simulation of flow in a pipe system.
► Error of almost all explicit approximations of the Colebrook relation is up to 3%.
► An explicit approximation of the Colebrook relation can be very complex but also it can be easily implemented in a computer code.