Turbulent Flow Friction Factor Calculation Using a Mathematically Exact Alternative to the Colebrook–White Equation

We present a novel, mathematically equivalent representation of the Colebrook–White equation to compute friction factor for turbulent flow in rough pipes. This new form is simple, no iterative calculations are necessary, and is well suited for accurate friction factor estimation. A limiting case of this equation provided friction factor estimates with a maximum absolute error of 0.029 and a maximum percentage error of 1% over a $20\times 500$ grid of $\epsilon ∕D$ and $\mathsf{R}$ values ( ${10}^{-6}⩽\epsilon ∕D⩽5\times {10}^{-2}$ ; $4\times {10}^3<\mathsf{R}<{10}^8$ ). This was more accurate than the best currently available noniterative approximation of the Colebrook–White equation (maximum absolute error of 0.058; maximum percentage error of 1.42%). The superior accuracy, however, was obtained at the expense of a 30% increase in computational effort over the noniterative approximation. The novel equation presented in this study is theoretical and eliminates the need for best fit parameters or complicated initial guesses that are a characteristic of various empirical approximations proposed to date. The simplicity with which this new equation can be solved, coupled with its smooth and predictable error behavior, should make it the method of choice for estimating turbulent flow friction factor in rough pipes.