Empirical explicit correlation for Darcy friction factor $\begin{array}{l}f\end{array}$ in non-smooth pipelines $\begin{array}{l}\epsilon > 0\end{array}$ which works for all pipe flow regimes with up to 2 % accuracy:

(1)	$\begin{array}{l}\displaystyle f = \frac{64}{\rm Re} \, \Bigg [ 1+ \frac{\big(\rm Re / 8 \big)^{12} }{ \big( \Theta_1 + \Theta_2 \big)^{1.5} } \Bigg]^{1/12}\end{array}$

(2)	$\begin{array}{l}\displaystyle \Theta_1 = \left[ 2.457 \, \ln \left( \left( \frac{7}{\rm Re} \right)^{0.9} + 0.27 \, \frac{\epsilon}{d} \right) \right]^{16}\end{array}$

(3)	$\begin{array}{l}\displaystyle \Theta_2 = \left( \frac{37530}{\rm Re} \right)^{16}\end{array}$

where

$\begin{array}{l}{\rm Re}\end{array}$	Reynolds number of a pipe fluid flow
$\begin{array}{l}d\end{array}$	Inner diameter of a pipe
$\begin{array}{l}\epsilon\end{array}$	inner pipe walls roughness

Fig. 1. Churchill correlation vs Colebrook–White correlation

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See also