Empirical implicit 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 non-laminar ( $\begin{array}{l}{\rm Re} > 2,000\end{array}$ ) flow:

(1)	$\begin{array}{l}\displaystyle \frac{1}{\sqrt{f}} = -2 \, \log \left( \frac{\epsilon}{3.7 \, d} + \frac{2.51}{{\rm Re} \sqrt{f}} \right)\end{array}$

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

There are numerous explicit approximations of Colebrook–White correlation, particularly (Monzon, Romeo, Royo, 2002):

(2)	$\begin{array}{l}\displaystyle f = 0.25 \, \left[ \log \left( \frac{\epsilon / d}{3.7065} - \frac{5.0272}{\rm Re} \log \Lambda \right) \right]^{-2}\end{array}$

where $\begin{array}{l}\Lambda\end{array}$ – is dimensionless parameter:

(3)	$\begin{array}{l}\displaystyle \Lambda = \frac{(\epsilon/d)}{3.827} - \frac{4.657}{\rm Re} \log \Bigg[ \bigg( \frac{\epsilon/d}{7.7918} \bigg)^{0.9924} + \bigg( \frac{5.3326}{208.815+Re} \bigg)^{0.9345} \Bigg]\end{array}$

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See also