Bellos friction factor correlation

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)	$\begin{array}{l}\displaystyle f = \frac{64}{\rm Re} \cdot \Phi\end{array}$

(2)	$\begin{array}{l}\displaystyle \Phi = \left( \frac{{\rm Re}}{64} \right)^{1-a} \cdot \left( 0.75 \cdot \ln \frac{{\rm Re}}{5.37} \right)^{-2 \,(1-a)\,b} \cdot \left( 0.83 \cdot \ln \frac{3.41}{\epsilon/d} \right)^{-2 \,(1-a)\,(1-b)}\end{array}$

(3)	$\begin{array}{l}\displaystyle a = \left[ 1+ \left( \frac{{\rm Re}}{2712} \right)^{8.4} \right]^{-1}\end{array}$

(4)	$\begin{array}{l}\displaystyle b = \left[ 1+ \left( \frac{{\rm Re} \cdot \epsilon/d}{150} \right)^{1.8} \right]^{-1}\end{array}$

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. Bellos correlation vs Colebrook–White correlation

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Bellos friction factor correlation