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For non-smooth pipelines 

LaTeX Math Inline
body\epsilon > 0
the Darcy friction factor 
LaTeX Math Inline
bodyf
  can be estimated from empirical Colebrook–White implicit  Colebrook–White correlation which works for non-laminar flow:

LaTeX Math Block
anchorf_CW
alignmentleft
\frac{1}{\sqrt{f}} = -2 \, \log \Bigg( \frac{\epsilon}{3.7 \, d}  + \frac{2.51}{{\rm Re} \sqrt{f}} \Bigg)


There are numerous approximation of 

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LaTeX Math Block Reference
anchorf_CW

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

LaTeX Math Block
anchorMOM9US0MZP
alignmentleft
f = 0.25 \, \biggleft[ \log \biggleft( \frac{\epsilon / d}{3.7065} - \frac{5.0272}{\rm Re} \log \Lambda \biggright)   \biggright]^{-2}


где

LaTeX Math Inline
body\Lambda
– безразмерный параметр, рассчитываемый по формуле:

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