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For laminar-turbulent transition and turbulent flow in pipeline the Nusselt number (Nu) becomes also dependant on friction with wall, quantifiable by Darcy friction factor 

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, and can be estimated through empirical correlation (Gnielinski

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{\rm Nu}=\frac{ (f/8) \, ({\rm Re} - 1000) {\rm Pr}  }{ 1 + 12.7 \, (f/8)^{1/2} \, ({\rm Pr}^{2/3} -1) }

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