When both forced and free convection are present simultaneously, the two contributions are combined using a fourth-power sum rule. The exponent of four gives a smooth transition between forced-dominated and buoyancy-dominated regimes.

\begin{equation}\label{eq:insulated-pipe-nusselt-combined} \mathrm{Nu} = \left(\mathrm{Nu}_{forced}^4 + \mathrm{Nu}_{free}^4\right)^{1/4} \end{equation}

Symbols

\(\mathrm{Nu}\)	Combined Nusselt number \([\unit{ \unitless}]\)
\(\mathrm{Nu}_{forced}\)	Forced-convection Nusselt number \([\unit{ \unitless}]\)
\(\mathrm{Nu}_{free}\)	Free-convection Nusselt number \([\unit{ \unitless}]\)