The Rayleigh number drives the free-convection heat transfer. It combines the buoyancy force (from the temperature difference between surface and air) with the fluid's resistance to motion (viscosity) and heat diffusion (thermal diffusivity).

\begin{equation}\label{eq:insulated-pipe-rayleigh} \mathrm{Ra} = \frac{g \beta \left|T_s - T_\infty\right| d^3}{\nu \alpha}\end{equation}

Symbols

\(\mathrm{Ra}\)	Rayleigh number \([\unit{ \unitless}]\)
\(g\)	Gravitational acceleration (9.81) \([\unit{ \meter\per\second\squared}]\)
\(\beta\)	Volumetric thermal expansion coefficient of air (1/ \(T_{film}\)) \([\unit{ \per\kelvin}]\)
\(T_s\)	Outer surface temperature \([\unit{ \kelvin}]\)
\(T_\infty\)	Ambient air temperature \([\unit{ \kelvin}]\)
\(d\)	Outer diameter of the surface \([\unit{ \meter}]\)
\(\nu\)	Kinematic viscosity of air at the film temperature \([\unit{ \meter\squared\per\second}]\)
\(\alpha\)	Thermal diffusivity of air at the film temperature \([\unit{ \meter\squared\per\second}]\)