The Rayleigh number characterises the strength of buoyancy-driven natural convection at the tank outer surface. It combines the buoyancy force (from the temperature difference between the surface and the ambient air) with the fluid's resistance to motion and heat diffusion.

\begin{equation}\label{eq:insulated-tank-rayleigh} \mathrm{Ra} = \frac{g \,\beta \,\lvert T_s - T_\infty \rvert\, d^3}{\nu\,\alpha} \end{equation}

Note

This formula uses gravitational acceleration g = 32.174 ft/s² (U.S. customary). The thermal expansion coefficient \(\beta\) is approximated as \(1 / T_\infty\) for an ideal gas.

Symbols

\(\mathrm{Ra}\)	Rayleigh number \([\unit{ \unitless}]\)
\(g\)	Gravitational acceleration (32.174) \([\unit{ \foot\per\second\squared}]\)
\(\beta\)	Volumetric thermal expansion coefficient of air (1/ \(T_\infty\)) \([\unit{ \per\degreeRankine}]\)
\(T_s\)	Outer surface temperature (jacket or bare tank wall) \([\unit{ \degreeRankine}]\)
\(T_\infty\)	Ambient air temperature \([\unit{ \degreeRankine}]\)
\(d\)	Tank outer diameter \([\unit{ \foot}]\)
\(\nu\)	Kinematic viscosity of air at ambient temperature \([\unit{ \foot\squared\per\second}]\)
\(\alpha\)	Thermal diffusivity of air at ambient temperature \([\unit{ \foot\squared\per\second}]\)