 Solution

The Rayleigh number compares destabilizing thermal buoyancy with viscous and thermal diffusion:The conductive heat flux is $k\Delta T/h$, and the Nusselt number is the actual upward heat flux divided by this value. The four-thirds convective heat-transfer law is the empirical large-Rayleigh closure $\mathrm{Nu}=C{\mathrm{Ra}}^{1/3}$. It givesThe layer depth cancels: thin thermal boundary layers control the resistance while convection mixes the interior. The dimensionless coefficient $C$ depends on the boundary and material-property regime; the two boundary-layer temperature drops can be absorbed into its convention. This is not a universal law for all Rayleigh-Bénard convection. Below convective onset the stable conductive solution instead has $\mathrm{Nu}=1$, and an impulsive initial conductive transient is not described by the steady large-Rayleigh closure.