Heat equation of cooling square plate

There are two ways to write the partial differential equation governing this heat transfer problem depending on whether you wish to consider temperature variations within the plate.

1. If you do wish to consider temperature variations within the plate, then you can perform an energy balance on a differential element (with volume $a,dx,dy$, where $a$ is the thickness) of the plate, which I'll assume lies in the x-y plane. You end up with a conduction term $knabla^2T$ corresponding to conductive heat transfer in the x and y directions (where $k$ is the thermal conductivity of the plate and T is the temperature change relative to ambient temperature) and a convective term $-2hT/a$ corresponding to convection to the surrounding air (where $h$ is the convection coefficient). The sum of energy inputs must be equal to the heat storage term, which is $rho c frac{partial T}{partial t}$, where $rho$ is the density and $c$ is the heat capacity. Thus, we have $$rho c frac{partial T}{partial t}=knabla^2 T-frac{2hT}{a}$$ with $T=T(x,y,t)$.

2. If you don't wish to consider temperature variations within the plate, then you can use a lumped-capacitance approach. Perform an energy balance on the entire plate (of area $A$) to obtain $rho c Aafrac{d T}{d t}=-2hAT$, or $$rho c afrac{d T}{d t}=-2hT$$ with $T=T(t)$.