Heat equation of cooling square plate I am trying to write down the PDE problem of cooling a square plate from 100 degrees. The air(25 degress) around the plate is going to cool the plate at the surface, but i do not know how to express this. My guess is
\begin{equation}
u_t(x,t)-au_{xx}(x,t)=25
\end{equation}
Is this correct? Note that this is a 2d plate but lives in 3d space where the air surrounds it everywhere.
 A: 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.


*

*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 $k\nabla^2T$ 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 $$\rho c \frac{\partial T}{\partial t}=k\nabla^2 T-\frac{2hT}{a}$$ with $T=T(x,y,t)$.

*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 $\rho c Aa\frac{d T}{d t}=-2hAT$, or $$\rho c a\frac{d T}{d t}=-2hT$$ with $T=T(t)$.
