Heat transfer in a finite cylinder This question has more to do with experimental physics than theoretical physics. I have a copper cylinder of radius $a$ and length $L$. On one side of the cylinder, let's say $x=0$, I have a source with a temperature that varies with time. Now, on the other side, $x=L$, the copper cylinder is exposed to air at atmospheric pressure and temperature.
Now, my problem is what happens at $x=L$. I know I can't consider that side as a source because it doesn't have a constant temperature. But if a wanted to solve the heat equation, which bonundary condition should I use for $x=L$?
 A: 
Now, my problem is what happens at $x=L$. I know I can't consider that side as a source because it doesn't have a constant temperature. But if a wanted to solve the heat equation, which boundary condition should I use for $x=L$?

Your boundary condition must express what is physically happening.
There are various possibilities.


*

*Assume that side of the cylinder loses heat through convection only:


If $T(r,x,t)$ is the temperature of the cylinder, then in that case:
$$\kappa\frac{\partial T(r,L,t)}{\partial x}=h[T(r,L,t)-T_{\infty}]$$
($\kappa T_x(r,L,t)=h[T(r,L,t)-T_{\infty}]$)
$T_{\infty}$ is the temperature of air at that end. This boundary condition means that the heat conducted though the end bit is equal to what is lost through convection at that end. $\kappa$ is the thermal diffusivity and $h$ the the convection coefficient.
In that case it's easier to define a new variable:
$$u(r,x,t)=T(r,x,t)-T_{\infty}$$
So:
$$\kappa \frac{\partial u}{\partial x}=hu$$
($\kappa u_x=hu$)


*Assume that side of the cylinder loses heat through radiative losses only:


Use Stefan-Boltzmann:
$$\kappa\frac{\partial T(r,L,t)}{\partial x}=\epsilon \sigma[T(r,L,t)^4-T_{\infty}^4]$$


*Combine $1.$ and $2.$, by assuming the end loses heat through convection and radiation:


$$\kappa\frac{\partial T(r,L,t)}{\partial x}=h[T(r,L,t)-T_{\infty}]+\epsilon \sigma[T(r,L,t)^4-T_{\infty}^4]$$


*Assume the tip $x=L$ is insulated:


$$\kappa\frac{\partial T(r,L,t)}{\partial x}=0$$
($u_x(r,L,t)=0$)
The latter is of course by far the easiest case to solve.
The choice is yours.
If you assume $\frac{\partial T}{\partial r}=0$, then $T$ reduces to $T(x,t)$ but the principle remains the same.
Your question is closely related to this question
