Heat equation: boundary conditions?

Question

Say we have a bar centered at $x=0$, that is heated. We have the 1D Heat equation which we can solve to find a parabolic temperature profile :

$$\kappa \frac{d^2 T}{d x^2}=-\frac{Q}{L S}$$

with

$$T(L/2)=T(-L/2)=T_{\infty}$$

$T_{\infty}$ is the ambient temperature at the end of the bar. This boundary condition alone is enough to solve the equation. However, now say we want to add a very tiny rod at the center of the bar that is connected to the environment to act as a path for heat flow. $R$ is the thermal resistance of the rod. $S$ is the cross sectional area of the bar. $\kappa$ is thermal conductivity of the system. $Q$ is the heat dissipated by joule heating. The corresponding boundary condition is:

$$\frac{T(0)-T_{\infty}}{R_{\rm thermal \ rod}}=\kappa S \left(\frac{d T}{dx}_{x=0^+}-\frac{d T}{d x}_{x=0^-} \right)$$

I have fiddled around with this math for a while but cannot figure out how to implement it.

In particular I am following this paper on the T-Bridge Method for thermal conductivity.

Answer 1

I think the problem is that while in the first case your differential equation applies to all your domain of interest and you can just use it, in the second situation the DE doesn't apply at x=0. This means you need to solve the DE at the domains $0<x<L/2$ and $-L/2<x<0$ separately, where it does still apply.

When applying it to those domains you should remember that the temperature $T$ is still continuous throughout all the domain because you don't want infinite $\frac{d T}{d x}$; that is, infinite heat fluxes. If you do that, after you solve the DE with $T(0)$ as an undeterminate boundary value, you should find that the right hand side of your third equation is a function of $T(0)$ and thus you can find it in terms of the problem constants $T_{\infty}$, $S$, $R_{\text{thermal rod}}$, etc.