Assume I have a bounded material with heat sources inside. The material is known (i.e. I know heat capacity and all relevant data) and the temperature of the boundary is fixed. I solved the (steady state) heat equation and hence know the temperature distribution.
How do I compute the heat flow through the boundary? I expect something like
$$ \int_\Gamma k\cdot \nabla T $$ where $\Gamma$ is the boundary and $T$ the temperature. Is this correct? If so, what value for $k$ do I have to use?