Heat balance with volumetric heat generation ($c_{\mathrm{vol}}{T}_t=\nabla \cdot (k\nabla T)+H$)

Integrating energy conservation over a fixed material volume gives storage equal to incoming conductive heat flux plus internally generated heat. Here $c_{\mathrm{vol}}$ is volumetric heat capacity, with units $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$. For a mass-specific heat capacity, the density factor must be included. Constant isotropic conductivity reduces the divergence to $k{\nabla }^{2}T$; variable conductivity must remain inside the divergence.