Dissipation due to heat conduction

[pere.balazs]

What is the dissipation due to heat conduction?

D = -k \frac{\vec q\cdot\nabla T}{T}

where q is the heat flux, k is the
coefficient of heat conduction and T is the absolut temperature.

What is the physical meaning of this?

 

[Mapes]

This is a peculiar term. According to Fourier's law of conduction,

\vec{q}\equiv -k\nabla T\mathrm{,}

so

-k \frac{\vec q\cdot\nabla T}{T}=k^2\frac{\nabla ^2 T}{T}

with units of W² m^-4 K^-1. In kinetic theory, this would correspond to Tk\dot{\sigma}, where \dot{\sigma} is the volumetric rate of entropy creation. To my knowledge, this term isn't used for anything useful; are you certain about it?