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?