Heat conduction in gravity field without convection

[lukas.suess]

I used to think that the heat does rise even in solid metals with no gas/liquid around
(No density argument is possible then.)
But couldn't find anything describing or even verifying it.
I'am pretty sure the gravitationally induced anharonicity in the atomic core potentials
should have at least a little effect.

The simplest description of heat transport in a thermally isolated rod is:
Power = diffTemp * ( thermConduct * Area /length)
My question would be wether it is necessary to consider gravity induced corrections
for technical purposes (9.81m/s**2) or not?

Could the formula be extended this way?
Power' = Power * (1 + inprod(e_rod, e_grav) * f)
and how could one determine the factor f?

 

[Chestermiller]

For all practical purposes, gravity has no effect on heat conduction in a rod.