# Temperature dependence for specific thermal diffusivity in the diffusion formula

I recently found this answer about the diffusion equation (nice one actually), but have one doubt about the temperature dependence of this formula.

If the "packet" of energy (terminology suggested here) is 6 degrees Celsius (i.e., temperature increase over time from 3 degrees C to 9 degrees Celsius), how much time would take to transport that temperature increase in a distance of 270 m?

I assume that it will take more time for that temperature to reach that depth (compared to a lower temperature). This "packet" of heat will be transported in material of thermal diffusivity of 10^-6 m2/s.

Any hints are appreciated, thanks in advance,

PS. This is not homework, it's part of a paper I'm reading (basically Fig. 1b), here the link: http://www.nature.com/scitable/knowledge/library/methane-hydrates-and-contemporary-climate-change-24314790

• @Danu well, it's not howework, it's part of a paper I'm reading and don't understand, that's why I'm asking here to get some support. I reword that phrase, so you can understand it. – Gery Sep 27 '14 at 10:52
• Temperature has nothing to do with energy. Heat is a well defined quantity even far from equilibrium, when temperature can not even be defined. Your question is using the wrong terminology and it is meaningless as asked, because heat flow problems depend on the boundary conditions. One can easily make steady state systems where two sides of e.g. a block of material are at vastly different temperatures, but the temperature distribution itself is stationary (even though plenty of heat flows from one end to the other). In that case the answer to your question would be "infinity". – CuriousOne Sep 27 '14 at 15:51
• @CuriousOne thanks for your answer, I had the same impression, though. Since this problem is in shallow marine sediments of the deep ocean, heat flow varies from 30 mW/m2 to 50 mW/m2 from 0 to 250 m of sediment. Would that help to get a rough estimate of how long time would be needed to heat up the sediment by 6 degrees C up to 250 m? – Gery Sep 27 '14 at 15:58
• Heat flow depends on temperature differences and, in case of sediments, we are certainly talking about inhomogeneous materials, in which the depth dependent thermal constants need to be established in detail experimentally. I will leave the estimation of these things to the folks who have done years of research on marine sediments and the thermal environment they were exposed to. For the purposes of climatology rough estimates are completely useless, anyway. Either someone does it right, or they may as well stay silent about it. – CuriousOne Sep 27 '14 at 16:05
• The scaling argument that you found is pretty good for linear diffusion (of heat, mass, whatever). Unless the properties of the system change (nonlinear) the propagation rate doesn't actually depend on the magnitude. For a change $\Delta T$, it will take the same time for a remote location to reach a certain fraction of that change. It's all relative. This reference is nice: web.ornl.gov/sci/diffusion/Theory/… – user3823992 Sep 28 '14 at 5:51