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The heat equation assumes that the heat transfer inside of a conductive solid is proportional to the temperature gradient.

However, we might imagine that every element of the solid radiates energy to the surrounding elements. In this case, does heat transfer become proportional to the temperature gradient times T⁴?

If we model this as infinitesimal elements absorbing radiation from their neighbors and reemitting it towards them, the steady-state equation would look something like:

3 (∇T)² + T ∇²T = 0

Here as well, it looks like when the temperature is high this would dominate the effect of conduction.

On the one hand, why would this happen inside an opaque medium? On the other hand, it seems seriously counterintuitive to think conduction could be increased by turning the solid into a sponge.

Does this actually happen? How is it usually modeled?

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Non-linear heat conduction is not unheard of - see, e.g., the first equation and references [1-3] in this article, which I found after a bit of googling (non-linear heat conduction/transfer are the keywords for more search)

Note however, that in most cases leading to heat equation (more generally to the diffusion equation) we consider situations where the gradient of temperature on relevant length scales is small compared to the value of temperature, i.e. the true heat flux is linearized $$ \nabla T^4(\mathbf{x},t)\approx T_0^4 + 4T_0^3\nabla T(\mathbf{x},t) $$ Expanding too straightforwardly to non-linear regime is risky, as there may be other physical effects that you do not atke into account, and which make the whole approach incorrect. Also, working with non-linear equations is much harder (unless you are interested in specific non-linear effects).

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  • $\begingroup$ Thank you. Without even thinking about large gradients, I'd have thought thermal conductivity would be given with graphs as a function of T, but everything I could find seems to indicate k is roughly independent of temperature. $\endgroup$
    – Arthur B
    Oct 21, 2021 at 11:49

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