Complex computational fluid dynamics software that can model material property changes

I need to simulate heat flow through a material whose properties change as a function of time and temperature. For example (this isn’t my exact application, but it gets to the heart of what I need): a wooden board heated from one side will undergo reactions which will change its R value (e.g. carbonization), and it will likely also change shape due to burning; I need to simulate the heat flow through materials undergoing property changes such as this.

Is there any CFD software that can handle this kind of work? I’ve tried searching around and I haven’t found anything, but of course this may be kind of a niche need. I found something somewhat similar in this paper, modeling phase change materials using ANSYS Fluent. However, I’m not sure if I can use their technique, since one of their core assumptions was that thermophysical properties of their materials were constant in temperature.

Gratis is preferred, since I won’t need to justify the cost to a supervisor, but I’ll look into commercial software as well if it seems promising. I have access to both Windows and Linux machines, so OS shouldn’t be a problem.