its not in a vacuum, its in air, it starts at the same temperature as the surroundings, the heat is applied to one full square side of the cube. If the sides of the cube were insulate then i could use Fourier's law however they are not so heat will be lost as it spreads throughout the cube at a different rate as the heated surface area will increase. It should be possible to calculate but I can't figure it out....
Since you're building a mathematical model for this,
- What's the differential which governs the temperature of the cube as a function of time and distance?
- What is the boundary condition at each of the faces of the cube? Is the heat influx constant at one face, or is the temperature kept constant?
Check the relevant chapters in your text. It should have what you're looking for.
I agree with siddarth. This is the basic heat conduction differential equation with some specified material properties and boundary conditions. Refer to your textbook on how to set up the problem.
My guess is that the resulting partial differential equation won't have an analytical solution and you will need to use a numerical solver. If you are familiar with matlab or mathematica they should be able to meet your needs.