Final answer:
The heat diffusion equation explains the spread of heat in a material by conduction and is derived from the conservation of energy, dEint = dQ. It's represented by Fourier's law, Q/t = -kA(dT/dx), and involves the thermal conductivity, area, and temperature gradient. For radiative heat transfer, the Stefan-Boltzmann law applies: Q/t = σA(T^4).
Step-by-step explanation:
The heat diffusion equation, also known as Fourier's law, describes how heat energy spreads within a given material. The derivation of the heat diffusion equation begins with the law of conservation of energy, which states that the change in internal energy, dEint, is equal to the heat added to the system, dQ, minus the work done by the system, dW. Therefore, for a system at constant volume, we have dEint = dQ, since dW is zero.
When applying this to a differential volume element, we assume that heat transfer is occurring by conduction, and thus, the rate of heat transfer by conduction can be given by Fourier’s law of thermal conduction stated as Q/t = -kA(dT/dx), where Q/t is the rate of heat transfer, k is the thermal conductivity of the material, A is the area through which the heat is flowing, and (dT/dx) is the temperature gradient in the direction of the heat flow.
Additionally, when considering heat transfer by radiation, the Stefan-Boltzmann law describes how an object radiates heat proportional to the fourth power of the absolute temperature, given by the equation Q/t = σA(T^4), where σ is the Stefan-Boltzmann constant, and T is the absolute temperature.