Fourier-Biot equation – Heat Conduction Equation
The heat conduction equation, known also as the Fourier-Biot equation, is a partial differential equation that describes the distribution of heat (or the temperature field) in a given body over time. Detailed knowledge of the temperature field is very important in thermal conduction through materials. Once this temperature distribution is known, the conduction heat flux at any point in the material or on its surface may be computed from Fourier’s law.
The heat equation is derived from Fourier’s law and conservation of energy. The Fourier’s law states that the time rate of heat transfer through a material is proportional to the negative gradient in the temperature and to the area, at right angles to that gradient, through which the heat flows.
A change in internal energy per unit volume in the material, ΔQ, is proportional to the change in temperature, Δu. That is:
∆Q = ρ.cp.∆T
Using these two equation we can derive the general heat conduction equation:
This equation is also known as the Fourier-Biot equation, and provides the basic tool for heat conduction analysis. From its solution, we can obtain the temperature field as a function of time.
In words, the heat conduction equation states that:
At any point in the medium the net rate of energy transfer by conduction into a unit volume plus the volumetric rate of thermal energy generation must equal the rate of change of thermal energy stored within the volume.