derivation of heat equation
Let us consider the heat conduction in a and specific heat capacity . Denote by the temperature in the point at the time . Let be a surface in the matter and the spatial region by it.
When the growth of the temperature of a volume element in the time is , the element releases the amount
of heat, which is the heat flux through the surface of . Thus if there are no sources and sinks of heat in , the heat flux through the surface in is
On the other hand, the flux through in the time must be proportional to , to and to the derivative of the temperature in the direction of the normal line of the surface element , i.e. the flux is
where is a positive (because the heat always from higher temperature to lower one). Consequently, the heat flux through the whole surface is
which is, by the Gauss’s theorem, same as
Equating the expressions (1) and (2) and dividing by , one obtains
Since this equation is valid for any region in the matter, we infer that
Denoting , we can write this equation as
This is the differential equation of heat conduction, first derived by Fourier.
- 1 K. Väisälä: Matematiikka IV. Handout Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).
|Title||derivation of heat equation|
|Date of creation||2013-03-22 18:45:04|
|Last modified on||2013-03-22 18:45:04|
|Last modified by||pahio (2872)|