In: Mechanical Engineering
Develop and explain each step the
general equation of heat transfer by conduction in
a constructive way starting from an element general differential in
the cartesian coordinate system and then reduce to get:
1) Laplace's equation (in R3)
Δ? = 0
2) The Poisson equation (with constant thermal conductivity ?)
Δ? + ? '''= 0
Δ is the Laplacian operator and ? ''' is a source term.
Note: start the analysis from a differential element with volume ??????. Make a energy balance over this volume considering that it can increase if temperature in time according to its thermal capacity. Give more generality to the model assuming there is a source term ? ''' which can be a source that produces thermal energy or consume it.