The Poisson’s equation the general conduction heat transfer applies to the case

Concept:

Generalized 3D conduction equation is given by Fourier equation which is: 

\(\frac{{{\partial ^2}T}}{{\partial {x^2}}} + \frac{{{\partial ^2}T}}{{\partial {y^2}}} + \frac{{{\partial ^2}T}}{{\partial {z^2}}} + \frac{{\dot q}}{k} = \frac{1}{\alpha }\left( {\frac{{\partial T}}{{\partial \tau }}} \right)\)

For Poisson's equation, the form of the equation should be ∇²T + a = 0;

For steady-state, \(\frac{{\partial T}}{{\partial \tau }} = 0\)

With heat generation, the equation takes the form

\({\nabla ^2}T + \frac{q}{k} = 0 \Rightarrow Poisson's \; equation\)

Without heat generation, the equation takes the form 

\({\nabla ^2}T = 0 \Rightarrow Laplace \; equation\)