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

150 views
in General
closed
The Poisson’s equation the general conduction heat transfer applies to the case
1. Steady state heat conduction with heat generation
2. Steady state heat conduction without heat generation
3. Unsteady state heat conduction without heat generation
4. Unsteady state heat conduction with heat generation

by (54.0k points)
selected

Correct Answer - Option 1 : Steady state heat conduction with heat generation

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

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