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Mathematical Model
General Heat Conduction Equation
The general equation of heat conduction in Cartesian coordinates $(x, y, z)$ is given by:
where is the thermal diffusivity, is heat generation, and is volumetric heat capacity.
Case A: Steady-State 2D Conduction
For steady-state conditions () with no internal heat generation () in a 2D plane (ignoring ), the equation simplifies to the Laplace equation:
Finite Difference Approximation
Using central difference approximations, the domain is discretized into a grid of nodes $(i, j)$. The nodal temperatures are solved iteratively:
where
Numerical Method
We use the **Gauss-Seidel** iterative method, which updates each node using the most recently computed values of its neighbors.