Heat Distribution Simulation in a Square Aluminum 7075 Plate Using Laplace Equation and MATLAB

The efficient management of heat transfer from aircraft engines to the wings is vital for maintaining thermal efficiency and structural integrity in modern aircraft design. Excessive heating of the wings, caused by engine-generated heat, can negatively impact aerodynamic performance and safety. This study focuses on analyzing heat distribution in a square aluminum 7075 plate to better understand heat transfer mechanisms. Using the Laplace equation, implemented through MATLAB (2023 Online Version), we aim to simulate and analyze heat distribution on the plate. The numerical method employed in this research involves solving the Laplace equation with Neumann boundary conditions, which represent insulated edges. The Liebmann method is used to iteratively reduce error to less than 1%. Simulations are conducted on an aluminum 7075 plate of dimensions 4x10⁻² m x 4x10⁻² m under various temperature conditions at the edges. Numerical results show that at the 9th iteration, the error reaches 0.71%, while MATLAB simulations yield an error of 0.4681% at the same iteration. The heat distribution across the plate is clearly visualized, and the analysis indicates that increasing the number of grids improves both the clarity and accuracy of the simulation results. In conclusion, this study demonstrates that applying the Laplace equation via MATLAB is an effective approach for analyzing heat distribution in aluminum 7075 plates. The results show that a finer grid resolution enhances accuracy, with a 101-grid system providing particularly clear and precise heat distribution patterns. These findings contribute to the optimization of thermal system designs, especially in aviation-related applications.