Chapter 4

This chapter covers two-dimensional steady state heat conduction, starting with Laplace's equation for systems with no internal heat generation. The analytical method introduced is separation of variables, which assumes the temperature distribution can be split into functions of x and y and then solved using boundary conditions. For more complex or irregular problems, the chapter introduces the numerical finite difference method, where the domain is divided into a uniform grid and temperature at each point is approximated by the average of its neighbors. This method is extended with the energy balance approach to include internal heat generation. The nodal equations for each point in the domain are then collected into a matrix system that can be solved using tools like MATLAB, Python, or Excel. The chapter provides a full problem solving strategy that includes identifying the type of problem, setting up the grid, applying boundary conditions, writing the nodal equations, and solving for the temperature distribution. A cheat sheet summarizes key forms of Laplace's equation, finite difference formulas, and matrix notation to guide the numerical process.