Consider steady two-dimensional heat transfer in a rectangular cross section \((60 \mathrm{~cm} \times 30 \mathrm{~cm})\) with the prescribed temperatures at the left, right, and bottom surfaces to be \(0^{\circ} \mathrm{C}\), and the top surface is given as \(100 \sin (\pi x / 60)\). Using a uniform mesh size \(\Delta x=\Delta y\), determine (a) the finite difference equations and \((b)\) the nodal temperatures.

Understanding the concept of 2D steady-state heat conduction is crucial for several engineering applications. This phenomenon occurs when heat is transferred within a material without any changes in temperature with time, creating a steady temperature field. At the steady state, the amount of heat entering any region of the material is equal to the heat leaving that region. In context to our exercise, we consider a rectangular cross-section where the heat transfer is two-dimensional and time-independent.

For instance, in our solved example, the edges of the rectangle are maintained at certain temperatures. The left, right, and bottom surfaces are kept at a constant temperature of 0°C, and the top surface follows a sinusoidal temperature distribution. The steady-state condition implies that the interior of the rectangle eventually reaches a temperature configuration that does not change over time, despite the complex boundary conditions.

Boundary conditions in heat transfer define how the heat behaves at the borders of the domain being studied. They are essential in setting up a heat transfer simulation or calculation because they directly affect the temperature distribution within the object.

There are generally three types of boundary conditions one may encounter:

• Dirichlet boundary condition specifies the value of temperature at the boundary.
• Neumann boundary condition defines the heat flux at the boundary.
• Robin boundary condition, a combination of Dirichlet and Neumann, where a heat transfer coefficient is specified.

Our example sets Dirichlet conditions by specifying temperatures at all sides of the rectangle. The uniformity of these conditions along the left, right, and bottom surfaces simplifies our calculations, while the varying top condition introduces complexity in the resulting temperature field.

The numerical solution of heat transfer problems involves solving complex differential equations that describe the behavior of temperature over space and, in transient problems, time. Since analytical solutions are often not feasible for complicated domains and boundary conditions, numerical methods such as the finite difference method (FDM) are employed.

In our example, the application of FDM allows us to approximate the continuous domain with a discrete grid, where the derivatives in the heat equation are replaced with differences. We then get a system of linear equations, which can be solved using computational methods to approximate the temperature field. These methods are quite powerful because they can handle varying material properties, complex geometries, and diverse boundary conditions, which makes them indispensable tools for engineers and scientists solving real-world heat transfer problems.

Calculating nodal temperatures is the essence of solving a heat transfer problem numerically. In our exercise, a grid is established over the rectangular domain, and we seek to find the temperature at each grid point or node. Using the finite difference approximations to the heat equation, a relationship is established between a node's temperature and its neighbors'.

This leads to a system of linear equations where each equation corresponds to a node inside the grid. The challenge is to resolve this system for temperatures at interior nodes while respecting the given boundary conditions. Various algorithms exist for this, including iterative methods like Gauss-Seidel and Jacobi, often favored for their simplicity and steady convergence characteristics. The resultant temperature at each node gives us a comprehensive look at how heat is distributed throughout the section, paving the way to predicting material behavior and designing efficient thermal systems.