Objectives: To formulate finite difference equations for 2-D heat conduction problems; To determine the unknown temperatures numerically using the Gauss-Seidel iteration method; To investigate the impact of the nodal size; To understand the effect of thermal boundary conditions. Approach: To meet the above objectives, you are asked to complete the following five parts of the project involving heat conduction in a two-dimensional solid. Part 1. Consider the process of heat conduction in a long square rod with isothermal boundaries under steady-state conditions shown in Figure 1 where the surface temperatures are kept constant with respect to time. If the entire domain of interest is divided into four equal regions resulting in only one interior node at the center, determine the temperature at this center node. (5%) Part 2. Part 3. Part 4. Part 5. Rather than four regions, now divide the domain of interest into nine equal regions resulting in four interior nodes as shown in Figure 2. Set up a system of finite difference equations using the Gauss-Seidel iteration method for the four interior nodes. Assuming an initial trial value of 450 K, determine numerically the four unknown temperatures, T1, T2, T3, and T4. Based on the argument of symmetry, explain the results you obtain. What is the temperature at the center of the rod? (20%) Now divide the domain of interest into 16 equal regions resulting in nine interior nodes as shown in Figure 3. Using the argument of symmetry, show that there are only four unknown temperatures among the nine interior nodes as illustrated in Figure 4. Set up a system of finite difference equations using the Gauss-Seidel iteration method for the four unknown temperatures. Assuming an initial trial value of 450 K, determine numerically the values of T, T2, T3, and Tc. How many numerical iterations are needed? (20%) To obtain a more detailed distribution of temperatures within the square rod, the domain of interest is further divided into 25 equal regions resulting in 16 interior nodes. Based on the argument of symmetry, there are only four unknown temperatures among these 16 interior nodes (see Figure 5). Set up a system of finite difference equations using the Gauss-Seidel iteration method for the four unknown temperatures. Assuming an initial trial value of 450 K, determine numerically the values of T1, T2, T3, and T4. How many numerical iterations are needed? What is the temperature at the center of the rod? From the results so obtained, shown schematically how heat is being conducted throughout the domain of interest. (25%) It is desired to understand the effect of thermal boundary conditions. Consider the case for which the right-hand side of the square rod is thermally insulated. If the domain of interest is divided into nine equal regions, there will be two surface nodes in addition to four interior nodes with unknown temperatures. Based on the argument of symmetry, show that there are only three unknown temperatures among these six nodes as shown in Figure 6. Set up a system of finite difference equations using the Gauss-Seidel iteration method for the three unknown temperatures. Assuming an initial trial value of 450 K, determine numerically the values of T1, T2, and T3. How many numerical iterations are needed such that the numerically determined values are within 0.1 K of the exact values? (30%) 500K 500K 12 T 400K 400K 400K 3 4 400K 500K Figure 1 500K 500K Figure 2 500K 1 2 2 1 4 5 6 3 T 3 400K 400K 400K 400K 00 8 9 2 500K Figure 3 500K 1 2 2 1 3 4 4 3 400K 400K 3 4 4 3. 400K 1 2 2 500K 500K Figure 4 500K 1 23 35 - 500K 23 Figure 5 Figure 6 Insulated surface