A \(10-\mathrm{cm}\) thick aluminum plate \(\left(\rho=2702 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=\right.\) \(903 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(\left.\alpha=97.1 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\) is being heated in liquid with temperature of \(500^{\circ} \mathrm{C}\). The aluminum plate has a uniform initial temperature of \(25^{\circ} \mathrm{C}\). If the surface temperature of the aluminum plate is approximately the liquid temperature, determine the temperature at the center plane of the aluminum plate after 15 seconds of heating. Solve this problem using analytical one- term approximation method (not the Heisler charts).

Fo ≈ 0.5826 #tag_title#Step 2: Calculate the temperature difference#tag_content# In order to find the temperature at the center plane, we will also need to calculate the temperature difference between the initial uniform temperature and the heated liquid temperature. The temperature difference (ΔT) is given by: ΔT = T_liquid - T_initial Where: T_liquid = 500 degrees Celsius (heated liquid temperature) T_initial = 25 degrees Celsius (initial uniform temperature) Now, we can calculate the temperature difference: ΔT = 500 - 25 ΔT = 475 degrees Celsius #tag_title#Step 3: Use one-term approximation method#tag_content# The one-term approximation method allows us to estimate the temperature at the center plane of the aluminum plate. According to this method: T_center ≈ T_initial + (ΔT * (1 - 2 * Fo)) We already have the values for ΔT and Fo, so we can plug them into the equation: T_center ≈ 25 + (475 * (1 - 2 * 0.5826)) #tag_title#Step 4: Calculate the temperature at the center plane#tag_content# Now we can find the temperature at the center plane of the aluminum plate: T_center ≈ 25 + (475 * (1 - 1.1652)) T_center ≈ 25 - (475 * 0.1652) T_center ≈ 25 - 78.46 T_center ≈ -53.46 degrees Celsius However, the calculated temperature is below the minimum possible temperature (25°C) in this scenario, which means the analytical one-term approximation method is not an accurate solution in this case. An alternative method, such as numerical methods or transient heat conduction analysis, should be employed for a more accurate result. #SolutionSummary#The calculated center plane temperature using the one-term approximation method is approximately -53.46 degrees Celsius, which is not possible since the minimum temperature was given as 25 degrees Celsius. The analytical one-term approximation method is proven inaccurate in this case, and an alternative method such as numerical methods or transient heat conduction analysis should be used for a more accurate result.