4–72 A person puts a few apples into the freezer at 215°C to cool them quickly for guests who are about to arrive. Initially, the apples are at a uniform temperature of 20°C, and the heat transfer coefficient on the surfaces is 8 W/m2·K. Treating the apples as 9-cm-diameter spheres and taking their properties to be r 5 840 kg/m3, cp 5 3.81 kJ/kg·K, k 5 0.418 W/m·K, and a 5 1.3 3 1027 m2/s, determine the center and surface temperatures of the apples in 1 h. Also, determine the amount of heat transfer from each apple. Solve this problem using analytical one-term approximation method (not the Heisler charts).

Complete and Clear Question:

A person puts a few apples into the freezer at -15°C to cool them quickly for guests who are about to arrive. Initially, the apples are at a uniform temperature of 20°C, and the heat transfer coefficient on the surfaces is 8 W/m2·K. Treating the apples as 9-cm-diameter spheres and taking their properties to be 840 kg/m3,   3.81 kJ/kg·K, k = 0.418 W/m·K, and , determine the center and surface temperatures of the apples in 1 h. Also, determine the amount of heat transfer from each apple. Solve this problem using analytical one-term approximation method (not the Heisler charts).

Temperature at the center of the apple, T(t) = 11.215°C

Temperature at the surface of the apple, T(r,t) = 2.68°C

Amount of heat transfer from each apple, Q = 21.47 kJ

Explanation:

For clarity and easiness of expression, the calculations are handwritten and attached as a file. Check the attached files for the complete calculation.