Problem 3 (30 pts) You are tasked to design a thermal process for treatment of a sphere of radius r = 5 mm, which is initially at 400 C in a furnace. The thermophysical properties of the sphere are p = 3000 kg/m3, k = 20 W/m-K, c = 1000 J/kg-K, and = 6.66x10-6 m/s. It is suddenly removed from the furnace and subjected to a two-step cooling process. Step 1 Cooling in air at 20C for a period of time ta until the center temperature reaches a critical value, Ta(0, ta) = 335C. For this step, the convective heat transfer coefficient is ha = 10 W/m-K. After the sphere has reached this critical temperature, the second step is initiated. Step 2 Cooling in a well-stirred water bath at 20C, with a convective heat transfer coefficient of hw 6,000 W/m-K. (a 15 pts) Calculate the time ta required for step 1 of the cooling process to be completed. (b 15 pts) Calculate the time tw required during step 2 of the process for the center of the sphere to cool from 335C (the condition at the completion of step 1) to 50C.