Consider Problem 5.9 except now the combined volume of the oil bath and the sphere is V_tot = 1m³. The oil bath is well mixed and well insulated. 

(a) Assuming the quenching liquid’s properties are that of engine oil at 380 K, determine the steadystate temperature of the sphere.

(b) Derive explicit finite difference expressions for the sphere and oil bath temperatures as a function of time using a single node each for the sphere and oil bath. Determine any stability requirements that might limit the size of the time step Δt.

(c) Evaluate the sphere and oil bath temperatures after one time step using the explicit expressions of part (b) and time steps of 1000, 10,000, and 20,000 s.

(d) Using an implicit formulation with Δt 100 s, determine the time needed for the coated sphere to reach 140°C. Compare your answer to the time associated with a large, well-insulated oil bath. Plot the sphere and oil temperatures as a function of time over the interval 0 h ≤ t ≤ 15 h.

Data From Problem 5.9

A solid steel sphere (AISI 1010), 300 mm in diameter, is coated with a dielectric material layer of thickness 2 mm and thermal conductivity 0.04 W/m · K. The coated sphere is initially at a uniform temperature of 500°C and is suddenly quenched in a large oil bath for which T∞ = 100°C and h = 3300 W/m² · K. Estimate the time required for the coated sphere temperature to reach 140°C.