A steel sphere with a diameter of 10 cm is to be hardened by first heating it to a uniform temperature of 850 oC and then quenching it in a large bath of water at 35 oC. The heat-transfer coefficient at the surface of the sphere is 500 W/m2K. Using the appropriate transient Conduction charts where necessary, answer the following questions with as much explanation as you can and with all working shown and assumptions made. a) Assuming the internal temperature of the sphere remains uniform (but changes with time), calculate the time for the sphere to cool to 250 oC. b) Show that the assumption in a) above is not realistic. c) Given b), calculate a more realistic time for the surface of the sphere to cool to 250 oC and for the centre of the sphere to cool to 250 oC. Compare your results to those obtained in a) and comment on the differences. d) In a) and c) above, what assumption is made regarding the temperature of the water in the bath? If the bath is rectangular, with width 0.25 m, length 0.5 and height 0.5 m and its walls are well insulated, how valid is this assumption? (hint, you will need to first find the total heat transfer from the sphere in c) above. e) Finally, given all the information above, estimate the final, uniform temperature of the sphere and water if it is left to cool for a very long time and all heat losses from the water to the surroundings can be neglected.