A 2-m 1.5-m section of wall of an industrial furnace burning natural gas is not insulated, and the temperature at the outer surface of this section is measured to be 80C. The temperature of the furnace room is 30C, and the combined convection and radiation heat transfer coefficient at the surface of the outer furnace is 10 W/m2 K.

(a) It is proposed to insulate this section of the furnace wall with glass wool insulation (k = 0.038 W/mK) in order to reduce the heat loss by 90 percent. Assuming the outer surface temperature of the metal section still remains at about 110C, determine the thickness of the insulation that needs to be used.

(b) The furnace operates continuously and has an efficiency of 78 percent. The price of the natural gas is $1.10/therm (1 therm = 105,500 kJ of energy content). If the installation of the insulation will cost $250 for materials and labor, determine how long it will take for the insulation to pay for itself from the energy it saves