A spherical shaped building is 112 meters in height and 157 m in diameter in outer dimensions. The thickness of the reinforced concrete shell of the sphere is 25 cm and the effective thermal conductivity of the material is k_wall = 0.5 W/m-K. On a cold windy December night in Vegas, the wind blows across the sphere at a speed of 20 m/s and the temperature of air was T_inf,o = 4 C. It is required that the temperature of air inside the Sphere be at a minimum of T_inf,i = 10 C in its unoccupied state. Large fans circulate the air within the Sphere at a h_i = 20 W/m2-K.

Estimate the heating energy rate (in watts) needed to maintain this interior air temperature.

Estimate the outer surface temperature of the sphere.

Estimate electricity cost needed to maintain the venue at this temperature through the night for a period of 8 hours. You may assume that the same steady state conditions exist through the night. The cost of electricity per kW-h is 0.18 cents.