In the design of buildings, energy conservation requirements dictate that the exterior surface area, A" be minimized. This requirement implies that, for a desired floor space, there may be optimum values associated with the number of floors and horizontal dimensions of the building. Consider a design for which the total floor space, A_f' and the vertical distance between floors, H_f' are prescribed.

(a) If the building has a square cross section of width W on a side, obtain an expression for the value of W that would minimize heat los to the surroundings. Heat loss may be assumed to occur from the four vertical side walls and from a flat roof. Express your result in terms of A_f and H_f'

(b) If A_f = 32,768 m² and H_f = 4 m. for what values of Wand N_f (the number of floors) is the heat loss minimized? If the average overall heat transfer coefficient is U = 1 W/m² ∙ K and the difference between the inside and ambient air temperatures is 25°C, what is the corresponding heat loss? What is the percent reduction in heat loss compared with a building for N_f = 2?