In the design of buildings, energy conservation requirements dictate that the exterior surface area, As, be mini- mized. 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, Af , and the vertical distance between floors, Hf , 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 loss 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 Af and Hf.

(b) If Af = 32,768 m2 and Hf = 4 m, for what values of W and Nf (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 percentage reduction in heat loss compared with a building for Nf = 2?