Chapter 3, Problem 3.22P

In the design of buildings, energy conservation requirementsdictate that the exterior surfacearea, $A_s,$ be minimized.This requirement implies that, for a desired floorspace, there may be optimum values associated withthe number of floors and horizontal dimensions of thebuilding. Consider a design for which the total floorspace, $A_f,$ and the vertical distance between floors, $H_f,$ are prescribed.

1. 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 fourvertical side walls and from a flat roof. Express yourresult in terms of $A_f$ and $H_f.$ If $A_f=32,768{\text{m}}^2$ and $H_f=4\text{m,}$ for what values ofW and $N_f$ (the number of floors) is the heat lossminimized? If the average overall heat transfercoefficient is $U=1{\text{W/m}}^2\cdot \text{K}$ and the differencebetween the inside and ambient air temperatures is $\text{25°C,}$ what is the corresponding heat loss? Whatis the percentage reduction in heat loss comparedwith a building for $N_f=2\text{?}$