Chapter 3, Problem 3.25P

A particular thermal system involves three objects of fixed shape with conduction resistances of $R_1=1\text{K/W,}{R}_2=2\text{K/W}$ and $R_3=4\text{K/W,}$ respectively. An objective is to minimize the total thermal resistance Riot associated with a combination of $R_1,R_2,$ and $R_3.$ The chief engineer is willing to invest limited funds to specify an alternative material for just one of the three objects; the alternative material will have a thermal conductivity that is twice its nominal value. Which object (1, 2, or 3) should be fabricated of the higher thermal conductivity material to most significantly decrease $R_{\text{tot}}?$ Hint: Consider two cases, one for which the three thermal resistances are arranged in series, and the second for which the three resistances are arranged in parallel.