Here we explore the trade-off between two bars made of different materials for the purpose of facilitating thermal contact between two ends at different temperatures. Heat conduction is in the axial...

Here we explore the trade-off between two bars made of different materials for the purpose of facilitating thermal contact between two ends at different temperatures. Heat conduction is in the axial direction along each bar. The length, cross sectional area, thermal conductivity and volume of the two bars are (L, A, k, V)1 and (L, A, k, V)₂. The overall end-to-end distance is fixed, L₁
+ L₂
= L. The objective is to find the ratio L₁∕L₂, or A₁∕A₂, such that the end-to-end thermal resistance is minimal. Show that this design solution is ruled by the dissimilarity ratio k₁V₁∕(k₂V₂). Show further that the bar with the higher thermal conductivity should be thinner and longer than the bar with lower thermal conductivity.