 Solution

In a steady state the heat equation says $(k{T}_s^{\prime }{)}^{\prime }=0$, so the positive-$x$ heat flux $q=-k{T}_s^{\prime }$ is the same constant $Q$ in both materials. Assume perfect thermal contact: temperature is continuous at $x=\ell$, and conservation of energy requires continuous heat flux there. Integrating from the right endpoint givesThe two formulas agree at the interface. Their slopes differ, but $-k_1{T}_s^{\prime }({\ell }^{-})=-k_2{T}_s^{\prime }({\ell }^{+})=Q$. The total temperature drop is $Q\ell (1/k_1+1/k_2)$, the sum of the two layers' contributions in steady heat conduction through layers.