 Steady heat conduction through layers

In one-dimensional steady heat conduction without internal heat sources, conservation of energy gives constant heat flux $q=-k{T}^{\prime }$. Through a layer of thickness $L_j$ and positive thermal conductivity $k_j$, the temperature drop is $q{L}_j/k_j$. If adjoining layers are in perfect thermal contact, temperature and heat flux are continuous at each interface, so summing the layer drops proves the displayed formula. Temperature is piecewise linear and its slope may jump when thermal conductivity changes.