### Engineering Motivation

This problem considers steady-state heat conduction in a square domain consisting of a regular array of 3 × 3 = 9 square blocks (which we also denote "regions") of different thermal conductivities. We impose uniform (unity) flux at the bottom; zero temperature at the top; and zero flux (insulated) on the vertical sides. Our interest is in the temperature distribution within the block and in particular the temperature of the bottom wall as a function of the block conductivities and conductivity distribution. The bottom wall temperature can be related to the overall thermal resistance of the domain, or equivalently the (inverse) effective conductivity of the "homogenized" composite medium.

### Physical Principles

From the physical point of view, this Worked Problem illustrates the fundamental aspects of steady conduction "thermal resistance": (*i*) the fundamental 1D conduction thermal resistance; (*ii*) the basic notions of thermal resistors connected in series and in parallel; (*iii*) the limitations of the 1D "lumped" resistance model in 2D configurations; and (*iv*) the development of rigorous upper and lower bound procedures that exploit the 1D thermal resistance concept. Other related Worked Problems will consider contraction/restriction resistance concepts, the 1D thermal fin approximation, and anisotropic conductivities.