Critical insulation thickness of a rectangular slab embedded with a periodic array of isothermal strips

We address the problem of two-dimensional heat conduction in a solid slab embedded with a periodic array of isothermal strips. The surfaces of the slab are subjected to a convective heat transfer boundary condition with a uniform heat transfer coefficient. Similar to the concept of critical insulation radius, associated with cylindrical and spherical configurations, we show that there exists a critical insulation thickness, associated with the slab, such that the total thermal resistance attains a minimum, i.e. a maximum heat transfer rate can be achieved. This result, which is not observed in one-dimensional heat conduction in a plane wall, is a consequence of the non-trivial coupling between conduction and convection that results in a 2D temperature distribution in the slab, and a non-uniform temperature on the surface of the slab. The findings of this work offer opportunities for improving the design of a broad range of engineering processes and products.