Abstract
We address the two-dimensional heat conduction problem due to a periodic array of isothermal pipes buried in a conductive medium. The upper surface of the medium is subjected to convection with a uniform heat transfer coefficient, and the lower surface is insulated. Similar to the concept of critical thickness associated with a slab embedded with isothermal strips, we show that there exists a critical depth such that the heat transfer rate is maximized. As the Biot number tends to infinity, the critical depth approaches zero for a single pipe buried in a semi-infinite medium. For a periodic array of isothermal pipes, there is also a critical Biot number beyond which the critical depth is zero. Furthermore, insulating the pipes reduces the critical depth, and the heat transfer rate does not vary significantly with respect to the depth.