On the “area to point” flow problem based on constructal theory

The study of the “area to point” flow problem, which generates heat uniformly, is conducted based on constructal theory in this paper. Bejan [Bejan A. Constructal-theory network of conducting path for cooling a heat generating volume. Int J Heat Mass Transfer 1997;40(4):799–816] analyzed the problem using an effective thermal conductivity, which simplified the optimization greatly, and deduced an approximate result. Ghodoossi and Egrican [Ghodoossi L, Egrican N. Exact solution for cooling of electronics using constructal theory. J Appl Phys 2003;93(8):4922–9] analyzed the problem without the simplification of an effective conductivity, obtained an exact result, found a great deviation from Bejan’s approximate result and stated that the simplification is the cause of the deviation in the approximate solution. It is proved in this paper that the cause of the deviation in the approximate solution is not the reasonable simplification of an effective conductivity but the mistakenly derived effective thermal conductivity. The approximate solution is revised, and the corresponding result, which is consistent with the exact solution, is obtained.