Optimal distribution of imperfection in conductive constructal designs of arbitrary configurations

In the last decade, different conductive networks for cooling heat generating bodies have been proposed, analyzed, and optimized. Links of a highly conductive material collect the heat generated in an area and conduct it to a heat sink. The important design point is how to dedicate the limited and expensive high conductivity material to each link. Although optimizations have been performed for heat generating bodies of different shapes separately, the lack of a general method is quite obvious in the literature. Therefore, an analytical general solution for optimal distribution of highly conductive material in networks of arbitrary configuration is developed in the present article. The proposed method gives the optimum thickness of each link with a simple expression, which only depends on the geometry of heat generating bodies. Moreover, it is mathematically proved that total thermal resistance of the optimal links is a function of Bejan's parameter k̂φ and a purely geometrical function. This achievement greatly simplifies the design and optimization of any conductive cooling network. Results of the present simple formulation for configurations of various geometries are completely consistent with the available analytical and numerical solutions.