The application of entransy dissipation theory in the volume-to-point heat conduction problem

Recent advances in semiconductor technologies are accompanied by an accelerated increase in power density level from high performance chips such as microprocessors. The thermal management of electronic devices, especially meeting the limitations on maximum operating temperature and ensuring temperature uniformity across the chip, becomes one of the most critical issues in the electronic industry. Much effort is devoted to devising the efficient electronic cooling technologies. One cooling strategy proposed by Bejan is that the heat generated in a finite volume may be removed to a point on the boundary through some embedded conducting paths with high conductivity in the substrate. The problem is to optimize the allocation of the conducting paths so that the generated heat can be most effectively dissipated and the highest temperature in the domain is minimized, which is called the volume-to-point conduction problem. Bejan developed the constructal method to optimize the high conductivity material allocation. However the uniformly distributed heat source is always assumed in Bejan's discuss, which is not consistent with the reality in the chips or other integrated circuits. In the present work, the volume-to-point conduction problem with non-uniform heat sources is formulated in the framework of the entransy dissipation theory and is solved by the variational method and finite element method. The tree-shape distribution of the conducting paths is obtained, which agrees with the constructal theory.