Geometric optimization of X-shaped cavities and pathways according to Bejan’s theory: Comparative analysis

Abstract This paper applies constructal design to study the geometry of a X-shaped isothermal cavity and a X-shaped high conductivity pathway that penetrate into a solid conducting wall. The objective is to minimize the maximal excess of temperature of the whole system, i.e. the hot spots, independent of where they are located. There is uniform heat generation on the solid body, which is insulated on the external perimeter. The total volume and the cavity volume, as well as the high conductivity material volume, are fixed, but the geometric lengths and thickness of both X-shaped configurations can vary. The emerged optimal configurations and performance are reported graphically and numerically. The results indicate that the increase of the complexity of the geometry can facilitate the access of heat currents and improve the thermal performance. The degree of freedom L 1 / L 0 proved to be significant on the performance of the X-shaped isothermal cavity, e.g. the once optimized ratio ( L 1 / L 0 ) o increases approximately 10% as the area fraction ϕ increases from ϕ  = 0.05 to 0.3. As for the X-shaped pathway case, it has been demonstrated that the dimensionless thermal conductivity of the path k p and the area fraction ϕ have a strong effect on the performance and configuration of the X-shaped blades: the twice minimized θ max,mm decreases approximately 70% as k p increases from 30 to 300 and it decreases approximately 84% as ϕ augments from 0.01 to 0.2. Furthermore, the X-shaped conductive pathways configuration increases its performance monotonically with the augmentation of the pathways thermal conductivity: in correspondence to the highest possible value of k p , the X-shaped conductive pathways present approximately the same heat removal capacity of the X-shaped cavities optimized in the first part of the paper.