This paper applies constructal design to study the geometry of a X-shaped cavity that penetrates into a solid conducting wall. The objective is to minimize the maximal dimensionless excess of temperature between the solid body and the cavity. There is uniform heat generation on the solid body. The total volume and the cavity volume are fixed, but the geometric lengths and thickness of the X-shaped cavity can vary. The cavity surfaces are isothermal while the solid body has adiabatic conditions on the outer surface. The emerged optimal configurations and performance are reported graphically. When compared to the Y- and C- and H-, the X-shaped cavity performs approximately 53% better than the Y-shaped cavity and 68% better than the C-shaped cavity for the area fraction f = 0.05, while its performance is 22% inferior to the performance of the H-shaped cavity for the area fraction f = 0.1. The results indicate that the increase of the complexity of the cavity geometry can facilitate the access of heat currents and improve the performance of the cavities.
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