This paper applies Constructal design to optimize the geometry of a T-Y-shaped cavity that penetrates into a solid conducting wall. The objective is to minimize the global thermal resistance between the solid and the cavity. There is uniform heat generation on the solid wall. The total volume and the cavity volume are fixed, but the geometric lengths of the T-Y-shaped cavity can vary. The shape of T-Y cavity is optimal under the following conditions: it penetrates the conducting wall almost completely, the shape of the area of the solid inserted into the cavity is long enough to cross completely the T-Y-shaped cavity (ending at the coordinate 0), and this area is 14% of the total area occupied by the body and the T-Y-shaped cavity. The results also show that the optimal T-Y-shaped cavity performs approximately 108% better when compared with an optimal C-shaped cavity. (C) 2009 Elsevier Ltd. All rights reserved.