In the present study, conductive cooling of a disc is done by means of incomplete constant and variable cross-section highly conductive inserts embedded in radial and tributary configurations. Variational calculus is invoked to determine the optimum shape of the cross-sections of the inserts. Firstly, it is tried to derive an equation for thermal resistance of the disc for radial configuration of inserts based on the procedure used in constructal studies. This is done by implementing the

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... ed thermal resistances of elemental sectors. Then, the computed elemental sectors are put together so that they make branching configuration of inserts in the disc. Out of the comparison between the obtained thermal resistances of the disc with constant and variable cross-sections, it is concluded that using variable cross-sections reduces thermal resistance, but this effect differs in radial and tributary configurations, i.e., increasing the complexity of tributary patterns does not always reduce the thermal resistance more effectively in comparison with radial configurations.