A comprehensive numerical study has been done to investigate two-dimensional, steady state, conjugate natural heat convection in the hemi spherical lower plenum of a fast breeder reactor under failed conditions. The continuity, momentum and energy equations are solved over the entire domain, using the corresponding properties for the solid and fluid regions. The control volume approach is employed in order to discretize the governing equations for their numerical solution. A parametric study has been done to study the variation of the velocity vectors and isotherms for different constant temperature of the heat source, simulating different heat generation rates. The actual problem in a nuclear reactor involves a volumetric heat generation in the debris falling over the heat shield plate under failed conditions of the reactor and heat is removed by a decay heat exchanger serving as a sink. In this study we have reduced this transient problem to a quasi-steady problem with a prescribed temperature on the heat shield plate. This makes the problem more tractable. The fluid flow pattern, variation of the temperature along the axis in and around the heat source are presented to show the overall heat transfer characteristics inside the plenum.