The mechanical design of the target module of an accelerator driven sub-critical nuclear reactor system (ADSS) calls for an analysis of the related thermal–hydraulic issues because of the sheer large amount of heat generation in its spallation target system during the course of nuclear interactions with the molten lead bismuth eutectic (LBE) target. The window of the target module is subject to high heat fluxes due to the direct impingement of high energy proton beam on its surface. A large amount of heat is deposited on the window and in the bulk of the LBE in the spallation region. Therefore, the problem of heat removal by the LBE is a challenging thermal–hydraulic issue. For this, one will need to examine the laminar/turbulent flows of low Prandtl number fluids (LBE) in a complex target module of an ADSS. In this study, the equations governing the flow and thermal energy are solved numerically using the streamline upwind Petrov–Galerkin (SUPG) finite element (FE) method. Special consideration has been given to the window under various thermal conditions, such as, isothermal, uniform and variable heat flux. The analysis has been extended to the case of heat generation in the liquid LBE. The principal purpose of the analysis is to trace the temperature distribution on the beam window and in the LBE. This also helps to check the suitability of the geometry in avoiding the recirculation or stagnation zones in the flow space that may lead to hot spots.