We extend a previous one-dimensional analytical study to consider two-dimensional transient effects of a cylindrical, sensible energy-storage system. The working fluid NaK is assumed to be incompressible. Energy transfer from the high temperature surrounding domain is due to conduction in the annulus and convection to the working fluid. We apply a two-dimensional, explicit numerical scheme to quantify transient temperature profiles within the two domains. In addition we consider second law aspects of the two subsystems and isolate irreversibilities which reduce the availability. Finally, we exploit the normalized figure of merit which characterizes the optimal ratio of exergy extraction to the total exergy change of the thermal storage system as a function of removal time.