An objective function combining the first and second laws of thermodynamics has been employed to delineate the thermodynamic performance on mixed convection around a vertical hollow, rotating cylinder within the laminar range with the variation of Rayleigh number (104 ≤ Ra ≤ 108), Reynolds number (ReD < 2100), and aspect ratio (1 ≤ L/D ≤ 20). Entropy generation in the system is predominantly triggered by heat transfer in comparison to fluid friction. The irreversibility incurred progressively increases with an increase in Ra and ReD. The variation pattern of has been demonstrated to find out the optimized regime where heat transfer is maximum within the laminar range. The contribution of fluid friction irreversibility toward total irreversibility rises abruptly with an increase in ReD for all cases of L/D and Ra. To demonstrate this study's thermodynamic characteristics, the static temperature contours as well as the contours of entropy generation have been represented pictorially. The estimation of cooling time has been reported by using the method of lumped capacitance.