Attainability of maximum work and the reversible efficiency from minimally nonlinear irreversible heat engines.

We use the general formulation of irreversible thermodynamics and study the minimally nonlinear irreversible model for heat engines operating between a time varying hot heat source of finite size and a cold heat reservoir of infinite size. We explicitly calculate the condition for obtaining optimized work output for this model once the system reaches the final thermal equilibrium state with that of the cold heat reservoir. We find that our condition resembles with the generalized condition to achieve an optimized work output for generalized irreversible heat engines in the nonlinear regime [Y. Wang, Phys. Rev. E {\bf 93}, 012120 (2016)]. We also find that the optimized efficiency obtained by this minimally nonlinear irreversible heat engine can reach the reversible efficiency under the tight coupling condition in which there is no heat leakage between the system and the reservoirs. Under this condition, we find that the reversible efficiency is obtain for any finite time interval with arbitrary power. We also calculate the efficiency at maximum power from the minimally nonlinear irreversible heat engine under the non-tight coupling condition. We find that the efficiency at maximum power is equal to the half of the reversible efficiency and the corresponding maximum work is half of the exergy for a specific choice of the heat leakage term. Our result matches exactly with the efficiency and the work at maximum power obtained in Ref. [Y. Izumida and K. Okuda, Phys. Rev. Lett. {\bf 112}, 180603 (2014)] for the exergy study of linear irreversible heat engines under the tight coupling condition.