The optimal configuration of the expansion process of a heated working fluid inside a cylinder for maximum work output with a movable piston and generalized radiative heat transfer law q̇∝Δ(Tn) is determined. The optimal process that maximizes the work output of the working fluid with fixed initial energy, initial volume, and final volume is obtained using optimal control theory. The optimal process consists of at most three stages, including an initial adiabatic branch, one intermediate Euler-Lagrange arc, and a final adiabatic branch. The analytical solutions of the intermediate Euler-Lagrange arc are obtained for different heat transfer laws, including n=−1, n=1, n=2, n=3, and n=4. Numerical examples for the optimal configurations are provided. Finally, the obtained results are compared with each other.

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Optimal configuration of a class of endoreversible heat engines with fixed duration, input energy, and linear phenomenological heat transfer law [q∝Δ(T−1)] has been determined. The optimal cycles that maximize the efficiency and the power output of the engine have been obtained using optimal-control theory, and the differential equations are solved by using Taylor series expansion. It is shown that the optimal cycle for maximum efficiency has eight branches including two isothermal branches, four maximum-efficiency branches, and two adiabatic branches, and that the optimal cycle for maximum-power output has six branches including two isothermal branches, four maximum-power branches, and without an adiabatic branch. The interval of each branch has been obtained, as well as the solutions of the temperatures of heat reservoirs and working fluid. Numerical examples are given. The obtained results are compared with those obtained with Newton’s heat transfer law [q∝Δ(T)] for maximum-efficiency and maximum-power output objectives.