Finite time thermodynamics is used to solve a new model of an extended Brayton cycle with variable-temperature heat reservoirs and finite size heat exchangers. The model takes into account external and internal entropy generation and handles heat recovery and heat leaks to the environment in a novel way. The extended system considerations are very important for minimizing entropy generation and maximizing second law efficiency, profit and ecological criterion. An optimization analysis was developed on this new model to determine its maximum power and minimum entropy generation, and amid the most important findings were the global maximum net power, global minimum entropy generation, optimum global heat exchangers size distribution, best working fluid specific heat ratio and optimal fluid heat capacities, some of these never having been published previously.