This paper presents the thermo-economic limits of ambient heat rejection in vertical fin arrays with buoyancy-driven flow enhancement through the chimney effect. A one-dimensional semi-analytical thermo-fluidic model is developed to assess the cooling power enhancement of the proposed heat sink design. A bi-objective optimization is performed utilizing genetic algorithm to present the tradeoffs between the cost and the thermal performance of a heat sink. For the considered baseplate geometry, the maximal cooling power without a chimney amounts 1540 W at a heat flux of 1.03 W/cm2. By adding a chimney up to 2.5 m high, the cooling power is increased by 46% to 2250 W at a heat flux of 1.50 W/cm2.