In this work the attainable region (AR) concept for process networks with outlet flow rate specifications is introduced for the first time. For process unit models to which the infinite dimensional State-space conceptual framework is applicable, it is shown that identification of AR boundary membership is equivalent to feasibility assessment of an infinite linear program (ILP). A number of important AR properties are then theoretically established, including AR convexity, and representation of the AR in a concentration state space of reduced dimension. Finite dimensional approximations of the aforementioned ILP are then employed in creating increasingly accurate approximations of the AR. A case study for the vapor-liquid equilibrium-based separation of a ternary azeotropic mixture is used to illustrate the proposed method. The quantified two- and four-Dimensional ARs indicate that acetone mole fractions above 0.79 (acetone/methanol azeotrope) are attainable for the considered outlet flow rate ratios. (c) 2013 American Institute of Chemical Engineers AIChE J, 60: 193-212, 2014