On the dimensionality and complexity of multiple-feed attainable regions

Much of the existing literature on attainable region (AR) theory has focused on finding it from a single feed. Yet, many industrial processes involve multiple feed streams and thus the total set of achievable states within a reactive process is affected by the number and type of feeds available. In this paper, we describe conditions for which multiple feeds change the dimension of the space belonging to the AR, allowing for an expansion of the region that would otherwise be infeasible from a single feed. Under specific conditions, we describe how multiple feeds may only alter the shape and size of the AR whilst preserving its dimension to that of a single feed AR. We determine the maximum number of the parallel reactors required to achieve these new states. This is obtained from the reaction stoichiometry and feeds alone, without consideration of reaction kinetics.