Minimal state representation for open fluid-fluid reaction systems

Reaction systems can be represented by first-principles models that describe the evolution of the states (typically concentrations, volume and temperature) by means of conservation equations of differential nature and constitutive equations of algebraic nature. The resulting models often contain redundant states since the various concentrations are not all linearly independent; indeed, the variability observed in the concentrations is caused by the reactions, the mass transferred between phases, the inlet and outlet streams. A minimal state representation is a dynamic model that exhibits the same behavior as the original model but has no redundant state. This paper considers the material balance equations associated with an open fluid-fluid reaction system that involves S_l species, R independent reactions, p_l independent inlets and one outlet in the first fluid phase (e.g. the liquid phase) and S_g species, p_g independent inlets and one outlet in the second fluid phase (e.g. the gas phase). In addition, there are p_m species transferring between the two phases. The (S_l+S_g)-dimensional model is transformed to q = R + 2p_m + p_l + p_g +2 variant states and S_l +S_g -q invariant states. Then, using the concept of accessibility of nonlinear systems, the conditions under which the transformed model is a minimal state representation are derived. It will be shown that the minimal number of concentration measurements needed to reconstruct the full state without kinetic information is R + p_m. The simulated chlorination of butanoic acid is used to illustrate the various concepts developed in the paper.