Minimal State Representation for Open Fluid-Fluid Reaction Systems
Reaction systems are typically 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 variability of the concentrations is not linked to the number of species, but to the number of independent reactions, the number of transferring species, and the number of 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 states. This paper considers the material balance equations associated with an open fluid--fluid reaction system that involves S_g species, p_g independent inlets and one outlet in the first fluid phase (e.g. the gas phase), S_l species, R independent reactions, p_l independent inlets and one outlet in the second fluid phase (e.g. the liquid phase). In addition, there are p_m species transferring between the two phases. Based on a nonlinear transformation that decomposes the (S_l+S_g) states of the original model into \sigma (= R+2p_m+p_l+p_g+2) variant states and (S_l+S_g-\sigma) invariant states, and on the concept of accessibility of nonlinear systems, the conditions under which the transformed model is a minimal state representation are derived. Furthermore, it is shown how to reconstruct unmeasured concentrations from measured concentrations and flow rates without knowledge of the reaction and mass--transfer rates. The minimal number of composition measurements needed to reconstruct the full state is (R + p_m). The simulated chlorination of butanoic acid is used to illustrate the various concepts developed in the paper.