DAEs in model reduction of chemical processes: An overview

Differential algebraic equation (DAE) systems of semi-explicit type arise naturally in the modeling of chemical engineering processes. The differential equations typically arise from dynamic conservation equations, while the algebraic constraints from constitutive equations, rate expressions, equilibrium relations, stoichiometric constraints, etc. Of particular interest are DAE systems of high index, i.e., those for which the algebraic constraints are singular and cannot be eliminated through appropriate substitutions. In this paper we provide an overview of generic classes of fast-rate chemical process models, which in the limit of infinitely fast rates, generate equilibrium-based models that are high-index DAE systems. These slow approximations of multi-time-scale systems can be obtained rigorously via singular perturbations. Two classes of nonstandard singularly perturbed systems leading to high-index DAEs are identified and analyzed. The first class arises in processes with fast rates of reaction or transport.We focus in particular on chemical reaction systems which often exhibit dynamics in multiple time-scales due to reaction rate constants that vary over widely different orders of magnitude. For such systems, we describe the sequential application of singular perturbations arguments for deriving nonlinear DAE models of the dynamics in the different time-scales. The second class arises in the modeling of tightly integrated process networks, i.e., those with large rates of recovery and recycle of material or energy. For such systems we describe a similar model reduction method for deriving DAE models of the slow network dynamics and discuss control-relevant considerations.