A general analysis of approximate nonlinear lumping in chemical kinetics. II. Constrained lumping

A general analysis of approximate constrained nonlinear lumping is presented for a chemical kinetic system described by an n-dimensional set of first order ordinary differential equations dy/dt=f(y). There is a one-to-one relation between the differential equation system and the linear partial differential operator A = Σ_i=1ⁿf_i(y) (∂/∂y_i). The algebraic method in nonlinear perturbation theory for lumping is extended to include constrained nonlinear lumping, in which the operator A is only transformed to a partially canonical form and some variables are left unlumped. A singular perturbation method is used to provide approximate analytical expressions for the solutions of the lumped variables. The resulting expressions can then be substituted into the equations describing the unlumped species, leading to a low dimensional system. The method is illustrated by application to a simple model describing the nonisothermal oxidation of hydrogen in a closed vessel. The results show that the method of constrained lumping leads to an accurate representation of the ignition features and maximum temperature rise given by the full model. The singular perturbation technique is proved to be only a special case of a general constrained lumping approach based on the algebraic method in nonlinear perturbation theory when the equations are linear in the deleted variables. Consequently the quasisteady-state approximation (QSSA) is the zeroth order approximation within the slow invariant manifold of the constrained approach. In cases where QSSA is not a good approximation, the first order correction generally provides significant improvement of the results.