On the error of the quasi-steady-state approximation

Abstract

Application of the quasi-steady-state approximation (QSSA) in chemical kinetics allows the concentration of some species (QSSA species) to be calculated not only via the solution of kinetic differential equations but also from the concentration of other species using algebraic equations. The difference in the concentrations of QSSA species obtained from the two calculations, at a single time point, is called the instantaneous QSSA error. This error represents a continuous perturbation of the calculated trajectory and causes an overall error in the concentrations of non-QSSA species as well. Two equations are given for the calculation of the instantaneous error. Initial selection of QSSA species can be based on the first equation, which predicts the instantaneous error of a single species. The second more involved error equation takes into account the interaction of errors of selected species and gives the instantaneous error for a group of QSSA species. Successful application of the QSSA requires that the overall error of important species be small. In some cases a small instantaneous error in the QSSA species can be magnified and results in large overall error. Such “pathological” cases can be detected by the calculation of the initial concentration sensitivity matrix. Those species, which induce large overall error, have to be excluded from the group of the QSSA species. The relation of the QSSA to the lifetime of species and to the stiffness of ODES is also discussed. The use of the error formulas is illustrated by the application of the QSSA for a propane pyrolysis mechanism and briefly for the combustion of H1.