| A general lumping analysis of exact and approximate lumping for an arbitrary reaction system under isothermal or nonisothermal conditions, coupled with or without diffusion, validated in the whole composition space or some region of it has been presented. The necessary and sufficient conditions for the existence of exact lumping under different conditions were obtained. These conditions are constructive so that the exact lumping schemes can be determined by them. Under isothermal conditions and without diffusion a reaction system with n components is exactly lumpable by an n {dollar}times{dollar} n(n {dollar}leq{dollar} n) constant matrix M with rank n in the whole composition space if and only if the transpose of the Jacobian matrix of the kinetic equations has common invariant subspaces. The lumping matrices are matrix representations of these invariant subspaces. Two approaches to determine the common invariant subspaces have been obtained. The kinetic equations of the lumped systems have a similar form as those of the unlumped ones. For realistic problems the lumping schemes must satisfy some restrictions and under these conditions the reaction systems most probably cannot be exactly lumped. Systematic approaches to determine approximate lumping schemes have been developed. A set of matrix equations were used to determine the unconstrained and constrained approximate lumping matrices valid in some region of the composition space. An optimization method and a direct method to determine the approximate lumping matrices validated in the whole composition space were constructed. The lumping analysis presented in this thesis can be applied not only in chemical kinetics but also in other areas, such as classical molecular dynamics, chemical engineering and control theory. |
