Approximate Nonlinear Lumping in Systems with Timescale Separation

5 Approximate Nonlinear Lumping in Systems with Timescale Separation 

Using the algebraic method in nonlinear perturbation theory, it is possible to find a transformation operator S such that the resultant operator 

The reactivity and physical properties of continuous species are defined as a function of a dimensionless variable x e [0, oo). This variable is usually related to a measurable physical quantity, such as molecular weight or boiling point. The fraction of a continuous species belonging to an interval of variable x can be calculated by integrating the time-dependent probability density function p(x, t) over this interval. According to its definition, the integral of this pdf is unit over the whole domain of definition of x at any time. 

The rate equations for lumped mixtures will now be discussed. If we consider an isothermal reaction system containing m different reactant types which react irreversibly with an n-th order rate, then the resulting rate equations become 

If the reaction order is assumed to be constant for all species, then the only species-dependent parameter is A ,-, and hence, a species can be defined by its concentration 

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