### Abstract

A general analysis of exact nonlinear lumping is presented. This analysis can be applied to the kinetics of any reaction system with n species described by a set of first-order ordinary differential equations dy/dt = f(y), where y is an n-dimensional vector and f(y) is an arbitrary n-dimensional function vector. We consider lumping by means of n̂ (n̂ ≤ n)-dimensional arbitrary transformation ŷ = h(y). The lumped differential equation system is d y ̂ Dt = _{y}(h(ŷ))f(h(ŷ)), where h_{y}(y) is teh Jacobian matrix of h(y), h is a generalized inverse transformation of h satisfying the relation h(h) = I_{n̂}. Three necessary and sufficient conditions of the existence of exact nonlinear lumping schemes have been determined. The geometric and algebraic interpretations of these conditions are discussed. It is found that a system is exactly lumpable by h only if h(y) = 0 is its invariant manifold. A linear partial differential operator A = Σ^{n}_{i=1} f_{i}(y)θ{symbol}/θ{symbol}y_{i} corresponding to dy dt = f(y) is defined. Using the eigenfunctions and the generalized eigenfunctions of A, the operator can be transformed to Jordan or diagonal canonical forms which give the lumped differential equation systems without determination of h. These approaches are illustrated by a simple example. The results of this analysis serve as a theoretical basis for the development of approaches for approximate nonlinear lumping.

Original language | English (US) |
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Pages (from-to) | 343-361 |

Number of pages | 19 |

Journal | Chemical Engineering Science |

Volume | 49 |

Issue number | 3 |

DOIs | |

State | Published - 1994 |

### All Science Journal Classification (ASJC) codes

- Chemistry(all)
- Chemical Engineering(all)
- Industrial and Manufacturing Engineering

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## Cite this

*Chemical Engineering Science*,

*49*(3), 343-361. https://doi.org/10.1016/0009-2509(94)87006-3