### Abstract

A general analysis of exact lumping is presented. This analysis can be applied to 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; f (y) is an arbitrary n-dimensional function vector. Here we consider lumping by means of an n̂ × n real constant matrix M with rank n̂ (n̂ < n). It is found that a reaction systems is exactly lumpable if and only if there exist nontrivial fixed invariant subspaces M of the transpose of the Jacobian matrix J^{T} (y) of f(y), no mater what value y takes, and the corresponding eignevalues are the same for J^{T}(y) and J^{T} (M̄ My). Here the rows of M are the basis vectors of M and M̄ is any generalized inverse of M satisfying MM̄ = I_{n̂} with I_{n̂} being the _{n̂}-identity matrix. The fixed invariant subspaces of J^{T}(y) can be obtained either from the simultaneously invariant subspaces of all A_{k}, where the A_{k}'s form the basis of the decomposition of J^{T}(y), or by determining the fixed Ker {Π_{i}(J^{T}(y)-λ_{i}I_{n}) ^{ri}Π_{j}[(σ^{2}_{ j} + τ^{2}_{ j})I_{n} - 2σ_{j}J^{T}(y) +(J^{T}(y))^{2}]^{rj}}, where λ_{i}, σ_{j}±_{i}τ_{j} are the real and nonreal eigenvalues of J^{T} (y) and λ_{i}, σ _{j} and τ_{j} are usually functions of y;r_{i},r_{j} are nonnegative integers. The kinetic equations of the lumped system can be described as dŷ/dt=Mf(M̄ŷ). This method is illustrated by some simple examples.

Original language | English (US) |
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Pages (from-to) | 1413-1430 |

Number of pages | 18 |

Journal | Chemical Engineering Science |

Volume | 44 |

Issue number | 6 |

DOIs | |

State | Published - 1989 |

### All Science Journal Classification (ASJC) codes

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

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## Cite this

*Chemical Engineering Science*,

*44*(6), 1413-1430. https://doi.org/10.1016/0009-2509(89)85014-6