A general analysis of exact lumping in chemical kinetics

Genyuan Li, Herschel Rabitz

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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 JT (y) of f(y), no mater what value y takes, and the corresponding eignevalues are the same for JT(y) and JT (M̄ My). Here the rows of M are the basis vectors of M and M̄ is any generalized inverse of M satisfying MM̄ = I with I being the -identity matrix. The fixed invariant subspaces of JT(y) can be obtained either from the simultaneously invariant subspaces of all Ak, where the Ak's form the basis of the decomposition of JT(y), or by determining the fixed Ker {Πi(JT(y)-λiIn) riΠj[(σ2 j + τ2 j)In - 2σjJT(y) +(JT(y))2]rj}, where λi, σj±iτj are the real and nonreal eigenvalues of JT (y) and λi, σ j and τj are usually functions of y;ri,rj 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 languageEnglish (US)
Pages (from-to)1413-1430
Number of pages18
JournalChemical Engineering Science
Issue number6
StatePublished - 1989

All Science Journal Classification (ASJC) codes

  • General Chemistry
  • General Chemical Engineering
  • Industrial and Manufacturing Engineering


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