Exact lumping

Kinetics of multiscale systems studied in this chapter and developed theory of dynamic limitation demonstrates that in multiscale limit lumping analysis can work (almost) exactly. Lumped concentrations are sums in groups, but these groups can intersect and usually there exist several intersections. 

So far, lumping has been defined, but nothing has been said concerning the dynamic behavior of the" system. Now we come to the definition of exact lumping A system is said to be exactly lumpable by the matrix L if there exists an N X N matrix K, enjoying the same properties as K does (i.e., off-diagonal elements of K are nonpositive, = 0, and there exists at least an m" =... 

The distinction of the original eigenvectors into a set of vanishing ones and a set of eigenvalue preserving ones forms the basis of the Ozawa (1973) analysis of exact lumping. 

For further details on the Wei and Kuo analysis of exact lumping, and for alternate mathematical descriptions of the same problem, the reader is referred to the original literature. We move here to a brief discussion of an extension due to Bailey (1972), who considered the case where the original system is a continuous one, so that Eq. (79) is written as... 

For exact lumping, one now again requires Eq. (85) to hold. Bailey (1972) has shown that the analog of Eq. (88) is, for the case at hand. 

Another type of extension of the Wei and Kuo results is to bilinear kinetic forms, an extension that was originally discussed by Li (1984), and later analyzed in much more detail by Li and Rabitz (1989, 1991a,b,c). Practical and experimental aspects of the general problem of exact lumping, some of which had already been discussed by Wei and Kuo (1969) and by Kuo and Wei (1969), are excellently reviewed by Weekman (1979). 

An important consideration about exact lumping is the following one, which is due to Coxson and Bischoff (1987a). If one considers reactor types other than a batch (or, equivalently, a plug flow) reactor, exact lumping carries over. In other words, the dynamics of the reduced system behave as if they were representative of true intrinsic kinetics. (This, as discussed in Section IV,C, is not true for overall kinetics, which may be regarded as nonexact lumping.) A somewhat similar result was proved by Wei and Kuo (1969) for the case of reactions with diffusion, such as occurs in porous catalysts. 

The constraints required for exact lumping are invariably very strict, and it is only natural that works have been published where the concept of approximate lumping is introduced. In its simplest possible form, approximate lumping is achieved when Eq. (88) [or its semicontinuous analog, Eq. (94)] holds only in some appropriately defined approximate sense, rather than exactly. 

This should be contrasted with the result discussed in the second paragraph after Eq. (101) Overall kinetics are far from being exact lumping, and they do not carry over to different reactor types. 

For exact lumping Mf(c) must be a function of c so that the reduced system can be expressed in terms of the new variables. Therefore, we need to know the inverse of M since... 

The inverse mapping from the c space to the c space is equally important as the forward mapping, not only because it provides a link between the lumped species and the original species, but because its existence is a necessary and sufficient condition for exact lumping. For a reduced system hstandard definition of an inverse will not apply. Therefore, we use the concept of a generalized inverse. The generalized inverse of an m X n matrix A satisfies the following criteria ... 

Show that condition (4.28) for exact lumping is necessary as well. 

Intuitively, one may expect that lumpability suggests that the system should have some degree of partial linearity, which is related to the Jacobian matrix J[c(t)] = df(c)/dc. Indeed, the system is exactly lumpable if and only if for any c in the composition space, the transpose of J[c(t)] has nontrivial fixed (i.e., c independent) invariant subspaces. It can be shown that the exactly lumped system is of the form... 

Let k be the eigenvalues and x, the eigenvectors of the reactivity matrix K. For an exact lumping matrix, M, and the corresponding lumped reactivity matrix, K, a vector Mx, will either vanish or be an eigenvector of K with the same corresponding eigenvalue ... 

The condition for exact lumping of a unimolecular scheme is the existence of M and X so that... 

Ftmction h is not unique, since several different functions h may belong to the same transformation function h. This inverse mapping is as important as the forward mapping not only because it provides the link between the lumped variables and the original species concentrations, but because its existence is a necessary condition of exact lumping. 

It is always possible to find matrices K and M that fulfil Eq. (7.33), but the solution is not unique. The equivalent problem is finding invariant subspaces of the original equations, i.e. invariant subspaces of the transpose of the Jacobian J (Y) so that the eigenvalues of J (Y) and J (M MY) are identical, which is fairly straightforward for this linear example where the Jacobian is a constant matrix. However, this is often a difficult task for more general nonlinear ODEs where applying the restrictions imposed by exact lumping may limit the level of reduction possible for the reduced scheme. 

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