Linear species lumping

If the function h is linear, then in chemical kinetics, this approach would be termed linear species lumping and is essentially a formalisation of the chemical lumping approach described in the previous section. In the linear case the transformation is simply a matrix multiplication operation ... 

A linear algebraic system of rate equations for the fast species results, which can be solved a priori. Hence a strongly reduced (in the number of species to be treated) system is obtained. This concept originates from astrophysical applications and from Laser physics. It is in some instances also referred to as collisional-radiative approximation , for the fast species, lumped species concept , bundle-n method or intrinsic low dimensional manifold (ILDM) method in the literature. We refer to [9,12,13] for further references on this. 

This example illustrates that chemical lumping is nothing other than a special case of linear lumping, and that this system is a good example of the application of exact linear lumping. In fact, we can express the relationship between the original and lumped species in the same formal way as we described in the previous section ... 

Due to the regular branched structure of this isomer, linear 1-alkenes heavier than 1-heptene are not present and the relative amount of propyl and butyl radicals is significantly different too. In other words, the lumped H-abstraction reaction of a single model component loses the variety of primary products obtained from the previous lumped A0C15. It seems relevant to observe that to improve ethylene selectivity prediction, alkene components heavier than hexenes can be conveniently described with two different species, respectively corresponding to the true component 1-C H2 and to a lumped mixture of the remaining normal and branched isomers. 

During polymer decomposition, whenever a species is formed with a molecular weight not equal to that of a pseudocomponent, its amount is linearly distributed between the adjacent bins (Bi,Bi+. This lumping procedure or discrete section method was successfully applied to the thermal degradation of PS (Faravelli et al., 2001). 

The presence of adsorbate interactions make this approach fail totally, as the isotherms are non-linear. The Newton-Raphson algorithm is a practical method of calculation for surface coverages of a 13 component FG isotherm. We make the further assumption that for species within a given lump, the adsorbate interaction energies are identical. In addition, we need to specify only the interaction energy between dissimilar pairs. Thus we have recourse to a Monte- Carlo Procedure(15)... 

This computer program carries out the reduction of detailed mechanisms using the Quasi- Stationary-State Approximation (QSSA). When the negligible species have been eliminated and the quasi-stationary species identified, the algorithm looks for a set of independent reactions allowing the quasi-stationary concentrations to be calculated by applying the QSSA. Lumped reactions are obtained by a linear combination of elementary reactions of the detailed mechanism. 

There are other lumping techniques for linear kinetics, such as cluster analysis observer theory , singular perturbation , and intrinsic lowdimensional manifolds. Recently, Androulakis treated kinetic mechanism reduction as an integer optimization problem with binary variables denoting the existence and nonexistence of reactions and species. The technique is amenable to uncertainty analysis. 

The mathematical systematization and explanation of lumping techniques are based on defining lumps either as an integral within a continuous set of reactants, defined by a continuous index variable, " or as linear combinations of discrete chemical species. ... 

We saw in the previous section that chemical lumping is often based on defining new species whose concentrations are linear combinations of those of the starting species within a mechanism. This approach can be generalised within a mathematical framework. The formal definition of lumping is the transformation of the original vector of variables Y to a new transformed variable vector Y using the transformation function h ... 

We include here a short example of linear lumping taken from Li and Rabitz (1989) in order to illustrate the approach. Consider the first-order reaction system involving reversible reactions between three species as follows ... 

Bellan et al. present the existence of self-similarity as an empirical observation resulting from the inspection of simulation results, and they do not provide a mathematical foundation to the method. Similar concentration curves may be a result of the existence of very different timescales, and the application of QSSA or partial equilibrium may result in linear relations between the concentrations. However, in these articles self-similarity was fotmd among long lifetime ( heavy ) species, and therefore, the existence of self-similarity seems to be a consequence of possible lumping relationships within the system variables. Although the self-similarity concept seems to be related to the lumping of species, it is not equivalent to it, since the derived hnear functions ccmtain the concentrations of all heavy species and not only a selection of them. 

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