Lumping of species

Lumping of species for hydrocarbon mixtures. [3rd Ed. P5-16] Prepare an experimental plan to find the rate law. (3rd Ed. P5-J7] Batch data on the liquid phase reaction... 

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. 

Solution of Eq. (2) is only part of the overall problem in mechanistic studies. Indeed, Eq. (2) represents only the first of many sequential inverse problems vthich we commonly lump together and which represent the overall inverse problem in chemical kinetics [110-112]. The next problem is the determination of the moles of all species, the elemental stoichiometries of species, the number of observable reactions, and the balanced stoichiometries for all observable reactions. 

The quantitative estimation of species by SEC-GC-MS technique requires a mathematical solution. Two types of approaches for the quantitative estimation can be envisioned. One for the estimation of one or more selected species of interest. The second approach is based on grouping of various species in coal liquids into a few chemical lumps and estimating the quantity of these lumps by using the data derived from the analysis is technique. 

The model includes fundamental hydrocarbon conversion kinetics developed on fresh catalysts (referred to as start-of-cycle kinetics) and also the fundamental relationships that modify the fresh-catalyst kinetics to account for the complex effects of catalyst aging (deactivation kinetics). The successful development of this model was accomplished by reducing the problem complexity. The key was to properly define lumped chemical species and a minimum number of chemical reaction pathways between these lumps. A thorough understanding of the chemistry, thermodynamics, and catalyst... 

An important part of the model development was first defining a set of lumped chemical species from the 300 identified species and then defining... 

Two different reduced kinetic models have been considered, both involving four lumped chemical species and three and four lumped reactions, respectively. These models describe the time evolution of the following components ... 

Semifield tests offer the ability to observe or measure a suite of biological and physicochemical variables, as with the analysis of natural ecosystems. Usually, the focus is on structural measures such as the number of species and number of individuals per population (often lumped into higher taxonomic units), physicochemical parameters (e.g., dissolved oxygen, and pH including toxicant residues), and some microclimate parameters (e.g., evaporation in TMEs). 

It is worth noting that some cyclic molecules play an important role in the mechanism. Another interesting feature results from the use of lumped compounds. The lumping of different species into a single pseudocompound causes a reduction in the complexity of the reacting system. For example, the 4 isomers of the branched C6 paraffins are lumped into a single pseudo-compound the 12 alkylbenzenes from C9 to C2o are described by three pseudo-compounds, C9, C1S and C20, etc. 

The description of reaction kinetics for circumstances where an exceptionally large number of species are involved or where well-defined molecular entities cannot be specified poses a particularly difficult challenge. Such situations can arise, for example, in the cracking of petroleum, the pyrolysis of biomass, and the liquefaction of coal. The best way to treat such systems is to either lump classes of reactions or follow the dynamics of functional groups rather than specific molecular species. 

Before assessing how a chemical moves in the environment, the relevant media, or compartments, must be defined. The environment can be considered to be composed of four broad compartments—air, water, soil, and biota (including plants and animals)—as shown in Fig. 6.6. Various approaches to modeling the environment have been described.14-16 The primary difference in these approaches is the level of spatial and component detail included in each of the compartments. For example, the most simplistic model considers air as a lumped compartment. A more advanced model considers air as composed of air and aerosols, composed of species such as sodium chloride, nitric and sulfuric acids, soil, and particles released anthropogenically.17 A yet more complex model considers air as composed of air in stratified layers, with different temperatures and accessibility to the earth s surface, and aerosols segmented into different size classes.16 As the model complexity increases, its resolution and the data demands also increase. Andren et al.16 report that the simplest of models with lumped air, water, and soil compartments is suitable for... 

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. 

With lumping of X, X2, and X3 into a single pseudo-species XL (shown in box), the reduced cycle now has only two members cat and X. The Briggs-Haldane equation 8.21 applies, without restriction to initial rate and with index X replaced by L ... 

Equation 8.65 is an alternative to lumping of the saturated and ligand-deficient species, as was done in the hydrocyanation Example 8.7 in the preceding section, and gives identical results with slightly less algebraic handling. 

Wei and Kuo (1969) have shown a direct construction of the matrix from the knowledge of the eigenvectors and eigenvalues of K. A simpler construction (for proper lumping) was proposed by Coxson and Bischoff (1987b) Let S be the diagonal A" X A matrix whose elements are the inverses of the number of species in each lump, that is, in each pseudospecies. Then K = [[,L ],S]. 

The remaining organic reactions in the simplified mechanism, 12 and 13, describe the oxidation of NO and NO2 by peroxy radicals and the formation of PAN. We emphasize that if CO and H2O are present, HOo and R02 are treated as separate species. In the absence of CO and H2O, however, only the single species RO2, which includes HO2 within it, is considered since Reactions 4-8 and 14 are omitted. In this cases 6 (Reaction 12) represents the fractional product of OH in the total lumped radical species RO2 . 

Each row of a can be regarded as a lumped reaction that yields a unit amount of the corresponding pivotal species. Thus ai, is the production of species i per mole produced of the first pivotal species, and a2i is the production of species i per mole produced of the second pivotal species. Adding these contributions, we get... 

Another important application of lumping is the lumping of continuous mixtures where an indefinitely large number of species are involved in a reaction [115-125]. Such techniques will be useful especially in processes which involve complex hydrocarbon mixtures such as petroleum feedstocks etc. The discrete system is then represented by a continuum and the lumping by an integration rather than a summation over the original variables. 

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