Lumped constituent models

Lumping constituents is clearly necessary when the reactant mixture is of such a complexity that it can only be characterized, for practical (e.g. analytical) reasons, by global properties that is notably the case for the pyrolysis of gas oils and crude oils. Examples of lumped constituent models can be found in refs. 35 and 36. In what follows, a few typical models will be briefly described, to bring to light the underlying principles of model building. 

Ross and Shu [38] report a model for naphtha pyrolysis. Naphtha feed is treated as a single lumped constituent A, which decomposes by a pseudo-elementary process of the first order into products By, viz. 

The first-order rate coefficient, k, of this pseudo-elementary process is assumed to vary with temperature according to an Arrhenius law. Model parameters are the stoichiometric coefficients v/ and the Arrhenius parameters of the rate coefficient, k. The estimation of the decomposition rate coefficient, k, requires a knowledge of the feed conversion, which is not directly measurable due to the complexity of analyzing both reactants and reaction products. Thus, a supplementary empirical relationship is needed to relate the feed conversion (conversion of A) to some experimentally accessible variable (Ross and Shu have chosen the yield of C3 and lighter hydrocarbons). It is observed that the rate coefficient, k, is not constant and decreases with increasing conversion. Furthermore, the zero-conversion rate coefficient depends on feed specifications (such as average carbon number, hydrogen content, isoparaffin/normal-paraffin ratio). Stoichiometric coefficients are also correlated with conversion. Of course, it is necessary to write supplementary empirical relationships to account for these effects. 

Leonard et al. [43] have used a similar pseudo-kinetic model to predict yields from cracking coils under any operating conditions. Model parameters are obtained by means of bench scale experiments on the given feedstock. Thus, it is very easy both to evaluate a priori a potential feedstock and to determine simultaneously the corresponding optimum operating conditions. The first-order rate coefficient is calculated for the model compound n — Ci6H34.  

Illes et al. [44] developed a reaction model for naphtha cracking which involves an nth-order decomposition of naphtha, considered as a single constituent. 

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Pseudo-kinetic Lumped constituents models Molecular reactions schemes... 

These authors [32, 33] have considered an alternative classification based on the nature of the variables involved in the model. They classify models by grouping them into opposite pairs deterministic vs. probabilistic, linear vs. non-linear, steady vs. non-steady state, lumped vs. distributed parameters models. In a lumped parameters model, variations of some variable (usually a spatial one) are ignored and its value is assumed to be uniform throughout the entire system. On the other hand, distributed parameters models take into account detailed variations of variables throughout the system. In the kinetic description of a chemical system, lumping concerns chemical constituents and has been widely used (see Sects. 2.4 and 2.5). 

Fernandez-Baujin and Solomon [45] have used a model involving two lumped constituents A and B. Pseudo-constituent A decomposes by a first-order process, whereas B disappears by a second-order process. 

The case to be considered here is that of a continuum mixture of chemical species in which chemical reactions occur. (A lumped-parameter model is given in (2).) The model for each different chemical compound is called a constituent. Were the constituents inert, then the amount of each would be an external constraint. However, since compounds can be produced or destroyed by chemical reactions in the mixtures being considered, the amount of each constituent is a non-conserved property. Then, if a mixture contains N constituents, and if R independent reactions are allowed to take place, N - R of the constituents can be selected as components and the amount of each component is a conserved property, i.e. its value can change only by being transported — it cannot be produced. Thus, every component is a constituent, but not every constituent is a component. However, the component amount, and the constituent amount of the same species are different. The amount of a constituent in a mixture at any instant represents the... 

This approach is applied naturally by the chemist on the basis of his knowhow and/or intuition. Several models in the literature describe lumping operations carried out according to distillation cuts (Verma et al, 1996), structural considerations (Quann and Jaffe, 1992) or functional considerations (paraffins, olefins, nitrogenated or sulphurated molecules). Lumping, often based on the available analyses, considerably reduces the number of constituents and can be used to produce a kinetic model comprising few parameters. 

In addition, since it is always tempting to refine the model by subdividing the lumped families, the number of kinetic constants between all these lumpings increases at an almost exponential rate with the number of pseudo-constituents considered, since they are apparent constants. 

We can use the solution x(t) = X) - k/(p.(()) (Pr - P/) t/L (assuming = 0) to illustrate the basic ideas behind time lapse invasion analysis, that is, we will take as our host model the equation k/(p.(t)) (Pr - P/)/L = - x(t)/t. Thus, if the position front x(t) can be monitored or measured as a function of the time t, say using resistivity, radioactive tracer, or CAT scan methods, it follows that the quotient x(t)/t yields information about the quantity k/( a.< )) (Pr - P )/L. Of course, the greater the value of x(t) or t, the smaller the experimental error. This invasion front measurement will provide, at most, the value of the lumped physical quantity k/(p.( )) (Pr - P/)/L. Thus, if any of its single constituent members k, p, ( ), P, P/, or L are required, values for the remaining quantities must first be found separately using other means. For example, if the pressure gradient (P,- - P/)/L and the porosity < ) is known, then the value of the mobility k/p is immediately available (but viscosity cannot be determined). 

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