Reaction Systems Model Reduction

Douglas, Conceptual Design of Chemical Processes , McGiaw-Hill, New York, 1988. 

Schembecker, W. Schilttenhelm, and K. H. Simmrock, Comput. Chem. Eng., 1993, 18, SuppL. S131-S135. 

PCA = principal components analysis SCI = single component identity. 

Reaction system in which each reaction has at most two reactants and at most two products, and the reaction rate orders correspond to the stoichiometries (so that 2A - is second-order in A, while A -I- B - is first-order in each of A and B), 

The number of first-order differential equations describing the behavior of the system. 

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Figure 1.7 Typical zero-order and corresponding second-derivative electronic absorption spectra of ethanol-reconstituted lipid/chloroform extracts of autoxidized model polyunsaturated fatty-acid compounds and inflammatory synovial fluid obtained after (1) reduction with NaBH4 and (2) dehydration with alcoholic H2S04- (a) Methyl linoleate subsequent to autoxidation in air at ambient temperature for a period of 72 h (—), or exposure to a Fenton reaction system containing EDTA (5.75 x 10 mol/dm ), H2O2 (1.14 X 10 mol/dm ) and Fe(ll) (5.75 x IO mol/dm ) as an aqueous suspension (—) (b) as (a) but with methyl linolenate (c) untreated rheumatoid knee-joint synovial fluid.

The performance of a chemical reactor can be described, in general, with a system of conservation equations for mass, energy, and momentum. To solve this system we must have a model for the reaction on the basis of which we can derive the intrinsic rate equation on one side, and a model of the reactor in which we want to run the reaction on the other side. Both tasks are, of course, interconnected and difficult to solve without reduction of more general equations to a suitable limiting reactor type to be used for each particular reaction system [4,26],... 

The obvious advantage is that the steady-state solution of an S-system model is accessible analytically. However, while the drastic reduction of complexity can be formally justified by a (logarithmic) expansion of the rate equation, it forsakes the interpretability of the involved parameters. The utilization of basic biochemical interrelations, such as an interpretation of fluxes in terms of a nullspace matrix is no longer possible. Rather, an incorporation of flux-balance constraints would result in complicated and unintuitive dependencies among the kinetic parameters. Furthermore, it must be emphasized that an S-system model does not necessarily result in a reduced number of reactions. Quite on the contrary, the number of reactions r = 2m usually exceeds the value found in typical metabolic networks. 

In a solid-fluid reaction system, the fluid phase may have a chemistry of its own, reactions that go on quite apart from the heterogeneous reaction. This is particularly true of aqueous fluid phases, which can have acid-base, complexation, oxidation-reduction and less common types of reactions. With rapid reversible reactions in the solution and an irreversible heterogeneous reaction, the whole system may be said to be in "partial equilibrium". Systems of this kind have been treated in detail in the geochemical literature (1) but to our knowledge a partial equilibrium model has not previously been applied to problems of interest in engineering or metallurgy. 

Klupinski et al. (2004) conclude that the reduction of nitroaromatic compounds is a surface-mediated process and suggest that, with lack of an iron mineral, reductive transformation induced only by Fe(II) does not occur. However, when C Cl NO degradation was investigated in reaction media containing Fe(II) with no mineral phase added, a slow reductive transformation of the contaminant was observed. Because the loss of C Cl NO in this case was not described by a first-order kinetic model, as in the case of high concentration of Fe(II), but better by a zero-order kinetic description, Klupinski et al. (2004) suggest that degradation in these systems in fact is a surface-mediated reaction. They note that, in the reaction system, trace amounts of oxidize Fe(II), which form in situ suspended iron oxide... 

Many methods have been developed for model analysis for instance, bifurcation and stability analysis [88, 89], parameter sensitivity analysis [90], metabolic control analysis [16, 17, 91] and biochemical systems analysis [18]. One highly important method for model analysis and especially for large models, such as many silicon cell models, is model reduction. Model reduction has a long history in the analysis of biochemical reaction networks and in the analysis of nonlinear dynamics (slow and fast manifolds) [92-104]. In all cases, the aim of model reduction is to derive a simplified model from a larger ancestral model that satisfies a number of criteria. In the following sections we describe a relatively new form of model reduction for biochemical reaction networks, such as metabolic, signaling, or genetic networks. 

Vora, N.P. (2000). Nonlinear Model Reduction and Control of Multiple Time Scale Chemical Processes Chemical Reaction Systems and Reactive Distillation Columns. PhD thesis, University of Minnesota - Twin Cities. 

One of the models proposed for a possible enzyme redox reaction mechanistic pathway suggests (Fig. 17.8) that the enzyme contains simultaneously a part that acts as a solution cathode containing a so-called cathodic system where reduction occurs, and another that acts as a solution anode where there is oxidation35. The total charge transfer for the whole chemical reaction is therefore zero. This model is not completely correct, but the concept of a total chemical reaction without electron transfer to the exterior of the enzyme, although controlled by electron transfer, is interesting. 

Further reduction of the constrained reaction path model is possible. Here we adopt a system-bath model in which the reaction path coordinate defines the system and all other coordinates constitute the bath. The use of this representation permits the elimination of the bath coordinates, which then increases the efficiency of calculation of the motion along the reaction coordinate. In particular. Miller showed that a canonical transformation of the reaction path Hamiltonian T + V) yields [38]... 

This model has been proved experimentally by studying the competition of the anodic decomposition reaction and the oxidation of Cu at p-GaAs in the dark and at n-GaAs under illumination [93]. This is a suitable redox system, because reduction and oxidation occur via the valence band, and because the anodic oxidation of Cu proceeds independently from the corrosion. Accordingly, the total current is given by... 

This example has shown how the procedures developed in earlier chapters can be used effectively for modeling. The reaction system has seventeen participants olefin, paraffin, aldehyde, alcohol, H2, CO, HCo(CO)3Ph, HCo(CO)2Ph, and nine intermediates. "Brute force" modeling would require one rate equation for each, four of which could be replaced by stoichiometric constraints (in addition to the constraints 11.2 to 11.4, the brute-force model can use that of conservation of cobalt). Such a model would have 22 rate coefficients (arrowheads in network 11.1, not counting those to and from co-reactants and co-products), whose values and activation energies would have to be determined. This has been reduced to two rate equations and nine simple algebraic relationships (stoichiometric constraints, yield ration equations, and equations for the A coefficients) with eight coefficients. Most impressive here is the reduction from thirteen to two rate equations because these may be differential equations. 

In a review article on oscillatory reactions (294), Sheintuch discusses the effect of introducing a heat balance for the catalyst rather than a mass balance for the reactor into the differential equation system for a surface reaction with oxidation/reduction cycles. Although the coverage equations alone can yield oscillatory behavior, as was the case for the models discussed in the previous section, Sheintuch s model is discussed in this section because introduction of the heat balance adds qualitatively new features. In this extended system complex, multiple peak behavior and quasiperiodicity was observed as shown in Fig. 8. Sheintuch also investigated the interaction of two oscillators. This work, however, will be treated in detail in Section V, were synchronization and chaos are discussed. 

Usually oscillations of this type are described by models of the general form depicted in Fig. 6h. At a high temperature and a high reaction rate, the catalyst begins to oxidize. This causes the temperature and rate to fall, and the system eventually reaches the low-temperature oxidized state. The catalyst then goes through a reduction process that raises the temperature and the reaction rate. The process then repeats to produce oscillations. The reaction systems for which these models have been developed are CO oxidation... 

Before the model discussed above was published, tha-e were three other suggestions of how to model spatiotemporal dynamics in electrochemical systems. The first attempt at a theoretical description of electrochemical pattern formation came from Jome. His model is based on a chemical instability in the reaction mechanism and only takes into account the concentrations of the reacting species as dependent variables, not the potential. This, of course, means that the model is not applicable to any of the systems exhibiting an electrical instability. This includes the examples treated by Jome, namely, anion reduction reactions or cation reduction in the presence of SCN . Meanwhile, both oscillators are unanimously classified as NDR oscillators [see Section n.2.(ii)] and hence their spatiotemporal description requires a different approach. 

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