Finding redundant reactions

Once the necessary species have been found, the second step in reducing a mechanism is the elimination of its non-important reactions. A classical and reliable method is the comparison of the contribution of reaction steps to the production rate of necessary species. A description of this method - without the pre-selection of redundant species - is given by Wamatz [4, 93]. A more recent application of this technique for methane flames is presented in [94]. According to this method a reaction is redundant if its contribution to the production rate of each necessary species is small. This rule sounds simple and obvious but there are, however, several drawbacks and pitfalls. First, the reaction contributions have to be considered at several reaction times (or at several heights in the case of steady flames). Second, all reaction contributions to each necessary species have to be considered, and it is not easy to analyze such huge matrices. The threshold of unimportance will vary from time to time, and from species to species. Applying a uniform threshold for each time and species (e.g., minimum 5% contribution) can either result in redundant reactions being left in the scheme, or an over-simplified mechanism. 

An alternative technique for the reduction of mechanisms, using reaction rates, is based on the sensitivity of production rates to changes in rate parameters [95]. If the parameters are the rate constants and the reactions are considered irreversible, then the normed rate sensitivities have the following form  

This equation shows that an element of the normed rate sensitivity matrix is the ratio of the rate of formation or consumption of species i in reaction j, to the production rate of species i. It is possible to consider the effect of each parameter on several production rates simultaneously using a least-squares objective function. This approach leads to the application of the following overall sensitivity type measure  

the summation is over the indices of all necessary species. This measure gives a rank order of reaction importances in the system at each of the reaction times considered. 

The method of the principal component analysis of the rate sensitivity matrix with a previous preselection of necessary species is a relatively simple and effective way for finding a subset of a large reaction mechanism that produces very similar simulation results for the important concentration profiles and reaction features. This method has an advantage over concentration sensitivity methods, in that the log-normalized rate sensitivity matrix depends algebraically on reaction rates and can be easily computed. For large mechanisms this could provide considerable time savings for the reduction process. This method has been applied for mechanism reduction to several reaction schemes [96-102]. 

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Generally, local concentration sensitivities are used for finding the parameters that have to be known with high precision, and for the identification of rate-limiting steps. An important achievement in the field of sensitivity analysis has been the introduction of principal component analysis as a method for the interpretation of the large amount of information contained in the sensitivity matrix. In the future, a more wide-spread application of principal component analysis is expected for the interpretation of concentration sensitivity results. This would help the extraction of further mechanistic details, or could be used for the detection of redundant reactions. The calculation of initial concentration sensitivities is very useful in some cases, but most simulation packages cannot calculate these sensitivities and there is no sign that such a feature will be incorporated. 

Practically in every general chemistry textbook, one can find a table presenting the Standard (Reduction) Potentials in aqueous solution at 25 °C, sometimes in two parts, indicating the reaction condition acidic solution and basic solution. In most cases, there is another table titled Standard Chemical Thermodynamic Properties (or Selected Thermodynamic Values). The former table is referred to in a chapter devoted to Electrochemistry (or Oxidation - Reduction Reactions), while a reference to the latter one can be found in a chapter dealing with Chemical Thermodynamics (or Chemical Equilibria). It is seldom indicated that the two types of tables contain redundant information since the standard potential values of a cell reaction ( n) can be calculated from the standard molar free (Gibbs) energy change (AG" for the same reaction with a simple relationship... 

The primary stage in finding an appropriate submechanism is the determination of redundant species. Species of chemical mechanisms can be classified into three categories. The reproduction of the concentration profiles of important species is the aim of the modelling process. Important species might, for example, include reaction products or initial reactants. Other species, termed necessary species, have to be present in the model to enable the accurate reproduction of the concentration profiles of important species, temperature profiles or other important reaction features. The remaining species are redundant species. If redundant species are on the lefthand side of a reaction, this reaction can then be eliminated from the mechanism without any effect on the output of the model. If such a species is on the righthand side, then the reaction may or may not be deleted. Even if the reaction has to be retained, the redundant species can be deleted from the list of products of the reaction. Of course the latter can only be done if preservation of atoms or mass is not a formal requirement for the mechanism. 

Finding a subset of a reaction mechanism with identical applicability to the full mechanism, should be the final step of every mechanism generation, and the first step of any mechanism reduction work. However, most published mechanisms contain plenty of species and reactions which are redundant over the range of experimental conditions they are intended to cover. A systematic search for redundant species is almost never carried out, and redundant species are usually identified either accidentally or on the basis of detailed chemical knowledge of the mechanism studied. Two techniques are described here which allow the identification of redundant species in a systematic way. 

Since in a redox reaction electrons are transferred, and since electrons have charge, there is an electric potential E associated with any redox reaction. The potentials for the oxidation component and reduction component of a reaction can be approximated separately based upon a standard hydrogen electrode (SHE) discussed later in this lecture. Each component is called a hall reaction. Of course, no half reaction will occur by itself any reduction half reaction must be accompanied by an oxidation half reaction. There is only one possible potential for any given half reaction. Since tire reverse of a reduction half reaction is an oxidation half reaction, it would be redundant to list potentials for both the oxidation and reduction half reactions. Therefore, half reaction potentials are usually listed as reduction potentials To find the oxidation potential for the reverse half reaction, the sign of the reduction potential is reversed. Below is a list of some common reduction potentials. 

Although this section may be a bit redundant after following the details enumerated in the illustration above, it is still probably worth the effort to find out what is obtained for the collision theory model that we have been using. It will be recalled that two hard spheres, A and B, which possess only kinetic energy react upon collision when their relative energy exceeds 77. In terms of TST we may write the reaction as... 

Sensitivity analysis method. To reduce the redundant steps in the kinetic model of chemical reactions, preference is mostly given to the method of sensitivity analysis, aheady described in Chapter 2. It was successfully applied in different areas of chemisty when studying the kinetic models of combustion reactions [21-28], cracking [28,29] atmospheric processes [30-33], self-oscillation reaction of Belousov-Zhabotinsky [52,53], and biological systems [54], In addition to the given referenees one can find solutions of similar problems in [55-59],... 

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