Redundant species

A number of different approaches have been suggested for systematic reduction of detailed reaction mechanisms [160,313], The most common approach involves a two-stage procedure. First, a skeletal mechanism is established by removing all redundant species and reactions. Second, the skeletal mechanism is further reduced by order-of-magnitude approximations, resulting in the analytically reduced mechanism. 

The skeletal or short mechanism is a minimum subset of the full mechanism. All species and reactions that do not contribute significantly to the modeling predictions are identified and removed from the reaction mechanism. The screening for redundant species and reactions can be done through a combination of reaction path analysis and sensitivity analysis. The reaction path analysis identifies the species and reactions that contribute significantly to the formation and consumption of reactants, intermediates, and products. The sensitivity analysis identifies the bottlenecks in the process, namely reactions that are rate limiting for the chemical conversion. The skeletal mechanism is the result of a trade-off between model complexity and model accuracy and range of applicability. 

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. 

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. 

The first stage of any reduction procedure should always be to establish which are the necessary species in the reaction mechanism over the range of conditions to be considered. In order to carry out the redundant species analysis, decisions must first be made about the important species and features which the reduced model must be able to reproduce accurately. In this example the important species were chosen as the primary reactants H2 and O2, and the product H2O. 

The analysis described in Section 4.6.1 was then applied over a range of ambient temperatures and reaction times for both the isothermal and non-isothermal model. Calculation of the 5, values revealed H2O2 and O3 to be redundant species for both models and at all reaction conditions tested. Table 4.4 shows examples of redundant species calculations from the full non-isothermal scheme at differing parts of the oscillatory trace... 

From the redundant species analysis it is clear that all reactions which consume H2O2 and O3 are redundant and can be removed automatically from the mechanism. In order to identify other redundant reactions the techniques of rate sensitivity analysis coupled with a principal component analysis of the resulting matrix can be used. The principal component analysis of the rate sensitivity matrix containing only the remaining important and necessary species will reveal the important reactions leading to reduced mechanisms applicable at various ambient temperatures. In principle it may be possible to produce a reduced scheme which models non-isothermal behaviour from analysis carried out on an isothermal model. An isothermal system is easier to model since thermodynamic and heat-transfer properties can be excluded from the calculations. However,... 

Isothermal model for low temperatures At a low temperature of TOOK, where the overall rate of reaction remains small, a subset of only 6 reactions involving the primary branching and termination routes are selected by the principal components. Selecting only the necessary species as part of the objective function, reactions 2, 3, 4, 7, 8 and 9 are chosen as important. However, if all species are included in the objective function then reactions 11, 36 and 37 are also selected by the principal component analysis, even though their removal from the scheme has little effect on the concentrations of the important species. This illustrates the importance of using the redundant species analysis prior to calculating rate sensitivities in choosing the optimum reduced scheme. Reactions 36 and 37 are fast-reversible reactions of O3 which has a low concentration. The present example demonstrates that in many cases such coupled reaction sets can be automatically removed from the model via the identification of redundant species. 

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. 

For the reduction of chemical mechanisms, reaction-rate analysis has probably the largest record of success. A novel way for the inspection of rates is based on the study of algebraic-rate sensitivities and the Jacobian matrix. These methods can be used for the automatic identification of redundant species and reactions, to produce a reduced mechanism consisting of a subset of the original mechanism. The use of algebraic manipulation in techniques such as the QSSA and lumping, make the production of a reduced mechanism essential and make subsequent calculations as simple as possible. 

Several methods have been suggested for the identification of redundant species. An early approach was introduced by Frenklach et al. (Frenklach et al. 1986 Frenklach 1991) who investigated the elimination of species from a detailed... 

Valorani et al. (2006) used the CSP method (see Sect. 6.4) for the identification of redundant species. They first define important species and check in which modes they are present. Reaction steps are then identified that have a significant contribution to these modes. These reaction steps may include further species, which will be considered as necessary species. Using an iterative procedure, the number of necessary species is continuously increased until at the end of the process no more important reactions are found. 

The connectivity method (CM) (Turanyi 1990c) identifies redundant species via the investigation of the Jacobian. Element (yjfj) dfjdy of the normalised Jacobian shows the percentage change of the production rate of species j due to a 1 % change in the concentration of species i. If the square of this effect is summed over all important species, then the value B,- shows the effect of a change in the concentration of each species on the concentrations of aU important species ... 

Figure 7.1 shows that starting from the group of important species, in an iterative procedure, all species can be identified that are necessary for the simulation of important species. Groups of species may be identified as redundant and can be eliminated, even if there are strong interactions between the redundant species. This type of approach was subsequently used by several other methods for the identification of redundant species as discussed later. 

Fig. 7.1 Relationships between species, as handled by several methods for the identifirration of redundant species. This is common in the connectivity method, the DRG family and the PFA methods. Starting from the important species, all other species are identified that are necessary for the calculation of the ctmcraitrations of the important species. The remaining redundant species are only loosely related to the group of important and necessary species...

So far we have discussed the removal of redundant species from a mechanism. It may also be useful to reduce the number of reactions for the remaining necessary species since the calculation of their rates at each time step can be computationally... 

Computational singular perturbation or CSP analysis also provides informatimi oti the contribution of the rates of the reaction steps to the various timescale modes within a model. It can therefore be used to identify redundant species and reactions as part of a model reduction procedure. The CSP methodology has been introduced in Sect. 6.4, and here we discuss aspects related to mechanism reduction. We continue the use of notations that were introduced in Sect. 6.4. 

Several of the mechanism reduction methods discussed so far (see Sects. 7.2—7.6) result in a smaller reaction mechanism, which is a subset of the original detailed mechanism obtained by the removal of redundant species and reactions. Other methods provide a smaller mechanism cruisisting of lumped species and/or lumped reaction steps (Sect. 7.7). A further group of methods was then discussed which identify fast timescales within the model (see Sects. 7.8 and 7.9), and the resulting reduced model is a new set of differential equations with accompanying algebraic equations. In some cases these equatiruis can be converted back to a reaction... 

The SEM program package (Nagy and Turanyi 2009 Nagy 2009) is able to detect effectively the redundant species and reactions within a reaction mechanism using the principle of simulation error minimisation (see Sect. 7.2.4). Programs SEM-CM and SEM-PCAF read CHEMKIN format mechanism files and carry out an automatic reduction. 

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