Necessary species

Develop a reaction mechanism for CO-H2-O2-N2O, using the H2-O2 mechanism in hydrogen.mec as starting mechanism. Add the necessary species and reactions, with rate constants obtained from an appropriate source (e.g., [256]). Because of the strongly oxidizing conditions, a number of reduced nitrogen species do not need to be considered. Only NO, NO2, N2O, NH, and N2 need to be included NH formed from N2O can be assumed to react to form NO or N2. Justify the choice of rate constants for the key reactions. 

If water is added to AA dissolved in benzene it too increases the % enol to 93.1% at 5.5 °C18). This observation, the opposite to that when water is the bulk solvent for AA, is due to a 1 1 association of AA and H20. This association is thought to be between water and the enol tautomer the equilibrium constant for the 1 1 complex was 2.4 (kg solvent mol-1)2. MAA and DMAA showed no association, but as they do and must exist only as the keto tautomers it was deduced that it was the enol tautomer that was the necessary species for interaction. This implies association via the enol OH as hydrogen bond donor. 

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. 

Here, 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]. 

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. 

Estimated effect of species for the rate of change of necessary species T = 2311 K, T = 818 K. A indicates species below the chosen threshold which are therefore redundant... 

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. 

A number of fluidized bed reactor model versions are based on the cross sectional averaged two-phase transport equations as presented in sect 3.4.7. Bue to the vigorous particle flow the fluidized beds are essentially isothermal, so no energy balance is generally required . In addition, the necessary species mass (mole) balance can be deduced from (3.498). The solids are considered... 

An overview of the methods used previously in mechanism reduction is presented in Tomlin et al. (1997). The present work uses a combination of existing methods to produce a carbon monoxide-hydrogen oxidation scheme with fewer reactions and species variables, but which accurately reproduces the dynamics of the full scheme. Local concentration sensitivity analysis was used to identify necessary species from the full scheme, and a principle component analysis of the rate sensitivity matrix employed to identify redundant reactions. This was followed by application of the quasi-steady state approximation (QSSA) for the fast intermediate species, based on species lifetimes and quasi-steady state errors, and finally, the use of intrinsic low dimensional manifold (ILDM) methods to calculate the mechanisms underlying dimension and to verify the choice of QSSA species. The origin of the full mechanism and its relevance to existing experimental data is described first, followed by descriptions of the reduction methods used. The errors introduced by the reduction and approximation methods are also discussed. Finally, conclusions are drawn about the results, and suggestions made as to how further reductions in computer run times can be achieved. 

The removal of reactions from the scheme was based on a principle component analysis of the rate sensitivity matrix that considers the local dependence of the rate of formation of a necessary species on rate parameters. In using this approach the first stage is to identify which species in the scheme are considered necessary for accurate prediction of the chosen important species and features. 

Those species that are considered to be important must be chosen first, and these are likely to include the main reactants and products. From this list of n important species, by use of a sensitivity analysis method, those species that require accurate concentrations in order for the important species to be modelled accurately are identified over a range of conditions and time-points. These species are referred to as necessary species, and include the important species, plus any species found to have an effect on the important species. The measure of the effect of changing the concentration of each species on the rate of production of an important species is defined as... 

Having defined a set of N necessary species it is now necessary to define a sub-set of reactions that still produces accurate concentrations of these species and temperature. This is achieved using a principle component analysis (Tomlin et al (1997) and references therein) of the normalised sensitivity matrix F, where... 

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. 

Species characterised by large B, values are closely connected to the important species and therefore are necessary species. In the next step, these necessary species are also included in the summation, and the B, values are recalculated. Species characterised by the largest B, values are again included in the summation, and this iteration is continued until all species that have close connection to the important species, directly or through other species, are identified. The rest of the species are considered to be redundant. 

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... 

The method of principal component analysis of matrix S (PCAS) was discussed in Sect. 5.3. The PCAS method allows the identification of the most important parameters related to selected simulation results. Therefore, if the objective function includes the concentrations of the important and necessary species (see Sect. 7.2) and the investigated parameters are the rate coefficients (or A-factors) of the reaction steps (Vajda et al. 1985 Vajda and Tur yi 1986 Turanyi 1990b Xu et al. 1999 Liu et al. 2005), it is also applicable for the generation of a reduced mechanism containing less reaction steps. A further development of the PCAS method is functional principal component analysis (fPCA) (Gokulakrishnan et al. 2006). This method facilitates the investigation of temporal and spatial changes in the importance of reaction steps in reaction—diffusiOTi systems. 

An iterative process is used to find the path fluxes of each selected species. Starting from the set of important species, using the relation Tab > , the set of other necessary species are identified. These are added to the investigated set, and the iterative process is continued until no new necessary species are found. Gou et al. (2013) used the PFA method to create a dynamic adaptive chemistry scheme for n-heptane and n-decane combustion mechanisms. 

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