Skeletal mechanism

Figure 7.5. Feedback diagram for skeletal mechanical regulation. When forces are applied to a whole bone, the stimulus that results is sensed by the bone cells in the tissue. The sensor cells then signal bone-forming and -removing cells to change the geometry and material properties of the bone.

The limitations encountered when obtaining an analytical solution to the conservation equations, as in the present work, differ from those encountered applying direct computational methods. For example, the cost of numerical computations is dependent on the grid and, especially, on the number of species for which conservation equations must be solved additional reactions do not add significantly to the computational effort. With RRA techniques, further limitations arise on the number of different reaction paths that can conveniently be included in the analysis. The analysis typically follows a sequence of reactions that make up the main path of oxidation, the most important reactions, while parallel sequences are treated as perturbations to the main solution and often are sufficiently unimportant to be neglected. The first step thus identifies a skeletal mechanism of 63 elementary steps by omitting the least important steps of the detailed mechanism [44]. 

Steady-state approximations are introduced into the skeletal mechanism to obtain the following six-step reduced mechanism ... 

Figure 25.4 Comparison between RRA predicted T° (curves) and peak temperatures from full numerical calculations. Hollow symbols show results with the detailed mechanism, while the solid symbols provide results using only the skeletal mechanism. Calculations for methane-air diffusion flames at p = 1 bar and fuel and oxidizer stream temperatures of Tf = To= 300 K... 

Figure 25.6 Comparison between integrated mole fractions of CH A = jXcu dZ) and CH2 B = f Xc dZ) from asymptotics and full numerical calculations. The calculations are for methane-air diffusion flames at p = 1 bar and oxidizer and fuel stream temperatures of Tp = To = 300 K. HoUow symbols represent results from numerical computations using the starting mechanism while filled symbols are results using only the skeletal mechanism the solid curves are the original RRA results and the dashed curves improvements [44]...

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. 

Once the skeletal mechanism is established, a reduced mechanism is developed by applying steady-state and partial-equilibrium assumptions. The criteria for assuming steady-state or partial-equilibrium are discussed in Section 13.2.5. The concentration of species, typically radicals, that can be assumed in steady state can be estimated based on concentrations of other species and rate constants for relevant reactions. Thereby the steady-state species can be eliminated from the reaction mechanism. After elimination of steady-state species, the required number of multi-step reactions is determined. The reaction rate for these multi-step reactions can be calculated from the reaction rates of the original mechanism. The multi-step reaction rates depend on the concentration of the eliminated steady-state species. Partial equilibrium assumptions are often applied to the fastest elementary reactions to simplify the estimation of the steady-state concentrations. 

Furthermore, mathematical procedures can be applied to the detailed mechanism or the skeletal mechanism which reduces the mechanism even more. These mathematical procedures do not exclude species, but rather the species concentrations are calculated by the use of simpler and less time-consuming algebraic equations or they are tabulated as functions of a few preselected progress variables. The part of the mechanism that is left for detailed calculations is substantially smaller than the original mechanism. These methods often make use of the wide range of time scales and are thus called time scale separation methods. The most common methods are those of (i) Intrinsic Low Dimensional Manifolds (ILDM), (ii) Computational Singular Perturbation CSF), and (iii) level of importance (LOl) analysis, in which one employs the Quasy Steady State Assumption (QSSA) or a partial equilibrium approximation (e.g. rate-controlled constraints equilibria, RCCE) to treat the steady state or equilibrated species. 

Species A is thereby connected to species B only if the pair wise error is above a user set error threshold. The starting point if the directed relation graph would be one of the important species, such as the fuel species. When species with low connectivity (below the threshold) is eliminated the result will be a skeletal mechanism similar to what is obtained by a necessity analysis. 

Fig. 9. Calculated temperatures and mole faction profiles for the end-gas of an Sl-engine. Top left temperature profile, top right HO2 molefraction representing a radical profile, bottom fuel profiles, the iso-octane and the n-heptane. Different reduced mechanisms are compared with the detailed and skeletal mechanisms (Soyhan et al., 2000)...

Lovas, T. (2009). Automatic generation of skeletal mechanisms based on level of importance analysis. Combustion and Flame Vol. 156, pp. 1348-1358. ... 

Niemeyer, K., Sung, C. Raju, M. (2010). Skeletal mechanism generation for surrogate fuels using directed relation graph with error propagation and sensitivity analysis. Combustion and Flame Vol. 157, pp 1760-1770. 

Soyhan, H., Amneus, P, Maufi, F. Sorusbay, C. (1999). A Skeletal Mechanism for the Oxidation of iso-Octane and n-Heptane Validated imder Engine Knock Conditions, SAE Technical Paper 1999-01-3484. 

Zeuch, T., Moerac, G., Ahmed, S.S. Mauss, F. (2008). A comprehensive skeletal mechanism for the oxidation of n-heptane generated by chemistry-guided reduction, Comb, and Flame Vol 155, pp 651-674... 

According to Pepiot and Pitsch (2005) [2], around 100 species are needed for the ignition of n-heptane. However, 12 species provide about 99% of mass of the final combustion products. Therefore, in order to reduce the computational cost, a skeletal mechanism with around 100 species is obtained and from these, reduced mechanisms are developed based on their reaction rates. 

L0vas, T., Houshfar, E., Bugge, M., Skreiberg, 0. Automatic generation of kinetic skeletal mechanisms for biomass combustion. Energy Fuels 27, 6979-6991 (2013)... 

Prager, J., Najm, H.N., Valorani, M., Goussis, D.A. Skeletal mechanism generation with CSP and validation for premixed n-heptane flames. Proc. Combust. Inst 32, 509-517 (2009)... 

Path flux analysis (PFA) is a method, similar to DRG, for the generation of skeletal mechanisms (Sun et al. 2010 Gou et al. 2013). In the PFA method, the production and consumption fluxes are used to identify the important reaction pathways. The first-generation production (Pa) and consumption (Ca) fluxes of species A are calculated according to equations... 

Sikalo et al. (2014) compared several options for the application of genetic algorithms to mechanism reduction, exploring the trade-off between the size and accuracy of the resulting mechanisms. Information on the speed of solution was also taken into account, so that, for example, the least stiff system (Sect. 6.7) could be selected. An automatic method for the reduction of chemical kinetic mechanisms was suggested and tested for the performance of reduced mechanisms used within homogeneous constant pressure reactor and burner-stabilised flame simulations. The flexibility of this type of approach has clear utility when restrictions are placed on the number of variables that can be tolerated within a scheme in the computational sense. However, the development of skeletal mechanisms is rarely the end point of any reductiOTi procedure since the application of lumping or timescale-based methods can be applied subsequently. These methods will be discussed in later sections. 

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