Introduction (C. Morley, M.J. Pilling)
1 Basic chemistry of combustion (R.W. Walker, C. Morley)
2 Elementary reactions (S.H. Robertson, P.W. Seakins, M.J. Pilling)
3 Kinetics databases (D.L. Baulch)
5 Global behaviour in the oxidation of hydrogen, carbon monoxide and simple hydrocarbons (S.K. Scott)
6 Experimental and numerical studies of oxidation chemistry and spontaneous ignition phenomena (J.F. Griffiths, C. Mohamed)
7 Autoignition in spark-ignition engines (D. Bradley, C. Morley)
Author Index
Subject Index
Alison S. Tomlin, Tamás Turányi, Michael J. Pilling
Chapter 4 (pp. 293-437) in: Low-temperature Combustion and Autoignition
M.J. Pilling (Editor)
Volume 35 of series 'Comprehensive Chemical Kinetics'
Elsevier, Amsterdam, 1997
ISBN 0-444-82485-5
248 references
Construction of combustion mechanisms
The principles for the automatic generation of reaction mechanisms seem to be well elaborated. Several research groups have independently written successful programs for the generation of pyrolysis and low-temperature alkane combustion mechanisms. However, while simulation programs are generally available, mechanism generation programs are not available either on a commercial basis or in the public domain. A possible reason for this might be that making such a program `fool-proof', i.e. suitable for use by an inexperienced user, is as time-consuming as the creation of the program itself. In spite of this, it is expected that such programs will be available in coming years.
The concept used in the generation of the GRI methane mechanism is applicable, not for the generation of a mechanism, but for the improvement of the agreement of an existing detailed mechanism with bulk experimental data. This concept has been discussed here as a mathematically assisted method for the generation of detailed mechanisms. The method may be used for the improvement of a mechanism, but care should be taken when employing this concept, since human errors may easily affect the final results.
Sensitivity and uncertainty analysis
For a long time the main topic of research in the area of sensitivity analysis was to find an accurate and effective method for the calculation of local concentration sensitivities. This question now seems to be settled, and the decoupled direct method (DDM) is generally considered the best numerical method. All the main combustion simulation packages such as CHEMKIN, LSENS, RUN1DL, and FACSIMILE calculate sensitivities as well as the simulation results, and therefore many publications contain sensitivity calculations. However, usually very little information is actually deduced from the sensitivity results. It is surprising that the application of principal component analysis is not widespread, since it is a simple post-processing method which can be used to extract a lot of information from the sensitivities about the structure of the kinetic mechanism. Also, methods for parameter estimation should always be preceded by the principal component analysis of the concentration sensitivity matrix.
Local concentration sensitivities can be used for mechanism investigation. As an example, it has been shown that rate limiting steps can be identified on the basis of the time derivative of the sensitivity matrix. This approach of rate limiting steps is in complete accordance with the classical definition and yet allows the identification of rate limiting steps in case of mechanisms of any size.
The question remains open as to whether the application of concentration sensitivity analysis for mechanism reduction is feasible. Various methods for the study of reaction rates, to be discussed in the next section, are applicable for mechanism reduction, with the advantage that they use much less computer time. However, with the increasing power of computers, this point of view may become less significant, and the application of concentration sensitivities might become important as a principally different way of finding unimportant reactions.
Application of formal uncertainty analysis to combustion systems has been very rare so far, and has been restricted to the utilisation of local sensitivities. Possible reasons are the limited knowledge about the uncertainty of parameters, and the fact that global sensitivity methods are computationally very intensive. In the future it is expected that both these limitations will be lifted, and detailed uncertainty analyses will appear for combustion calculations. On the other hand, one of the main applications of sensitivity analysis has been to form a qualitative picture about which parameters should be known precisely in order to reproduce accurately a set of experimental observations.
Mechanism reduction without time-scale analysis
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 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.
Comparison of reaction rates, called rate-of-production analysis, is a frequently applied technique and it is the basis of limiting the size of a newly created mechanism. However, this technique requires a lot of manual effort. Algebraic rate sensitivities are the partial derivatives of production rates with respect to rate parameters. These measures are equal to normed reaction rate contributions. Inspection of algebraic rate sensitivities based on either the sum of squares of the coefficients (overall sensitivities) or principal component analysis, is a simpler and more automatic way for the identification of redundant reactions than that based on rate-of-production analysis.
Formal lumping techniques have been sought which will provide methods of reducing any general reaction system to a lower dimensional system of equations. This is achieved by representing information about groups of species using a single variable. General methods for both linear and nonlinear lumping have been investigated. Linear methods, where the new species are represented as linear combinations of the original ones, work well for linear systems and also provide some degree of reduction for nonlinear schemes. Nonlinear methods are much more general, but can involve complicated algebraic theory which might limit their use.
The main advantage of using lumping approaches for mechanism reduction is that searching for the lumping functions can be done systematically based only on the form of the equations rather than any intuitive knowledge of the chemical processes. There are however problems associated with some lumping methods. For example, in linear lumping there are large numbers of possible lumping matrices for each dimension of reduced scheme and the the number of possible lumped schemes increases with the number of original variables. A combinatorical explosion may occur, and there has to be some way of choosing the optimum lumped scheme for each problem. The calculation of the inverse matrix M' provides further problems and is not unique although the choice of inverse matrices does not affect the form of the lumped equations.
Rather than testing each reduced scheme to check whether its dynamics matches that of the original scheme there is a need to combine lumping with physical information about the system in order to select appropriate lumping matrices. This has been carried out to some extent in chemical lumping, but what are needed are general rules which can be applied to any reaction mechanism. One way forward is to incorporate knowledge about the time-scales involved in the reaction into the lumping process. The criteria on which we base the success of the lumped scheme is usually its agreement with the full scheme for the important variables. This agreement may be within certain error limits and so the existence of exact lumping schemes may not be necessary. The development of approaches for approximate lumping is therefore important, and as we shall see, when coupled with a time-scale analysis, can provide a practical way forward for nonlinear lumping.
Reduction based on the investigation of time-scales
Methods based on the separation of time-scales between chemical species have shown a high degree of success in reducing chemical systems. Such methods may also reduce the stiffness of the equation system by removing fast equilibrating processes, thus allowing the use of less expensive integration methods.
The QSSA is the oldest method in use and perhaps has the best proven success rate. The selection of QSSA species can now be automated using recently developed methods such as CSP, and techniques to evaluate QSSA errors. Once identified the QSSA species can be eliminated from the reaction scheme, either by solving the QSSA expressions explicitly or by using methods for the generation of global reactions such as used by Peters etc. There are two remaining problems in the application of the QSSA to general systems. The first is the algebraic solution of the non-linear equation systems or their truncation to some linear form. Some methods have been used such as the application of partial equilibrium assumptions, but the criteria for their application often relies on trial and error. The technique of inner iteration provides a solution to the remaining nonlinear QSSA expressions, though will not be as computationally efficient as an explicit solution would be. The second problem is the limitations of the approach associated with the fact that it equates species directly with the time-scales of the associated linear system. If there are more fast time-scales (as identified by an investigation of the eigenvalues) than QSSA species, then the optimum reduced model may not be produced by the application of the QSSA. In some cases the fast variables of the model will not be individual species, but will be functions of the original species, as shown by the application of CSP.
The ILDM technique proposed by Maas and Pope overcomes this problem by describing geometrically the optimum slow manifold of a system. The criteria for reduction is based on the time-scales of linear combinations of variables and not on species themselves. The main advantage of the technique is that it requires no information concerning which reactions are to be assumed in equilibrium or which species in quasi-steady-state. The only inputs to the system are the detailed chemical mechanism and the number of degrees of freedom required for the simplified scheme. The ILDM method then tabulates quantities such as rates of production on the lower dimensional manifold. For this reason it is necessarily better suited to numerical problems since it does not result in sets of rate equations which are directly related to the inputs of the model. The CSP method can formulate such a set of equations but because it relies on a linear approach, these equations change with time and cannot therefore be written down explicitly for nonlinear systems. In the next Section we describe techniques under development which address these problems based on the application of lumping to systems with time-scale separation. These methods will be based on the idea of transforming the system to reveal purely fast variables, each of which will be associated with a single time-scale.
Approximate lumping in systems with time-scale separation
The method of lumping using time-scale separation provides a general method of reducing nonlinear systems. New fast variables are defined which can be nonlinear combinations of the original variables and these fast variables can be eliminated from the scheme using singular perturbation methods. The resulting system of equations essentially describes the motion along the slow manifold of the system and is perfectly good for long time behaviour. These general methods improve on the QSSA in two ways. Firstly, the new fast variables are combinations of the original species, so that more fast variables than QSSA species may be found. Secondly, a higher order accuracy can be achieved so that the methods may apply in regions of composition space where the QSSA does not. The main problem with the general approximate nonlinear lumping technique is that it can lead to complicated algebraic expressions which might be difficult to manipulate automatically. The resulting equation system, whilst being more accurate than that generated by lower order methods such as the QSSA, may contain complex right hand sides. Increasing generality and accuracy is therefore gained at the expense of ease of application and efficiency.
Fitting small mathematical models to kinetic data is an alternative to the reduction of large kinetic models. The mathematical models can be systems of ordinary differential equations (ODEs) or algebraic equations (including systems of difference equations) and the kinetic data may be either simulation results of a detailed model or experimental data. When simulation results are fitted, the original experimental information might be distorted by the inaccurate large model. However, experimental data are usually too sparse and are not appropriate as a basis for fitting a small model that is intended for use as an extrapolation and interpolation tool.
The key problem in making a small fitted ODE model is not the determination of the values of the parameters but finding a small ODE with optimal structure. So far the main approach has been to set up a skeleton mechanism that corresponds to chemical kinetic knowledge about the system. Arrhenius type expressions are used for the description of the temperature dependence of the reaction rates, and the powers of concentrations in the rate expressions are parameters to be fitted. This way of setting up the small systems of ODEs is heuristic, but fitting parameters has been an automatic process based on the least-squares method.
Fitted kinetic ODEs have been less successful so far than reduced mechanisms. One reason for this limited success may be a result of the hybrid chemical-mathematical approach for finding the ODE models. It is expected that in the future, fully mathematical methods will be used for finding optimal small systems of ODEs which can accurately describe the kinetic data fitted, and which can be used as an extrapolation tool.
Fitted algebraic models offer very high simulation speed and also good accuracy over the parameter space for which they are fitted. In spite of some successful applications, it is not yet clear if the high simulation speed is accompanied by robustness and flexibility. If algebraic models prove to be robust enough, they may have wide applications in combustion simulations. In combustion models, thermodynamic data such as the heat capacities of each species, are not calculated from first principles, but stored in the form of high order polynomials as a function of temperature. Maybe in the foreseable future complex kinetic models for many fuels and conditions will also be available in the form of polynomials. It will then be possible for combustion simulations to concentrate on reproducing complex fluid behaviour, rather than spending a large amount of computational effort on chemical kinetic calculations.
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