Analysis of Kinetic Reaction Mechanisms

Chemical processes can be described by detailed kinetic reaction mechanisms consisting of several hundreds or even thousands of reaction steps. Such reaction mechanisms are used in many fields of science and technology, including combustion, atmospheric chemistry, environmental modelling, process engineering, and systems biology. This book describes methods for the analysis of reaction mechanisms that are applicable in all these fields. The book will address topics such as the importance of model evaluation as well as the need for model reduction under situations where the kinetic model is coupled with models describing complex physical processes where computational expense becomes a critical issue. It includes topics such as: the basic foundations of chemical kinetic models; methods for the automatic generation of kinetic mechanisms; sources of thermodynamic and kinetic data; methods for uncertainty and sensitivity analysis; timescale analyses; similarities in model sensitivities; and chemical model reduction. Within the introduction we discuss the motivations behind the text as well as providing a brief summary of key reference texts on similar topics from the current literature.

This chapter provides an introduction to the basic concepts of reaction kinetics simulations. The level corresponds mainly to undergraduate teaching in chemistry and in process, chemical and mechanical engineering. However, some topics are discussed in more detail and depth in order to underpin the later chapters. The section “parameterising rate coefficients” contains several topics that are usually not present in textbooks. For example, all reaction kinetics textbooks discuss the pressure dependence of the rate coefficients of unimolecular reactions, but usually do not cover those of complex-forming bimolecular reactions. The chapter contains an undergraduate level introduction to basic simplification principles in reaction kinetics. The corresponding sections also discuss the handling of conserved properties in chemical kinetic systems and the lumping of reaction steps.

The creation of a kinetic reaction mechanism involves the definition of stoichiometries for each of the reaction steps and also the provision of values for all kinetic and thermodynamic parameters. Whilst this sounds like a simple task, in reality, it is extremely complicated. Reaction mechanisms often undergo updates and revisions over time, as the quantification of input parameters is improved through new kinetic studies or as new reaction steps are identified as being important. Recently developed mechanisms describing a range of kinetic problems in combustion, pyrolysis, atmospheric chemistry and biochemistry tend to be very large, and it is almost impossible to generate such mechanisms by hand. Fortunately, several mathematical methods and computational tools have been elaborated for the automatic generation of reaction mechanisms in each of these fields. These computer codes are able to handle various sources of chemical kinetic and thermodynamic data and will be described in this chapter. We also describe the variety of data sources which are used to help quantify the parameters within developed mechanisms.

Chemical changes that occur within reaction kinetic models are traditionally depicted by reaction pathways. One possible approach is to investigate the flow of a conserved property (such as the number of carbon atoms) from one species to another within the reaction scheme. In this way, element flux diagrams can be generated, which can be used for visualising the main reaction pathways within a mechanism (e.g. by representing the strength of fluxes through arrow thickness). These may also be useful within the context of the reduction of reaction mechanisms by highlighting which are the major and minor channels within the scheme. Another possibility is to explore the reaction chain that shows how other species contribute to the generation of a chosen species under investigation. Pathways leading to the consumption or production of a species can be generated in an algorithmic way, and in this chapter, we discuss methods to perform such reaction pathway analyses.

The aim of sensitivity and uncertainty analysis methods is to determine the influence of changes in model input parameters on the output of mathematical models. Such methods can help to highlight key model inputs that drive uncertainties in model predictions. Here we describe a range of mathematical tools for sensitivity and uncertainty analysis which may assist in the evaluation of large kinetic mechanisms. Approaches based on local sensitivity, local uncertainty and global uncertainty analysis are covered, as well as examples of their application to a variety of chemical kinetic models. Local sensitivity analysis is a routinely used method for the investigation of models and the theory behind it is discussed. Uncertainty analysis reveals the uncertainty of the simulation results caused by the uncertainty of model input parameters. Such uncertainties can be estimated using local sensitivity coefficients, but global uncertainty methods based on sampling approaches usually provide more realistic results. Global sensitivity methods can then be applied which determine how each input parameter contributes to the overall output uncertainty based on measures such as output variance. Various global methods for sensitivity analysis are discussed here, including the Morris screening method, the calculation of sensitivity indices based on random sampling, the Fourier Amplitude Sensitivity Test (FAST) method and the different surface response methods. All of these methods can be applied generally to mathematical models, but we also include a discussion of topics specifically related to reaction kinetics such as uncertainties in rate coefficients and the characterisation of the uncertainty of Arrhenius parameters.

A very characteristic feature of chemical kinetic models (in common with many other models in science) is that they contain a wide range of different timescales. This may have consequences for model behaviour and also for the selection of appropriate solution methods for the resulting equation systems. Several aspects of timescales of models are therefore discussed within this chapter. The discussion begins with the definition of various simple quantities used to measure timescales, such as species half-life and species lifetime, and explores their relationship to the time-dependent behaviour of the model. Timescales are closely related to the dynamic behaviour of the model following a perturbation within the chemical kinetic system, e.g., by suddenly altered concentrations. Systematic investigation of such perturbations can be achieved for large systems using computational singular perturbation (CSP) theory which is introduced here. Another common feature of chemical kinetic models is that the chemical kinetics relaxes the system to lower and lower-dimensional attractors until either a stationary point or chemical equilibrium (zero-dimensional attractor) or other low-dimensional attractor (e.g. a limit cycle) is reached. This leads to the importance of slow manifolds in the space of variables which will be investigated within this chapter. One practically important consequence of the presence of very different timescales is the stiffness of reaction kinetic models. Methods for dealing with stiffness within numerical models are therefore discussed.

Increases in both chemical kinetics knowledge and the capacity of computers have led to the availability of very large detailed kinetic mechanisms for many problems. These mechanisms may contain up to several thousand species and several ten thousand reaction steps. For computational reasons, however, large mechanisms still cannot be used in spatially 2D or 3D computational fluid dynamics simulations, where the applied mechanism typically requires less than 100 species. Also, within such large mechanisms, the key processes can be masked by the presence of many reaction steps of only marginal importance. A first step to reducing the size of a kinetic mechanism is to identify species and reaction steps which do not need to be included in order to accurately predict the key target outputs of the model. Such methods lead to so-called “skeletal” schemes. This chapter discusses many different methods for the identification of redundant species and reaction steps within a mechanism, including those based on sensitivity and Jacobian analyses, the comparison of reaction rates, trial and error and calculated entropy production. Another family of methods for the development of skeletal schemes is based on the investigation of reaction graphs. We discuss here the directed relation graph (DRG) method and its derivatives, and the path flux analysis (PFA) method. Mechanism reduction may be also based on optimisation methods which minimise an objective function related to the simulation error between the full and reduced models, subject to a set of constraints (e.g. numbers of species required). Integer programming and genetic algorithm-based methods have been used for such an optimisation and are discussed here. From these skeletal schemes, subsequent reductions can be achieved via either species or reaction lumping. Chemical and mathematical approaches to lumping are discussed with applications in combustion, atmospheric and biological systems. Reduction methods based on timescale separation are then introduced starting with the classic quasi-steady-state approximation (QSSA). Computational singular perturbation (CSP) methods are then described as a means of informing the derivation of analytically reduced models. Further efficiency gains can also be obtained by using a numerical approximation of a function in place of more traditional descriptions of chemical source terms within simulation models. The generation of such numerical reduced models can be based on the original differential equations and the thermodynamics of the problem or deduced from the simulation results. Using any of these methods, the applied function has to meet special requirements, such as the need to be evaluated quickly and to provide an accurate approximation. We discuss a series of approaches, tabulation methods, artificial neural networks (ANNs) and various types of polynomials, that all have been tested and applied within the context of kinetic modelling.

If a model is strongly autocatalytic and very different timescales are present, both of which are characteristic features of many reaction kinetic models, then the calculated local sensitivity functions are usually similar to each other. An implication of this is that in many cases, by changing a number of input parameters simultaneously according to certain ratios, almost identical simulation results can be obtained for output variables of kinetic models, over quite wide ranges of concentrations or reaction conditions. The similarity relations can be sorted into categories of local similarity, scaling relationships and global similarity. Such similarity relations have been found in models of combustion systems (explosions and flames) and molecular biological models. The theory of the origin of all these similarity relations is discussed in this chapter. The similarity of sensitivity functions is related to several important topics, such as discrimination between models, uniqueness of a model and robustness of biological systems.

This book discusses many complicated algorithms for the investigation and reduction of simulation models based on detailed reaction mechanisms. Fortunately, computer codes are readily available to facilitate the application of most of the methods described in this book. A large number of these codes have been made freely available for teaching and academic research. Many commercial codes (usually with good support) are also offered for these tasks, and for most commercial codes, academic licences are available at a lower cost than commercial ones. In this chapter we introduce a range of such computer programs which are organised according to the following categories: (1) general simulation codes in reaction kinetics, (2) special codes for the simulation of gas kinetic systems, (3) programs for the analysis and reduction of reaction mechanisms, (4) programs for the investigation of biological reaction kinetic systems (“systems biology codes”) and finally (5) codes for global uncertainty analysis. In all cases the basic features of the codes are discussed and a reference to the availability is given.