A functional-PCA approach for analyzing and reducing complex chemical mechanisms
Introduction
The use of detailed chemical kinetic models of combustion and pyrolysis is becoming more widespread in the development, analysis, and control of combustion processes. However, coupling detailed kinetic models with heat, mass and momentum transfer balance equations to simulate reactive flows can produce large systems of equations that are not practical to solve with current computing technology (Hilbert, Tap, El-Rabii, & Thevenin, 2004). Much of the problem arises from the fact that, to describe the chemistry that occurs during combustion, the required number of elementary reactions in a detailed kinetic mechanism varies from several hundred to a few thousand, including a large number of intermediate species. In many cases, the main reaction pathway is largely dictated by a subset of the entire mechanism. Hence a major objective in combustion modelling is to identify these important reactions. First-order, elementary sensitivity analysis is a popular method for doing this (Yetter, Dryer, & Rabitz, 1985). Sensitivity analysis involves perturbing the important parameters in a model (e.g., the rate constants) and seeing how the predictions change. When applied to combustion problems, or any complicated chemical scheme, challenges arise because these sensitivities will vary with reaction conditions, location in a reactor, and/or time. The scale of the problem can increase dramatically. For example, in a reaction mechanism consisting of 500 reactions and 100 species in a reactor for which values are calculated at even only 20 time steps, one ends up with 1 × 106 sensitivity coefficients to evaluate the sensitivity of each species to each reaction at each time step. It is difficult to obtain a clear picture of what is important in this reaction mechanism looking at individual sensitivity coefficients, so multivariate analysis techniques, such as principal component analysis (PCA) are often used to extract the important information from the sensitivities (Vajda, Valko, & Turányi, 1985). PCA can be used to identify important reactions by decomposing the covariance matrix of the sensitivities, thereby highlighting the patterns in the sensitivity behaviour. Conventional PCA approaches produce an averaged set of principal components because they work with a large sensitivity matrix in which the sensitivity matrices at each time step are stacked vertically. In this paper, we demonstrate the use of functional principal components analysis (fPCA) (Ramsay & Silverman, 1997), which allows for decomposition of the sensitivity behaviour with time. The fPCA approach explicitly takes into account the time-varying behaviour of the sensitivities, and allows for the possibility that the dominant set of reactions can change with time.
The distinction between the work reported in this paper and the pioneering work of Turanyi lies in this treatment of the sensitivity profiles in the PCA analysis. Vajda et al. (1985) formed a large sensitivity matrix in which the sensitivities at each time step are stacked in a matrix which is subsequently decomposed. The result is an average PCA over the time horizon, and principal components that are static. This is the approach that has been used in most work that has followed on from Turanyi's original work. Researchers have also performed PCA on sensitivity matrices at given time steps—an individual PCA analysis that can be repeated at each time step along the trajectory. The resulting principal components are then examined and the significant reactions plotted at each time step. Finally, another approach that has been adopted is to integrate the sensitivity matrices over the time horizon, and apply a decomposition to this integrated sensitivity matrix. The fPCA approach that we are using treats the sensitivity trajectories as time functions and seeks to come up with a set of eigenfunctions which can be examined to identify which reactions are significant. Since we are using a discretized approach to perform the calculations, we take the sensitivity matrices at each time step and concatenate them to produce a large, “horizontal”, sensitivity matrix. This matrix is then decomposed using a PCA analysis. The loadings in the principal components reflect how sensitivities with respect to certain reactions, at possibly different time steps, produce large components. In this way, the time evolution of reactions can be taken more into account. Other researchers have also integrated or summed the sensitivity matrices at each time step, producing a time-averaged sensitivity matrix.
This paper begins with a brief review of some common model reduction techniques, focussing on sensitivity analysis and using PCA to analyse the results, pointing out the limitations with current techniques. The application of the fPCA technique is then explained in detail, and is illustrated with a well-known combustion example considered under two different operating scenarios.
Section snippets
Model reduction techniques
The available techniques for reducing kinetic mechanisms can be classified into two major groups: (1) consolidation methods and (2) exclusion methods. In exclusion methods, the unimportant or redundant reactions and species are eliminated, i.e., the reduced mechanism is a subset of the detailed one. In consolidation methods, on the other hand, some of the species and reactions are expressed in terms of the others or via pseudo-species and reactions. Thus, the main difference between the two
Consolidation methods
Kinetic model reduction methods using time-scale analysis and lumping are two commonly used consolidation methods in combustion chemistry modelling. Time-scale analysis exploits the difference between fast and slow reactions to reduce the number of reactions and species in a detailed kinetic mechanism. Traditional reduction techniques such as the quasi-steady-state assumption (QSSA) (Chapman & Underhill, 1913) and the partial-equilibrium assumption (PEA) (Michaelis & Menten, 1913) fall under
Exclusion methods
Although sensitivity analysis remains the popular method for identifying redundant reactions and species, integer programming based optimization methods (Petzold & Zhu, 1999; Edwards, Edgar, & Manousiouthakis, 2000; Androulakis, 2000) have been employed in recent years to develop reduced kinetic schemes. The use of non-linear optimization methods such as non-linear, integer programming preserves the non-linearity of a reaction system, as opposed to sensitivity analysis, in which linear,
Principal component analysis
Principal Component analysis uses a singular value decomposition (SVD) of the matrix of sensitivities of species concentrations with respect to kinetic coefficients to identify linear combinations of sensitivities that explain the major variation in the sensitivity matrix. A reduced set of these transformed sensitivities is identified that approximates the sensitivity behaviour. The linear combinations of sensitivities represent combinations of the original reactions; reactions that are not
Functional PCA
The local sensitivities of the species concentrations to perturbations in the kinetic parameters are time dependent, and follow a trajectory over time or spatial position in the reactor. Consequently, it is reasonable to expect that the dominant reactions might evolve over time. In particular, one could expect that the linear combinations that explain the major variation in the sensitivities would change with time. In other words, the most appropriate SVD is one in which the eigenvectors
Example-reduction of CO oxidation scheme
A detailed kinetic scheme for CO oxidation, shown in Table 1, is used as an example to demonstrate the application of functional PCA. The CO oxidation model has 26 reversible, elementary reactions and 12 species (Yetter et al., 1985). Two plug-flow reactor simulations are performed using the same initial conditions (0.2% CO, 1% H2O, 2.8% O2, and 96% N2) but the user-defined temperature profiles differ slightly, as shown in Fig. 1, Fig. 2. The residence time used is 0.8 s. As can be seen in Fig. 1
Conclusions
The application of functional principal component analysis for identifying important reactions in a combustion kinetic mechanism has been demonstrated using a well-known example evaluated under two different operating profiles. The results demonstrate that PCA applied to a large sensitivity coefficient matrix can be used to quickly identify the key reactions in a reaction mechanism, however, the importance of the reactions is averaged over the residence time or length of the reactor. Locally or
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Present address: Combustion Science and Engineering, Inc., 8940 Old Annapolis Road, Suite L, Columbia, MD 21045, USA.