(698g) Dynamical Model Reduction Using the Green's Function Matrix

Concerns about climate change and global oil crisis have given motivation for applied and industrial chemistry research to better understand detailed mechanisms of chemical kinetics in related systems, such as atmosphere, combustion or pyrolysis. Such mechanistic knowledge is necessary for the design and optimization of these systems. Rapid growth of kinetic knowledge in the past decades in this area has contributed significantly in the development of huge reaction networks with hundreds or thousands of participating chemical reactions and species [1]. Furthermore, mathematical models of ordinary differential equations (ODEs) have commonly been created to simulate the dynamical behavior of spatially homogeneous reaction networks, to which a quantitative model analysis can be applied in order to gain insights [2]. In such large reaction networks, there often exists a number of species and reactions, which have little contribution to the system behavior, but their inclusion in the model results in increased computational time. Hence, it is often desired to reduce model size, especially for end-application such as optimization [1].

Reduction of linear ODE models (linear in states) can be done using established methods, such as balanced truncation or generalized Grammians [3]. Except for networks involving only unimolecular reactions, ODE models of chemical systems however are generally nonlinear. In this case, model reduction are often carried out using information from parametric sensitivities [1] or reaction rates [4], in which a reduced model is constructed by eliminating reactions that are not affecting desired system outputs. Consequently, there is one fundamental difference between reduction methods for linear and nonlinear models: model truncation is done by removing states in linear case, while reduction is carried out by removing reactions in nonlinear. In the latter, removal of a state is only a result of eliminating all reactions connecting that species to the rest of the system.

As system dimensionality is related to the number of states, we propose a new method for reducing nonlinear kinetic ODE models that directly eliminates unimportant states. The method is again based on sensitivity analysis. Traditionally, parametric sensitivity coefficients are computed as the ratio of the changes in the system behaviour with respect to an infinitesimal change in the parameter from its nominal value [5]. These coefficients give information regarding the contribution of individual parameter to the system outputs. The magnitude of sensitivities thus indicates the degree of importance of the corresponding reactions, based on which the reactions are retained or eliminated [1]. In contrast, the model reduction here is based on the Green's function matrix (GFM) as sensitivity coefficients with respect to initial concentrations [6]. The GFM analysis gives a dynamical, species-by-species insight on how a system output behavior is accomplished and complementarily how (impulse) perturbation to a particular species concentration at a given time propagates through the network. Based on this information, the model reduction is done by eliminating species or even whole pathway (a set of species) to which the outputs of interest show low sensitivities.

The efficacy of the proposed method is demonstrated through applications to a well-known low-temperature propane pyrolysis model of Edelson and Allara [7] with 98 reactions and 36 species and an industrial steam cracking of ethane pyrolysis model with 146 reactions and 26 species [8]. A dynamic model reduction was carried out in both the models resulting in reduced models having nearly half the number of original species, which confer more than five times reduction in computational cost and give 10% relative error from the original model outputs.

References:

1. Turanyi, T., Reduction of Large Reaction Mechanisms. New Journal of Chemistry, 1990. 14: p. 795-803.

2. Turányi, T., Sensitivity analysis of complex kinetic systems. Tools and applications. Journal of Mathematical Chemistry, 1990. 5(3): p. 203-248.

3. Dullerud, G.E. and F.G. Paganini, A course in robust control theory : a convex approach. 2000, New York: Springer. 417 p.

4. Bhattacharjee, B., et al., Optimally-reduced kinetic models: Reaction elimination in large-scale kinetic mechanisms. Combustion and Flame, 2003. 135(3): p. 191-208.

5. Varma, A., M. Morbidelli, and H. Wu, Parametirc Sensitivity in Chemical Systems, ed. A. Varma, M. Morbidelli, and H. Wu. 1999: Cambridge University Press, Cambridge, UK.

6. Perumal, T.M., Y. Wu, and R. Gunawan, Dynamical analysis of cellular networks based on the Green's function matrix. J Theor Biol, 2009. 261(2): p. 248-59.

7. Edelson, D. and D.L. Allara, A computational analysis of the alkane pyrolysis mechanism: Sensitivity analysis of individual reaction steps. International Journal of Chemical Kinetics, 1980. 12(9): p. 605-621.

8. Sun, W. and M. Saeys, Construction of an ab initio Kinetic Model for Industrial Ethane Pyrolysis. AIChE Journal, 2010. submitted