(673d) Range-Constrained Simultaneous Reaction and Species Elimination in Kinetic Mechanisms

Chemical kinetic models have become indispensible in making policy and business decisions. In industry, models are being used to design chemical processes and combustion engines. In public policy, these models have been used to inform legislation such as the Montreal Protocol and the Clean Air Act of 1990. Many chemical processes occur under spatially inhomogeneous conditions in which chemical reactions are strongly coupled to thermal, mass, and momentum transport processes. Simulation of first-principles models of reacting flows is computationally demanding since it requires the solution of a nonlinear partial differential equation for each chemical species being modeled. The strong nonlinearities, stiffness, and large number of PDEs (typically on the order of hundreds) require model reduction techniques to make industrially relevant problems tractable by simplifying the equations in the model in an automated, controlled way

to approximate the solution of the model.

The primary objective of model reduction in the context of reducing computational effort is to minimize the CPU effort needed to solve the reduced model, subject to constraints that ensure that the error between the numerical solution of the detailed model and the solution of the reduced model is bounded to within acceptable limits over specified reaction conditions (temperatures, pressures, and species mass fractions). In order to limit this error when using the reduced model in a numerical integration routine, it is necessary to bound the difference between the chemical source term of the detailed model and the chemical source term of the reduced model for each species. Although many methods (such as [4], [5], and [1]) have some form of

error control (usually in the form of an adjustable parameter), few methods attempt to determine rigorous error bounds on the chemical source term of the reduced model with respect to the detailed model. Without these error bounds, it is extremely difficult to estimate the error in the solution obtained by applying a model reduction technique when solving a detailed model numerically.

It is also worth noting that bounding the error due to model reduction at a finite number of points in state space does not suffice [8]. Oluwole et al. have demonstrated that error bounds satisfied by a reduced model at points in state space do not necessarily hold within their convex hull. For this reason, a reduced model with error bounds satisfied at a finite collection of points in state space may not satisfy its stated error tolerances after one time step of numerical integration. Consequently, it is absolutely critical that reduced models be generated with error bounds on the time derivatives of state variables, and that these error bounds are satisfied over ranges in state space.

To generate a reduced model satisfying error bounds over a range of reaction conditions, we propose the method of range-constrained simultaneous reaction and species elimination. This method extends previous work on range-constrained reaction elimination by Oluwole et al. [8] by introducing additional logical variables that account for the inclusion or exclusion of chemical species, where the exclusion of a given species is equivalent to treating the species as inert. It is also an extension of point-constrained simultaneous reaction and species elimination by Mitsos et al. [6] in that it replaces the point constraints on error bounds between full and reduced models with error constraints over a Cartesian product of intervals in mass fraction-temperature space. The resulting formulation is an integer linear semi-infinite program (SIP). No algorithm currently exists to solve integer SIPs to global optimality. Instead of solving to global optimality, we use part of an algorithm by Bhattacharjee et al. [2] to determine guaranteed feasible points of our

integer linear SIP by using interval extensions [7] to formulate an integer linear program (ILP) restriction. DAEPACK [9] is used to generate interval extensions to formulate the ILP, which is then solved to global optimality using CPLEX [3]. The solution of this ILP corresponds to a reduced model that is guaranteed to satisfy error bounds on the time derivatives of state variables over the range of reaction conditions specified in the Cartesian product of intervals. Since this reduced model corresponds to a feasible point of our original integer linear SIP, it is not necessarily the reduced model with the fewest number of reactive species that satisfies the stated error bounds on the time derivatives of state variables. Nevertheless, the method yields significant reductions in the number of reactive species in the kinetic model for many problem instances, as can be seen by results for benchmark problems.

Having obtained a reduced model from the solution of the ILP restriction, this model can then be used within a numerical method to approximate faithfully the solution of the detailed model from which it was generated. Error bounds on the reduced model could then be propagated through the numerical method in order to obtain error bounds on the solution of the detailed model, yielding a more computationally efficient means of obtaining a numerical solution for currently intractable reacting flow problems to within known numerical error.

References:

[1] I. P. Androulakis. Kinetic mechanism reduction based on an integer programming approach. AIChE Journal, 46(2):361?371, 2000.

[2] B. Bhattacharjee, W. H. Green, and P. I. Barton. Interval methods for semi-infinite programs. Computational Optimization and Applications, 30:63?93, 2005.

[3] ILOG. ILOG CPLEX 11.0 User's Manual.

[4] S. H. Lam. Using CSP to understand complex chemical kinetics. Combustion Science and Technology, 89(5):375?404, 1993.

[5] T. Lu and C. K. Law. Systematic approach to obtain analytic solutions of quasi steady state species in reduced mechanisms. Journal of Physical Chemistry A, 110:13202?13208, 2006.

[6] A. Mitsos, G. M. Oxberry, P. I. Barton, and W. H. Green. Optimal automatic reaction and species elimination in kinetic mechanisms. Combustion and Flame, 155(1-2):118?132, 2008.

[7] R. E. Moore. Methods and Applications of Interval Analysis. Society for Industrial and Applied Mathematics, Philadelphia, 1987.

[8] O. O. Oluwole, B. Bhattacharjee, J. E. Tolsma, P. I. Barton, and W. H. Green. Rigorous valid ranges for optimally reduced kinetic models. Combustion and Flame, 146:348?365, 2006.

[9] J. Tolsma and P. I. Barton. DAEPACK: An open modeling environment for legacy models. Industrial and Engineering Chemistry Research, 36(6):1826?1839, 2000.