The Role of Thermodynamics in Model Reduction When Using Invariant Grids

Abstract

Abstract. In the present work, we develop in detail the process leading to reduction of models in chemical kinetics when using the Method of Invariant Grids (MIG). To this end, reduced models (invariant grids) are obtained by refining initial approximations of slow invariant manifolds, and used for integrating smaller and less stiff systems of equations capable to recover the detailed description with high accuracy. Moreover, we clarify the role played by thermodynamics in model reduction, and carry out a comparison between detailed and reduced solutions for a model hydrogen oxidation reaction.

Method of invariant grid for model reduction of hydrogen combustion

“…[3,4,5,6,7]. In the sequel, we focus on the aspects related to MIG application to non-isothermal cases.…”

“…As suggested in [3], the notion of a quasiequilibrium manifold (QEM) can be used for initializing the MIG procedure (see also [6]). In general, a q-dimensional QEM represents a manifold in the concentration space which is given by minimizing the Lyapunov function (6) under a set of q linear constraints expressing a re-parametrization of the original variables [X i ] in terms of some new variables ξ j .…”

“…Assuming that d elements are involved in the reaction, we can construct another Lyapunov function (6) where µ ki represent the number of atoms of the k-th chemical element in species i and W i is the molecular weight of species i. The fixed parameters λ k and λ are chosen such that ∇G|ē = 0 at the equilibrium, with the gradient of G computed under fixed e. Because of the conservation of atoms and density, the time derivative of (6) is non-positive:…”

“…In particular, the vector l 1 appearing in (15) was taken as the left eigenvector of the Jacobian matrix J evaluated at the equilibrium point corresponding to the eigenvalue with the smallest absolute value [5,6]. The initial grid, as well as any subsequent grid, is …”

“…Where h is the dimension of the latter intersection subspace. Following the MIG approach, once an initial grid G 0 (in general, non-invariant) is given, it can be refined by solving at each node the following system of h equations [6] …”