One-dimensional slow invariant manifolds for spatially homogenous reactive systems

Abstract

A reactive system's slow dynamic behavior is approximated well by evolution on manifolds of dimension lower than that of the full composition space. This work addresses the construction of one-dimensional slow invariant manifolds for dynamical systems arising from modeling unsteady spatially homogeneous closed reactive systems. Additionally, the relation between the systems' slow dynamics, described by the constructed manifolds, and thermodynamics is clarified. It is shown that other than identifying the equilibrium state, traditional equilibrium thermodynamic potentials provide no guidance in constructing the systems' actual slow invariant manifolds. The construction technique is based on analyzing the composition space of the reactive system. The system's finite and infinite equilibria are calculated using a homotopy continuation method. The slow invariant manifolds are constructed by calculating attractive heteroclinic orbits which connect appropriate equilibria to the unique stable physical equilibrium point. Application of the method to several realistic reactive systems, including a detailed hydrogen-air kinetics model, reveals that constructing the actual slow invariant manifolds can be computationally efficient and algorithmically easy.

The Rate-Controlled Constrained-Equilibrium Approach to Far-From-Local-Equilibrium Thermodynamics

“…This is consistent with the theoretical proof by Ziegler [67] as well as a recent numerical study [17] that maximal entropy production rate is not a general feature of the standard equations of chemical kinetics.…”

“…these are then inverted to express the multipliers in terms of the values of the constraints, (17). Moreover, it is noteworthy that as a result of the last inversion, we have the identity…”

One-Dimensional Slow Invariant Manifolds for Fully Coupled Reaction and Micro-scale Diffusion

“…closely follows that of Al-Khateeb et al [18], where superscripts (o), ( * ), and (e) denote evaluation at reference pressure, initial state, and equilibrium, respectively, and quantities presented with an overbar (¯) denote evaluation on a per-mole basis. The governing equations for our reaction-diffusion system are the species evolution equations,…”

“…Other similar techniques such as minimal entropy production trajectory [8] and invariant constrained equilibrium edge [9] also construct invariant manifolds using equilibrium thermodynamic potentials. The slow invariant manifold (SIM) is another invariant manifold based on consideration of the local times scales of the system, which can be constructed using various techniques [10,11,12,13,14,15,16,17,18,19]. Al-Khateeb et al [18] give a detailed discussion of these spatially homogeneous invariant manifold methods, which are applied to ordinary differential equations (ODEs).…”

“…These can be categorized into two types: (i) methods which use spatially homogeneous manifolds and then account for reaction-transport coupling, and (ii) methods that build manifolds based on the governing PDEs, which include transport. Methods of the first type include the Maas-Pope projection (MPP) [20,21], the close-parallel assumption [22,23], andstudy is largely an extension of [18] to include effects of diffusion.…”