Entropy production and timescales

Abstract

Spatially homogeneous reactive systems are characterised by the simultaneous presence of a wide range of timescales. When the dynamics of such reactive systems develop very slow and very fast timescales separated by a range of active timescales, with large gaps in the fast/active and slow/active timescales, then it is possible to attain a multi-scale adaptive model reduction along with the integration of the governing ordinary differential equations using the G-Scheme framework. The G-Scheme assumes that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. We derive expressions that reveal the direct link between timescales and entropy production by resorting to the estimates of the contributions of the fast and slow subspaces provided by the G-Scheme. With reference to a constant pressure adiabatic batch reactor, we compute the contribution to entropy production by the four subspaces. These numerical experiments show that, as indicated by the theoretical derivation, the contribution to entropy production of the fast subspace is of the same magnitude as the error threshold chosen for the numerical integration, and that the contribution of the slow subspace is generally much smaller than that of the active subspace. We explicitly exploit this property to identify the slow and fast subspace dimensions differently from the method adopted in the G-Scheme, where the dimensions of the subspaces are defined on the basis of the asymptotic approximations of the contributions of the fast and slow subspaces. Comparison of the outcome of the analyses performed using two types of criteria underlines the substantial equivalence of the two. This property opens the door to a number of possible applications that will be explored in future work. For example, it is possible to utilise the information on entropy production associated with reactions within each subspace to define an entropy participation index that can be utilised for model reduction.

Keywords:

• entropy production
• G-Scheme
• timescale analysis

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. G stands for Grappa, an Italian liquor produced by distillation followed by an analogous reduction.

2. In chemical kinetics, a reversible reaction is a reaction where the reactants form products that react together to give the reactants back, whereas an irreversible reaction implies only reactants forming products. In the context of this paper, we use the term ‘irreversible reactions’ to address the non-isentropicity of the chemical reaction processes.

3. The use of time as independent variable produces a near-singular functional behaviour at the explosion time.

Additional information

Funding

Support for this work has been provided by the Clean Combustion Research Center, King Abdullah University of Science & Technology (CCRC/KAUST), Saudi Arabia [1975-03 Center Competitive Funding (CCF) Subaward Agreement]; MV acknowledges the support of the Italian Ministry of University and Research (MIUR).