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Journal of Non-Equilibrium Thermodynamics

Founded by Keller, Jürgen U.

Editor-in-Chief: Hoffmann, Karl Heinz

Managing Editor: Prehl, Janett / Schwalbe, Karsten

Ed. by Michaelides, Efstathios E. / Rubi, J. Miguel

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1437-4358
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Volume 43, Issue 2

Issues

Systematic Constraint Selection Strategy for Rate-Controlled Constrained-Equilibrium Modeling of Complex Nonequilibrium Chemical Kinetics

An Automatable and Thermodynamically Consistent, Quasi-Equilibrium Model of Far Nonequilibrium States of Complex Reacting Systems Based on Probing the Fully Detailed Model and Taking a Truncated Singular Value Decomposition of the Resulting Evolution of the Degrees of Disequilibrium

Gian Paolo Beretta / Luca Rivadossi / Mohammad Janbozorgi
Published Online: 2018-03-17 | DOI: https://doi.org/10.1515/jnet-2017-0055

Abstract

Rate-Controlled Constrained-Equilibrium (RCCE) modeling of complex chemical kinetics provides acceptable accuracies with much fewer differential equations than for the fully Detailed Kinetic Model (DKM). Since its introduction by James C. Keck, a drawback of the RCCE scheme has been the absence of an automatable, systematic procedure to identify the constraints that most effectively warrant a desired level of approximation for a given range of initial, boundary, and thermodynamic conditions. An optimal constraint identification has been recently proposed. Given a DKM with S species, E elements, and R reactions, the procedure starts by running a probe DKM simulation to compute an S-vector that we call overall degree of disequilibrium (ODoD) because its scalar product with the S-vector formed by the stoichiometric coefficients of any reaction yields its degree of disequilibrium (DoD). The ODoD vector evolves in the same (S-E)-dimensional stoichiometric subspace spanned by the R stoichiometric S-vectors. Next we construct the rank-(S-E) matrix of ODoD traces obtained from the probe DKM numerical simulation and compute its singular value decomposition (SVD). By retaining only the first C largest singular values of the SVD and setting to zero all the others we obtain the best rank-C approximation of the matrix of ODoD traces whereby its columns span a C-dimensional subspace of the stoichiometric subspace. This in turn yields the best approximation of the evolution of the ODoD vector in terms of only C parameters that we call the constraint potentials. The resulting order-C RCCE approximate model reduces the number of independent differential equations related to species, mass, and energy balances from S+2 to C+E+2, with substantial computational savings when C ≪ S-E.

Keywords: model reduction in nonequilibrium thermodynamics; rate-controlled constrained equilibrium; RCCE constraints; degrees of disequilibrium; singular value decomposition; principal component analysis

References

  • [1]

    S. Vajda, P. Valko, and T. Turanyi, Principal component analysis of kinetic models, Int. J. Chem. Kinet. 17 (1985), 55–81.CrossrefGoogle Scholar

  • [2]

    S. H. Lam and D. A. Goussis, Understanding complex chemical kinetics with computational singular perturbation, Symp., Int., Combust. 22 (1988), 931–941.Google Scholar

  • [3]

    S. J. Fraser, The steady state and equilibrium approximations: A geometrical picture, J. Chem. Phys. 88 (1988), 4732–4738.CrossrefGoogle Scholar

  • [4]

    R. Law, M. Metghalchi, and J. C. Keck, Rate-controlled constrained equilibrium calculation of ignition delay times in hydrogen-oxygen mixtures, Symp., Int., Combust. 22 (1989), 1705–1713.CrossrefGoogle Scholar

  • [5]

    J. C. Keck, Rate-controlled constrained-equilibrium theory of chemical reactions in complex systems, Prog. Energy Combust. Sci. 16 (1990), 125–154.CrossrefGoogle Scholar

  • [6]

    M. R. Roussel and S. J. Fraser, On the geometry of transient relaxation, J. Chem. Phys. 94 (1991), 7106–7113.CrossrefGoogle Scholar

  • [7]

    U. Maas and S. B. Pope, Simplifying chemical kinetics – Intrinsic low dimensional manifolds in composition space, Combust. Flame 88 (1992), 239–264.CrossrefGoogle Scholar

  • [8]

    G. Li, A. S. Tomlin, H. Rabitz, and J. Tóth, Determination of approximate lumping schemes by a singular perturbation method, J. Chem. Phys. 99 (1993), 3562–3574.CrossrefGoogle Scholar

  • [9]

    S. Singh, J. M. Powers, and S. Paolucci, On slow manifolds of chemically reactive systems, J. Chem. Phys. 117 (2002), 1482–1496.CrossrefGoogle Scholar

  • [10]

    E. L. Haseltine and J. B. Rawlings, Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetic, J. Chem. Phys. 117 (2002), 6959–6969.CrossrefGoogle Scholar

  • [11]

    A. N. Gorban and I. V. Karlin, Method of invariant manifold for chemical kinetics, Chem. Eng. Sci. 58 (2003), 4751–4768.CrossrefGoogle Scholar

  • [12]

    D. Lebiedz, Computing minimal entropy production trajectories: An approach to model reduction in chemical kinetics, J. Chem. Phys. 120 (2004), 6890–6897.CrossrefGoogle Scholar

  • [13]

    M. Valorani, F. Creta, D. A. Goussis, J. C. Lee, and H. N. Najm, An automatic procedure for the simplification of chemical kinetic mechanisms based on CSP, Combust. Flame 146 (2006), 29–51.CrossrefGoogle Scholar

  • [14]

    E. Chiavazzo, A. N. Gorban, and I. V. Karlin, Comparison of invariant manifolds for model reduction in chemical kinetics, Commun. Comput. Phys. 2 (2007), 964–992.Google Scholar

  • [15]

    A. N. Al-Khateeb, J. M. Powers, S. Paolucci, A. J. Sommese, J. A. Diller, J. D. Hauenstein, et al.,One-dimensional slow invariant manifolds for spatially homogenous reactive systems, J. Chem. Phys. 131 (2009), 024118.Google Scholar

  • [16]

    G. P. Beretta, J. C. Keck, M. Janbozorgi, and H. Metghalchi, The rate-controlled constrained-equilibrium approach to far-from-local-equilibrium thermodynamics, Entropy 14 (2012), 92–130.CrossrefGoogle Scholar

  • [17]

    E. P. Gyftopoulos and G. P. Beretta, Entropy generation rate in a chemically reacting system, J. Energy Resour. Technol. 115 (1993), 208–212.CrossrefGoogle Scholar

  • [18]

    J. C. Keck and D. Gillespie, Rate-controlled partial-equilibrium method for treating reacting gas mixtures, Combust. Flame 17 (1971), 237–241.CrossrefGoogle Scholar

  • [19]

    J. C. Keck, Rate-controlled constrained equilibrium method for treating reactions in complex systems, in: R. D. Levine, M. Tribus (Eds.), The Maximum Entropy Formalism, MIT Press, Cambridge, MA, 1979, pp. 219–245. Available online at www.jameskeckcollectedworks.org/.Google Scholar

  • [20]

    G. P. Beretta and J. C. Keck, The constrained-equilibrium approach to nonequilibrium dynamics, in: R. A. Gaggioli (Ed.), Second Law Analysis and Modeling, ASME Book H0341C-AES, Vol. 3, ASME, New York, 1986, pp. 135–139. Available online at www.jameskeckcollectedworks.org/.Google Scholar

  • [21]

    K. C. Chen, A. Csikász-Nagy, B. Gyorffy, J. Val, B. Novák, and J. J. Tyson, Kinetic analysis of a molecular model of the budding yeast cell cycle, Mol. Biol. Cell 11 (2000), 369–391.CrossrefGoogle Scholar

  • [22]

    A. Lovrics, A. Csikász-Nagy, I. G. Zsély, J. Zádor, T. Turányi, and B. Novák, Time scale and dimension analysis of a budding yeast cell cycle model, BMC Bioinform. 7 (2006), 494.CrossrefGoogle Scholar

  • [23]

    I. Surovtsova, N. Simus, T. Lorenz, A. König, S. Sahle, and U. Kummer, Accessible methods for the dynamic time-scale decomposition of biochemical systems, Bioinformatics 25 (2009), 2816–2823.CrossrefGoogle Scholar

  • [24]

    A. I. Karpov, Minimal entropy production as an approach to the prediction of the stationary rate of flame propagation, J. Non-Equilib. Thermodyn. 17 (1992), 1–10.CrossrefGoogle Scholar

  • [25]

    V. Yousefian, A rate-controlled constrained-equilibrium thermochemistry algorithm for complex reacting systems, Combust. Flame 115 (1998), 66–80.CrossrefGoogle Scholar

  • [26]

    Q. Tang and S. B. Pope, Implementation of combustion chemistry by in situ adaptive tabulation of rate-controlled constrained equilibrium manifolds, Proc. Combust. Inst. 29 (2002), 1411–1417.CrossrefGoogle Scholar

  • [27]

    Q. Tang and S. B. Pope, A more accurate projection in the rate-controlled constrained equilibrium method for dimension reduction of combustion chemistry, Combust. Theory Model. 8 (2004), 255–279.CrossrefGoogle Scholar

  • [28]

    S. Rigopoulos and T. Løvås, A LOI-RCCE methodology for reducing chemical kinetics, with application to laminar premixed flames, Proc. Combust. Inst. 32 (2009), 569–576.CrossrefGoogle Scholar

  • [29]

    T. Løvås, S. Navarro-Martinez, and S. Rigopoulos, On adaptively reduced chemistry in large eddy simulations, Proc. Combust. Inst. 33 (2011), 1339–1346.CrossrefGoogle Scholar

  • [30]

    V. Hiremath and S. B. Pope, A study of the rate-controlled constrained-equilibrium dimension reduction method and its different implementations, Combust. Theory Model. 17 (2013), 260–293.CrossrefGoogle Scholar

  • [31]

    F. Hadi and M. R. H. Sheikhi, A comparison of constraint and constraint potential forms of the Rate-Controlled Constraint-Equilibrium method, J. Energy Resour. Technol. 138 (2015), 022202.Google Scholar

  • [32]

    F. Hadi, M. Janbozorgi, M. R. H. Sheikhi, and H. Metghalchi, A study of interactions between mixing and chemical reaction using the Rate-Controlled Constrained-Equilibrium method, J. Non-Equilib. Thermodyn. 41 (2016), 257–278.Google Scholar

  • [33]

    F. Hadi, V. Yousefian, M. R. H. Sheikhi, and H. Metghalchi, Time scale analysis for Rate-Controlled Constrained-Equilibrium constraint selection, in: Proceedings of the 10th U.S. National Combustion Meeting, Eastern States Section of the Combustion Institute, College Park, Maryland, April 23–26, 2017, 1–6.Google Scholar

  • [34]

    G. P. Beretta, M. Janbozorgi, and H. Metghalchi, Degree of Disequilibrium Analysis for Automatic Selection of Kinetic Constraints in the Rate-Controlled Constrained-Equilibrium Method, Combust. Flame 168 (2016), 342–364.CrossrefGoogle Scholar

  • [35]

    H. C. Ottinger, General projection operator formalism for the dynamics and thermodynamics of complex fluids, Phys. Rev. E 57 (2015), 1416–1420.Google Scholar

  • [36]

    G. P. Beretta, Steepest Entropy Ascent model for far-non-equilibrium thermodynamics. Unified implementation of the Maximum Entropy Production Principle, Phys. Rev. E 90 (2014), 042113.Google Scholar

  • [37]

    A. Montefusco, F. Consonni, and G. P. Beretta, Essential equivalence of the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) and steepest-entropy-ascent models of dissipation for nonequilibrium thermodynamics, Phys. Rev. E 91 (2015), 042138.Google Scholar

  • [38]

    S. Cano-Andrade, G. P. Beretta, and M. R. von Spakovsky, Steepest-entropy-ascent quantum thermodynamic modeling of decoherence in two different microscopic composite systems, Phys. Rev. A 91 (2015), 013848.Google Scholar

  • [39]

    G. Li and M. R. von Spakovsky, Steepest-entropy-ascent quantum thermodynamic modeling of the relaxation process of isolated chemically reactive systems using density of states and the concept of hypoequilibrium state, Phys. Rev. E 93 (2016), 012137.Google Scholar

  • [40]

    G. Lebon, D. Jou, and M. Grmela, Extended reversible and irreversible thermodynamics: A Hamiltonian approach with application to heat waves, J. Non-Equilib. Thermodyn. 42 (2017), 153–168.Google Scholar

  • [41]

    G. Li, M. R. von Spakovsky, and C. Hin, Steepest entropy ascent quantum thermodynamic model of electron and phonon transport, Phys. Rev. B 97 (2018), 024308.Google Scholar

  • [42]

    G. P. Beretta and E. P. Gyftopoulos, What is a chemical equilibrium state? J. Energy Resour. Technol. 137 (2015), 021008.Google Scholar

  • [43]

    G. P. Beretta and J. C. Keck, Energy and entropy balances in a combustion chamber. Analytical solution, Combust. Sci. Technol. 30 (1983), 19–29.CrossrefGoogle Scholar

  • [44]

    G. P. Beretta, M. Janbozorgi, and H. Metghalchi, Use of degree of disequilibrium analysis to select kinetic constraints for the Rate-Controlled Constrained-Equilibrium (RCCE) method, in: Proceedings of ECOS 2015 – The 28th International Conference On Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems, Pau, France, June 30–July 3, 2015. Available online at www.gianpaoloberetta.info/.Google Scholar

  • [45]

    L. Rivadossi and G. P. Beretta, Validation of the ASVDADD constraint selection algorithm for effective RCCE modeling of natural gas ignition in air, in: Proceedings of IMECE2016 – the ASME 2016 International Mechanical Engineering Congress and Exposition, November 11–17, 2016, Phoenix, Arizona, USA – paper IMECE2016-65323. Available online at www.gianpaoloberetta.info/ https://doi.org/10.1115/IMECE2016-65323.Google Scholar

  • [46]

    G. P. Beretta, Nonlinear quantum evolution equations to model irreversible adiabatic relaxation with maximal entropy production and other nonunitary processes, Rep. Math. Phys. 64 (2009), 139–168.CrossrefGoogle Scholar

  • [47]

    M. Valorani, D. A. Goussis, F. Creta, and H. N. Najm, Higher order corrections in the approximation of inertial manifolds and the construction of simplified problems with the CSP method, J. Comput. Phys. 209 (2005), 754–786.CrossrefGoogle Scholar

  • [48]

    D. Lebiedz, V. Reinhardt, and J. Siehr, Minimal curvature trajectories: Riemannian geometry concepts for slow manifold computation in chemical kinetics, J. Comp. Physiol. 229 (2010), 6512–6533.CrossrefGoogle Scholar

  • [49]

    D. Lebiedz, J. Siehr, and J. Unger, A variational principle for computing slow invariant manifolds in dissipative dynamical systems, SIAM J. Sci. Comput. 33 (2011), 703–720.CrossrefGoogle Scholar

  • [50]

    C. D. Martin and M. A. Porter, The extraordinary SVD, Am. Math. Mon. 119 (2012), 838–851.CrossrefGoogle Scholar

  • [51]

    M. Janbozorgi and H. Metghalchi, Rate-Controlled Constrained-Equilibrium Modeling of H-O Reacting Nozzle Flow, J. Propuls. Power 28 (2012), 677–684.CrossrefGoogle Scholar

  • [52]

    L. Rivadossi and G. P. Beretta, Validation of the ASVDADD constraint selection algorithm for effective RCCE modeling of natural gas ignition in air, J. Energy Resour. Technol. 140 (2018), 052201.Google Scholar

  • [53]

    P. D. Kourdis, R. Steuer, and D. A. Goussis, Physical understanding of complex multiscale biochemical models via algorithmic simplification: Glycolysis in saccharomyces cerevisiae, Physica D 239 (2010), 1798–1817.CrossrefGoogle Scholar

  • [54]

    V. Damioli, G. P. Beretta, A. Salvadori, C. Ravelli, and S. Mitola, Multi-physics interactions drive VEGFR2 relocation on endothelial cells, Scientific Reports 7 (2017), 16700.Google Scholar

  • [55]

    E. A. Piana, S. Uberti, A. Copeta, B. Motyl, and G. Baronio, An integrated acoustic–mechanical development method for off-road motorcycle silencers: from design to sound quality test, Int. J. Interact. Des. Manuf. (2018). https://doi.org/10.1007/s12008-018-0464-x.Google Scholar

  • [56]

    E. A. Piana, B. Grassi, F. Bianchi, and C. Pedrotti, Hydraulic balancing strategies: A case-study of radiator-based central heatig systems, Building Serv. Eng. Res. Technol. (2018). https://doi.org/10.1177/0143624417752830.Google Scholar

  • [57]

    G. P. Beretta, Modeling non-equilibrium dynamics of a discrete probability distribution: General rate equation for maximal entropy generation in a maximum-entropy landscape with time-dependent constraints, Entropy 10 (2008), 160–182.CrossrefGoogle Scholar

  • [58]

    L. M. Martyushev and V. D. Seleznev, Maximum entropy production: application to crystal growth and chemical kinetics, Current Opinion in Chemical Engineering 7 (2015), 23–31.CrossrefGoogle Scholar

About the article

Received: 2017-09-28

Revised: 2018-02-08

Accepted: 2018-02-26

Published Online: 2018-03-17

Published in Print: 2018-04-25


Citation Information: Journal of Non-Equilibrium Thermodynamics, Volume 43, Issue 2, Pages 121–130, ISSN (Online) 1437-4358, ISSN (Print) 0340-0204, DOI: https://doi.org/10.1515/jnet-2017-0055.

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