Skip to main content
Article thumbnail
Location of Repository

Slow Invariant Manifold and its approximations in kinetics of catalytic reactions

By Muhammad Shahzad


Equations of chemical kinetics typically include several distinct time scales. There exist many methods which allow to exclude fast variables and reduce equations to the slow manifold. In this thesis, we start by studying the background of the quasi equilibrium approximation, main approaches to this approximation, its consequences and other related topics.\ud We present the general formalism of the quasi equilibrium (QE) approximation with the proof of the persistence of entropy production in the QE approximation. We demonstrate how to apply this formalism to chemical kinetics and describe the difference between QE and quasi steady state (QSS) approximations. In 1913 Michaelis and Menten used the QE assumption that all intermediate complexes are in fast equilibrium with free substrates and enzymes. Similar approach was developed by Stuekelberg (1952) for the Boltzmann kinetics. Following them, we combine the QE (fast equilibria) and the QSS (small amounts) approaches and study the general kinetics with fast intermediates present in small amounts. We prove the representation of the rate of an elementary reaction as a product of the Boltzmann factor (purely thermodynamic) and the kinetic factor, and find the basic relations between kinetic factors. In the practice of modeling, a kinetic model may initially not respect thermodynamic conditions. For these cases, we solved a problem: is it possible to deform (linearly) the entropy and provide agreement with the given kinetic model and deformed thermodynamics ?\ud We demonstrate how to modify the QE approximation for stiffness removal in an example of the CO oxidation on Pt. QSSA was applied in order to get an approximation to the One dimensional Invariant Grid for oxidation of CO over Pt. The method of intrinsic low dimension manifold (ILDM) was implemented over the same example (CO oxidation on Pt) in order to automate the process of reduction and provide more accurate simplified mechanism (for one-dimension), yet at the cost of a significantly more complicated implementation

Publisher: University of Leicester
Year: 2011
OAI identifier:

Suggested articles


  1. A general analysis of exact lumping in chemical kinetics. doi
  2. A lumping analysis in monomolecular reaction systems: Analysis of the exactly lumpable system, doi
  3. A lumping analysis in monomolecular reaction systems. Analysis of the approximately lumpable system. doi
  4. A note on the kinetics of enzyme action, doi
  5. (2006). An efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems, doi
  6. Asymptotic behaviour of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives. doi
  7. Asymptotology of chemical reaction networks, doi
  8. Chaos Near Resonance Springer-Verlag doi
  9. (2006). Combustion: Physical and Chemical Fundamentals, Modeling and Simulation, Experiments, Pollutant Formation. Hardcover,
  10. (1972). Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, doi
  11. (2004). Constructive methods of invariant manifolds for kinetic problems, doi
  12. (2001). Corrections and enhancements of quasi–equilibrium states, doi
  13. Description of non-isothermal reactions in terms of Marcelin-De-Donder Kinetics and its generalizations, doi
  14. Development of Non-stiff Reduced Mechanisms for Direct Numerical Simulations, doi
  15. Die Kinetik der Intervintwirkung, doi
  16. (1986). Dissipation in many–body systems: A geometric approach based on information theory, doi
  17. Dynamic and static limitation in reaction networks, revisited, doi
  18. Eine Theorie der Photochemischen Reaktionsgeschwindigkeiten,
  19. Entropy: The Markov Ordering Approach. doi
  20. (1984). Equilibrium encircling. Equations of chemical kinetics and their thermodynamic analysis,
  21. (1986). Essays on chemical relaxation, doi
  22. G.W.Elementary Principles in Statistical Mechanics, doi
  23. (1972). General mass action kinetics, Archive for Rational Mechanics and Analysis, doi
  24. Gorban 2 and Iliya V. Karlin 1, Comparison of Invariant Manifolds for Model Reduction in Chemical Kinetics(2007). doi
  25. Information theory and statistical mechanics, in: Statistical Physics. Brandeis Lectures, doi
  26. (1965). Introduction to the Analysis of Chemical Reactors, doi
  27. (2004). Invariant grids for reaction kinetics, doi
  28. (2009). Invariant Manifolds and Lattice Boltzmann Method for Combustion, doi
  29. (2005). Invariant manifolds for physical and chemical kinetics, doi
  30. Invariant manifolds for physical and chemical kinetics, volume 660 of Lect. Notes Phys. doi
  31. (1991). Kinetic models of catalytic reactions. doi
  32. Lumping strategy. 2. System theoretic approach, doi
  33. (2003). Method of invariant manifold for chemical kinetics, doi
  34. (1973). Nonlinear irreversible processes, doi
  35. (1984). ODE methods for the solution of differential/algebraic systems, doi
  36. (2006). On a modified version of ILDM approach: Asymptotic analysis based on integral manifolds, doi
  37. (1972). On chemical kinetics of a certain class, Archive for Rational Mechanics and Analysis 46, doi
  38. (1945). On Onsager’s principle of microscopic reversibility, doi
  39. On the Kinetics of Complex Reactions, doi
  40. Optimal component lumping: Problem formulation and solution techniques, doi
  41. Partitioning techniques and lumping computation for reducing chemical kinetics. APLA: An automatic partitioning and lumping algorithm, doi
  42. (2008). Quasi-Steady-State Laws in Enzyme Kinetics, doi
  43. Reciprocal relations in irreversible processes. doi
  44. Removing the stiffness from interfacial flow with surface-tension, doi
  45. Russo Sterling Winthrop Inc.,
  46. (1992). Simplifying chemical kinetics: Intrinsic lowdimensional manifolds in composition space, doi
  47. Simplifying Principles for Chemical and Enzyme Reaction Kinetics, doi
  48. (1986). Singular perturbation theory for open enzyme reaction networks, doi
  49. Singular perturbations on the infinite interval. doi
  50. (2010). Solving Ordinary Differential Equations II: doi
  51. (2004). Spanning trees and optimization problems, doi
  52. Systematic approach to elucidation of multistep reaction networks, doi
  53. Systems of differential equations containing small parameters multiplying some of the derivatives.
  54. (1994). The CSP method for simplifying kinetics, doi
  55. The Effect of Lumping and Expanding on Kinetic Differential Equations, doi
  56. (1953). The Elucidation of Reaction Mechanisms by the Method of doi
  57. The quasi-steady-state assumption: A case study in perturbation. doi
  58. (1971). The simultaneous numerical solution of differential-algebraic equations, doi
  59. The structure and analysis of complex reaction systems. doi
  60. (1952). Theoreme H et unitarite de
  61. (1992). Thermodynamic parametrization, doi
  62. Thermodynamics and an Introduction to Themostatistics (2nd ed.). doi
  63. Thermodynamics of driven systems, doi
  64. (1865). Über vershiedene für die Anwendungen bequeme Formen der Hauptgleichungen der Wärmetheorie. doi
  65. (2004). Uniqueness of thermodynamic projector and kinetic basis of molecular individualism, doi
  66. (1991). V.I.Bykov, A.N.Gorban, V.I.Elohin, Kinetic Models of Catalytic Reactions, Elsevier, R.G. Compton (Ed.) Series "Comprehensive Chemical Kinetics", doi
  67. (2003). Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitzungsberichte der keiserlichen Akademie der Wissenschaften (1872), 66, 275–370. Translation: Further studies on the thermal equilibrium of gas molecules,

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.