Global Optimization for the Chemical and Phase Equilibrium Problem using Interval Analysis

  • K. I. M. Mckinnon
  • C. Millar
  • M. Mongeau
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 7)


This paper addresses the problem of minimizing the Gibbs free energy in the n c -component, multi-phase chemical and phase equilibrium problem involving different thermodynamic models. The algorithmic approach used is based on the tangent-plane criterion of Gibbs: the global optimization problem considered, which involves a search space of n(n + 1) dimensions, is reduced to a finite sequence of global optimization steps in n-space, and local optimization steps in nK-space, Kn + 1.

We describe an algorithm performing the global optimization step involved in this lower-dimensional search space using techniques from interval analysis. We report good numerical results on instances of the Gibbs free energy minimization problem.


Global optimization interval analysis tangent-plane criterion Gibbs free energy chemical and phase equilibrium non-convex optimization 


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  1. 1.
    L. Baker, A. Pierce, and K. Luks. Gibbs energy analysis of phase equilibria. Society of Petroleum Engineers Journal, 22: 731–742, 1982.CrossRefGoogle Scholar
  2. 2.
    J. Gibbs. Graphical methods in thermodynamics of fluids. Trans. Connecticut Acad, 2: 311, 1873.Google Scholar
  3. 3.
    J. Gibbs. A method of geometrical representation of the thermodynamic properties of substances by means of surfaces. Trans. Connecticut Acad, 2: 382, 1873.Google Scholar
  4. 4.
    E. Hansen. Global Optimization Using Interval Analysis, volume 165 of Pure and Applied Mathematics. Marcel Dekker, New York, 1992.Google Scholar
  5. 5.
    Y. Jiang, W. Smith, and G. Chapman. Global optimality conditions and their geometric interpretation for the chemical and phase equilibrium problem. SIAM Journal on Optimization, 1995. To appear.Google Scholar
  6. 6.
    C. McDonald and C. Floudas. GLOPEQ: A new computational tool for the phase and chemical equilibrium problem. 1994. Submitted.Google Scholar
  7. 7.
    C. McDonald and C. Floudas. Global optimization for the phase and chemical equilibrium problem: Application to the NRTL equation. Computers (amp) Chemical Engineering, 1995. To appear.Google Scholar
  8. 8.
    K. I. M. McKinnon and M. Mongeau. A generic global optimization algorithm for the chemical and phase equilibrium. Technical Report MS94/1, Dept. of Mathematics and Statistics, University of Edinburgh, U.K., 1994.Google Scholar
  9. 9.
    M. L. Michelsen. The isothermal flash problem. Part I. Stability. Fluid Phase Equilibria, 9: 1–19, 1982.CrossRefGoogle Scholar
  10. 10.
    M. L. Michelsen. The isothermal flash problem. Part II. Phase-split calculation. Fluid Phase Equilibria, 9: 21–40, 1982.CrossRefGoogle Scholar
  11. 11.
    H. Ratschek and J. Rokne. New Computer Methods for Global Optimization. Wiley, New York, 1982.Google Scholar
  12. 12.
    W. Smith and R. Missen. Chemical Reaction Equilibrium Analysis: Theory and Algorithms. Wiley & Sons, 1982.Google Scholar
  13. 13.
    J. A. Trangenstein. Customized minimization techniques for phase equilibrium computations in reservoir simulation. Chemical Engineering Science, 42 (12): 2847–2863, 1987.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • K. I. M. Mckinnon
    • 1
    • 2
  • C. Millar
    • 1
    • 2
  • M. Mongeau
    • 1
    • 2
  1. 1.Dept of Mathematics and StatisticsUniversity of EdinburghUK
  2. 2.Labo Approximation & OptimisationUniversité Paul SabatierToulouse cedexFrance

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