Reliable Computing

, Volume 12, Issue 6, pp 427–450 | Cite as

Reliable Modeling and Optimization for Chemical Engineering Applications: Interval Analysis Approach

  • Youdong Lin
  • C. Ryan Gwaltney
  • Mark A. Stadtherr
Article

Abstract

In many applications of interest in chemical engineering it is necessary to deal with nonlinear models of complex physical phenomena, on scales ranging from the macroscopic to the molecular. Frequently these are problems that require solving a nonlinear equation system and/or finding the global optimum of a nonconvex function. Thus, the reliability with which these computations can be done is often an important issue. Interval analysis provides tools with which these reliability issues can be addressed, allowing such problems to be solved with complete certainty. This paper will focus on three types of applications: 1) parameter estimation in the modeling of phase equilibrium, including the implications of using locally vs. globally optimal parameters in subsequent computations; 2) nonlinear dynamics, in particular the location of equilibrium states and bifurcations of equilibria in ecosystem models used to assess the risk associated with the introduction of new chemicals int the environment; 3) molecular modeling, with focus on transition state analysis of the diffusion of a sorbate molecule in a zeolite.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adjiman, C. S., Androulakis, I. P., and Floudas, C. A.: A Global Optimization Method, Alpha-BB, for General Twice-Differentiable Constrained NLPs—II. Implementation and Com putational Results, Comput. Chem. Eng. 22 (1998), p. 1159.CrossRefGoogle Scholar
  2. 2.
    Adjiman, C. S., Dallwig, S., Floudas, C. A., and Neumaier, A.: A Global Optimization Method, Alpha-BB, for General Twice-Differentiable Constrained NLPs—I. Theoretical Advances, Com put, Chem. Eng.22 (1998), p. 1137.CrossRefGoogle Scholar
  3. 3.
    Baker, J.: An Algorithm for The Location of Transition-States, J. Comput. Chem. 7(1986), pp. 385–395.CrossRefGoogle Scholar
  4. 4.
    Bischof, C. H., Lang, B., Marquardt, W., and Monnigmann, M.: Verified Determination of Sin gularities in Chemical Processes, in: Proceedings SCAN 2000, 9th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics,Karlsruhe, 2000.Google Scholar
  5. 5.
    Brennecke, J. F. and Maginn, E. J.: Ionic Liquids: Innovative Fluids for Chemical Processing, AIChEJ.47 (2001), pp. 2384–2389.CrossRefGoogle Scholar
  6. 6.
    Burgos-Solorzano, G. I., Brennecke, J. F., and Stadtherr, M. A.: Validated Computing Approach for High-Pressure Chemical and Multiphase Equilibrium, Fluid Phase Equilib.219 (2004), pp. 245–255.CrossRefGoogle Scholar
  7. 7.
    Doedel, E. J., Champneys, A. R., Fairgrieve, T. F, Kuznetsov, Y. A., Sandstede, B., and Wang, X. J.: AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equa tions,Technical report, Department of Computer Science, Concordia Univeristy, Montreal, 1997.Google Scholar
  8. 8.
    Freemantle, M.: Meeting Briefs: Ionic Liquids Separated from Mixtures by CO2, Chem. Eng. News80 (36) (2002), pp. 44 5.Google Scholar
  9. 9.
    Gau, C.-Y., Brennecke, J. F., and Stadtherr, M. A.: Reliable Parameter Estimation in VLB Modeling, Fluid Phase Equilib.168 (2000), pp. 1–18.CrossRefGoogle Scholar
  10. 10.
    Gau, C.-Y. and Stadtherr, M. A.: Deterministic Global Optimization for Data Reconciliation and Parameter Estimation Using Error-in-Variables Approach, in: Luus, R. (ed.): Optimization and Optimal Control in Chemical Engineering,Research Signpost, Trivandrum, 2002.Google Scholar
  11. 11.
    Gau, C.-Y. and Stadtherr, M. A.: Deterministic Global Optimization for Error-in-Variables Param eter Estimation, AIChEJ.48 (2002), pp. 1191–1197.CrossRefGoogle Scholar
  12. 12.
    Gau, C.-Y. and Stadtherr, M. A.: New Interval Methodologies for Reliable Chemical Process Modeling, Comput. Chem. Eng.26 (2002), pp. 827–840.CrossRefGoogle Scholar
  13. 13.
    Gau, C.-Y. and Stadtherr, M. A.: Reliable Nonlinear Parameter Estimation Using Interval Analysis Error-in-Variable Approach, Comput. Chem. Eng.24 (2000), pp. 631–638.Google Scholar
  14. 14.
    Gerke, V and Marquardt, W.: A Singularity Theory Approach to the Study of Reactive Distillation, Comput. Chem. Eng.21 (1997), pp. S1001–S1006.Google Scholar
  15. 15.
    Gmehling, J., Onken, U., and Arlt, W.: Vapor-Liquid Equilibrium Data Collection, Chemistry Data Series, Vol. I, Parts 1–8,DECHEMA, Frankfurt/Main, 1977–1990.Google Scholar
  16. 16.
    Gragnani, A., De Feo, O., and Rinaldi, S.: Food Chains in the Chemostat: Relationships between Mean Yield and Complex Dynamics, B. Math. Biol.60 (4) (1998), pp. 703–719.Google Scholar
  17. 17.
    Gwaltney, C. R. and Stadtherr, M. A.: Reliable Computation of Equilibrium States and Bifurca tions in Nonlinear Dynamics, in: Proceedings PARA’04 Workshop on State-of-the-art in Scientific Computing,Lyngby, 2004.Google Scholar
  18. 18.
    Gwaltney, C. R., Styczynski, M. P., and Stadtherr, M. A.: Reliable Computation of Equilibrium States and Bifurcations in Food Chain Models, Comput. Chem. Eng.28 (2004), pp. 1981–1996.CrossRefGoogle Scholar
  19. 19.
    Hansen, E. R. and Walster, G. W.: Global Optimization Using Interval Analysis,Marcel Dekker, New York, 2004.Google Scholar
  20. 20.
    Hooke, R. and Jeeves, T. A.: Direct Search Solution of Numerical and Statistical Problems, J. Assoc. Comput. Mach.8 (1961), pp. 212–229.MATHGoogle Scholar
  21. 21.
    Hua, J. Z., Brennecke, J. F, and Stadtherr, M. A.: Enhanced Interval Analysis for Phase Stability Cubic Equation of State Models, Ind. Eng. Chem. Res.37 (1998), p. 1519.Google Scholar
  22. 22.
    Hua, J. Z., Brennecke, J. F., and Stadtherr, M. A.: Reliable Computation of Phase Stability Using Interval Analysis Cubic Equation of State Models, Comput. Chem. Eng.22 (1998), p. 1207.Google Scholar
  23. 23.
    Jastorff, B., Stormann, R., Ranke, J., Molter, K., Stock, F., and Oberheitmann, B.: How Haz ardous Are Ionic Liquids? Structure-Activity Relationships and Biological Testing as Important Elements for Sustainability Evaluation, Green Chemistry5 (2003), pp. 136–142.Google Scholar
  24. 24.
    Jaulin, L., Kieffer, M., Didrit, O., and Walter, E: Applied Interval Analysis,Springer-Verlag, London, 2001.Google Scholar
  25. 25.
    June, R. L., Bell, A. T., and Theodorou, D. N.: Transition-State Studies of Xenon and SF6Diffusion in Silicalite, J. Phys. Chem.95 (1991), pp. 8866–8878.CrossRefGoogle Scholar
  26. 26.
    Karger, J. and Ruthven, D. M.: Diffusion in Zeolites and Other Microporous Solids,Wiley, New York, 1992.Google Scholar
  27. 27.
    Kaupe, A. R: Algorithm 178 Direct Search, Commun,ACM 6 (1963), p. 313.Google Scholar
  28. 28.
    Kearfott, R. B.: Rigorous Global Search: Continuous Problems,Kluwer Academic Publishers, Dordrecht, 1996.Google Scholar
  29. 29.
    Kiselev, A. V., Lopatkin, A. A., and Shulga, A. A.: Molecular Statistical Calculation of Gas Adsorption by Silicalite, Zeolites5 (1985), pp. 1508–1516.CrossRefGoogle Scholar
  30. 30.
    Kuznetsov, Y. A.: Elements of Applied Bifurcation Theory,Springer-Verlag, New York, 1998.Google Scholar
  31. 31.
    Lin, Y. and Stadtherr, M. A.: Advances in Interval Methods for Deterministic Global Optimization in Chemical Engineering, J. Global Optim. 29 (2004), pp. 281–296.Google Scholar
  32. 32.
    Lin, Y. and Stadtherr, M. A.: Locating Stationary Points of Sorbate-Zeolite Potential Energy Surfaces Using Interval Analysis, J, Chem. Phys.121 (2004), pp. 10159–10166.Google Scholar
  33. 33.
    Lin, Y. and Stadtherr, M. A.: LP Strategy for Interval-Newton Method in Deterministic Global Optimization, Ind. Eng. Chem. Res. 43(2004), pp. 3741–3749.Google Scholar
  34. 34.
    Maier, R. W., Brennecke, J. P., and Stadtherr, M. A.: Reliable Computation of Homogeneous Azeotropes, AIChEJ. 44(1998), p. 1745.Google Scholar
  35. 35.
    Maier, R. W, Brennecke, J. P., and Stadtherr, M. A.: Reliable Computation of Reactive Azeotropes, Comput. Chem. Eng.24 (2000), pp. 1851–1858.CrossRefGoogle Scholar
  36. 36.
    Maier, R. W. and Stadtherr, M. A.: Reliable Density-Functional-Theory Calculations of Adsorp tion inNanoporous Materials, AIChEJ.47 (2001), pp. 1874–1884.Google Scholar
  37. 37.
    McKinnon, K. I. M., Millar, C. G., and Mongeau, M.: Global Optimization for the Chemical and Phase Equilibrium Problem Using Interval Analysis, in: Floudas, C. A. and Pardalos, P. M. (eds): State of the Art in Global Optimization Computational Methods and Applications,Kluwer Academic Publishers, Dordrecht, 1996.Google Scholar
  38. 38.
    Moghadas, S. M. and Gumel, A. B.: Dynamical and Numerical Analysis of a Generalized Food- chain Model, Appl. Math. Comput.142 (1) (2003), pp. 35 19.Google Scholar
  39. 39.
    Monnigmann, M. and Marquardt, W.: Normal Vectors on Manifolds of Critical Points for Para metric Robustness of Equilibrium Solutions of ODE Systems, J. Nonlinear Sci.12 (2002), pp. 85–112.MathSciNetCrossRefGoogle Scholar
  40. 40.
    Neumaier, A.: Interval Methods for Systems of Equations,Cambridge University Press, Cam bridge, 1990.Google Scholar
  41. 41.
    Olson, D. H., Kokotailo, G. T., Lawton, S. L., and Meier, W. M.: Crystal Structure and Structure- Related Properties of ZSM-5, J. Phys. Chem.85 (1981), pp. 2238–2243.CrossRefGoogle Scholar
  42. 42.
    Rohn, J. and Kreinovich, V: Computing Exact Componentwise Bounds on Solution of Linear Systems with Interval Data Is NP-Hard, SIAMJ. Matrix. Anal.16 (1995), pp. 415 120.Google Scholar
  43. 43.
    Schnepper, C. A. and Stadtherr, M. A.: Robust Process Simulation Using Interval Methods, Comput. Chem. Eng.20 (1996), p. 187.CrossRefGoogle Scholar
  44. 44.
    Scurto, A. M., Xu, G., Brennecke, J. R, and Stadtherr, M. A.: Phase Behavior and Reliable Computation of High-Pressure Solid-Fluid Equilibrium with Cosolvents, Ind. Eng. Chem. Res.42 (2003), pp. 6464–6475.Google Scholar
  45. 45.
    Siirola, J. D., Hauen, S., and Westerberg, A. W.: Agent-Based Strategies for Multiobjective Optimization,Paper 265g, AIChE Annual Meeting, Indianapolis, 2002.Google Scholar
  46. 46.
    Stadtherr, M. A., Schnepper, C. A., and Brennecke, J. F.: Robust Phase Stability Analysis Using Interval Methods, AIChE Symp. Ser.91 (304) (1995), p. 356.Google Scholar
  47. 47.
    Stradi, B. A., Brennecke, J. P., Kohn, J. P., and Stadtherr, M. A.: Reliable Computation of Mixture Critical Points, AIChE J.47 (2001), pp. 212–221.CrossRefGoogle Scholar
  48. 48.
    Stradi, B. A., Xu, G., Brennecke, J. P., and Stadtherr, M. A.: Modeling and Design of an Environmentally Benign Reaction Process, AIChE Symp. Ser. 96(323) (2000), pp. 371–375.Google Scholar
  49. 49.
    Tessier, S. R., Brennecke, J. R, and Stadtherr, M. A.: Reliable Phase Stability Analysis for Excess Gibbs Energy Models, Chem. Eng. Sci.55 (2000), p. 1785.Google Scholar
  50. 50.
    Trefethen, N.: A Hundred-Dollar Hundred-Digit Challenge, SIAMNews35 (2002), p. 1.Google Scholar
  51. 51.
    Tsai, C. J. and Jordan, K. D.: Use of An Eigenmode Method to Locate The Stationary-Points on The Potential-Energy Surfaces of Selected Argon And Water Clusters, J, Phys. Chem.97 (1993), pp. 11227–11237.CrossRefGoogle Scholar
  52. 52.
    Ulas, S., Diwekar, U. M., and Stadtherr, M. A.: Uncertainties in Parameter Estimation and Optimal Control in Batch Distillation, Comput, Chem, Eng, 29(2005), pp. 1805–1814.CrossRefGoogle Scholar
  53. 53.
    Westerberg, K. M. and Ploudas, C. A.: Locating All Transition States and Studying the Reaction Pathways of Potential Energy Surfaces, J. Chem. Phys. 110(1999), pp. 9259–9295.CrossRefGoogle Scholar
  54. 54.
    Xu, G., Brennecke, J. P., and Stadtherr, M. A.: Reliable Computation of Phase Stability and Equilibrium from the SAPT Equation of State, Ind, Eng, Chem, Res, 41(2002), pp. 938–952.CrossRefGoogle Scholar
  55. 55.
    Xu, G., Scurto, A. M., Castier, M., Brennecke, J. P., and Stadtherr, M. A.: Reliable Computation of High Pressure Solid-Fluid Equilibrium, Ind. Eng. Chem. Res.39 (2000), pp. 1624–1636.Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  • Youdong Lin
    • 1
  • C. Ryan Gwaltney
    • 1
  • Mark A. Stadtherr
    • 1
  1. 1.Department of Chemical and Biomolecular EngineeringUniversity of Notre DameNotre DameUSA

Personalised recommendations