Interval Mathematics Applied to Critical Point Transitions

• Stradi, Benito A. ^[1]
  1. [1] Institute of Technology of Costa Rica, Department of Materials Science and Engineering
• Localización: Revista de Matemática: Teoría y Aplicaciones, ISSN 2215-3373, ISSN-e 2215-3373, Vol. 12, Nº. 1-2, 2005, págs. 29-44
• Idioma: inglés
• DOI: 10.15517/rmta.v12i1-2.248
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    La determinación de puntos críticos de mezclas es importante tanto por razones prácticas como teóricas en el modelamiento del comportamiento de fases, especialmente a presiones altas. Las ecuaciones que describen el comportamiento de mezclas complejas cerca del punto crítico son significativamente no lineales y con multiplicidad de soluciones para las ecuaciones del punto crítico. Aritmética de intervalos puede ser usada para localizar con confianza todos los puntos críticos de una mezcla dada. El método también verifica la no–existencia de un punto crítico si una mezcla de composición dada no tiene dicho punto. Este estudio usa un algoritmo denominado Newton–Intervalo/Bisección–Generalizada que provee una garantía matemática y computacional de que todos los puntos críticos de una mezcla han sido localizados.Estos problemas cubren los modelos de ecuaciones cúbicas de estado; sin embargo, la técnica es de propósito general y puede ser aplicada en el caso de otros problemas no lineales.

  • English

    The determination of critical points of mixtures is important for both practical and theoretical reasons in the modeling of phase behavior, especially at high pressure. The equations that describe the behavior of complex mixtures near critical points are highly nonlinear and with multiplicity of solutions to the critical point equations. Interval arithmetic can be used to reliably locate all the critical points of a given mixture. The method also verifies the nonexistence of a critical point if a mixture of a given composition does not have one. This study uses an interval Newton/Generalized Bisection algorithm that provides a mathematical and computational guarantee that all mixture critical points are located. The technique is illustrated using several example problems. These problems involve cubic equation of state models; however, the technique is general purpose and can be applied in connection with other nonlinear problems.

• Referencias bibliográficas
  • Baker, L. E.; Kraemer, D. L. (1980) “Critical points and saturation pressure calculations for multicomponent systems”, Soc. Pet. Eng. J. February:...
  • Boberg, T. C.; White, R. R. (1962) “Prediction of critical mixtures”, Ind. Eng. Chem. Fund. 1(1): 40–44.
  • Boshkov, L. Z.; Yelash, L. V. (1997) “Closed-loops of liquid-liquid immiscibility in binary mixtures predicted from the Redlich-Kwong equation...
  • Debenedetti, P. G. (1996) Metastable Liquids: Concepts and Principles. Princeton University Press, New Jersey.
  • Deiters, U.; Schneider, G. M. (1976) “Fluid mixtures at high pressures. computer calculations of the phase equilibria and the critical phenomena...
  • Enick, R.; Holder, G. D.; Morsi, B. I. (1985) “Critical and three phase behavior in the carbon dioxide/tridecane system”, Fluid Phase Equil....
  • Grieves, R. B.; Thodos, G. (1962) “The critical temperature of ternary hydrocarbon systems”, Ind. Eng. Chem. Fundam. 1(1): 45–48.
  • Hansen, E. (1992) Global Optimization using Interval Analysis. Marcel-Dekker, New York.
  • Heidemann, R. A. (1975) “The criteria for thermodynamic stability”, AIChE J. 21(4): 824–826.
  • Heidemann, R. A. (1983) “Computation of high pressure phase equilibria.”, Fluid Phase Equil. 14: 55–78.
  • Heidemann, R. A.; Khalil, A. M. (1980) “The calculation of critical points”, AICHE J. 5: 769–779.
  • Hicks, C. P.; Young, C. L. (1977) “Theoretical prediction of phase behavior at high temperatures and pressures for non-polar mixtures”, J....
  • Hua, J. Z.; Brennecke, J.; Stadtherr, M. A. (1998) “Enhanced interval analysis for phase stability: Cubic equation of state models”, Ind....
  • Hua, J. Z.; Brennecke, J. F.; Stadtherr, M. A. (1996) “Reliable phase stability analysis for cubic equation of state models”, Computer Chem....
  • Hurle, R. L.; Jones, F.; Young, C. L. (1977) “Theoretical prediction of phase behavior at high temperatures and pressures for non-polar mixtures.”,...
  • Hurle, R. L.; Toczylkin, L.; Young, C. L. (1977) “Theoretical prediction of phase behavior at high temperatures and pressures for non-polar...
  • Hytoft, G.; Gani, R. (1996) IVC-SEP Program Package. Danmarks Tekniske Universitet, Lyngby, Denmark.
  • Imre, A.; Martinas, K.; Rebelo, L. P. N. (1998) “Thermodynamics of negative pressures in liquids”, J. Non-Equilib. Thermodyn. 23: 351–375.
  • Kearfott, R. B. (1987) “Some tests of generalized bisection”, ACM Transactions on Mathematical Software 13(3): 197–220.
  • Kearfott, R. B. (1989) “Interval arithmetic methods for non-linear systems and non-linear optimization: An outline and status”, in: Sharda,...
  • Kearfott, R. B. (1990) “Interval arithmetic techniques in the computational solution of non-linear systems of equations: Introduction, examples,...
  • Kearfott, R. B. (1996) Rigorous Global Search: Continuous Problem. Kluwer Academic Publishers, Dordrecht, The Netherlands.
  • Kearfott, R. B.; Novoa, M. (1990) “Algorithm 681: Intbis. a portable newton/bisection package”, ACM Trans. Math. Software 16: 152–157.
  • Knuppel, O. (1994) “PROFIL/BIAS- A Fast Interval Library”, Computing 53: 277–287.
  • Luks, K. D.; Turek, E. A.; Baker, L. E. (1987) “Calculation of minimum miscibility pressure”, SPE Res. Eng. J. November: 501–506.
  • Michelsen, M. L. (1982) “The isothermal flash problem. Part II, Phase split calculation”, Fluid Phase Equil. 9: 21–40.
  • Michelsen, M. L. (1984) “Calculation of critical points and phase boundaries in the critical region”, Fluid Phase Equil. 16: 57–76.
  • Michelsen, M. L.; Heidemann, R. A. (1981) “Calculation of critical points from cubic two-constant equations of state”, AIChE J. 3(27): 521–523.
  • Modell, M.; Reid, R. C. (1983) Thermodynamics and its Applications. Prentice-Hall, Englewood Cliff, New Jersey.
  • Moore, R. E. (1966) Interval Analysis. Prentice-Hall, Englewood Cliffs, New Jersey.
  • Munoz, F.; Chimowitz, E. H. (1993) “Critical phenomena in mixtures. I. Thermodynamic theory for the binary critical azeotrope”, J. Chem. Phys....
  • Nagarajan, N. R.; Cullick, A. S.; Griewank, A. (1991) “New strategy for phase equilibrium and critical point calculations by thermodynamic...
  • Nagarajan, N. R.; Cullick, A. S.; Griewank, A. (1991) “New strategy for phase equilibrium and critical point calculations by thermodynamic...
  • Neaumaier, A. (1990) Interval methods for systems of equations. Cambridge University Press, Cambridge.
  • Reid, J. M. P. R. C.; Poling, B. E. (1987) The Properties of Gases and Liquids. McGraw-Hill, New York.
  • Reid, R. C.; Beegle, B. L. (1977) “Critical point criteria in Legendre transform notation”, AIChE J. 5: 726–731.
  • Rochocz, G. L.; Castier, M.; Sandler, S. I. (1997) “Critical point calculations for semi-continuos mixtures”, Fluid Phase Equilibria 139:...
  • Rowlinson, J. S.; Swinton, F. L. (1982) Liquid and Liquid Mixtures. Butterworth Scientific, London.
  • Sadus, R. J. (1994) “Calculating critical transitions of fluid mixtures: Theory vs experiment”, AIChE J. 40 (8): 1376–1403.
  • Schnepper, C. A.; Stadtherr, M. A. (1996) “Robust process simulation using interval methods”, Computers Chem. Engng. 20 (2): 187–199.
  • Scott, R. L.; van Konynenburg, P. H. (1970) “Static properties of solutions”, Discussions of the Faraday Society 49: 87–97.
  • Spear, R. R.; Robinson, R. L.; Chao, K.-C. (1971) “Critical states of ternary mixtures and equations of state”, Ind. Eng. Chem. Fundam. 10...
  • Stockfleth, R.; Dohrn, R. (1998) “An algorithm for calculating critical point in multicomponent mixtures which can be easily implemented in...
  • Stradi, B. A. (2000) Measurement and Modeling of the Phase Behavior of High-Pressure Reaction Mixtures and the Computation of Mixture Critical...
  • Teja, A. S.; Kropholler, H. W. (1975) “Critical states fo mixtures in which azeotropic behavior persists in the critical region”, Chem. Engng....
  • Teja, A. S.; Rowlinson, J. S. (1973) “The prediction of the thermodynamic properties of fluids and fluid mixtures-IV. Critical and azeotropic...
  • van Konynenburg, P. H. (1968) Critical Lines and Phase Equilibria in Binary Mixtures. PhD thesis, University of California at Los Angeles,...