(672d) Parameter Estimation for Phase Equilibrium Problems Via Bilevel Programs

In this talk a formulation for parameter estimation with activity coefficient models is proposed based on bilevel programs with nonconvex lower-level programs. The upper-level program is the minimization of prediction error. The lower-level programs represent the thermodynamic stability requirement. Case studies from liquid-liquid and vapor-liquid equilibria are given. Special emphasis is placed on formulation and solution methodology.

Good activity coefficient models are necessary for the reliable design of separation systems. There is a variety of activity coefficient models, such as the modified Wilson, NRTL and UNIQUAC models [1]. These models contain adjustable parameters and, provided that suitable parameter values are used, the models can accurately predict the phase behavior of virtually any relevant system. No direct method exists for the calculation of the parameters, and therefore these must be estimated based on measured phase-splits. In separations design temperature is not fixed, and therefore a temperature dependence for the parameters is needed, based on some postulated function, e.g., a linear or second order polynomial. The actual parameter estimation is to find the coefficients of this polynomial.

Current methods for this parameter estimation have several limitations. One limitation is that the isoactivity method can give unstable predictions with a ``perfect fit''. In other words, current methods can furnish parameter values for which the measured phase split is a local minimum of the Gibbs free energy, but not the global one, which is not physical. There are methods to check for this case in retrospect, but when stability is not satisfied, these methods don't offer a way of proceeding. A second limitation is that due to their flexibility the activity coefficient models can predict more phases than actually exist in the system [2,3,4]. Clearly, this case must also be excluded. There are methods to detect this situation a posteriori, but not a generic reliable method of excluding it a priori. Third, typically parameters are estimated at a fixed temperature and then the temperature coefficients are estimated in a second calculation; this is tedious and not guaranteed to give a good fit. Finally, since local optimization methods are used, it cannot be guaranteed that the best possible fit was obtained, which is particularly important in the case of model-experiment mismatch.

In this presentation parameter estimation for phase equilibrium problems is cast as a bilevel optimization problem. Bilevel programs are programs where one optimization problem (upper-level program) is constrained by the (global) solutions of another optimization problem (lower-level program). A major challenge in bilevel programs is nonconvexity in the lower-level program and only recently methods have been proposed to deal with this case rigorously [5]. In the proposed formulation, the upper-level program minimizes the error in predictions by adjusting the model parameters; typically a least squares error or absolute error objective function is used here. The lower-level programs represent the stability requirement, i.e., that the predictions must be the global minimum of the Gibbs free energy for the chosen parameter values; for computational reasons specialized stability criteria [6,7] are used instead. The lower-level programs are nonconvex and their global solution is necessary. The talk first motivates the formulations and then gives a formal statement. The solution method based on [5] is detailed and then case studies are presented. These examples chosen are known to be challenging. It is shown that existing methods can lead to spurious solutions that are avoided by the proposed approach.

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[2] R. A. Heidemann and J. M. Mandhane. Some properties of the NRTL equation in correlating liquid-liquid equilibrium data. Chemical Engineering Science, 28(5):1213--1221, 1973.

[3] A. C. Mattelin and L. A. Verhoeye. Correlation of binary miscibility data by means of NRTL equation. Chemical Engineering Science, 30(2):193--200, 1975.

[4] D. Tassios. Number of roots in NRTL and LEMF equations and effect on their performance. Industrial & Engineering Chemistry Process Design and Development, 18(1):182--186, 1979.

[5] A., P. Lemonidis, and P. I. Barton. Global solution of bilevel programs with a nonconvex inner program. In press: Journal of Global Optimization, October 12, 2007.

[6] L. E. Baker, A. C. Pierce, and K. D. Luks. Gibbs energy analysis of phase equilibria. Soc. Petrol. Engrs. J., 22:731--742, 1982.

[7] A. Mitsos and P. I. Barton. A dual extremum principle in thermodynamics. AIChE Journal, 53(8):2131--2147, 2007.