It is notable that thermodynamic models are nowadays being used for the modeling of very complex phase behaviors at extreme conditions. For instance, in the modern thermochemical cycles being proposed for efficient water splitting into oxygen and hydrogen, thermodynamic models are being applied well outside the region of conditions used for the fitting of their parameters [1]. The flexibility of local composition models comes from their capability to predict excess Gibbs free energy surfaces with many inflection points, which in thermodynamics is translated into multiple stable or meta-stable phase splits. Furthermore, the dependence of the model parameters on temperature makes parameter estimation a difficult optimization task and the final model predictions, in terms of phase stability, very uncertain. Correspondingly, the questions that need to be addressed are: a) what are the implications of not using a phase stability test when estimating the parameters of these models using phase equilibrium data, and b) is there a way of implementing a phase stability test a priori in the parameter estimation strategy?
Taking as an example the NRTL model, many studies can be found in the literature where the estimation of parameters using the so-called iso-activity method results in model predictions for the number of liquid phases considerably different from those experimentally measured. Heidemann and Mandhane [2] report sets of binary interaction parameters for the NRTL model, with which they show that the model satisfies the iso-activity criterion, but the predicted stable phase split is considerably different from the measured LLE. Mattelin and Verhoeye [3] and Tassios [4] show that the number of roots of the iso-activity equations, using the NRTL model, is higher than one, which is the reason for most of the problems. By using a polynomial expression for the temperature dependence of the NRTL binary parameters, Segura et al. [5] show that without a phase stability test the estimation of the temperature dependent parameters results in the prediction of stable liquid splits different from those used for the parameter estimation. For binary systems Sørensen and Arlt [6], the authors of the DECHEMA collection of LLE data, address this problem by plotting the Gibbs free energy function vs. the liquid composition for each one of the systems examined and discarding parameter values that lead to more inflection points than absolutely necessary for the modeling of the phase split. However, it is unclear what one should do if during the fitting of the model parameters a solution is found that results in more complex phase behaviors than those observed. Moreover, there are systems for which the best fit is achieved if more inflection points are allowed to exist, yet only one stable phase split is predicted by the model. Simoni et al. [7] present an interval Newton algorithm that can be used to find all the possible solutions for the binary parameters that satisfy a measured liquid split at a specified temperature. However, the modeling of phase equilibria at a specified temperature is not of much use for real engineering problems, where temperature is a design and operating variable of the process and can and will vary during process operation. The inclusion of the dependence of the model parameters on temperature results in an optimization problem, to which there is not a solution but “best fit” values.
The difficulties in fitting parameters of local composition models are not limited to the NRTL model or to LLE problems. Maier et al. [8] show that even when using the Wilson model for the modeling of VLE of a simple binary system, one could get local solutions for the parameter estimation objective function, which results in a failure to predict an azeotrope, a crucial issue in the modeling and design of separation processes. Xu et al. [9] present cases in which inappropriate fitting of the NRTL parameters results in the prediction of a homogeneous azeotrope as heterogeneous, which is also a crucial difference for process design and modeling. Many examples can be found in the DECHEMA VLE data collection [10] where the interaction parameters proposed predict homogeneous azeotropes as heterogeneous, or the prediction of heterogeneous azeotropes is not at all in agreement with the LLE data for the systems studied.
It is, therefore, imperative to formulate the parameter estimation strategy for phase equilibrium problems in such a way that phase stability is used as a constraint in the formulation. This can be done by formulating the parameter estimation as a bilevel program, in which the upper-level program corresponds to the minimization of the discrepancy between predicted and measured properties, whereas the lower-level program is a classical Gibbs free energy minimization problem subject to constraints given by mass balance and the number of phase splits allowed. This bilevel program can be solved globally based on an algorithm by Mitsos et al. [11]. This way, the calculated parameters of any of the aforementioned models automatically satisfy the necessary and sufficient conditions that guarantee that the number of phases modeled is in agreement with those experimentally observed. Many examples from the literature will be presented, in which problems are identified with the parameter values reported and for which the proposed formulation leads to the estimation of parameter values that result in accurate and valid model predictions. Finally, the issue of extrapolating the models to ranges outside the measured ones and the precautions that have to be taken will be discussed.
References
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