(747k) Constrained Best Subset Selection Methodology for the Regression of Helmholtz Energy Equations

Accurate thermodynamic properties are important for the development of new processes and technologies and simulating an optimized flowsheet. Unfortunately, equations of state typically used, such as Peng Robinson and Soave-Redlich Kwong, are inaccurate in critical regions to be able to provide accurate models. A recent development in equations of state uses first principles to determine one unifying equation explicit in Helmholtz energy to overcome these inaccuracies. These equations have been developed for over 100 pure substances and have been expanded to mixtures [1, 2, 3, 4].

A major challenge in the development of Helmholtz energy equations is the fitting of large datasets containing correlated data to the main unifying equation [3]. The current fitting procedures depend on cycling between linear and nonlinear regression techniques that are restricted to equality constraints and result in local solutions requiring multiple starts or an experienced user manually selecting an initial starting point [4]. Advancing these techniques to eliminate the need for multistart heuristics and enforce inequality constraints on the resulting models would allow us to control the thermodynamic behavior, improve extrapolation behavior, and maintain thermodynamically feasible limitations on the regressed equation while simultaneously fitting the data [5].

In order to address these challenges associated with fitting data to Helmholtz energy equations, we have developed a new model-building procedure from thermodynamic data. The main idea is to apply best subset selection to allow for the controlling of thermodynamic slopes with inequality constraints while fitting all the data simultaneously. A bank of terms is chosen to best represent the different phase regions and thermodynamic behavior. In order to avoid overfitting, our procedure systematically selects a subset of these terms to optimally fit the multiple thermodynamic property data sets. We rely on a global optimization solver [6] to find an optimal solution that optimizes the fit as well as the number of terms in the model according to an information criterion.

The proposed approach was used to fit a new Helmholtz energy equation for toluene. The resulting models were validated against results from the literature [7]. With the same data and bank of terms, the new methodology results in accuracy within the range of the experimental error of the thermodynamic properties incorporated. Our methodology can be applied to other pure and pseudo-pure chemical data sets to automatically generate new Helmholtz energy equations for use in simulations and optimization. References cited

[1] Lemmon, E. W.; Huber, M. L.; McLinden, M. O. NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 9.1, National Institute of Standards and Technology. 2013; https://www.nist.gov/srd/refprop.

[2] Span, R. Multiparameter equations of state: An accurate source of thermodynamic property data; Springer-Verlag, 2000.

[3] Span, R.; Wagner, W.; Lemmon, E. W.; Jacobsen, R. T. Multiparameter equations of state - recent trends and future challenges. Fluid Phase Equilibria 2001, 183-184, 1-20.

[4] Wagner, W.; Pruß, A. The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. J. Phys. Che. Ref.Data 2002, 387.

[5] Lemmon, E. W.; Jacobsen, R. T. A New Functional Form and New Fitting Techniques for Equations of State with Application to Pentafluoroethane (HFC-125). J. Phys. Chem. Ref. Data 2005, 34, 69−108.

[6] Tawarmalani, M.;Sahinidis, N. V. Global optimization of mixed-integer nonlinear programs: A theoretical and computational study, Mathematical Programming, 99, 563-591, 2004.

[7] E. W. Lemmon. Correlations short fundamental equations of state for 20 industrial fluids. J. Chem. Eng. Data, (51):785{850, 2006.