Comment on “Reliable Correlation for Liquid–Liquid Equilibria outside

Comment/Reply Cite This: J. Chem. Eng. Data XXXX, XXX, XXX-XXX

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Comment on “Reliable Correlation for Liquid−Liquid Equilibria outside the Critical Region” Moll Glass and Alexander Mitsos* AVT - Aachener Verfahrenstechnik, Process Systems Engineering, RWTH Aachen University, Forckenbeckstraße 51, D-52074 Aachen, Germany S Supporting Information *

ABSTRACT: We comment on work by Ruszczyński et al., who extended the regression method of Cunico et al. (Fluid Phase Equilibria, 2014) and proposed certain Gibbs free energy models for binary liquid−liquid equilibria. We find that their regression method considers only necessary criteria for stability, and we derive constraints to be imposed on the thermodynamic parameters during regression such that the criteria become sufficient and the models thus become numerically easy to use.

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s opposed to using the same energy model for both phases α and β in equilibrium, one can superimpose two (convex) submodels, e.g., Wilson with ideal gas for vapor−liquid equilibria and asymmetric models for ionic liquid systems.1 As long as the Gibbs free energy of each submodel Gα,β is partially convex and min{Gα, Gβ} is locally nonconvex with respect to composition, isopotential is not only necessary but becomes both necessary and sufficient for satisfaction of Baker’s criterion,2 i.e., thermodynamic stability, as well as the correct number of phases. (For the importance of these criteria, in addition to the criterion for the correct number of phase splits, which is related to convexity properties of min{Gα, Gβ}, see refs 3−5) We observe that the proposed Gibbs free energy models6 exhibit this advantageous property under certain assumptions on the regressed parameters. From eq 15 in ref 6 it follows that the Gibbs energies of the proposed model can be written as Gα = xα(ln xα + ln γα1 ) + (1 − xα)[ln(1 − xα) + ln γα,∞ + ln γα,2 *] 2 β β β β β,∞ β, and G = x (ln x + ln γ1 + ln γ1 *) + (1 − x )[ln(1 − xβ) + ln γβ2], where xα,β ∈ X = (0, 1) are the mole fractions, T ∈ > 0 is the temperature in kelvins, and γ(α,β), * are the rational activity i are the activity coefficients at infinite coefficients and γ(α,β),∞ i dilution of species i. Since with eqs 9−14 in ref 6 2c α , β(x α , β − (x α , β)2 ) + T ∂ 2G α , β = ∂(x α , β)2 Tx α , β(1 − x α , β) α,β

α,β 2

should be noted, however, that this does not guarantee the correct number of phase splits.) Consequently, the demanding nonstandard programs for parameter estimation3−5 can be simplified by imposing cα,β > −2T as a constraint in the regression formulation. It should be noted that the condition is sufficient but not necessary for satisfaction of Baker’s criterion; should physically meaningful parameter values be excluded, Baker’s criterion will need to be satisfied by some other probably more involved means during regression, and at least, a posteriori checks are required. Finally, in our opinion, the reported regression method6,8 could also benefit from regressing all of the parameters simultaneously in addition to using global solvers to obtain the best possible fit upon convergence. Regressing only a subset of parameters while holding the remaining parameters fixed might help to reveal unidentifiable parameters.6 However, this heuristic should be carried out at most to render the model identifiable prior to regressing the full parameter set of the identifiable model. The reason for this is that regressing a subset does not guarantee even a locally optimal fit. Also, it should be noted that whether a model is identifiable is not a question of choosing an appropriate regression technique but rather of selecting an appropriate model. In the Supporting Information, we provide the corresponding GAMS code using the established deterministic global solver BARON 17.4.19 for the liquid−liquid equilibrium (LLE) of octanol/water, which is comparable to the case studies of Ruszczyński et al.6 in terms of broadness of the miscibility gap. The octanol/water example demonstrates that constraining the parameter space to −2T < cα,β < 0, regressing all of the parameters simultaneously, and using a global solver impose only low modeling and computational effort while resulting in an excellent fit for the proposed Gibbs model.6

(1)

α,β

and maxxα,β∈X x − (x ) = 0.25, G (·,T) is partially convex with respect to xα,β on X iff cα,β > −2T (also see, e.g., Dahm and Visco7 for a comparable result for the Porter model). This is complementary to the condition cα,β < 0 proposed to ensure the desired sign of derivatives of activity coefficients with respect to composition (mentioned as “phase stability” in the caption of Table 1 in ref 6). Provided that cα,β > −2T, flash calculations are facilitated since isopotential is sufficient for Baker’s criterion, giving a system of nonlinear equations as opposed to a nonlinear program. The use of the Gibbs models proposed in ref 6 for process simulation is thus simplified. Perhaps more importantly, during parameter estimation the correct number of phases and Baker’s criterion are met as long as isopotential is satisfied. (It © XXXX American Chemical Society

Received: July 14, 2017 Accepted: September 28, 2017

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DOI: 10.1021/acs.jced.7b00642 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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Comment/Reply

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.7b00642. GAMS file for regression of the LLE of octanol/water to the model of Ruszczyński et al., subject to the constraint −2T < cα,β < 0 (TXT)

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AUTHOR INFORMATION

Corresponding Author

*Fax: +49 (0) 241 80−92326. Phone: +49 (0) 241 80-94704. E-mail: [email protected], [email protected]. ORCID

Alexander Mitsos: 0000-0003-0335-6566 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.G. thanks Studienstiftung des deutschen Volkes for financial support and Łukasz Ruszczyński and co-workers for valuable suggestions on the Supporting Information.

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REFERENCES

(1) Simoni, L. D.; Brennecke, J. F.; Stadtherr, M. A. Ind. Eng. Chem. Res. 2009, 48, 7246−7256. (2) Baker, L. E.; Pierce, A. C.; Luks, K. D. SPEJ, Soc. Pet. Eng. J. 1982, 22, 731−742. (3) Mitsos, A.; Bollas, G. M.; Barton, P. I. Presented at the 19th European Symposium on Computer Aided Process Engineering ESCAPE 19, 2009. (4) Mitsos, A.; Bollas, G. M.; Barton, P. I. Chem. Eng. Sci. 2009, 64, 548−559. (5) Bollas, G. M.; Barton, P. I.; Mitsos, A. Chem. Eng. Sci. 2009, 64, 1768−1783. (6) Ruszczyński, Ł.; Zubov, A.; O’Connell, J. P.; Abildskov, J. J. Chem. Eng. Data 2017, 62, 2842. (7) Dahm, K. D.; Visco, D. P., Jr. Fundamentals of Chemical Engineering Thermodynamics, 1st ed.; Amundson, N. R., Ed.; Cengage Learning Engineering: Stamford, CT, 2014. (8) Special Issue on PPEPPD 2013: Cunico, L. P.; Ceriani, R.; Sarup, B.; O’Connell, J. P.; Gani, R. Fluid Phase Equilib. 2014, 362, 318−327. (9) Tawarmalani, M.; Sahinidis, N. V. Math. Program. 2005, 103, 225−249.

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DOI: 10.1021/acs.jced.7b00642 J. Chem. Eng. Data XXXX, XXX, XXX−XXX