Unphysical Critical Curves of Binary Mixtures Predicted with GERG Models
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Unphysical Critical Curves of Binary Mixtures Predicted with GERG Models Ulrich K. Deiters1 · Ian H. Bell2 Received: 7 August 2020 / Accepted: 21 September 2020 © The Author(s) 2020
Abstract When applied to asymmetric binary mixtures (e.g., methane + pentane or heavier alkanes, hydrogen-containing mixtures), the GERG equation of state (GERG-2004 or GERG-2008) predicts critical curves with physically unreasonable temperature maxima above the critical temperature of the heavier component. These maxima are associated with physically impossible vapor–liquid equilibria. The phenomenon is probably caused by corrections for critical anomalies that were built into the empirical pure-fluid equations of state forming the foundation of the GERG model. These corrections ensure that the model represents thermodynamic data of pure fluids quite well even close to their critical points. For mixtures, however, the corrections can cause artifacts. Keywords Asymmetric fluid mixtures · Critical curves · GERG equation of state · Vapor–liquid equilibria Symbols A Helmholtz energy BX Critical amplitude for property X Cp , CV Isobaric, isochoric heat capacity c, w, pk Parameters of critical-anomaly correction kij Adjustable parameter in mixing rule Eq. (6) N Number of components p Pressure R Gas constant S Entropy T Temperature T ∗ Characteristic temperature * Ulrich K. Deiters ulrich.deiters@uni‑koeln.de 1
Institute of Physical Chemistry, University of Cologne, Greinstr. 4–6, 50939 Cologne, Germany
2
Applied Chemicals and Materials Div., National Institute of Science and Technology, Boulder, CO 80305, USA
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Vol.:(0123456789)
169
Page 2 of 19
International Journal of Thermophysics
(2020) 41:169
umin Eigenvector associated with 𝜆min v∗ Characteristic volume x Vector of mole fractions, x = (x1 , … , xN ) 𝛼, … , 𝛿 Critical exponents r 𝛼 r Dimensionless residual Helmholtz energy ( 𝛼0i : of component i, 𝛼ijr : of component pair ij within the departure function) 𝛼p Isobaric thermal expansivity 𝛽X,ij , 𝛾X,ij GERG parameters for the reducing property Xr ( X = T, V ) 𝜅T Isothermal compressibility 𝜆min Lowermost eigenvalue of 𝚿 𝜌 Molar density 𝝆 Vector of molar concentrations, 𝝆 = 𝜌x 𝜏 Reciprocal reduced temperature Ψ Helmholtz energy density, Ψ = 𝜌Am 𝚿 Hessian matrix of Ψ(𝝆) 𝜔 Reduced density
1 Introduction In 1989 Wagner and Setzmann published an equation of state for methane [1] that had an exceptionally wide range of validity and was able to represent almost all existing experimental data on methane within the error of the experiments. It became not only the reference equation for methane, but also the template for the reference equations of several other substances. Later Wagner, Span, and Lemmon generated simplified equations of state [2–5], to be used when computing speed was important or the experimental data were too scarce to permit the construction of a full reference equation. The GERG-2004 equation of state and its upgrade GERG-2008 extended these pure-fluid equations to mixtures [6, 7
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