Empirical Multiparameter Equations of State – an Accurate Source of Thermodynamic-Property Data for Pure Fluids and Mixtures

For well measured, technically and scientifically relevant fluids and fluid mixtures empirical multiparameter formulations in form of fundamental equations of state have been established as reference for thermodynamic properties. Well known examples for reference equations of state are those for carbon dioxide[1], nitrogen[2], and water[3] – equations of state for fluids with excellent data sets, which are frequently applied not only in technical applications but also for calibration purposes and as scientific references. A number of fluids that are only relevant for technical applications are described with very high accuracy today, too. In particular this is true for some refrigerants, see e.g. the well-known work of Lemmon and Jacobsen[4]. Modern developments regarding mixture models based on empirical multiparameter equations of state essentially go back on an approach by Lemmon and Tillner-Roth[5]. While work on mixtures was largely related to refrigerants in the beginning, recent developments focused on natural gas[6],[7] and CO₂-rich[8] mixtures. Some of these models were formally accepted as international standards[9],[10], others have been established as de facto standards by the scientific community and by internationally used software products.[11]

The main advantage of empirical multiparameter equations of state is their high accuracy, which is essentially limited only by the accuracy of the best available experimental data, and their consistency – all thermodynamic properties are calculated from derivatives of a single thermodynamic potential. This way, highly accurate data for density and speed of sound yield information for all other properties as well. Studies on suitable mathematical structures for fundamental equations of state, the use of algorithms optimizing their mathematical structure, and finally the use of constraints in nonlinear fitting [4] have significantly improved the numerical stability of multiparameter equations of state[12]. They extrapolate well and yield reasonably accurate results in (limited) regions without data. And it has been shown that multiparameter equations of state yield favourable results even when being used in conjunction with models for systems as complex as hydrates.[13]

However, empirical multiparameter equations of state do have a number of drawbacks as well. They still depend on the availability of accurate experimental data in broad ranges of states, and estimates for the uncertainty of property values calculated from such equations can only be established by comparison to experimental data. In multicomponent mixtures, they rely on an independent description of all binary subsystems. Subsystems with poor data situation may affect in particular the representation of phase equilibria in multicomponent mixtures, where the predictive capabilities of multiparameter mixture models are very limited. And the way in which states in the instable part of the fluid surface are described restricts their applicability for example in density functional theory calculations.

The presentation aims at a fair description of both strengths and weaknesses of empirical multiparameter equations of state. Ongoing work on principal improvements and extensions of the range of applicability of empirical multiparameter equations of state will be referred to.

[1] R. Span and W. Wagner: A new equation of state for carbon dioxide covering the fluid region from the triple point temperature to 1100 K at pressures up to 800 MPa. J. Phys. Chem. Ref. Data 25, 1509 - 1596 (1996).

[2] R. Span, E. W. Lemmon, R. T Jacobsen, W. Wagner and A. Yokozeki: A reference equation of state for the thermodynamic properties of nitrogen for temperatures from 63.151 K to 1000 K and pressures to 2200 MPa. J. Phys. Chem. Ref. Data, 29, 1361 - 1433 (2000).

[3] W. Wagner and A. Pruß: The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. J. Phys. Chem. Ref. Data 31, 387 - 535 (2002).

[4] E. W. Lemmon and R. T Jacobsen: A new functional form and new fitting techniques for equations of state with application to pentafluoroethane (HFC-125). J. Phys. Chem. Ref. Data 34, 69 - 108 (2005).

[5] E. W. Lemmon and R. Tillner-Roth: A Helmholtz energy equation of state for calculating the thermodynamic properties of fluid mixtures. Fluid Phase Equilibria 165, 1 - 21 (1999).

[6] O. Kunz and W. Wagner: The GERG-2008 wide-range equation of state for natural gases and other mixtures: An expansion of GERG-2004. J. Chem. Eng. Data 57, 3032 - 3091(2012).

[7] M. Thol, M. Richter, E. F. May, E. W. Lemmon and R. Span: EOS-LNG: a fundamental equation of state for the calculation of thermodynamic properties of liquefied natural gases. J. Phys. Chem. Ref. Data 48 (3), 033102 (2019).

[8] J. Gernert and R. Span: EOS–CG: A Helmholtz energy mixture model for humid gases and CCS mixtures. J. Chem. Thermodyn. 93, 274 - 293 (2016).

[9] International Association for the Properties of Water and Steam (IAPWS): Revised release on the IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use (2014).

[10] International Organization for Standardization (ISO): ISO 20765-2:2015, Natural gas - Calculation of thermodynamic properties - Part 2: Single-phase properties (gas, liquid, and dense fluid) for extended ranges of application (2015).

[11] E. W. Lemmon, I. H. Bell, M. L. Huber, and M. O. McLinden: NIST Standard Reference Database 23: Reference fluid thermodynamic and transport properties - REFPROP, version 10.0. National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, 2019.

[12] R. Span and W. Wagner: On the extrapolation behavior of empirical equations of state. Int. J. Thermophys., 18, 1415 - 1443 (1997).

[13] S. Hielscher, B. Semrau, A. Jäger, V. Vinš, C. Breitkopf, J. Hrubý and R. Span: Modification of a model for mixed hydrates to represent double cage occupancy. Fluid Phase Equilibria 490, 48-60 (2019).