Multiparameter EoS

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The following multiparameter EoS will be available in TP-Cloud:

• The most updated multiparameter EoS
• EoS-CG
• GERG-2008
• EOS-LNG

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Introduction to multiparameter EoS

The desire to represent the available experimental data in a compact and precise manner has motivated the development of state-of-the-art multiparameter EoS. These EoS are developed on the basis of a comprehensive analysis of experimental data and a diligent optimization procedure, with functional forms optimized for accuracy. Multiparameter EoS have been developed for single-component fluids and mixtures ¹. For pure components, the most general way to formulate the residual Helmholtz energy of multiparameter EoS is:

$$\alpha_{\text{meos}}=\frac{A_{\text{meos}}^{\text{r}}}{NRT} = \sum_{i=1}^In_i\left(\frac{T_c}{T}\right)^{t_i}\left(\vphantom{\frac{T_c}{T}}\frac{v_c}{v}\right)^{d_i}\exp{\left(-\eta_i\left(\vphantom{\frac{T_c}{T}}\frac{v_c}{v} - \epsilon_i\right)^{p_i}-\beta_i\left(\vphantom{\frac{T_c}{T}}\frac{v_c}{v} - \gamma_i\right)^{p_i}\right)}$$

where subscript $c$ refers to values at the critical point, and $n_i, t_i, d_i, p_i, \eta_i, \epsilon_i, \beta_i$ and $\gamma_i$ are fitted constants. The parameter $\eta_i$ and $\beta_i$ is equal to 0 for polynomial terms. For mixtures (subscript mix), the description is:

$$\frac{A_{\text{meos,mix}}^{\text{r}}}{NRT} = \sum_{j=1}^{N_c}x_j\alpha_{\text{meos},j}+\Delta\alpha_{\text{meos,mix}}^{\text{r}}$$

where the index $j$ runs over all components, $N_c$, and the last term is called a departure function, which accounts for non-ideality in mixtures, and can be formulated as:

$$\Delta\alpha_{\text{meos,mix}}^{\text{r}}=\sum_{i=1}^{N_c-1}\sum_{j=i+1}^{N_c}x_ix_jF_{ij}\alpha_{\text{meos},ij}^{\text{r}}$$

where $F_{ij}$ is a scaling factor for the interaction between component i and j, and $\alpha_{\text{meos},ij}$ is a binary specific departure function.

The most updated multiparameter EoS (MEOS)

For a wide selection of pure components, the has been developed dedicated multiparameter EoS, which serve as references for the calculation of thermodynamic properties of the pure fluid. Very rarely are these single-component multiparameter EoS updated. For instance, the reference for CO$_2$ was published in 1996 ² and the reference for propane was published in 2009 ³. The development of departure functions of mixtures, however, is an ongoing work in progress. Significant improvements are published yearly on different of specific binary mixture parameters are published on a yearly basis.

From time to time, a collection of pure-component descriptions and binary departure functions are bundled and named. Examples of such collections are GERG-2004, GERG-2008, EoS-LNG and EoS-CG. The most updated reference in TP-Cloud refers to the most recent and accurate multiparameter EoS formulation, which now is called "EoS-CG-2021" ⁴ updated with the most accurate binary departure functions that have been published since then (⁵, ⁶). New improvements are immediately added to the most updated multiparameter EoS in TP-Cloud once published in the open literature.

Below you see a reference lost of the available MEOS fluids:

GERG-2008

GERG-2008 EoS was adopted as an ISO Standard (ISO 20765-2) ⁷ for natural gases and similar mixtures. Dedicated Helmholtz energy equations of state with fewer terms are introduced in the GERG-2008 EoS for the description of the pure fluid contributions to make the EoS less computationally demanding. As a consequence, the EoS is less accurate than the most updated reference. The full set of parameters for the pure component and mixtures can be found in Refs. ⁸ and ⁹. The EoS is included in TP-Cloud as a reference, but it is less accurate than the most update reference multiparameter EoS, in particular for CO$_2$-rich mixtures, where an example showing prediction of vapor-liquid phase equilibria in the CO$_2$-argon mixture is shown below.

EOS-LNG

EOS-LNG uses the pure fluid descriptions of the GERG-2008 EoS, but with mixing rules improved for the descritpion of liquified natural gas (LNG). ¹⁰

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References

Footnotes

1. Span, Roland. Multiparameter equations of state: an accurate source of thermodynamic property data. Springer Science & Business Media, 2013. ↩

2. Span, Roland, and Wolfgang 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." Journal of physical and chemical reference data 25.6 (1996): 1509-1596. ↩

3. Lemmon, Eric W., Mark O. McLinden, and Wolfgang Wagner. "Thermodynamic properties of propane. III. A reference equation of state for temperatures from the melting line to 650 K and pressures up to 1000 MPa." Journal of Chemical & Engineering Data 54.12 (2009): 3141-3180. ↩

4. Neumann, Tobias, et al. "EOS-CG-2021: A Mixture Model for the Calculation of Thermodynamic Properties of CCS Mixtures." International Journal of Thermophysics 44.12 (2023): 178. ↩

5. Beckmüller, R., et al. "New fundamental equations of state for binary hydrogen mixtures containing argon, helium, and neon." Cryogenics 140 (2024): 103817. ↩

6. Tkaczuk, Jakub, et al. "Equations of state for the thermodynamic properties of binary mixtures for helium-4, neon, and argon." Journal of Physical and Chemical Reference Data 49.2 (2020). ↩

7. ISO-Standard 20765-2. Natural gas-Calculation of thermodynamic properties – Part 2: Single-phase properties (gas, liquid, and dense fluid) for extended ranges of application, Reference number ISO 20765-2: 2015 (E). ↩

8. Kunz, O., Klimeck, R., Wagner, W., Jaeschke, M. The GERG-2004 wide-range equation of state for natural gases and other mixtures. GERG TM15 2007. Fortschr.-Ber. VDI, Reihe 6, Nr. 557, VDI Verlag, Düsseldorf, 2007. ↩

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

10. Thol, M., Richter, M., May, E. F., Lemmon, E. W., Span, R. EOS-LNG: A Fundamental Equation of State for the Calculation of Thermodynamic Properties of Liquefied Natural Gases. J. Phys. Chem. Ref. Data 48 (2019), 033102. ↩