Extended corresponding state EoS

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The following extended corresponding state EoS are available in TP-Cloud:

• SPUNG
• Lee-Kesler

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The extended corresponding state principle has been remarkably successful in capturing both equilibrium and non-equilibrium properties. For computationally demanding applications, such as computational fluid dynamics or process simulations, computationally efficient EoS are needed. The extending corresponding state principle offers an excellent compromise between high accuracy and computational speed.

Corresponding state principles are well-known, and a detailed description can be found in Ref. ¹. The basis for the theory comes from dimensional analysis of the configurational part of the statistical mechanical partition function that leads to expressions for residual thermodynamic properties as dimensionless groups. Originally it was developed as a simple two-parameter principle, which expressed the residual compressibility factor as a function of the reduced temperature and the reduced molar volume, and thus required only the critical temperature and density for the reference fluid and the real fluid in order to estimate the thermodynamic properties. Experiments have shown that this is a good approximation for heavy noble gases and for approximately spherical molecules such as oxygen and methane. For other fluids, the corresponding state principle needs to be extended. This explains the naming; the Extended corresponding state principle.

SPUNG

The SPUNG methodology was touted as the "Ultimate Two-Parameter" EoS by Michelsen and Mollerup ². The SPUNG EoS makes use of a very accurate EoS, such as a multiparameter EoS, to describe the thermodynamic properties of the reference fluid. The properties of the reference fluid are then linked to the thermodynamic properties of other fluids and fluid mixtures through:

$$\tilde{v} = \frac{v}{v_c\psi}$$

and

$$\tilde{T} = \frac{T}{T_c\theta}$$

where the shape factors $\psi$ and $\theta$, link the reduced molar volume and temperature to that of those of the reference fluid $\tilde{v}$ and $\tilde{T}$. The SPUNG principle uses analytical values for shape factors derived from a cubic EoS. The idea behind SPUNG is that even though cubic EoS are not exceedingly accurate for any single component, they describe very accurately the difference between fluids and fluid mixtures. This idea is supported by the excellent phase-equilibrium predictions obtained with cubic EoS. The table below shows the percentage deviation for properties of CO$_2$, using the multiparameter EoS of propane as reference, with SRK to calculate shape factors (SPUNG-SRK), compared to SRK ³.

The main advantage of the SPUNG methodology, however, becomes evident when the computational speed of calculating properties in multicomponent systems is compared to the SRK EoS. The table below ⁴ shows how the ratio of computational time relative to SRK increases with the number of components for SPUNG-SRK and the multiparameter EoS, GERG-2008.

The computational time of SRK is about double that of SPUNG-SRK regardless of the number of components in the mixture. Since a separate EoS is used for each component in GERG-2008, the computational time scales more poorly, and quickly escalates with a large number of components in the mixture.

Lee-Kesler

Lee and Kesler also developed an extended corresponding state EoS, which has been named after them ⁵. It uses a mBWR EoS, which is a less accurate ancestor of the modern multiparameter EoS, as a reference, and the acentric factor as a scaling parameter to account for the departure due to the non-spherical shape of molecules. As it gave better liquid-phase predictions than cubic EoS, it was a popular EoS in the oil and gas industry 30 years ago. Plocker later extended the applicability of the Lee-Kesler EoS to asymmetric mixtures, developing what was later known as the Lee Kesler Plocker EoS ⁶. The Lee Kesler EoS is available in TP-Cloud for comparative purposes.

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References

Footnotes

1. Ely, J. F.; Marrucho, I. M. F. In Equations of State for Fluid Mixtures; Sengers, J. V.; Kayser, R. F.; Peters, C. J; White, H. J, Eds.; IUPAC, 2000; Chapter The Corresponding-states Principle, pp 289-320. ↩

2. Michelsen, Michael L., and Jørgen M. Mollerup. Thermodynamic models: fundamentals & computational aspects. Holte, Denmark: Tie-Line Publications, 2004. ↩

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. Wilhelmsen, Øivind, et al. "Thermodynamic modeling with equations of state: present challenges with established methods." Industrial & Engineering Chemistry Research 56.13 (2017): 3503-3515. ↩

5. Lee, Byung Ik, and Michael G. Kesler. "A generalized thermodynamic correlation based on three‐parameter corresponding states." AIChE Journal 21.3 (1975): 510-527. ↩

6. Plocker, Ulf, Helmut Knapp, and John Prausnitz. "Calculation of high-pressure vapor-liquid equilibria from a corresponding-states correlation with emphasis on asymmetric mixtures." Industrial & Engineering Chemistry Process Design and Development 17.3 (1978): 324-332. ↩