Overview of EoS

Equations of state (EoS) are the cornerstones of any flow assurance software. An EoS is a mathematical relation between thermodynamic variables such as temperature, pressure, volume and internal energy.

The need for a wide selection of EoS

There does not exist an EoS with superior accuracy for all purposes¹; different types of EoS are preferred in different applications.

TP-Cloud has a comprehensive collection of EoS available. The advantage of this is that it allows us to carefully test a wide selection of models on an equal basis to select the best choice of EoS and parameters for your application.

Categories of EoS in TP-Cloud

There are seven categories of EoS available in TP-Cloud. Some of the categories, such as cubic EoS, have much extra functionality to improve their accuracy and extend their applicability. An illustration of how the different categories of EoS are connected is shown below.

• Cubic EoS: Cubic EoS are the simplest type of equation of state that can simultaneously represent properties of vapor and liquids. Even though they are simple, they are surprisingly accurate, in particular in the representation of phase equilibria. Their simplicity results in high computational speed and robustness. This is an ideal combination in process simulations or computational fluid dynamics calculations, where computational speed is an important criterion. Their liquid-phase density predictions can be greatly improved by using a Péneloux shift, and their accuracy in the calculation of phase equilibria can be enhanced by use of tailored $\alpha$-correlations or mixing models.

• Cubic plus association EoS: In cubic plus association (CPA) EoS, an additional contribution is added to a cubic EoS in order to represent hydrogen bonds (association). These EoS are excellent for prediction of water solubility and hydrate formation limits.

• SAFT EoS: Statistical Associating Fluid Theory (SAFT) EoS are founded in statistical mechanics and thermodynamic perturbation theory. This means that they have a molecular basis, and often excellent predictive abilities, in particular in the representation of complex molecules and polymers.

• Extended corresponding state EoS: The extended corresponding state approach is an excellent compromise between accuracy and computational speed. The methodology uses a very accurate multiparameter EoS to describe the reference component, and makes use of the extended corresponding state principle to predict the thermodynamic properties of other components and fluid mixtures.

• Electrolyte EoS: EoS from both the CPA and SAFT category can be further extended to include terms that describe the polar interactions in electrolytes. Electrolyte EoS can be used to calculate the properties of both vapors and liquids containing electrolytes.

• Activity coefficient models: These models can provide a very accurate description of electrolyte solutions, but describe only the liquid-phase.

• Multiparameter EoS: Multiparameter EoS are the most accurate EoS for representing PVT properties of fluids without electrolytes for pure components and mixtures where experimental data are available. They are considered the reference for calculation of most thermodynamic properties in both natural gas and CCS applications.

By following the link of each category, you find a detailed description of the EoS that will be available in TP-Cloud. Please note that not all of these EoS are fully implemented in the software yet. Furthermore, it is a continuous, ongoing work to expand TP-Cloud with new, improved parameter-sets.

The table below lists the EoS that are available today, and how many components each one covers.

Ideal gas properties, reference enthalpy and entropy and critical properties

Equations of state always describe residual thermodynamic properties with respect to the ideal gas state. To arrive at a complete description of the fluid, it is necessary to complement the equation of state with the ideal gas heat capacity of the fluid in question and a reference value of the enthalpy and entropy and some specified temperature and pressure. Having the perfect EoS is of no use if the ideal gas properties are inaccurate.

To emphasize the importance of this issue, we have evaluated isentropic compression of pure CO$_2$ from 1 to 2 and 5 bar respectively. The first row in the table below shows that the standard ideal gas heat capacity used in a different, established process modelling software results in an outlet temperature that is 1 K and 1.5 K lower than the most accurate correlation used in TP-Cloud² after a compression to 2 and 5 bar respectively.

Using an inaccurate ideal gas capacity can give errors of several degrees of Kelvin on the outlet temperature from valves, compressors or heat exchangers.

Furthermore, in order to correctly capture both the equilibrium compositions and the enthalpy of reaction, it is crucial to have accurate and consistent reference enthalpies and entropies for the components involved. Many EoS, such as cubic EoS, use the critical temperature and pressure as input.

In TP-Cloud, we use the most accurate values from literature for all of the above properties, regardless of the choice of EoS. We use the standard enthalpy and entropy of formation of the pure components at a pressure of one bar and a temperature of 298.15K as a reference, unless otherwise specified.

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References

Footnotes

1. Kontogeorgis, Georgios M., et al. "Equations of state in three centuries. Are we closer to arriving to a single model for all applications?" Chemical Engineering Science: X, 7 (2020): 100060. ↩

2. 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. ↩