#4: Introducing CPA in PVTsim Nova 2.0
With PVTsim Nova 2.0 Calsep is introducing the CPA equation of state as one of the EoS alternatives. The CPA equation of state was presented by Kontogeorgis et al. (1996). CPA stands for Cubic Plus Association, which refers to a classic cubic equation of state with an association term. More specifically CPA combines the SRK equation with the association term from SAFT to account for hydrogen bonding between molecules of either the same component or different components. In the implementation in PVTsim Nova 2.0 CPA reduces to the SRK-Peneloux EoS model if no associating components are present.
Expressed in terms of the Z factor, the CPA model can be written as
The contribution from the cubic part takes the form
The parameter is given by
For non-polar components, such as hydrocarbons, the parameters a0, b and c1 can be determined from expressions in Tc, Pc and ω.
The contribution from the association part can be written as (Michelsen and Hendriks (2001))
where, i is the index for components and Ai is the index of association sites on component i. XAi is the fraction of sites of type A on component i, which are not bonded to other sites. g is the simplified expression of the radial distribution function as given by Kontogeorgis et al. (1999)
The term ‘association site’ refers to a region of the constituent molecule that may potentially form a hydrogen bond with a region (or site) on a neighboring molecule. PVTsim Nova 2.0 assumes that aqueous components can self-associate, i.e. form H bonds between their own molecules. This includes water and hydrate inhibitors as for example methanol (Figure 1).
Figure 1. Association sites (one positive and one negative) on a Methanol molecule allowing self-association
It is further assumed that the aqueous components can cross-associate with each other (Figure 2).
Figure 2. Example of cross-associating component pair (Water and Methanol)
The fraction of sites XAi is given by
where ΔAiBj is the association strength between site A on component i and site B on component j. It is calculated from the association energy, εAiBj and the association volume, βAiBj from
For polar components such as water and hydrate inhibitors a0, b and c1 are estimated together with εAiBi and βAiBi by fitting to experimental data, typically the saturated liquid densities and vapor pressure.
Cross association parameters are in PVTsim Nova 2.0 calculated using the CR-1 combining rule (Voutsas et al. (1999))
Components like CO2 and H2S are best treated as solvating components in aqueous mixtures i.e. they do not self-associate (εAiBi and βAiBi are 0), but form hydrogen bonds with water (Figure 3). For solvating components a modified CR-1 combining rule is used (Folas et al., 2006).
Figure 3. CO2, a solvating component, cross-associates but does not self-associate
In general, to use CPA for a component, one needs:
- a0, b and c1 (or Tc, Pc and ω of the component assuming it is not self-associating)
- Association parameters εAiBi and βAiBi
Initially PVTsim will rely on literature values for the pure component parameters. Calsep will continue to carry out verifications of the parameters with literature data. Verification has already been carried out using the same data material as used to update the HV-parameters (TechTalk July 2015). Figures 4 to 6 show the results of the verification with water-methane mutual solubility data.
Figure 4. Water solubility in Methane at low temperatures
Figure 5. Water solubility in Methane at higher temperatures
Figure 6. Methane solubility in water
Calsep A/S, “Huron-Vidal Mixing Rule. A Unique Combination of a Cubic Equation and a GE Model”, www.calsep.com, TechTalk July 2015.
Chapman, W. G., Gubbins, K. E., Jackson, G. and Radosz, M., "New Reference Equation of State for Associating Liquids", Ind. Eng. Chem. Res. 29(8), 1990, pp. 1709-1721.
Folas, G. K., Kontogeorgis, G. M., Michelsen, M. L and Stenby, E. H., “Application of the Cubic-Plus-Association (CPA) Equation of State to Complex Mixtures with Aromatic Hydrocarbons”, Ind. Eng. Chem. Res. 45(4), 2006, pp. 1527-1538.
Huang, S. H.and Radosz, M., "Equation of State for Small, Large, Polydisperse, and Associating Molecules", Ind. Eng. Chem. Res. 29(11), 1990, pp. 2284-2294.
Kontogeorgis, G. M., Voutsas, E. C., Yakoumis, I. V. and Tassios, D. P., “An Equation of State for Associating Fluids”, Ind. Eng. Chem. Res. 35(11), 1996, pp. 4310-4318.
Kontogeorgis, G. M., Yakoumis, I. V., Meijer, H., Hendriks, E. and Moorwood, T., “Multicomponent Phase Equilibrium Calculations for Water–Methanol–Alkane Mixtures”, Fluid Phase Equilibria 158, 1999, pp. 201-209.
Michelsen, M. L. and Hendriks, E. M., ”Physical Properties from Association Models”, Fluid phase equilibria 180(1), 2001, pp. 165-174.