Cubic equations of state CEoS are particularly popular due
Cubic equations of state (CEoS) are particularly popular due to their simplicity and relatively good accuracy in predicting vapour-liquid equilibrium and liquid-liquid equilibrium for small, non-polar, non-associating molecules, such as light hydrocarbons. CEoS can match the phase boundaries (L/LL and L/VL) for heavy oil mixed with paraffinic solvents and dissolved gases such as carbon dioxide [4,7]. However, they are unable to match the yield or composition of asphaltene-rich phases without the use of composition dependent binary interaction parameters [2,33]. CEoS cannot account for self-association and this lack may limit their ability to model asphaltene related phase behaviour.
Two equations of state (EoS) that account for molecular self-association are statistical association fluid theory (SAFT)  and cubic plus association (CPA) . A modified version of SAFT was used to model the onset of asphaltene precipitation from a crude oil after CO2 and CH4 injection [8,9] and the yield of precipitated asphaltenes as a function of the gas-to-oil ratio . The Perturbed Chain form of the SAFT (PC-SAFT) was used to match the yield of asphaltene precipitation from n-alkane diluted bitumens . The self-association term within the SAFT model was not required or used in these studies. Similarly, CPA was used to model the onset of asphaltene precipitation during the depressurization of live Fosinopril sodium australia with and without gas injection [, , , , ], and the onset and yield for n-alkane diluted bitumens . The asphaltene self-association term of the CPA was used in these studies and was tuned to match onset and yields. While both SAFT and CPA have shown promising results, this study focuses on the CPA approach because it could be constructed started from a previously developed oil characterization for a cubic equation of state [22,39].
Although the CPA-EoS has been used to match asphaltene yield for different n-alkane diluted bitumen, the following issues with the oil characterization and modeling have yet to be addressed:
The objective of this study was to develop a methodology to model the phase behavior of diluted bitumens with the Cubic Plus Association equation of state and to evaluate the strengths and weakness of this modeling approach. The methodology was developed by fitting the model to propane and n-pentane diluted bitumen phase boundaries, asphaltene precipitation data, and phase composition data from the literature at temperatures and pressures up to 150 °C and 10 MPa, respectively [2,33]. The bitumen characterization was based on a distillation assay rather than a SARA or GC assay because boiling points are more representative of the molecular interactions that define the phase behaviour of the system . The asphaltene fraction was characterized such that its highest boiling point cuts were less soluble than its lower boiling point cuts. Two different characterization approaches were proposed, CPA-C5 and CPA-C3, where “C5” and “C3” denote the material insoluble in n-pentane and propane, respectively; that is, asphaltenes. The CPA-C5 approach is similar to previous characterizations . The CPA-C3 approach was proposed because the CPA-C5 approach was unable to fit asphaltene yield data from propane diluted bitumen. Both models were further tested on asphaltene precipitation data from the literature  for other bitumens diluted with precipitants such as n-pentane, n-hexane and n-octane at temperatures from 0 to 50 °C at atmospheric pressure. The ability of the proposed approaches to predict other thermodynamic derivative properties was beyond the scope of this study and was not evaluated.
The CPA equation of state The version of CPA used in this study was proposed by Kontogeorgis et al.  and combines the physical terms from the Soave-Redlich-Kwong (SRK) EoS with the association term derived by Michelsen and Hendriks  as follows:where P is pressure, R is the universal gas constant, T is the temperature, V is the molar volume, a and b are constants, ρ is molar density, g is the radial distribution function, x is the mole fraction of component i, subscript A denotes a type of site on the molecules in component i, and X is the fraction of Site A in Component i that is not bonded to other sites.