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# The radial distribution function is given by For

The radial distribution function, , is given by:
For pure components and pseudo-components, the constants a and b are calculated as:whereand Tc, Pc and ω are the critical temperature, critical pressure and acentric factor, respectively.
Parameters a and b for mixtures are calculated from those of the pure components or pseudo-components using the classic van der Waals mixing rules:
For the association term (the last term in Eq. (1)), each component must be classified into an association type with a specified number of association sites per molecule (up to 4). The sites must be defined as either positive representing proton donors, or negative representing proton accepters. By default, only sites with an opposite sign can associate with each other. The fraction of unoccupied sites, X, is then related (analogously to the Langmuir adsorption isotherm) to the association strength between two sites (Site A and B) on two different molecules (i and j) as follows:
When i ≠ j, the interaction is termed cross-association and when i=j, the interaction represents self-association. The self-association strength can be expressed as follows:where is the association Captopril and is the association volume. In this study, the self-association and cross-association energies were adjusted manually or calculated from the self-association terms using the Elliot mixing rule given by:
The association energy and volume are the two defining parameters for the association term in the CPA-EoS; that is, all of the interaction terms can be determined from the and of the components and pseudo-components. Hence, once the number and type of association sites for the associating components has been defined, the CPA-EoS requires a total of 5 parameters for each component: Tc, Pc, and ω for the SRK physical terms, and, and for the association term. Binary interaction parameters in Eq. (6) must also be defined for each pair of components or pseudo-components.

Modeling methodology
The “trick” to modeling asphaltene precipitation with the CPA-EoS is to recognize that the components that primarily form a heavy phase and contribute to the maximum yield (the yield at high precipitant contents) are the self-associating components within the characterization. The self-association was necessary to partition components preferentially to the heavy phase even at high solvent contents in the feed. In other words, the self-association counters the effect of dilution something a cubic EOS cannot model accurately without the use of composition dependent binary interaction parameters [39]. The CPA-EoS can be tuned to accurately predict yields below the input amount of self-associating components but not above. Therefore, the amount of self-associating components must be defined to equal or exceed the maximum yield to be modeled. The problem is Twisting number the maximum yield is different for different solvents. One solution is to define the amount of self-associating components based on the highest yield expected to be encountered in any application. For solvent diluted bitumens, that amount is the “asphaltene” yield in propane diluted bitumen (C3-asphaltenes). Note that the C3-asphaltenes include many components that would typically be defined as resins or maltenes.

Results and discussion
First, the results for n-pentane diluted and propane diluted bitumen with CPA-C5 and CPA-C3 characterization approaches are discussed and compared. Then, the extension of the model to other bitumens and solvents is examined. Finally, the limiting number of pseudo-components is identified and the strengths and limitations of the proposed methodology are discussed.

Conclusions
Two oil characterizations were developed for the CPA model applied to the phase behavior of mixtures of bitumen and n-alkanes: 1) the CPA-C5 approach with the fraction of self-associating components based on the n-pentane insoluble content (19 wt% of the bitumen); 2) the CPA-C3 approach based on the propane insoluble content (50 wt% of the bitumen). The CPA-C5 model was less complex and ran 40% faster but could not match the yield and phase composition data for propane diluted bitumen. The key to matching yield data is that self-associating components must equal or exceed the content on the oil that is insoluble in the given solvent.