Relationship Between the Binary Interaction Parameters (kij) of the Peng-Robinson and Those of the Soave-Redlich-Kwong Equations of State. Application to the Definition of the PR2SRK Model

Chemical engineers from petroleum companies widely use cubic equations of state (EoS) and especially the ones proposed by Peng and Robinson (Peng-Robinson EoS noted PR EoS afterwards) and by Soave (Soave-Redlich-Kwong EoS, noted SRK EoS afterwards). These equations of state directly stem from the Van der Waals theory and can be written for a given mixture under the general form:

P = RT/(v-b) – a/Q(v)

where P is the pressure, T, the absolute temperature and v, the molar volume of the fluid. Q(v) is a second order polynomial in v. Classical Van der Waals mixing rules are frequently used to relate the a and b parameters of a mixture to the composition and to the a_i and b_i parameters of the pure components i. Parameter a is taken to be a quadratic function of the mole fractions and parameter b, a linear function. A binary interaction parameter (BIP), classically noted k_ij, is usually involved in the a-parameter expression to provide more flexibility to the EoS. This k_ij parameter can be either chosen constant or temperature-dependent (a mole fraction dependence is more scarcely introduced in the k_ij expression). It is however important to note that numerical values of BIPs are specific (i) to the considered EoS, (ii) to the alpha-function (Soave, Twu, Mathias-Copeman, etc.) involved in the mathematical expression of the a_i parameter.

The key point when using such cubic EoS to describe complex mixtures like petroleum fluids is thus to give appropriate values to the binary interaction parameters. We however know by experience that the BIPs, suitable for the PR-EoS (k_{ij_PR}) cannot be used for the SRK EoS (k_{ij_SRK} different from k_{ij_PR}). This assessment makes it impossible for petroleum engineers to mix the use of these two equations of state. Indeed, they usually have tables containing the numerical values of the BIPs only for the most widely used EoS in their company. To overcome this limitation, a relationship between the SRK BIP (k_{ij_SRK}) and the PR BIP (k_{ij_PR}) is established in this study. This objective could be reached thanks to the rigorous equivalence between the classical mixing rules with temperature-dependent k_ij and the combination at constant packing fraction of a Van Laar-type excess Gibbs energy model with a cubic EoS. This equivalence makes it possible to find out a relationship between the E_ij(T) parameters issued from the Van Laar function and the k_ij(T) of the classical mixing rules. Our key idea was to make the hypothesis that the infinite pressure residual molar excess Gibbs energy (g^E_res_inf) was independent of the used EoS. Doing so, a simple relationship between the E_ij suitable for the PR EoS (E_{ij_PR}) and those suitable for the SRK EoS (E_{ij_SRK}) can be obtained. Using this relationship and the one linking the k_ij and the E_ij, it was possible to find out a simple and general equation connecting the k_ij of a given EoS to the k_ij of any other EoS. This approach was then used to deduce k_{ij_SRK} from a known k_{ij_PR}. In addition, using the previously mentioned mathematical equation relating k_{ij_PR} and k_{ij_SRK}, the PPR78 model which may be seen as a group contribution method for the estimation of the temperature-dependent BIPs of the widely spread PR EoS, was used to generate k_ij for the SRK EoS. It is shown how the group interaction parameters initially determined for the PR EoS can be simply used to predict the temperature-dependent BIPs for the SRK EoS. This new predictive model has been called PR2SRK (here PR means predictive but also means Peng-Robinson since all the used parameters come from the PPR78 model). The results obtained with the PR2SRK model are in many cases very accurate in both the sub-critical and critical regions. We can thus conclude that the hypothesis we made (g^E_res_inf independent of the EoS) is pertinent.