(191f) 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.