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# Accurate equilibrium predictions of CO containing mixtures

Accurate equilibrium predictions of CO2 containing mixtures, however, are challenging with traditional equations of state. The reason for this may be that CO2 has a large quadrupole moment. The large quadrupole moment of CO2, as well as size asymmetry, are for instance believed to be the reasons for the liquid-liquid equilibrium (LLE) between CO2 and heavy hydrocarbons, and the reason for the low temperature azeotrope formed between mixtures of CO2 and lighter hydrocarbons. Cubic equations of state such as the Soave–Redlich–Kwong (SRK) equation of state (EoS), however, treat CO2 as an inert compound. Even in a modern EoS such as the Statistical Association Fluid Theory (SAFT) only dispersive forces are usually considered for CO2. The continued use of these models may be attributed to the fact that several mixtures, such as CO2+hydrocarbons, are described quite well, when a single binary interaction parameter (k), is used. To deal with polar and quadrupolar interactions a number of multipolar terms have been suggested in the literature. These terms are mainly based on a third order perturbation theory developed from statistical mechanics by Stell and co-workers [2], [3], [4], [5]. The perturbation theory was originally developed for pure fluids using the Stockmayer potential or the conceptually simpler hard sphere model with a central point dipole or quadrupole. Using the former potential, Gubbins and Twu [6], [7] developed directly applicable expressions for polar and quadrupolar fluid mixtures. During the last decade a number of quadrupolar terms have been included in the SAFT framework. Inspired by Stell and co-workers [2], [3], [4], [5] Gross [8] developed a new quadrupolar c-kit inhibitor using the two center Lennard–Jones pair potential as the reference fluid. The resulting quadrupolar contribution was added to the Perturbed-Chain SAFT (PC-SAFT) to give the Perturbed-Chain Polar SAFT (PCP-SAFT). This EoS can be used without any additional adjustable parameters. Karakatsani et al. [9], [10] and Karakatsani and Economou [11] introduced two quadrupolar terms to the PC-SAFT framework. Both terms are based on the perturbation terms from by Larsen et al. [5], which use the pure hard sphere fluid as the reference fluid. The authors suggest two quadrupolar terms; an expression which employs the full correlation integrals from Larsen et al. [5], this term does not use any additional adjustable parameters, and a simpler version where the correlation integrals are truncated at the zeroth order term. Later, NguyenHuynh et al. [12] extended a group contribution SAFT EoS to quadrupolar (and polar) fluid mixtures. The quadrupolar term used by the authors is based on the theory of Gubbins and Twu [6], [7]. The term is extended to chain molecules using a procedure suggested by Jog et al. [13] and Jog and Chapman [14]. It seems that improved predictions and correlations (smaller k) are typically obtained for binary vapor liquid equilibrium (VLE) when a quadrupolar term is coupled to SAFT or PC-SAFT. The quadrupolar models, however, have some limitations; The models are only applicable to molecules with a highly symmetric quadrupole moment, so that the quadrupole moment reduces to a single scalar value. It has furthermore been shown that false liquid–liquid splits may be predicted by the models [15]. Mixtures of several quadrupolar molecules are challenging, and the results are often better if only one component is assumed to have a quadrupole moment [8]. Another more pragmatic approach to account for the quadrupole moment of CO2 is to consider CO2 to be a self-associating compound. Such procedures often work well resulting in improved predictions and correlations with small interaction parameters [16], [17], [18], [19]. Unfortunately the improvement is obtained at the cost of additional pure component parameters and, in some cases, an extra adjustable parameter for the binary mixtures. In this work, inspired by the recent advances within the SAFT-family, and in an effort to obtain a physically more correct and predictive model, a quadrupolar term is proposed and combined with the well-known cubic plus association (CPA) EoS (Kontogeorgis et al. [20]). The term is based on the explicit expressions developed by Larsen et al. [5] for a hard sphere fluid with a point quadrupole. To simplify the expressions we truncate the correlation integrals at the zeroth order term, similar to the approach used in the truncated Perturbed-Chain Polar SAFT (tPC-PSAFT) [10], [11].