Prediction of liquid–liquid equilibrium from the Peng–Robinson+COSMOSAC equation of state
Introduction
Liquid–liquid extraction is an important separation process in chemical industry, especially when distillation is not suitable such as mixtures which form an azeotrope or contain compounds with a very high boiling temperature. A reliable knowledge of the liquid–liquid equilibria (LLE) is crucial for the design and optimization of extraction process (Treybal, 1951). Conventionally, activity coefficient models such as NRTL (Renon and Prausnitz, 1968) and UNIQUAC (Abrams and Prausnitz, 1975) are used to describe the LLE with binary interaction parameters obtained from regression of experimental data (Asseline and Renon, 1970; Prausnitz et al., 2004). For systems without experimental LLE data, infinite dilution activity coefficients can be used instead to estimate the binary interaction parameters (Hartwick and Howat, 1995) at the cost of a potential loss of accuracy (Lin et al., 2009). Escobedo-Alvarado and Sandler (Escobedo-Alvarado and Sandler, 1998) and Matsuda et al. (Matsuda et al., 2002) use a different approach which combines an equation of state (EOS) with a liquid model through Gex-based mixing rule to predict LLE at high pressure with the binary interaction parameters fitted to low pressure LLE data. However, prediction of LLE with the molecular structure as the only input is much more challenging than those of vapor–liquid equilibria (VLE).
Group contribution methods such as UNIFAC (Fredenslund et al., 1977) and its modifications (Ferreira et al., 2005; Fu et al., 1996; Gmehling et al., 1993; Magnussen et al., 1981; Peres and Macedo, 1997; Weidlich and Gmehling, 1987) have also been geared to improve the accuracy in LLE prediction. Two good examples are the UNIFAC-LLE (Magnussen et al., 1981) whose parameters are optimized from experimental LLE data and the modified UNIFAC (Gmehling et al., 1993, Gmehling et al., 1998; Hansen et al., 1991; Weidlich and Gmehling, 1987) which considers the interaction parameters to be temperature-dependent. The accuracy of UNIFAC-LLE in describing LLE have been demonstrated (Gupte and Danner, 1987), but it is suggested to be used in a limited temperature range from 10–40 °C.
COSMO-based methods, such as COSMO-RS (Klamt, 1995; Klamt et al., 1998), COSMO-SAC (Lin and Sandler, 2002; Wang et al., 2007), and COSMO-RS(O1) (Grensemann and Gmehling, 2005), are another approach having a great potential for predicting LLE. These methods use the results from quantum mechanical calculations to describe the interaction between molecules in condensed phase. Recently, COSMO-RS is used to examine its ability in predicting LLE for aromatic extraction systems (Banerjee et al., 2007) and systems containing ionic liquids (Freire et al., 2007, Freire et al., 2008; Jork et al., 2005; Lei et al., 2007).
In this work, we show that the combination of Peng–Robinson (PR) EOS (Peng and Robinson, 1976) with the results of COSMO solvation calculation, referred to as the PR+COSMOSAC model, is an effective way for predicting liquid–liquid equilibria. In this method, the energy parameter and volume parameter b(x) of the PR EOS are obtained from solvation charging free energy and the cavity volume of solvation calculation, respectively. This approach had already been used to predict the vapor pressures and critical properties of pure substances, VLE of binary mixtures, and 1-octanol-water partition coefficient and infinite dilution activity coefficient in water (Hsieh and Lin, 2008, Hsieh and Lin, 2009a, Hsieh and Lin, 2009b; Lin et al., 2007). However, this original method is found to have limited prediction capability for highly nonideal mixtures, such as the LLE. By refining the description of the strength of hydrogen-bonding interactions between different types of species, we show that the miscibility gap and its temperature variation of highly nonideal mixtures can be well determined without reducing its accuracy in predicting properties of pure substances and VLE of mixtures. The results of LLE predictions are compared to those from the modified UNIFAC, UNIFAC-LLE, and COSMO-SAC models. This approach does not require the use of any binary interaction parameter and can serve as a complementary to conventional group contribution methods.
Section snippets
The Peng–Robinson+COSMOSAC equation of state
The Peng–Robinson EOS provides a mathematical relation between the temperature T, pressure P, and molar volume of a pure fluid as followswhere a(T) is a temperature-dependent energy parameter and b is a covolume parameter. Conventionally these two parameters must be determined from the critical properties and the acentric factor of the chemical substance. To apply the PR EOS to mixtures, a mixing rule, such as the van der Waals one-fluid mixing rule, Wong–Sandler
Computational procedure
The first step of using PR EOS is to determine the energy parameter and volume parameter of the system considered. The determination of these two parameters consists of four steps. First, perform the quantum mechanical solvation (DFT/COSMO) calculations. This is usually the most time-consuming step. However, a large database of DFT/COSMO calculation results developed and maintained by Liu's group at the Virginia Polytechnic Institute and State University [denoted as the VT
Results and discussion
The binary mixture systems considered here can, according to the species involved in the formation of hydrogen bonds, be classified into three groups: (I) no hydrogen bond (e.g., nitroethane+hexane); (II) hydrogen bonds between like species (e.g., pentane+water (hydrogen bonds between water only)); (III) hydrogen bonds between unlike and like species (e.g., nitroethane+water (hydrogen bonds between nitroethane and water, and between water molecules)).
Conclusion
With the parameters determined from first-principles solvation calculations, it is shown here that the Peng–Robinson EOS is capable of reliable prediction for thermodynamic properties of pure fluids, vapor–liquid equilibrium, and liquid–liquid equilibrium. A refined model for hydrogen-bonding interaction is introduced in order to account for the fact that hydrogen-bonding strength varies with the type of species that form the hydrogen bond. The resultant model allows for the prediction of LLE
Acknowledgement
The author would like to thank the financial support from Grant NSC 97-2221-E-002-085 and NSC 97-2917-I-002-118 by the National Science Council of Taiwan and computation resources from the National Center for High-Performance Computing of Taiwan.
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