Vapor liquid equilibrium modeling of alkane systems with Equations of State: “Simplicity versus complexity”

Two types of Equations of State (EoS), which are characterized here as “simple” and “complex” EoS, are evaluated in this study. The “simple” type involves two versions of the Peng–Robinson (PR) EoS: the traditional one that utilizes the experimental critical properties and the acentric factor and the other, referred to as PR-fitted (PR-f), where these parameters are determined by fitting pure compound vapor pressure and saturated liquid volume data. As “complex” EoS in this study are characterized the EoS derived from statistical mechanics considerations and involve the Sanchez–Lacombe (SL) EoS and two versions of the Statistical Associating Fluid Theory (SAFT) EoS, the original and the Perturbed-Chain SAFT (PC-SAFT).

The evaluation of these two types of EoS is carried out with respect to their performance in the prediction and correlation of vapor liquid equilibria in binary and multicomponent mixtures of methane or ethane with alkanes of various degree of asymmetry. It is concluded that for this kind of systems complexity offers no significant advantages over simplicity. Furthermore, the results obtained with the PR-f EoS, especially those for multicomponent systems that are encountered in practice, even with the use of zero binary interaction parameters, indicate that this EoS may become a powerful tool for reservoir fluid phase equilibria modeling.     

Vapor–liquid equilibria of selected binary mixtures formed by 2-methylpyrazine at 94.7 kPa

Vapor–liquid equilibria at 94.7?kPa, over the entire composition range are obtained for the binary mixtures formed by 2-methylpyrazine with 1,2-dichloroethane, 1,1,1-trichloroethane, 1,1,2,2-tetrachloroethane, trichloroethylene, tetrachloroethylene, N,N-dimethylformamide and N,N-dimethylacetamide. A Swietoslawski type ebulliometer is used to measure the bubble point temperatures necessary to determine the vapor–liquid equilibria. The Wilson equation is used to represent measured liquid phase composition versus temperature data.     

The Cubic-Two-State Equation of State: Cross-associating mixtures and Monte Carlo study of self-associating prototypes

A new equation of state for associating fluids has recently been presented by Medeiros and Tellez-Arredondo, the Cubic-Two-State Equation of State (CTS EoS) [Ind. Eng. Chem. Res. 47 (2008) 5723]. This equation arises from the coupling of the Soave–Redlich–Kwong EoS (SRK) with an association term from a two-state association model. The CTS EoS is polynomial in volume and it is able to describe vapor pressures and molar volume of associating fluids such as water, alcohol and phenol, among others. The equation is also able to describe the liquid–vapor equilibria of their mixtures with alkanes. In this paper, the physical and thermodynamic foundations of the CTS EoS are further investigated. In order to verify its applicability for cross-associating systems, the equation was employed in the prediction of phase equilibria behavior of binary alcohol–alcohol and water–alcohol mixtures. Very good agreement between predictions and experimental phase equilibria data was obtained with very simple combining rules and only one adjustable binary parameter. No additional parameters were necessary to describe ternary systems. With the purpose of checking the model's hypothesis and limitations, the two-state association term was coupled with the hard sphere Carnahan–Starling EoS, forming the CS-TS equation and the association characteristic parameters were determined theoretically for prototype association fluids. Monte Carlo NPT simulations of such fluids were performed and the results were compared with the equation's predictions. The CS-TS was able to describe qualitatively the pvT$pvT$ behavior of the prototype; nevertheless, it is not as accurate as those predictions obtained from the combination CS with Wertheim's association term. It seems that, when adjusting parameters of the CTS EoS to real substances, the discrepancies between the predicted and the real association contribution are dissipated among other adjustable parameters, specially on the dispersive term of the SRK equation. Finally, it is shown that CTS EoS isotherms can only have one or three real bigger roots than the co-volume for positive pressures, similar to cubic equations of state, and then it has the desirable form to describe vapor–liquid phase equilibria of associating compounds mixtures.     

Vapor–liquid equilibria predictions of hydrogen–hydrocarbon mixtures with the Huron–Vidal mixing rule

An excess Gibbs-equation of state (G^E-EoS) framework based on the Huron–Vidal mixing rule, has been applied to study vapor–liquid equilibria (VLE) of hydrogen–hydrocarbon mixtures. The mixing rule couples the Peng–Robinson–Stryjek–Vera (PRSV) EoS with a local composition solution model. The solution model is based on one-fluid theory treatment and assigns a single energy parameter to each binary pair. This energy parameter relates to the preference of the molecules for like to unlike interactions. The allocation of a system's number of interactions to the individual species in a binary mixture, incorporates the use of size parameters which gain significance only in the liquid phase. In a two parameter form, the framework has been used for the simultaneous data reduction of a large number of binary and several ternary hydrogen–hydrocarbon mixtures. These systems were taken over an extended range of pressures and temperatures. Results from the data reduction are reported in both tabular and graphical forms. Correlations for the model parameters have been identified with the acentric factor of the hydrocarbon in hydrogen–hydrocarbon binary mixtures. In a fully predictive mode, the model has shown to describe well VLE of binary hydrogen–linear alkane systems. Comparisons of these results with calculations from the Peng–Robinson (PR) EoS and the classical mixing rule (vdW) are included.     

The shapes of critical loci of binary fluid mixtures on the p–T plane

The relations between the shapes of the critical loci of binary fluid mixtures and the positions of the critical points of their component fluids on the p–T plane were considered theoretically. The consideration was carried out for vapor–liquid phase equilibria in the model systems without liquid–liquid equilibria. We focused only the relations between the shapes and the tangent of the straight lines connecting the critical points of the component fluids on the p–T plane, and so the effects of the distance between the critical points were out of the scope of this work. The changes in the shapes of the critical loci were classified logically into three patterns. Literature survey has shown that the two patterns exist in the experimental results for several types of mixtures. The shapes of the critical loci of the binary fluid mixtures on the p–T plane are closely related to the relative positions between the critical points of their component fluids.     

Thermodynamics of mixtures containing a very strongly polar compound: IV – application of the DISQUAC,UNIFAC and ERAS models to DMSO+ organic solvent systems

Binary mixtures of dimethylsulfoxide (DMSO) with alkane, benzene, toluene 1-alkanol, or 1-alkyne have been investigated in terms of DISQUAC. The corresponding interaction parameters are reported. ERAS parameters for 1-alkanol + DMSO mixtures are also given. ERAS calculations were developed considering DMSO as a not self-associated compound.

DISQUAC represents fairly well a complete set of thermodynamic properties: molar excess enthalpies, molar excess Gibbs energies, vapor–liquid equilibria, natural logarithms of activity coefficients at infinite dilution, or partial molar excess enthalpies at infinite dilution. DISQUAC improves UNIFAC calculations for H ^E . Both models yield similar results for VLE. In addition, DISQUAC also improves, ERAS results for 1-alkanol + DMSO mixtures. This may be due to ERAS cannot represent the strong dipole–dipole interactions present in such solutions.     

Phase equilibria study of Lennard–Jones mixtures by an analytical equation of state

The recently developed equation of state (EOS) for Lennard–Jones mixtures [Y. Tang, B.C.-Y. Lu, Fluid Phase Equilibria 146 (1998) 73.] is further investigated in this work for describing phase equilibria of these mixtures. The investigation covers vapor–liquid equilibria (VLE), liquid–liquid equilibria (LLE), vapor–liquid–liquid equilibria (VLLE) and vapor–vapor equilibria (VVE) over a wide range of temperatures, pressures and molecular characteristic parameters. Results from the van der Waals one-fluid (VDW1) theory are included for comparison. The newly proposed theory performs very well for VLE and LLE and the performance is better than the VDW1 theory; but both theories yield only qualitative results for VVE. It is also found that one system should exhibit VLLE, which was not noticed in previous investigations. Results from two other perturbation theories are also compared in some cases.     

Liquid–liquid equilibria of copolymer mixtures based on an equation of state

Liquid–liquid equilibria of copolymer mixtures were studied by an equation of state (EoS) for chain-like fluids. The equation consists of a reference term for hetero-nuclear hard-sphere chain fluids developed by Hu et al. where the next-to-nearest-neighbor correlations have been taken into account; and a perturbation term from Alder et al.’s square-well attractive potential. The segment parameters, including number of segments, segment diameter and interaction energy between segments, are obtained by fitting pVT data of pure homopolymer. For the case of different species in the same copolymer, the interaction parameters for unlike segment pairs are obtained by fitting pVT data of pure copolymer. For the interaction between segment of homopolymer and different species in copolymer, the parameters are treated as adjustable by fitting liquid–liquid equilibria data. In the latter case, the difference between different species in a copolymer is simply neglected as an approximation. Therefore, in general, only one pair of adjustable interaction parameter is determined from LLE data. To model miscibility maps of copolymer mixtures having two or three kinds of species, the interaction parameters are obtained from the boundary between miscible and immiscible regions. The EoS used in this work can correlate phase behavior including coexistence curves, miscibility windows and miscibility maps.     

High pressure phase equilibrium for ethylene + 1-propanol system at 283.65 K

Phase equilibria and saturated densities for ethylene+1-propanol system at high pressures were measured using a static-circulation apparatus at 283.65 K. The equilibrium composition and saturated density of each phase were determined by using gas chromatograph and vibrating tube density meters, respectively. The saturated points near the critical region are further measured by the conventional indirect method. The present experimental results include vapor–liquid equilibria (VLE), liquid–liquid equilibria (LLE), and vapor–liquid–liquid equilibria (VLLE). The experimental data were correlated with various equations of state.     

Calculation of isentropic compressibility and sound velocity in two-phase fluids

Derivative properties from equations of state (EoS) are well defined for homogeneous fluid systems. However, some of these properties, such as isothermal and isentropic (or adiabatic) compressibilities and sound velocity need to be calculated at conditions for which a homogeneous fluid splits into two (or more) phases, liquid or vapor. The isentropic compressibility and sound velocity of thermodynamically equilibrated fluids exhibit important discontinuities at phase boundaries, as noticed long ago by Landau and Lifschitz in the case of pure fluids. In this work, the two-phase isentropic compressibility (or inverse bulk modulus) is expressed in terms of the two-phase isothermal compressibility, two-phase thermal expansivity and an apparent heat capacity, defined as the partial derivative of total enthalpy with respect to temperature at constant pressure and composition. The proposed method is simple (simpler than previous approaches), easy to implement and versatile; it is not EoS-dependent and it requires only a flash routine and the expression of total enthalpy at given pressure, temperature and composition. Our approach is applied to a variety of fluid systems representative of reservoir applications and geophysical situations, including petroleum fluids (oil and gas condensate) and mixtures of water and gas (methane or CO₂). For low gas content in the two-phase fluid, i.e., near bubble point conditions, we obtain significantly lower bulk moduli and sound velocities than predicted within Wood's conventional approach, in which the liquid and gas phases are considered to be “frozen” at the passage of the acoustic wave.     

Isothermal (vapour + liquid) equilibria for binary mixtures of diisopropyl ether with (methanol,or ethanol,or 1-butanol): Experimental data,correlations, and predictions

A glass dynamic recirculating still was employed for the measurement of isothermal (vapour + liquid) equilibrium (VLE) data for the binary mixtures of diisopropyl ether (DIPE) + alcohol, viz. (DIPE + methanol), (DIPE + ethanol), and (DIPE + 1-butanol) at T = (305.15, 315.15, and 325.15) K, T = (313.15, 323.15, and 333.15) K and T = (318.15, and 338.15) K, respectively. The combined standard uncertainties in the reported system pressures, temperatures and phase compositions are ±0.2 kPa, ±0.1 K and ±0.003, respectively. Maximum pressure azeotropes were observed for all isotherms of the (DIPE + methanol) and (DIPE + ethanol) systems. The experimental results were correlated using both the γ–ϕ and ϕ–ϕ approaches. For the correlation of the VLE data with the γ–ϕ approach, the Wilson, NRTL and UNIQUAC G^E models with the truncated two-term virial equation of state (Hayden and O’Connell correlation for second virial coefficient computation) were used. In the ϕ–ϕ correlation approach, the Peng–Robinson equation of state was used with the Wong–Sander mixing rules incorporating the same G^E models used in the γ–ϕ approach. Comparisons between the experimental values and predictions using UNIFAC (Dortmund) and the Predictive Soave–Redlich–Kwong (PSRK) model were performed to test the predictive capabilities of these models for the experimental data measured here. The thermodynamic consistency of the experimental data was checked with the Herington area test.     

Application of the Perturbed-Chain SAFT equation of state to polar systems

The Perturbed-Chain SAFT (PC-SAFT) equation of state is applied to pure polar substances as well as to vapor–liquid and liquid–liquid equilibria of binary mixtures containing polar low-molecular substances and polar co-polymers. For these components, the polar version of the PC-SAFT model requires four pure-component parameters as well as the functional-group dipole moment. For each binary system, only one temperature-independent binary interaction k_ij is needed. Simple mixing and combining rules are adopted for mixtures with more than one polar component without using an additional binary interaction parameter. The ability of the model to accurately describe azeotropic and non-azeotropic vapor–liquid equilibria at low and at high pressures, as well as liquid–liquid equilibria is demonstrated for various systems containing polar components. Solvent systems like acetone–alkane mixtures and co-polymer systems like poly(ethylene-co-vinyl acetate)/solvent are discussed. The results for the low-molecular systems also show the predictive capabilities of the extended PC-SAFT model.     

Modeling of thermodynamic behavior of PVT properties and cloud point temperatures of polymer blends and polymer blend + carbon dioxide systems using non-cubic equations of state

In recent years, many factors influencing phase behavior of polymer blends have been studied because of their widely technological importance, as a simple method of formulating new materials with tailored properties which make them suitable for a variety of applications. This work has three main goals which were reached by using the Perturbed Chain Statistical Associating Fluid Theory (PC-SAFT) and the Sanchez–Lacombe (SL) non-cubic equations of state (EoS), which in previous works have shown their ability to handle long chain and associating interactions. First, both equations of state were tested with the correlation of the specific volumes of pure blends (PBD/PS, PPO/PS, PVME/PS, PEO/PES) and the prediction of the specific volumes for blends; second, the modeling of blend miscibilities in the liquid–liquid equilibria (LLE) of PBD/PS, PPG/PEGE, PVME/PS, PEO/PES, and PnPMA/PS blends; third, the modeling of the phase behavior of PS/PVME blends at various compositions in the presence of CO₂. PC-SAFT and SL pure-component parameters were regressed by fitting pure-component data of real substances (liquid pressure–volume–temperature, PVT, data for polymers and vapor pressure and saturated liquid molar volume for CO₂) and the fluid phase behavior of blend systems were simulated fitting one binary interaction parameter (k_ij) by regression of experimental data using the modified likelihood maximum method. Results were compared with experimental data obtained from literature and an excellent agreement was obtained with both EoS, which were also capable of predicting the fluid phase behavior corresponding to the critical solution temperatures (LCST: lower critical solution temperature, UCST: upper critical solution temperature) of blends.     

High pressure phase equilibrium properties for ethane+1-butanol system at 313.15 K

Phase equilibria and saturated densities for ethane+1-butanol system at high pressures were measured using a static-circulation apparatus at 313.15 K. The experimental apparatus equipped with three Anton Paar DMA 512S vibrating tube density meters was previously developed for measuring vapor–liquid–liquid equilibrium (VLLE) at high pressures. Co-existing phase composition and saturated density of each phase can be measured by means of the apparatus with a maximum temperature and pressure of 400 K and 20 MPa, respectively. The present experimental results include vapor–liquid equilibria (VLE), liquid–liquid equilibria (LLE), and VLLE. The equilibrium composition and density of each phase were determined by gas chromatography and density measurements, respectively. The experimental data were correlated with various equations of state.     

A Computationally Efficient Approach to Applying the SAFT Equation for CO2 + H2O Phase Equilibrium Calculations

The statistical associating fluid theory equation of state (EoS) is employed in a time efficient way for the correlation and prediction of vapor–liquid equilibrium of the CO₂ + H₂O binary system for the temperature (10–100 °C) and pressure (1–600 bar) ranges suitable for simulation of CO₂ geologic sequestration. The effective number of segments and energy parameter are correlated with the reduced temperature. Simple mixing rules are applied to obtain binary interaction parameters. Assigning a fixed H₂O composition in the mixing rule makes the phase equilibrium calculations relatively fast compared to other EoS’s. The results obtained by the model used were found to be in satisfactory agreement with the literature data.     

On the thermodynamics of alcohol solutions. Phase equilibria of binary and ternary mixtures containing any number of alcohols

Nagata, I., 1985. On the thermodynamics of alcohol solutions. Phase equilibria of binary and ternary mixtures containing any number of alcohols. Fluid Phase Equilibria, 19: 153–174.Binary vapor—liquid and liquid—liquid equilibrium data for alcohol solutions includin one or two alcohols are correlated with the UNIQUAC associated solution theory (Nagata and Kawamura). The theory uses pure liquid association constants determined by the method of Brandani and a single value of the enthalpy of the hydrogen bond equal to ?23.2 kJ mol ^?1 for pure alcohols. For alcohol-active nonassociating component mixtures and alcohol—alcohol mixtures the theory involves additional solvation constants. The theory is extended to contain ternary mixtures with any number of alcohols. Ternary predictions of vapor—liquid and liquid—liquid equilibria are performed using only binary parameters. Good agreement is obtained between calculated and experimental results for many representative mixtures.     

Vapor–liquid equilibria and density measurement for binary mixtures of o-xylene + NMF, m-xylene + NMF and p-xylene + NMF at 333.15 K, 343.15 K and 353.15 K from 0 kPa to 101.3 kPa

Isothermal vapor–liquid equilibria at 333.15 K, 343.15 K and 353.15 K for three binary mixtures of o-xylene, m-xylene and p-xylene individually mixed with N-methylformamide (NMF), have been obtained at pressures ranged from 0 kPa to 101.3 kPa over the whole composition range. The Wilson, NRTL and UNIQUAC activity coefficient models have been employed to correlate experimental pressures and liquid mole fractions. The non-ideal behavior of the vapor phase has been considered by using the Peng–Robinson equation of state in calculating the vapor mole fraction. Liquid and vapor densities were measured by using two vibrating tube densitometers. The excess molar volumes of the liquid phase were also determined. Three systems of o-xylene + NMF, m-xylene + NMF and p-xylene + NMF mixtures present large positive deviations from the ideal solution and belong to endothermic mixings because their excess Gibbs energies are positive. Temperature dependent intermolecular parameters in the NRTL model correlation were finally obtained in this study.     

Phase‐equilibrium behavior of vinyl chloride/n‐butane and its application in determination of vinyl chloride heterogeneous polymerization kinetics

Studies of the phase‐equilibrium behavior of vinyl chloride (VCM)/n‐butane mixtures and the kinetics of VCM heterogeneous polymerization, using n‐butane as a reaction medium, were carried out using a 1‐L glass autoclave. The vapor composition was measured by gas chromatography, showing that the vapor pressure of the VCM/n‐butane mixture was located above the line connecting the points for pure VCM and n‐butane. The concentration of VCM in the vapor phase was greater than that in the corresponding liquid phase. It was confirmed that the presence of poly(vinyl chloride) (PVC) resin had no significant influences on the phase equilibrium of VCM/n‐butane mixtures. Thus, the phase‐equilibrium equations were applied to determine the conversion of VCM during heterogeneous polymerization. The conversions calculated from the variations of vapor pressure or composition agreed with those determined by the weighing method. The conversion–time and polymerization rate–time curves obtained for VCM heterogeneous polymerization showed that the polymerization accelerated at low initiator concentration, but the polymerization rate decreased with an increase of conversion at relatively high initiator concentrations. The chain‐transfer reaction to n‐butane was confirmed by a decrease of the molecular weight and broadening of the molecular weight distribution of PVC. © 2001 John Wiley & Sons, Inc. J Polym Sci Part A: Polym Chem 39: 2179–2188, 2001