Vapor-liquid equilibria for polyisobutylene solutions

Vapor sorption isotherms in binary solutions of polyisobutylene (PIB), (M_η = 4.7 × 10⁶ g/mol) in hydrocarbons (cyclopentane; cyclohexane; n-heptane; 2,2-dimethyl butane; and 2,2,4-trimethyl pentane) and chlorinated methanes [carbon tetrachloride (CCI₄) and chloroform (CHCI₃)] have been determined at 23.5°C using the piezoelectric sorption method. The polymer-solvent interaction parameter χ obtained agrees with previously published values determined by using gas-liquid chromatography and a quartz-helix vapor sorption apparatus. The Flory theory of corresponding-states has been applied to the experimental results through the χ parameter and affords a good prediction of the concentration dependence of χ for solutions of chloroform, carbon tetrachloride, n-heptane, and 2,2-dimethyl butane in PIB. The experimental values of \documentclass{article}\pagestyle{empty}\begin{document}$ [\partial (a_1 /\psi _1 )/\partial a_1 ]_{T,P} $\end{document} for the PIB solutions are constant over the measured concentration range, for example \documentclass{article}\pagestyle{empty}\begin{document}$ [\partial (a_1 /\psi _1 )/\partial a_1 ]_{T,P} $\end{document} = ?4.1 for CCI₄, ?3.65 for CHCI₃, ?3.0 for 2,2-dimethyl butane and n-heptane, ?2.7 for 2,2,4-trimethyl pentane, ?2.7 for cyclohexane, and ?1.7 for cyclopentane, where a₁ is the solvent activity and ψ₁ is the solvent segment fraction. The correlations between the values of \documentclass{article}\pagestyle{empty}\begin{document}$ [\partial (a_1 /\psi _1 )/\partial a_1 ]_{T,P} $\end{document} and the theories of Guggenheim, Miller, Huggins, and Flory are discussed.     

Vapor-liquid equilibria for polybutadiene and polyisoprene solutions

Vapor-liquid equilibria in binary solutions of hydrocarbons (n-hexane, benzene, toluene, cyclohexane) and chlorinated methanes [carbon tetrachloride (CCl₄), chloroform (CHCl₃), and dichloromethane (CH₂Cl₂)] in polybutadiene (PBD) and polyisoprene have been determined at 23.5°C by using the piezoelectric sorption method. The weight-fraction activity coefficient of solvent (a₁/w₁) in cis-PBD (98% cis-1,4 addition) and random cis-trans-PBD (r-PBD, 34.3% cis-1,4 addition; 54.3% trans-1,4 addition; 11.4% vinyl-1,2 addition) are almost equal for CCI₄, CHCI₃, CH₂CI₂, benzene, and toluene solutions, while the values of a₁/w₁ in n-hexane and cyclohexane solutions in cis-PBD are larger than those in r-PBD solutions. The values of a₁/w₁ for solutions of hydrocarbons and chlorinated methanes in cis-1,4 polyisoprene (95% cis-1,4 addition) have been compared with those for cis-PBD.     

Vapor-liquid equilibria for the nitrogen-isobutane system

Vapor-liquid equilibria have been investigated experimentally for the nitrogen-isobutane system at temperatures from 120 K to 220 K and at pressures up to 150 bar. Below 126.5 K, two liquid phases were observed as pressure was increased to near the vapor pressure of pure nitrogen. The equilibrium ratio of nitrogen in the binary system and the Henry’s law constants for nitrogen in isobutane were determined from experimental data. The experimental phase equilibrium data were correlated by means of the Peng-Robinson equation of state.     

Modelling of phase equilibria for associating mixtures using an equation of state

In the present work, the group contribution with association equation of state (GCA-EoS) is extended to represent phase equilibria in mixtures containing acids, esters, and ketones, with water, alcohols, and any number of inert components. Association effects are represented by a group-contribution approach. Self- and cross-association between the associating groups present in these mixtures are considered. The GCA-EoS model is compared to the group-contribution method MHV2, which does not take into account explicitly association effects. The results obtained with the GCA-EoS model are, in general, more accurate when compared to the ones achieved by the MHV2 equation with less number of parameters. Model predictions are presented for binary self- and cross-associating mixtures.     

Vapor-liquid equilibria for alcohol/hydrocarbon systems using the CPA Equation of state

The recently developed Cubic-Plus-Association Equation of State (CPA EoS) is extended in this study to binary systems containing one associating compound (alcohol) and an inert one (hydrocarbon). CPA combines the Soave-Redlich-Kwong (SRK) equation of state for the physical part with an association term based on perturbation theory. The classical van der Waals one-fluid mixing rules are used for the attractive and co-volume parameters, and b, while the extension of the association term to mixtures is rigorous and does not require any mixing rules. Excellent correlation of Vapor-Liquid Equilibria (VLE) is obtained using a small value for the interaction parameter (k_ij) in the attractive term of the physical part of the equation of state even when it is temperature-independent. CPA yileds much better results than SRK and its performance is similar to that of other association models, like the Anderko EoS, and the more complex SAFT and Simplified SAFT EoS.     

Calculation and prediction of fluid phase equilibria from an equation of state

Liquids or compressed gases consisting of light molecules show deviations from classical mechanics, which are caused by the discontinuity of energy levels. From the assumption that each molecule is confined to a cell with a size depending on the free volume, a quantum correction is derived which extends any van der Waals type equation of state to quantum gases. The correction is applied to a semiempirical equation of state developed by the author. The extended equation yields reasonable critical compressibility factors and gives a better representation of PVT data than the uncorrected equation. Furthermore high pressure phase equilibria in mixtures containing helium and hydrogen have been calculated. Again the agreement with experimental data is improved; the adjustable binary interaction parameters have values close to the Berthelot-Lorentz rules and are less temperature dependent.     

Vapor-liquid equilibrium of hexadecapolar fluids from a perturbation-based equation of state

In this work a numerically tractable expression for the interaction potential between two point hexadecapoles with octahedral symmetry and a molecular-based equation of state derived by perturbation theory for hexadecapolar fluids are presented. The polar system is modeled by square-well particles with a point hexadecapole with octahedral symmetry at their centers. This equation of state is analytical in the state variables and in the potential parameters and allows us to study the effects of the hexadecapolar moment strength on the thermodynamic properties and liquid-vapor phase diagram. The equation presented here is applied to the thermodynamics of sulfur hexafluoride and gives very good predictions for the saturation pressures and the vapor-liquid phase diagram.     

Solid–liquid equilibria based on an equation of state for chain fluids

Solid–liquid equilibria were studied using an equation of state previously developed for fluids containing chain-like molecules. The method was used to correlate solubilities of normal alkanes and aromatic compounds with high molecular mass in hydrocarbon solvents. With one temperature independent parameter for the interaction energy, good agreement can be obtained between calculated results and experimental data for selected systems.     

Vapor-liquid and vapor-solid phase equilibria for united-atom benzene models near their triple points: the importance of quadrupolar interactions

Gibbs ensemble Monte Carlo simulations were used to calculate the vapor-liquid and vapor-solid coexistence curves for benzene using two simple united-atom models. An extension of the Gibbs ensemble method that makes use of an elongated box containing a slab of the condensed phase with a vapor phase along one axis was employed for the simulations of the vapor-solid equilibria and the vapor-liquid equilibria at very low reduced temperatures. Configurational-bias and aggregation-volume-bias Monte Carlo techniques were applied to improve the sampling of particle transfers between the two simulation boxes and between the vapor and condensed-phase regions of the elongated box. An isotropic united-atom representation with six Lennard-Jones sites at the positions of the carbon atoms was used for both force fields, but one model contained three additional out-of-plane partial charge sites to explicitly represent benzene's quadrupolar interactions. Both models were fitted to reproduce the critical temperature and density of benzene and yield a fair representation of the vapor-liquid coexistence curve. In contrast, differences between the models are very large for the vapor-solid coexistence curve. In particular, the lack of explicit quadrupolar interactions for the 6-site model greatly reduces the energetic differences between liquid and solid phases, and this model yields a triple point temperature that is about a factor of 2 too low. In contrast, the 9-site model predicts a triple point of benzene at T = 253 +/- 6 K and p = 2.3 +/- 0.8 kPa in satisfactory agreement with the experimental data (T = 278.7 K and p = 4.785 kPa).     

Cubic equation of state for calculation of phase equilibria in association systems

Chemical potential of an associating component is derived from cubic equation of state. It is separated in the rigorous way into effective physical part and excess of chemical contribution. The both parts of the chemical potential are given in form of explicit expressions, which does not need any cumbersome integration or differentiation. The simple expression for association equilibrium is derived. It depends only on repulsive term of EOS. The proposed method is applied for correlating VLE equilibria in binary mixtures and for prediction of pressure in ternary mixtures on basis of the constituent binary VLE data.     

A modification of the martin equation of state for calculating vapour-liquid equilibria

A modification of the Martin (1979) equation of state suitable for vapour-liquid equilibrium calculations is presented. The temperature dependence of this modification is determined from the second virial coefficient correlation of Tsonopoulos (1974) and the pure-component vapour pressures. Excellent predictions of phase equilibria have been obtained for a number of binary systems, including a system containing hydrogen. For the systems tested, the equation of this study is more accurate than the Peng-Robinson (1976) equation of state.     

A semi-empirical equation of state applied to vapor-liquid equilibria calculations

A five-parameter equation of state is proposed to calculate the vapor-liquid equilibria of compounds in binary and multicomponent mixtures. This equation is closely related to a previous equation of state proposed by the author, the main modification being in the entropic term where the parameter m assumes a constant value for all compounds. It is shown that the van der Waals conditions at the critical point and the Morbidelli-Carra' algorithm enable the calculation of three other constants. Rules are given to calculate the remaining constant K which pertains to the enthalpic term. The proposed method only requires knowledge of the critical constants and of the normal boiling temperature as input parameters. A wide application of the new equation to both polar and non-polar binary systems indicates the following: the proposed method is predictive for ideal or nearly ideal mixtures; the correlation of mixtures of hydrocarbons having very different molar volumes can be obtained by optimizing only the binary interaction parameter linked to the enthalpic term; the new equation also correlates well with strongly non-ideal systems which exhibit a miscibility gap; the prediction of multicomponent vapor-liquid equilibria from the binary data alone is also reliable for both polar and non-polar mixtures.     

A modified clausius equation of state for calculation of multicomponent refrigerant vapor-liquid equilibria

Gow, A.S., 1993. A modified Clausius equation of state for calculation of multicomponent refrigerant vapor-liquid equilibria. Fluid Phase Equilibria, 90: 219-249.

A modified Clausius equation of state with a single temperature dependent energy-volume parameter a(T) in the attractive term was designed to describe the vapor pressure vs. temperature relationship of 39 pure refrigerant fluids including elementary cryogenic materials (e.g. He, Ar, N₂, CO₂, CH₄, etc.), chlorofluorocarbons (CFCs), hydrofluorocarbons (HFCs), hydrochlorofluorocarbons (HCFCs), fluorocarbons (FCs), and various other simple cryogenic compounds. The equation developed represents the vapor-liquid coexistence dome, and the superheated vapor compressibility factor and enthalpy for pure refrigerants.

The vapor-liquid equilibrium for refrigerant mixtures is calculated using a “phi-phi” method with “one fluid” van der Waals mixing and combining rules for the equation of state parameters a_M(T), b_M and c_M. A single interaction constant k₁₂ is used to describe non-ideal behavior of each binary. The binary interaction constant, which is a strong function of temperature, and the sign of which signifies the type of deviations from Raoult's law, is obtained by correlating experimental bubble point data for isothermal binary refrigerant mixtures. The proposed equation of state generally describes binary P-x,y data more accurately the higher the temperature for a given system. The method presented is extended to predict vapor-liquid equilibria for the R14-R23-R13 ternary system at 198.75 K using binary interaction constants at this temperature for the three binaries involved.     

Vapor-liquid equilibria of polymer solutions determined by molecular mechanics

A method of calculating interaction parameters used in phase equilibrium calculations was extended for predicting solvent activities of polymer solutions. A pair of interaction parameters are determined by calculating interaction energies between all pairs of molecules in the solution of interest with a molecular mechanics method called the consistent force field (CFF). The conformational space of a pair of molecules is sampled with a Monte Carlo algorithm followed by energy minimizations. The method is used to calculate solvent activities for the diethylketone/polypropylene system, giving results in good agreement with experimental values. In addition, solvent activities are predicted for the diethylketone/polyethylenesystem for which no experimental data are available. The method is fully predictive, as no fitting to experimental phase-equilibrium data is carried out.     

Vapor-liquid equilibria for binary mixtures of carbon dioxide and fatty acid ethyl esters

Vapor-liquid equilibria were measured and correlated using the Peng-Robinson equation of state for five binary systems of carbon dioxide and fatty acid ethyl esters (ethyl stearate, ethyl oleate, ethyl linoleate, ethyl eicosapentanoate, ethyl docosahexanoate) at 313.15 K, 323.15 K and 333.15 K. Solubility in CO₂ of fatty acid ethyl esters of equal chain length but of various degrees of unsaturation was compared. Although there was no distinct difference in solubility at lower pressures, at higher pressures (more than 15 MPa), those with a higher degree of unsaturation showed a slightly higher solubility. When the solubility in CO₂ of methyl esters and the corresponding ethyl esters were compared, it was noted that the former showed a slightly higher solubility at all system pressures measured in this work.     

Vapor-liquid equilibria forn-tetradecane-benzene mixtures at 25 and 50°C

Vapor pressures for the benzene+n-tetradecane system have been measured using a static apparatus. The excess Gibbs free energy was calculated using a modified Barker method and a Pade approximant equation. The selection of the most adequate approximation is discussed. Results are compared with those corresponding to other thermodynamic properties. The activity coefficient of benzene at infinite dilution calculated from these data agrees well with the value obtained from the literature.     

Analysis and applications of a group contribution sPC-SAFT equation of state

A group contribution (GC) method for estimating pure compound parameters for the molecular-based perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) is proposed in a previous work [A. Tihic, G.M. Kontogeorgis, N. von Solms, M.L. Michelsen, L. Constantinou, Ind. Eng. Chem. Res. 47 (2008) 5092–5101]. In this paper, an investigation of the predictive capability of the GC sPC-SAFT EoS through comparison of the method’s predictions for compounds with high molecular weights and several selected binary mixtures of industrial significance with experimental data such as thiols, sulphides and polynuclear aromatics is presented. Additionally, predictions of activity coefficient at infinite dilution for athermal systems are compared with the results using existing activity coefficient models. The results show that calculated pure compound parameters using the proposed GC method allow satisfactory representation of experimental data of investigated systems with the sPC-SAFT EoS. Moreover, the variety of functional groups in the available GC scheme ensures broad applications of the GC sPC-SAFT EoS.     

Vapor-liquid equilibria for binary mixtures of carbon dioxide with benzene,toluene and p-xylene

Vapor-liquid equilibrium data for carbon dioxide - benzene, carbon dioxide - toluene, and carbon dioxide - p-xylene were measured for pressures up to 6.5 MPa, and at temperatures of 353 K, 373 K, and 393 K. The solubility of benzene in the dense carbon dioxide vapor phase is higher than that of either toluene or p-xylene. In the liquid phase, carbon dioxide is more soluble in p-xylene than in toluene or benzene. The experimental data obtained were compared with calculations from three correlations: the Peng-Robinson equation, the UNIFAC activity coefficient correlation, and the Perturbed-Anisotropic- Chain Theory (PACT). All three correlations predict phase compositions in good agreement with the experimental data.     

New regularities and an equation of state for liquids

Three regularities have been introduced for liquids (T < T_C and ρ > ρ_C) based on average potential energy. The experimental data have been used to show the validity of the regularities. First, there exists near-linearity relation between and ρ for all isotherms of a liquid, where P_i and ρ are internal pressure and density, respectively. Second, changes linearly with ρ for each isotherm of any liquid, where Z and V_m are compressibility factor and molar volume, respectively. Third, a new regularity using the definition of bulk modulus and our new equation of state between reduced bulk modulus and density has been introduced, that is versus ρ must be linear for all isotherms of a liquid where B_r is the reduced bulk modulus.

A new equation of state has been also derived. The density of some liquids in the extensive ranges of temperature and pressure has been calculated using the new equation of state. The densities calculated from this equation agree with experiment to better than 0.3%. The new equation of state can predict internal pressure, thermal expansion coefficient, and isothermal compressibility of liquids within experimental error.     

Modeling and simulation of equation of state for nonadditive hard disk mixtures

We present exact results for mixtures of nonadditive hard disks and use some of them to derive a consistent model for the equation of state. We also performed molecular dynamics simulation for hard disks over a wide range of size ratios. Comparison of the model to the data shows that the model is accurate for all densities in the case of additive and slightly nonadditive (nonadditivity parameter within ±0.1) mixtures. For large nonadditivity, the model is accurate for low to moderate densities only, and starts to deteriorate at high densities.