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.     

Isobaric vapour–liquid equilibria of methyl cellosolve–aliphatic alcohol systems

Isobaric vapour–liquid equilibrium data are determined at 40 and 95 kPa for six binary mixtures consisting of methyl cellosolve (2-methoxyethanol) as a common component and aliphatic alcohols as non-common components. The non-common components include ethanol, 1-propanol, 1-butanol, 2-methyl 1-propanol, 2-methyl 2-propanol and 1-pentanol. The experimental T–x data are correlated using Wilson and NRTL equations for the liquid phase activity coefficients using a non-linear regression approach based on maximum likelihood principle. Excess free energy of mixing, computed from the activity coefficients, is positive in all the systems over the entire range of composition.     

Isobaric vapour–liquid equilibria for the binary systems of fuel oxygenates with aromatic hydrocarbons

Isobaric vapour–liquid equilibrium data at 720?mm?Hg for the binary systems of diisopropyl ether with o-xylene and m-xylene and dimethoxymethane with benzene and toluene are determined. A Swietoslawski type ebulliometer is used for the measurements. The experimental T?x data are used to estimate Wilson parameters and the parameters in turn are used to calculate the vapour compositions and activity coefficients. The activity coefficients are used to calculate molar excess Gibbs free energy (G ^E). All the systems studied here do not exhibit azeotropes. Excess Gibbs free energy values are positive over the entire range of composition for all the systems.     

Molecular dynamics investigations on Lennard–Jones systems near the gas–liquid critical point

NVT simulations on Lennard–Jones (L–J) systems near the gas–liquid critical point were performed by a direct approach. As a result, the two necessary conditions for simulating the systems in accordance with the thermodynamic limit were proposed: (i) L/ξ≳20 (L: the box-length, ξ: the correlation length), (ii) the total time of evolution, t_E>500 L–J units, for ξ≈3.5. The proposed conditions are probably very close to the sufficient ones. The influence of finite-size effects on pressure and density of small systems was qualitatively predicted. The prediction was confirmed by the simulations but only for L markedly lower than the length of typical critical wave, 2πξ. For L markedly higher, the evolutions were dominated by an effect called here the instability effect. The effect became negligible just when the condition for L/ξ was fulfilled. The ξ₀′ constant for L–J fluid was estimated from direct measurements of ξ to be 0.27±0.02 (L–J units). The thermodynamic parameters of the critical point, obtained from extrapolation, were in agreement with the results of other authors. The β_C exponent was estimated from minimization for a high range of temperatures to be 0.346. A comparison of the efficiency of NVT and NpT methods was also performed and no distinct differences were noted.     

Molecular simulation of the phase behaviour of ternary fluid mixtures: the effect of a third component on vapour–liquid and liquid–liquid coexistence

The Gibbs ensemble algorithm is implemented to determine the vapour–liquid and liquid–liquid phase coexistence of dilute ternary fluid mixtures interacting via a Lennard–Jones potential. Calculations are reported for mixtures with a third component characterised by different intermolecular potential energy parameters. Comparison with binary mixture data indicates that the choice of energy parameter for the third component affects the composition range of vapour–liquid substantially. The addition of a third component lowers the energy of liquid phase while slightly increasing the energy of the vapour phase.     

Second virial coefficients of Lennard–Jones chains

The second virial coefficients B₂ of Lennard–Jones chain fluids were calculated through Monte Carlo integration as a function of chain length m (up to 48 segments) and temperature. We found that at a fixed temperature the second virial coefficient decreases with chain length. At low temperatures, the virial coefficient changes sign from positive to negative as m increases. The simulation data also provide an estimate for the theta temperature T_Θ at which the attractive and repulsive interactions cancel each other for dilute solutions. It is found that the theta temperature T_Θ for Lennard–Jones chains with m>32 is 4.65 independent of chain length m. A comparison of simulated values of B₂ with those evaluated from two different perturbation theories for chain fluid shows that these approximate theories underestimate the second virial coefficients of Lennard–Jones chains.     

Molecular dynamics study on homonuclear and heteronuclear chains of Lennard–Jones segments

We report molecular dynamics (MD) simulation data for three simulated fluids: a homopolymer with 16 tangent Lennard–Jones (LJ) segments at the reduced temperature of 1.25, an equimolar binary homopolymer fluid with eight tangent LJ segments at 15 state points, and three corresponding copolymers with equimolar segment fraction and varying segment distribution at 15 state points. We find that the compressibility factors and energies do not change as the segment distribution varies in the copolymer example. The simulation data are compared with thermodynamic perturbation theory (TPT1) calculations. The TPT1 compressibility factors compare favorably with the MD data at high reduced temperatures but differ significantly at lower temperatures.     

A new correlation on predicting self- and mutual-diffusion coefficient of Lennard–Jones chain fluid

Based on the Chapman–Enskog theory of diffusion and molecular dynamics simulation data for Lennard–Jones chain (LJC) fluid, a new semi-empirical correlation for calculating the self-diffusion coefficient of LJC fluid is proposed. The new correlation introduces in two correction functions with six fitting parameter to modify the impact of intermolecular repulsive and attractive potential energy on molecular friction coefficient. The new correlation represents the experimental self-diffusion coefficients with an average absolute deviation (AAD) of 3.46% for 23 polyatomic compounds (1102 experimental data points) over wide ranges of temperature and pressure. On this basis, the van der Waals mixing rule is adopted to calculate the mutual-diffusion coefficient of binary LJC fluid. By comparison of calculated results with the experimental data of 12 binary LJC systems over wide range of temperature and composition, the average absolute deviation is 6.98% which verifies the accuracy and the effectiveness of the new correlation.     

New equipment using a static analytic method for the study of vapour–liquid equilibria at temperatures down to 77 K

A new apparatus designed for the study of vapour–liquid equilibria (VLE) of small molecules at temperatures down to 77 K and pressures up to 40 MPa was built, set-up and used up to 2.8 MPa. It is based on the `static-analytic' method using two pneumatic capillary sampling devices, which allow taking small and reliable in-situ samples and performing direct analyses through gas chromatography. Accurate P, x, y isothermal data obtained for the O₂–N₂ and Ar–N₂ mixtures are well represented by the Soave–Redlich–Kwong equation of state (SRK-EoS). The conclusions which can be drawn from the comparison with literature data are the following: maximum deviations on pressure, temperature and vapour phase composition, between literature experimental data and the calculated ones (using our binary interaction parameters) are not larger than 3.3%, 0.12 K and 4.2%. We also point out inconsistencies in several literature data.     

High pressure vapour–liquid equilibria for the mixture dl-γ-tocopherol/methanol

dl-γ-Tocopherol is one of the four homologues compounds also known as vitamin E. Tocopherols are present in all four forms (α-, β-, γ-, δ-tocopherol) in many natural sources and extraction design of purification processes used for their enrichment needs detailed knowledge of the phase equilibrium behaviour. Experimental vapour–liquid equilibrium data (pressure–composition isotherms) for the binary system dl-γ-tocopherol/methanol at four temperatures 473 K, 498 K, 523 K, 543 K and pressures up to 12 MPa are presented. The measurements were carried out in a static constant volume high-pressure apparatus. Different cubic equations of state and mixing rules were used to model the data. A modified Redlich–Kwong–Soave EOS gives the better correlation. On the basis of the new experimental data and those for the system dl-α-tocopherol/methanol and reasonable agreement between the two systems, using for both the same thermodynamic model, behaviour of the ternary system dl-α-tocopherol/dl-γ-tocopherol/methanol has been predicted.     

Modelling of high-pressure phase equilibria using the Sako–Wu–Prausnitz equation of state: II. Vapour–liquid equilibria and liquid–liquid equilibria in polyolefin systems

Phase equilibria in binary and ternary polyolefin systems are calculated using the cubic equation of state proposed by Sako–Wu–Prausnitz (SWP). Calculations were done for high-pressure phase equilibria in ethylene/polyethylene (LDPE) systems and for liquid–liquid equilibria (LLE) in systems containing either high-density polyethylene or poly(ethylene-co-propylene). The calculations for the copolymer/solvent systems are compared with those using the SAFT EOS. The two equations of state can describe UCST, LCST as well as U-LCST behaviour with similar accuracy. Whereas, the binary interaction parameter is temperature-independent for SAFT, it is found to be a function of temperature for the SWP model. Moreover, the influence of an inert gas on the LCST of the polyethylene/hexane system is investigated using the SWP EOS. The polydispersity of the different polyethylenes is considered in the phase equilibrium calculations using pseudocomponents chosen by the moments of the experimental molecular weight distributions.     

Thermal diffusion in Lennard–Jones fluids in the frame of the law of the corresponding states

This work is related to the definition of a reduced thermal diffusion coefficient thanks to numerical microscale molecular dynamics simulations. This cross transport process, also called Soret effect, couples mass flux and thermal gradient and is still largely misunderstood. For this study, we have applied a boundary driven non-equilibrium molecular dynamics algorithm on Lennard–Jones spheres mixtures. Simulations have been performed at a constant reduced supercritical state, using a van der Waals’ one fluid approximation in order to fulfil the law of the corresponding states. In binary mixtures, we have studied the molecular parameters and the molar fraction influences on thermal diffusion separately and then combined. It is shown that, on pressure and on thermal conductivity, the corresponding states law is fulfilled for a wide range of molecular parameters ratios. In this frame, we have then constructed simple correlations which relate thermal diffusion factor to the mixture parameters. Combining the relations obtained, a reduced thermal diffusion factor taking into account all the various contributions has been defined. Finally, it is shown that this relation enables us to estimate thermal diffusion in various binary and ternary mixtures of Lennard–Jones spheres representing alkanes with a maximum deviation of 15%.     

Equation of state for Lennard–Jones flexible ring fluids

We present a new equation of state for Lennard–Jones (LJ) flexible ring fluids. We perform Monte-Carlo simulations for freely-jointed Lennard–Jones chain fluids (3-, 6- and 8-mer) in the canonical ensemble and obtain the intramolecular end-to-end pair correlation function data under extensive density and temperature conditions. We correlate these as a function of the density, the temperature and the number of segments in a chain. We apply this function to thermodynamic perturbation theory (TPT) and obtain a new equation of state for Lennard–Jones flexible ring fluids. We also compare existing simulation data [J. Chem. Phys. 104 (1996) 1729] with the results obtained using the newly derived equation of state.     

Onsager heat of transport at the aniline liquid–vapour interface

The heat of transport for passage of matter through a gas–liquid interface has been determined for the aniline liquid–vapour system, by measuring stationary-state pressure differences produced by known temperature differences over a distance of 2 mm on the vapour side of the interface. For the range of pressures used, 2 mm is between 6 and 34 mean free paths. Coupling of the heat and matter fluxes is significant over the whole of this range. At the higher pressures the heat of transport is more than 20% of the heat of condensation; at the lower pressures it is more than 50%.     

Molecular dynamics simulation study of friction and diffusion of a tracer in a Lennard–Jones solvent

The friction and diffusion coefficients of a tracer in a Lennard–Jones (LJ) solvent are evaluated by equilibrium molecular dynamics simulations in a microcanonical ensemble. The solvent molecules interact through a repulsive LJ force each other and the tracer of diameter σ₂ interacts with the solvent molecules through the same repulsive LJ force with a different LJ parameter σ. Positive deviation of the diffusion coefficient D of the tracer from a Stokes–Einstein behavior is observed and the plot of 1/D versus σ₂ shows a linear behavior. It is also observed that the friction coefficient ζ of the tracer varies linearly with σ₂ in accord with the prediction of the Stokes formula but shows a smaller slope than the Stokes prediction. When the values of ratios of sizes between the tracer and solvent molecules are higher than 5 approximately, the behavior of the friction and diffusion coefficients is well described by the Einstein relation D = k _B T/ζ, from which the tracer is considered as a Brownian particle.     

Exact solution of the Schrödinger equation with a Lennard–Jones potential

The Schrödinger equation with a Lennard–Jones potential is solved by using a procedure that treats in a rigorous way the irregular singularities at the origin and at infinity. Global solutions are obtained thanks to the computation of the connection factors between Floquet and Thomé solutions. The energies of the bound states result as zeros of a function defined by a convergent series whose successive terms are calculated by means of recurrence relations. The procedure gives also the wave functions expressed either as a linear combination of two Laurent expansions, at moderate distances, or as an asymptotic expansion, near the singular points. A table of the critical intensities of the potential, for which a new bound state (of zero energy) appears, is also given.     

Predicting vapor–liquid equilibria of fatty systems

In the present work, a group contribution method is proposed for the estimation of the vapor pressure of fatty compounds. For the major components involved in the vegetable oil industry, such as fatty acids, esters and alcohols, triacylglycerols (TAGs) and partial acylglycerols, the optimized parameters are reported. The method is shown to be accurate when it is used together with the UNIFAC model for estimating vapor–liquid equilibria (VLE) of binary and multicomponent fatty mixtures comprised in industrial processes such as stripping of hexane, deodorization and physical refining. The results achieved show that the group contribution approach is a valuable tool for the design of distillation and stripping units since it permits to take into account all the complexity of the mixtures involved. This is particularly important for the evaluation of the loss of distillative neutral oil that occurs during the processing of edible oils.The combination of the vapor pressure model suggested in the present work with the UNIFAC equation gives results similar to those already reported in the literature for fatty acid mixtures and oil–hexane mixtures. However, it is a better tool for predicting vapor–liquid equilibria of a large range of fatty systems, also involving unsaturated compounds, fatty esters and acylglycerols, not contemplated by other methodologies. The approach suggested in this work generates more realistic results concerning vapor–liquid equilibria of systems encountered in the edible oil industry.     

Isothermal vapour–liquid equilibria in the binary and ternary systems consisting of an ionic liquid, 1-propanol and CO2

Vapour–liquid equilibrium measurements for binary and ternary systems containing carbon dioxide, 1-propanol, and 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide or 1-decyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide or 1-ethyl-3-methylimidazolium trifluoromethanesulfonate ionic liquids are presented in this work. The binary CO₂ + 1-decyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide system at 313.15 K at pressure range from 2 to 14.4 MPa was examined. The obtained phase envelop shows that even at low pressure of CO₂ the solubility of the gas in the ionic liquid is high. The ternary phase equilibria were studied at 313.15 K and pressures in the range from 9 to 12 MPa. The ternary phase diagrams show that higher CO₂ pressure diminishes the miscibility gap.     

Molecular simulation of vapor–liquid equilibria of toxic gases

In this paper, we derived the potential parameters for three toxic gases, hydrogen sulfide, phosgene and nitrous oxide, modeled by the effective Stockmayer potential model proposed by Gao et al. [Fluid Phase Equilib. 137 (1997) 87]. The vapor–liquid equilibria (VLE) of these substances have been extensively investigated over a wide range of temperatures by the Gibbs ensemble Monte Carlo (GEMC) technique. The simulated saturated densities and pressures are in good agreement with experimental data. The critical properties obtained by regression of the simulated data also agree well with the experimental values. The present work demonstrates that the effective Stockmayer potential can describe well the toxic gases concerned.