Equation of state for square-well chain molecules with variable range. I: Application for pure substances

An equation of state (EOS) for square-well chain molecules with variable range developed on the basis of statistical mechanics for chemical association in our previous work is employed for the calculations of pVT properties and vapor–liquid equilibria (VLE) of pure non-associating fluids. The molecular parameters for 73 normal substances and 46 polymers are obtained from saturated vapor pressure and liquid molar volume data for normal fluids or pVT data for polymers. Linear relations are found for the molecular parameters of normal fluids with their molecular weight of homologous compounds. This indicates that the model parameters of homologous series, subsequently pVT and VLE, can be predicted when experimental data are not available. The predicted saturated vapor pressures and/or liquid volumes are satisfactory through the generalized model parameters. The calculated VLE and pVT for normal fluids and polymers by this EOS are compared with those from other engineering models, respectively.     

A new development of equation of state for square-well chain-like molecules with variable width 1.1 ≤ λ ≤ 3

A new equation of state (EOS) for square-well chain molecules and their mixtures with variable well-width range (SWCF-VR-EOS) has been developed based on the sticky-point model for chemical association. Two important modifications have been made. Firstly, a new dispersion contribution to the Helmholtz function of monomers due to square-well potential with variable well-width range of 1.1 ≤ λ ≤ 3 was established by combining the second-order perturbation theory and Chiew's PY2 approximation of the integral equation. Secondly, the contribution of chain formation to the Helmholtz function is divided into two parts: One is from the hard sphere, and the other is from the effect of square-well potential described via the nearest-neighbor and next-to-nearest-neighbor residual cavity correlation functions (CCFs). The predicted compressibility factors and vapor–liquid coexistence curves for square-well fluids as well as for their mixtures are in good agreement with simulations. The new EOS has been applied to real non-associating fluids and the corresponding mixtures by adopting one-fluid mixing rule. The pVT and vapor–liquid equilibria (VLE) can be correlated satisfactorily. The model parameters for some homologous compounds are found to be linear with the molar mass indicating that the pVT and VLE of those homologous compounds can be predicted even if no accurate data are available.     

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.     

Phase equilibria involved in extractive distillation of dipropyl ether + 1-propyl alcohol using 2-ethoxyethanol as entrainer

Consistent vapour–liquid equilibrium data at 101.3 kPa have been determined for the ternary system dipropyl ether + 1-propyl alcohol + 2-ethoxyethanol and two constituent binary systems: dipropyl ether + 2-ethoxyethanol and 1-propyl alcohol + 2-ethoxyethanol. The dipropyl ether + 2-ethoxyethanol system shows positive deviations from ideal behaviour and 1-propyl alcohol + 2-ethoxyethanol system exhibits no deviation from ideal behaviour. The activity coefficients and the boiling points were correlated with their compositions by the Wilson, NRTL and UNIQUAC equations. It is shown that the models allow a very good prediction of the phase equilibria of the ternary system using the pertinent parameters of the binary systems. The parameters obtained from binary data were utilized to predict the phase behaviour of the ternary system. The results showed a good agreement with the experimental values. Moreover, the entrainer capabilities of 2-ethoxyethanol were compared with 1-pentanol, butyl propionate and N,N-dimethylformamide, concluding that N,N-dimethylformamide is the best entrainer.     

Prediction of phase equilibria in the systems carbon dioxide (1)–fatty acids (2) by two cubic EOS models and classical mixing rules without binary adjustable parameters

This study evaluates the accuracy of estimating data in the series of systems carbon dioxide (1)–fatty acids (2) by two cubic equations of state, namely the EOS of Peng and Robinson in its original form and the recently proposed cubic EOS. The classical mixing rules are implemented in entirely predictive manner, i.e. without binary adjustable parameters. It is demonstrated that both models may yield reliable predictions of the data. However the EOS of Peng and Robinson fails in predicting the topology of phase behavior of the heavy homologues. The second cubic EOS predicts the Global Phase Behavior in the homologous series under consideration satisfactorily accurate, which in particular means qualitatively correct estimation of the liquid–liquid equilibria. The recently proposed EOS has no significant advantage over the EOS of Peng and Robinson in predicting the vapour–liquid equilibria data under consideration.     

Vapour pressure and excess Gibbs energy of binary 1,2-dichloroethane + cyclohexanone,chloroform + cyclopentanone and chloroform + cyclohexanone mixtures at temperatures from 298.15 to 318.15 K

The vapour pressures of the binary systems 1,2-dichloroethane + cyclohexanone, chloroform + cyclopentanone and chloroform + cyclohexanone mixtures were measured at temperatures between 298.15 and 318.15 K. The vapour pressures vs. liquid phase composition data for three isotherms have been used to calculate the activity coefficients of the two components and the excess molar Gibbs energies, G^E, for these mixtures, using Barker's method. Redlich–Kister, Wilson, NRTL and UNIQUAC equations, taking into account the vapour phase imperfection in terms of the 2-nd virial coefficient, have represented the G^E values. No significant difference between G^E values obtained with these equations has been observed. Our data on vapour–liquid equilibria (VLE) and excess properties of the studied systems are examined in terms of the DISQUAC and modified UNIFAC (Dortmund) predictive group contributions models.     

Calculation of pVT and vapor–liquid equilibria of copolymer systems based on an equation of state

A molecular thermodynamic model for copolymers and their mixtures has been established by adopting the hard-sphere-chain fluid as a reference and a square-well (SW) term as well as an association term as a perturbation. The latter is introduced to consider various associating functions in a chain-like molecule based on the shield-sticky model of chemical association. The model adopts five molecular parameters, i.e. r_i, σ_ii ε_ii/k, δε_ii/k and ω_ii, for a polymer species i, where the last two are responsible for association. These parameters can be obtained from the pVT data of the corresponding molten homopolymer i. The model can be used to correlate pVT data for molten copolymers with an adjustable parameter describing the interaction between different polymer species. The model can also be used to calculate vapor–liquid equilibria (VLE) for copolymer solutions with three adjustable interaction parameters.     

Modeling phase equilibria of alkanols with the simplified PC-SAFT equation of state and generalized pure compound parameters

The simplified PC-SAFT equation of state has been applied to liquid–liquid, vapor–liquid and solid–liquid equilibria for mixtures containing 1- or 2-alkanols with alkanes, aromatic hydrocarbons, CO₂ and water. For the alkanols we use generalized pure compound parameters. This means that two of the physical pure compound parameters, m (segment number) and σ (segment diameter), are obtained from linear extrapolations, since m and mσ³, increase linearly with respect to the molar mass, and moreover, the two association parameters (association energy and association volume) were assumed to be constant for all alkanols. Only the dispersion energy is fitted to experimental data. Thus it is possible to estimate parameters for several 1- and 2-alkanols. The final aim is to develop a group contribution approach for PC-SAFT which is suitable for complex compounds, considering that the motivation of this project is to obtain a thermodynamic model which can be used in the development of sophisticated products such as pharmaceuticals, polymers, detergents or food ingredients. One of the severe limitations in applying SAFT-type equations of state to these compounds is that the procedure for obtaining the pure compound parameters is usually based on fitting to saturated vapor pressure and liquid density data over an extended temperature range. However, such data are rarely available for complex compounds. To verify the new pure compound parameters, comparisons to ordinary optimized alkanol parameters, where all five pure compound parameters were fitted to experimental liquid density and vapor pressure data, were made. The results show that the new generalized alkanol parameters from this work perform at least as well as other alkanol parameter sets.     

Reliable evaluation of temperature-independent parameters in correlation of vapour–liquid equilibrium data

A new algorithm/program has been elaborated for simultaneous processing of different sets of vapour–liquid equilibrium data. The program was tested with six binary hexane + isomeric pentanol systems, each of them measured at three different isobaric conditions and one isothermal system of tert-butyl-methyl-ether + 2-methyl-2-propanol measured at three different temperatures. The correlation uses the maximum likelihood method, taking into account real behaviour of vapour phase. The parameters obtained are valid within the whole temperature range of the data, and are consistent in comparison with those obtained from individual correlations of isobars or isotherms. Results are presented for the Wilson and NRTL equations.     

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.     

Investigation of the surface tension of methane and n-alkane mixtures by perturbed-chain statistical associating fluid theory combined with density-gradient theory

The perturbed-chain statistical associating fluid theory (PC-SAFT) and density-gradient theory are used to construct an equation of state to describe the phase behavior of binary methane–n-alkane mixtures. With the molecular parameters and influence parameters regressed from bulk properties and surface tensions of pure fluids, respectively as input, both the bulk and interfacial properties are investigated. The surface tension of the binary systems methane–propane, methane–pentane, methane–heptane and methane–decane are predicted, and the results are satisfactory compared with the experimental data. Our results show that PC-SAFT combined with density-gradient theory is able to describe the interfacial properties of binary methane–n-alkane mixtures in wide temperature and pressure ranges, and illustrate the influence of the equilibrium bulk properties and chain length of n-alkane molecule on the interfacial properties.     

Phase equilibrium for the systems diisopropyl ether,isopropyl alcohol + 2,2,4-trimethylpentane and +n-heptane at 101.3 kPa

Consistent vapour–liquid equilibrium data for the ternary systems diisopropyl ether + isopropyl alcohol + 2,2,4-trimethylpentane and diisopropyl ether + isopropyl alcohol + n-heptane are reported at 101.3 kPa. The vapour–liquid equilibrium data have been correlated by Wilson, NRTL and UNIQUAC equations. The ternary systems do not present ternary azeotropes.     

Measurement and correlation of vapor–liquid equilibria for methanol + methyl laurate and methanol + methyl myristate systems near critical temperature of methanol

A flow-type method was adopted to measure the vapor–liquid equilibria for methanol + methyl laurate and methanol + methyl myristate systems at 493–543 K, near the critical temperature of methanol (T_c = 512.64 K), and 2.16–8.49 MPa. The effect of temperature and fatty acid methyl esters to the phase behavior was discussed. The mole fractions of methanol in liquid phase are almost the same for both systems. In vapor phase, the mole fractions of methanol are very close to unity at all temperatures. The present vapor–liquid equilibrium data were correlated by PRASOG. A binary parameter was introduced to the combining rule of size parameter. The binary parameters of methanol + fatty acid methyl ester systems were determined by fitting the present experimental data. The correlated results are in good agreement with the experimental data. The vapor–liquid equilibria for methanol + methyl laurate + glycerol and methanol + methyl myristate + glycerol ternary systems were also predicted using the methanol + fatty acid methyl ester binary parameters. The mole fractions of methanol in vapor phase are around unity even if glycerol is included in the systems.     

(Vapour + liquid) equilibria for (2,2-dimethoxypropane + methanol) and (2,2-dimethoxypropane + acetone)

The isothermal and isobaric (vapour + liquid) equilibria for (2,2-dimethoxypropane + methanol) and (2,2-dimethoxypropane + acetone) measured with an inclined ebulliometer are presented. The experimental results are analysed using the UNIQUAC equation with the temperature-dependent binary parameters with satisfactory results. Isobaric (vapour + liquid) equilibria data for these systems at p=99.99 kPa are compared with the literature data. Experimental vapour pressure of 2,2-dimethoxypropane are also included.