Calculation of critical lines of hydrocarbon/water systems by extrapolating mixing rules fitted to subcritical equilibrium data

The phase behavior of water/hydrocarbon mixtures in a wide range of pressures is important for various applications ranging from reservoir engineering to environmental engineering. In this work, mutual solubility and critical loci of hydrocarbon/water systems are calculated using the Peng–Robinson–Stryjek–Vera cubic equation of state with four mixing rules: (1) van der Waals mixing rules with one binary interaction parameter (vdW-1), (2) van der Waals mixing rules with asymmetric composition dependent binary interaction parameter (vdW-A), (3) Wong–Sandler mixing rules (WS) and (4) second-order modified Huron–Vidal mixing rules (MHV2). It was found that the parameters obtained from correlating liquid–liquid equilibria using different mixing rules would lead to prediction of completely different forms of critical behavior. Unusual branches of critical loci were found with WS and MHV2 mixing rules. Therefore, equation of states models must be used with extreme caution when applied for predicting phase behavior over wide ranges of temperatures and pressures.     

A note on excess Gibbs energy models,equations of state and the local composition concept

A density-dependent local composition expression for the residual energy is derived from a generalized NRTL expression for the excess energy and the van der Waals fluid theory. Integration of this expression yields a volume-dependent expression for the Helmholtz energy from which equations of state utilizing the local composition concept are derived and which in the high-density limit contain the well-known activity coefficient models.The local composition versions of the Carnahan—Starling—van der Waals, the Redlich—Kwong—Soave and the Peng—Robinson equations of state are derived. It is further shown that the group contribution versions of the NRTL, the Wilson and the UNIQUAC excess models may be derived from the generalized NRTL expression for the residual energy when applied to groups instead of molecules.It is thus demonstrated that all current local composition activity-coefficient models can be derived from a local composition version of the van der Waals equation of state using different sets of assumptions. In the same way the van Laar, the Scatchard—Hildebrand and the Flory—Huggins activity coefficient models are obtained from the van der Waals equation of state using the original mixing rules.     

Thermodynamic consistency of high pressure ternary mixtures containing a compressed gas and solid solutes of different complexity

A thermodynamic consistency test applicable to high pressure binary gas–solid mixtures is extended to ternary mixtures containing a compressed gas and two solid solutes. A high pressure mixture containing carbon dioxide as solvent and two chemically similar solutes (2,3 dimethylnaphthalene and 2,6 dimethylnaphthalene) and a high pressure mixture containing carbon dioxide as solvent and two chemically different solutes (capsaicin and β-carotene), are considered in the study. Several sets of isothermal solubility data for binary and ternary mixtures are considered in the study. The Peng–Robinson equation of state with the mixing rules of Wong and Sandler have been employed for modeling the solubility of the solid in the case of binary mixtures, while the classical van der Waals mixing rules were used for modeling the ternary mixtures containing two solid solutes. Then the proposed thermodynamic consistency test has been applied. The results show that the thermodynamic test for ternary mixtures can be applied with confidence determining consistency or inconsistency of the experimental data used.     

Computing all the azeotropes in refrigerant mixtures through equations of state

Azeotropic mixtures of fluorocarbon (FC) and hydro fluorocarbon (HFC) with hydrocarbons are gaining popularity as drop-in substitutes for CFCs and HCFCs. A method to compute all the azeotropes in a refrigerant mixture through the equation of state approach is described. The method allows prediction of all the azeotropes in a refrigerant mixture and is in close agreement with the experimental data. Both the vapor and the liquid phase non-idealities are incorporated through fugacity coefficients modeled using Peng–Robinson–Stryjek–Vera equation of state with Wong-Sandler and van der Waals mixing rules. Homotopy continuation based methodology guarantees computation of all the solutions of necessary and sufficient condition of azeotropy in multicomponent refrigerant mixtures. The method establishes the pressure dependency of azeotropic composition allowing prediction of bifurcation pressure where refrigerant azeotropes may appear or disappear and predicts azeotropes at elevated pressures. The approach is independent of equation of state and mixing rules but rely on their ability to represent the phase behavior. The approach is tested with R23–R13, propane–R227ea binary mixtures and a ternary mixture of R32–R125–R143a.     

Bubble pressures and saturated liquid molar volumes of trichlorofluoromethane—chlorodifluoromethane mixtures. Representation of refrigerant-mixtures vapor—liquid equilibrium data by a modified form of the Peng—Robinson equation of state

Experimental vapor—liquid equilibrium data and saturated liquid molar volumes of chlorodifluoromethane—trichlorofluoromethane binary mixtures have been obtained at four temperatures (298.15, 323.15, 348.15 and 373.15 K) using apparatus described previously.The experimental vapor—liquid equilibria are represented well by a modified form of the Peng—Robinson equation of state with one interaction parameter, but the mean deviation between the calculated and experimental densities is 5%.Vapor—liquid data for binary refrigerant mixtures from the literature are treated using the modified form of the Peng—Robinson equation of state with one adjusted interaction parameter in the mixing rule for a. The representation is fair and is not improved by introducing an additional parameter in the mixing rule for b.     

Application of a generalized van der Waals equation of state to several nonpolar mixtures

A generalized van der Waals equation of state, applied recently (Nguyen Van Nhu and Kohler, 1995) to the calculation of excess properties and phase equilibria for the mixture methane + ethane, is now extended to several nonpolar binary mixtures.Improved mixing rules for the van der Waals attractive term and for the correction term are proposed. With these mixing rules, the equation gives good agreement for vapour-liquid and liquid-liquid equilibria over a large temperature range for 29 binary mixtures. The agreement of mixture volumes and cross second virial coefficients is also satisfactory.     

Vapor–liquid equilibrium of carbon dioxide with diethyl methylmalonate,diethyl ethylmalonate and diethyl n-butylmalonate at elevated pressures

Vapor–liquid equilibrium (VLE) data for binary mixtures of CO₂ with homologous esters of diethyl methylmalonate, diethyl ethylmalonate, and diethyl n-butylmalonate at 308.2, 318.2, and 328.2 K, respectively, over the pressure range 1.4–8.4 MPa were measured using a semi-flow apparatus. New gas solubility data for CO₂ in esters are presented, and the Henry’s law constants for CO₂ in these esters are evaluated by employing the Krichevsky–Ilinskaya (KI) equation. The VLE data were also correlated using the Soave–Redlich–Kwong and the Peng–Robinson equations of state (EOSs) with various types of mixing rules. It is shown that EOS with both the van der Waals mixing rules and the two adjustable parameters yield satisfactory correlation results.     

A generalized technique to obtain pure component parameters for two-parameter equations of state

A generalized technique is presented for the calculation of the pure component parameters for use in a general two-parameter equation of state. The method requires as input data the vapor pressure and saturated liquid volume of a component at a given temperature, and is both accurate and simple to use.Correlations for the calculation of the parameters are presented for the van der Waals, Redlich—Kwong and Peng—Robinson forms of cubic equations of state. A comparison is made between the new method and the corresponding-states approach.     

A local composition extension of the van der Waals mixing rule for PR cubic equation of state

New mixing rules (VWLC-I and II) capable of connecting van der Waals (VDW) to CEOS/A^E mixing rule models were developed. These models are able to incorporate the same multi-component mixture parameters obtained for the van der Waals and CEOS/A^E models simultaneously. The VWLC mixing rules directly incorporate local compositions into the cubic equations of state (CEOS). The energy parameters required for the local compositions are calculated from the CEOS parameters. The Peng–Robinson (PR) CEOS was used for this study. Binary interactions parameters were obtained by adjusting the vapor pressure of the binary mixture for several low and high-pressure systems. The predictive capabilities of the VWLC mixing rules were tested by vapor–liquid equilibria calculations for low and high-pressure multicomponent systems. The results were compared with the predictions of the VDW mixing rule and a Huron–Vidal (HV) kind of CEOS/A^E-NRTL mixing rule. The VWLC mixing rules are consistent models giving good results in a broad range of pressures and temperatures in binary and multicomponent mixtures. They compare favorably with the CEOS/A^E-NRTL mixing rule for low-pressure systems. In high-pressure ternary systems VWLC-I and II give good predictions, much better, in fact, than the CEOS/A^E-NRTL mixing rule.     

A three-parameter cubic equation of state for asymmetric mixture density calculations

A new three-parameter cubic equation of state of the van der Waals type with one parameter temperature dependent, P = RT/(V − b) − a(T)/[V(V + c) + b(3V + c)], has been developed for representation of liquid volumes (or densities) for asymmetric mixtures such as CO₂C₁₉ and C₁C₁₀. The calculated results are better than those obtained from the two-parameter Peng—Robinson equation, the three parameter Schmidt—Wenzel equation, the volume-translated Soave—Redlich—Kwong equation proposed by Peneloux et al., and the volume-translated Peng—Robinson equation developed in this work. The parameters of the new equation have been generalized in terms of the acentric factor ω and reduced temperature T_r.     

Applicability of cubic equation of state mixing rules on correlation of excess molar volume of non-electrolyte binary mixtures – Part II

The excess molar volume (V?^E) data of the 24 binary highly non-ideal mixtures containing dicyclic ethers (593 data points) were correlated by the Peng–Robinson–Stryjek–Vera (PRSV) cubic equation of state (CEOS) coupled with two different classes of mixing rules: (i) the composition dependent van der Waals (vdW) mixing rule and (ii) the excess free energy mixing rules (CEOS/G?^E) based on the approach of the Gupta–Rasmunssen–Fredenslund (GRF), as well as the Twu–Coon–Bluck–Tilton (TCBT) mixing rule; both rules with the NRTL equation as the G?^E model. The results obtained by these models show that the type of applied mixing rules, including the number and position of interaction parameters are of great importance for a satisfactory correlation of V?^E data. The GRF mixing rules gave mostly satisfactory results for V?^E correlation of the non-ideal binary systems available at one isotherm of 298.15?K, while for the correlation in temperature range from 288.15 to 308.15?K the TCBT model can be recommended.     

The experiments and correlations of the solubility of ethylene in toluene solvent

Polymer cyclic olefin copolymer (COC) is produced from the reaction of attaching ethyl groups to the norbornene monomer in liquid phase. The first step of process is dissolving ethylene in a liquid phase where toluene is present as a cosolvent. Thus, the solubility of ethylene in liquid toluene is the most important factor affecting the production of COC. In this study, the solubility of ethylene in toluene was measured in the temperature range from 323.15 to 423.15 K and pressure range from 5 to 25 bar. The experiments were conducted by the method of pressure decaying with a newly designed apparatus. The experimental results show that the solubility of ethylene in toluene increases with increasing pressure but decreases with increasing temperature.The experimental solubility data were expressed in the vapor–liquid equilibrium relationship and correlated fairly well by the bubble–pressure calculation with the Peng–Robinson equation of state (PR EOS) incorporated with the van der Waals one-fluid and the Zhong–Masuoka mixing rules with the consideration of binary interaction parameters. The results showed the van der Waals (vdW-1) mixing rule is slightly better than the Z–M mixing rule for pressure correlation but the Z–M mixing rule is slightly better for vapor composition correlation.A semi-empirical solubility equation with four parameters for the present binary system was proposed in this study. This proposed model estimates the solubility easier and as accurate as the PR EOS does for the present system.     

Applicability of cubic equation of state mixing rules on correlation of excess molar volume of Non-Electrolyte binary mixtures

The excess molar volume V ^E data of the binary liquid systems were correlated by the Peng–Robinson–Stryjek–Vera equation of state coupled with two different types of mixing rules: composition dependent van der Waals mixing rule (vdW) and the mixing rule based on the Gupta–Rasmussen–Fredenslund method (GRF), with the NRTL equation as G ^E model. The results obtained by these models show that type of applied mixing rule, a number and position of interaction parameters are of great importance for a satisfactory correlation of V ^E data. The GRF mixing rules coupled with the NRTL model gave mostly satisfactory results for V ^E correlation of the nonideal binary systems of diverse complexity.     

A novel equation of state for the prediction of thermodynamic properties of fluids

This work proposes a new equation of state (EOS) based on molecular theory for the prediction of thermodynamic properties of real fluids. The new EOS uses a novel repulsive term, which gives the correct hard sphere close packed limit and yields accurate values for hard sphere and hard chain virial coefficients. The pressure obtained from this repulsive term is corrected by a combination of van der Waals and Dieterici potentials. No empirical temperature functionality of the parameters has been introduced at this stage. The novel EOS predicts the experimental volumetric data of different compounds and their mixtures better than the successful EOS of Peng and Robinson. The prediction of vapor pressures is only slightly less accurate than the results obtained with the Peng-Robinson equation that is designed for these purposes. The theoretically based parameters of the new EOS make its predictions more reliable than those obtained from purely empirical forms.     

Low pressure vapor–liquid equilibrium in ethanol + congener mixtures using the Wong–Sandler mixing rule

Phase equilibrium in binary ethanol mixtures found in alcoholic beverage production has been analyzed using a cubic equation of state (EoS) and suitable mixing and combining rules. The main objective of the study is the accurate modeling of the congener concentration in the vapor phase (substances different from ethanol), considered to be an important enological parameter in the alcohol industry. The Peng–Robinson (PR) equation of state has been used and the Wong–Sandler (WS) mixing rules, that include a model for the excess Gibbs free energy, have been incorporated into the equation of state constants. In the Wong–Sandler mixing rules the van Laar (VL) model for the excess Gibbs energy has been used. This combination of equations of state, mixing rules and combining rules are commonly applied to high pressure phase equilibrium and have not yet been treated in a systematic way to complex low pressure ethanol mixtures as done in this work. Nine binary ethanol + congener mixtures have been considered for analysis. Comparison with available literature data is done and the accuracy of the calculations is discussed, concluding that the model used is accurate enough for engineering applications.     

Equations of state for strongly nonideal fluid mixtures: Application of local compositions toward density-dependent mixing rules

A local-composition, two-fluid model has been developed for equation-of-state calculations of fluid-phase equilibria for asymmetric mixtures; it is applicable to any equation of state of the van der Waals form. A modification of the quasichemical theory of Guggenheim is applied to mixtures at all fluid densities. Desirable boundary conditions are met at low densities, at high densities, and at high temperatures.In effect, the local-composition model uses density-dependent mixing rules. It contains no new adjustable binary parameters and can be extended to multicomponent mixtures without ternary (or higher) parameters. It appears that, when compared to conventional one-fluid models, significant improvement may be obtained in predictions for vapor-liquid equilibria of typical asymmetric mixtures.     

Vapor‐Liquid Phase Equilibria for Carbon Dioxide+Isopentanol Binary System at Elevated Pressure

High‐pressure vapor‐liquid phase equilibrium data for carbon dioxide+isopentanol were measured at temperatures of 313.2, 323.1, 333.5 and 343.4 K in the pressure range of 4.64 to 12.71 MPa in a variable‐volume high‐pressure visual cell. The experimental data were well correlated with Peng‐Robinson equation of state (PR‐EOS) together with van der Waals‐2 two‐parameter mixing rule, and the binary interaction parameters were obtained. Henry coefficients and partial molar volumes of CO₂ at infinite dilution were estimated based on Krichevsky‐Kasarnovsky equation, and Henry coefficients increase with increasing temperature, however, partial molar volumes of CO₂ at infinite dilution are negative and the magnitudes decrease with temperature.     

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.     

High-pressure phase equilibria for the carbon dioxide–2-methyl-1-butanol,carbon dioxide–2-methyl-2-butanol,carbon dioxide–2-methyl-1-butanol–water,and carbon dioxide–2-methyl-2-butanol–water systems

High-pressure vapor–liquid equilibria for the binary carbon dioxide–2-methyl-1-butanol and carbon dioxide–2-methyl-2-butanol systems were measured at 313.2 K. The phase equilibrium apparatus used in this work is of the circulation type in which the coexisting phases are recirculated, on-line sampled, and analyzed. The critical pressure and corresponding mole fraction of carbon dioxide for the binary carbon dioxide–2-methyl-1-butanol system at 313.2 K were found to be 8.36 MPa and 0.980, respectively. The critical point of the binary carbon dioxide–2-methyl-2-butanol was also found 8.15 MPa and 0.970 mole fraction of carbon dioxide. In addition, the phase equilibria of the ternary carbon dioxide–2-methyl-1-butanol–water and carbon dioxide–2-methyl-2-butanol–water systems were measured at 313.2 K and several pressures. These ternary systems showed the liquid–liquid–vapor phase behavior over the range of pressure up to their critical point. The binary equilibrium data were all reasonably well correlated with the Redlich–Kwong (RK), Soave–Redlich–Kwong (SRK), Peng–Robinson (PR), and Patel–Teja (PT) equations of state with eight different mixing rules the van der Waals, Panagiotopoulos–Reid (P&R), and six Huron–Vidal type mixing rules with UNIQUAC parameters.     

Vapor–liquid equilibria for the carbon dioxide + propane system over a temperature range from 253.15 to 323.15 K

The isothermal vapor–liquid equilibrium (VLE) data were measured for the binary system of the carbon dioxide + propane at eight temperatures ranging from 253.15 to 323.15 K. Since the blends are natural refrigerants and have good thermophysical properties, they are considered as promising alternative refrigerants. The VLE measurement was performed at pressures up to 7.2 MPa in the circulation type equipment with a view cell. The binary system was found to be a zeotropic mixture in the tested temperature range and could be correlated with a sufficient accuracy by using the Peng–Robinson equation of state (PR EoS) with the van der Waals one fluid mixing rule. A comparison with published experimental VLE data has been carried out by means of the PR equation of state.