Critical points of hydrocarbon mixtures with the Peng–Robinson,SAFT, and PC-SAFT equations of state

The phase behavior of fluids at high pressures can be rather complex, even for mixtures of relatively simple molecules, such as hydrocarbons. In this work, we use the Hicks and Young algorithm to calculate mixture critical points, comparing five modeling options: Peng–Robinson EOS: (1) original and (2) with parameters fitted from molar volume and vapor pressure data; (3) SAFT EOS; and PC-SAFT EOS: (4) original and (5) with refitted parameters to match pure component critical data. Calculations were carried out for binary hydrocarbon mixtures and 29 multicomponent mixtures. The SAFT EOS provided the worst representation of the systems tested and, interestingly, the conventional cubic EOS provided, in general, the best representation.     

Volume translation in equations of state as a means of accurate property estimation

Molar volumes predicted by cubic volumetric equations of state (EOS)s over a range of reduced temperatures and pressures from T_r = 0.5–2.0 and P_r = 0.5–10.0 were compared to reference molar volume values for water, carbon dioxide, hydrogen sulfide, ammonia, nitrogen, oxygen, argon, and the first eight normal paraffin alkanes. A similarity of molar volume residuals was observed for predictions by the Soave–Redlich–Kwong EOS, indicating that volume translation can be used to improve the accuracy of these molar volume predictions. The necessity of both density and temperature dependence in the volume translation is established; guidelines for maintaining thermodynamic consistency and a correlation for identifying the maxima in molar volume residuals are provided.     

Quintic equation of state for pure substances in sub- and supercritical range

A new quintic equation of state (EOS) for pure substances and mixtures is proposed. The equation is based on critical parameters and one saturation point. The proposed Q5EOS is a generalisation of many cubic equations of state. Equation Q5 has five parameters, four of which are temperature-independent. The temperature-dependent parameter a is expressed by a relation based on a simple power function. Parameters defining this function can be calculated from saturation data, Boyle temperature and supercritical data.     

A comparison of mixing rules for the combination of COSMO-RS and the Peng–Robinson equation of state

In this work, the COSMO-RS model is combined with a volume-translated Peng–Robinson equation of state (EOS) via a G^E$G^{\text{E}}$-based mixing rule. The performance of several mixing rules previously published for this purpose is compared and semi-empirical modifications to one of them are introduced to improve its performance in our application. The new mixing rule contains three internal parameters that are adjusted to achieve consistency between the mixing rule and COSMO-RS. No experimental binary data is needed for our EOS. The new COSMO-RS-based, predictive EOS introduces a density dependence into COSMO-RS and extends its applicability to higher pressures and to mixtures containing supercritical components.     

A modification to the Peng–Robinson-fitted equation of state for pure substances

In this work we present two modifications to the Peng–Robinson-Fitted equation of state where pure component parameters are regressed to vapor pressure and saturated liquid density data. The first modification (PR-f-mod) is a method that enhances the equation of state pure component property predictions through simple temperature dependent pure component parameters. In the second modification (PR-f-prop) we propose a temperature dependency for co-volume b in the repulsive parameter of the EoS, and revise the temperature function in the attractive term. The agreement with experimental data for 72 pure substances, including highly polar compounds, is remarkably good. We obtain average absolute deviations in saturated liquid density of less than 1% for all substances studied.     

Experimental study of high pressure phase equilibrium of (CO2 + NO2/N2O4) mixtures

Experimental bubble pressure, as well as liquid density of (CO₂ + NO₂/N₂O₄) mixtures are reported at temperatures ranging from (298 to 328.45) K. Experiments were carried out using a SITEC high-pressure variable volume cell. Transition pressures were obtained by the synthetic method and liquid density was deduced from measurement of the cell volume. Correlation of experimental results was carried out without considering chemical equilibrium of NO₂/N₂O₄ system. (Liquid + vapour) equilibrium was found to be accurately modelled using the Peng–Robinson equation of state with classical quadratic mixing rules and with a binary interaction coefficient k_ij equal to zero. Nevertheless, modelling of liquid density values was unsatisfactory with this approach.     

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.     

Thermodynamic consistency of experimental VLE data for asymmetric binary mixtures at high pressures

A thermodynamic consistency of isothermal vapor–liquid equilibrium data for 9 non-polar and 8 polar binary asymmetric mixtures at high pressures has been evaluated. A method based on the isothermal Gibbs–Duhem equation was used for the test of thermodynamic consistency using a Φ–Φ approach. The Peng–Robinson equation of state coupled with the Wong–Sandler mixing rules were used for modeling the vapor–liquid equilibrium (VLE) within the thermodynamic consistency test. The VLE parameters calculations for asymmetric mixtures at high pressures were highly dependent on bubble pressure calculation, making more convenient to eliminate the data points yielding the highest deviations in pressure. However the results of the thermodynamic consistencies test of experimental data for many cases were found not fully consistent. As a result, the strategies for solving these problems were discussed in detailed.     

High-pressure phase behavior of CO2 + 1-butyl-3-methylimidazolium chloride system

The phase behavior of carbon dioxide (CO₂) and the ionic liquid (IL) 1-butyl-3-methylimidazolium chloride ([bmim][Cl]) was measured and correlated at high pressures up to ∼40 MPa and at temperatures between 353.15 K and 373.15 K. The solubility data of CO₂ in [bmim][Cl] were obtained by observing the bubble point pressure at specific temperatures. A variable-volume view cell, which is a high-pressure equilibrium apparatus, was used to measure the CO₂ + [bmim][Cl] system solubility under varying pressure and temperature conditions. In addition, liquid–liquid–vapor (LLV) three-phase behavior was investigated using the equilibrium cell to be able to determine the classification of phase-behavior type by Scott and Van Konynenburg. Based on the LLV phase behavior, this system most likely has type III phase-behavior which is common for IL + CO₂ systems. The resulting data showed that CO₂ dissolved well in the IL at low CO₂ concentrations, but that the pressure derivative of CO₂ solubility dramatically decreased as the mole fraction of CO₂ was increased. The experimental data were well fitted by the Peng–Robinson equation of state with a quadratic mixing rule and cubic parameters estimated by the Joback method.     

Evaluation of co-volume mixing rules for bitumen liquid density and bubble pressure estimation

The Peng–Robinson cubic equation of state (CEOS) is widely used to predict thermodynamic properties of pure fluids and mixtures. The usual implementation of this CEOS requires critical properties of each pure component and combining rules for mixtures. Determining critical properties for components of heavy asymmetric mixtures such as bitumen is a challenge due to thermolysis at elevated temperatures. Group contribution (GC) methods were applied for the determination of critical properties of molecular representations developed by Sheremata for Athabasca vacuum tower bottoms (VTB). In contrast to other GC methods evaluated, the Marrero–Gani GC method yielded estimated critical properties with realistic, non-negative values, followed more consistent trends with molar mass and yielded normal boiling points consistent with high temperature simulated distillation data. Application of classical mixing rules to a heavy asymmetric mixture such as bitumen yields saturated liquid density and bubble pressure estimates in qualitative agreement with experimental data. However the errors are too large for engineering calculations. In this work, new composite mixing rules for computing co-volumes of asymmetric mixtures are developed and evaluated. For example, composite mixing rules give improved bubble point predictions for the binary mixture ethane + n-tetratetracontane. For VTB and VTB + decane mixtures the new composite mixing rules showed encouraging results in predicting bubble point pressures and liquid phase densities.     

Simple modification of the temperature dependence of the Sanchez–Lacombe equation of state

In this work, the interaction energy term of the Sanchez–Lacombe equation of state (SL EOS) was modified to take into account the temperature dependence of hydrogen bonding and ionic interactions. A simple function was used in the form of the Langmuir equation that reduces to the original SL EOS at high temperature. Comparisons are shown between the ?*-modified SL EOS and the original SL EOS. The ?*-modified SL EOS could represent volumetric data for the group of non-polar fluids, polar fluids and ionic liquids to within an absolute average deviation (AAD) of 0.85%, 0.51%, and 0.054%, respectively whereas, the original Sanchez–Lacombe EOS gave AAD values of 0.99%, 1.2%, and 0.21%, respectively. The ?*-modified SL EOS provides remarkably better PVT representation and can be readily applied to mixtures.     

Empirical correction to the Peng–Robinson equation of state for the saturated region

This work presents an empirical correction to improve the Peng–Robinson equation of state (PR EOS) for representing the densities of pure liquids and liquid mixtures in the saturated region using the volume translation method. A temperature-dependent volume correction is employed to improve the original PR EOS so that it can match the true critical point of pure fluids. The volume correction is generalized as a function of the critical parameters and the reduced temperature. The volume translation PR (VTPR) EOS with the generalized volume correction accurately represents the saturated liquid densities for different polar and non-polar fluids, including alkanes, cycloparaffins, halogenated hydrocarbons, olefins, cyclic olefins, aromatics and inorganic molecules. The average relative deviations for 91 pure compounds was 1.37%. The generalized VTPR EOS was also used to predict the saturated liquid density of 53 binary mixtures with a relative deviation of 0.98%. The generalized VTPR EOS can also be extended to other materials. The accuracy of the generalized VTPR EOS compares well with other methods and equations of state.     

Parameters estimation and VLE calculation in asymmetric binary mixtures containing carbon dioxide + n-alkanols

In the present work, the estimation of the parameters for asymmetric binary mixtures of carbon dioxide + n-alkanols has been developed. The binary interaction parameter k₁₂ of the second virial coefficient and non-random two liquid model parameters τ₁₂ and τ₂₁ were obtained using Peng–Robinson equation of state coupled with the Wong–Sandler mixing rules. In all cases, Levenberg–Marquardt minimization algorithm was used for the parameters optimization employing an objective function based on the calculation of the distribution coefficients for each component. Vapor–liquid equilibrium for binary asymmetric mixtures (CO₂ + n-alkanol, from methanol to 1-decanol) was calculated using the obtained values of the mentioned parameters. The agreement between calculated and experimental values was satisfactory.     

Vapor–liquid equilibrium measurements and modeling of the n-butane + ethanol system from 323 to 423 K

Vapor–liquid equilibrium (VLE) data are presented for the n-butane + ethanol system in the temperature range from 323 to 423 K. Measurements were performed using a “static-analytic” apparatus, equipped with two electromagnetic ROLSI™ capillary samplers, and thermally regulated via an air bath. This work presents vapor compositions which have not been explicitly measured previously. The modeling of the data was performed using two models: the Peng–Robinson equation of state with the Wong and Sandler mixing rule and NRTL excess function (PR/WS/NRTL); and the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state. To assess the effect of dipole–dipole interactions present, a dipolar contribution developed by Jog and Chapman (1999) [20] was tested with the second model. Temperature dependent binary interaction parameters have been adjusted to the new data. The PR/WS/NRTL equation of state shows good correlation with the results, while the PC-SAFT is slightly less accurate.     

Application of a volume-translated Peng-Robinson equation of state on vapor-liquid equilibrium calculations

A volume-translated Peng-Robinson (VTPR) equation of state (EOS) is developed in this study. Besides the two parameters in the original Peng-Robinson equation of state, a volume correction term is employed in the VTPR EOS. In this equation, the temperature dependence of the EOS energy parameter was regressed by an improved expression which yields better correlation of pure-fluid vapor pressures. The volume correction parameter is also correlated as a function of the reduced temperature. The VTPR EOS includes two optimally fitted parameters for each pure fluid. These parameters are reported for over 100 nonpolar and polar components. The VTPR EOS shows satisfactory results in calculating the vapor pressures and both the saturated vapor and liquid molar volumes. In comparison with other commonly used cubic EOS, the VTPR EOS presents better results, especially for the saturated liquid molar volumes of polar systems. VLE calculations on fluid mixtures were also studied in this work. Traditional van der Waals one-fluid mixing rules and other mixing models using excess free energy equations were employed in the new EOS. The VTPR EOS is comparable to other EOS in VLE calculations with various mixing rules, but yields better predictions on the molar volumes of liquid mixtures.     

Joule–Thomson inversion curves of mixtures by molecular simulation in comparison to advanced equations of state: Natural gas as an example

Molecular modelling and simulation as well as four equations of state (EOS) are applied to natural gas mixtures regarding Joule–Thomson (JT) inversion. JT inversion curves are determined by molecular simulation for six different natural gas mixtures consisting of methane, nitrogen, carbon dioxide and ethane. These components are also regarded as pure fluids, leading to a total of 10 studied systems. The results are compared to four advanced mixture EOS: DDMIX, SUPERTRAPP, BACKONE and the recent GERG-2004 Wide-Range Reference EOS. It is found that molecular simulation is competitive with state-of-the-art EOS in predicting JT inversion curves. The molecular based approaches (simulation and BACKONE) are superior to DDMIX and SUPERTRAPP.     

跨接VTSRK方程计算烃类的pVT性质

烃类pVT性质的精细表征对能源动力、化工等领域应用有重要价值,临界区热力性质描述是难点之一.本文建立了烷烃（C1-C20）的跨接比容平移Soave-Redlich-Kwong（SRK）（跨接VTSRK）状态方程,在SRK状态方程的基础上引入了比容平移和跨接方法,以改善饱和液相密度和近临界区域热力学性质的计算精度,方程参数被表达为物质临界参数和偏心因子的函数. 比较结果表明,跨接方程对烷烃（C1-C20）饱和蒸气压、饱和气相密度、饱和液相密度的计算平均偏差分别为1.01%、1.83%和0.93%,显著优于原方程,单相区和近临界区的pVT性质计算精度也比原状态方程有较大改善. 进一步将方程推广到环烷烃（环丙烷、环戊烷和环己烷）和苯、甲苯的计算,也获得了较好效果,验证了方程的预测能力.     

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.