Peng-Robinson equation of state for vapor-liquid equilibrium calculations for carbon dioxide + hydrocarbon mixtures

Lin, H.-M., 1984. Peng-Robinson equation of state for vapor-liquid equilibrium calculations for carbon dioxide + hydrocarbon mixtures. Fluid Phase Equilibria, 16: 151–169.Binary interaction parameters δ_ij in the Peng-Robinson equation of state have been determined from vapor-liquid equilibrium data for binary mixtures of carbon dioxide with a variety of hydrocarbons. A constant value of δ_ij ? 0.125 appears to represent the experimental data well in most cases. Comments are made on the recent work of Kato, Nagahama and Hirata, who correlated δ_ij as a function of temperature for CO₂ + n-paraffin binary mixtures.     

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.     

Crossover Peng-Robinson equation of state with introduction of a new expression for the crossover function

In this research, we use the original Peng-Robinson (PR) equation of state (EOS) for pure fluids and develop a crossover cubic equation of state which incorporates the scaling laws asymptotically close to the critical point and it is transformed into the original cubic equation of state far away from the critical point. The modified EOS is transformed to ideal gas EOS in the limit of zero density. A new formulation for the crossover function is introduced in this work. The new crossover function ensures more accurate change from the singular behavior of fluids inside the regular classical behavior outside the critical region. The crossover PR (CPR) EOS is applied to describe thermodynamic properties of pure fluids (normal alkanes from methane to n-hexane, carbon dioxide, hydrogen sulfide and R125). It is shown that over wide ranges of state, the CPR EOS yields the thermodynamic properties of fluids with much more accuracy than the original PR EOS. The CPR EOS is then used for mixtures by introducing mixing rules for the pure component parameters. Higher accuracy is observed in comparison with the classical PR EOS in the mixture critical region.     

Gas solubility calculations. II. Application of a new group-contribution equation of state

A new equation of state has been developed for polar as well as nonpolar components. It is based on the generalized van der Waals partition function and uses local-composition mixing rules. The group-contribution version of this equation of state (the GC-EOS) is described and tables containing parameters for 14 solvent groups and 9 gases (H₂, N₂, CO, O₂, CH₄, C₂H₄, CO₂, C₂H₆ and H₂S) are presented.The GC-EOS predicts vapor-liquid equilibria well for all kinds of systems involving the groups considered. The method requires only information concerning readily accessible pure-component properties. Calculations for multicomponent systems show that the method suggested provides very good predictions of multicomponent high-pressure vapor-liquid equilibria and fairly good predictions of Henry's constants in mixed solvents.     

Phase equilibrium for surfactant Ls-54 in liquid CO(2) with water and solubility estimation using the Peng-Robinson equation of state

It is known that the commercial surfactant Dehypon^® Ls-54 is soluble in supercritical CO₂ and that it enables formation of water-in-CO₂ microemulsions. In this work we observed phase equilibrium for the Ls-54/CO₂ and Ls-54/water/CO₂ systems in the liquid CO₂ region, from 278.15 to 298.15 K. In addition, the Peng–Robinson equation of state (PREOS) was used to model the phase behavior of Ls-54/CO₂ binary system as well as to estimate water solubilities in CO₂. Ls-54 in CO₂ can have solubilities as high as 0.086 M at 278.15 K and 15.2 MPa. The stability of the microemulsion decreases with increasing concentration of water, and lower temperatures favor increased solubility of water into the one-phase microemulsion. The PREOS model showed satisfactory agreement with the experimental data for both Ls-54/CO₂ and water/CO₂ systems.     

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.     

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.     

Representation of the vapor-liquid equilibrium of the ternary system carbon dioxide - propane - methanol and its binaries with a cubic equation of state : a new mixing rule

Erroneous liquid phase splitting is often predicted when correlating the vapor-liquid equilibrium of alcohol-alkane systems with a cubic equation of state. It is shown that even the local-composition mixing rules are not sufficiently flexible for an accurate representation of the propane-methanol system, and a new empirical three-parameter mixing rule is introduced. The ternary system carbon-dioxide - propane - methanol is well-predicted from the binary data.     

Computation of multiphase equilibrium phenomena with an equation of state

This paper presents a general approach to multiphase equilibrium computations using an equation of state. Various types of calculations are described: flash calculations, saturation-pressure and saturation-temperature calculations, and calculations of the feed composition corresponding to a given saturation pressure and temperature.A stagewise procedure for flash calculations using the QNSS method allows efficient computation of phase equilibria for systems with any number of phases. The initial guess of the composition of an extra phase which is required to start the iteration is obtained from an analysis of the Gibbs-energy surface.Newton's method is used to construct multiphase boundaries on various types of diagrams (pressure-temperature, pressure-composition and temperature-composition). Multiphase critical points on these boundaries are also estimated by interpolation.The algorithms developed are tested for a typical reservoir-oil-CO₂ mixture which exhibits three-phase liquid 1-liquid 2-vapor (L₁L₂V) separation. The predictions obtained using the Peng-Robinson of state are satisfactory.     

Simulations of vapor water clusters at vapor-liquid equilibrium

The Gibbs-ensemble Monte Carlo methods based on the extended single point charge [H. J. C. Berendsen, J. R. Grigera, and T. P. Straatsma, J. Phys. Chem. 91, 6269 (1987)] potential-energy surface have been used to study the clustering of vapor phase water under vapor-liquid equilibrium conditions between 300 and 600 K. It is seen that the number of clusters, as well as the cluster size, increase with temperature. This is primarily due to the increase in vapor density that accompanies the temperature increase at equilibrium. In addition, due to entropic effects, the percentage of clusters that have linear (or open) topologies increases with temperature and dominates over the minimum-energy cyclic topologies at the temperatures studied here. These results are insensitive to the number of molecules used in the simulations and the criterion used to define a water cluster.     

Determination and correlation of binary vapor-liquid equilibrium data

The Chao modification of the Redlich-Kister equation for correlating liquid activity coefficients is further modified to facilitate the evaluation of the constants. The determination of liquid activity coefficients at infinite dilutions by means of an ebulliometer with the consideration of vapor hold-up in the condenser is proposed. Both of these proposals are equally applicable to isothermal data and to isobaric data.For the purpose of illustration, vapor-liquid equilibrium data for the binary system acetone-methanol together with the azeotropic point and the liquid activity coefficients at infinite dilutions were determined at 760 Torr. The validity of the correlated data was confirmed with the enthalpies of mixing data at saturation, available in the literature.     

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.     

Modeling associating hydrocarbon + alcohol mixtures using the Peng-Robinson equation of state and the Wong-Sandler mixing rules

The main objective of the study is the accurate modeling of the bubble pressure and of the vapor phase concentration in associating hydrocarbon + alcohol mixtures and the correct comparison with results from the literature. A relatively simple equation of state is used and comparison is done considering various factors that affect the accuracy of the results, so fair and correct conclusions can be drawn. The mathematical complexity of the model, the type and amount of basic properties and the number of adjustable parameters used by the model, among other factors are discussed. The Peng-Robinson equation of state including the Wong-Sandler mixing rules was used. This combination of equations of state and mixing rules have not yet been applied in a systematic way to alcohol + hydrocarbon mixtures at low and moderate pressure, as done in this work, although other complex equation of state models have been used for some selected systems. It is concluded that simple and well-founded models can correlate equilibrium data in these complex mixtures with similar accuracy than other more sophisticated models.     

用非线性统计理论方程计算气-液平衡

用非线性统计理论方法计算气－ 液平衡组成,参数有明确的物理意义,并可用纯组分的物性参数进行估算．用此方法计算了气－ 液平衡计算中最困难的二元含水体系的平衡组成．结果表明该方法简单,计算精度高,有理论和实际意义．     

Monte Carlo simulation of equilibrium reactions at vapor-liquid interfaces

Chemical reactions are known to behave differently, depending upon their local environment. While the interactions with neighboring molecules may alter both the kinetics of chemical reactions and the overall equilibrium conversion, we have performed simulations of the latter. The particular environment that we address is the vapor-liquid interface, since only a few, limited studies have explored the influence of an interface on equilibrium reaction behavior. Simple dimerization reactions are modeled, as well as more complex multicomponent reactions, using the reactive Monte Carlo (RxMC) simulation technique. We find that the conversion of a reaction can be markedly different at an interface as compared to the bulk vapor and liquid phases, and these trends are analyzed with respect to specific intermolecular interactions. In conjunction, we calculate the surface tension of the reacting fluids at the interface, which is found to have unusual scaling behavior, with respect to the system temperature.     

Calculation of vapour-liquid equilibrium of associated systems by an equation of state

A recently proposed equation of state of the van der Waals type is applied to calculate phase equilibria in hydrogen-bonding, non-electrolytic systems. Association is accounted for by treating alcohols, acids, etc., as mixtures of associated species formed by up to 14 monomers. The method involves essentially one weakly temperature-dependent adjustable parameter per binary system.The calculations cover vapour-liquid equilibrium both at low and elevated pressure in binary systems formed by an associating substance and a non-associating compound such as a hydrocarbon or halogenated hydrocarbon. An attempt has been made to include all experimental data available for these systems in the literature. A number of calculations for ternary systems as well as of liquid-liquid equilibria are included, and a limited number of solvated systems are also treated.Owing to its single adjustable parameter, the method may be used to test existing experimental data and to predict such data.     

Vapor-liquid equilibrium predictions of refrigerant mixtures from a cubic equation of state with a G-EoS mixing rule

A modified excess Gibbs energy model which is based on the local composition concept and assigns a single energy parameter per pair of components, is incorporated into the G^E—EoS thermodynamic formalism for vapor-liquid equilibrium (VLE) calculations of simple and complex refrigerant mixtures. One temperature set of data close to 273 K is used to obtain the model's parameters, which are used to extrapolate the VLE at other temperatures and pressures. A one-parameter form of the model based on the Wong-Sandler mixing rule is presented for several simple systems. The physical significance of the model's energy parameter is connected to the preference of the mixture for like to unlike interactions. The model is applied for VLE predictions of the ternary system R14-R23-R13, and the results are compared to calculations using the 3PWS model [H. Orbey. S.I. Sandler, Ind. Eng. Chem. Res. 34 (1995) 2520–2525] and the van der Waals mixing rule. Modelling of a few complex systems with only three data points given at each temperature is shown with a two parameter version of our model on the basis of the Huron-Vidal mixing rule.     

Modeling of gas adsorption equilibrium over a wide range of pressure: a thermodynamic approach based on equation of state

A thermodynamic approach based on the Bender equation of state is suggested for the analysis of supercritical gas adsorption on activated carbons at high pressure. The approach accounts for the equality of the chemical potential in the adsorbed phase and that in the corresponding bulk phase and the distribution of elements of the adsorption volume (EAV) over the potential energy for gas-solid interaction. This scheme is extended to subcritical fluid adsorption and takes into account the phase transition in EAV. The method is adapted to gravimetric measurements of mass excess adsorption and has been applied to the adsorption of argon, nitrogen, methane, ethane, carbon dioxide, and helium on activated carbon Norit R1 in the temperature range from 25 to 70 degrees C. The distribution function of adsorption volume elements over potentials exhibits overlapping peaks and is consistently reproduced for different gases. It was found that the distribution function changes weakly with temperature, which was confirmed by its comparison with the distribution function obtained by the same method using nitrogen adsorption isotherm at 77 K. It was shown that parameters such as pore volume and skeleton density can be determined directly from adsorption measurements, while the conventional approach of helium expansion at room temperature can lead to erroneous results due to the adsorption of helium in small pores of activated carbon. The approach is a convenient tool for analysis and correlation of excess adsorption isotherms over a wide range of pressure and temperature. This approach can be readily extended to the analysis of multicomponent adsorption systems.     

Monte Carlo simulation of equilibrium reactions at modified vapor-liquid interfaces

The equilibrium conversion of a chemical reaction is known to be affected by its local environment. Various factors may alter reaction equilibria, including shifts in pressure or temperature, solvation, adsorption within porous materials, or the presence of an interface. Previously, reactive Monte Carlo simulations have been used to predict the equilibrium behavior of chemical reactions at vapor-liquid interfaces. Here, a route is tested for tuning the interfacial conversion of a Lennard-Jones dimerization reaction by adding surfactants to the vapor-liquid interface. Several temperatures are explored as well as several different surfactant models. Even with the addition of a small concentration of surfactants, the simulations predict significant shifts in the conversion at the interface. In general, the shifts in the conversion tend to be related to the values of the interfacial tension.     

Modeling of phase equilibrium and vapor adsorption on carbon black based on a combination of a lattice theory and equation of state

A thermodynamic approach is developed in this paper to describe the behavior of a subcritical fluid in the neighborhood of vapor-liquid interface and close to a graphite surface. The fluid is modeled as a system of parallel molecular layers. The Helmholtz free energy of the fluid is expressed as the sum of the intrinsic Helmholtz free energies of separate layers and the potential energy of their mutual interactions calculated by the 10-4 potential. This Helmholtz free energy is described by an equation of state (such as the Bender or Peng-Robinson equation), which allows us a convenient means to obtain the intrinsic Helmholtz free energy of each molecular layer as a function of its two-dimensional density. All molecular layers of the bulk fluid are in mechanical equilibrium corresponding to the minimum of the total potential energy. In the case of adsorption the external potential exerted by the graphite layers is added to the free energy. The state of the interface zone between the liquid and the vapor phases or the state of the adsorbed phase is determined by the minimum of the grand potential. In the case of phase equilibrium the approach leads to the distribution of density and pressure over the transition zone. The interrelation between the collision diameter and the potential well depth was determined by the surface tension. It was shown that the distance between neighboring molecular layers substantially changes in the vapor-liquid transition zone and in the adsorbed phase with loading. The approach is considered in this paper for the case of adsorption of argon and nitrogen on carbon black. In both cases an excellent agreement with the experimental data was achieved without additional assumptions and fitting parameters, except for the fluid-solid potential well depth. The approach has far-reaching consequences and can be readily extended to the model of adsorption in slit pores of carbonaceous materials and to the analysis of multicomponent adsorption systems.