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.     

Equation of state for the NH3−H2O system

An equation of state (EOS) for the NH₃–H₂O system has been developed. This EOS incorporates a highly accurate end-member EOS and on an empirical mixing rule. The mixing rule is based on an analogy with high order contributions to the virial expansion for mixtures. Comparison with experimental data indicates that the mixed system EOS can predict both phase equilibria and volumetric properties for this binary system with accuracy close to that of the experimental data from 50°C and 1 bar to critical temperatures and pressures.     

New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures

Huron, M.-J. and Vidal, J., 1979. New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures. Fluid Phase Equilibria, 3: 255-271.Good correlations of vapour-liquid equilibria can be achieved by applying the same two-parameter cubic equation of state to both phases. The results primarily depend on the method used for calculating parameters and, for mixtures, on the mixing rule. True parameters are the covolume b and the energy parameter a/b. For this latter one, deviations from a linear weighting rule are closely connected to the excess free energy at infinite pressure. Thus any mixing rule gives a model for the excess free energy, or any accepted models for this property can be used as mixing rules.From the above, an empirical polynomial mixing rule is used for data smoothing and evaluation, while for practical work a local composition model is used. The mixing rule thus obtained can be reduced to the classical quadratic rule for some easily predicted values of the interaction energies. For highly polar systems, it includes three adjustable parameters. Using literature data, the new mixing rule is applied, in the low and high pressure range, to binary mixtures with one or two polar compounds, giving good data correlation and sometimes avoiding false liquid-liquid immiscibility.     

Calculation of high pressure vapour—liquid equilibria for systems containing polar substances with the use of an augmented van der Waals equation of state

An augmented van der Waals equation of state based on a perturbation theory has been applied to the calculation of high pressure vapour—liquid equilibria for systems containing polar substances. The equation of state comprises four terms, which imply the contributions from repulsion, symmetric, non-polar asymmetric, and polar asymmetric interactions. The characteristic parameters of each pure substance have been determined by three methods with the use of vapour pressures and saturated liquid densities. Mixing models for the terms of the repulsion, symmetric, and non-polar asymmetric interactions are the same as used previously. Two types of mixing models based on a three-fluid model and/or a one-fluid model are developed for the polar asymmetric term. The polar asymmetric term has a large effect on the prediction of the vapour—liquid equilibrium. With the introduction of a binary interaction parameter, the equation is found to be useful in correlating the vapour—liquid equilibria for a system containing a polar substance except near a critical region.     

Equation of state for square-well chain molecules with variable range II. Extension to mixtures

An equation of state (EOS) developed in our previous work for square-well chain molecules with variable range is further extended to the mixtures of non-associating fluids. The volumetric properties of binary mixtures for small molecules as well as polymer blends can well be predicted without using adjustable parameter. With one temperature-independent binary interaction parameter, satisfactory correlations for experimental vapor–liquid equilibria (VLE) data of binary normal fluid mixtures at low and elevated pressures are obtained. In addition, VLE of n-alkane mixtures and nitrogen + n-alkane mixtures at high pressures are well predicted using this EOS. The phase behavior calculations on polymer mixture solutions are also investigated using one-fluid mixing rule. The equilibrium pressure and solubility of gas in polymer are evaluated with a single adjustable parameter and good results are obtained. The calculated results for gas + polymer systems are compared with those from other equations of state.     

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.     

An equation of state for hydrogen fluoride

Using a similar approach as Lencka and Anderko [AIChE J. 39 (1993) 533], we developed an equation of state for hydrogen fluoride (HF), which can correlate the vapor pressure, the saturated liquid and vapor densities of it from the triple point to critical point with good accuracy. We used an equilibrium model to account for hydrogen bonding that assumes the formation of dimer, hexamer, and octamer species as suggested by Schotte [Ind. Eng. Chem. Process Des. Dev. 19 (1980) 432]. The physical and chemical parameters are obtained directly from the regression of pure component properties by applying the critical constraints to the equation of state for hydrogen fluoride. This equation of state together with the Wong–Sandler mixing rule as well as the van der Waals one-fluid mixing rule are used to correlate the phase equilibria of binary hydrogen fluoride mixtures with HCl, HCFC-124, HFC-134a, HFC-152a, HCFC-22, and HFC-32. For these systems, new equation of state with the Wong–Sandler mixing rule gives good results.     

Comparison of methods for calculating thermodynamic properties of binary mixtures in the sub and super critical state: Lee–Kesler and cubic equations of state for binary mixtures containing either CO2 or H2S

The (ρ,T,p) and (vapor + liquid) equilibria for fluid mixtures containing either CO₂ or H₂S have been determined from 13 equations of state. The estimated values have been compared with published experimental results. CO₂ and H₂S were used to represent non-polar and polar fluids, respectively. The equations of state investigated were as follows: (a) the Lee–Kesler equation; (b) two equations that included new reference fluids for the Lee–Kesler method; (c) three so-called extended equations of state; and (d) seven cubic equations of state. After adjustment of the binary interaction parameters the predicted values differed from the experimental data by about 0.8% for CO₂ mixtures while for H₂S mixtures the uncertainty was about ±2.8%. Somewhat larger errors, although still lower than ±5%, were obtained for co-existing phase densities; the Lee–Kesler method provided results of the highest accuracy. The cubic equations proposed by Schmidt and Wenzel and Valderrama provide the most reliable predictions of both single and co-existing phase densities. Comparison of the predicted (vapor + liquid) equilibrium with experiment shows that each of the seven cubic equations provides results of similar accuracy and all within ±6%.     

Mixing rules for hardsphere chain mixtures and their extension to heteronuclear hardsphere polyatomic fluids and mixtures

Mixing rules are very important for the calculation of fluid properties using different equations of state. In order to find the theoretical lead of the mixing rule for the size parameter, a mixing rule [1] for hardsphere mixtures has been proposed on the basis of Carnahan-Starling equation and Boublik-Mansoori equation. As its extension, mixing rules for hardsphere chain mixtures are proposed in this work. A mixing rule for the segment number (or chain length) is derived on the limitation of the equality of segment diameters, from the first order thermodynamic perturbation theories (TPT1) for pure chain fluids and for chain mixtures. Meanwhile, the mixing rule for the segment diameter is the same as the mixing rule for hardsphere mixtures on the limitation of monomer mixtures. The two mixing rules are checked together over wide ranges of conditions for hardsphere chain mixtures and compared with the first order thermodynamic perturbation theory (TPT1) and also with simulation data available in literature. An another interesting usage of new mixing rules is to describe the heteronuclear hardsphere polyatomic pure fluids, which consist of hardspheres with different segment diameters as in methane and ethane in which carbon and hydrogen atoms are looked as bonded spheres, and heteronuclear hardsphere polyatomic mixtures. The comparison with simulation data shows the validity of the mixing rules.     

Theoretical Studies on Critical Properties of Binary System with Supercritical Carbon Dioxide

A new expression of mixing rule is suggested according to the Mayson‘s mixing rule in this paper,which adopts the Redlich-Kwong cubic equation of state and the modifed Chueh-Prausnitz method to calculate the experiment critical points of six binary mixtures CO2 toluene,CO2 cyclohexane,CO2 n-butanal,CO2 i-butanal,CO2 methanol,CO2 ethanol.The coefficients of interaction parameter in the expression of mixing rule were optimized from experimental data.The calculated results of critical temperature and critical pressure meet the experiment data well.The maximum relative errore of temperature and pressure between the calculation results and experiment data are 1.493% and 5.2236% respectively,indicating that the proposed expression of mixing rule is reasonable.This may provide a fundamental method for studying and predicting the properties of supercritical fluids.     

Phasepy: A Python based framework for fluid phase equilibria and interfacial properties computation

Phasepy is a Python based package for fluid phase equilibria and interfacial properties calculation from equation of state (EoS). Phasepy uses several tools (i.e., NumPy, SciPy, Pandas, Matplotlib) allowing use Phasepy under Jupyter Notebooks. Phasepy models phase equilibria with the traditional ϕ –γ and ϕ –ϕ approaches, where ϕ (fugacity coefficient) can be modeled as a perfect gas, virial gas or EoS fluid, whereas γ (activity coefficient) can be described by conventional models (NRTL, Wilson, Redlich-Kister expansion, and the group contribution modified-UNIFAC). Interfacial properties are based on the square gradient theory couple to ϕ –ϕ approach. The available EoSs are the cubic EoS family extended to mixtures through the quadratic, modified-Huron-Vidal, and Wong-Sandler mixing rules. Phasepy allows to analyze phase stability, compute phase equilibria, interfacial properties, and optimize their parameters for vapor–liquid, liquid–liquid, and vapor–liquid–liquid equilibria for multicomponent mixtures. Phasepy implementation, and robustness are illustrated for binary and ternary mixtures.     

Extension of Huron-Vidal-type Mixing Rule to Three-parameter Harmens-Knapp Cubic Equation of State

In this paper was extended the HV-type mixing rules to Harmens-Knapp cubic equation state (HK CEOS). The new HV-type mixing rule with HK CEOS was tested for Vapor-liquid equilibrium(VLE) of different polar and nonpolar systems. The tested results are in good agreement with existing experimental data within a wide range of temperatures and pressures. In comparison with the VDW mixing rule, the new mixing rule gives much better predictions for the VLE of nonpolar and polar systems.     

A new three-parameter cubic equation of state for calculation physical properties and vapor–liquid equilibria

A new three-parameter cubic equation of state is presented by combination of a modified attractive term and van der Waals repulsive expression. Also a new alpha function for the attractive parameter of the new EOS is proposed. The new coefficients of alpha function and the other parameters of the attractive term are adjusted using the data of the saturated vapor pressure and liquid density of almost 60 pure compounds including heavy hydrocarbons. The new EOS is adopted for prediction of the various thermophysical properties of pure compounds such as saturated and supercritical volume, enthalpy of vaporization, compressibility factor, heat capacity and sound velocity. Following successful application of the new EOS for the pure components, using vdW one-fluid mixing rules, the new EOSs are applied to prediction of the bubble pressure and vapor mole fraction of the several binary and ternary mixtures. The accuracy of the new EOS for phase equilibrium calculation is demonstrated by comparison of the results of the present EOSs with the PT, PR, GPR and SRK cubic EOSs.     

Density improvement of the SRK equation of state

The SRK equation of state has been modified by volume translation in order to improve its accuracy both in- and outside the critical region. A temperature dependent volume correction is proposed which can match the true critical point of the pure component, and provides accurate densities for polar and non-polar pure substances both near to and far from the critical point. It can also be easily extended to mixtures, and the calculation results show that it can shift the critical locus towards experimental values and gives good results for the liquid densities of mixtures.     

Application of the Perturbed-Chain SAFT equation of state to polar systems

The Perturbed-Chain SAFT (PC-SAFT) equation of state is applied to pure polar substances as well as to vapor–liquid and liquid–liquid equilibria of binary mixtures containing polar low-molecular substances and polar co-polymers. For these components, the polar version of the PC-SAFT model requires four pure-component parameters as well as the functional-group dipole moment. For each binary system, only one temperature-independent binary interaction k_ij is needed. Simple mixing and combining rules are adopted for mixtures with more than one polar component without using an additional binary interaction parameter. The ability of the model to accurately describe azeotropic and non-azeotropic vapor–liquid equilibria at low and at high pressures, as well as liquid–liquid equilibria is demonstrated for various systems containing polar components. Solvent systems like acetone–alkane mixtures and co-polymer systems like poly(ethylene-co-vinyl acetate)/solvent are discussed. The results for the low-molecular systems also show the predictive capabilities of the extended PC-SAFT model.     

Shape factors in equations of state. Part II. Repulsion phenomena in multicomponent chain fluids

An equation of state for the multicomponent fluid phase of nonattracting rigid particles of arbitrary shape is presented. The equation is a generalization of a previously presented equation of state for pure fluids of rigid particles; the approach describes the volumetric properties of a pure fluid in terms of a shape factor, zeta, which can be back calculated by scaling the volumetric properties of pure fluids to that of a hard sphere. The performance of the proposed equation is tested against mixtures of chain fluids immersed in a "monomeric" solvent of hard spheres of equal and different sizes. Extensive new Monte Carlo simulation data are presented for 19 binary mixtures of hard homonuclear tangent freely-jointed hard sphere chains (pearl-necklace) of various lengths (three to five segments), with spheres of several size ratios and at various compositions. The performance of the proposed equation is compared to the hard-sphere SAFT approach and found to be of comparable accuracy. The equation proposed is further tested for mixtures of spheres with spherocylinders. In all cases, the equation proved to be accurate and simple to use.     

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 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.     

A new model and extension of Wong–Sandler mixing rule for prediction of (vapour + liquid) equilibrium of polymer solutions using EOS/GE

The cubic equation of state (CEOS) is a powerful method for calculation of (vapour + liquid) equilibrium (VLE) in polymer solutions. Using CEOS for both the vapour and liquid phases allows one to calculate the non-ideality of polymer solutions based on a single EOS approach. However, the traditional mixing rules are not appropriate to extend the CEOS to non-ideal mixtures such as polymer solutions. Several authors have applied the EOS/G^E approach to predict (vapour + liquid) equilibria in polymer solutions, however, incorporating an appropriate excess Gibbs free energy for the new mixing rule is a major step. In this research, the NRTL-NRF model was extended in terms of volume fraction of polymer and solvent (instead of mole fraction), then equilibrium calculations were carried out using PRSV EOS and Wong–Sandler mixing rules. Using the adjustable parameters as a function of solution temperature, the NRTL-NRF model can be used as a predictive model. In comparison with NRTL model, the results of the new NRTL-NRF model show better accuracy.     

A three-parameter cubic equation of state for prediction of thermodynamic properties of fluids

A new cubic three-parameter equation of state has been proposed for PVT and VLE calculations of simple, high polar and associating fluids. The parameters are temperature dependent in sub-critical region, but temperature independent in super-critical region. The results for 42 simple and 14 associative pure compounds indicate that the calculated saturation properties and volumetric properties over the whole temperature range, up to high pressures, by the proposed equation of state (EOS), were in better agreement with the experimental data, compared with those obtained by the five well-known EOSs (P–R, P–T, Adachi et al., Yu–Lu, and M4). Two derivative properties, molar enthalpy and heat capacity of water and ammonia have been calculated, and demonstrated the thermodynamic consistency of the EOS parameters. Also VLE calculations have been performed for 41 binary mixtures of different type of fluids, including those of interest in petroleum industry. The results indicated the high capability of the proposed EOS for calculating the thermodynamic properties of pure and fluid mixtures.