Modification of the Peng–Robinson equation of state for helium in low temperature regions

Helium shows the nearest behaviour to ideal gas in the room conditions. In contrast, thermodynamic behaviour of helium in the critical region, in which its liquefaction is possible, is extremely complicated. The equation of state (EOS), which is in common use for helium, is the modified Benedict–Webb–Rubin (MBWR) EOS developed by McCarty and Arp which is a 13th-order equation with 32 substance-dependent parameters. MBWR is a complicated EOS and its use is time consuming. In this work, the modified Peng–Robinson EOS introduced by Feyzi et al. is customised with 10 adjustable parameters for helium in the temperature range of 2.20–15.20 K and pressures up to 16 bar. The proposed EOS is able to predict the properties of helium in the vapour–liquid equilibrium (VLE) conditions and in the single gas-phase region. In addition, a temperature-dependent correlation for constant pressure heat capacity of helium from very low up to normal temperatures is proposed. The liquefaction process of helium, which is being done by cooling it to very low temperatures by passing through a Joule–Thomson valve, is predicted by the proposed EOS. Very accurate results are observed.     

Simulation of double retrograde vaporization using the Peng–Robinson equation of state

Double retrograde vaporization is a phenomenon characterized by an unexpected retrograde dew point curve at compositions approaching nearly the pure volatile component (component A) and at temperatures very close to the critical temperature of the more volatile component (Tc_A). On the p–x–y diagram, instead of the single-domed dew point curve in the familiar “single” retrograde vaporization, double retrograde vaporization shows two “domes” at temperatures above but close to Tc_A. At temperatures below but close to Tc_A, the dew point curve has an “S”-shape. This results respectively in quadruple- or triple-valued dew points at a specific composition. In this work, the phenomenon of double retrograde vaporization has been simulated using a cubic equation of state. Both the “double-dome” and the “S”-shape curves for the binary systems (ethane+linalool) and (ethane+d-limonene) were successfully modelled, even without the use of binary interaction parameters. Results are also obtained by optimizing interaction parameters using experimental bubble point data. Even though double retrograde vaporization has rarely been observed in literature, we believe that it is the normal behaviour that always occurs in binary mixtures in which the two components differ largely enough in molecular symmetry to produce a very steep dew point curve. To further verify this generality, simulations were performed on a number of binary mixtures of different families. Double retrograde vaporization was estimated in every system with a steep dew point curve.     

Interaction parameter for hydrogen-containing mixtures in the Peng—Robinson equation of state

A new correlation for the Peng—Robinson interaction parameter δ_ij of hydrogen-containing mixtures is proposed here. Values of δ_ij obtained from literature vapour—liquid equilibrium data are represented by a cubic polynomial in terms of the temperature. The correlation predicts better values than other correlations proposed in the literature for the same systems, and is applicable to wider ranges of temperature. Applications of the proposed equations to VLE are presented.     

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.     

Vapor pressures of pure compounds using the Peng–Robinson equation of state with three different attractive terms

Accurate representation of pure compounds vapor pressures is required to increase the robustness of equations of state when predicting phase equilibria for mixtures. Using cubic equations of state, this representation largely depends on improving the temperature-dependent attractive term of the equation of state (EOS) to cover data from the triple to critical points. With this purpose, the Peng–Robinson equation of state is applied with three different attractive terms: Mathias, Mathias–Copeman, and Carrier–Rogalski–Péneloux. Experimental vapor pressures for 311 pure compounds (9000 experimental values) have been fitted. The studied compounds include nitrogen compounds, oxides, sulfides, chlorides, oxyhalides, inorganic compounds, alkanes, cycloalkanes, alkenes, alkadienes, alkynes, aromatic hydrocarbons, halogenated alkanes, halogenated cycloalkanes, halogenated alkenes, halogenated aromatic hydrocarbons, alcohols, ethers, aldehydes, ketones, alkanoic acids, esters, phenols, heterocyclic oxygen compounds, heterocyclic nitrogen compounds, hydrocarbon nitrogen compounds, and sulfur compounds. Overall average absolute deviations of 0.416, 0.214 and 0.276% have been found for the resulting Peng–Robinson–Mathias (PRM), Peng–Robinson–Mathias–Copeman (PRMC), and Peng–Robinson–Carrier–Rogalski–Péneloux (PRCRP) equations of state, respectively.     

A Modified Peng–Robinson Equation of State for Heavy Hydrocarbons

Russian Journal of Physical Chemistry A - A new simple and accurate functional form for an attractive parameter α is introduced for Peng–Robinson equation of state. The modified...     

Calculation of the second virial coefficients of alkali metals by modified Peng–Robinson equation

In this work, the Peng–Robinson (P–R) equation of state has been modified by proposing a new α function for calculating the second virial coefficients of alkali metals. The relationship between α^0.5 and (1???T _r ^0.5 ) is a nonlinear function. The correlation between the second virial coefficient and P–R equation was presented by expanding the P–R equation into its Taylor series form. For P–R equation, the linear correlation between parameters C₁ and C₂ of α function and acentric factors \( \omega \) of alkali metals was proposed. The new α function and its first, second and third derivatives are continuous. The average standard deviations of compressibility factor which calculated by modified P–R equation are less than 4.3%. The second virial coefficients of alkali metals were calculated over the temperature range 600–3000 K by using the modified P–R equation. Comparison with literature data, the new equation provides more reliable and accurate second virial coefficient predictions for alkali metals than the original P–R equation. It is useful to guide and improve calculation of the second virial coefficients of other metal vapors for design and operation of separation processes in vacuum metallurgy.     

The Kwak–Mansoori approach to the Peng–Robinson equation for determining the thermodynamic consistency of VLE in ethanol + congener mixtures

The modified Peng–Robinson equation of state proposed by Kwak and Mansoori (PR/KM) is used in a thermodynamic consistency test of phase equilibrium data for binary ethanol + congener mixtures found in alcoholic distillation processes. Congener substances are those components in a must that are present at very low concentration, but their presence is necessary to give the distilled liquor their particular aromatic and tasting characteristics. The congener substances considered in this study are: acetic acid, ethyl acetate, furfural, methanol, 2-methyl-1-propanol, 1-pentanol, 1-propanol and methyl acetate. A flexible area test method is applied to analyse 25 isothermal P–x–y data of ethanol + congener mixtures available in the open literature. The consistency method determines the value of three integral expressions derived from the Gibbs–Duhem equation; one integral is calculated using experimental data only and the other two by using values calculated with the PR/KM model. For all cases, the method gives a clear answer about consistency or inconsistency of a set of isothermal P–x–y data.     

A theoretical form of the Martin-Hou equation of state

A new equation of state is derived from the Barker-Henderson hard-sphere perturbation theory. It has the form similar to the Martin-Hou equation of state. The numerical values of the characteristic constants in the equation can be calculated by the method of Martin and Hou. The equation can be used to predict P-V-T properties accurately for fluids when the critical parameters (T_c, P_c and V_c) and one point on the vapor pressure cure are given. By using the functional relationships between the characteristic constants and the microscopic parameters, the molecular microscopic parameters of the substance can be obtained.     

A saturated liquid density equation in conjunction with the Predictive-Soave–Redlich–Kwong equation of state for pure refrigerants and LNG multicomponent systems

An equation and a set of mixing rules for the prediction of liquid density of pure refrigerants and liquified natural gas (LNG) multicomponent systems have been developed. This equation uses the parameters of Mathias and Copeman [P.M. Mathias, T.W. Copeman, Fluid Phase Equilib. 13 (1983) 91–108] temperature dependent-term for the Predictive-Soave–Redlich–Kwong [T. Holderbaum, J. Gmehling, Fluid Phase Equilib. 70 (1991) 251–265] equation of state and hence it could be used together with this equation. The equation uses a characteristic parameter for each refrigerant; however, if it is not available, a value of zero is recommended. This model gives an average of absolute errors less than 0.42% for the prediction of liquid density of 28 pure refrigerants consisting of 2489 data points and 0.33% for 18 multicomponent LNG systems involving 132 data points. The model parameters were determined from pure component properties and reported. These parameters were then used without any adjustment to predict liquid density of multicomponent LNG mixtures and excellent results were obtained. The model was also compared with other available methods.     

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.     

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.     

Development of a new form for the alpha function of the Redlich–Kwong cubic equation of state

The dependence on temperature and acentric factor of the attractive term of the Redlich–Kwong equation of state has been modified. A new alpha function is expressed in a generalized form. The new equation allows a good representation of vapor pressure data of a great variety of compounds, as well as thermodynamic properties such as the enthalpy of vaporization and the entropy of vaporization.     

An effective modification of the Benedict–Webb–Rubin equation of state

A modification of the BWR equation of state is proposed, which is a simplified form of a previously proposed one. It applies to systems formed by hydrocarbons and related compounds, with particular attention to the critical conditions. The range of treatable compounds was extended to a value 0.9 of the acentric factor, corresponding to C₂₀ hydrocarbons. The critical compressibility factor Z_c was made independent of the acentric factor, for a more accurate prediction of pure-component properties (the previous equation did not give the same improvement). Mixing rules require one binary interaction constant for each component pair. Zero binary constants can be used for methane–alkane and alkane–alkane pairs. Examples of applications to pure hydrocarbons and their mixtures are given.     

Comparison of different mixing rules for prediction of density and residual internal energy of binary and ternary Lennard–Jones mixtures

Mixing rules are necessary when equations of state for pure fluids are used to calculate various thermodynamic properties of fluid mixtures. The well-known van der Waals one-fluid (vdW1) mixing rules are proved to be good ones and widely used in different equations of state. But vdW1 mixing rules are valid only when molecular size differences of components in a mixture are not very large. The vdW1 type density-dependent mixing rule proposed by Chen et al. [1] is superior for the prediction of pressure and vapor–liquid equilibria when components in the mixture have very different sizes. The extension of the mixing rule to chain-like molecules and heterosegment molecules was also made with good results. In this paper, the comparison of different mixing rules are carried out further for the prediction of the density and the residual internal energy for binary and ternary Lennard–Jones (LJ) mixtures with different molecular sizes and different molecular interaction energy parameters. The results show that the significant improvement for the prediction of densities is achieved with the new mixing rule [1], and that the modification of the mixing rule for the interaction energy parameter is also necessary for better prediction of the residual internal energy.     

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.     

Calculation of solid's solubilities in mixed liquid solvents by the λh equation using mixing rules

Based on the proposed mixing rules for λ and h, the λh equation was extended to calculate the solubilities of solids in mixed liquid solvents. The correlation results for 89 pairs of binary solvent systems and 4 pairs of quaternary solvent systems showed that the extended equation had good agreement with experimental data. Furthermore, a prediction method was given with no parameter adjustment, which was suitable for nonalcohol-solvent systems.     

Solid–liquid equilibria for binary and ternary systems with the Cubic-Plus-Association (CPA) equation of state

A systematic investigation of the CPA model’s performance within solid–liquid equilibria (SLE) in binary mixtures (methane + ethane, methane + heptane, methane + benzene, methane + CO₂, ethane + heptane, ethane + CO₂, 1-propanol + 1,4-dioxane, ethanol + water, 2-propanol + water) is presented. The results from the binary mixtures are used to predict SLE behaviour in ternary mixtures (methane + ethane + heptane, methane + ethane + CO₂). Our results are compared with experimental data found in the literature.