Three-parameter Cubic Equation Of State Research Articles

Patel-Teja cubic equation of state – A review of modifications and applications till 2022

The Patel-Teja equation of state (PT EOS) is extensively researched, modified, and applied three-parameter cubic equation of state for the prediction of volumetric and thermodynamic properties by both academia and industry. As a result, we have presented a review article on PT EOS covering its history and progress up to 2022. We attempted to compile all the modifications along with applications to PT EOS for both pure fluids and mixtures. The prediction of a similar critical compressibility factor for all fluids, followed by unreliable liquid density estimation using two-parameter equations of state, prompts researchers to seek a better alternative. Therefore, three-parameter equations of state were introduced with substance dependent critical compressibility factor which led to a better representation of liquid density. We divided modifications to PT EOS into three categories: (a) Modify expression of parameters (a,b, and c) or alpha functions (b) Introduce new terms in the expression of the equation of state (c) Propose new mixing rules or modify them for the application to mixtures.The generalized Patel-Teja-Valderrama equation of state (PTV EOS) is one of the most successful and widely used modifications of PT EOS. Despite significant efforts to improve the accuracy and performance of PT EOS, an accurate equation of state that applies to all situations remains elusive, necessitating further study and development in the field.

Ionic liquid excess molar volume prediction: A conceptual comparison

In this study, the prediction of excess molar volume of ionic liquid + solvent binary system is described using modified two-parameter and three-parameter cubic equation of state (CEoS). The studied binary systems are consisted of 41 ionic liquids, 40 solvents and 5293 data point in the wide temperature range (278.15–353.15 K), ionic liquid mole fraction (0.0036–0.9920) and atmospheric pressure i.e. P = 1 bar. At the first step, the volume-translated parameter (c) was obtained based on three linear, distance function and exponential-type temperature-dependent cases and using experimental pure density. After that, and due to easy-to-use of c-parameter, various formula was presented based on pure component thermodynamic features for all three cases (IL and solvent, separately). Finally, to investigate the accuracy of presented c-parameter formula, the excess molar volume of 119 various binary systems was molded using volume-translated PR (VTPR) EoS and volume-translated SRK (VTSRK) EoS without using any binary interaction parameter and the obtained results was compared with the original three-parameter CEoS, Patel-Teja (PT) EoS. The obtained results indicated that the performed two-parameter CEoS modification leads to increase the accuracy of the conventional PR EoS and SRK EoS in comparision to original two-parameter CEoS version and also, PT EoS. The average absolute percent deviation of the VTSRK EoS (c-parameter based on distance function) and the VTPR EoS (c-parameter based on exponential-type temperature-dependent volume-translated) were fewer values through all studied cases and it was 2.0443 and 2.8927, respectively.

Evaluation of translated-consistent equations of state compared for the prediction of the Joule–Thomson effect at high pressures and high temperatures

In this research, Joule–Thomson coefficients for pure, binary, ternary, and multicomponent systems are predicted using translated and consistent versions of Peng-Robinson and Redlich-Kwong equations of state. The former is claimed to be certainly the most accurate and the safest three-parameter cubic equation of state ever published. The obtained results are compared with the predictions provided by Peng-Robinson, Soave-Redlich-Kwong, Volume Translated Peng-Robinson, Volume Translated Soave-Redlich-Kwong, and Patel-Teja-Valderrama equations of state along with different alpha functions, including Peng – Robinson original form (1976), Coquelet et al. (2004), Haghtalab et al. (2011), Saffari – Zahedi (2013), Soave – Redlich – Kwong original form (1972), Ozokwelu – Erbar (1987), Trebble – Bishnoi (1987), Nasrifar – Bolland (2004), and Pina – Martinez et al. (2019). Furthermore, the obtained results are compared to the available experimental data. Moreover, the performance of translated-consistent Peng-Robinson and translated-consistent Redlich-Kwong equations of state is determined at low pressures (60 atm) up to high pressures (360 atm) in comparison to the other equations of state. The dependency of the Joule – Thomson coefficient as a function of the mole fraction has been investigated. To compare better, Joule–Thomson inversion curves are also depicted for a variety of systems, containing pure, binary, ternary, and multicomponent mixtures. The obtained results are compared to the experimental data, correlated results, and molecular simulations.

Vapor-Liquid Equilibrium of the Ethylene–Butane Mixture

The article deals with the investigation of the ethylene-butane natural gases mixture that is promising for practical application. A unit with a small volume cell created to obtain data on phase equilibria and thermal properties of binary mixtures of little investigated and rare substances using several experimental methods was presented. It is shown that it was possible to use non-standard measuring equipment to obtain sufficiently reliable results in a wide range of temperatures, pressures and compositions. Application of a new method to determine two-component gas mixtures concentration was demonstrated. The method of experimentation was described as well. The experimental results of ethylene–butane mixture thermal properties and vapor-liquid equilibrium that have expanded our knowledge in relation to this two-component natural refrigerant behavior in the low-temperature region and at high dilutions were tabulated. The comparison of the experimental and calculated data showed the possibility of applying the three-parameter cubic equation of state to simulate the thermodynamic properties of the ethylene–butane mixture without using test data of the mixture.

Prediction of thermodynamic properties of refrigerant fluids with a new three-parameter cubic equation of state

To describe the thermodynamic properties of refrigerant fluids, it is important to use a reliable thermodynamic model able to predict accurate results for both pure compounds and mixtures. In this study, a new three-parameter cubic equation of state is presented, based on the modification of the well-known Patel–Teja equation of state. The new equation of state is associated with the Mathias–Copeman alpha function.By only knowing the acentric factor ω and the experimental critical compressibility factor Zc of pure compounds, it is possible to predict the thermodynamic properties of both pure compounds and mixtures by means of the new equation of state. No binary interaction parameter kij is needed for the prediction of mixture properties.The results obtained with the new equation of state show a good agreement with experimental data for vapor–liquid equilibrium and density properties. The obtained results are particularly satisfying for liquid density, and in the vicinity of the critical point, by comparison with the results obtained using the Peng–Robinson and the Patel–Teja equations of state.

A new equation of state for studying the thermodynamic properties of real gases

In this study, based on the compressibility effect of gas molecules, a new three-parameter cubic equation of state (EOS) is derived. To validate this EOS, density predictions of methane, ethane, carbon dioxide and oxygen have been studied using the new EOS at the temperature of 373 K and at the pressures up to 100 MPa. The results show a good agreement with reference data and this suggests that the proposed EOS would help to improve the study of thermodynamic properties for real gases.

Prediction of compressibility factor for gas condensate under a wide range of pressure conditions based on a three-parameter cubic equation of state

The gas compressibility factor as an essential thermodynamic parameter is often used to analyze PVT behavior in natural gas engineering. To accurately predicate it for various gas condensates, a data base containing 916 data sets covering a wide range of experimental conditions is employed to establish and test the prediction model based on a three-parameter cubic equation of state (EoS). The presented model which based on EPT EoS combines with Elliott–Daubert binary interaction coefficients correlation, Ahmed et al. splitting and Hosseinifar–Jamshidi characterization methods is superior to conventional empirical correlations with three mixing rules and two-parameter EoSs as SRK, PR, PRSV and MPR2. Statistical error analysis shows that it outperforms empirical correlations with average absolute relative errors of 1.45% and coefficient of determination of 0.989. At pressure and temperature up to 95.04 MPa and 429.3 K, respectively, the model also outperforms two-parameter EoSs with average absolute relative errors of 1.65% and coefficient of determination of 0.992. The presented model is efficient and practical for predicting the compressibility factor of gas condensate.

An improved generalized three-parameter cubic equation of state for pure fluids

An improved simple generalized three-parameter cubic equation of state for pure fluids is proposed to estimate pure component fluid properties by incorporating substance dependent epsilon-function and almost substance independent sigma-function in the generic cubic equation of state, followed by temperature and acentric factor dependent attractive term. The proposed cubic equation of state is utilized to predict critical compressibility factor, liquid molar volume, vapor molar volume, and pure component vapor pressure for several simple hydrocarbons and fatty acid of methyl esters, and the predicted properties are compared with the actual values. In this study, most of the pure fluid properties are considered from literature; however experimentation is carried out for few fatty acid of methyl esters to determine pure fluid vapor pressure at elevated temperature and pressure. The proposed cubic equation of state is found to be accurate in predicting both literature reported and experimentally determined pure fluid properties. The overall performance of the proposed cubic equation of state is also compared with commonly used cubic equations of state and is found to be superior for predicting critical compressibility factors and other pure fluid properties.

Phase Equilibria of CO2 + n-Alkane Binary Systems in Wide Ranges of Conditions: Development of Predictive Correlations Based on Cubic Mixing Rules

The phase equilibria of binary CO2 + n-alkane mixtures have been studied by an important number of authors, both experimentally and using different types of thermodynamic models. Modeling studies of the phase behavior of such highly nonideal systems have generally achieved only partially accurate results in the correlation of phase equilibrium data when considering wide ranges of temperature, pressure, and n-alkane molecular weight. In this study, a predictive correlation for the phase behavior of CO2 + n-alkane systems, based on a three-parameter cubic equation of state (EOS), that is, the RK-PR EOS, coupled to cubic mixing rules (CMRs), is developed and tested. CMRs have been shown to be capable of an accurate correlation of the phase equilibria asymmetric CO2 + n-alkane binary systems, in wide ranges of temperature and pressure, when using system-specific interaction parameters. For developing the predictive correlation a critical review of published experimental data for the series was carried out, covering a total of about 100 references. An important degree of inaccuracy or scatter is often found when comparing data sets from different laboratories, specially for the more asymmetric systems (CO2 + a long chain n-alkane). Tables of references covering CO2 + n-alkane systems from C1 to C36 are presented for different types of experimental data, including critical end points (CEPs), critical points, liquid–liquid–vapor (LLV) equilibrium, and isobaric (Txy), isothermal (Pxy), and isoplethic (PT) two-phase equilibrium data sets. Examples of disagreement between different sets of data are presented and discussed. In some cases, a decision concerning the identification of the set that should be regarded as the most reliable, can be based on the experimental method employed, on the purity of the n-alkane, and on the observation of other data for conditions, and/or systems in the series, which are close to those of the data set under scrutiny. Nevertheless, the availability of such information is not enough, in other cases, to assess the quality of a given data set, where we have either different data sets in disagreement or a unique set, for which we are in doubt about its accuracy. In such situation, a predictive correlation for the whole series of binary systems is helpful to make a decision on the possible level of reliability of a given phase equilibrium data set. The present study is useful both to make decisions on conflicts between contradictory phase equilibrium data sets and to predict the phase equilibria of binary systems that have no experimental information available in the literature.

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.

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.

Thermodynamic Properties of Light Synthetic Natural Gas Mixtures Using the RK−PR Cubic Equation of State

In this work, we test the recently developed generalized Redlich−Kwong/Peng−Robinson (RK−PR) three-parameter cubic equation of state for predicting thermodynamic properties of light, synthetic natural gas mixtures using the simple one-fluid mixing rule. The required binary interaction parameters (considered temperature-independent) come from correlation of literature vapor−liquid equilibrium data for the involved binary systems covering wide temperature and pressure ranges. Densities and phase envelopes of natural gas mixtures calculated with the equation of state are compared to highly accurate literature data to check the validity of the model. We also report comparisons with the predictions obtained from the Patel−Teja (PT) and the molecular-based, perturbed-chain SAFT (PC−SAFT) equations of state. RK−PR has good predictive ability for these systems considering its simplicity.

A new cubic equation of state for reservoir fluids

A new three-parameter cubic equation of state is developed with special attention to the application for reservoir fluids. One parameter is taken temperature dependent and others are held constant. The EOS parameters were evaluated by minimizing saturated liquid density deviation from experimental values and satisfying the equilibrium condition of equality of fugacities simultaneously. Then, these parameters were fitted against reduced temperature and Pitzer acentric factor. For calculating the thermodynamic properties of a pure component, this equation of state requires the critical temperature, the critical pressure, the acentric factor and the experimental critical compressibility of the substance. Using this equation of state, saturated liquid density, saturated vapor density and vapor pressure of pure components, especially near the critical point, are calculated accurately. The average absolute deviations of the predicted saturated liquid density, saturated vapor density and vapor pressure of pure components are 1.4%, 1.19% and 2.11%, respectively. Some thermodynamic properties of substances have also been predicted in this work.

Prediction of PVT and VLE in polymer solutions using a cubic-perturbed equation of state

A three-parameter cubic equation of state has been proposed for predicting PVT for polymers and VLE in polymer solutions. The pure component parameters a ii, b i, and c i for seven polymers have been correlated using the PVT data. The correlated accuracy of PVT at temperatures from 308 to 491 K and pressures up to 200 MPa is 0.05%. The solvent activities in polymer solutions have been predicted for 45 binary data sets using the excess Gibbs free energy mixing rule combined with the group solution model. The pure solvent parameters a ii and b i necessary for predicting the activities have been evaluated directly using the correlation previously generalized using the critical temperature, critical pressure and acentric factor. The total average deviation of activities is 11.10%.

98/02382 A new three-parameter cubic equation of state of refrigeration engineering calculations : Zhang, H.-L. et al. Int. J. Refrig., 1997, 20, (6), 421–440

98/02382 A new three-parameter cubic equation of state of refrigeration engineering calculations : Zhang, H.-L. et al. Int. J. Refrig., 1997, 20, (6), 421–440

Two- and three-phase equilibria in systems containing benzene derivatives, carbon dioxide, and water at 373.15 K and 10–30 MPa

Phase equilibria in binary and ternary systems containing benzene derivatives, carbon dioxide, and water have been investigated experimentally at a temperature of 373.15 K and pressures ranging from 10 to 30 MPa. The experimental results and literature data for further systems in cluding benzene derivatives, carbon dioxide, and water provide a basis of experimental phase equilibrium data, clearly showing the influence of the number and the position of methyl and hydroxy groups of benzene derivatives that are in phase equilibria with carbon dioxide and water. The experimental data have been correlated using the equations of state by Peng and Robinson [D.-Y. Peng, D.B. Robinson, A new two-constant equation of state, Ind. Eng. Chem., Fundam., 15 (1976) 59–64] and Yu and Lu [J.-M. Yu, B.C.-Y. Lu, A three-parameter cubic equation of state for asymmetric mixture density calculations, Fluid Phase Equilibria, 34 (1987) 1–19] with the mixing rule proposed by Mathias et al. [P.M. Mathias, H.C. Klotz, J.M. Prausnitz, Equation-of-State mixing rules for multicomponent mixtures: The problem of invariance, Fluid Phase Equilibria, 67 (1991) 31–44]. The phase behavior of ternary systems determined experimentally, can be predicted well, using two binary interaction parameters per binary system, optimized to reproduce the phase equilibria of the binary subsystems.

Extension of the Wong-Sandler mixing rule to the three-parameter Patel-Teja equation of state: Application up to the near-critical region

The Wong-Sandler mixing rule was extended to the Patel-Teja three-parameter cubic equation of state by introducing suitable corresponding parameters. The new model, PT–WS model, has been extensively tested on various classes of binary/ternary mixtures, special attention was paid to the effects on the representation of mixture phase behavior in the near-critical region. The test results indicate that the new model is capable of improving both the vapor–liquid equilibrium and volumetric properties significantly, especially for highly asymmetric systems. The proposed method of implementing the Wong-Sandler mixing rule can be applied to other multi-parameter cubic equations of state.

A new three-parameter cubic equation of state for refrigeration engineering calculations

A new three-parameter cubic equation of state is proposed in which the repulsion pressure term of the semiempirical van der Waals equation has been modified based on the hard-sphere potential and a functional form similar to Redlich-Kwong attraction term is adopted for the attraction term. All three parameters are treated as functions of temperature. For the purpose of examining the applicability of the new equation, it was applied to describe the thermodynamic properties of difluoromethane (R-32) and pentafluoroethane (R-125). It has been found that the essential thermodynamic properties are represented by the new equation as accurately as a conventional far-complicated modified Benedict-Webb-Rubin equation in the working range of refrigeration equipment. The new cubic equation was also compared with the Peng-Robinson equation and the Carnahan-Starling-De Santis equation.

A new cubic equation of state

Abstract Two-parameter cubic equations of state are analyzed and a new three-parameter cubic equation of state is proposed with the critical compressibility factor taken as substance dependent. An apparent critical compressibility factor is determined by optimizing the calculation of saturated liquid densities while the equality of fugacities along the saturation curve is imposed. The new cubic equation of state, combined with a previously proposed α function and mixing rule, proves to be a powerful equation for predicting properties of pure components and mixtures. The new equation not only improves volumetric property calculations relative to earlier efforts, but also achieves high degree of accuracy in the calculation of vapor-liquid equilibria for highly nonideal systems such as polar/nonpolar mixtures. Alcohol/hydrocarbon mixtures are used as an extreme test of the new equation of state.

VLE correlation for binary 1-alkanol + n-alkane mixtures

The three-parameter cubic equation of state previously proposed for polar fluids and fluid mixtures by Iwai et al. has been refined for its determination of the pure component parameters, and used to correlate the vapor-liquid equilibrium values for 15 1-alkanol + n-alkane mixtures. Three different mixing rules were used for obtaining the parameter “a” for mixtures. The results obtained from bubble-point calculations are compared with those obtained with the Peng-Robinson equation and the Trebble-Bishnoi equation. The proposed approach together with the mixing rule proposed by Schwartzentruber et al. yielded the lowest deviations in the calculated total pressures and vapor compositions.