Two-parameter Cubic Equation Of State Research Articles

Comprehensive study on deep eutectic solvent density based on various EoSs: SRK, PT, VTSRK, sPC-SAFT

In this communication, the capability of a two-parameter cubic equation of state (EoS) i.e. Soave-Redlich-Kwong (SRK) EoS to predict the density behavior of deep eutectic solvents (DESs) in the wide range of temperature conditions and at atmospheric pressure, was improved using the volume-translated concept. At first, the c-parameter of 139 DESs (1793 experimental data points) was obtained referring to volume-translated SRK (VTSRK) EoS. The obtained results were compared with the original SRK and Patel-Teja (PT) EoSs and sPC-SAFT (simplified perturbed chain statistical associating fluid theory) EoS. Afterward, three c-parameter correlations were recommended based on molecular weight and acentric factor (Case 1), acentric factor (Case 2), and critical compressibility factor (Case 3). The capability of recommended c-parameter correlations was investigated by density modeling of 67 new DESs systems (419 experimental data). The AARD% (average absolute relative deviation) of Case 1, Case 2, Case 3 and sPC-SAFT EoS was 4.24%, 3.24%,4.19% and 0.5677%, respectively.

Vapor phase and two-phase PvTz measurements of difluoromethane + 2,3,3,3-tetrafluoroprop-1-ene

This paper provides 217 PvTz points for the binary systems of difluoromethane (R32) and 2,3,3,3-tetrafluoroprop-1-ene (R1234yf) both in the two-phase and superheated vapor regions. The R32 + R1234yf data were measured along eleven isochores (0.029794, 0.029936, 0.030724, 0.032113, 0.061323, 0.063915, 0.102150, 0.111536, 0.116592, 0.144796, 0.280627) m3·kg−1 for temperatures from (263 to 383) K for eleven R32 mole fractions (0.1214, 0.1330, 0.1792, 0.2712, 0.3432, 0.5229, 0.6231, 0.6489, 0.7020, 0.8122, 0.9460). From the measurements in the two-phase region, the vapor-liquid equilibrium behavior of the studied binary systems was derived by applying the flash method with the Carnahan–Starling–De Santis equation of state, the Peng–Robinson equation of state and a two-parameter cubic equation of state proposed by Stryjek. The representation of the vapor – liquid equilibrium for the studied binary systems obtained from the isochoric data is consistent with the experimental vapor – liquid equilibrium data available in the literature. The superheated vapor data were correlated using the virial equation of state, in addition to the aforementioned equations of state. The vapor phase data showed good agreement with the values calculated with the equations of state and the predictions provided by REFPROP 10.0.

Phase behavior of multicomponent alkane mixtures and evaluation of predictive capacity for the PR and RKPR EoS's

The RKPR EoS was developed in the past decade as a three-parameter model, with the intention to overcome some of the limitations of classic two-parameter cubic equations of state. More recently, parameter correlations were presented for alkanes and their mixtures, showing excellent capacity to predict the fluid phase behavior of binary mixtures up to the more asymmetric combinations and in wide ranges of temperature and pressure.In this work, different types of phase equilibrium data for multicomponent mixtures are considered, in order to see whether the good performance previously demonstrated for binary systems is also transferable to multicomponent mixtures like the ones typically encountered in synthetic fluids designed in the lab as model reservoir fluids.In the case of ternary systems with good availability of constant temperature and pressure data sets, the type and behavior of these Gibbs triangle diagrams are analyzed with reference to the general or global behavior of the system, based on pure compound saturation and binary critical lines, something we had not found in the literature preceding this work.The results and comparisons presented in this article allow concluding that RKPR2015 can be a useful tool with very good predictive capacity to describe the fluid phase equilibria of hydrocarbon mixtures. Advantages over the Peng-Robinson equation appear more clearly for the more asymmetric fluids, including synthetic gas condensates.

A simple and generalized model to represent the vapor-liquid equilibria and the liquid-molar-volume of alcohol-alkane binary mixtures

A simple thermodynamic model to describe the vapor-liquid equilibria and liquid densities of alcohol-alkane type mixtures is developed. The proposed model is based on a two-parameter cubic equation of state translated in volume and the Huron-Vidal-NRTL mixing rules. Generalized expressions in terms of the critical temperature and the critical pressure of alcohols and alkanes are developed for the binary energy interaction parameters (A12, A21) and the non-random mixing constant (α12) that appears in the NRTL activity model.To obtain the generalized expressions, the homologous series concept is used as follows: in first place, four series composed by an alcohol between ethanol and 1-pentanol with some n-alkanes between butane and n-undecane are selected. For each series, it is supposed that each binary parameter (A12, A21 and α12) can be described as a linear function in terms of the ratio RTc/Pc of n-alkanes. The slope and the intercept of generalized expressions for each series are determined by minimizing the total absolute relative deviation in bubble pressure. Finally, it is found that all the slopes and the intercepts for parameters A12 and A21 can been correlated as quadratic functions in terms of the RTc/Pc ratio of alcohols. Also, it is found that for all the series, the same generalized expression can be used to get parameter α12. In total, 22 mixtures are used to develop the model. The average absolute relative deviation in bubble pressure (AADP) estimated is 1.65%. Also, the average absolute deviation in the vapor phase molar fraction (DY) is 0.007.To validate the proposed model, calculations are performed for 31 binary mixtures that include branched alcohols and branched alkanes that were not used during the development of the model. In general, predictions are adequate and results obtained are similar to those reported in literature for associating models like the Cubic-Plus-Association (CPA), the Statistical Association Fluid Theory (SAFT) and the Group Contribution Association (GCA) equations of state. The AADP for the new model is 2.80%, while the DY value is 0.009.Finally, liquid densities predictions are performed for 22 alcohol-alkane mixtures and it is found that calculated values are in good agreement with experimental data. The average absolute relative deviation calculated in liquid density is 1.16%. In total 645 densities and 1030 bubble pressures are predicted satisfactorily with the proposed model.

Phase equilibrium calculations with quasi-Newton methods

The phase split problem, formulated as an unconstrained minimization of the Gibbs free energy, is commonly solved by the second-order Newton method, preceded by a number of first-order successive substitutions. For difficult problems, the convergence radius of the Newton method may be small and a high number of successive substitution iterations may be required before the switch, or in repeated switch-backs. An interesting alternative is given by the quasi-Newton methods, representing a good compromise between complexity and convergence speed. The quasi-Newton BFGS method exhibits a super-linear convergence rate (in some cases without step length control) and a rank two update of the Hessian matrix guarantees a hereditary positive definiteness. In this work, a scaling methodology is proposed for finding the appropriate change of variables for phase equilibrium problems; applied to the two-phase split problem, the resulting change of variables leads to a Hessian matrix of the form H=I+D+ND, where I is the identity matrix, D is a diagonal matrix with elements vanishing at the solution, and ND is an effective low-rank matrix. The results of numerical experiments carried out on several test cases show that the BFGS method using the proposed variables is more robust and efficient than previous implementations (from the literature and open source codes). A two-parameter cubic equation of state was used in this work, but any equation of state can be used. The quasi-Newton methods are particularly suited for thermodynamic models for which the Hessian matrix is difficult or costly to obtain.

Volume-translated Peng-Robinson equation of state for liquid densities of diverse binary mixtures

Two-parameter cubic equations of state (CEOS) are widely used in process engineering calculations. However, inaccurate liquid density predictions remain a significant deficiency in these equations. To remedy this problem, a volume translation of the CEOS is frequently employed. In a recent work, we presented a volume-translated Peng-Robinson equation of state (VTPR EOS) that is capable of providing accurate density predictions for both saturated- and single-phase regions of pure fluids at high pressures. In the current work, we present an extension of that approach, employing conventional mixing rules, to predict densities of liquid mixtures over large ranges of pressure and temperature. For this purpose, two databases were compiled for vapor–liquid equilibrium and liquid density measurements of 73 binary systems composed of diverse chemical species. The molecular species in the databases ranged widely in terms of molecular size, shape, asymmetry and polarity and, thus, were well suited to test the efficacy of our approach. Overall, the databases contained more than 5000 data points for vapor–liquid equilibrium measurements and over 13,000 data points for liquid density measurements of mixtures.Results indicate that extension of the VTPR EOS to liquid mixtures is capable of providing reliable density predictions for diverse binary mixtures over large ranges of pressure. The VTPR EOS predictions of mixture liquid densities yield errors that are three to five times lower than the corresponding predictions from the untranslated Peng-Robinson equation of state (PR EOS). Specifically, the overall percentage average absolute deviations (%AAD) from the VTPR EOS varied from 1.5 to 3 for binary mixtures. This represents a substantial improvement relative to the untranslated PR EOS, for which errors ranged from 2 to 15%AAD.

Efficient and reliable mixture critical points calculation by global optimization

Multicomponent systems may exhibit several critical points or no critical point at all. Local methods can find only one critical point for a given initial guess. Recently, several global methods have been proposed for finding all the solutions of the problem. In the present work, we propose a gradient-based calculation method using global optimization, with temperature and molar volume as primary variables, and with analytical partial derivatives calculated from a two-parameter cubic equation of state. The Tunneling global optimization method is used for finding all the global minima. The implementation is based on a unique feature of the Tunneling method, which is able to find efficiently and reliably multiple minima at the same level. Several mixtures from binaries to petroleum reservoir fluids are used to test the proposed method. Numerical experiments proved the efficiency and reliability of the Tunneling method for finding all mixture critical points.

Numerical properties of equations involving high-order derivatives of pressure with respect to volume

AbstractThis paper presents some unexpected features related to the solution of equations containing a high-order derivative of pressure with respect to volume equated to zero. For pure components, such equations define, in the pressure-temperature plane, nodal curves similar in shape to mixture spinodal curves. The analysis was made for a general form of two-parameter cubic equations of state and various numerical aspects for the Redlich-Kwong equation of state are exemplified.

Isochoric Phase Stability Testing for Hydrocarbon Mixtures

A method for global phase stability testing at isotherm-isochoric specifications has been developed. The formulation of the tangent plane distance (TPD) criterion in terms of the Helmholtz free energy density (calculated from a general two-parameter cubic equation of state) with component molar density as independent variables is used. An unconstrained minimization problem of a smooth function is solved using the tunneling global optimization method. Calculations are carried out for several representative mixtures; special attention is paid to the performance of the algorithm for near-critical conditions and the vicinities of phase boundaries. The test examples show the robustness and efficiency of the proposed method, and its high reliability in solving the isochoric phase stability problem.

Pseudocomponent Delumping for Multiphase Systems with Waxy Solid Phase Precipitation

Modeling of wax precipitation from hydrocarbon mixtures requires extended compositional data for the heavy fractions; as a result, the number of components in the mixture is usually large and phase equilibrium calculations are computationally expensive and may be prohibitive. A lumping procedure which reduces the dimensionality of phase equilibrium calculations without affecting the location of solid phase transitions is used. By lumping into pseudocomponents, some information on mixture behavior is lost; this information is recovered by a delumping (inverse lumping) procedure. Delumping acts as an interface between simulation tools and results involving different fluid representation levels. The key in the delumping procedure is to relate equilibrium constants of the detailed fluid to some quantities evaluated from lumped fluid flash results. A general form of a two-parameter cubic equations of state is used for vapor and liquid phases, and heavy components are assumed to precipitate in a single solid solution. The proposed lumping/delumping procedures are successfully tested for two synthetic mixtures and a typical reservoir fluid (gas-condensate). Very good agreement is obtained between phase distributions and component mole fractions of detailed and delumped systems. This is the first time that a delumping procedure is proposed for multiphase systems with solid phase precipitation.

Consistent delumping of multiphase flash results

An analytical and consistent delumping method, recently formulated and implemented for processes involving two-phase flash calculations, is extended to flash calculations involving any number of coexisting phases. The equation of state considered is a two-parameter cubic equation of state (EoS) with van der Waals mixing rules and non-zero binary interaction parameters (BIPs). The method is used for calculating three- and four-phase equilibria in a series of hydrocarbon mixtures containing different amounts of carbon dioxide, nitrogen, hydrogen sulphide and water. In all examples, the location, compositions and proportions of the coexisting phases obtained by the delumping procedure are extremely close to those obtained from calculations with the detailed fluid, even in cases where the multi-phase region is very tiny. To the best of our knowledge, this is the first time that a delumping procedure is formulated and successfully used for multiphase equilibria.

A new two-parameter cubic equation of state for predicting phase behavior of pure compounds and mixtures

In this work, a new two-parameter cubic equation of state is presented based on perturbation theory for predicting phase behavior of pure compounds and of hydrocarbons and non-hydrocarbons. The parameters of the new cubic equation of state are obtained as functions of reduced temperature and acentric factor. The average deviations of the predicted vapor pressure, liquid density and vapor volume for 40 pure compounds are 1.116, 5.696 and 3.083%, respectively. Also the enthalpy and entropy of vaporization are calculated by using the new equation of state. The average deviations of the predicted enthalpy and entropy of vaporization are 2.393 and 2.358%, respectively. The capability of the proposed equation of state for predicting some other thermodynamic properties such as compressibility, second virial coefficient, sound velocity in gases and heat capacity of gases are given, too. The comparisons between the experimental data and the results of the new equation of state show the accuracy of the proposed equation with respect to commonly used equations of state, i.e. PR and SRK. The zeno line has been calculated using the new equation of state and the obtained result compared with quantities in the literatures. Bubble pressure and mole fraction of vapor for 16 binary mixtures are calculated. Averages deviations for bubble pressure and mole fraction of vapor are 9.380 and 2.735%, respectively.

A new cubic equation of state for simple fluids: pure and mixture

A two-parameter cubic equation of state is developed. Both parameters are taken temperature dependent. Methods are also suggested to calculate the attraction parameter and the co-volume parameter of this new equation of state. For calculating the thermodynamic properties of a pure compound, this equation of state requires the critical temperature, the critical pressure and the Pitzer’s acentric factor of the component. Using this equation of state, the vapor pressure of pure compounds, especially near the critical point, and the bubble point pressure of binary mixtures are calculated accurately. The saturated liquid density of pure compounds and binary mixtures are also calculated quite accurately. The average of absolute deviations of the predicted vapor pressure, vapor volume and saturated liquid density of pure compounds are 1.18, 1.77 and 2.42%, respectively. Comparisons with other cubic equations of state for predicting some thermodynamic properties including second virial coefficients and thermal properties are given. Moreover, the capability of this equation of state for predicting the molar heat capacity of gases at constant pressure and the sound velocity in gases are also illustrated.

An Extension of CEOS/AE Zero-Pressure Mixing Rules for an Optimum Two-Parameter Cubic Equation of State

Cubic equations of state are widely used in refinery and petroleum reservoir industries for the prediction of fluid-phase behavior. Two of the most well-known cubic equations of state broadly accepted in industry are the SRK and PR equations. However, neither the SRK nor the PR equation of state yields the best results for the prediction of liquid densities of polar components and heavy hydrocarbons. An approach is proposed to locate an optimum two-parameter cubic equation of state. A methodology is also proposed to modify Twu's CEOS/AE zero-pressure mixing rules to extend the range of application of these mixing rules. The mixing rule developed in this work is incorporated into our new cubic equation of state for the prediction of phase equilibria of highly nonideal chemical mixtures.

Thermodynamics of semicontinuous mixtures using equations of state with group contributions

In the chemical engineering design of industrial distillation columns data on the vapor-liquid equilibrium of complex mixtures are very important. Because of the expense of the experimental determination of such data, there is interest in their accurate prediction. Multicomponent mixtures such as petroleum or petroleum fractions contain a large number of similar chemical species. The method of continous thermodynamics is suitable for the description of the vapor-liquid equilibria of these systems. Continuous thermodynamics is based on the application of a continuous distribution function for describing the composition of multicomponent mixtures. Coupled with activity coefficient models, this method is shown to be convenient for the prediction of vapor-liquid equilibria. However, applications have been limited to low pressures. Therefore, in this paper a model for the prediction of vapor-liquid equilibria based on a two-parameter cubic equation of state is developed. These parameters are calculated using the mixing rule of Wong and Sandler [1] which allows the inclusion of group contribution models (e.g. UNIFAC, modified UNIFAC or ASOG).

A new mixing rule for cubic equations of state and its application to vapor-liquid equilibria of polymer solutions

A new mixing rule for van der Waals-type two-parameter cubic equations of state (EOSs) has been proposed in this work, which is particularly suitable for highly asymmetric systems. The new mixing rule has been applied to the PRSV and SRK EOSs to calculate vapor-liquid equilibria (VLE) of polymer solutions. Compared to the other two mixing rules which have been used for VLE calculation of polymer solutions, the new mixing rule with only one temperature-independent parameter is very accurate for a wide range of temperatures. As the new mixing rule is very simple and accurate and only one temperature-independent parameter is needed, it enables the two-parameter cubic EOSs to be used in the practical calculations needed in the industrial processing of polymer solutions.

Use of the binary interaction parameter Lij( T,P,xi) function for van der Waals-type equations of state in vapor-liquid equilibrium calculations

The predictive accuracy of the newly developed binary interaction parameter function L ij = L ij ( T,P,x i ) of Lielmezs and co-workers has been studied further by means of the Redlich-Kwong-Soave (RKS) and Peng-Robinson (PR) equations of state in the vapor-liquid equilibria calculations over a wide range of P-T-x-y values. Sets of coefficients for use in the binary interaction parameter function L ij = L ij ( T,P,x i ) have been established for 29 selected binary systems. The results calculated of the vapor-liquid equilibria (bubble point) for the binary systems considered (L-RKS, L-PR) are in excellent agreement with the experimental data and the values obtained using the F-function of two-parameter cubic equations of state (F-RKS, F-PR). The predictions made in this work (L-RKS, L-PR) have a slight edge over the results calculated via the original Redlich-Kwong-Soave (RKS) and Peng-Robinson (PR) methods.

The use of the F-function of the two-parameter cubic equation of state in vapour—liquid equilibrium calculations of binary mixtures

The recently developed F-function of Shaw and Lielmezs, Mak and Lielmezs, and Liu, Lim and Lielmezs, modified to fit the van der Waals type cubic equations of state, has been used to assess the vapour—liquid equilibria calculations of binary mixtures over a wide range of P– T– x– y values. Sets of coefficients for use in the binary interaction parameter function L ij = L ij ( T, P, x i ) were determined for 23 representative binary mixtures. The prediction (this work) of the vapour—liquid equilibria (bubble point calculations) of the binary systems considered are in excellent agreement with the experimental data, and show slight improvement over the values obtained by means of the original Redlich—Kwong—Soave (RKS) and Peng—Robinson (PR) equations.

Use of F-function of two-parameter cubic equation of state in P- V- T calculations of binary mixtures

The applicability of the recently proposed F-function of Shaw and Lielmezs and Mak and Lielmezs, modified to fit the van der Waals type cubic equation of state, has been studied over a wide range of P-V-T-x values by means of available gas and liquid state experimental data for pure substances and binary mixtures. New optimum binary interaction parameters L ij = L ij( T, P, x), functionally dependent on temperature ( T), pressure ( P) and composition ( x), were introduced for 18 selected binary mixtures. The predicted (this work) saturated vapour pressure and liquid and vapour volume values for pure substance and liquid and vapour compressibility factor values for binary systems are in excellent agreement with the experimental data, and show a slight edge over the values calculated by means of the original Redlich-Kwong-Soave (RKS) and Peng-Robinson (PR) equations.

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.