Density-dependent Mixing Rules Research Articles

PVTx properties of H 2O–H 2S fluid mixtures at elevated temperature and pressure based on new experimental data

The volumetric properties of H 2O–H 2S fluid mixtures have been determined experimentally at temperatures of 150 to 400 °C and pressures up to 240 bar. Using these data and existing equations of state, we have developed a thermodynamic model for H 2O–H 2S fluid mixtures. This model is based on an asymmetric description of phases, which includes an activity model and a P–T-dependent Henry’s law constant for the liquid, and equations of state with mixing rules for the vapour. The fugacity of the vapour was calculated using the cubic equations of state of Peng and Robinson (1976) and Patel and Teja (1982) with density-dependent and composition-dependent mixing rules. Sets of binary interaction parameters for these equations were fitted to the experimental data obtained in this study supplemented by high-temperature PVTx data for H 2O–H 2S fluid mixtures reported in the literature. The Peng–Robinson equation used in conjunction with density-dependent mixing rules was found to be the most accurate of the available equations in representing the properties of the vapour phase. The errors in the pressure of the homogeneous vapour mixtures estimated using the above equations of state (relative deviation from the experimentally determined pressure) were comparatively low, ∼5% to 8%. However, the errors were significantly higher for the estimated pressure of vapour saturated with liquid, i.e., along the vapour–liquid phase boundary (11–15%), due to the polar nature of H 2O and H 2S and the resulting highly non-ideal behaviour of the fluid mixtures. The results of this study make it possible to reliably estimate the volumetric properties of aqueous fluids containing H 2S at temperatures and pressures up to 400 °C and 240 bar, i.e., for conditions commonly encountered in natural hydrothermal systems.

Estimation of solid solubilities in supercritical carbon dioxide: Peng–Robinson adjustable binary parameters in the near critical region

The density dependence of the binary parameters of the Peng–Robinson equation of state in near the critical region was examined. Published solubility data of eight compounds in pure CO 2 have been fitted to the Peng–Robinson equation in combination with one and two parameters van der Waals mixing rules and in combination with the three parameter density dependent mixing rule of Mohamed and Holder. A systematic study has been done to determine the influence of different terms in the mixing rules. In order to obtain density dependence, binary parameters were calculated for each isotherm at particular experimental point separately in the way to equalise experimental and calculated solubility data. The system was formulated as an equation-oriented model and solved by means of a nonlinear programming optimisation algorithm. For all compounds the binary interaction parameters thus obtained were found to vary strongly with pressure in the range from 75 bar to approximately 150 bar, i.e. near the critical end point (CEP) of the low temperature branch of the three phase solid–liquid–gas (SLG) curve. At higher pressures, the parameter is practically independent on pressure. In general, for the systems investigated, k ij increases linearly with increasing density and reaches a constant value at higher densities in the range from 700 to 800 kg/m 3, depending on the system under investigation.

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.

EXTRAÇÃO DO COLESTEROL COM MISTURAS DE DIÓXIDO DE CARBONO E ETANO SUPERCRÍTICO

There are nutritional and dietary factors, such as those linking excess blood cholesterol to heart diseases and cancer, that justify the development of a technology for the reduction of cholesterol levels in consumed meals. Previous studies have demonstrated the viability of using CO2 for the removal of cholesterol from butter oil. Cholesterol has higher solubility in ethane than in CO2. The use of CO2/ethane mixtures, therefore, presents as an attractive alternative, as a compromise between the higher ethane cost better cholesterol removal efficiency obtained and envolved. Using a high pressure experimental extraction apparatus that allows the independent control of temperature and pressure, cholesterol solubilities in supercritical CO2/ethane mixtures with 8%, 16%, 34%, 76%, 88% e 96,5% of ethane were determined at 328,1 K and pressures from 120 to 190 bar. Experimental results show an increase in solubility with pressure and a composition of ethane in the mixture. Experimental data were correlated with a thermodynamics model that uses Peng-Robinson equation with two different mixing rules: the normally used van der Waals and a rule that considers the interaction parameter density dependent resulting in a quartic equation of state. The density dependent mixing rule reveal more successful in the correlation of experimental data.

Describing vapor-liquid equilibria in methanol +n-alkane systems by means of an equation of state with association

The ability of an equation of state incorporating association to describe vapor-liquid equilibrium of methanol +n-alkane systems has been tested. An equation of state with association is divided into “physical” and “chemical” parts, where the Patel-Teja cubic equation of state is used for the description of the physical part and an association tenn forms the chemical part. Five two- and threeparameter mixing rules were tested and compared with an equation of state without association. On the average for the first eight n-alkanes, the improvement in fit of pressure using association was 30–50 % and the calculated vapor compositions were also much improved. In addition, there were lewer points with unstable liquid phase (false liquid-liquid split). Both with and without association, density-dependent mixing rules based on a local-composition concept gave the best result.

A modified perturbed hard-sphere equation of state

A modified perturbed hard-sphere equation of state, represented by the Carnahan—Starling equation with the van der Waals attractive part, has been investigated using the Mollerup approach for the energy parameter a for pure substances. The equation has been extended to mixtures, for both vapor—liquid equilibria and solid—supercritical fluid equilibria, using the Boublik—Mansoori equation with the same van der Waals attractive part. The vapor pressure, density, second virial coefficient, specific heat capacity and enthalpy were examined for pure substances. The new equation was also able to predict accurately the vapor pressures of fatty acids. The predictions of both vapor—liquid equilibria and solid—supercritical fluid equilibria, using the one fluid van der Waals mixing rules with one binary interaction parameter, are significantly better than those obtained by cubic equations of state, especially for large molecules. The predictions for solid—supercritical fluid systems were improved by incorporating solute—solute interaction parameters. Density dependent mixing rules were also examined.

A partial molar fugacity coefficient useful for changing mixing rules of cubic equations of state

Abstract Lawal, A.S. and van der Laan, E.T., 1994. A partical molar fugacity coefficient useful for changing mixing rules of cubic equation of state. Fluid Phase Equilibria, 95: 109-121. A cubic equation of state (EOS) is used to develop a cohesive technique for the partial molar fugacity coefficient of a component in a mixture. The technique simplifies the determination of fugacity coefficients for any type of mixing rules by the introduction of a differential operator and logarithmic transformation of a mixture property. The technique is very attractive in that it introduces functional modifications into previously established mixing rules of pressure-explicit equations of state, and offers a unique opportunity for a unified fugacity routine in PvT packages and process simulation models. The technique accommodates density-dependent mixing rules as well as non-cubic equations; any type of mixing rule with varying levels of difficulty is admissible by the differential operator, without starting the derivation of mixing fugacity coefficient afresh. Several mixture rules are presented to show the judicous use of the differential operator for the partial fugacity coefficient.

Prediction of VL and VLL equilibria of mixtures containing petroleum reservoir fluids and methanol with a cubic EoS

Abstract Avlonitis, D., Danesh, A. and Todd, A.C., 1994. Prediction of VL and VLL equilibria of mixtures containing petroleum reservoir fluids and methanol with cubic EoS. Fluid Phase Equilibria, 94: 181-216. A cubic equation of state, with simple non-density-dependent mixing rules, is used to represent the phase behaviour of multicomponent mixtures containing hydrocarbons up to n -octane, nitrogen, carbon dioxide, hydrogen sulphide, water and methanol. The reliability of the model is tested against diverse experimental VL and VLL equilibrium data of multicomponent systems. The proposed correlation can be applied for efficient design and operation of transfer lines and processing units. The practical value of the theoretically correct concept of density dependence is tested on a density-dependent version of the above mixing rules. It is demonstrated that while both mixing rules produce very similar results when only one polar component is present, the density-dependent mixing rules may be inferior for mixtures containing both water and methanol, owing to an inherent deficiency. It is also demonstrated that an invariant version of the proposed non-density-dependent mixing rule may not lead to indisputably improved predictions for the studied systems.

Vapor-liquid equilibria in the systems toluene/naphthalene and cyclohexane/naphthalene

In this study, the authors report isothermal vapor-liquid equilibrium (VLE) data for the toluene/naphthalene and cyclohexane/naphthalene systems which can be considered as model compounds for coal liquids. Vapor-liquid equilibrium data for the binary systems toluene/naphthalene and cyclohexane/naphthalene were measured at 0-1,300 kPa and 370-500 K using a 1-L stirred autoclave system. All pure components and binary P-T data were well fitted with a three-constant Antoine equation. The data can bee accurately correlated with the modified Peng-Robinson equation of state using density-dependent mixing rules to describe both the vapor and liquid phases. The binary interaction parameters and correction factors for the equation of state are reported at each isotherm.

The experimental determination and modelling of VLE for binary subsystems of the quaterhary system N 2 + CH 4 + C 4H 10 + C 14H 30 UP to 1000 bar and 440 K

As a part of a laboratory study on nitrogen displacement in light-oil reservoirs, in which a mixture of methane, butane and tetradecane is used as a model oil, the phase behaviour of the binary subsystems of the quaternary system nitrogen + methane + butane + tetradecane is investigated in the temperature and pressure range of interest. The phase behaviour of the binary systems nitrogen + methane and methane + butane is known from literature. VLE of the binary systems nitrogen + tetradecane and methane + tetradecane were determined experimentally at 320 – 440 K according to the synthetic method using a high pressure cell with sapphire windows. VLE of the binary system butane + tetradecane were determined experimentally in the same temperature range according to the same method but now using the so-called Cailletet equipment. The experimental data were fitted using several equations of state in combination with a number of mixing rules. It is found that local composition mixing rules of the Huron-Vidal type and density dependent mixing rules as used by Mohammed and Holder and a new density dependent mixing rule, are superior to the conventional mixing rules with one or two binary interaction parameters. With these improved mixing rules a satisfactory description of the experimental data, even at pressures as high as 1200 bar can be realized.

Vapor-liquid equilibria in the system of toluene/aniline, aniline/naphthalene, and naphthalene/quinoline

This paper reports on vapor-liquid equilibria for the aniline/naphthalene, toluene/aniline, and naphthalene/quinoline systems that have been determined at 0-1500 kPa and 490-623 K by using a static equilibrium cell. The data can be accurately correlated with the modified Peng-Robinson equation of state by using density-dependent mixing rules. The binary interaction parameters and correction factors for the equation of state are reported at each isotherm. The presence of coal-derived solids in these binary systems did not influence any of the binary bubble pressures.

Vapor-liquid equilibria in the systems of n-decane/tetralin, n-hexadecane/tetralin, n-decane/1-methylnaphthalene, and 1-methylnaphthalene/tetralin

Vapor-liquid equilibrium data for the binary systems of n-decane/tetralin, n-hexadecane/tetralin, n-decane/1-methyinaphthalene, and 1-methyinaphthalene/tetralin were measured at a low to moderate pressure (0 - 1123 kPa) by using a static equilibrium cell. The binary P-x data were isothermally correlated using the modified Peng-Robinson equation of state to describe both vapor and liquid phases at 473.15, 533.15, and 573.15 K. In this paper interaction parameters for density-dependent mixing rules are reported at each isotherm.

Practical equation of state and activity coefficient models based on the exp-6 fluid

Dodd L.R. and Sandler S.I., 1991. Practical equation of state and activity coefficient models based on the exp-6 fluid. Fluid Phase Equilibria, 69: 99-139. In this paper we test a partition function model recently developed for the exp-6 fluid mixture against experimental data for real fluids. This model leads to a unique size and density-dependent mixing rule providing non-quadratic density-dependence which satisfies the low-density and pure fluid boundary conditions. The equation of state calculations with this model gave good results, especially for highly asymmetric systems. Applications of a similar mixing rule to cubic engineering equations of state for real fluids yielded better predictions than the van der Waals one-fluid mixing rule. However, except in the near critical region, any clear advantage of the new density-dependent mixing rule for real fluids is lost when adjustable parameters are added. We also show that an activity coefficient model based on this exp-6 model leads to good correlations for low pressure vapor-liquid equilibria of both simple and very complex mixtures, indeed mixtures that could not easily be described by other models with the same number of parameters.

Vapor-liquid equilibrium data for the ammonia-water system and its description with a modified cubic equation of state

New pressure-temperature-overall composition vapor-liquid equilibrium (VLE) data ar reported fot the ammonia-water system at five temperatures between 20 and 140°C and up to 500 psia. These data have been converted to T-P-x data, and vapor-phase compositions have been calculated by means of a Redlich-Kwong equation of state modified to include Peneloux's volume translation and a density-dependent mixing rule. In order to achieve agreement of calculated vapor-phase compositions with previous literature results, it was necessary to use different k ij values in the vapor and liquid phases

Phase equilibria of essential oil components and supercritical carbon dioxide

The solubility of pure essential oil components and their binary mixtures in supercritical carbon dioxide was correlated using an equation of state with density-dependent mixing rules. This equation was then used to predict the phase diagram of the two binary carbon dioxide—limonene and carbon dioxide—citral systems at temperatures from 308.2 to 323.2 K and over the whole range of pressure and composition. In addition, the phase behaviour of ternary carbon dioxide—limonene—citral system was calculated at pressures from 9.0 to 10.9 MPa at 323.2 K and over the whole range of composition.

Molecular thermodynamics of fluid mixtures at low and high densities

A molecular-thermodynamic framework is proposed to describe phase equilibria over a wide range of densities and compositions. The proposed framework is expressed through a model for the Helmholtz energy that incorporates arbitrary mixing rules at high densities, while at low densities the model reduces to the correct second-virial-coefficient limit. In effect, this framework provides density-dependent mixing rules. The proposed framework is illustrated with an equation of state of the Boublik—Mansoori—van der Waals form. Special attention is given to high-pressure vapor—liquid equilibria for systems containing water and hydrocarbons.

The generalized van der waals partition function. V. Mixture of square-well fluids of different sizes and energies

In this communication we report the results of Monte Carlo simulations for mixtures of square-well molecules of different sizes and interaction energies. We use these results to test various local composition models which have been suggested, and find that only the model proposed earlier is in reasonable agreement with the simulations. This local composition model is then used in the generalized van der Waals partition function to derive a new density-dependent equation of state mixing rule. This mixing rule increases the order of the volume dependence of any equation to which it is applied, though a simple modification to the solution algorithm proposed here easily handles this problem. We find that this new density-dependent mixing rule leads to more accurate predictions than the van der Waals one-fluid mixing rule when no binary interaction parameters are used, and retains only the advantage of smaller values of these parameters and slightly better critical region predictions when binary interaction parameters are used.

Isochoric collinearity studies with density dependent mixing rules

Collinearity of isochores passing through the cricondentherm (CT) of a mixture has been proven theoretically and demonstrated with experimental data (Rowlinson et al., 1986). Also, recent research has shown that cubic EOS in combination with their standard quadratic MR predict the phenomena of isochoric collinearity at the CT (Barrufet and Eubank, 1987). In addition, several researchers have proposed density dependent MR for use with non—simple mixtures. This work is designed to investigate whether introducing density dependence in the MR used with cubic EOS will affect the prediction of isochoric collinearity at the CT. To accomplish this, the Redlich—Kwong equation of state with recently proposed density dependent mixing rules is used to predict the cricondentherm (CT) of six hypothetical binary mixtures. The slope of the isochore passing through the CT is then calculated on both the two—phase and the single—phase sides of the CT and compared. This check of the physical phenomena of isochoric collinearity at the CT is then compared with similar calculations from the Redlich—Kwong equation of state with van der Waals' mixing rules. The density dependent mixing rules used are found to violate the collinearity constraint at the CT, while use of the quadratic mixing rules do not. The effect of the density dependence in the mixing rules upon predicted quantities and the size of the retrograde region is also observed.

Extended 1-fluid theory for mixtures containing non-spherical molecules

An approximative method is proposed that relates the thermodynamic properties of mixtures of non-spherical molecules to those of mixtures of spherical molecules. It is formulated as a 1-fluid theory with density-dependent mixing rules containing two non-integer exponents. The spherical exponent retains the functional dependence on concentration and density that it has in the case of mixtures of spheres. The non-spherical exponent has been obtained by Monte Carlo simulation of mixtures of hard spheres and fused spheres (di- and tri- atomics); it depends little on density, molecular shape, or composition. For a long-ranged pair potential its value is close to 1; for a square-well potential of range 1.5 the value is close to 0.8. Therefore the effects of molecular size and of shape can be separated. The new mixing rule has been used in connection with a non-cubic equation of state for the calculation of phase equilibria in binary fluid mixtures under high pressure. The representation of isotherms at different temperatures and of critical curves has been improved significantly.

Correlation of thermodynamic properties of fluids by means of equations of state

Equations of state of different families are shortly described and reviewed. Simple models, such as the cubic equations of state, are considered in their performance to more sophisticated ones. Furthermore, different mixing rules are discussed, including the simple, random, and the complicate density-dependent mixing rules. Attention has been focused on the problem of the parameter estimation in equations of state: calculations are presented showing how an equation of state can be tuned for representing thermodynamic properties. Limitations of multiproperty and mixture prediction are also put forward.