Rules For Equations Of State Research Articles

Prediction of pair interactions in mixtures by matrix completion.

Molecular simulations enable the prediction of physicochemical properties of mixtures based on pair-interaction models of the pure components and combining rules to describe the unlike interactions. However, if no adjustment to experimental data is made, the existing combining rules often do not yield sufficiently accurate predictions of mixture data. To address this problem, adjustable binary parameters ξij describing the pair interactions in mixtures (i + j) are used. In this work, we present the first method for predicting ξij for unstudied mixtures based on a matrix completion method (MCM) from machine learning (ML). Considering molecular simulations of Henry's law constants as an example, we demonstrate that ξij for unstudied mixtures can be predicted with high accuracy. Using the predicted ξij significantly increases the accuracy of the Henry's law constant predictions compared to using the default ξij = 1. Our approach is generic and can be transferred to molecular simulations of other mixture properties and even to combining rules in equations of state, granting predictive access to the description of unlike intermolecular interactions.

Measurement and modelling of surface tensions of systems containing n-hexadecane, n-heptane and n-pentane

The surface tension of (n-hexadecane+n-heptane), (n-hexadecane+n-pentane) and (n-hexadecane+n-heptane+n-pentane) mixtures were measured at 91.3 kPa and the 293.15–323.15 K temperature range by the maximum bubble pressure method. The surface tension of pure n-hexadecane was measured at 91.3 kPa and the 303.15–343.15 K temperature range. The measured surface tension data of pure n-hexadecane was correlated by SPTM3 model, which was obtained from coupling scaled particle theory (SPT) and MMM equation of state. For the extension of the SPTM3 model to mixture calculations, the commonly used mixing rules for the cubic equations of state were applied. The comparisons between the obtained surface tension data and those calculated by the SPTM3 model indicate that the data can be described with deviations less than 1.56%. Therefore, the SPTM3 model is applicable for surface tension calculations of the studied mixtures.

Present status and potential of group contribution methods for process development

Abstract The development of chemical processes can be performed with the help process simulators. However, for the simulation, besides kinetic data, a reliable knowledge of the pure component and mixture properties is required. Since often 60–80% of the total costs arise in the separation step, a reliable knowledge of the phase equilibrium behaviour of the system to be separated is of special importance. The models for excess Gibbs energy g E and equations of state allow the calculation of the phase equilibrium behaviour of multicomponent systems using binary information alone. But often, the required binary experimental data are missing. For these systems, a reliable predictive thermodynamic model with a large range of applicabilities would be most desirable. With a view to the development of a powerful predictive method, work on the development of a group contribution method was started in Dortmund 1973. A prerequisite for this work is a large database. Therefore, all published pure component properties, phase equilibrium data, and excess properties were stored in a computerized form. At the same time, experimental techniques for the systematic measurements were built up to measure the missing values. The information stored was used for the development of the group contribution method UNIFAC. The weaknesses of this method were removed with the development of modified UNIFAC. With the introduction of g E -mixing rules in equations of state, the usage of the “solution of groups”-concept lead to group contribution equations of state. In combination with electrolyte models, the influence of strong electrolytes on the phase equilibrium behaviour can be taken into account. In this paper, the continuous development, present status, and potential of group contribution methods for the development of chemical processes will be shown.

高圧混合流体の相平衡測定

Phase equilibrium relations at high pressures are required in practical use, such as in the design and operation of supercritical fluid extraction. High-pressure phase equilibrium measurements of fluid mixtures cannot simply generate the reliable data. In the present review, the reference data are first mentioned. For the development of mixing rules in equations of state for fluid mixtures, reliable standard phase equilibrium data of fluid mixtures at high pressures are required as reference of standard. Next, recent researches on the phase equilibrium measurements at high pressures are reviewed by classifying the types of the experimental apparatus: static, circulation, static-circulation, flow, synthetic, and mass balance. All experimental methods and apparatus seem consistent, with the exception of some special cases. The experimental data obtained by the different experimental methods and apparatus should agree one another. In the high-pressure phase equilibrium measurements of fluid mixtures just a minor poor treatment in the experiment causes a large error in the final experimental results, and all experimental treatments should be carefully consistent to establish the reliable consistent data.

Mixing rules in cubic equations of state applied to refrigerant mixtures

Two equations of state and four mixing rules are applied to the calculation of the bubble point pressure of several refrigerant mixtures. An equation of state proposed by one of the authors, known as the PTV equation, and a modified Soave-Redlich-Kwong equation of state, known as Predictive Soave-Redlich-Kwong (PSRK), have been used. The mixing rules considered in the study include the classic van der Waals mixing rule with one and with two parameters for the PTV equation, and a model that includes the excess Gibbs free energy for the PSRK equation. Eighteen data sets from the literature for five binary refrigerant mixtures containing R134A (CH2FCF3) were considered for analysis. A special program was implemented for these calculations in which the fundamental equation for phase equilibria, the equality of fugacity of each component in all phases, was applied. Correlations for the interaction parameters as functions of the acentric factor are proposed. We present conclusions about usefulness of the different models and a recommendation for the best equation of state + mixing rule combination for correlating vapor-liquid equilibrium in the refrigerant mixtures studied.

Estimation of Henry's constants for aqueous systems at elevated temperatures

The treatment and removal of dilute organic contaminants from water require methods for estimating the vapor–liquid equilibrium behavior of these mixtures. Highly hydrophobic solutes may exhibit a maximum in Henry's law constant that is difficult to predict using traditional activity coefficient models. This paper examines the application of an activity coefficient model fitted specifically for dilute aqueous systems (UNIFAC-ENV) to make predictions for the infinite dilution activity coefficients at ambient conditions. The use of activity coefficient based mixing rules for equations of state to extend the range of the UNIFAC-ENV predictions to the temperatures typical for steam stripping conditions (80–140°C) is also evaluated. Application of this approach to Henry's law data for trichloroethylene in water indicates that these models require temperature dependent interaction coefficients to accurately predict the highly non-ideal solution behavior observed.

Associating models and mixing rules in equations of state for water/hydrocarbon mixtures

Equations of state (EoS) for associating fluids calculate explicitly the effect of hydrogen bonding by using chemical theory, perturbation theory or quasi-chemical theory. In all cases, the number of hydrogen bonding sites per molecule is an input parameter to the model. Specifically for water, two-, three- or four-site models have been used in the past by different investigators. In this work, the associated-perturbed-anisotropic-chain-theory (APACT) and the statistical-associating-fluid-theory (SAFT) are applied to predict the phase equilibrium of water/hydrocarbon mixtures, with emphasis on liquid-liquid equilibria (LLE). The accuracy of the different models to describe association in water is investigated. Different mixing rules are examined for SAFT. The original mixing rules proposed based on the van der Waals one-fluid theory work well for vapor-liquid equilibria (VLE) predictions, but not for water/hydrocarbon LLE, which are primarily influenced by the large difference in intermolecular interactions between water and hydrocarbon. For water/hydrocarbon mixtures, the one-fluid theory fails to predict reliably the hydrocarbon solubility because the local composition is very different from the bulk composition. A more realistic mixing rule for these systems is proposed that is based on the asymmetric mixing rule introduced originally for the perturbed-hard-chain-theory (PHCT). This mixing rule is used here for SAFT predictions of water/hydrocarbon LLE resulting in an improvement of the hydrocarbon solubility predictions. Nevertheless, the agreement of all the models investigated with the experimental data is, in general, poor. For comparison, calculations with the Redlich-Kwong-Joffe-Zudkevitch (RKJZ) cubic EoS are also reported.

A note on a modified Huron-Vidai mixing rule consistent with the second virial coefficient condition

Using the procedure by Wong and Sandler (1992), zero pressure mixing rules for cubic equations of state can be made consistent with the quadratic composition dependence of the second virial coefficient. The MHV1 mixing rule extended on the second virial coefficient condition has been applied to 6 binary and 3 ternary systems. No significant differences have been observed from predictions of the original MHV1 model which uses the linear mixing rule for equation of state parameter b.

Quadratic mixing rules for equations of state: Origins and relationships to the virial expansion

Hall, K.R., Iglesias-Silva, G.A. and Mansoori, G.A., 1993. Quadratic mixing rules for equations of state. Origins and relationships to the virial expansion. Fluid Phase Equilibria, 91: 67-76. Quadratic mixing rules, originally proposed by van der Waals, are used extensively in mixture calculations involving equations of state. Several ways exist to derive such mixing rules from the principles of statistical mechanics or merely classical thermodynamic arguments. In this report we relate the quadratic mixing rules to the rigorous mixing rules for virial coefficients of mixtures. The virial equation of state having a precise basis in statistical mechanics provides theoretical guidance for formulations of equations of state. In addition, the mixture combining rules for the virial coefficients are rigorous. The thesis of this paper is that, if multibody interactions occur in certain ways, the mixing rules for all virial coefficients become identical to the quadratic mixing rule for the second virial coefficient. This implies that quadratic mixing rules should be universal for coefficients (or their combinations) of equations of state.

New applications of Kahl's VLE analysis to engineering phase behavior calculations

Abstract Nitsche, J.M., 1992, New applications of Kahls' VLE analysis to engineering phase behavior calculations. Fluid Phase Equilibria , 78: 157-190. In a little known but profound paper, G.D. Kahl (1967) (Phys. Rev., 155: 78-80) developed a compelling argument questioning the validity of isothermal integration of the van der Waals equation between the spinodal points in accordance with Maxwell's equal-area construction. This work generalizes Kahl's approach and shows conclusively that it is entirely possible to describe vapor-liquid equilibria without lending credence to the thermodynamic properties of unstable states. The analysis is structured around a new free energy parameter Δ( T ) that allows adjustment of calculated vapor pressures without alteration of the equation of state itself, as is demonstrated using a novel modification of the Redlich-Kwong equation. The analysis opens the door to greater flexibility in using excess free energy models to devise mixing rules for equations of state. An extension of the theoretical development leads to a new type of VLE procedure whereby unrelated equations for liquid and vapor phases can be combined for essentially exact volumetric description of the pure fluids comprising a mixture. Introduction of a parameter Δ( T ) also makes it possible to formulate a conjecture explaining the apparent nonexistence of liquid-solid critical points.

Equation-of-State mixing rules for multicomponent mixtures: the problem of invariance

Mathias P.M., Klotz H.C. and Prausnitz J.M., 1991. Equation-of-state mixing rules for multicomponent mixtures: the problem of invariance. Fluid Phase Equilibria, 67: 31-44. Several researchers have recently extended the usual quadratic mixing rules in equations of state by using higher-order polynomials in component mole fraction. Most of these higher-order polynomials exhibit a serious deficiency for multicomponent systems since they are not invariant when a component is divided into two or more identical subcomponents. In this paper we investigate the source of this deficiency and propose a formulation to overcome it. While our analysis here is illustrated with the Peng-Robinson equation of state, it is applicable to other equations of state and to excess-Gibbs-energy models for calculating activity coefficients.

One-fluid mixing rules for cubic equations of state : III. Solubility of pure solids in supercritical mixed solvents

Three cubic equations of state are carefully examined to evaluate their capability for correlating the solubility of pure solids in supercritical mixed solvents. After obtaining the pure solute-solvent interaction parameters, the solid solute-cosolvent interaction parameters were directly calculated from the experimental ternary solubility data. The Redlich-Kwong, Soave, and Peng-Robinson equations of state correlate well the solubility data of 32 ternary systems producing AAD(%) errors of 12.54, 12.50 and 12.67% respectively.

One-fluid mixing rules for cubic equations of state: II. Solubility of solid mixtures in supercritical fluids

Three cubic equations of state are carefully examined to evaluate their capability for correlating the solubility of solid mixtures in supercritical fluids. After obtaining the pure solute-solvent interaction parameters, the solute-solute interaction parameters were directly calculated from the experimental ternary solubility data. The Redlich-Kwong, Soave, and Peng-Robinson equations of state correlate well the ternary data. However, the Redlich-Kwong equation of state gives the best result among them.

One-fluid mixing rules for cubic equations of state: I. Solubility of pure solids in supercritical fluids

Three cubic equations of state are carefully examined to evaluate their capability for correlating the supercritical solubility data of solid solutes in compressed gases. The overall average absolute deviatons between experimental and predicted values for 35 binary systems are 17.33% (Redlich-Kwong), 22.92% (Soave), and 22.93% (Peng-Robinson) using one binary parameter and 12.55% (Redlich-Kwong), 15.57% (Soave), and 13.71% (Peng-Robinson) using two binary parameters.

Correlation of ammonia-water equilibrium data with various modified Peng-Robinson equations of state

Extensive new data on the phase behavior of the ammonia-water binary system were recently obtained by us in a comprehensive experimental study. These new data were correlated using a number of recent proposals for modifying binary mixing rules in equations of state. The Peng-Robinson equation was taken as the basis for the study. Additional parameters were introduced into the mixing rules by making parameter b quadratic in composition, by adopting suggestions of Huron and Vidal and Soave that permit the dense fluid to be described by activity coefficient models, by introducing modified local composition concepts as proposed by Mollerup, Whiting and Prausnitz, and Mathias and Copeman, or by presuming the presence of two hydrated ammonia species in reaction equilibrium with the two substances. Increasing the numbers of parameters resulted in a marginally improved correlation.

Thermodynamic calculation of supercritical-fluid equilibria: New mixing rules for equations of state

Simple cubic equations of state with conventional mixing rules have played an important role in the calculation of phase equilibria and other thermodynamic properties of non-polar fluid mixtures. In the application of supercritical fluids to separation processes, volumetric as well as phase equilibrium properties are very important for rational process design. Heyen (1980) proposed a cubic equation of state which shows better accuracy in the calculation of volumetric properties, compared to the Peng-Robinson equation of state. In order to apply his equation to polar mixtures, Heyen recently proposed a density-independent mixing rule, but this does not obey the universally-observed quadratic mixing rule of the second virial coefficient in the low-density limit. This paper proposes a new density-dependent mixing rule for the Heyen equation of state. The Heyen equation of state with our new mixing rule appears to calculate the phase equilibria and the volumetric properties of CO 2-containing non-polar as well as polar mixtures with good accuracy.