Calculation Of Solid-liquid Equilibria Research Articles

Measurements and calculations of solid-liquid equilibria in two quaternary systems: LiCl–NaCl–SrCl2–H2O and LiCl–KCl–SrCl2–H2O at 298 K

Using the method of isothermal dissolution equilibrium, stable phase equilibria of unreported strontium-containing systems LiCl–NaCl–SrCl2–H2O and LiCl–KCl–SrCl2–H2O at 298 K have been studied in accordance with the compositions of the underground brine in western Sichuan Basin. The experimental results show that there are no solid solution and complex salt in two quaternary systems. The phase diagram of quaternary system LiCl–NaCl–SrCl2–H2O consists of two invariant points, five isothermal dissolution curves and four crystallization regions. The quaternary system LiCl–KCl–SrCl2–H2O belongs to a simple type and the phase diagram consists of two invariant points, five isothermal dissolution curves and four crystallization regions. The solubilities of salts in the above systems were accurately calculated with Pitzer electrolyte solution theory. It was found that the calculated phase diagrams coincided basically with the experimental phase diagrams by evaluated the accuracy through the experimental results. Meanwhile, the applicability of Pitzer parameters selected in this paper was validated.

Measurements and calculations of solid-liquid equilibria in the ternary systems NaBr–Na2SO4–H2O and KBr–K2SO4–H2O at 348 K

According to the compositions of the underground brine resources in the west of Sichuan Basin, solubilities of the ternary systems NaBr–Na2SO4–H2O and KBr–K2SO4–H2O were investigated by isothermal method at 348 K. The equilibrium solid phases, solubilities of salts, and densities of the solutions were determined. On the basis of the experimental data, the phase diagrams and the density-composition diagrams were plotted. In the two ternary systems, the phase diagrams consist of two univariant curves, one invariant point and two crystallization fields. Neither solid solution nor double salts were found. The equilibrium solid phases in the ternary system NaBr–Na2SO4–H2O are NaBr and Na2SO4, and those in the ternary system KBr–K2SO4–H2O are KBr and K2SO4. Using the solubilities data of the two ternary subsystems at 348 K, mixing ion-interaction parameters of Pitzer’s equation θxxx, Ψxxx and Ψxxx were fitted by multiple linear regression method. Based on the chemical model of Pitzer’s electrolyte solution theory, the solubilities of phase equilibria in the two ternary systems NaBr–Na2SO4–H2O and KBr–K2SO4–H2O were calculated with corresponding parameters. The calculation diagrams were plotted. The results showed that the calculated values have a good agreement with experimental data.

Measurements and calculations of solid-liquid equilibria in the quaternary system KBr–CaBr2–MgBr2–H2O at (298 and 323) K

Through using the isothermal dissolution equilibrium method, phase equilibria in the quaternary system KBr–CaBr2–MgBr2–H2O were studied at (298 and 323) K. Based on the experimental data, the phase diagrams were obtained. It can be seen that two phase diagrams all contain complex salts. In quaternary system KBr–CaBr2–MgBr2–H2O at 298 K, there are two invariant points, five univariant curves, and four crystallization fields, corresponding to KBr, CaBr2·6H2O, MgBr2·6H2O and KBr·MgBr2·6H2O. At 323 K, a new double salt 2MgBr2·CaBr2·12H2O could be found in this quaternary system. As a result, it has three invariant points, seven univariant curves, and five crystallization regions fields, corresponding to KBr, CaBr2·4H2O, MgBr2·6H2O, KBr·MgBr2·6H2O and 2MgBr2·CaBr2·12H2O. Moreover, the phase diagrams for the quaternary system do not consist new precipitated solids at 298 K and 323 K in comparision with the phase diagrams for ternary subsystems. Using Pitzer theoretical and mathematical calculation tool, the solubilities of the quaternary system were calculated under the condition of (298 and 323) K. And make a comparison with the experimental data. It shows that the calculation results are in good agreement with experimental values.

Measurements and calculations of solid-liquid equilibria in quaternary system Li2SO4-Na2SO4-K2SO4-H2O at 288 K

The isothermal dissolution equilibrium method has been used to investigate the solid and liquid equilibria of quaternary system Li2SO4-Na2SO4-K2SO4-H2O at 288 K. The phase diagram and water diagram were plotted according to the experiment results. It was found that Li2SO4-Na2SO4-K2SO4-H2O quaternary system contains four double salts at 288 K. There are six invariant points, twelve univariant curves and seven areas of crystallization: Li2SO4 H2O, Na2SO4 10H2O, K2SO4, Li2SO4 3Na2SO4 12H2O, Li2SO4 K2SO4, Na2SO4 3K2SO4, 2Li2SO4 Na2SO4 K2SO4. Using the equilibrium solubility data of the quaternary system at 288 K, the solubility product(lnKsp) of solid phase 2Li2SO4 Na2SO4 K2SO4, which is not reported was acquired. Based on the chemical model of Pitzer’s electrolyte solution theory, the solubilities for the quaternary system Li2SO4-Na2SO4-K2SO4-H2O at 288 K were calculated with corresponding parameters. Compared the results of experimental measurement with those of calculation, it was showed that the calculated values had a good agreement with the experimental ones.

Measurements and calculations of solid-liquid equilibria in the quaternary system NaBr–KBr–Na2SO4–K2SO4–H2O at T = 348 K

Using an isothermal dissolution equilibrium method, solid-liquid equilibria in the quaternary system NaBr–KBr–Na2SO4–K2SO4–H2O were studied at T = 348 K. The phase diagram, water diagram and the density-composition diagram were plotted according to the experiment results. One double salt Na2SO4·3K2SO4 was found in the quaternary system at T = 348 K. There are five areas of crystallization, seven univariant solubility curves, and three invariant points. Respectively, the five crystallization regions are Na2SO4·3K2SO4, NaBr, KBr, Na2SO4 and K2SO4. Based on the chemical model of Pitzer's electrolyte solution theory, the solubilities of the quaternary systems NaCl–KCl–Na2SO4–K2SO4–H2O, NaBr–KBr–Na2SO4–K2SO4–H2O have been calculated at T = 348 K. The calculation diagrams were plotted. Comparing experimental measurement with calculation, the results showed that the calculated results are in accordance with experiments.

Vapor–Liquid and Solid–Liquid Modeling with a Universal Quasichemical Segment-Based Activity Coefficient Model

A segment-based activity coefficient model is developed through the Universal Quasichemical (UNIQUAC-SAC) approach and is applied for vapor–liquid and solid–liquid equilibrium calculations. We incorporated the conceptual segment approach of the NRTL-SAC activity coefficient model with modification of binary interaction parameters for the conceptual segments. The conceptual segments are represented by n-hexane, dimethyl sulfoxide, nitromethane, and water as hydrophobic, polar attractive, polar repulsive, and hydrophilic, respectively. The UNIQUAC-SAC molecular parameters are obtained for 82 common solvents, 62 of which are common solvents in the pharmaceutical industry. The UNIQUAC-SAC and NRTL-SAC models are employed to compute the vapor–liquid equilibrium of binary, ternary, and quaternary systems. In most cases, the UNIQUAC-SAC model gives a better prediction than the NERL-SAC model. Moreover, the UNIQUAC-SAC model is extended to modeling the solubility of organic nonelectrolyte solute in common solvent...

The anomalous melting behavior of water in aqueous PVME solutions

The Wertheim lattice thermodynamic perturbation theory is used to predict the liquid–liquid and solid–liquid coexistence data for a model polymer solution. The theory predicts bimodal LCST phase behavior and an unusual step with composition in the solid–liquid equilibrium of the solvent. The theoretical solid–liquid equilibrium calculations are used to interpret experimental data obtained for aqueous solutions of poly(vinyl methyl ether) (PVME), which is known to show bimodal LCST phase behavior. An experimental method is proposed, employing Fourier transform infrared (FTIR) spectroscopy to determine the equilibrium melting line of water in the presence of PVME. In addition, the complete melting line of water is obtained by partial integration of the melting endotherm observed using modulated temperature differential scanning calorimetry (MTDSC). Both, the FTIR and MTDSC methods are in good agreement, experimentally confirming the predicted step with composition in the solid–liquid equilibrium. This peculiar concentration dependence of the melting curve of ice provides a new explanation for the inhibited crystallization of water in aqueous PVME solutions, since the actual supercooling (at high polymer concentration) is smaller than it could be anticipated for a conventional course of the melting curve. Hence, the vicinity of the glass transition region in these highly concentrated polymer mixtures leads to a dramatic slowing down of the nucleation rate and thus the subsequent crystallization. Moreover, the atypical shape of the equilibrium melting line also provides a new explanation for the double melting endotherm observed in (MT)DSC experiments, which is conventionally attributed to the melting at different temperatures of bound and free water.

Molecular dynamic simulations and global equation of state of square-well fluids with the well-widths from λ = 1.1 to 2.1

We present the results of extensive new molecular dynamic (MD) simulations in the one-phase region for square well fluids with well widths λ = 1.10, 1.15, 1.20, 1.25, 1.375, 1.50, 1.75, 1.90, 2.0, and 2.10. These data have been used in developing a crossover equation of state (CR EOS) for square-well fluids with well widths 1.1 ≤ λ ≤ 2.1. The CR EOS incorporates non-analytic scaling laws in the critical region, and in the limit of low densities yields the exact second and third virial coefficients. Also in the high-temperature region, it reproduces first-order perturbation theory results. The CR EOS was tested against our new MD simulations, and earlier MD and Monte-Carlo (MC) simulations reported by other authors as well. Excellent agreement between calculated values and simulation data for all SW fluids is observed. In combination with the density-functional theory, the CR EOS is also capable of reproducing surface tension simulations with high accuracy. Application of the CR EOS for solid–liquid equilibrium calculations in combination with the Lennard–Jones and Devonshire cell model for the solid phase, is also discussed.

Direct calculation of solid-liquid equilibria from density-of-states Monte Carlo simulations

A density-of-states Monte Carlo method is proposed for simulations of solid-liquid phase equilibria. A modified Wang-Landau density-of-states sampling approach is used to perform a random walk in regions of potential energy and volume relevant to solid-liquid equilibrium. The method provides a direct estimate of the relative density of states [Omega(U,V)] and thus the relative free energy within these regions, which is subsequently used to determine portions of the melting curve over wide ranges of pressure and temperature. The validity and usefulness of the method are demonstrated by performing crystallization simulations for the Lennard-Jones fluid and for NaCl.

Solid–liquid phase equilibria of binary indole mixtures with some aromatic compounds using a solid–liquid–vapor equation-of-state

The separation of useful chemicals, such as indole from coal tar, is an important subject of research in the field of fossil energy, renewable energy, and utilization of byproducts and waste materials. High-pressure crystallization is one of the proposed separation methods. To understand the process requires detailed knowledge of solid–liquid phase behaviors. Solid–liquid phase equilibrium data of various binary mixtures of indole have been reported in the literature. In the present report, we analyze some of the experimental data with our unified solid–liquid–vapor equation-of-state to see whether our model can be useful for the calculation of solid–liquid equilibria in these organic systems. Good correlations of the data and predictions of phase behavior, at very high pressures, are demonstrated.

New solid equation of state combining excess energy mixing rule for solid–liquid equilibria

AbstractCrude oil is a mixture of light and heavy hydrocarbon components. The light components in crude oil keep the heavy components soluble in solution. However, the cooling of crude oil below its solidification point leads to the formation of a solid‐wax phase. A number of thermodynamically based models have been proposed in the literature for the prediction of wax formation in both crude oils and fuels. Some of these models assume ideal liquid solution behavior. Others are a modification of the regular solution theory or the Flory‐Huggins model for activity coefficients, with varying degrees of success. The absence of a proper model in the petroleum industry to describe the liquid‐phase nonideality for mixtures of hydrocarbons with large differences in size is still one of the major problems in solid–liquid equilibrium calculations. A simple methodology is proposed to develop a cubic equation of state (CEOS) for the solid phase to allow the accurate calculation of solid–liquid equilibria. The methodology is successfully applied to the Twu‐Sim‐Tassone (TST) cubic equation of state. The solid TST equation of state is then combined with the TST excess Helmholtz energy mixing rule to handle highly nonideal systems for the accurate prediction of solid–liquid phase equilibria.

Improved models for the prediction of activity coefficients in nearly athermal mixtures Part II. A theoretically-based GE-model based on the van der Waals partition function

This work, which is a continuation of a recent publication on the same topic (G.M. Kontogeorgis, P. Coutsikos, P. Tassios and Aa. Fredenslund, Fluid Phase Equilibria, 92 (1994) 35), presents a new theoretically-based G E-model (Chain-FV) for alkane systems. The model is based on a modified form of the generalized van der Waals partition function and attempts to account for all non-energetic effects of solutions of both short- and long-chain alkanes, including alkane polymers. Both the free-volume effects and the density-dependent rotational degrees of freedom are considered. The resulting G E-model which, despite its derivation from a partition function resembles the Flory-Huggins formula, is suitable for vapor-liquid and solid-liquid equilibrium calculations for nearly athermal polymer solutions as well as for alkane systems. We show that using plausible assumptions for the free-volume and the external-degree-of-freedom parameter, very good predictions are obtained for activity coefficients of asymmetric alkane systems at both concentration ends, for solid-liquid equilibrium calculations, as well as in extreme cases (polymer solutions, activity coefficients of heavy model alkane polymers in short-chain compounds recently available from molecular simulation studies). The predictions of the Chain-FV model are better than the Flory-Huggins and Elbro terms and equivalent to the recently proposed empirical R-UNIFAC and p-FV models. However, unlike the latter two models, the one proposed in this work offers a successful combination of simplicity and accuracy, with a solid theoretical basis.

Evaluation of activity coefficient models in prediction of alkane solid-liquid equilibria

In the petroleum industry, the absence of a proper model to describe the liquid phase non-ideality for mixtures of alkanes with large size differences is still one of the main problems in solid-liquid equilibrium calculations. A search is made for a reliable model for the prediction of activity coefficients in these systems. The models investigated were originally developed for the polymer mixtures. The performances of original Flory-Huggins, modified UNIFAC, GCFLORY model, Flory free-volume, Entropic free-volume and some empirical modifications of these last two models are analysed extensively and compared using the deviations between experimental and predicted values of solid appearance temperature as criteria. A comprehensive experimental SLE data base with around 60 binary systems and more than 1000 data points for mixtures of linear, branched and cyclic alkanes is used. An analysis of the errors introduced by the simplifying assumptions more commonly used is also performed. Activity coefficient models that have been used in wax formation predictions, but which are not included in this comparison, are briefly discussed. It is shown that the original Flory-Huggins activity coefficient model, the regular solution theory and the ideal solution behavior, used in several wax formation models, as well as the modified UNIFAC and original Entropic free-volume models, are not appropriate for the description of the liquid phase in alkane systems. The importance of a free-volume contribution to the phase behavior description of liquid mixtures whose components have significant size differences is evident. The Flory free-volume and a modified version of Entropic free-volume seem to be the simplest and most reliable models.

Calculation of solid-liquid equilibria in aqueous solutions of nitrate salts using an extended UNIQUAC equation

An extended UNIQUAC-Debye-Hückel model is used to predict solid-liquid equilibria for aqueous solutions of nitrate salts with the cations K +, Mg 2+ and Ca 2+. In addition to a model for activity coefficients in mixed electrolyte solutions, the method requires knowledge of the solubility products of all possible solid phases. Furthermore, some ternary solid-liquid equilibrium data are required. It is shown that the extended UNIQUAC equation can be applied to the calculation of solid-liquid equilibria in the K +-Mg 2+-Ca 2+-NO 3 −-H 2O system with good results. A new algorithm for calculation of solid-liquid equilibrium in ionic solutions has been proposed. The number and amount of phases in equilibrium are found by minimization of the total Gibbs function of the system.

Calculation of solid-liquid equilibria using the modified BWR equation of state of Lee and Kesler.

The solubility of solid hydrocarbons and inorganics (carbon dioxide and hydrogen sulfide) in liquid hydrocarbons was correlated using the modified BWR equation of state of Lee and Kesler and the pseudocritical method to compute the fugacity coefficient of the solute in subcooled liquid. This method requires two binary parameters, k12 and s12, for each binary pair to be computed from experimental solid-liquid equilibrium data. These parameters are tabulated for a large number of binary systems. The correlation resulting from this investigation will be valuable in the design of cryogenic processes.