Solid-liquid equilibrium using the SAFT-VR equation of state: Solubility of naphthalene and acetic acid in binary mixtures and calculation of phase diagrams

A solid-liquid equilibrium (SLE) thermodynamic model based on the SAFT-VR equation of state (EOS) is presented. The model allows for the calculation of solid-liquid phase equilibria in binary mixtures at atmospheric pressure. The fluid (liquid) phase is treated with the SAFT-VR approach, where molecules are modelled as associating chains of tangentially bonded spherical segments interacting via square-well potentials of variable range. The equilibrium between the liquid and solid phase is treated following a standard thermodynamic method that requires the experimental temperature and enthalpy of fusion of the solute. The model is used to calculate the solubilities of naphthalene and acetic acid in common associating and non-associating organic solvents and to determine the solid-liquid phase behaviour of binary mixtures with simple eutectics. The SAFT-VR pure component model parameters are determined by comparison to experimental vapour pressure and saturated liquid density data with the choice of association models according to the nature of the molecule; in addition, an unlike adjustable parameter (k_ij) is used to model the solutions. The solubility data of naphthalene and acetic acid in both associating and non-associating solvents are reproduced essentially within the accuracy of the experimental measurements. The phase boundaries and the position of the eutectic points in the binary mixtures considered are, in most cases, reproduced with the accuracy commensurate with the industrial applications. Overall, the results presented show that the SAFT-VR EOS can be used with confidence for the prediction of the SLE of binary systems at atmospheric pressure.     

Understanding the fluid phase behaviour of crude oil: Asphaltene precipitation

We present a simplified but consistent picture of asphaltene precipitation from crude oil from a thermodynamic perspective, illustrating its relationship to the familiar bubble curve via the calculation of constant-composition p-T phase diagrams that incorporate both the bubble curve and the asphaltene precipitation boundary. Using the statistical associating fluid theory (SAFT) we show that the position of the precipitation boundary can be explained using a very simple fluid model including relatively few components. Our results support the view that the precursor to asphaltene precipitation is a liquid-liquid phase separation due to a demixing instability in the fluid. Moreover, the bubble curve for these systems is seen to represent a boundary between regions of two-phase (liquid-liquid) and three-phase (vapour-liquid-liquid) equilibria.     

Modeling of phase equilibria with CPA using the homomorph approach

For association models, like CPA and SAFT, a classical approach is often used for estimating pure-compound and mixture parameters. According to this approach, the pure-compound parameters are estimated from vapor pressure and liquid density data. Then, the binary interaction parameters, k_ij, are estimated from binary systems; one binary interaction parameter per system. No additional mixing rules are needed for cross-associating systems, but combining rules are required, e.g. the Elliott rule or the so-called CR-1 rule. There is a very large class of mixtures, e.g. water or glycols with aromatic hydrocarbons, chloroform-acetone, esters-water, CO₂-water, etc., which are classified as “solvating” or “induced associating”. The classical approach cannot be used and the cross-association interactions are difficult to be estimated a priori since usually no appropriate experimental data exist, while the aforementioned combining rules cannot capture the physical meaning of such interactions (as at least one of the compounds is non-self-associating). Consequently, very often one or more of the interaction parameters are optimized to experimental mixture data. For example, in the case of the CPA EoS, two interaction parameters are often used for solvating systems; one for the physical part (k_ij) and one for the association part (β^cross). This limits the predictive capabilities and possibilities of generalization of the model. In this work we present an approach to reduce the number of adjustable parameters in CPA for solvating systems. The so-called homomorph approach will be used, according to which the k_ij parameter can be obtained from a corresponding system (homomorph) which has similar physical interactions as the solvating system studied. This leaves only one adjustable parameter for solvating mixtures, the cross-association volume (β^cross). It is shown that the homomorph approach can be used with success for mixtures of water and glycols with aromatic hydrocarbons as well as for mixtures of acid gases (CO₂, H₂S) with alcohols and water. The homomorph approach is less satisfactory for mixtures with fluorocarbons as well as for aqueous mixtures with ethers and esters. In these cases, CPA can correlate liquid-liquid equilibria for solvating systems using two adjustable parameters. The capabilities and limitations of the homomorph approach are discussed.     

Application of a renormalization-group treatment to the statistical associating fluid theory for potentials of variable range (SAFT-VR)

An accurate prediction of phase behavior at conditions far and close to criticality cannot be accomplished by mean-field based theories that do not incorporate long-range density fluctuations. A treatment based on renormalization-group (RG) theory as developed by White and co-workers has proven to be very successful in improving the predictions of the critical region with different equations of state. The basis of the method is an iterative procedure to account for contributions to the free energy of density fluctuations of increasing wavelengths. The RG method has been combined with a number of versions of the statistical associating fluid theory (SAFT), by implementing White's earliest ideas with the improvements of Prausnitz and co-workers. Typically, this treatment involves two adjustable parameters: a cutoff wavelength L for density fluctuations and an average gradient of the wavelet function Φ. In this work, the SAFT-VR (variable range) equation of state is extended with a similar crossover treatment which, however, follows closely the most recent improvements introduced by White. The interpretation of White's latter developments allows us to establish a straightforward method which enables Φ to be evaluated; only the cutoff wavelength L then needs to be adjusted. The approach used here begins with an initial free energy incorporating only contributions from short-wavelength fluctuations, which are treated locally. The contribution from long-wavelength fluctuations is incorporated through an iterative procedure based on attractive interactions which incorporate the structure of the fluid following the ideas of perturbation theories and using a mapping that allows integration of the radial distribution function. Good agreement close and far from the critical region is obtained using a unique fitted parameter L that can be easily related to the range of the potential. In this way the thermodynamic properties of a square-well (SW) fluid are given by the same number of independent intermolecular model parameters as in the classical equation. Far from the critical region the approach provides the correct limiting behavior reducing to the classical equation (SAFT-VR). In the critical region the β critical exponent is calculated and is found to take values close to the universal value. In SAFT-VR the free energy of an associating chain fluid is obtained following the thermodynamic perturbation theory of Wertheim from the knowledge of the free energy and radial distribution function of a reference monomer fluid. By determining L for SW fluids of varying well width a unique equation of state is obtained for chain and associating systems without further adjustment of critical parameters. We use computer simulation data of the phase behavior of chain and associating SW fluids to test the accuracy of the new equation.     

Semiclassical approach to model quantum fluids using the statistical associating fluid theory for systems with potentials of variable range

Thermodynamic properties of quantum fluids are described using an extended version of the statistical associating fluid theory for potentials of variable range (SAFT-VR) that takes into account quantum corrections to the Helmholtz free energy A, based on the Wentzel-Kramers-Brillouin approximation. We present the theoretical background of this approach (SAFT-VRQ), considering two different cases depending on the continuous or discontinuous nature of the particles pair interaction. For the case of continuous potentials, we demonstrate that the standard Wigner-Kirkwood theory for quantum fluids can be derived from the de Broglie-Bohm formalism for quantum mechanics that can be incorporated within the Barker and Henderson perturbation theory for liquids in a straightforward way. When the particles interact via a discontinuous pair potential, the SAFT-VR method can be combined with the perturbation theory developed by Singh and Sinha [J. Chem. Phys. 67, 3645 (1977); and ibid. 68, 562 (1978)]. We present an analytical expression for the first-order quantum perturbation term for a square-well potential, and the theory is applied to model thermodynamic properties of hydrogen, deuterium, neon, and helium-4. Vapor-liquid equilibrium, liquid and vapor densities, isochoric and isobaric heat capacities, Joule-Thomson coefficients and inversion curves are predicted accurately with respect to experimental data. We find that quantum corrections are important for the global behavior of properties of these fluids and not only for the low-temperature regime. Predictions obtained for hydrogen compare very favorably with respect to cubic equations of state.     

The Area method for pure fluids and an analysis of the two-phase region

The Area method, developed recently for solving multicomponent phase equilibrium problems, has been extended to pure fluids. The method is based on maximizing a single objective function in the Helmholtz-volume surface along any given isotherm, which reduces the number of independent variables to only two: the saturated liquid and vapour volumes. Two techniques are employed to find the maximum of the objective function, the integral and iterative. The integral always finds the thermodynamically stable solution without any prior assumptions about the values of the molar volumes. This factor distinguishes the integral from the iterative technique and also from methods based on the Maxwell equal-area principle. The method has been applied to a group of high accuracy non-cubic equations of state and some of the thermodynamic inconsistencies which occur inside the two-phase region are explored. A new inequality constraint which eliminates these inconsistencies during the development of new equations of state is proposed, and initial results with fitting a preliminary Helmholtz equation of state for benzene are encouraging.     

Application of stochastic algorithms for parameter estimation in the liquid–liquid phase equilibrium modeling

In this work the Simulated Annealing (SA) and Particle Swarm Optimization (PSO) algorithms were employed to modeling liquid–liquid phase equilibrium data. For this purpose, some strategies for stochastic algorithms were investigated from common test functions and used in LLE parameter estimation procedure. The strategy used for the flash calculation was based on the isoactivity criteria associated with phase stability test and interpolation function for the initial estimate to improve reliability of phase equilibria calculations. It is shown that both algorithms SA and PSO were capable of estimating the parameters in models describing liquid–liquid phase behavior of binary and multicomponent systems with a good representation of the experimental data.     

LIRIC 3.2 an updated model for iodine behaviour in the presence of organic impurities

Library of iodine reactions in containment (LIRIC) is a comprehensive mechanistic model for the chemical and mass transport behaviour of iodine in containment under postulated nuclear reactor accident conditions. The LIRIC model has successfully simulated the iodine behaviour in experiments performed under conditions relevant to post-accident containment, and some of the results have been published previously. This document describes the latest version of the model, LIRIC 3.2, and presents simulation results of a few integrated tests performed in the Radioiodine Test Facility.     

两相平衡范特霍夫方程的应用

应用相平衡常数讨论了范特霍夫方程在简化一些两相平衡热力学公式推导时的应用。     

A phase diagram approach to determine the composition of vapor from a microemulsion base

The tangents to the evaporation path curves in the W/O microemulsion base of water, (W), pentanol, (P), and sodium dodecyl sulfate, (S), were extended to the W/P axis to establish the relative composition, (W_V, P_V), of the vapor leaving the liquid.The composition of the vapor, with which the microemulsion is in contact includes also the contribution from the relative humidity of the surrounding atmosphere. The difference between the composition of these gases is clarified using the algebraic expressions from the phase diagram, but the quantitative composition of the equilibrium vapor is not available without further numerical information. The limits of the vapor for evaporation direction under different relative humidities were clarified.     

A modern approach to the solution and analysis of the Simha-Somcynsky model for the description of pressure-volume-temperature properties of polymer fluids

We revisit the Simha-Somcynsky model of polymer fluids with the purpose of developing novel theoretical and computational approaches to simplify and speed up its solution as well as the fitting of experimental data, and decrease its level of mathematical complexity. We report a novel method that allows us to solve one of the two equations of the model exactly, thus putting the level of mathematical difficulty on a par with the one of other models for polymer fluids. Moreover, we describe a computational algorithm capable of fitting all five parameters of the model in an unbiased way. The results obtained reproduce literature results and fit experimental pressure-volume-temperature and solubility parameter data for three polymers very accurately. Moreover, the new techniques allow for the investigation of the model at very low temperatures. Unexpectedly, the model predicts behaviors that could be interpreted as a glass transition, as routinely observed in dilatometry and differential scanning calorimetry, and a glass phase. We compared the predicted and experimental T g’s for cis poly(1,4-butadiene) and found an excellent quantitative agreement.     

A stepwise simulation approach to the chemical potential of fluids

A new approach to the computation of the chemical potential of fluids is presented. In this method the particle-insertion operation in the conventional test particle method is replaced by the growth of a specific particle. Application of the new technique to hard sphere and Lennard-Jones fluids shows that it is capable of providing reliable estimates of the chemical potential, even at high density where the conventional test particle methods are difficult to apply.     

An efficient model for the prediction of CO2 hydrate phase stability conditions in the presence of inhibitors and their mixtures

A thermodynamic model for the prediction of CO₂ hydrate phase stability conditions in the presence of pure and mixed salts solutions and various ionic liquids (ILs) is developed. In the proposed model van der Waals and Platteeuw model is used to compute the hydrate phase, Peng–Robinson equation of state (PR-EoS) for the gas phase and the Pitzer–Mayorga–Zavitsas-Hydration model is employed to calculate the water activity in the liquid water phase. This model is an extension of the model developed by Tumba et al. (2011) for the prediction of methane and CO₂ hydrate phase stability conditions in the presence of tributylmethylphosphonium methylsulfate IL solution. Shabani et al. (2011) mixing rule is modified by incorporating the water–inhibitor (salt/IL) interaction parameter to calculate the water activity in mixed salt solutions. The model predictions are also calculated using the Pitzer–Mayorga model separately and compared with predictions of the developed model. The model predictions are compared with experimental results on the phase stability of CO₂ hydrate in the presence of ILs, pure and mixed salts as reported in literatures. The ILs are chosen from imidazolium cationic family with various anion groups such as bromide (Br), tetrafluoroborate (BF₄), trifluoromethanesulfonate (TfO), and nitrate (NO₃) and the common salts such as NaCl, KCl and CaCl₂. Good agreement between the developed model predictions and the literature data is observed. The overall average absolute deviation (AARD%) with Pitzer–Mayorga–Zavitsas-Hydration model is observed to be within ±1.36% while Pitzer–Mayorga model accuracy were about ±1.44 %. Further, the model is extended to calculate the inhibition effect of selected inhibitors (ILs and salts) on CO₂ hydrate formation.     

The Differential Relationships of Thermodynamic Variables for the Equilibrium of Heterogeneous Substances

For the system without adiabatic walls, rigid walls or semi-permeable walls and without chemical reactions or without other restrictions except restrictions of phase equilibrium conditions, if the number of components of the system is k and the number of phases is φ, the degree of freedom of the system at equilibrium is f=k-φ+2. Because the degree of freedom is incapable of being negative, f=k-φ+2≥0, viz.φ≤k+2. For the heterogeneous equilibrium, the number of phases is at least 2, so φ=k+2-f≥2, viz. f≤k. Hence the range of change of φ and f is 2≤φ≤k+2,0≤f≤k, respectitvely. If φ=k+2, there are no independent variables in the system at equilibrium. If φ=k+1, there is one independent variable; if the temperature is selected as the independent variable, the other dependent variables can be expressed as the function of the temperature. If φ=k, there are two independent variables; if the temperature and pressure are selected as the independent variables, the other dependent variables can be expressed as the function of the temperature and pressure. If 2≤φ≤k-1, there are more than two independent variables; if the temperature, pressure and some concentrations are selected as independent variables, the other dependent variables can be expressed as the function of the temperature, pressure and these concentrations. The differential relationships of dependent variables and independent variables are educed out according to the principle of phase equilibriums for 2≤φ≤k-1. In any phase the number of the variables is(k+1), viz. temperature T, pressure p and (k-1) mole fractions x1, x2,…, xk-1. The temperature and pressure are common variables of every phase. The number of independent variables is at best k for the heterogeneous equilibriums of k components. The temperature, pressure and (k-2) concentrations are selected as independent variables. The independent concentration variables are selected entirely from the first phase and the concentration variables of the other phases all act as dependent variables. There is at least one dependent concentration variable in the first phase.     

Phase behavior of ionic fluids in slitlike pores: a density functional approach for the restricted primitive model

We present a density functional theory of nonuniform ionic fluids. This theory is based on the application of the electrostatic contribution to the free energy functional arising from mean spherical approximation for a bulk restricted primitive model and from the energy route bulk equation of state. In order to employ this functional we define a reference fluid and additional averaged densities, according to the approach introduced by Gillespie, Nonner and Eisenberg [J. Phys.: Condens. Matter 14, 12129 (2002)]. In the case of bulk systems the proposed theory reduces to the mean spherical approximation equation of state, arising from the energy route and thus it predicts the first-order phase transition. We use this theory to investigate the effects of confinement on the liquid-vapor equilibria. Two cases are considered, namely an electrolyte confined to the pore with uncharged walls and with charged walls. The dependence of the capillary evaporation diagrams on the pore width and on the electrostatic potential is determined.