Modeling the vapor–liquid equilibria of polymer–solvent mixtures: Systems with complex hydrogen bonding behavior

The vapor–liquid equilibria of binary polymer–solvent systems was modeled using the Non-Random Hydrogen Bonding (NRHB) model. Mixtures of poly(ethylene glycol), poly(propylene glycol), poly(vinyl alcohol) and poly(vinyl acetate) with various solvents were investigated, while emphasis was put on hydrogen bonding systems, in which functional groups of the polymer chain can self-associate or cross-associate with the solvent molecules. Effort has been made to explicitly account for all hydrogen bonding interactions. The results reveal that the NRHB model offers a flexible approach to account for various self- or cross-associating interactions. In most cases model's predictions (using no binary interaction parameter k_ij = 0) and model's correlations (using one temperature independent binary interaction parameter, k_ij ≠ 0) are in satisfactory agreement with the experimental data, despite the complexity of the examined systems.     

Modeling phase equilibria for acid gas mixtures using the CPA equation of state. Part II: Binary mixtures with CO2

In Part I of this series of articles, the study of H₂S mixtures has been presented with CPA. In this study the phase behavior of CO₂ containing mixtures is modeled. Binary mixtures with water, alcohols, glycols and hydrocarbons are investigated. Both phase equilibria (vapor-liquid and liquid-liquid) and densities are considered for the mixtures involved. Different approaches for modeling pure CO₂ and mixtures are compared. CO₂ is modeled as non self-associating fluid, or as self-associating component having two, three and four association sites. Moreover, when mixtures of CO₂ with polar compounds (water, alcohols and glycols) are considered, the importance of cross-association is investigated. The cross-association is accounted for either via combining rules or using a cross-solvation energy obtained from experimental spectroscopic or calorimetric data or from ab initio calculations. In both cases two adjustable parameters are used when solvation is explicitly accounted for. The performance of CPA using the various modeling approaches for CO₂ and its interactions is presented and discussed, comparatively to various recent published investigations. It is shown that overall very good correlation is obtained for binary mixtures of CO₂ and water or alcohols when the solvation between CO₂ and the polar compound is explicitly accounted for, whereas the model is less satisfactory when CO₂ is treated as self-associating compound.     

Modeling the high-pressure phase equilibria of carbon dioxide–triglyceride systems: A parameterization strategy

The group contribution equation of state (GC-EOS) has been used in several published works to correlate or predict the high-pressure phase equilibria of a variety of systems of practical interest. Nevertheless, quantitative and even qualitative disagreement among predictions and experimental data has been detected in mixtures of CO₂ with heavy compounds, such as triglycerides, when operating at high pressure. For instance, phase split up to indefinitely high pressures has been computed, when the observed experimental behavior shows full miscibility at sufficiently high pressure. In the present work, we study the influence on calculated critical lines and solubilities (Pxy diagrams) of the group-based interaction parameters k_ij, for the interactions of CO₂ with both, the triglyceride (TG) group and the paraffinic groups. Based on such study, we propose a parameterization procedure that improves upon the conventional parameter regression practice. The distinguishing feature of such procedure is the repeated observation of the global phase equilibrium behavior, studying in particular the effect of the group–group interaction parameters on critical lines, on the composition of the phases at equilibrium along liquid–liquid–vapor lines, and on selected isothermal or isobaric phase equilibrium diagrams. For the case of the non-randomness parameter, we use a universal positive value, more consistent with its physical meaning.     

Modelling of phase equilibria for associating mixtures using an equation of state

In the present work, the group contribution with association equation of state (GCA-EoS) is extended to represent phase equilibria in mixtures containing acids, esters, and ketones, with water, alcohols, and any number of inert components. Association effects are represented by a group-contribution approach. Self- and cross-association between the associating groups present in these mixtures are considered. The GCA-EoS model is compared to the group-contribution method MHV2, which does not take into account explicitly association effects. The results obtained with the GCA-EoS model are, in general, more accurate when compared to the ones achieved by the MHV2 equation with less number of parameters. Model predictions are presented for binary self- and cross-associating mixtures.     

Modeling the phase equilibria of hydrogen sulfide and carbon dioxide in mixture with hydrocarbons and water using the PCP-SAFT equation of state

The perturbed-chain polar statistical associating fluid theory (PCP-SAFT) equation of state is applied to correlate phase equilibria for mixtures of hydrogen sulfide (H₂S) and carbon dioxide (CO₂) with alkanes, with aromatics, and with water over wide temperature and pressure ranges. The binary mixtures of H₂S–methane and CO₂–methane are studied in detail including vapor–liquid, liquid–liquid and fluid–solid phase equilibria. Very satisfying results were obtained for the binary mixtures as well as for the ternary mixture of H₂S–CO₂–methane using the (constant) interaction parameters of the binary pairs.     

Carbon dioxide-water phase equilibria results from the Wong-Sandler combining rules

A model based upon the Peng-Robinson equation of state with the Wong-Sandler mixture combining rule (W-S MCR) can correlate phase equilibria in CO₂ + H₂O. The W-S MCR requires two energy parameters for liquid behavior and one interaction parameter for gas behavior, k_ij. In this paper, we present expressions for the energy parameters which cover a wide temperature range, and we use a new procedure to obtain k_ij by relating it to experimental cross second virial coefficients, B_ij. The three-phase pressures calculated for this system using our proposed model agree with the experimental data within a fraction of 1 bar. The correlated phase behavior of CO₂ + H₂O appears to be accurate over the ranges 1 – 1000 bar and 298.15–623.15 K. The proposed model also confirms the advantage of using the W-S MCR for phase equilibrium calculations.     

Prediction of phase equilibria in hydrocarbon + near-critical solvent systems

In this paper the Hole Quasichemical Group-Contribution model (HM) is used to describe phase diagram peculiarities of multicomponent aqueous systems containing hydrocarbons and alcohols. The model correctly describes liquid-liquid (LL), liquid-liquid-vapour (LLV), liquid-vapour (LV) and vapour-vapour equilibria (VVE), and critical and azeotrope curves. The potential use of the HM in the prediction of the solubilities of hydrocarbons in near-critical water and the densities of coexisting phases in the critical region of the system n-hexane-water are presented.     

An algebraic method that includes Gibbs minimization for performing phase equilibrium calculations for any number of components or phases

The most widely used technique for performing phase equilibria calculations is the K-value method (equality of chemical potentials). This paper proposes a more efficient algorithm to achieve the results that includes Gibbs minimization when we know the number of phases. Using the orthogonal derivatives, the tangent plane equation and mass balances, it is possible to reduce the Gibbs minimization procedure to the task of finding the solution of a system of non-linear equations. Such an operation is easier and faster than finding tangents or areas, and appears to converge as fast as the K-value method. Examples illustrate application of the new technique to two and three phases in equilibrium for binary and ternary mixtures.     

Modeling the hydrogen solubility in methanol,ethanol, 1-propanol and 1-butanol

Modeling hydrogen solubility in primary normal alcohols (methanol, ethanol, 1-propanol and 1-butanol) has been studied in this article. Equations of state (EOS), simple correlations and Artificial Neural Networks (ANN) have been compared to find the best modeling technique. Utilizing an equation of state requires an iterative calculation procedure and optimized interaction parameters. Iterative calculation is not suitable when time is important and optimized interaction parameters are not always available. In addition, selection of proper equation of state and mixing rules are serious problems. Simple correlations can be applied to avoid iterative calculations but they have limited flexibility.     

High-pressure vapor–liquid and vapor–liquid–liquid equilibria in the carbon dioxide + 1-heptanol system

Vapor–liquid equilibria (VLE) and vapor–liquid–liquid equilibria (VLLE) data for the carbon dioxide + 1-heptanol system were measured at 293.15, 303.15, 313.15, 333.15 and 353.15 K. Phase behavior measurements were made in a high-pressure visual cell with variable volume, based on the static-analytic method. The pressure range under investigation was between 0.58 and 14.02 MPa. The Soave–Redlich–Kwong (SRK)-EOS coupled with Huron–Vidal (HV) mixing rules and a reduced UNIQUAC model, was used in a semi-predictive approach, in order to represent the complex phase behavior (critical curve, LLV line, isothermal VLE, LLE, and VLLE) of the system. The topology of phase behavior is qualitatively correct predicted.     

Isothermal phase equilibria and structural phase transition in the carbon dioxide + cyclopropane mixed-gas hydrate system

The isothermal phase equilibria of the carbon dioxide + cyclopropane mixed-gas hydrate system were investigated by means of static temperature measurement and Raman spectroscopic analysis. Raman spectra indicated that the crystal structure of the carbon dioxide + cyclopropane mixed-gas hydrate changes from structure-I to structure-II and back to structure-I with an increase of the equilibrium carbon dioxide composition at 279.15 K, while each simple gas hydrate belongs to structure-I at the temperature. Whereas, unlike 279.15 K, no structural phase transition occurs along the isothermal stability boundary at 284.15 K.     

Solution of the isochoric–isoenergetic flash problem by direct entropy maximization

A new algorithm to find the phase equilibrium conditions in systems with specified values of internal energy, volume, and number of moles of each component present (isochoric–isoenergetic flash problem) is proposed. The core of the procedure consists of maximizing the system entropy by iterating on the values, in each phase, of internal energy, volume, and number of moles of each component. Analytical expressions for the physical properties and derivatives required by the calculations were generated by computer algebra. The algorithm tests for the possible need to add or remove phases during the course of iterations. The paper discusses possible numerical difficulties during application of the procedure and how to overcome them. The algorithm has shown to be robust and capable of solving multiphase equilibrium problems, avoiding trivial solutions.     

A global investigation of phase equilibria using the perturbed-chain statistical-associating-fluid-theory approach

The recently developed perturbed-chain statistical-associating-fluid theory (PC-SAFT) is investigated for a wide range of model parameters including the parameter m representing the chain length and the thermodynamic temperature T and pressure p. This approach is based upon the first-order thermodynamic perturbation theory for chain molecules developed by Wertheim [M. S. Wertheim, J. Stat. Phys. 35, 19 (1984); ibid. 42, 459 (1986)] and Chapman et al. [G. Jackson, W. G. Chapman, and K. E. Gubbins, Mol. Phys. 65, 1 (1988); W. G. Chapman, G. Jackson, and K. E. Gubbins, ibid. 65, 1057 (1988)] and includes dispersion interactions via the second-order perturbation theory of Barker and Henderson [J. A. Barker and D. Henderson, J. Chem. Phys. 47, 4714 (1967)]. We systematically study a hierarchy of models which are based on the PC-SAFT approach using analytical model calculations and Monte Carlo simulations. For one-component systems we find that the analytical model in contrast with the simulation results exhibits two phase-separation regions in addition to the common gas-liquid coexistence region: One phase separation occurs at high density and low temperature. The second demixing takes place at low density and high temperature where usually the ideal-gas phase is expected in the phase diagram. These phenomena, which are referred to as "liquid-liquid" and "gas-gas" equilibria, give rise to multiple critical points in one-component systems, as well as to critical end points and equilibria of three fluid phases, which can usually be found in multicomponent mixtures only. Furthermore, it is shown that the liquid-liquid demixing in this model is not a consequence of a "softened" repulsive interaction as assumed in the theoretical derivation of the model. Experimental data for the melt density of polybutadiene with molecular mass Mw=45,000 gmol are correlated here using the PC-SAFT equation. It is shown that the discrepancies in modeling the polymer density at ambient temperature and high pressure can be traced back to the liquid-liquid phase separation predicted by the equation of state at low temperatures. This investigation provides a basis for understanding possible inaccuracies or even unexpected phase behavior which can occur in engineering applications of the PC-SAFT model aiming at predicting properties of macromolecular substances.     

Hydrate-containing phase equilibria for mixed guests of carbon dioxide and nitrogen

Hydrate-containing phase equilibria for mixtures composed of carbon dioxide, nitrogen and water are of potential importance for the flow assurance in the transportation of captured carbon dioxide. Literature data for such mixture tend to be reported on water free basis. In this study three- and four-phase equilibria were experimentally studied at the pressure ranging from 5 to 20 MPa. Isobaric dissolution temperatures of formed hydrates were measured and reported for accurately determined loading compositions. Complex phase behaviors composed of two-, three- and four- phases were observed and they were analyzed by comparing with calculations using GSMGem program developed by Sloan and Koh [1]. Phase equilibria were found to be sensitive to water contents in water dominating mixtures. CSMGem and HYSYS (version 7.1) from AspenTech calculations were found in general agreements with present and literature data.     

Predicting mixture phase equilibria and critical behavior using the SAFT-VRX approach

The SAFT-VRX equation of state combines the SAFT-VR equation with a crossover function that smoothly transforms the classical equation into a nonanalytical form close to the critical point. By a combinination of the accuracy of the SAFT-VR approach away from the critical region with the asymptotic scaling behavior seen at the critical point of real fluids, the SAFT-VRX equation can accurately describe the global fluid phase diagram. In previous work, we demonstrated that the SAFT-VRX equation very accurately describes the pvT and phase behavior of both nonassociating and associating pure fluids, with a minimum of fitting to experimental data. Here, we present a generalized SAFT-VRX equation of state for binary mixtures that is found to accurately predict the vapor-liquid equilibrium and pvT behavior of the systems studied. In particular, we examine binary mixtures of n-alkanes and carbon dioxide + n-alkanes. The SAFT-VRX equation accurately describes not only the gas-liquid critical locus for these systems but also the vapor-liquid equilibrium phase diagrams and thermal properties in single-phase regions.     

Thermodynamic modelling of asphaltene precipitation and related phenomena

Asphaltenes are considered to be the heaviest and most polar fractions of crude oils and are frequently implicated in problems encountered during production and refining as a result of phase separation. In recent years, considerable effort has been given to understanding the phase behaviour of these structurally heterogeneous materials from both experimental and computational perspectives. Various experimental studies have confirmed the long-advanced colloidal behaviour of asphaltenes in organic media, and this has inspired a number of modelling strategies. The present review is specifically concerned with advances in modelling asphaltene phase behaviour with emphasis on the use of the statistical associating fluid theory (SAFT), which it attempts to place into the wider context of thermodynamic treatments.     

Fluid phase equilibria in the system n-butane + water

Although the phase equilibria in the system n-butane + water have been studied frequently, a review of the experimental results has revealed serious disagreement among the various investigators. In this work, the data from the literature are supplemented with some new solubility data. These data are then used construct a model, based on Henry's law, for the phase equilibria.