Use of monomer fraction data in the parametrization of association theories

Association theories such as the CPA (cubic-plus-association), NRHB (non-random hydrogen bonding) equations of state and the various variants of SAFT (statistical associating fluid theory) have been extensively applied to phase equilibrium calculations. Such models can also be used for estimating the monomer fraction of hydrogen bonding compounds and their mixtures. Monomer fraction data are obtained from spectroscopic measurements and they are available for a few compounds such as pure water and alcohols as well as for some alcohol–alkane and similar mixtures. These data are useful for an understanding of the capabilities and limitations of association models. The purpose of this work is two-fold: (i) to compare the performance of three models, CPA, NRHB and sPC-SAFT, in predicting the monomer fraction of water, alcohols and mixtures of alcohol-inert compounds and (ii) to investigate whether “improved” model parameters can be obtained if monomer fraction data are included in the parameter estimation together with vapor pressures and liquid densities. The expression “improved” implies parameters which can represent several pure compound properties as well as monomer fraction data for pure compounds and mixtures. The accuracy of experimental monomer fraction data is discussed, as well as the role of monomer fraction data in clarifying which association scheme should be used in these equations of state. The results reveal that the investigated association models (CPA, sPC-SAFT and NRHB) can predict, at least qualitatively correct, monomer fractions of associating compounds and mixtures. Only, small differences are observed between the models. In addition, it has been shown that, using a suitable association scheme, a single set of parameters can describe satisfactorily vapor pressures, liquid densities and monomer fractions of water and alcohols. The 4C scheme is the best choice for water, while for methanol there is small difference between the 2B and 3B association schemes.     

Prediction of the surface tension of mixed electrolyte solutions based on the equation of Patwardhan and Kumar and the fundamental Butler equations

The predictive equation of Patwardhan and Kumar for the water activity of mixed electrolyte solutions has been used together with the fundamental Butler equations to establish a new simple predictive equation for the surface tension of mixed electrolyte solutions. This newly proposed equation can provide the surface tensions of multicomponent solutions using only the data of the corresponding binary subsystems of equal ionic strength. No binary interaction parameters are required. The predictive capability of the equation has been tested with the experimental data for 26 concentrated multicomponent electrolyte solutions at different temperatures and compared with the model of Li et al. Both equations agree well with the experimental results of systems examined over entire experimental composition ranges, but the new equation generally gives better predictions for most 1:1 electrolyte systems examined, and considerable improvement in predictions has been achieved for all the mixtures containing 1:2 and 2:2 electrolytes and for 1:1 electrolyte systems at higher temperatures.     

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.     

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.     

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.     

Application of the Perturbed-Chain SAFT equation of state to polar systems

The Perturbed-Chain SAFT (PC-SAFT) equation of state is applied to pure polar substances as well as to vapor–liquid and liquid–liquid equilibria of binary mixtures containing polar low-molecular substances and polar co-polymers. For these components, the polar version of the PC-SAFT model requires four pure-component parameters as well as the functional-group dipole moment. For each binary system, only one temperature-independent binary interaction k_ij is needed. Simple mixing and combining rules are adopted for mixtures with more than one polar component without using an additional binary interaction parameter. The ability of the model to accurately describe azeotropic and non-azeotropic vapor–liquid equilibria at low and at high pressures, as well as liquid–liquid equilibria is demonstrated for various systems containing polar components. Solvent systems like acetone–alkane mixtures and co-polymer systems like poly(ethylene-co-vinyl acetate)/solvent are discussed. The results for the low-molecular systems also show the predictive capabilities of the extended PC-SAFT model.     

Peng-Robinson equation of state for vapor-liquid equilibrium calculations for carbon dioxide + hydrocarbon mixtures

Lin, H.-M., 1984. Peng-Robinson equation of state for vapor-liquid equilibrium calculations for carbon dioxide + hydrocarbon mixtures. Fluid Phase Equilibria, 16: 151–169.Binary interaction parameters δ_ij in the Peng-Robinson equation of state have been determined from vapor-liquid equilibrium data for binary mixtures of carbon dioxide with a variety of hydrocarbons. A constant value of δ_ij ? 0.125 appears to represent the experimental data well in most cases. Comments are made on the recent work of Kato, Nagahama and Hirata, who correlated δ_ij as a function of temperature for CO₂ + n-paraffin binary mixtures.     

The Effect of Functional Groups on the Phase Behavior of Carbon Dioxide Binaries and Their Role in CCS

In recent years we have focused our efforts on investigating various binary mixtures containing carbon dioxide to find the best candidate for CO₂ capture and, therefore, for applications in the field of CCS and CCUS technologies. Continuing this project, the present study investigates the phase behavior of three binary systems containing carbon dioxide and different oxygenated compounds. Two thermodynamic models are examined for their ability to predict the phase behavior of these systems. The selected models are the well-known Peng–Robinson (PR) equation of state and the General Equation of State (GEOS), which is a generalization for all cubic equations of state with two, three, and four parameters, coupled with classical van der Waals mixing rules (two-parameter conventional mixing rule, 2PCMR). The carbon dioxide + ethyl acetate, carbon dioxide + 1,4-dioxane, and carbon dioxide + 1,2-dimethoxyethane binary systems were analyzed based on GEOS and PR equation of state models. The modeling approach is entirely predictive. Previously, it was proved that this approach was successful for members of the same homologous series. Unique sets of binary interaction parameters for each equation of state, determined for the carbon dioxide + 2-butanol binary model system, based on k₁₂–l₁₂ method, were used to examine the three systems. It was shown that the models predict that CO₂ solubility in the three substances increases globally in the order 1,4-dioxane, 1,2-dimethoxyethane, and ethyl acetate. CO₂ solubility in 1,2-dimethoxyethane, 1.4-dioxane, and ethyl acetate reduces with increasing temperature for the same pressure, and increases with lowering temperature for the same pressure, indicating a physical dissolving process of CO₂ in all three substances. However, CO₂ solubility for the carbon dioxide + ether systems (1,4-dioxane, 1,2-dimethoxyethane) is better at low temperatures and pressures, and decreases with increasing pressures, leading to higher critical points for the mixtures. By contrast, the solubility of ethyl acetate in carbon dioxide is less dependent on temperatures and pressures, and the mixture has lower pressures critical points. In other words, the ethers offer better solubilization at low pressures; however, the ester has better overall miscibility in terms of lower critical pressures. Among the binary systems investigated, the 1,2-dimethoxyethane is the best solvent for CO₂ absorption.     

Application of the CPA equation of state to reservoir fluids in presence of water and polar chemicals

The complex phase equilibrium between reservoir fluids and associating compounds like water, methanol and glycols has become more and more important as the increasing global energy demand pushes the oil industry to target reservoirs with extreme or complicated conditions, such as deep or offshore reservoirs. Conventional equation of state (EoS) with classical mixing rules cannot satisfactorily predict or even correlate the phase equilibrium of those systems. A promising model for such systems is the Cubic-Plus-Association (CPA) EoS, which has been successfully applied to well-defined systems containing associating compounds. In this work, a set of correlations was proposed to calculate the CPA model parameters for the narrow cuts in ill-defined C₇₊ fractions. The correlations were then combined with either the characterization method of Pedersen et al. or that of Whitson et al. to extend CPA to reservoir fluids in presence of water and polar chemical such as methanol and monoethylene glycol. With a minimum number of adjustable parameters from binary pairs, satisfactory results have been obtained for different types of phase equilibria in reservoir fluid systems and several relevant model multicomponent systems. In addition, modeling of mutual solubility between light hydrocarbons and water is also addressed.     

New (p, ρ, T) data for carbon dioxide – Nitrogen mixtures from (250 to 400) K at pressures up to 20 MPa

Comprehensive (p, ρ, T) measurements on two binary mixtures (0.10 CO₂ + 0.90 N₂ and 0.15 CO₂ + 0.85 N₂) were carried out in the gas phase at seven isotherms between (250 and 400) K and pressures up to 20 MPa using a single sinker densimeter with magnetic suspension coupling. A total of 69 (p, ρ, T) data for the first mixture and 69 (p, ρ, T) data for the second are presented in this article. The uncertainty in density was estimated to be (0.02 to 0.15)%, while the uncertainty in temperature was 3.9 mK and the uncertainty in pressure was less than 0.015% (coverage factor k = 2). Experimental results were compared with densities calculated from the GERG equation of state and with data reported by other authors for similar mixtures. Results yielded that, while deviations between experimental data and values calculated from the GERG equation were lower than 0.05% in density for low pressures, the relative error at high pressures and low temperatures increased to about (0.2 to 0.3)%. The main aim of this work was to contribute to an accurate density data base for CO₂/N₂ mixtures and to check or improve equations of state existing for these binary mixtures.     

Comparison of methods for calculating thermodynamic properties of binary mixtures in the sub and super critical state: Lee–Kesler and cubic equations of state for binary mixtures containing either CO2 or H2S

The (ρ,T,p) and (vapor + liquid) equilibria for fluid mixtures containing either CO₂ or H₂S have been determined from 13 equations of state. The estimated values have been compared with published experimental results. CO₂ and H₂S were used to represent non-polar and polar fluids, respectively. The equations of state investigated were as follows: (a) the Lee–Kesler equation; (b) two equations that included new reference fluids for the Lee–Kesler method; (c) three so-called extended equations of state; and (d) seven cubic equations of state. After adjustment of the binary interaction parameters the predicted values differed from the experimental data by about 0.8% for CO₂ mixtures while for H₂S mixtures the uncertainty was about ±2.8%. Somewhat larger errors, although still lower than ±5%, were obtained for co-existing phase densities; the Lee–Kesler method provided results of the highest accuracy. The cubic equations proposed by Schmidt and Wenzel and Valderrama provide the most reliable predictions of both single and co-existing phase densities. Comparison of the predicted (vapor + liquid) equilibrium with experiment shows that each of the seven cubic equations provides results of similar accuracy and all within ±6%.     

Representation for binary mixtures of n-alcohols + sub and supercritical CO2 by a group-contribution method

In order to represent vapour–liquid equilibria of binary n-alcohol–carbon dioxide mixtures the excess function-equation of state method is used in which carbon dioxide is described by the IUPAC equation of state and alcohols by a Peng–Robinson type equation where the attractive parameter is estimated by a group-contribution method. The excess function is of the Van Laar type in which the interaction parameters are calculated by a group-contribution method. This approach allows to correlate and predict with quite good accuracy VLE of binary systems of alcohols and CO₂, even for heavier alcohols.     

Application of the perturbed chain SAFT equation of state to complex polymer systems using simplified mixing rules

A simplified perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state is applied to polymer systems that include a variety of non-associating (esters, cyclic hydrocarbons), polar (ketones) as well as associating (amines, alcohols) solvents. The solvent pure-component parameters that are not available in the literature are estimated by correlating vapor-pressure and liquid-density data. The performance of the simplified PC-SAFT is compared to the original PC-SAFT equation of state for polymer systems of varying complexity. It is shown that the applied simplification is not at the expense of the accuracy of equation of state, while the computational time and complexity are significantly reduced, especially for associating systems. With no binary interaction parameter, simplified PC-SAFT is successfully able to predict vapor–liquid equilibria of polymers with non-associating solvents. In the case of associating solvents, a small binary interaction parameter k_ij is usually needed for the satisfactory correlation of the experimental data.     

Gas solubility calculations. II. Application of a new group-contribution equation of state

A new equation of state has been developed for polar as well as nonpolar components. It is based on the generalized van der Waals partition function and uses local-composition mixing rules. The group-contribution version of this equation of state (the GC-EOS) is described and tables containing parameters for 14 solvent groups and 9 gases (H₂, N₂, CO, O₂, CH₄, C₂H₄, CO₂, C₂H₆ and H₂S) are presented.The GC-EOS predicts vapor-liquid equilibria well for all kinds of systems involving the groups considered. The method requires only information concerning readily accessible pure-component properties. Calculations for multicomponent systems show that the method suggested provides very good predictions of multicomponent high-pressure vapor-liquid equilibria and fairly good predictions of Henry's constants in mixed solvents.     

Experimental determination and calculation of the critical curves for the binary systems of CO2 containing ketone, alkane, ester and alcohol, respectively

A set of variable-volume autoclave with a quartz window was used for the experimental determination of the high-pressure phase equilibria and critical curves. The critical temperatures, pressures, densities and mole volumes in the region near the critical point of CO₂ were examined for eleven binary systems of supercritical CO₂ (SC CO₂) with different kinds of substances (ketone, alkane, ester and alcohol), respectively. The critical curves of the above binary systems were also calculated using an equation of state. The equation consists of a hard body repulsion term and an additive perturbation term, which takes care of the attractive molecular interaction. The calculated data were compared with the experimental data, and yielded good agreements. At the same time, the values of the adjustable parameters, λ, k_σ and k_? were obtained. The critical curves of the above eleven binary systems at higher temperatures and pressures all belong to type I.     

Application of association models to mixtures containing alkanolamines

Two association models, the CPA and sPC-SAFT equations of state, are applied to binary mixtures containing alkanolamines and hydrocarbons or water. CPA is applied to mixtures of MEA and DEA, while sPC-SAFT is applied to MEA-n-heptane liquid-liquid equilibria and MEA-water vapor-liquid equilibria. The role of association schemes is investigated in connection with CPA, while for sPC-SAFT emphasis is given on the role of different types of data in the determination of pure compound parameters suitable for mixture calculations. Moreover, the performance of CPA and sPC-SAFT for MEA-containing systems is compared. The investigation showed that vapor pressures and liquid densities were not sufficient for obtaining reliable parameters with either CPA or sPC-SAFT, but that at least one other type of information is needed. LLE data for a binary mixture of the associating component with an inert compound is very useful in the estimation. The simple 4-site scheme is suitable for both CPA and sPC-SAFT and little is gained by using more complex association schemes. Finally, the results of CPA and sPC-SAFT are overall similar and whatever differences are seen appear to be more related to details in the parametrization rather than the different functional forms of the two equations of state.     

Prediction of non-isothermal ternary gas-phase breakthrough experiments based on binary data

The results of breakthrough experiments in an adsorption column packed with commercial activated carbon for three binary CO₂/N₂ mixtures as well as for two ternary CO₂/N₂/H₂ mixtures are presented. The experiments were carried out at two different temperatures (25 and 45 °C), four different pressures (1, 5, 10 and 20 bar) and three different flow rates. To analyze the experiments, the breakthrough profiles are simulated using a one-dimensional model consisting of material and energy balances together with the necessary constitutive equations. Transport parameters such as the heat and mass transfer coefficients are fitted to the results from the experiments with the binary mixtures (CO₂/N₂) and then compared to parameters obtained in a previous work (Adsorption 18: 143–161, 2012) for binary CO₂/H₂ mixtures. Furthermore, the parameters obtained for binary mixtures are used to predict the outcome of breakthrough experiments with ternary CO₂/N₂/H₂ mixtures. These simulations are then tested by experiments, showing that their prediction capability is rather satisfactory for a large range of experimental conditions.     

SRK equation of state: Predicting binary interaction parameters of hydrocarbons and related compounds

The parameter mixing rules of the Soave–Redlich–Kwong (SRK) equation of state are rewritten as Huron–Vidal mixing rules, where infinite-pressure activity coefficients are predicted by group contributions. Alkanes are treated as composed by one group type and aromatics by two types, aliphatic and aromatic. Hydrocarbon mixtures can be treated using one universal interaction parameter. Light compounds like methane, N₂, CO₂, H₂S, etc. are treated as separate groups; each one requires a pair of parameters for its interactions with aliphatic and aromatic groups. Group interaction parameters were determined from experimental VLE data. From them, binary interaction constants of the classical quadratic mixing rules can directly be derived.     

The phase envelopes of alternative solvents (ionic liquid,CO2) and building blocks of biomass origin (lactic acid,propionic acid)

Solid–liquid, liquid–liquid and vapour–liquid equilibrium measurements for binary and ternary systems containing building blocks of biomass origin such as propionic acid, lactic acid and alternative solvents like carbon dioxide and 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ionic liquid have been carried out at 313.15 K. The binary solid–liquid and liquid–liquid equilibrium measurements were performed at ambient pressure. The vapour–liquid equilibrium was studied in the range of pressure from 3.54 to 12 MPa while ternary systems were examined at 9, 10 and 12 MPa. The samples from the coexisting phases were taken and the compositions of both liquid and vapour phases were determined experimentally. The three-phase system was observed for lactic acid + ionic liquid + CO₂ as well. The achieved results were correlated using the Peng–Robinson equation of state with the Mathias–Klotz–Prausnitz mixing rule. The set of interaction parameters for the employed equations of state and the mixing rule for the investigated systems were obtained.     

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.