Density-and-temperature-dependent volume translation for the SRK EOS: 1. Pure fluids

A mathematical framework for applying a density-and-temperature-dependent volume translation in a thermodynamically consistent manner was developed. Volumetric equations of state (EOS)s that incorporate this translation procedure can be used to generate derived properties, such as fugacity and enthalpy departure, that are based on isothermal departure or residuals from ideal gas state conditions. This kind of translation serves to improve the original EOS and not simply act as a correlation for molar volumes. A density-and-temperature-modified translation of this type was applied to the Soave–Redlich–Kwong EOS and was shown to possess accuracy for saturation pressure predictions equivalent to the untranslated EOS, as well as greatly improved density predictions compared to what is available when using only constant valued translation. The EOS translated in this manner retains many of the important features of the untranslated EOS, such as explicit calculation of volume roots, while having the representation capabilities of substantially more complicated models, such as the extended virial equation of Benedict, Webb, Rubin, and Starling.     

A three-parameter cubic equation of state for prediction of thermodynamic properties of fluids

A new cubic three-parameter equation of state has been proposed for PVT and VLE calculations of simple, high polar and associating fluids. The parameters are temperature dependent in sub-critical region, but temperature independent in super-critical region. The results for 42 simple and 14 associative pure compounds indicate that the calculated saturation properties and volumetric properties over the whole temperature range, up to high pressures, by the proposed equation of state (EOS), were in better agreement with the experimental data, compared with those obtained by the five well-known EOSs (P–R, P–T, Adachi et al., Yu–Lu, and M4). Two derivative properties, molar enthalpy and heat capacity of water and ammonia have been calculated, and demonstrated the thermodynamic consistency of the EOS parameters. Also VLE calculations have been performed for 41 binary mixtures of different type of fluids, including those of interest in petroleum industry. The results indicated the high capability of the proposed EOS for calculating the thermodynamic properties of pure and fluid mixtures.     

A comparison of mixing rules for the combination of COSMO-RS and the Peng–Robinson equation of state

In this work, the COSMO-RS model is combined with a volume-translated Peng–Robinson equation of state (EOS) via a G^E$G^{\text{E}}$-based mixing rule. The performance of several mixing rules previously published for this purpose is compared and semi-empirical modifications to one of them are introduced to improve its performance in our application. The new mixing rule contains three internal parameters that are adjusted to achieve consistency between the mixing rule and COSMO-RS. No experimental binary data is needed for our EOS. The new COSMO-RS-based, predictive EOS introduces a density dependence into COSMO-RS and extends its applicability to higher pressures and to mixtures containing supercritical components.     

Simple modification of the temperature dependence of the Sanchez–Lacombe equation of state

In this work, the interaction energy term of the Sanchez–Lacombe equation of state (SL EOS) was modified to take into account the temperature dependence of hydrogen bonding and ionic interactions. A simple function was used in the form of the Langmuir equation that reduces to the original SL EOS at high temperature. Comparisons are shown between the ?*-modified SL EOS and the original SL EOS. The ?*-modified SL EOS could represent volumetric data for the group of non-polar fluids, polar fluids and ionic liquids to within an absolute average deviation (AAD) of 0.85%, 0.51%, and 0.054%, respectively whereas, the original Sanchez–Lacombe EOS gave AAD values of 0.99%, 1.2%, and 0.21%, respectively. The ?*-modified SL EOS provides remarkably better PVT representation and can be readily applied to mixtures.     

Data driven temperature dependence of EOS parameters

Using three kinds of experimental data (p_c, V_c and T_c at the critical state or P_s, V_s^v and V^sL at different temperatures for saturation data) three parameters of the EOS may be directly determined and the temperature dependence of the parameters may be established from thermodynamic conditions of vapour-liquid critical point or vapour-liquid phase equilibria respectively. The principle was demonstrated on the BACK EOS. Examples of argon and n-alkanes were used to demonstrate the idea. It was found, that there are two parameter sets of the BACK equation that satisfy the critical or saturation conditions for certain pure compounds. The BACK equation is able to reproduce experimental Z_c values for compounds above Z_c = 0.2764 (that is, for argon, methane, ethane), but it is improper for higher alkanes. In case of n-alkanes we found that there is no simple function for T dependence of BACK parameters. Parameter values obtained in the way demonstrated may be useful to give initial values for parameter estimation from experimental (e.g., VLE) data.     

A new development of equation of state for square-well chain-like molecules with variable width 1.1 ≤ λ ≤ 3

A new equation of state (EOS) for square-well chain molecules and their mixtures with variable well-width range (SWCF-VR-EOS) has been developed based on the sticky-point model for chemical association. Two important modifications have been made. Firstly, a new dispersion contribution to the Helmholtz function of monomers due to square-well potential with variable well-width range of 1.1 ≤ λ ≤ 3 was established by combining the second-order perturbation theory and Chiew's PY2 approximation of the integral equation. Secondly, the contribution of chain formation to the Helmholtz function is divided into two parts: One is from the hard sphere, and the other is from the effect of square-well potential described via the nearest-neighbor and next-to-nearest-neighbor residual cavity correlation functions (CCFs). The predicted compressibility factors and vapor–liquid coexistence curves for square-well fluids as well as for their mixtures are in good agreement with simulations. The new EOS has been applied to real non-associating fluids and the corresponding mixtures by adopting one-fluid mixing rule. The pVT and vapor–liquid equilibria (VLE) can be correlated satisfactorily. The model parameters for some homologous compounds are found to be linear with the molar mass indicating that the pVT and VLE of those homologous compounds can be predicted even if no accurate data are available.     

Equation of state for square-well chain molecules with variable range. I: Application for pure substances

An equation of state (EOS) for square-well chain molecules with variable range developed on the basis of statistical mechanics for chemical association in our previous work is employed for the calculations of pVT properties and vapor–liquid equilibria (VLE) of pure non-associating fluids. The molecular parameters for 73 normal substances and 46 polymers are obtained from saturated vapor pressure and liquid molar volume data for normal fluids or pVT data for polymers. Linear relations are found for the molecular parameters of normal fluids with their molecular weight of homologous compounds. This indicates that the model parameters of homologous series, subsequently pVT and VLE, can be predicted when experimental data are not available. The predicted saturated vapor pressures and/or liquid volumes are satisfactory through the generalized model parameters. The calculated VLE and pVT for normal fluids and polymers by this EOS are compared with those from other engineering models, respectively.     

Crossover Peng-Robinson equation of state with introduction of a new expression for the crossover function

In this research, we use the original Peng-Robinson (PR) equation of state (EOS) for pure fluids and develop a crossover cubic equation of state which incorporates the scaling laws asymptotically close to the critical point and it is transformed into the original cubic equation of state far away from the critical point. The modified EOS is transformed to ideal gas EOS in the limit of zero density. A new formulation for the crossover function is introduced in this work. The new crossover function ensures more accurate change from the singular behavior of fluids inside the regular classical behavior outside the critical region. The crossover PR (CPR) EOS is applied to describe thermodynamic properties of pure fluids (normal alkanes from methane to n-hexane, carbon dioxide, hydrogen sulfide and R125). It is shown that over wide ranges of state, the CPR EOS yields the thermodynamic properties of fluids with much more accuracy than the original PR EOS. The CPR EOS is then used for mixtures by introducing mixing rules for the pure component parameters. Higher accuracy is observed in comparison with the classical PR EOS in the mixture critical region.     

Extension of the group-contribution lattice-fluid equation of state

This work focuses on the extension of the numbers of group parameters and application of the group-contribution lattice-fluid equation of state (GCLF EOS). The new group parameters of the GCLF EOS were evaluated by means of the volume translated Peng–Robinson equation of state (VTPR EOS) and the UNIFAC model. Values for 20 main groups and 33 subgroups are added into the current parameter matrix. The procedure used in this work can also be used to evaluate group parameters for the groups not present in the current matrix. Some examples are given to show the reliability of the new group parameters. Two new applications of the GCLF EOS are present: the effect of polymeric additive to solvents in extractive distillation and prediction of the crystallinity of polymers in the presence of gas.     

A novel approach to thermophysical properties prediction for chloro-fluoro-hydrocarbons

In this paper we present a new procedure, based on quantum/molecular (QM/MM) mechanics and molecular dynamics (MD) computer simulations, for estimating the Perturbed Hard Sphere Chain Theory (PHSCT) equation of state (EOS) parameters of a set of 14 alternative chloro-fluoro-hydrocarbons (CFH) of industrial relevance and to predict their thermophysical properties and PVT behavior. Force field and quantum-mechanical techniques were employed in molecular modeling and for the calculation of geometrical and chemico-physical parameters. The Connolly surface algorithm, corrected for quantum-mechanical effects, was used in the evaluation of molecular surfaces and volumes. From these data, the parameters of the PHSCT EOS, V* and A*, were obtained. The third parameter, E*, was calculated from extensive MD simulations under NPT conditions. The new, original method proposed in this work gives good results, is relatively inexpensive, is absolutely general and can be applied in principle to any EOS, provided the parameters have a physical meaning. The tuning of the energetic parameter to a generated data set accounts for the degree of empiricism introduced at a certain stage in the development of any EOS.     

Parameters estimation and VLE calculation in asymmetric binary mixtures containing carbon dioxide + n-alkanols

In the present work, the estimation of the parameters for asymmetric binary mixtures of carbon dioxide + n-alkanols has been developed. The binary interaction parameter k₁₂ of the second virial coefficient and non-random two liquid model parameters τ₁₂ and τ₂₁ were obtained using Peng–Robinson equation of state coupled with the Wong–Sandler mixing rules. In all cases, Levenberg–Marquardt minimization algorithm was used for the parameters optimization employing an objective function based on the calculation of the distribution coefficients for each component. Vapor–liquid equilibrium for binary asymmetric mixtures (CO₂ + n-alkanol, from methanol to 1-decanol) was calculated using the obtained values of the mentioned parameters. The agreement between calculated and experimental values was satisfactory.     

Thermodynamic phase equilibria of wax precipitation in crude oils

Economic loss due to wax precipitation in oil exploitation and transportation has reached several billion dollars a year recently. Development of a model for better understanding of the process of wax precipitation is therefore very important to reduce the loss. In this paper, a new thermodynamic model for predicting phase equilibriums of crude oils is proposed. The modified SRK EOS and the UNIQUAC equations are used to describe the vapor, liquid phase and the wax, respectively. New correlations have been introduced to calculate the volume parameter, c, in SRK EOS and the heat of vaporization in UNIQUAC equation. The model can be used to describe the systems which contain paraffin, naphthene and aromatic fractions. New correlations for the enthalpies, temperatures of solid–solid transitions and fusion enthalpies of paraffins are established in this paper based on data obtained from open literature. By using the proposed modified model, the wax precipitation in hydrocarbon fluids has been predicted for three crude oil systems. The calculation results have been compared with experimental observations and those results obtained using regular solution models. It is found that wax precipitation in complex systems can be better predicted by using this new model.     

A new three-parameter cubic equation of state for calculation physical properties and vapor–liquid equilibria

A new three-parameter cubic equation of state is presented by combination of a modified attractive term and van der Waals repulsive expression. Also a new alpha function for the attractive parameter of the new EOS is proposed. The new coefficients of alpha function and the other parameters of the attractive term are adjusted using the data of the saturated vapor pressure and liquid density of almost 60 pure compounds including heavy hydrocarbons. The new EOS is adopted for prediction of the various thermophysical properties of pure compounds such as saturated and supercritical volume, enthalpy of vaporization, compressibility factor, heat capacity and sound velocity. Following successful application of the new EOS for the pure components, using vdW one-fluid mixing rules, the new EOSs are applied to prediction of the bubble pressure and vapor mole fraction of the several binary and ternary mixtures. The accuracy of the new EOS for phase equilibrium calculation is demonstrated by comparison of the results of the present EOSs with the PT, PR, GPR and SRK cubic EOSs.     

Estimation of solubility parameter by the modified ER equation of state

The applications of the solubility parameter in chemical, petroleum and polymer engineering industries have been cleared up along the past 50 years. In this article, the Hildebrand solubility parameter of over 250 substances were calculated by the modified ER (Esmaeilzadeh–Roshanfekr) equation of state and some others (the Peng–Robinson, Soave–Redlich–Kwong, Patel–Teja and Schmidt–Wenzel) and compared with the experimental data. Once the less average errors of the mER method predictions were satisfied in subcritical and some supercritical fluids region, a correlation based on this EOS was presented in order to calculate the total HSP (Hansen solubility parameter) of various types of organic components categorized in 13 distinct groups including paraffins, olefins, aromatics, naphthenes, alcohols, aldehydes, ketones, ethers, esters, amines, carboxylic acids and two petroleum sub-fractions (resins and asphaltenes). The optimal values of the model parameters were obtained applying the DE (differential evolution) optimization method. The absolute average deviations of the proposed correlations results from the experimental ones lied between 0.09 and 6%.     

Prediction of phase equilibria in the systems carbon dioxide (1)–fatty acids (2) by two cubic EOS models and classical mixing rules without binary adjustable parameters

This study evaluates the accuracy of estimating data in the series of systems carbon dioxide (1)–fatty acids (2) by two cubic equations of state, namely the EOS of Peng and Robinson in its original form and the recently proposed cubic EOS. The classical mixing rules are implemented in entirely predictive manner, i.e. without binary adjustable parameters. It is demonstrated that both models may yield reliable predictions of the data. However the EOS of Peng and Robinson fails in predicting the topology of phase behavior of the heavy homologues. The second cubic EOS predicts the Global Phase Behavior in the homologous series under consideration satisfactorily accurate, which in particular means qualitatively correct estimation of the liquid–liquid equilibria. The recently proposed EOS has no significant advantage over the EOS of Peng and Robinson in predicting the vapour–liquid equilibria data under consideration.     

Mixing rules for van-der-Waals type equation of state based on thermodynamic perturbation theory

A concept based on the thermodynamic perturbation theory for a `simple fluid' has been applied to the attractive term of a van-der-Waals type equation of state (EOS) to derive a simple mixing rule for the a parameter. The new mixing rule is a small correction to the original one-fluid approximation to account for the influence of particles of j-type on the correlation function of ii-type in a mixture consisting of particles of i and j types. The importance of the correction has been shown by comparison of the calculated results for binary mixtures of Lennard–Jones fluids with the data obtained by numerical method (Monte-Carlo simulation). The new mixing rules can be considered as a flexible generalization of the conventional mixing rules and can be reduced to the original v-d-W mixing rules by defaulting the extra binary parameters to zero. In this way the binary parameters already available in the literature for many systems can be used without any additional regression work. Extension of the new mixing rules to a multicomponent system do not suffer from `Michelsen–Kistenmacher syndrome' and provide the correct limit for the composition dependence of second virial coefficients. Their applicability has been illustrated by various examples of vapor–liquid and liquid–liquid equilibria using a modified Patel–Teja EOS. The new mixing rules can be applied to any EOS of van-der-Waals type, i.e., EOS containing two terms which reflect the contributions of repulsive and attractive intermolecular forces.     

A new equation of state for studying the thermodynamic properties of real gases

In this study, based on the compressibility effect of gas molecules, a new three-parameter cubic equation of state (EOS) is derived. To validate this EOS, density predictions of methane, ethane, carbon dioxide and oxygen have been studied using the new EOS at the temperature of 373 K and at the pressures up to 100 MPa. The results show a good agreement with reference data and this suggests that the proposed EOS would help to improve the study of thermodynamic properties for real gases.     

Evaluation of the parameters of the Bender equation of state for low acentric factor fluids and carbon dioxide

Volumetric properties of several low acentric factor fluids (Ar, CH₄, C₂H₆, Kr, N₂, Ne, O₂, Xe) as well as CO₂ are modeled using the Bender equation of state. This equation is a linear function of 19 adjustable parameters, which are evaluated from properties data, using a linear numerical procedure. The validity of the EOS is tested by calculating the Joule-Thomson inversion curve. A simple model is in particular used to correlate the inversion properties predicted by the Bender equation, expressed in term of reduced pressure as a function of reduced temperatures ranging from 0.8 to 6. The simple correlation reproduces accurately the used data. We employ data on state behaviour ρ(P,T) of homogeneous fluid phases, vapour-liquid equilibrium, second virial coefficient and the coordinates of the critical point.