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Application of the corresponding-state law to the parameterization of SAFT-type models: generation and use of “generalized charts”
Abstract
Most of SAFT and cubic equations of state (EoS) for non-associating pure components obey the 3-parameter corresponding-state law meaning that the knowledge of 3 component-specific properties is a prerequisite to apply an EoS to a given pure species. The way used to attribute values to EoS parameters is called “parameterization procedure” here. In this article, it is shown that generalized charts, derived from the corresponding-state theory, can be used advantageously to parameterize SAFT models. SAFT EoS is a well-established class of models frequently used for process simulation. Depending on the EoS parameter set selected, most of SAFT EoS are likely to predict unrealistic phenomena (mainly, the presence of multiple critical points and 3-fluid phase equilibrium regions in pure-component phase diagrams). Generalized charts can be used to detect whether such unrealistic phenomena affect pure-component phase diagrams in the temperature and pressure domains of interest. As a second application, it is proposed to parameterize SAFT EoS in the same spirit as cubic EoS: the 3 SAFT input parameters are fixed in order to exactly reproduce the experimental critical temperature, critical pressure and one point of the vaporization curve through the acentric factor. It is shown that the procedure is simple to implement and opens the door to a large industrialization of SAFT EoS (i.e., the possibility to determine SAFT input parameters for any compound using a universal and univocal method).
... Notably, following the concept of the Cubic EoS, Polishuk [33] implemented a standardized parameterization method based on critical point properties in order to develop a preliminary version of industrialized SAFT model (CP-PC-SAFT). Moine et al. [28] and Privat et al. [34] proposed a robust and simple parameterization method using Generalized Charts based on the correspondingstate law (I-PC-SAFT). These Generalized Charts systematically constructed the relation between three SAFT input parameters and the corresponding properties, namely critical temperature, critical pressure and acentric factor and inspired an alternative parameterization approach for SAFT EoS. ...
... Further, when the critical point is reached, a critical fluid loses another one degree of freedom, compared with VLE system. Therefore, the reduced critical temperature, pressure and density, in agreement with the research results by Polishuk et al. [52] and Privat [34], depend directly on m: ...
... A conclusion could be made that through the above-mentioned section, SAFT-type models used for nonassociating pure components obey the three-parameter corresponding-state law. The observation that T Ã c , P Ã c and x can be expressed as univocal functions of m as discussed and illustrated by Privat et al. [34] To facilitate the application, a following polynomial correlations, as shown below, for segment number m ranging from 1 to 20 could be applied to estimate the PC-SAFT parameters: ...
The residual entropy scaling (RES) theory integrated with different thermodynamic Equation of States (EoS) have obtained several accurate and simple viscosity models of working fluids, which is crucial for designing and optimizing of the energy utilization system. However, the influence of parameterization methods on RES’s viscosity calculations is rarely mentioned in a systematic way. Hence, a robust parameterization method based on generalized relation between the Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) EoS’s input parameters of 11 pure refrigerants (including hydrocarbons (HCs), CO2, hydrofluorocarbons (HFCs), hydrofluoroolefins (HFOs)) and their critical properties is introduced and coupled with a robust formulation of the reduced viscosity term in RES theory recently proposed by Dehlouz et al. Accurate vapor pressure calculations and near-constant difference between calculated liquid densities and high-accuracy correlation model in saturated state could be obtained with this parameterization method. Viscosities in vapor and liquid phases are exactly reproduced with an overall error of 3.14% applying the model proposed in this work, which is comparable to those from three RES models in literature and the extended corresponding states (ECS) model with overall errors of 3.36%, 3.22%, 3.41% and 1.77%, respectively. In addition, the divergence behavior in dilute region and the component-specific curves of dimensionless reduced viscosity and entropy variables are improved than other RES models under study to a certain extent. Meanwhile, as a supplement, the limitation of the above RES models on predicting viscosities in the near critical region is also discussed in detail.
... Moine et al. 18 proposed a second industrialized version of the PC-SAFT EoS, in which the three non-associating parameters of the PC-SAFT EoS are obtained through exact reproduction of the experimental critical temperature, critical pressure and acentric factor. They used the approach of Privat et al. 94 , whom established universal relationships between the segment parameter, and the three aforementioned experimental properties by: ...
... -(1) the Cubic EoS (CEoS), derived from the original Van der Waals model which include the reference EoS by Peng and Robinson 34 or Soave 33 , -(2) the SAFT-type EoS (SAFT stands for Statistical Associating Fluid Theory 6 ) which include, among others, the well-known SOFT SAFT 9 , SAFT-VR 8 and the PC-SAFT 12 models. To make their model accepted by the scientific community, EoS developers must guarantee not only a sound theoretical background but also a correct parametrization strategy in order to obtain reliable values for the various thermodynamic properties of interest 18,94 . By parametrization, it is here meant the methodology used to assign values to model parameters that are not fixed by the theory. ...
Thermodynamic models are at the heart of modeling tools in process engineering that provide, on the one hand, the description, both qualitative and quantitative, of phase equilibria of complex mixtures, and on the other hand, a measure of energetic properties (enthalpies, entropies, heat capacities) necessary to establish energy and exergy balances. This work is particularly interested in the equations of state from the statistical associating fluid theory (SAFT) applied to pure compounds. Our main objective is to propose universal and systematic parameterization methods for these models including, in particular, the association parameters which essentially reflect the influence that hydrogen bonds have on the thermodynamic properties of a pure fluid.Using one of the most popular variants of the SAFT family (the PC-SAFT model), we have investigated which experimental properties are most relevant to consider when regressing the pure body parameters (m,σ,ϵ/k) that are sufficient to describe the non-associated compounds. We have shown that only the saturating vapor pressure and the density of the liquid are found to be strictly necessary to obtain a reliable description of the thermodynamics of a non-associated pure body (i.e., its phase equilibrium properties and energy properties). This work has also produced a database of parameters for 1800 pure bodies, divided into 1252 non-associated species and 548 self-associated species (treated as non-associated species). Using this parameterization, we were able to accurately describe over 70% of all compounds, and of these, nearly 60% of the self-associated compounds were correctly represented.The influence of the association term on the performance of the equation of state was then studied for a family of strongly associated molecules: the linear alcohols. This allowed us to draw three major conclusions: 1) the association term significantly improves the efficiency of SAFT models for associated molecules; 2) in the event that the same association parameters are used for all compounds in the family, saturation vapor pressure and liquid density data are sufficient to estimate the association parameters and 3) if component-specific association parameters are chosen, derived property data must be included in the regression (e.g., liquid heat capacity).In a second part, we present a methodology for the evaluation of the performance of equations of state of mixtures in the development of which we participated, and its application to the UMR-PRU model.
... More details on NEMD simulations to produce quasi-experimental thermophysical data are provided in refs. (Privat et al., 2019;Galliero et al., 2017;Galliero and Montel, 2009). To make the problem manageable by the NEMD simulations, we used a thermodynamic lumping technique, reducing the number of components of the studied fluids (8 pseudo-components in the oil case 1, as shown in Tables 1 and 7 pseudo-components in case 2 as shown in Table 2). ...
In many petroleum reservoirs, the fluid properties vary through the reservoir thickness. Variation of the composition, i.e. compositional grading, can affect reserve estimation, production and enhanced oil recovery strategies. Apart from gravity, the geothermal gradient may also contribute to the fluid distribution. Thermodiffusion is the governing phenomenon determining the contribution of the geothermal gradient. The non-equilibrium thermodynamics models are applied for the calculation of the compositional gradients under the varying temperature.
In order to determine the variations in pressure and composition with depth and to be able to indicate if/where a gas-oil contact exists, we have developed a model based on the principles of irreversible thermodynamics, within the approach to thermodiffusion in porous media proposed by Montel et al. (2019). Based on the relationships where pressure, chemical potentials, and thermal gradient are linked, the distribution of hydrocarbons in a petroleum reservoir is described. A computational algorithm accounting for non-ideality of the mixture, characterization, and phase transitions has been developed. The model and the computational procedure have been validated by comparison with the case studies reported in the literature for a North Sea reservoir, and with sample component distributions produced by application of molecular dynamics simulations. It has been shown that the model is capable of predicting the fluid distributions with depth with no or a minimum of adjustable parameters. The thermal gradient modifies the predicted fluid distributions making them closer to the observed data points. Depending on the mixture composition, the thermal diffusion may either enforce the effect of gravity or counteract it.
The performance of the perturbed‐chain statistical associating fluid theory‐type equations of state (PC‐SAFT‐type EOSs) is compromised in predicting properties of pure compounds in the critical region. In our previous research, we introduced an improved volume‐translated rescaled PC‐SAFT EOS (VTR‐PC‐SAFT EOS) by incorporating a dimensionless distance‐function. Such VTR‐PC‐SAFT EOS is built based on a critical point‐based PC‐SAFT EOS that can reproduce the critical temperature, critical pressure, and critical molar volume of pure compounds. VTR‐PC‐SAFT EOS is found to significantly improve the accuracy of phase behavior predictions in both critical and noncritical regions for pure compounds. In this study, we assess the performance of VTR‐PC‐SAFT EOS in reproducing the critical and noncritical properties of 251 pure compounds, encompassing 20 distinct chemical species. The testing results indicate that, compared to the other three PC‐SAFT‐type EOSs, the VTR‐PC‐SAFT EOS can consistently provide more accurate representations of critical and noncritical properties of 251 pure compounds.
Statistical associating fluid theory (SAFT)-type equations of states (EOSs) have been widely used to model the phase behavior of pure compounds (e.g., carbon dioxide). One issue with conventional SAFT-type EOSs is that they fail to accurately reproduce the critical pressure and critical temperature of pure compounds. To address this limitation, the three parameters in SAFT-type EOSs (i.e., m, σ, and ε/k) must be adjusted to exactly reproduce the critical temperature and critical pressure of pure compounds, leading to the development of the critical-point perturbed-chain SAFT (CPPC-SAFT). However, this is achieved by sacrificing the accuracy of the liquid-density predictions. As such, a fourth parameter, that is, the volume translation parameter, is indispensable for improving the accuracy of liquid density predictions. In this study, a new nonlinear temperature-dependent volume translation model was introduced into the perturbed-chain SAFT EOS (PC-SAFT EOS) for more accurate density predictions of CO2. The newly developed volume-translated and critical-point PC-SAFT EOS (VT-CPPC-SAFT EOS) was found to be superior to previously developed models, such as the original PC-SAFT (Gross and Sadowski, 2001), CPPC-SAFT (Anoune et al., 2021), and industrialized version of PC-SAFT EOS (I-PC-SAFT) (Moine et al., 2019), because it can provide more accurate density predictions for CO2 in different phase states. In addition, the VT-CPPC-SAFT EOS accurately reproduced the critical pressure and critical temperature of CO2.
SAFT-VR Mie is an established equation of state that offers excellent holistic descriptions of fluid behavior. However, the model does not account for the long-wavelength density fluctuations encountered in the critical region and can therefore not describe this region accurately. To this end, SAFT-VR Mie is treated with renormalization corrections, yielding SAFT-VR Mie + RG. This procedure is based on the ideas of White [ J. Chem. Phys. 1999, 111(20), 9352-9356; J. Chem. Phys. 2000, 112(7), 3236-3244; J. Chem. Phys. 2000, 113(4), 1580-1586 ] where the fluid structure is incorporated into the renormalization corrections. SAFT-VR Mie + RG improves the description of pure-component properties in the critical region, without losing accuracy outside the critical region. The model is extended to mixtures using the isomorphism approach and applied to binary n-alkane systems. Remarkable predictions of critical loci are obtained without binary interaction parameters. SAFT-VR Mie + RG also describes the equilibrium phase behavior of these systems outside the critical region accurately, thus achieving the overarching aim of developing a model for global fluid phase equilibria.
This chapter discusses thermodynamic properties and phase equilibria using equations of states. Focus is thereby placed on the perturbed-chain statistical associating fluid theory (PC-SAFT) as a representative for physically-based equations of state. It is a model that is suitable for nonspherical molecules and hydrogen-bonding species or mixtures. Thermodynamic models are regularly used in process simulations to provide properties like enthalpies, densities, pressures, coexisting compositions of mixtures, chemical potentials for chemical equilibria. With modern equations of state, it is also possible to predict interfacial tensions, adsorption isotherms, or contact angles using density functional theory or density gradient theory. Moreover, transport coefficients, such as shear viscosity, thermal conductivity, and diffusion coefficients can be correlated or predicted using excess entropy scaling approaches. This chapter is also concerned with determining meaningful model parameters. The predictive capability of the model, once pure-component and binary interaction parameters were adjusted to a given set of thermodynamic properties, for predicting further properties relevant for a flowsheet simulation is discussed. Group-contribution approaches are presented to address the frequently encountered case of absent experimental data for parameter adjustment.
Deep eutectic solvents (DES) are recently synthesized to cover limitations of conventional solvents. These green solvents have wide ranges of potential usages in real-life applications. Precise measuring or accurate estimating thermophysical properties of DESs is a prerequisite for their successful applications. Density is likely the most crucial affecting characteristic on the solvation ability of DESs. This study utilizes seven machine learning techniques to estimate the density of 149 deep eutectic solvents. The density is anticipated as a function of temperature, critical pressure and temperature, and acentric factor. The LSSVR (least-squares support vector regression) presents the highest accuracy among 1530 constructed intelligent estimators. The LSSVR predicts 1239 densities with the mean absolute percentage error (MAPE) of 0.26% and R² = 0.99798. Comparing the LSSVR and four empirical correlations revealed that the earlier possesses the highest accuracy level. The prediction accuracy of the LSSVR (i.e., MAPE = 0. 26%) is 74.5% better than the best-obtained results by the empirical correlations (i.e., MAPE = 1.02%).
There are high demands for reliable hydrogen-alcohol phase equilibria in separation and conversion-related industrial processes. Since experimental measurements cannot be directly included in the computer-aided handling of these processes, this study utilizes various computational techniques for estimating hydrogen solubility in seven alcoholic solvents (methanol, ethanol, 1-propanol, 2-propanol, allyl alcohol, 1-butanol, and furfuryl alcohol). Ranking analysis shows that the adaptive-neuro fuzzy inference system having genfis2 (ANFIS2) is the best choice for this purpose. The model predictions are in excellent agreement with the 194 laboratory measurements (RAD = 3.32%, MSE = 6.9 × 10⁻⁴, and R² = 0.998896). Statistical uncertainty analysis confirms that the ANFIS2 model is superior to the previously proposed equations of state and empirical correlations in the literature. Simulation results confirm that 1-butanol and furfuryl alcohol has the highest and lowest hydrogen absorption tendency, respectively. Furthermore, the ANFIS2 justifies that the solubility of hydrogen in all alcohols obeys Henry's law and decreases by decreasing temperature and pressure.
Parameterization is defined as the method used to assign values to component-dependent parameters of an Equation of State (EoS). For SAFT-type EoS, historically, little attention has been paid to these methods, while higher importance was devoted to the theoretical foundations of the model. In this study, our ultimate objective was defined as the accurate reproduction of VLE and energetic properties (vapor pressure Psat, saturated-liquid density ρliqsat, enthalpy of vaporization ΔvapH, saturated-liquid heat-capacity cP,liqsat). Obviously, fitting the EoS parameters to these four property data immediately ensures meeting our objective. However, noticing that these four property data are not always available in practice and that some property data may have a marginal impact on the results of the fitting procedure, we searched for the necessary and sufficient properties against which EoS parameters should be regressed to meet our objective. Twelve combinations of property data among Psat, ρ liqsat, ΔvapH, and cP,liqsat were tested. For each property combination, the three PC-SAFT molecular parameters (m, σ, ϵ/k) were estimated for 544 nonassociating species disposing of experimental data for the four target properties. We concluded that it is necessary and sufficient to fit the PC-SAFT parameters to both Psat and ρ liqsat data. Finally, a molecular-parameter database for the PC-SAFT EoS was proposed for 1252 nonassociating compounds disposing of at least Psat and ρ liqsat data. The performance of these predictions is contrasted against other SAFT (I-PC-SAFT) and cubic EoS (tc-PR and gen-tc-PR).
The most commonly used probabilistic model in reliability studies is the Perfect Renewal Process (PRP), which is characterized by the condition or type of maintenance represented: once the maintenance activities are executed, the equipment is restored to its original condition, leaving it “as good as new.” It is widely used since it represents an optimistic state when an item is replaced, assuming a perfect operational condition of the item after the maintenance. Some models have been developed for determining optimum preventive maintenance (PM) based on different criteria, and almost all aimed at PRP reliability modeling. The contribution of this paper is to analyze a model for determining the optimal preventive maintenance policy for a long time run under PRP and developing a general and chart-based tool for the problem, making it easier to solve the day-to-day practice and operation of equipment. As a result, a generalized chart was developed to support maintenance decisions through the elaboration of an original isometric table and complemented with a step-by-step methodology to determine the optimum time in which the preventive maintenance activities must be implemented. In most cases, these types of maintenance activities will consider a replacement activity.
The 2nd degree polynomial volume function (v2+ubv+wb2), involved in the attractive term of cubic equations of state, has a strong influence on the calculated volumetric properties and, to a lesser extent, influences the calculated vapor-liquid equilibrium properties such as vapor pressure and enthalpy of vaporization. This function contains two parameters (u and w) that were either selected constant or constituent-dependent in the literature. In this work, it is analyzed through a systematic methodology how parameters u and w influence the accuracy of predicted volumetric and vapor-liquid equilibrium properties, both for untranslated and translated cubic equations of state. It was thus possible to determine the optimum values of such parameters and to discuss if the values selected in well-known cubic equations of state are the most relevant.
The SAFT and cubic models are often compared in the open literature. Usually, limitations and strengths of both types of equations of state (EoS) are ascribed to their theoretical foundations. Little attention has been paid until now to the influence of parameterization choices on the performance of these EoS. By parameterization, it is here meant the way or method used to attribute values to EoS component-dependent parameters. SAFT-type and cubic EoS for pure compounds use two different parameterization methods: SAFT parameters are generally regressed on vapor-pressure and liquid-density data while cubic models use as input parameters experimental values of the critical temperature, pressure and acentric factor and are generally parameterized in order to exactly reproduce these three properties. As a consequence of these parameterization methods, SAFT EoS overestimate critical pressures while cubic EoS fail in reproducing accurately liquid-density data. In order to discuss the influence of parameterization methods on EoS performances, three different parameterization methods are considered and applied to both SAFT and cubic models. It is shown that at a fixed parameterization method, SAFT and cubic EoS are equivalent as they exhibit similar accuracies for the correlation of vapor-pressure, enthalpy of vaporization, heat capacity and liquid-density data. This study led us to conclude that EoS requiring three component-dependent parameters cannot provide simultaneous accurate prediction for all the properties of a pure fluid. It is highlighted that the key point to get high accuracy (for non-associating pure compounds) is to work with an EoS equivalent to a four-parameter corresponding-states principle, i.e., that requires the specification of four macroscopic properties for a given pure compound. Eventually, we have considered a volume-translated PC-SAFT EoS version involving 4 input parameters: the three classical ones ( m, σ, ε) and a volume-translation parameter (c). Such an EoS shows excellent performances: for no less than 587 non-associating compounds vapor-pressure data are predicted with an error of 1.5 %, liquid-heat capacity and enthalpy of vaporization are predicted with an error of 3 % while liquid density data are predicted with an error of 4 %; critical temperatures and pressures are exactly reproduced.
A modified Redlich-Kwong equation of state is proposed. Vapor pressures of pure compounds can be closely reproduced by assuming the parameter a in the original equation to be temperature-dependent. With the introduction of the acentric factor as a third parameter, a generalized correlation for the modified parameter can be derived. It applies to all nonpolar compounds. With the application of the original generalized mixing rules, the proposed equation can be extended successfully to multicomponent-VLE calculations, for mixtures of nonpolar substances, with the exclusion of carbon dioxide. Less accurate results are obtained for hydrogen-containing mixtures.
The PC-SAFT equation of state is a very popular and promising model for fluids that employs a complicated pressure-explicit mathematical function (and can therefore not be solved analytically at a specified pressure and temperature, contrary to classical cubic equations). In this work, we demonstrate that in case of pure fluids, the PC-SAFT equation may exhibit up to five different volume roots whereas cubic equations give at the most three volume roots (and yet, only one or two volume roots have real significance). The consequence of this strongly atypical behaviour is the existence of two different fluid–fluid coexistence lines (the vapour-pressure curve and an additional liquid–liquid equilibrium curve) and two critical points for a same pure component, which is obviously physically inconsistent. In addition to n-alkanes, nearly 60 very common pure components (branched alkanes, cycloalkanes, aromatics, esters, gases, and so on) were tested out and without any exception, we can claim that all of them exhibit this undesired behaviour. In addition, such similar phenomena (i.e. existence of more than three volume roots) may also arise with mixtures. From a computational point of view, most of the algorithms used for solving equations of state only search for three roots at the most and are thus likely to be inefficient when an equation of state gives more than three volume roots. To overcome this limitation, a simple procedure allowing to identify all the possible volume roots of an equation of state is proposed.
In this paper, translated and consistent versions of the Peng-Robinson (tc-PR) and Redlich-Kwong (tc-RK) cubic equations of state (CEoS) are developed. The adjective consistent means that the α-function used in such models passes the consistency test we recently developed and which guarantees safe extrapolation in the supercritical region and safe VLE calculation in multi-component systems. The adjective translated means that a volume translation aimed at exactly reproducing the experimental saturated liquid volume at a reduced temperature of 0.8 was used. The key conclusion of this paper is that the tc-PR EoS is certainly the safest and the most accurate 3-parameter cubic EoS ever published. Indeed the average deviations over roughly 1000 compounds belonging to different chemical families are: ΔPsat<1 %, and .
This study highlights that α-functions used in cubic equations of state must be of class C2 i.e. that their first and second derivatives must exist and must be continuous, positive , monotonically decreasing , convex and also verify , for any value of the temperature T. Our proposed “consistency test for α-functions” gathers all these conditions. The non-respect of one of them can entail low-accuracy prediction of binary phase diagrams involving at least one supercritical compound (this statement is illustrated through the case-studies of the CO2-argon and CO2-decane systems) as well as improper variations of pure-component supercritical properties ( and ) with respect to the temperature. Finally, an extensive study of the mostly used α-functions described in the open literature is performed and shows that all of them fail this test. Some component-dependent α-functions may however pass this test but only if mathematical constraints are added to their parameters.
The study of phase equilibria is one of the most important sources of information about the nature of intermolecular forces in liquids and their mixtures and is of the highest importance for designing and optimizing processes. Many of the main features of vapor-liquid and liquid-liquid phase behavior were well characterized experimentally during the early part of the 20th century, and many equations of state were developed to reproduce the many types of phase diagrams observed for binary systems. In spite of the quasi-infinite number of possible configurations and rearrangements of fluid-fluid equilibrium phase diagrams, this paper presents a near-exhaustive classification scheme of fluid phase equilibria in binary systems. It starts from the one proposed by Van Konynenburg and Scott and brings it up-to-date by detailing the progress carried out on this topic since their classification. scheme was first proposed. The second part of this paper is devoted to describing the transitions between the various types of systems.
The phase behavior of water/hydrocarbon mixtures in a wide range of concentrations, temperatures, and pressures is important in a variety of chemical engineering applications. For this reason, the physical understanding and mathematical modeling of these aqueous–organic mixtures constitute a challenging task, both for scientists and for applied engineers. In this work, mutual solubilities, critical loci, and mixing enthalpies of water + hydrocarbon, water + carbon dioxide, water + nitrogen, water + hydrogen sulfide, and water + hydrogen binary mixtures are predicted using the PPR78 cubic equation of state (EoS). The extremely nonideal behavior of these systems produces unusual and complex thermodynamic behavior. As an example, such mixtures often exhibit type III phase behavior in the classification scheme of Van Konynenburg and Scott and are characterized by a vapor–liquid critical line which first exhibits a temperature minimum and then extends to temperatures above the critical point of pure water. Such a behavior, called gas–gas equilibria of the second kind is a consequence of the large degree of immiscibility of the two components. The selected PPR78 model combines the Peng–Robinson cubic EoS and a group-contribution method aimed at predicting the temperature-dependent binary interaction parameters, kij(T), involved in the Van der Waals one-fluid mixing rules. Although, it is acknowledged that cubic EoS with a constant kij are not suitable to predict phase equilibria of such highly nonideal systems, the addition of the H2O group to the PPR78 model makes it possible to conclude that the use of temperature-dependent binary interaction parameters not only results in qualitatively accurate predictions over wide pressure and temperature ranges but also leads to quantitatively reasonable predictions for many of the studied systems.
In this work, using molecular dynamics simulations, we have investigated how simple could be a coarse grained molecular model yielding simultaneously equilibrium densities, surface tensions and transport properties of some n-alkanes (from methane to n-decane) along the vapor-liquid equilibrium curve. For that purpose, as an initial model, the fully flexible Lennard-Jones Chain (3 “molecular” parameters) model has been chosen. Using this simple molecular model, good results have been obtained on equilibrium properties of all tested n-alkanes despite a systematic slight overestimation of critical temperature, critical pressure and surface tensions. In addition, concerning transport properties, good results have been obtained for methane and n-butane except for thermal conductivity in the gas state. For n-heptane and n-decane it has been found that thermal conductivity is systematically underestimated while viscosity is well estimated except at low temperatures. Concerning thermal conductivity, this misevaluation can be corrected if the zero-density thermal conductivity is known. Concerning shear viscosity, it is found that an additional “rigidity” fourth parameter is required to improve the results when dealing with the longest chains at the lowest temperatures.
In a previous work, some irregular behaviours of the PC-SAFT EoS – and more generally of SAFT-type EoS – were pointed out for pure components at low temperatures. In particular, it was shown that for pure fluids at a fixed temperature and pressure, these equations of state were likely to exhibit up to five molar volumes, thus leading to the simultaneous existence of pseudo liquid–liquid and liquid–vapour phase equilibria at a same temperature. In this work, low-temperature calculations are performed using the PC-SAFT EoS which is combined with a fugacity model for the solid phase in order to predict solid–liquid phase equilibria. It is shown that when varying the binary-interaction parameter of the PC-SAFT equation, non-feasible phase diagrams may be calculated including for instance, the presence of a liquid–liquid azeotrope, of a liquid–liquid–solid–solid four-phase equilibrium or the simultaneous existence of several eutectic points.
Two-parameter equations of state like PR or SRK predict a universal critical compressibility factor for pure compounds, being intrinsically unable to reasonably describe the PVT properties of different fluids and their asymmetric mixtures, which require at least three component specific parameters in the density dependence. In this paper, we have explored the capabilities of the Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) and the Simplified Perturbed Hard Chain Theory (SPHCT) equations of state to represent pure compound properties. Original parameters published by their respective authors overestimate critical temperatures, which is detrimental for the study and design of supercritical separation processes. Therefore, we have investigated here the performance of rescaled parameters for each model, i.e. those matching the experimental critical temperature and pressure of pure compounds.The rescaling of the size and energetic parameters as implemented in recent works in the literature was found to provide poor results, and instead a simple procedure to obtain optimized rescaled parameters from only Tc, Pc and the acentric factor is proposed. These parameters show a regular behaviour for n-alkanes with PC-SAFT and their performance is tested in this work also for carbon dioxide, using the recent equations by Span and Wagner as reference data. The proposed parameters reproduce vapour pressures with the same accuracy of PC-SAFT original parameters but assure an exact representation of the experimental critical pressure and temperature. However, this is achieved at the expense of underestimated liquid densities, this limitation being common to different models and becoming more pronounced for larger molecules.
The current study demonstrates that the pure-component critical temperatures generated by SAFT-type equations of state (EoS) are independent of the hard-core diameter (). This remarkable feature leads to the development of a simple tool, called a global map, for a comprehensive investigation of the sophisticated SAFT-models under consideration. Practically, a net of critical isotherms in the m (chain length), or (segment energy parameter) plane is plotted not only for the ordinary but also for the additional unrealistic critical temperatures. Such a map makes it possible to check whether unrealistic phase equilibria will be predicted by the considered SAFT model.
In addition, the present study points out that the numerical pitfalls encountered at ambient temperature with the CK-SAFT and the PC-SAFT EoS can be eliminated by refitting the values of the universal constants.
A general strategy for global phase equilibrium calculations (GPEC) in binary mixtures is presented in this work along with specific methods for calculation of the different parts involved. A Newton procedure using composition, temperature and volume as independent variables is used for calculation of critical lines. Each calculated point is analysed for stability by means of the tangent plane distance, and the occurrence of an unstable point is used to determine a critical endpoint (CEP). The critical endpoint, in turn, is used as the starting point for constructing the three-phase line. The equations for the critical endpoint, as well as for points on the three-phase line, are also solved using Newton's method with temperature, molar volume and composition as the independent variables.The different calculations are integrated into a general procedure that allows us to automatically trace critical lines, critical endpoints and three-phase lines for binary mixtures with phase diagrams of types from I to V without advance knowledge of the type of phase diagram. The procedure requires a thermodynamic model in the form of a pressure-explicit EOS but is not specific to a particular equation of state.
The van der Waals equation of state is used to determine phase diagrams
for a wide variety of binary fluid mixtures. The locus of the critical
line in pressure-temperature-composition space is determined exactly by
solving a set of equations with the aid of a computer. The van der Waals
constants am and bm for the mixture depend
quadratically and linearly upon the mole fractions xi:
am = sumisumj xi
xjaij and bm =
sumixibii. Mixtures are characterized
by three non-dimensional parameters: ξ =
(b22-b11)/(b11+b22), zeta =
(a22b22-2-
a11b11-
2)/(a11b11-
2+a22b22-2) and Λ =
(a11b11-2-
2a12/b11b22+a22b22-2)/(a11b11-
2+a22b22-2). The parameter
Λ can be related to the low-temperature enthalpy of mixing and
the parameter zeta to the difference between the gas-liquid critical
pressures of the pure fluids. Most of the calculations are for molecules
of equal size (ξ = 0), but calculations for a size ratio of two (ξ
= 1/3) are also reported. Nine characteristic types of critical lines
are distinguished and these correspond to nine separate regions on a
Λ , zeta -diagram. Isobaric temperature-composition diagrams and
pressure-temperature projections are given for one example from each
region to illustrate the possible types of phase equilibrium. Special
attention is given to the details of lower critical solution temperature
behaviour (type IV) such as is found in the system methane + n-hexane,
to tricritical points (symmetrical and unsymmetrical), to azeotropy, and
to the possibility of double azeotropy. The phase diagrams calculated
from the van der Waals equation seem to account, at least qualitatively,
for all but one of the varieties of phase equilibria found in binary
fluid mixtures: the low-temperature lower critical solution points in
some highly structured aqueous solutions of alcohols and amines.
Predictions of critical lines and partial miscibility of binary mixtures of hydrocarbons have been made by using a modified version of the statistical associating fluid theory (SAFT). The so-called soft-SAFT equation of state uses the Lennard-Jones potential for the reference fluid, instead of the hard-sphere potential of the original SAFT, accounting explicitly for the repulsive and dispersive forces in the reference term. The mixture behavior is predicted once an adequate set of molecular parameters (segment size, dispersive energy, and chain length) of the pure fluid is available. We use two sets of such parameters. The first set is obtained by fitting to the experimental saturated liquid density and by equating the chemical potential in the liquid and vapor phases for a range of temperatures and pressures. The second set is obtained from the previous one, by rescaling the segment size and dispersive energy to the experimental critical temperature and pressure. Results obtained from the theory with these parameters are compared to experimental results of hydrocarbon binary mixtures. The first set gives only qualitative agreement with experimental critical lines, although the general trend is correctly predicted. The agreement is excellent, however, when soft-SAFT is used with the rescaled molecular parameters, showing the ability of SAFT to quantitatively predict the behavior of mixtures. The equation is also able to predict transitions from complete to partial miscibility in binary mixtures containing methane. © 1998 American Institute of Physics.
The high-pressure phase equilibria of water + n-alkane mixtures are characterized by vapor-liquid critical lines which first exhibit a temperature minimum and then extend to temperatures above the critical point of pure water; this so-called “gas−gas” coexistence is a consequence of the large degree of immiscibility of the two components. We use a simplified version of the SAFT equation of state, which is based on the thermodynamic perturbation theory of Wertheim for associating fluids: the original SAFT equation of state treats the molecules as chains of Lennard-Jones segments while the simplified SAFT-HS equation treats molecules as chains of hard-sphere segments with van der Waals interactions. The water molecules are modeled as spherical repulsive cores with four association sites which mediate the hydrogen-bonding interactions. It turns out that a simple relationship for the parameters of the various mixtures can be used with the SAFT-HS treatment. The nonspherical nature of the alkanes is incorporated into the theory by treating the molecules as chains formed from united-atom spherical segments. The parameters for the pure components of the water + n-butane mixture are fitted to the critical points of each component; the strength and range of the hydrogen-bonding interaction between water molecules were obtained in a separate study by fitting to the vapor pressure and saturated liquid density of pure water. The parameters for the unlike interactions are fitted to the minimum of the high-pressure gas−liquid critical line of the water + n-butane mixture. We use a simple relationship between the number of segments in the united-atom chain models of the n-alkanes and the number of carbon atoms to predict the properties of mixtures of water with other n-alkane homologues without the recourse to further fitting. The phase equilibria of the mixtures obtained using this transferable interaction parameter approach are in excellent agreement with the experimental data even though the parameters are fitted to just one mixture. The water + methane system is the exception to this: the pure component parameters have to be refitted to the anomalous critical point of methane, as does the unlike mean-field interaction. We also predict that the type III phase behavior exhibited by water + n-alkane mixtures persists even for very long n-alkane chains, i.e., water + n-eicosane.
The perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state is applied to binary and ternary mixtures of polymers, solvents, and gases. The three pure-component parameters required for nonassociating molecules were identified for six polymer compounds. The phase equilibrium of polymer systems, which often involves high-pressure liquid−liquid mixtures as well as vapor−liquid mixtures at lower pressures, was investigated. Using a constant binary interaction parameter (kij), the PC-SAFT equation of state gives good correlations of the appropriate phase behavior over wide ranges of conditions. Comparisons to an earlier version of SAFT reveal an improvement of the proposed model.
A new set of molecular transferable parameters for the n-alkane series is proposed. n-Alkanes are modeled as homonuclear chainlike molecules formed by tangentially bonded Lennard-Jones segments of equal diameter and the same dispersive energy. Phase equilibria calculations of heavy pure members of the series, up to n-octatetracontane (n-C48H98), and of ethane/n-decane and ethane/n-eicosane mixtures are performed with the soft-SAFT (statistical associating fluid theory) equation of state. This SAFT-type equation explicitly accounts for repulsive and dispersive forces in the reference term through a Lennard-Jones interaction potential, and it has been proven to accurately describe the phase behavior of light n-alkanes. Using the new set of parameters, the soft-SAFT equation is able to accurately predict the phase behavior of pure heavy n-alkanes. The dependence of the critical properties of pure n-alkanes with the carbon number is also predicted to be in quantitative agreement with experimental data, validating, at the same time, some recent simulation results of heavy members of the series. For the mixtures, the use of simple Lorentz−Berthelot combining rules provides quantitative agreement with experimental data over a broad range of temperatures and pressures. The physical meaning and transferability of these parameters are also discussed.
A modified SAFT equation of state is developed by applying the perturbation theory of Barker and Henderson to a hard-chain reference fluid. With conventional one-fluid mixing rules, the equation of state is applicable to mixtures of small spherical molecules such as gases, nonspherical solvents, and chainlike polymers. The three pure-component parameters required for nonassociating molecules were identified for 78 substances by correlating vapor pressures and liquid volumes. The equation of state gives good fits to these properties and agrees well with caloric properties. When applied to vapor−liquid equilibria of mixtures, the equation of state shows substantial predictive capabilities and good precision for correlating mixtures. Comparisons to the SAFT version of Huang and Radosz reveal a clear improvement of the proposed model. A brief comparison with the Peng−Robinson model is also given for vapor−liquid equilibria of binary systems, confirming the good performance of the suggested equation of state. The applicability of the proposed model to polymer systems was demonstrated for high-pressure liquid−liquid equilibria of a polyethylene mixture. The pure-component parameters of polyethylene were obtained by extrapolating pure-component parameters of the n-alkane series to high molecular weights.
The volumetric and thermodynamic functions correlated by Pitzer and co-workers analytically represented with improved accuracy by a modified BWR equation of state. The representation provides a smooth transition between the original tables of Pitzer et al. and more recent extensions to lower temperatures. It is in a form particularly convenient for computer use.
The statistical associating fluid theory for potentials of variable range (SAFT-VR) is used to examine the phase behaviour in the CO2+n-alkane homologous series. A unique set of transferable parameters for the unlike interactions are used which allow the prediction of the phase behaviour of different members of the series with little experimental data. A change in phase behaviour from type II in the van Konynenburg scheme (continuous gas–liquid critical line and liquid–liquid immiscibility at low temperatures) for CO2+n-dodecane, to type IV (discontinuous gas–liquid critical line and liquid–liquid immiscibility) for CO2+n-tridecane, and to type III (continuous transition from gas–liquid to liquid–liquid critical behaviour) for CO2+n-tetradecane is observed in agreement with experimental data.
Incluye bibliografía e índice
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