Thermodynamic modeling of vapor–liquid equilibrium for binary polyethylene glycol/solvent solutions using cubic equations of state: optimization and comparison of CEoS models

The cubic equation of state (CEoS) is a powerful method for calculation of (vapor + liquid) equilibrium in polymer solutions. Using CEoS for both the vapor and liquid phases allows one to calculate the non-ideality of polymer solutions based on a single EoS approach. In this research, vapor–liquid equilibrium calculations of Polyethylene glycol(Polyethylene oxide)/solvent solutions were carried out. In this approach eight models containing PRSV and SRK CEoS separately combined with four mixing rules namely one-parameter van der Waals one-fluid, two-parameter van der Waals one-fluid (vdW2), Wong–Sandler, and Zhong–Masuoka were applied to calculations of bubble point pressure. For the better prediction, the adjustable binary interaction parameters existing in any mixing rule were optimized. The results were very acceptable and satisfactory. The results of absolute average deviations between predicted results and experimental bubble point pressure data were calculated and presented. Although the capability of two cubic equations of state had a good agreement with experimental data and predict the correct type of phase behavior in all cases, the performance of the PRSV+vdW2 was more reliable than the other models.     

方型状态方程在高压相平衡中的研究及应用

对近年来状态方程的研究及其在高压相平衡计算中的应用状况作了综述分析。以ＰＲＳＶ方程为基础，通过实际计算考察和评价不同混合规则在复杂体高压汽液平衡、超临界流体／固体体系相平衡计算中的性能     

Extending the range of COSMO‐SAC to high temperatures and high pressures

The range of the predictive Gibbs energy of solvation model, COSMO‐SAC, is extended to large ranges of density, pressure, and temperature for very nonideal mixtures by combining it with an equation of state (EOS) using the Wong‐Sandler mixing rule. The accuracy of isothermal vapor‐liquid equilibria (VLE) calculations based on using the predictive COSMO‐SAC model and separately the correlative NRTL model is compared, each combined with three different forms of the Peng‐Robinson equation of state. All the models considered require the value of the EOS mixing rule binary parameter k_ij. The NRTL model also requires three other parameters obtained from correlation low pressure VLE data. The PRSV + COSMO‐SAC model is showed, with its one adjustable parameter obtained from low temperature data leads good predictions at much higher temperatures and pressures. © 2017 American Institute of Chemical Engineers AIChE J, 64: 1806–1813, 2018     

PRSV: An improved peng—Robinson equation of state for pure compounds and mixtures

The temperature and acentric factor dependence of the attractive term of the Peng—Robinson equation of state have been modified. The introduction of a single pure compound parameter allows the accurate reproduction of the vapor pressure data for a wide variety of substances. Nonpolar, polar nonassociating and associating compounds are equally well represented by the cubic PRSV equation of state. The conventional one-binary-parameter mixing rule allows the correlation of the vapor—liquid equilibrium data for a wide variety of binary systems. Only for systems formed by a polar compound (associating or not) and a saturated hydrocarbon, are results poorer than those obtained with conventional excess Gibbs energy functions.     

Phase behavior of mixture of supercritical CO2 + ionic liquid: Thermodynamic consistency test of experimental data

Various models have been applied composed of the Peng‐Robinson equation of state (PR‐EoS) and the Soave‐Redlich‐Kwong equation of state (SRK‐EoS) associated with three mixing rules including the following: Wong‐Sandler (WS), van der Waals one (vdW1), and van der Waals two (vdW2) for phase behavior modeling of mixtures of supercritical CO₂ + different ionic liquids in vapor–liquid equilibrium (VLE) region. It has been found that the PR EoS implying the WS mixing rule can be used as a reliable thermodynamic model to perform a thermodynamic consistency test on the experimental data of phase behaviors of the supercritical CO₂ + ionic liquid systems (19 commonly‐used ionic liquids have been studied). The results show that 40% of the experimental data seem to be thermodynamically consistent, 55.5% seem to be thermodynamically inconsistent, and 4.5% seem to be not fully consistent. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3892–3913, 2013     

Measurement and calculation of solubility of quinine in supercritical carbon dioxide

Solubility of quinine in supercritical carbon dioxide(SCCO_2) was experimentally measured in the pressure range of 8 to 24 MPa, at three constant temperatures: 308.15 K, 318.15 K and 328.15 K. Measurement was carried out in a semi-dynamic system. Experimental data were correlated by iso-fugacity model(based on cubic equations of state, CEOS), Modified Mendez–Santiago–Teja(MST) and Modified Bartle semi-empirical models. Two cubic equations of state: Peng–Robinson(PR) and Dashtizadeh–Pazuki–Ghotbi–Taghikhani(DPTG) were adopted for calculation of equilibrium parameters in CEOS modeling. Interaction coefficients(k_(ij) l_(ij)) of van der Waals(vdW) mixing rules were considered as the correlation parameters in CEOS-based modeling and their contribution to the accuracy of model was investigated. Average Absolute Relative Deviation(AARD) between correlated and experimental data was calculated and compared as the index of validity and accuracy for different modeling systems. In this basis it was realized that the semi-empirical equations especially Modified MST can accurately support the theoretical studies on phase equilibrium behavior of quinine–SCCO_2 media. Among the cubic equations of state DPGT within two-parametric vd W mixing rules provided the best data fitting and PR within one-parametric vd W mixing rules demonstrated the highest deviation respecting to the experimental data. Overall, in each individual modeling system the best fitting was observed on the data points attained at 318 K, which could be perhaps due to the moderate thermodynamic state of supercritical phase.     

Modeling high pressure vapor–liquid equilibrium of ternary systems containing supercritical CO2 and mixed organic solvents using Peng–Robinson equation of state

High pressure vapor–liquid equilibrium (VLE) of CO₂-expanded organic solvents was investigated using Peng–Robinson-LCVM-UNIFAC equation of state. Bubble pressure of several ternary mixtures was predicted using this model and correlations were developed based only on binary experimental data. A sensitivity study of the LCVM parameter numerical value was done by considering the coherence between the mathematical features of the mixing rule and the quality of the simulation. The results provided by PR-LCVM-UNIFAC were compared with those ones given by Peng–Robinson equation of state using the classical quadratic mixing rules (PR-CMR). Despite the use of two adjustable parameters for each binary system, PR-CMR is not able to provide good results when applied to ternary systems. The capability of PR-LCVM-UNIFAC model to predict liquid mixture density for ternary systems using parameters regressed only from bubble pressure experimental data was also investigated. Due to the lack of liquid density experimental data, it was possible to perform only a qualitative assessment of the density curves calculated by this equation of state.     

On the application of GE models to high-pressure vapor—liquid equilibrium calculations

This paper demonstrates that the two excess Gibbs free energy G^E models with temperature-dependent parameters in the modified Huron—Vidal first-order (MHV1) mixing rule allow the Peng—Robinson equation of state modified by Stryjek and Vera (PRSV) to be used for calculation of vapor—liquid equilibria over wide ranges of temperature and pressure. Good results are obtained for strongly non-ideal binary and ternary systems.     

Thermodynamic modeling of vapor-liquid equilibria and excess properties of the binary systems containing diethers and n-alkanes by cubic equation of state

A comparison of the performances of two different approaches of cubic equations of state models, based on a classical van der Waals and mixing rules incorporating theG ^E equation, was carried out for correlation of Vapor-Liquid Equilibria (VLE), H^E and C _P ^E data alone, and simultaneous correlation of VLE+H^E, VLE+C _P ^E , H^E +C _P ^E and VLE+H^E +C _P ^E data for the diethers (1,4-dioxane or 1,3-dioxolane) with n-alkane systems. For all calculations the Peng-Robinson-Stryjek-Vera cubic equation of state (PRSV CEOS) was used. A family of mixing rules for the PRSV CEOS based on the Modified van der Waals one-fluid mixing rule (MvdW1) and two well-known CEOS/G^E mixing rules (MHV1 and MHV2), was considered. The NRTL equation, as the G^E model with linear or reciprocal temperature dependent parameters, was incorporated in the CEOS/G^E models. The results obtained by the CEOS/G^E models exhibit significant improvement in comparison to the MvdW1 models.     

Calculation of pure compound saturated enthalpies and saturated volumes with the PRSV equation of state. Revised k1 parameters for alkanes

The Stryjek-Vera (1986a) modification of the Peng-Robinson (1976) equation of state has been used to predict heats of vaporization, enthalpies of saturated phases and saturated volumes for pure compounds ranging in molecular complexity from diatomic gases to compounds showing polarity and association. Heats of vaporization at the normal boiling temperature are reproduced within 2% for the majority of the compounds tested. Similar results were obtained for the enthalpy of the saturated phases. Revised k₁ parameters of the PRSV equation of state are reported for alkanes. An empirical correction has been developed for calculating saturated liquid volumes from the liquid volumes given by the PRSV equation of state. For the range of reduced temperatures from 0.5 to 0.99, deviations from experimental values are smaller than 4%. Saturated vapor volumes calculated from the PRSV equation of state were found to be satisfactory for most technical calculations.     

立方扰动硬链方程

本文通过在SPHC方程中引入拟合Carnahan-Starling硬球方程所得的斥力项,建立了立方型扰动硬链状态方程.由拟合正构烷烃的饱和蒸汽压和液相密度所确定的3个纯组分参数与碳原子数呈良好线性关系.对19种纯物质饱和蒸汽压和液相密度数据计算的平均相对误差分别为1.52%和2.97%;对混合物提出了一种新的混合规则,应用于预测含短链和长链烃类及二氧化碳、硫化氢混合物的高压汽液平衡,优于采用SPHC、SRK和PT方程所得的计算结果.     

PRSV — An improved peng-Robinson equation of state with new mixing rules for strongly nonideal mixtures

Two new composition dependent mixing rules for cubic equations of state are proposed. Both mixing rules contain two adjustable binary parameters and reduce to the conventional one parameter mixing rule when the parameters are equal. Vapor-liquid equilibrium data for mixtures of polar (associated or not) compounds with saturated hydrocarbons and for systems water/alcohol have been used to test the new mixing rules applied to the PRSV cubic equation of state. For these highly nonideal systems, correlation of the data with a new mixing rule of the Van Laar type for the PRSV equation gives better results than those obtained using excess Gibbs energy functions like the Wilson equation, NRTL and UNIQUAC.     

Measurement and correlation of solid drugs solubility in supercritical systems

A dynamic experimental set-up was utilized to measure ibuprofen solubility in supercritical CO2 at the pressure range of 8-13 MPa and the temperatures of 308, 313 and 318 K. Mole fraction values varied from 0.015＆#215;10^-3 to 3.261＆#215;10^-3 and correlated by using seven different semi empirical equations of state （Bartle, Modi-fied Bartle, Mendez-Teja, Modified Mendez-Teja, Kumar-Johnson, Sung-shim and Gordillo） as well as seven cubic equations of state （van der Waals, Redlich-Kwong, Soave-Redlich-Kwong, Peng-Robinson, Stryjek-Vera, Patel-Teja-Valderana and Pazuki）. Single and twin-parametric van der Walls mixing rules （vdW1, vdW2） were ap-plied in order to estimate the supercritical solution properties. The physicochemical properties were also obtained using Joback, Lydersen and Ambrose methods. Absolute average relatives deviation （AARD） were calculated and compared for all the correlating systems. Results showed that among the cubic equations of state （EOSs） the Pazuki equation （AARD=19.85% using vdW1 and AARD=8.79% using vdW2） and SRK equation （AARD=19.20%using vdW1 and AARD=10.03%using vdW2） predicted the ibuprofen solubility in supercritical CO2 with the least error in comparison to the others. Among the semi-empirical EOSs the most desirable deviation （AARD〈10%） was obtained by using Modified Bartle and Modified Mendez-Teja equations in all the studied temperatures.     

Fluid phase behavior modeling of CO2 + molten polymer systems using cubic and theoretically based equations of state

Phase equilibrium data of CO₂ + molten polymer systems are of great relevance for chemical engineers because these are necessary for the optimal design of polymer final‐treatment processes. This kind of processes needs information about gas solubilities in polymers at several temperatures and pressures. In this work, CO₂ solubilities in molten polymers were modeled by the perturbed chain‐statistical associating fluid theory (PC‐SAFT) equation of state (EoS). For comparison, the solubilities were also calculated by the lattice gas theory (LGT) EoS, and by the well‐known Peng‐Robinson (PR) cubic EoS. To adjust the interactions between segments of mixtures, there were used classical mixing rules, with one adjustable temperature‐dependent binary parameter for the PC‐SAFT and PR EoS, and two adjustable binary parameters for the LGT EoS. The results were compared with experimental data obtained from literature. The results in terms of solubility pressure deviations indicate that the vapor–liquid behavior for CO₂ + polymer systems is better predicted by the PC‐SAFT model than by LGT and PR models. POLYM. ENG. SCI., 2008. © 2008 Society of Plastics Engineers.     

Modeling of solid–liquid equilibria for polyethylene and polypropylene solutions with equations of state

We modeled solid–liquid equilibria (SLEs) in polyethylene and polypropylene solutions with a Soave–Redlich–Kwong (SRK) cubic equation of state (EOS) and a perturbed‐chain statistical associating fluid theory (PC‐SAFT) EOS. Two types of mixing rules were used with SRK EOS: The Wong–Sandler mixing rule and the linear combination of the Vidal and Michelsen mixing rules (LCVM), both of which incorporated the Bogdanic and Vidal activity coefficient model. The performance of these models was evaluated with atmospheric‐pressure and high‐pressure experimental SLE data obtained from literature. The basic SLE equation was solved for the equilibrium melting temperature instead of for the composition. The binary interaction parameters of SRK and PC‐SAFT EOS were estimated to best describe the experimental equilibrium behavior of 20 different polymer–solvent systems at atmospheric pressure and 31 other polymer–solvent systems at high pressure. A comparison with experimental data showed that SRK–LCVM agreed very well with the atmospheric SLE data and that PC‐SAFT EOS was more efficient in high‐pressure conditions. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2011     

Rapid expansion of cellulose triacetate from ethyl acetate solutions

Rapid‐expansion‐of‐supercritical‐solution (RESS) experiments were performed for solutions of cellulose triacetate [CTA; weight‐average molecular weight = 145,700, polydisperity = 2.07] in ethyl acetate [EA] over a range of concentrations and conditions of polymer–solvent phase behavior. Solid‐solubility experiments were carried out beforehand and identified hot and compressed liquid EA at approximately 175°C as a good solvent for CTA. Cloud‐point measurements were then used to locate the region of liquid–liquid equilibrium (LLE) for this system. The RESS results indicate that the phase state of the pre‐expansion mixture determines the product size, and the overall concentration of the pre‐expansion mixture determines the product morphology. However, we have also discovered a new constraint: for the production of well‐formed fibers and particles, the rapid‐expansion path must include penetration into a region of LLE, which must exist over a sufficient pressure range so that a separate, polymer‐rich liquid phase has time to develop before the onset of vapor–liquid equilibrium (VLE). If rapid expansions are carried out at temperatures near or below the lower critical end point, the expansion path leads directly into a region of VLE, and a disruptive vapor expansion occurs within the continuous liquid phase. In this case, hollow particles, hollow structures, and even foams are produced. By the proper choice of operating conditions, we were able to produce continuous CTA fiber from a 5 wt % solution in EA at 250°C and 69 bar. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 95: 290–299, 2005     

A four parameter cubic equation of state with temperature dependent covolume parameter

A four-parameter, Ghoderao–Dalvi–Narayan 2 cubic equation of state(GDN2 CEOS), is presented which incorporates the following: 1. The experimental value of the critical compressibility factor has been used as a fixed input parameter for calculations; 2. All the parameters(a, b, c, d) of CEOS are temperature dependent functions in the subcritical region and are temperature independent functions in the supercritical region and; 3. A new α function is introduced with two compound specific parameters which are estimated by matching saturated vapor pressure at two fixed temperature points T_r= 0.5, 0.7. Our formalism enables us to cast three of the four parameters of the CEOS as a function of the remaining parameter. The proposed CEOS is used to predict properties of 334 pure compounds, including saturated vapor pressure and liquid density, compressed liquid density, heat capacities at the constant pressure and volume, enthalpy of vaporization, sound velocity. To calculate thermodynamic properties of a pure compound, the present CEOS require the critical temperature, the critical pressure, the Pitzer's acentric factor, the critical compressibility factor, and two parameters of the alpha function. The saturated liquid density predictions for pure fluids are very accurate when compared with GDN1(Ghoderao–Dalvi–Narayan 1),MPR(Modified Peng–Robinson), and PT(Patel–Teja) equations of state. Unlike MPR EOS, the proposed temperature dependent covolume parameter b in the present work satisfies all the constraints mentioned in the literature to avoid thermodynamic inconsistencies at the extreme temperature and pressure. Using van der Waals one-fluid mixing rule, the present CEOS is further used to predict bubble pressure and the vapor mole fraction of binary mixtures.     

二氧化碳与丙酮,乙醚以及甲醇二元体系饱和液体密度的测定和关联

根据作者建立的一套实验装置,对二氧化碳-丙酮、二氧化碳-乙醚及二氧化碳-甲醇三个含极性混合物二元体系进行了饱和液体密度和泡点压力的测定,测定结果用立方型状态方程进行了关联。     

应用Excel进行泡点与露点计算

泡点与露点计算是化工热力学课程中气液平衡计算的重要内容,也是教学中的难点。目前,教科书中只讲了计算方法,缺少实际计算的展示。本文介绍了微软电子表格Excel在泡点与露点计算中的应用。将已知量、待求量初值与计算公式输入电子表格的各单元格中,求出各组分的气相逸度与液相逸度,根据气液平衡准则,采用Excel多变量规划求解的强大计算功能,即可计算出混合物的泡点压力、泡点温度、露点压力与露点温度。计算过程简单明了,无需编程,方便教学演示,是一种值得推广的教学与实用计算方法。     

Bubble point pressures of binary methanol/water mixtures in fine‐mesh screens

Binary methanol/water mixture bubble point tests involving three samples of fine‐mesh, stainless steel screens as porous liquid acquisition devices are presented in this article. Contact angles are measured as a function of methanol mass fraction using the Sessile Drop technique. Pretest predictions are based on a Langmuir isotherm fit. Predictions and data match for methanol mole fractions greater than 50% when pore diameters are based on pure liquid tests. For all three screens, bubble point is shown to be a maximum at a methanol mole fraction of 50%. Model and data are in disagreement for mole fractions less than 50%, which is attributed to variations between surface and bulk fluid properties. A critical Zisman surface tension value of 23.2 mN/m is estimated, below which contact angles can be assumed to be zero. Solid/vapor and solid/liquid interfacial tensions are also estimated using the equation of state analysis from Neumann and Good. Published 2013 American Institute of Chemical Engineers AIChE J 60: 730–739, 2014