Virial Equation Of State Research Articles

Isothermal Vapor–Liquid Equilibrium Measurements for Alcohol + Water/n-Hexane Azeotropic Systems Using Both Dynamic and Automated Static-Synthetic Methods

Isothermal vapor–liquid equilibrium measurements (pressure– temperature–liquid and vapor composition) were conducted for water (1) + propan-1-ol (2), n-hexane (1) + butan-2-ol (2), and n-hexane (1) + 2-methyl-propan-1-ol (2) at three temperatures, each using a dynamic moderate pressure equilibrium still. Additionally, pressure–temperature-overall composition measurements for the first two systems were conducted for the intermediate temperatures using a newly automated static-synthetic apparatus to confirm the operability of the automation scheme and to compare the model predictions of the vapor-phase compositions from the processed pressure–temperature–liquid composition data by the static-synthetic method with the experimental vapor compositions measured by the dynamic method. The automation scheme was developed and implemented on the original static total pressure vapor–liquid equilibrium apparatus of Raal et al. (Fluid Phase Equilib. 2011, 310, 156–165) using the LabVIEW graphical programming language. In comparison to the manual operating mode, this scheme improved the efficiency of operation by reducing the man-hours involved and minimized the associated uncertainty with the measurement procedure with respect to liquid composition. Once executed, the control scheme requires approximately 2 days to produce a single 40-data point (pressure–temperature–liquid composition) isotherm and minimizes human intervention to 2–3 h in comparison to a 2-week long measurement procedure in the nonautomated operating mode and in excess of 3 weeks using the dynamic method with analysis by gas chromatography. All experimental data were modeled using the γ–Φ approach with the NRTL and UNIQUAC activity coefficient models and the virial equation of state with the Hayden and O’Connell correlation. For the pressure–temperature–liquid and vapor composition data measured, thermodynamic consistency testing was performed. The data sets passed the point test with 0.01 tolerance and the area test with 10% tolerance. Experimental vapor-phase compositions obtained by phase sampling in the dynamic method compared reasonably well with the predictions from the pressure–temperature–liquid composition data measured by the automated static-synthetic method.

Gaseous [formula omitted] measurements for propane (R290) + trifluoroiodomethane (R13I1) system

An experimental study of the pressure–volume–temperature-composition (PVTx) properties in the gas phase for the R290 + R13I1 system was performed with an isochoric method. The measurements of R290 + R13I1 mixtures were conducted in the range of temperatures from 283.37 K to 353.29 K, pressures from 242.2 kPa to 1065.0 kPa, densities from 0.1091 mol dm−3 to 0.4133 mol dm−3, and compositions from 0.4502 to 0.8524 mol fraction of propane. The uncertainties in the present work were estimated to be within ±10 mK for temperature, ± 2.4 kPa for pressure, ± 0.1 mol% for mole composition and ±0.16% for density, respectively. On the basis of the experimental PVTx property data, a truncated virial equation of state was developed for the R290 + R13I1 system. This equation reproduces the experimental data in gas phase with relative deviation within ±0.23% in pressure and within ±0.25% in density.

Vapor phase pvTx measurements of binary mixtures of difluoromethane (R32) and 2,3,3,3-tetrafluoroprop-1-ene (R1234yf)

The pvTx properties of binary mixture of difluoromethane (R32) and 2,3,3,3-tetrafluoroprop-1-ene (R1234yf) in the vapor phase were measured at temperatures from (298 to 383) K and at pressures from (475 to 5686) kPa covering the mole fractions of xR32 = (0.119, 0.259, 0.449, 0.698, 0.874) by using a Burnett apparatus. The combined expanded uncertainties of temperatures, pressures, mole fraction and density are estimated to be 24 mK, 1.6 kPa, 0.01 and within 0.2% (k = 2) over the entire temperature and pressure ranges, respectively. Based on the experimental data of this work, a virial equation of state was developed with the absolute average deviation of compressibility factor lower than 0.19% for each composition and this equation shows physically reasonable behaviour of the second and third virial coefficients. Furthermore, the relative deviations of the experimental data from the multi-parameter equation of state of REFPROP 10.0 are less than 0.6% for the mixtures.

Analytical alpha functions in bulk phase and nanopores

Alpha functions are of critical importance to the cubic equations of state (EOSs) while most existing alpha functions are correlations in the empirical/semi-empirical formulations and limited to the bulk phase calculations. In this study, a new analytical alpha function and its nanoscale-extended formulation, which considers the strong intermolecular interactions under confinement effects, are developed based on the virial EOS and statistical thermodynamics. The proposed alpha functions, coupled with a nanoscale-extended EOS, are validated for a general use and be accurate to calculate the phase and thermodynamic properties of a wide variety of components in bulk phase and nanopores.

A numerical study of natural convection properties of supercritical water (H2O) using Redlich–Kwong equation of state

In this article, the Crank-Nicolson implicit finite difference method is utilized to obtain the numerical solutions of highly nonlinear coupled partial differential equations (PDEs) for the flow of supercritical fluid (SCF) over a vertical flat plate. Based on the equation of state (EOS) approach, suitable equations are derived to calculate the thermal expansion coefficient (\( \beta \)) values. Redlich–Kwong equation of state (RK-EOS), Peng-Robinson equation of state (PR-EOS), Van der Waals equation of state (VW-EOS) and Virial equation of state (Virial-EOS) are used in this study to evaluate \( \beta \) values. The calculated values of \( \beta \) based on RK-EOS is closer to the experimental values, which shows the greater accuracy of the RK-EOS over PR-EOS, VW-EOS and Virial-EOS models. Numerical simulations are performed for \( {\text{H}}_{2} {\text{O}} \) in three regions namely subcritical, supercritical and near critical regions. The unsteady velocity, temperature, average heat and momentum transport coefficients for different values of reduced pressure and reduced temperature are discussed based on the numerical results and are shown graphically across the boundary layer.

Vapor–Liquid Equilibrium Measurements of Ether Alcohol Blends for Investigation on Reformulated Gas

New isothermal binary vapor–liquid equilibrium (VLE) systems of 1-pentanol + 2,2,4-trimethylpentane at 350.2, 360.2, and 370.2 K, methyl-tert-butyl-ether + 2,2,4-trimethylpentane at 307.2, 317.2, and 327.2 K, and methyl-tert-butyl-ether + 1-pentanol at 317.2 and 327.2 K were measured. The vapor–liquid equilibrium data were measured using a dynamic low pressure glass still. The experimental data were regressed using the γ–Φ approach (combined method) with the Wilson, NRTL, and UNIQUAC activity coefficient models for description of the liquid phase nonideality. The virial equation of state with the Hayden and O’Connell correlation for the second virial coefficient was used to describe the vapor phase nonideality. The system of 2,2,4- trimethylpentane + 1-pentanol exhibited a minimum boiling azeotrope. The methyl-tert-butyl-ether + (2,2,4-trimethylpentane/1-pentanol) systems showed positive deviation from Raoult’s law. The aforementioned activity coefficient models correlated the experimental data well. The ...

Solubility of Syngas Components in Water, Acetic Acid, and Alcohol Using New Standard Fugacity Methodology

The UNIQUAC model in combination with the Peng–Robinson and the Hayden–O’Connell Virial equation of state was used for correlating vapor–liquid equilibrium for a gas–solvent system containing the following: CO2–water, CO2–ethanol, CO2–acetic acid, CO–water, CO–ethanol, CO–acetic acid, CH4–water, CH4–ethanol, CH4–acetic acid, N2–water, N2–ethanol, N2–acetic acid, H2–water, and H2–ethanol. Prausnitz and Shair correlation, for the fugacity of the hypothetical liquid solute at a pressure of 1 atm, was implemented in the parametrization procedure as additional objective function so as to reduce the uncertainty on the constants of the model. The results show that the UNIQUAC equation, coupled with an appropriate equation of state to model the vapor phase nonideality, is able to represent the binaries in the range of temperatures and pressures considered in the study, i.e., from 0 to 310 °C and from 1 to 400 bar, respectively. In addition to model parameter estimation, a new generic method to evaluate gas solubi...

Methods for calculating equilibrium properties of pure substances, considering the critical point features

This paper deals with a fundamental equation of state for matter (FEoS). When constructing FEoS, the method of pseudocritical points (PCP) is used. The PCP method is based on the Benedek hypothesis and a new representation of the scale hypothesis of the critical point. We show the possibility to construct FEoS of argon based on the PCP method with the following characteristics: (i) FEoS is transformed into the virial equation of state in the region of small densities; (ii) FEoS is transformed into the Widom equation in the region of the critical point; (iii) FEoS has its working area with pressure (0 ≤ p/pc ≤ 740), density (0 ≤ p/pc ≤ 3.2) and temperature (83.8058 ≤ T ≤ 2300∼K) in. We compare FEOS with a number of known equations of argon state and discuss the results.

A new slip velocity suitable for natural gas flows in energy applications

ABSTRACTFiguring out the natural gas behavior through micro/nanochannels is vital for accurate prediction of the rarefied flows and design optimization of Micro electro mechanical systems (MEMS) and Nano electro mechanical systems (NEMS) in energy applications. Continued with the previous work, introduction of a modified kinetic-based slip velocity for natural gas flows suitable for energy applications is the purpose of the current paper. In comparison with the previous work, this model includes an additional term due to compressibility factor gradient. We used the virial equation of state to calculate the compressibility factor gradient along the flow direction. The new model is verified with previously published results.

Isothermal Vapor–Liquid Equilibrium Measurements of Butan-2-one + 2-Methyl-propan-1-ol/Pentan-1-ol

Isothermal vapor–liquid equilibrium (VLE) data were measured for the systems of butan-2-one (1) + 2-methyl-propan-1-ol (2)/pentan-1-ol (2) using a dynamic low-pressure apparatus. Measurements were undertaken at approximately 333, 343, and 348 K. The VLE data were correlated by minimizing the pressure residual using the combined method with the Wilson and UNIQUAC activity coefficient models being employed for the liquid phase nonideality, using the virial equation of state with the Hayden and O’Connell correlation for the vapor phase nonideality. The Wilson model provided a marginally superior fit of the VLE data to the UNIQUAC model. The thermodynamic consistency of the data was tested, and it was found that all data sets were thermodynamically consistent according to the area and point tests. Excess enthalpies were calculated from the modeled VLE data and were extrapolated to conditions of experimental data from the literature. A good correlation was found between the experimental literature and calculat...

Micellization and adsorption modeling of single and mixed nonionic surfactants

The surface tension and heats of dilution of aqueous solutions of octyl-β-d-glucopyranoside (C8G1), tetraethylene glycol monooctyl ether (C8E4), pentaethylene glycol monooctyl ether (C8E5) and C8E5-C8G1 mixtures at 298.2 K were studied by tensiometry (ST) and isothermal titration calorimetry (ITC). Single surfactant micelle formation and growth were modeled using a multiple chemical equilibrium approach under the assumption that micelles are sphero-cylinders that can be thermodynamically described by three parameters, namely spherical and cylindrical contributions to the micelle molar Gibbs energy and the enthalpy of micellization. The two-dimensional virial equation of state was employed to correlate the surface tensions, which required two additional parameters: a 2D second virial coefficient and a reference monomer concentration. For the binary surfactant solutions, combining and mixing rules were suggested with no further parameters. For the single surfactants, the model correlated very well both ITC and ST experimental data. For mixed surfactants, ST data were also very well predicted, but this was not the case for ITC data. For the critical micelle concentrations, the model enthalpograms presented both positive and negative deviations from the experimental results, depending on the surfactant relative concentrations. Very low C8G1 concentrations promoted the demicellization of C8E5, which suggests strong interactions between monomer hydrophilic heads and the possible formation of small aggregates.

Accurate Experimental (p, ρ, and T) Data for the Introduction of Hydrogen into the Natural Gas Grid (II): Thermodynamic Characterization of the Methane–Hydrogen Binary System from 240 to 350 K and Pressures up to 20 MPa

Most of the experimental density data of the methane–hydrogen binary system available at the time of the development of the equation of state for natural gases and related mixtures, GERG-2008, at temperatures above 270 K were limited to hydrogen contents higher than 0.20 (amount-of-substance fraction). On the contrary, for mixtures with hydrogen contents lower than 0.20, experimental data were limited to temperatures below 270 K. This work intends to close the gap and provides accurate experimental (p, ρ, and T) data for three binary mixtures of methane and hydrogen, (0.95 CH4 + 0.05 H2), (0.90 CH4 + 0.10 H2), and (0.50 CH4 + 0.50 H2), at temperatures of 240, 250, 260, 275, 300, 325, and 350 K, thus extending the range of available experimental data to higher temperatures for mixtures with hydrogen contents lower than 0.20 and, accordingly, to lower temperatures for mixtures with hydrogen contents higher than 0.20. The density measurements were performed by using a single-sinker densimeter with magnetic suspension coupling at pressures up to 20 MPa. Experimental data were compared to the corresponding densities calculated from the GERG-2008 and the AGA8-DC92 equations of state, respectively. The experimental data are within the uncertainty of both equations of state, except at the lower temperatures of 240 and 250 K and pressures over 14 MPa for the mixtures with a hydrogen content of 0.05 and 0.10, respectively. The virial coefficients B(T, x) and C(T, x), as well as the second interaction virial coefficient B12(T) for the methane–hydrogen binary system, were also calculated from the experimental data set at temperatures from 240 to 350 K using the virial equation of state.

Vapor–Liquid Equilibrium Measurements of Ethanol and n-Nonane or n-Decane Binary Mixtures with Large Relative Volatility

A specialized still was used for the measurement of vapor–liquid equilibrium (VLE) data by the dynamic-analytic approach at low pressures. The specialized apparatus improves the measurement of VLE in high relative volatility systems. Isothermal measurements were conducted for the ethanol (1) + n-nonane (2) systems at 323.21, 333.21, and 343.21 K and the ethanol (1) + n-decane (2) systems at 328.17, 338.17, and 348.16 K. The experimental data was modeled by the combined approach using the NRTL activity coefficient model for the liquid phase and with the virial equation of state for the vapor phase with the Hayden and O’Connell correlation. The experimental data was fitted by minimizing the pressure residual. UNIFAC predictions were also conducted, and there was significant discrepancy between experimental and predicted data, highlighting the importance of precise experimental data in these types of systems. Thermodynamic consistency tests showed all data to be consistent.

Gaseous PvTx measurements of HFO-1234yf + HFC-32 binary mixture by single-sinker magnetic suspension densimeter

The measurements of PvTx properties of HFO-1234yf/HFC-32 binary mixture in the vapor phase were carried out by single-sinker magnetic suspension densimeter in the range of temperatures from 279.8 to 347.9 K, pressures from 0.10 to 2.66 MPa, densities from 2.30 to 66.02 kg m−3, and mole fractions of HFO-1234yf from 0.062 to 0.610. The experimental expanded uncertainty (k = 2) in temperature, pressure, and mole fraction are estimated to be 12 mK, 1.4 kPa, and 0.0002, respectively. The relative expanded uncertainty (k = 2) in density is 0.029–0.104%. On the basis of the experimental PvTx properties data, a truncated virial equation of state was developed for the binary HFO-1234yf/HFC-32 system. It reproduces the experimental values with average absolute deviations of 0.10% and 0.11% for pressure and density, respectively.

A new variant of a scaling hypothesis and a fundamental equation of state based on it

This paper deals with a fundamental equation of state (FEOS) for substances. We have suggested a new method. It allows constructing FEOS that is based on the scaling theory of critical phenomena and describes thermodynamic properties related to liquid and gas phases of a substance in a wide range of the pressures and temperatures. In the framework of the methodological approach, we have provided: (i) a transition of FEOS in a virial equation of state in the low density region; (ii) a transition of FEOS in a Widom equation of state in the critical region. The method has been tested on the example of FEOS of R218. The area of applicability of FEOS is 0 < ρ/ρc < 3.2 in the density and 133 < T < 440 K in the temperature. We have compared FEOS with some equations of state and discussed the results.

Measurements of vapor pressure and gaseous pvT property for trans-1,3,3,3-tetrafluoropropene (HFO-1234ze(E))

The saturated vapor pressures and gaseous pvT properties for trans-1,3,3,3-tetrafluoropropene (HFO-1234ze(E)) were measured using a Burnett apparatus. For vapor pressure, 15 data points were obtained in the temperature range from (303–373) K and pressure range from (578–3027) kPa. The experimental data was correlated by a Wagner type equation with the average absolute deviation of 0.003%. For gaseous pvT property, 101 data points were obtained in the temperature range from (313–373) K, pressure range from (117–2831) kPa and density range from (0.05–1.6) mol·dm−3. The second and third virial coefficients along 13 isotherms were obtained. The density data were correlated by a three-term truncated virial equation of state with a relative standard deviation of 0.06%.

Divergence of activity expansions: Is it actually a problem?

For realistic interaction models, which include both molecular attraction and repulsion (e.g., Lennard-Jones, modified Lennard-Jones, Morse, and square-well potentials), the asymptotic behavior of the virial expansions for pressure and density in powers of activity has been studied taking power terms of high orders into account on the basis of the known finite-order irreducible integrals as well as the recent approximations of infinite irreducible series. Even in the divergence region (at subcritical temperatures), this behavior stays thermodynamically adequate (in contrast to the behavior of the virial equation of state with the same set of irreducible integrals) and corresponds to the beginning of the first-order phase transition: the divergence yields the jump (discontinuity) in density at constant pressure and chemical potential. In general, it provides a statistical explanation of the condensation phenomenon, but for liquid or solid states, the physically proper description (which can turn the infinite discontinuity into a finite jump of density) still needs further study of high-order cluster integrals and, especially, their real dependence on the system volume (density).

PvT Properties of 2,3,3,3-Tetrafluoroprop-1-ene (HFO-1234yf) in the Gaseous Phase

2,3,3,3-Tetrafluoroprop-1-ene (HFO-1234yf) is considered as the promising alternative low-global warming potential refrigerant to replace HFC-134a. In this study, the gaseous pvT properties of HFO-1234yf were measured at temperatures from 252 to 345 K and pressures up to 1.91 MPa using a single-sinker densimeter. The density range is from 3.732 to 112.152 kg/m3. The standard uncertainties of the experimental data are estimated as 6 mK and 0.7 kPa for the temperature and pressure, respectively. The relative uncertainty in density is 0.014–0.034% in this study. The average absolute deviation and relative deviation of density from reported equations of state are 0.024 kg/m3 and 0.157%, respectively. A virial equation of state was developed for HFO-1234yf. The calculated results show good agreement with the experimental data.

Mass transfer and thermodynamic modeling of carbon dioxide absorption into MEA aqueous solution

Abstract In this research, thermodynamic and absorption rate of carbon dioxide in monoethanolamine (MEA) solution was investigated. A correlation based on both liquid and a gas phase variable for carbon dioxide absorption rate was presented using the π-Buckingham theorem. The correlation was constructed based on dimensionless numbers, including carbon dioxide loading, carbon dioxide partial pressure, film parameter and the ratio of liquid phase film thickness and gas phase film thickness. The film parameter is used to apply the effect of chemical reactions on absorption rate. A thermodynamic model based on the extended-UNIQUAC equations for the activity coefficients coupled with the Virial equation of state for representing the non-ideality of the vapor phase was used to predict the CO2 solubility in the CO2-MEA-H2O system. The average absolute error of the results for the correlation was 6.4%, which indicates the accuracy of the proposed correlation.

Direct numerical simulations of supersonic turbulent channel flows of dense gases

The influence of dense-gas effects on compressible wall-bounded turbulence is investigated by means of direct numerical simulations of supersonic turbulent channel flows. Results are obtained for PP11, a heavy fluorocarbon representative of dense gases, the thermophysics properties of which are described by using a fifth-order virial equation of state and advanced models for the transport properties. In the dense-gas regime, the speed of sound varies non-monotonically in small perturbations and the dependency of the transport properties on the fluid density (in addition to the temperature) is no longer negligible. A parametric study is carried out by varying the bulk Mach and Reynolds numbers, and results are compared to those obtained for a perfect gas, namely air. Dense-gas flow exhibits almost negligible friction heating effects, since the high specific heat of the fluids leads to a loose coupling between thermal and kinetic fields, even at high Mach numbers. Despite negligible temperature variations across the channel, the mean viscosity tends to decrease from the channel walls to the centreline (liquid-like behaviour), due to its complex dependency on fluid density. On the other hand, strong density fluctuations are present, but due to the non-standard sound speed variation (opposite to the mean density evolution across the channel), the amplitude is maximal close to the channel wall, i.e. in the viscous sublayer instead of the buffer layer like in perfect gases. As a consequence, these fluctuations do not alter the turbulence structure significantly, and Morkovin’s hypothesis is well respected at any Mach number considered in the study. The preceding features make high Mach wall-bounded flows of dense gases similar to incompressible flows with variable properties, despite the significant fluctuations of density and speed of sound. Indeed, the semi-local scaling of Patelet al.(Phys. Fluids, vol. 27 (9), 2015, 095101) or Trettel &amp; Larsson (Phys. Fluids, vol. 28 (2), 2016, 026102) is shown to be well adapted to compare results from existing surveys and with the well-documented incompressible limit. Additionally, for a dense gas the isothermal channel flow is also almost adiabatic, and the Van Driest transformation also performs reasonably well. The present observations open the way to the development of suitable models for dense-gas turbulent flows.