Systematic prediction of critical point coordinates from molecular parameters of equations of state and interaction potentials

Based on the literature data of PVT measurements, the amplitudes of the equations of the critical isotherm D0(Zk), the critical isochore Г0(Zk), the phase boundaries В0(Zk) are expressed in terms of the critical factor of compressibility of the substance Zk=PkVk/RTk in the entire fluctuation region near the critical point. By doing so, a phenomenological method has been used for calculating the values of the critical exponents of the fluctuation theory of phase transitions based on the introduction of small parameters into the equations of the fluctuation theory. It has been shown that, within the limits of the PVT measurement errors, these dependences D0(Zk) and В0(Zk) on the compressibility factor are linear, and Г0 practically does not depend on the compressibility factor Zk. The relationship of these amplitudes with the amplitudes a and k of the linear model of the system of parametric scale equations of state of substance near the critical point has been established. It has been shown that the dependences k(Zk) and а(Zk) are also linear in the entire fluctuation region near the critical point. The obtained dependences k(Zk) and а(Zk) agree with the known relationship between the amplitudes of the critical isotherm D0(Zk), critical isochore Г0(Zk), phase boundaries В0(Zk) Aerospace Institute of the National Academy of Sciences of Ukrainewithin the framework of the system of parametric scaling equations. The relations а(Zk), k(Zk) make it possible, on the basis of a linear model of the system of parametric scale equations of state of substance, to determine such important characteristics of the critical fluid as the temperature and field dependences of the correlation length Rc(T,m) and the fluctuation part of the thermodynamic potential Ф(T,m) in the entire fluctuation region near the critical point. Then, based on the form of the fluctuation part of the thermodynamic potential Ф(T,m)~Rc(T,m)-3, the results obtained allow one to calculate the field and temperature dependences of the thermodynamic quantities for a wide class of molecular liquids in the close vicinity of the critical point (DP<10-3, Dr<10-2, t<10-4), where precision experiments are significantly complicated, and its can also be used when choosing the conditions for the most effective practical application of the unique properties of the critical fluid in the newest technologies.

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Implementation of scaling and extended scaling equations of state for the critical point of fluids

Journal of Research of the National Bureau of Standards ◽

10.6028/jres.083.021 ◽

1978 ◽

Vol 83 (4) ◽

pp. 329 ◽

Cited By ~ 6

Author(s):

M.R. Moldover

Keyword(s):

Critical Point ◽

Equations Of State

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One-dimensional refraction properties of compression shocks in non-ideal gases

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.10 ◽

2017 ◽

Vol 814 ◽

pp. 185-221 ◽

Cited By ~ 12

Author(s):

Nicolas Alferez ◽

Emile Touber

Keyword(s):

Critical Point ◽

Equations Of State ◽

Organic Rankine Cycle ◽

Ideal Gas ◽

Acoustic Properties ◽

Compressible Turbulence ◽

Isentropic Compression ◽

Volume Expansivity ◽

Ideal Gases ◽

Mach Numbers

Non-ideal gases refer to deformable substances in which the speed of sound can decrease following an isentropic compression. This may occur near a phase transition such as the liquid–vapour critical point due to long-range molecular interactions. Isentropes can then become locally concave in the pressure/specific-volume phase diagram (e.g. Bethe–Zel’dovich–Thompson (BZT) gases). Following the pioneering work of Bethe (Tech. Rep. 545, Office of Scientific Research and Development, 1942) on shocks in non-ideal gases, this paper explores the refraction properties of stable compression shocks in non-reacting but arbitrary substances featuring a positive isobaric volume expansivity. A small-perturbation analysis is carried out to obtain analytical expressions for the thermo-acoustic properties of the refracted field normal to the shock front. Three new regimes are discovered: (i) an extensive but selective (in upstream Mach numbers) amplification of the entropy mode (hundreds of times larger than those of a corresponding ideal gas); (ii) discontinuous (in upstream Mach numbers) refraction properties following the appearance of non-admissible portions of the shock adiabats; (iii) the emergence of a phase shift for the generated acoustic modes and therefore the existence of conditions for which the perturbed shock does not produce any acoustic field (i.e. ‘quiet’ shocks, to contrast with the spontaneous D’yakov–Kontorovich acoustic emission expected in 2D or 3D). In the context of multidimensional flows, and compressible turbulence in particular, these results demonstrate a variety of pathways by which a supplied amount of energy (in the form of an entropy, vortical or acoustic mode) can be redistributed in the form of other entropy, acoustic and vortical modes in a manner that is simply not achievable in ideal gases. These findings are relevant for turbines and compressors operating close to the liquid–vapour critical point (e.g. organic Rankine cycle expanders, supercritical $\text{CO}_{2}$ compressors), astrophysical flows modelled as continuum media with exotic equations of state (e.g. the early Universe) or Bose–Einstein condensates with small but finite temperature effects.

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Equations of State of a Pure Material, Binodal, Spinodal, Critical Point, and Sample Problem

Lectures in Classical Thermodynamics with an Introduction to Statistical Mechanics ◽

10.1007/978-3-030-49198-7_17 ◽

2020 ◽

pp. 159-169

Author(s):

Daniel Blankschtein

Keyword(s):

Critical Point ◽

Equations Of State ◽

Sample Problem ◽

Pure Material

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Critical Point and Saturation Pressure Calculations for Multipoint Systems (includes associated paper 8871 )

Society of Petroleum Engineers Journal ◽

10.2118/7478-pa ◽

1980 ◽

Vol 20 (01) ◽

pp. 15-24 ◽

Cited By ~ 31

Author(s):

L.E. Baker ◽

K.D. Luks

Keyword(s):

Equation Of State ◽

Critical Point ◽

Critical Region ◽

Equations Of State ◽

Saturation Pressure ◽

Direct Calculation ◽

Iterative Solution ◽

Dew Point ◽

K Value ◽

Bubble Point

Abstract Calculation of fluid properties and phase equilibria isimportant as a general reservoir engineering tool andfor simulation of the carbon dioxide or rich gasmultiple-contact-miscibility (MCM) mechanisms. Of particular interest in such simulations is thenear-critical region, through which the compositionalpath must go in an MCM process.This paper describes two mathematical techniquesthat enhance the utility of an equation of state forphase equilibrium calculations. The first is animproved method of estimating starting parameters(pressure and phase compositions) for the iterativesaturation pressure (bubble-point or dew-point)solution of the equation of state. Techniquespreviously have been presented for carrying out thisiterative solution; however, the previously describedprocedure for obtaining initial parameter values wasnot satisfactory in all cases. The improved methodutilizes the equation of state to estimate theparameter values. Since the same equation then isused to calculate the saturation pressure, the methodis self-consistent and results in improved reliability.The second development is the use of the equationof state to calculate directly the critical point of afluid mixture, based on the rigorous thermodynamic criteria set forth by Gibbs. The paper presents aniterative method for solving the highly nonlinearequations. Methods of obtaining initial estimates ofthe critical temperature and pressure also arepresented.The techniques described are illustrated withreference to a modified version of the Redlich-Kwong equation of state (R-K EOS); however, theyare applicable to other equations of state. They havebeen used successfully for a wide variety of reservoirfluid systems, from a simple binary to complexreservoir oils. Introduction MCM processes such as CO2 or rich gas miscibledisplacements (conducted at pressures below thecontact-miscible pressure) traverse a compositional path that goes through the near-critical region. Thishas been described in several papers. Simulationof an MCM process requires the use of an equation of state to describe the liquid- and vapor-phasesaturations and compositions. Fussell and Yanosikdescribed an MVNR (minimum-variableNewton-Raphson)method for solution of a version of theR-K EOS. They discussed some of the difficulties ofobtaining solutions to the equation of state in the near-critical region and showed that the MVNRmethod gave improved results.Experience with the MVNR method hasdemonstrated a need for an improved estimate ofinitial iteration parameters (pressure, phasecompositions)for an iterative solution of saturation(dew-point and bubble-point) pressures. It waslearned that the semitheoretical K-value correlationused for initial estimates usually gave satisfactoryresults when the fluid system contained significantamounts of heavy components (C7+) but was oftenunsatisfactory for fluid systems containing only lightcomponents. This type of system is exemplified bythe fluids in a dry gas-rich gas mixing zone or bymixtures rich in CH4, CO2, or N2.Experience also has demonstrated a need for direct calculation of the critical point. While the MVNRsolution technique discussed by Fussell and Yanosikexhibits improved convergence in the near-critical region, it often is difficult to obtain convergedsolutions of the equation of state at compositionswithin a few percent of the critical composition. SPEJ P. 15＾

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