Helmholtz Energy Equation Research Articles

Phase equilibrium modelling for multi-component mixtures using highly accurate Helmholtz energy equation of state

Phase stability of a mixture at given conditions is studied. The modelling of equilibrium phase compositions is considered when the original phase of the mixture is unstable. The tangent plane distance criterion is used for stability analysis and the Gibbs energy minimisation is employed for phase equilibrium calculation when the successive substitution does not work well. With the Helmholtz free energy equation of state, thermodynamic properties in single phase and two-phase equilibrium were calculated for methane-ethane mixtures. Phase envelopes were also calculated.

PHASE ENVELOPE CONSTRUCTION FOR MIXTURES USING HIGHLY ACCURATE HELMHOLTZ ENERGY EQUATION OF STATE

Phase envelope construction for mixtures is very important in the oil and gas industry. The most widely used model for phase envelope construction is Peng — Robinson equation of state (PR-EOS) due to its simplicity. In order to construct phase envelopes of natural gas, a highly accurate Helmholtz energy equation of state GERG-2008 was proposed. In this work, the accuracy of phase envelopes calculated by a highly accurate equation of state GERG-2008, simplified GERG-2008 and traditional cubic PR-EOS was analyzed. The pressure-based algorithm is used to calculate phase envelopes. Phase envelopes of the methane—ethane mixtures were constructed and compared with reference data. The results show that phase envelopes can be constructed with GERG-2008 in high accuracy. PR-EOS has good accuracy in phase envelope construction under low pressure. The simplified GERG-2008 also works well under low pressure; however, as pressure increases, it performs worse than the simpler PR-EOS, especially in the vicinity of critical point. Besides, a modified density solver for the complicated GERG-EOS was proposed. Calculations show that the proposed density solver can provide reliable results.

Thermodynamic speed of sound of xenon

Thermodynamic speed of sound data are an important basis for the development of Helmholtz energy equations of state because of their thermal and caloric nature. Moreover, they can be measured rapidly and with a high accuracy. Xenon is sampled with the pulse-echo technique to provide speed of sound data covering a temperature range from 217 K to 500 K with a pressure of up to 100 MPa. The measurement cell’s path length is calibrated with water and validated with the reference equation of state by Wagner and Pruß (2002). At a confidence level of 95% (k=2), the data have an overall expanded uncertainty of up to 0.17% near the critical point and less than 0.1% in the liquid and supercritical regions. The results are in good agreement with the Helmholtz energy equation of state by Lemmon and Span (2006) with a maximum deviation of up to 1.1%. The present data are also used to optimize the parameters of the Lennard-Jones potential and its truncated and shifted form for xenon. This parameterization leads to a convincingly better performance for the speed of sound calculation, but the representation of other properties, like the vapor-liquid two phase region, is significantly deteriorated.

Thermodynamic Speed of Sound Data for Liquid and Supercritical Alcohols

Because of their caloric and thermal nature, speed of sound data are vital for the development of fundamental Helmholtz energy equations of state for fluids. The present work reports such data for ...

Aplicación de ecuaciones de estado cúbicas y desarrollo de correlaciones numéricas para el cálculo de propiedades termodinámicas en saturación de refrigerantes para su uso en microcontroladores

The state-of-the-art of computer libraries in the field of thermodynamic properties are based on the Helmholtz energy equation. Unfortunately, the use of high-level programming languages and its complex implementation often exceeds the memory limit of most microcontrollers. In this context, after examination of the behavior of five variants of the Peng-Robinson equation, a set of simple equations for the saturation properties of common refrigerants that exceeded the precision of the former ones were developed. The proposed correlations are directly programmable in C language, facilitating the implementation in microcontrollers. Several prototype equations were fitted to the thermodynamic data in the reference RefProp library for a few thermodynamics properties by means of a non-linear multiparametric least-squares fitting procedure. Finally, a discussion of the best function for each property, as well as the average and local coefficients and uncertainties have been reported for several common refrigerants.

High-Pressure Speed of Sound Measurements of trans-1-Chloro-3,3,3-trifluoropropene (R1233zd(E)) in Liquid Region for Temperature from (273.15 to 353.15) K

This paper presents speed of sound measurements of liquid trans-1-chloro-3,3,3-trifluoropropene (R1233zd(E)) along six isotherms over the temperature range from (273.15 to 353.15) K for pressures up to 35 MPa by means of the double pulse–echo method. The expanded uncertainty of the speed of sound measurements at a confidence level of 95% is 0.1%. The experimental results were compared with predictions from the state-of-the-art Fundamental Helmholtz Energy equation of state [Mondejar, M.E.; McLinden, M.O.; Lemmon, E.W., Thermodynamic Properties of trans-1-Chloro-3,3,3-trifluoropropene (R1233zd(E)): Vapor Pressure, (p,ρ,T) Behavior, and Speed of Sound Measurements, and Equation of State. J. Chem. Eng. Data, 2015, 60, 2477–2489].

Thermodynamic properties of (R1234yf + R290): Isochoric pρTx and specific heat capacity cv measurements and an equation of state

In this paper, isochoric pρTx and specific heat capacity cv for (R1234yf + R290) binary mixtures were measured using an adiabatic batch calorimeter with intermittent heating. A total of 42 pρTx data points over temperatures from (254.28 to 348.30) K and 89 isochoric specific heat capacity data points over temperatures from (255.48 to 347.55) K were obtained for liquid (R1234yf + R290) with mole fractions of R1234yf at (0.825, 0.607, 0.521 and 0.285). The standard uncertainties were estimated to be 10 mK for temperature, 5 kPa for pressure, 0.3% for density and 1.0% for isochoric specific heat capacity. The experimental pρTx data were correlated by an empirical Tait equation with average absolute relative deviation of 0.19%. A Helmholtz energy equation of state based on the multi-fluid approximations model was developed for (R1234yf + R290) using the present and available experimental data. Eleven mixture rules are employed and the optimal Helmholtz energy equation of state calculates the density, VLE and isochoric specific heat capacity properties with sufficient accuracy. The compressed liquid density and isochoric specific heat capacity data in this work are well represented with average absolute relative deviation of 0.21% and 0.66%, respectively.

Measurement and modelling of the vapor–liquid equilibrium of (CO2 + CO) at temperatures between (218.15 and 302.93) K at pressures up to 15 MPa

Abstract Precise knowledge of vapor–liquid equilibrium (VLE) data of (CO2 + diluent) mixtures is crucial in the design and operation of carbon capture, transportation and storage processes. VLE measurements of the (CO2 + CO) system are reported along seven isotherms at temperatures ranging from just above the triple-point temperature of CO2 to 302.93 K and at pressures from the vapor pressure of pure CO2 to approximately 15 MPa, including near-critical mixture states for all isotherms. The measurements are associated with estimated standard uncertainties of 0.006 K for temperature, 0.009 MPa for pressure and 0.011x(1 − x) for mole fraction x. The new VLE data have been compared with two thermodynamic models: the Peng-Robinson equation of state (PR-EOS) and a multi-fluid Helmholtz-energy equation of state known as EOS-CG. The PR-EOS was used with a single temperature-dependent binary interaction parameter, which was fitted to the experimental data. In contrast, EOS-CG was used in a purely-predictive mode with no parameters fitted to the present results. While PR-EOS generally agrees fairly well with the experimental data, EOS-CG showed significantly better agreement, especially close to the critical point.

Prediction of vapour-liquid and vapour-liquid-liquid equilibria of nitrogen-hydrocarbon mixtures used in J-T refrigerators

Nitrogen-hydrocarbon mixtures are widely used as refrigerants in J-T refrigerators operating with mixtures, as well as in natural gas liquefiers. The Peng-Robinson equation of state has traditionally been used to simulate the above cryogenic process. Multi parameter Helmholtz energy equations are now preferred for determining the properties of natural gas. They have, however, been used only to predict vapour-liquid equilibria, and not vapour-liquid-liquid equilibria that can occur in mixtures used in cryogenic mixed refrigerant processes. In this paper the vapour-liquid equilibrium of binary mixtures of nitrogen-methane, nitrogen-ethane, nitrogen-propane, nitrogen-isobutane and three component mixtures of nitrogen-methane-ethane and nitrogen-methane-propane have been studied with the Peng-Robinson and the Helmholtz energy equations of state of NIST REFPROP and compared with experimental data available in the literature.

How to predict the vapor slope of temperature-entropy saturation boundary of working fluids from molecular groups?

The vapor slope of temperature-entropy saturation boundary of working fluids has a significant effect on the efficiency and the safety of thermodynamic processes. Therefore, how to predict the vapor slope efficiently and quickly is worth studying. In this work, the vapor slope represented by the slope angle is firstly correlated with the reduced temperature, the molecular weight, 10 molecular groups and a topological index. The 10 molecular groups can cover most working fluids employed in thermodynamic cycles. The chosen topological index is able to distinguish all the isomers of working fluids. The relationship between the slope angle and these variables is established by an artificial neural network. A total of 4338 data points for 54 working fluids are used to create this network. These data are calculated from the highly accurate Helmholtz energy equation at the reduced temperature range of 0.6–1. The predicted results from molecular groups show a good agreement with desired slope angles. The average absolute deviations for training, validation and test sets are 0.68%, 0.64%, 0.68%, respectively. In the selection and design of working fluids, the established model can be easily employed to predict the saturated vapor slope only from the molecular structures.

Numerical investigation of bubble dynamics using tabulated data

An explicit density-based solver of the compressible Euler equations suitable for cavitation simulations is presented, using the full Helmholtz energy Equation of State (EoS) for n-Dodecane. Tabulated data are derived from this EoS in order to calculate the thermodynamic properties of the liquid, vapour and mixture composition during cavitation. For determining thermodynamic properties from the conservative variable set, bilinear interpolation is employed; this results to significantly reduced computational cost despite the complex thermodynamics model incorporated. The latter is able to predict the temperature variation of both the liquid and the vapour phases. The methodology uses a Mach number consistent numerical flux, suitable for subsonic up to supersonic flow conditions. Finite volume discretization is employed in conjunction with a second order Runge–Kutta time integration scheme. The numerical method is validated against the Riemann problem, comparing it with the exact solution which has been derived in the present work for an arbitrary EoS. Further validation is performed against the well-known Rayleigh collapse of a pure vapour bubble. It is then used for the simulation of a 2-D axisymmetric n-Dodecane vapour bubble collapsing in the proximity of a flat wall placed at different locations from the centre of the bubble. The predictive capability of the incorporated Helmholtz EoS is assessed against the widely used barotropic EoS and the non-isothermal Homogeneous Equilibrium Mixture (HEM).

New knowledge on the temperature-entropy saturation boundary slope of working fluids

The slope of temperature-entropy saturation boundary of working fluids has a significant effect on the thermodynamic performance of cycle processes. However, for the working fluids used in cycles, few studies have been conducted to analyze the saturated slope from the molecular structure and mixture composition. Thus, in this contribution, an analytical expression on the slope of saturated curve is obtained based on the highly accurate Helmholtz energy equation. 14 pure working fluids and three typical binary mixtures are employed to analyze the influence of molecular groups and mixture compositions on the saturated slope, according to the correlated parameters of Helmholtz energy equation. Based on the calculated results, a preliminary trend is demonstrated that with an increase of the number of molecular groups, the positive liquid slope of pure fluids increases and the vapor slope appears positive sign in a narrow temperature range. Particularly, for the binary mixtures, the liquid slope is generally located between the corresponding pure fluids', while the vapor slope can be infinity by mixing dry and wet fluids ingeniously. It can be proved through the analysis of mixtures' saturated slope that three types of vapor slope could be obtained by regulating the mixture composition.

Recent trends in the development of Helmholtz energy equations of state and their application to 3,3,3-trifluoroprop-1-ene (R-1243zf)

Recent trends in the development of Helmholtz energy equations of state are briefly reviewed. Optimization procedures have been improved over the last two decades, and now nonlinear least-square fitting is effectively used in the optimization. This fitting technique makes it possible to develop a reliable equation of state for fluids with limited experimental data. The fitting is demonstrated for 3,3,3-trifluoroprop-1-ene (R-1243zf; CAS number 677-21-4). Experimental data for the critical parameters, vapor pressures, and liquid and vapor densities, including those at the saturation state, are available for this refrigerant. The fitting results in an equation with 17 terms. The equation of state is valid for temperatures from 234 to 376 K and for pressures up to 35 MPa. Comparisons to experimental data and verification of extrapolation behavior show that the equation of state has potential for most technical applications.

A Helmholtz Energy Equation of State for Sulfur Dioxide

A Helmholtz energy equation of state with independent variables of temperature and density was developed for sulfur dioxide (SO2) based on thermodynamic property data from the literature. The equation of state is valid from the triple-point temperature of 197.7 to 525 K, with pressures up to 35 MPa and densities up to 25.4 mol·dm–3. The uncertainties in density of the equation of state are 0.1% in the liquid phase, 0.25% in the vapor phase, and 1% in the critical region. The uncertainty in vapor pressure is 0.2% and the uncertainty in saturated liquid density is 0.2% below 410 K. The uncertainty in isobaric heat capacity is 2% between 200 and 290 K. In the critical region, the uncertainties are higher for all properties except for vapor pressure. The behavior of the equation of state is correct not only within the region of validity, but also at high temperatures and pressures, and far below the triple-point temperature.

Compressed Liquid Densities and Helmholtz Energy Equation of State for Fluoroethane (R161)

In this study, compressed liquid densities of Fluoroethane (R161, CAS No. 353-36-6) were measured using a high-pressure vibrating-tube densimeter over the temperature range from (283 to 363) K with pressures up to 100 MPa. A Helmholtz energy equation of state for R161 was developed from these density measurements and other experimental thermodynamic property data from the literature. The formulation is valid for temperatures from the triple point temperature of 130 K to 420 K with pressures up to 100 MPa. The approximate uncertainties of properties calculated with the new equation of state are estimated to be 0.25 % in density, 0.2 % in saturated liquid density between 230 K and 320 K, and 0.2 % in vapor pressure below 350 K. Deviations in the critical region are higher for all properties. The extrapolation behavior of the new formulation at high temperatures and high pressures is reasonable.

Speed-of-sound measurements in (argon + carbon dioxide) over the temperature range from (275 to 500) K at pressures up to 8 MPa

The speed of sound of two (argon+carbon dioxide) mixtures was measured over the temperature range from (275 to 500)K with pressures up to 8MPa utilizing a spherical acoustic resonator. The compositions of the gravimetrically prepared mixtures were (0.50104 and 0.74981) mole fraction carbon dioxide. The vibrational relaxation of pure carbon dioxide led to high sound absorption, which significantly impeded the sound-speed measurements on carbon dioxide and its mixtures; pre-condensation may have also affected the results for some measurements near the dew line. Thus, in contrast to the standard operating procedure for speed-of-sound measurements with a spherical resonator, non-radial resonances at lower frequencies were taken into account. Still, the data show a comparatively large scatter, and the usual repeatability of this general type of instrument could not be realized with the present measurements. Nonetheless, the average relative combined expanded uncertainty (k=2) in speed of sound ranged from (0.042 to 0.056)% for both mixtures, with individual state-point uncertainties increasing to 0.1%. These uncertainties are adequate for our intended purpose of evaluating thermodynamic models. The results are compared to a Helmholtz energy equation of state for carbon capture and storage applications; relative deviations of (−0.64 to 0.08)% for the (0.49896 argon+0.50104 carbon dioxide) mixture, and of (−1.52 to 0.77)% for the (0.25019 argon+0.74981 carbon dioxide) mixture were observed.

Gradient Theory simulations of pure fluid interfaces using a generalized expression for influence parameters and a Helmholtz energy equation of state for fundamentally consistent two-phase calculations

The fidelity of Gradient Theory simulations depends on the accuracy of saturation properties and influence parameters, and require equations of state (EoS) which exhibit a fundamentally consistent behavior in the two-phase regime. Widely applied multi-parameter EoS, however, are generally invalid inside this region. Hence, they may not be fully suitable for application in concert with Gradient Theory despite their ability to accurately predict saturation properties. The commonly assumed temperature-dependence of pure component influence parameters usually restricts their validity to subcritical temperature regimes. This may distort predictions for general multi-component interfaces where temperatures often exceed the critical temperature of vapor phase components. Then, the calculation of influence parameters is not well defined. In this paper, one of the first studies is presented in which Gradient Theory is combined with a next-generation Helmholtz energy EoS which facilitates fundamentally consistent calculations over the entire two-phase regime. Illustrated on pentafluoroethane as an example, reference simulations using this method are performed. They demonstrate the significance of such high-accuracy and fundamentally consistent calculations for the computation of interfacial properties. These reference simulations are compared to corresponding results from cubic PR EoS, widely-applied in combination with Gradient Theory, and mBWR EoS. The analysis reveals that neither of those two methods succeeds to consistently capture the qualitative distribution of obtained key thermodynamic properties in Gradient Theory. Furthermore, a generalized expression of the pure component influence parameter is presented. This development is informed by its fundamental definition based on the direct correlation function of the homogeneous fluid and by presented high-fidelity simulations of interfacial density profiles. The new model preserves the accuracy of previous temperature-dependent expressions, remains well-defined at supercritical temperatures, and is fully suitable for calculations of general multi-component two-phase interfaces.

Calculation of phase equilibria for multi-component mixtures using highly accurate Helmholtz energy equations of state

To test the thermodynamic stability and to determine the equilibrium phase compositions in case the original phase is found unstable is one of the greatest challenges associated with calculating thermodynamic properties of multi-component mixtures. The minimization of the tangent plane distance function is a widely used method to check for stability, while different approaches can be chosen to minimize the Gibbs energy in order to find the phase equilibrium. While these two problems have been applied to several different thermodynamic models, very little work has been published on such algorithms using multi-parameter Helmholtz energy equations of state. In this work, combined stability and flash calculation algorithms at given pressure and temperature (p,T), pressure and enthalpy (p,h), and pressure and entropy (p,s) are presented. The algorithms by Michelsen et al. (1982, 1982, 1987) are used as basis and are adapted to multi-parameter Helmholtz energy models. In addition, a robust and sophisticated density solver is proposed which is necessary for the calculation of properties from the Helmholtz energy model at given state variables other than temperature and density. All partial derivatives necessary to solve the isothermal, isenthalpic and isentropic flash problems using numerical methods based on the Jacobian matrix are derived analytically and given in the supplementary material to this article. Results for some multi-component systems using the GERG-2008 model (Kunz and Wagner, 2012) are shown and discussed.

Analytical calculation and evolutionary regression method for isentropic exponent of hydrogen gas at the throat of critical nozzle

As one of the most promising renewable energy resources, the hydrogen has been used in the fields such as aerospace, industry, and fuel cells. Critical nozzles are widely used for mass flow-rate measurement of high hydrogen gas, since the flow measurement process is not affected by its downstream flow disturbance. The flow rule of real hydrogen gas through a critical nozzle is complicated and the thermophysical property of hydrogen at the nozzle throat is vital to the accurate measurement of hydrogen flow. In this paper, based on explicit Helmholtz energy and entropy-enthalpy equations, the basic flow parameter and isentropic volume change exponent are analytically calculated. In addition, an accurate explicit equation is determined by the nonlinear regression analysis where the ways of selection, exchange and mutation derived from evolutionary algorithm are introduced to search for optimal population individual. The regression standard deviation is 0.0089%, mean residual deviation is 0.0285%, and maximum residual deviation is 0.1781%. The result shows that it not only can rapidly find the optimal solution which has the lowest number of equation items and the great overfitting suppression capability, but also has a high computation accuracy. This algorithm can also be applied to modeling flow characteristic parameters for every other flow device.

G113 状態方程式開発における最近の動向

This paper reviews the recent trend in the developments of Helmholtz energy equations of state. In I990s the equations were developed mainly using a linear least square method, and a step wise regression analysis was employed to select an optimum set of terms from a large bank of terms. This method is fast, but requires a large amount of experimental data. Equations obtained from this method consist of 25-50 terms. Recent approaches are based on a nonlinear least square algorithm, where not only deviations in calculated values but departures from thermodynamic requirements are included in the sum of square. These approaches are very time consuming, but can result in equations with 15-20 terms.