Multiparameter Equations Of State Research Articles

A new method to calculate and predict thermodynamic properties of nonpolar, polar and quantum fluids at high temperatures

As the only known equation of state (EOS) with the rigid theoretical foundation, the virial equation of state (VEOS) can be reliable enough to describe real-gas imperfection for low-to-moderate densities when truncated after the second or third virial coefficient. In our previous work, on the basis of the corresponding state principle, the generalized second and third virial coefficient models were proposed for nonpolar, polar and quantum fluids in a wide temperature range. In this work, combined with the ideal heat capacities, the high-temperature performance of the truncated VEOSs was evaluated on derived thermodynamic properties including speed of sound, isobaric heat capacity, entropy and internal energy. The criterion for being “valid” is defined as 1% relative deviation from the multiparameter EOSs, and this work presents the valid density and pressure regions of the truncated VEOSs for the derived thermodynamic properties. The truncated VEOSs have valid densities of the saturated densities below the critical temperatures, and have valid densities that tend to be constant beyond the critical temperatures. The SRK EOS is chosen as the representative of generalized EOSs. By the comparisons with the SRK EOS, the truncated VEOSs show the advantage in stability and universality. The truncated VEOSs can give a reliable extrapolation with wide applicable pressure ranges at high temperatures for nonpolar, polar and quantum fluids. The applicable density regions of the truncated VEOSs for derived thermodynamic properties are recommended to be the valid density regions of the pvT property. The applicable temperature ranges of the truncated VEOSs for derived thermodynamic properties are determined by the temperature ranges of the available ideal heat capacity data.

Extrapolating into no man’s land enables accurate estimation of surface properties with multiparameter equations of state

Thermodynamic properties of homogeneous fluids in the metastable and unstable regions are needed to describe confined fluids, interfaces, nucleating embryos and estimate critical mass flow rates. The most accurate equations of state (EoS) called multiparameter EoS, have a second, non-physical Maxwell loop that renders predictions unreliable in these regions. We elaborate how information from the stable region can be used to reconstruct the metastable and unstable regions. For a simple interaction potential, comparison to results from molecular simulations reveals that isochoric expansion of the pressure from stable states reproduces simulation results in the metastable regions. By constructing a dome that extends above the critical point, we obtain an extrapolated pressure from multiparameter EoS that is free of second Maxwell loops. A reconstructed EoS is developed next, by integrating the extrapolated pressure from a stable state to obtain the Helmholtz energy. The consistency of the reconstructed EoS is gauged by computing phase equilibrium densities, pressures, and enthalpies of evaporation, which are in reasonable agreement with experimental values. Combined with density gradient theory, the reconstructed EoS yields surface tensions of water, carbon dioxide, ammonia, hydrogen and propane that deviate, on average, 4.4%, 1.6%, 6.0%, 0.7% and 5.4% from experimental values respectively. The results reveal a potential to develop more accurate extrapolation protocols, which can be leveraged to obtain prediction of metastable properties, surface properties or used as constraints in fitting multiparameter EoS.

Superancillary Equations for the Multiparameter Equations of State in REFPROP 10.0

Superancillary equations have been developed for the recommended (by NIST) multiparameter equations of state (EOS) for all 147 pure fluids in NIST REFPROP 10.0. These superancillary equations represent the orthobaric densities and saturation pressure of the EOS as a function of temperature by Chebyshev expansions to an accuracy better than the iterative calculations in REFPROP and are hundreds to thousands of times faster to evaluate than a full iterative solution of Maxwell’s criteria. The C++ code required to develop and test the superancillary equations is provided as open-source material. The methodology is straightforwardly extensible to new multiparameter EOS, establishing a new paradigm for the evaluation of vapor–liquid equilibria for pure fluids.

Modeling Thermodynamic Properties of Mixtures of CO2 + O2 in the Allam Cycle by Equations of State

Different equations of state (EOS) were applied for describing thermodynamic properties of the system CO2+O2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {CO}_{2} + \ ext {O}_{2},$$\\end{document} which is important for the Allam cycle: cubic EOS (Soave–Redlich–Kwong, Peng–Robinson), molecular-based EOS (PC-SAFT, PCP-SAFT, SAFT-VR Mie, polar soft-SAFT, BACKONE, sCPA), and multiparameter EOS (GERG-2008, EOS-CG). The pure component models were taken from the literature. The results for the mixture were compared to experimental data from the literature for two cases: (i) the thermodynamic properties of the mixture were modeled using predictive mixing and combination rules; (ii) additional binary interaction parameters were fitted to experimental data for improving the performance. In the predictive mode (i), the best results were obtained with molecular-based EOS. In the adjusted mode (ii) the best results were obtained with the multi-parameter GERG-2008 EOS and EOS-CG.

A General Model for Thermodynamic Properties of Fluid Mixtures Based on Helmholtz Energy Formulations for the Components. Virial Expansion and Reduction to van der Waals Mixing Rules

Over the recent decades, Helmholtz energy formulations became available for a broad range of fluids. These multiparameter equations of state (R. Span, Springer 2000) allow computation of thermodynamic properties essentially within the experimental errorbars. Corresponding states-based model by Lemmon and Tillner-Roth (Fluid Phase Equilib 165:1, 1999) enabled construction of Helmholtz energy formulations for mixtures. However, we show that this model generates a non-physical dependence of virial coefficients on composition, which can be strong when the components are dissimilar. We propose a new mixture model that overcomes this deficiency. It has two main ingredients: (i) Quadratic mixing of “Helmholtz volumities”. This quantity with units of molar volume is introduced as a ratio of the molar residual Helmholtz energy to a product of gas constant, thermodynamic temperature, and molar density. It reduces to the second virial coefficient in the zero-density limit. Helmholtz volumities are considered for components and “cross-components”, hypothetical fluids representing the binary interactions. (ii) Replacing the variables—reduced reciprocal temperatures and reduced densities—with temperature and density scaling functions. Different scaling functions can be used for different components and cross-components, thus providing a highly flexible framework for representing the properties of mixtures. The scaling functions must be expandable into Taylor series in terms of molar concentrations in the zero-density limit. For the proposed mixture model, we develop formulas for computing virial coefficients up to the fourth order. Furthermore, we show that when the proposed mixture model is applied to a cubic equation of state, the conventional van der Waals mixing rules can be retrieved. These findings allow to consider the new model as a viable alternative to the corresponding states method of modeling thermodynamic properties of fluid mixtures.

Short nozzles design for real gas supersonic flow using the method of characteristics

Finding a supersonic nozzle minimum length is a classical application of the method of characteristics (MOC) for aerospace propulsion technology among many other applications, such as air and vapor handling processes. That methodology allows design contoured nozzle free of oblique shock waves created at sharp area changes in short nozzles. While the designing technique is well established in textbooks for ideal gas flow, the use of the MOC technique for real gas is problematic due to the highly complex fluid behavior, captured by modern real gas equations-of-state (EOS). This work presents the MOC devised for real gases using multi-parameter equations-of-state (MPEOS) for different substances and compositions. This paper also compares the MPEOS solution obtained from other classical EOS, such as Peng–Robinson, and the ideal gas equation to establish the optimal application range for each EOS and their effect on the nozzle wall construction. The study was carried out for a commercial fluid refrigerant, pure carbon dioxide, and a CO2−CH4 mixture. The methodology can be used for designing industrial pieces of equipment, such as turbo-machineries and supersonic gas separators or supersonic ejectors, to evaluate the isentropic flow expansion within those devices.

Optimal interstage pressures of multistage compression with intercooling processes

This study develops a method to determine the optimal interstage pressures for two-stage and three-stage compressions with intercoolers. The developed method can also be applied using multi-loop calculation for compression involving more than three stages. The developed relationship is expressed in terms of temperatures, densities and the derivative of entropy with respect to pressure. The properties used in this study are calculated from the multiparameter equations of state. Equations of optimal interstage pressures for the nonequal inlet temperature of compressors based on the ideal gas assumption, called an ideal gas model, are found and investigated. The study results show that for the high pressure and low temperature region, the ideal gas model cannot predict the interstage pressure that minimizes the total specific compression work. High pressure and low temperature are considered by comparing the critical pressure and temperature of the considered gas. The developed model can provide the optimal interstage pressure which causes minimum total compression work even if the compressions are conducted in a high pressure and low temperature region. For two-stage compression of CO2 from 6.25 MPa, 305 K to 20 MPa with isentropic efficiencies of 0.9, the developed method can save 16.63% of specific compression work, comparing to the ideal gas model. For the developed method, if the isentropic efficiency of the stage compressor increases, the calculated optimal interstage pressure raised by that stage also increases. The increase in outlet temperature of the intercooler causes increasing optimal value of interstage pressure raised by the previous compression stage.

Equation of state based analytical formulation for optimization of sCO2 Brayton cycle

Real gas thermodynamic cycles employ state-of-the-art property routines to compute the state properties as ideal gas relations are inaccurate. Multiparameter equation of state (EOS), though accurate, is computationally time consuming due to iterative solutions needed for complex cycles. Alternatively, look-up tables alleviate the computational time but compromise the accuracy due to interpolation and extrapolation schemes. This paper proposes a novel analytical formulation to overcome the tradeoff between accuracy and computation cost. Helmholtz energy-based EOS in combination with Jacobian transformations and real gas thermodynamics is used to arrive at a single equation linking the state points of a thermodynamic cycle. The efficacy of the formulation is demonstrated to deduce the optimum pressure ratio while maximizing specific work or efficiency of a simple recuperated supercritical CO2 Brayton cycle. The difference in optimum pressure ratio predicted by the analytical formulation and conventional property-routine is inconsequential compared to an order of gain in the computational time.

Efficient and Precise Representation of Pure Fluid Phase Equilibria with Chebyshev Expansions

In this work, it is shown that Chebyshev expansions can be used to provide a representation of pure fluid phase equilibria models that is approximately 400 times faster to evaluate than the full phase equilibrium calculation with multiparameter equations of state. The use of extended precision for the phase equilibrium calculation allows for the construction of a numerical representation approaching the numerical precision of double-precision arithmetic.

Turbulence Quantification in Supercritical Nitrogen Injection: An Analysis of Turbulence Models

In Liquid Rocket Engines, higher combustion efficiencies come at the cost of the propellants exceeding their critical point conditions and entering the supercritical domain. The term fluid is used because, under these conditions, there is no longer a clear distinction between a liquid and a gas phase. The non-conventional behavior of thermophysical properties makes the modeling of supercritical fluid flows a most challenging task. In the present work, a RANS computational method following an incompressible but variable density approach is devised on which the performance of several turbulence models is compared in conjunction with a high accuracy multi-parameter equation of state. Also, a suitable methodology to describe transport properties accounting for dense fluid corrections is applied. The results are validated against experimental data, becoming clear that there is no trend between turbulence model complexity and the quality of the produced results. For several instances, one- and two- equation turbulence models produce similar and better results than those of Large Eddy Simulation (LES). Finally, considerations about the applicability of the tested turbulence models in supercritical simulations are given based on the results and the structural nature of each model.
 Keywords: Supercritial fluids, RANS turbulence modeling, Liquid rocket engines

Thermodynamic properties of a model hydrocarbon ternary mixture in the vicinity of critical point: Measurements and modeling with crossover equation of state

In the framework of the proposed theoretical model crossover and scaling equations of state for near-critical multicomponent hydrocarbon mixture were obtained. To elaborate the equations of state the experimental data measured on the serial setup intended for comprehensive pressure-volume-temperature (PVT) research of a natural gas-condensate systems were used. The information concerning the mixture composition was not involved. The obtained equations describe the experimental data with the accuracy not exceeding the measurement inaccuracy. It has been shown that in extended range of thermodynamic parameters the crossover equation of state looks more reliable. The crossover model describes the transition from the scaling behavior to the mean-field behavior, which is typical for cubic and multiparameter equations of state. The model was extended for calculation of the volume fractions of gas and liquid phases needed for practical applications. In addition, the expression for isothermal compressibility has been derived. It was demonstrated that the isothermal compressibility has a cusp-like anomaly in the close vicinity of the critical point.

Isobaric Heat Capacity Measurements of Supercritical R1234yf

R1234yf (2,3,3,3-tetrafluoropropene) is a low global warming potential (GWP) replacement for R134a, a fluid commonly used as a refrigerant in domestic and automotive air conditioning systems and as a working fluid in organic Rankine cycles. The use of R1234yf in energy conversion cycles requires an accurate determination of its thermophysical properties, including the isobaric heat capacity (cP). Using a custom-made flow calorimeter, experimental cP measurements of supercritical R1234yf were made at temperatures T = 373.15 to 413.5 K and absolute pressures P = 3.5 to 10 MPa. The experimental apparatus was calibrated using R134a to increase the measurement accuracy. The heat capacity measurements of R1234yf were compared to available published experimental data and to values obtained from a multiparameter equation of state (EOS). The measured cP values agreed well with the EOS, providing an average absolute deviation (AAD) of 1.6%.

Thermodynamic Modeling with Equations of State: Present Challenges with Established Methods

Equations of state (EoS) are essential in the modeling of a wide range of industrial and natural processes. Desired qualities of EoS are accuracy, consistency, computational speed, robustness, and predictive ability outside of the domain where they have been fitted. In this work, we review present challenges associated with established models, and give suggestions on how to overcome them in the future. The most accurate EoS available, multiparameter EoS, have a second artificial Maxwell loop in the two-phase region that gives problems in phase-equilibrium calculations and excludes them from important applications such as treatment of interfacial phenomena with mass-based density functional theory. Suggestions are provided on how this can be improved. Cubic EoS are among the most computationally efficient EoS, but they often lack sufficient accuracy. We show that extended corresponding state EoS are capable of providing significantly more accurate single-phase predictions than cubic EoS with only a doublin...

The spinodal of single- and multi-component fluids and its role in the development of modern equations of state

The spinodal represents the limit of thermodynamic stability of a homogeneous fluid. In this work, we present a robust methodology to obtain the spinodal of multicomponent fluids described even with the most sophisticated equations of state (EoS) available. We elaborate how information about the spinodal and its uncertainty can contribute both in the development of modern EoS and to estimate their uncertainty in the metastable regions. Inequality constraints are presented that can be exploited in the fitting of modern EoS of single-component fluids to avoid inadmissible pseudo-stable states between the vapor and liquid spinodals. We find that even cubic EoS violate some of these constraints.With the use of a selection of EoS representative of modern applications, we compare vapor and liquid spinodal curves, superheat and supersaturation limits from classic nucleation theory (CNT), and available experimental data for the superheat limit. Computations are performed with pure species found in natural gas, binary mixtures, as well as a multi-component natural gas mixture in order to demonstrate the scalability of the approach. We demonstrate that there are large inconsistencies in predicted spinodals from a wide range of EoS such as cubic EoS, extended corresponding state EoS, SAFT and multiparameter EoS. The overall standard deviation in the prediction of the spinodal temperatures were 1.4 K and 2.7 K for single- and multi-component liquid-spinodals and 6.3 K and 26.9 K for single- and multi-component vapor spinodals.The relationship between the measurable limit of superheat, or supersaturation, and the theoretical concept of the spinodal is discussed. While nucleation rates from CNT can deviate orders of magnitude from experiments, we find that the limit of superheat from experiments agree within 1.0 K and 2.4 K with predictions from CNT for single- and multi-component fluids respectively. We demonstrate that a large part of the metastable domain of the phase diagram is currently unavailable to experiments, in particular for metastable vapor. Novel techniques, experimental or with computational simulations, should be developed to characterize the thermodynamic properties in these regions, and to identify the thermodynamic states that define the spinodal.

Helmholtz Energy Transformations of Common Cubic Equations of State for Use with Pure Fluids and Mixtures.

Comprehensive sets of derivatives of Helmholtz energy transformations of several common cubic equations of state are presented. These derivatives can be used for implementing cubic equations of state into complex multi-fluid mixture models when no multiparameter equation of state is available for a mixture component. Thus, pure fluids (or mixtures) for which no accurate model exists in literature can be modeled with a relatively small set of fluid property data. Composition derivatives have been calculated for the cases where the last mole fraction is either an independent or dependent variable. Analytic derivatives are presented up to fourth order in the independent variables combined with composition derivatives up to third order; this set covers the common requirements for derivatives needed in the state-of-the-art thermophysical property libraries. A C++ implementation of the presented analyses, data for computer code validation, and information about the computer algebra tools used to calculate the intermediate derivatives are provided as supplementary information.

Reference Quality Vapor–Liquid Equilibrium Data for the Binary Systems Methane + Ethane, + Propane, + Butane, and + 2-Methylpropane, at Temperatures from (203 to 273) K and Pressures to 9 MPa

A specialized cell designed for vapor liquid equilibrium (VLE) measurements at cryogenic temperatures and high pressures was used to measure new (p,T,x,y) data for binary mixtures of methane + ethane, + propane, + 2-methylpropane (isobutane), and + butane from (203 to 273) K at pressures up to 9 MPa. A literature review of VLE data for these binary mixtures indicates that a significant number have large uncertainties; however, because estimates of uncertainties in measured phase compositions are often not quantified sufficiently, it can be difficult for equation of state (EOS) developers to identify which data sets are of poor quality. Robust quantitative uncertainties were estimated for the VLE data acquired in this work, which allowed the identification of literature data sets that should not be included in future EOS development. The new data were compared with the predictions of the Peng–Robinson (PR) EOS and the Groupe European de Recherche Gaziere (GERG-2008) multiparameter EOS. The former describes the new data measured at low pressures within experimental uncertainty but deviates systematically from the data as the bubble point pressure is increased; for the binary mixtures containing either of the butanes, the maximum relative deviation of the data from the PR EOS amounted to nearly 10 % of the methane liquid mole fraction. The GERG-2008 EOS was better able to describe the new high-pressure data for the CH4 + C3H8 system than the PR EOS. However, for both the CH4 + C4H10 mixtures, the GERG EOS deviated from these data by an amount twice as large as the PR EOS because of ambiguity about which VLE literature data sets should be used in model development. The new data resolve these ambiguities and should facilitate the development of improved EOS as needed, for example, in simulations of low temperature natural gas separation processes.

Computational Fluid Dynamic Simulation of a Supercritical CO2 Compressor Performance Map

The performance map of a radial compressor operating with supercritical CO2 is computed by means of three-dimensional steady state Reynolds-averaged Navier–Stokes simulations. The geometry investigated is part of a 250 kW prototype which was tested at Sandia National Laboratories (SNL). An in-house fluid dynamic solver is coupled with a lookup table algorithm to evaluate the fluid properties. Tables are generated using a multiparameter equation of state, which ensures high accuracy in the fluid characterization. The compressor map is calculated considering three different rotational speeds (45 krpm, 50 krpm, and 55 krpm). For each speed-line, several mass flow rates are simulated. Numerical results are compared to experimental data from SNL to prove the potential of the methodology.

Vapor–liquid equilibria for the binary systems ethylene + water, ethylene + ethanol, and ethanol + water, and the ternary system ethylene + water + ethanol from Gibbs-ensemble molecular simulation

The conceptual design of a reactive separation process for the hydration of ethylene to ethanol requires reliable vapor–liquid equilibrium (VLE) data for the ternary system ethylene+water+ethanol. Due to the paucity of experimental data points in the VLE phase diagrams that have been reported for that system, molecular simulation looks appealing in order to predict such data. In this work, the Gibbs-ensemble Monte Carlo (GEMC) method was used to calculate the VLE of the pure components (ethylene, water, and ethanol), the binary subsystems (ethylene+water, ethylene+ethanol, and ethanol+water), and the ternary system (ethylene+water+ethanol). A set of previously validated Lennard-Jones plus point-charge potential models were chosen for the pure components, and the validity of these models was corroborated from the good agreement of the GEMC simulation results for the vapor pressure and the VLE phase diagrams of those components with respect to calculations carried out by means of the most accurate (reference) multiparameter equations of state currently available for ethylene, water, and ethanol. These potential models were found to be capable of predicting the available VLE phase diagrams of the binary subsystems: ethylene+water at 200 and 250°C, ethylene+ethanol at 150, 170, 190, 200, and 220°C, and ethanol+water at 200, 250, 275, and 300°C. Molecular simulation predictions for the VLE phase diagrams of the ternary system at 200°C and pressures of 30, 40, 50, 60, 80, and 100atm, were found to be in very good agreement with predictions previously made by use of a thermodynamic model that combines the Peng–Robinson–Stryjek–Vera equation of state, the Wong–Sandler mixing rules, and the UNIQUAC activity coefficient model. The agreement between the predictions of these two independent approaches gives confidence for the subsequent use of molecular simulation to predict the combined phase and chemical equilibrium of the ternary system and check the validity of predictions previously made by means of the thermodynamic model.

Equations of state for HFO-1234ze(E) and their application in the study on refrigeration cycle

This paper aims to describe the thermodynamic properties of the HFO-1234ze(E) with the molecular based BACKONE and PC-SAFT equations of state (EOS) that can be easily extended to mixtures. This paper also evaluates the accuracies of the BACKONE, PC-SAFT, and multi-parameter EOSs. For the data used in the construction of the EOS, the average absolute deviations (AAD) between experimental vapour pressures, experimental saturated liquid densities and values from the BACKONE EOS are 0.34% and 0.42%, respectively. The AAD between experimental vapour pressures, experimental saturated liquid densities and those from the PC-SAFT EOS are 0.16% and 1.51%, respectively. The AAD between predicted compressed liquid densities, predicted isobaric heat capacities, predicted pressures in gaseous phase from the BACKONE EOS and experimental data are 0.60%, 2.29%, and 0.99%, respectively. The AAD between predicted compressed liquid densities, predicted isobaric heat capacities, predicted pressures in gaseous phase from the PC-SAFT EOS and experimental data are 1.35%, 2.91%, and 1.84%, respectively. An independent investigation for data used in the construction of the multi-parameter EOS shows that the AAD between experimental vapour pressures, experimental saturated liquid densities, experimental densities in compressed liquid, experimental isobaric heat capacities, experimental pressures in gaseous phase and those from the multi-parameter EOS are 0.55%, 0.17%, 0.07%, 2.21%, and 0.11%, respectively. The BACKONE, PC-SAFT, and multi-parameter EOSs are used in the investigation of a refrigeration cycle using HFO-1234ze(E) as refrigerant for air-conditioning application. The study shows that the relative differences of all investigated characteristics calculated from thermodynamic properties from these EOSs are mostly within 1%. The thermodynamic properties of HFO-1234ze(E), HFO-1234yf, R-134a, R22, and R32 from the most accurate EOSs are used in the investigation of a refrigeration cycle. The results show that the cycle using R1234ze(E) as refrigerant has the highest coefficient of performance.

Evaluation of SPUNG* and Other Equations of State for Use in Carbon Capture and Storage Modelling

In this work, Equations of State (EoS) relevant for carbon capture and storage modelling have been evaluated for pure CO2 and CO2-mixtures with particular focus on the extended corresponding state approach, SPUNG/SRK. Our work continues the search of an EoS which is accurate, consistent and computationally fast for CO2-mixtures. These EoS have been evaluated: Soave-Redlich-Kwong (SRK), SRK with Peneloux shift, Peng-Robinson, Lee-Kesler, SPUNG/SRK and the multi-parameter approach GERG-2004. The EoS were compared to the accurate reference EoS by Span and Wagner for pure CO2. Only SPUNG/SRK and GERG-2004 predicted the density accurately near the critical point (< 1.5% Absolute Average Deviation (AAD)). For binary mixtures, Lee-Kesler and SPUNG/SRK had similar accuracy in density predictions. SRK had a sufficient accuracy for the gas phase below the critical point (<2.5%), and Peng Robinson had a decent accuracy for liquid mixtures (<3%). GERG-2004 was the most accurate EoS for all the single phase density predictions. It was also the best EoS for all the VLE predictions except for mixtures containing CO2 and O2, where it had deviations in the bubble point predictions (∼20% AAD). Even though multi-parameter EoS such as GERG-2004 are state-of-the-art for high accuracy predictions, this work shows that extended corresponding state EoS may be an excellent compromise between computational speed and accuracy. The SPUNG approach combines high accuracy with a versatile and transparent methodology. New experimental data may easily be taken into account to improve the predictive abilities in the two phase region. The approach may be improved and extended to enable applications for more difficult systems, such as polar mixtures with CO2 and H2O.