Thermodynamic Equation Of State Research Articles

Thermal conductivity prediction of pure refrigerants and mixtures based on entropy-scaling concept

The residual entropy scaling (RES) concept builds up connection between the transport properties and thermodynamic equation of state directly. However, the low-density divergence issue of the reduced dimensionless thermal conductivity might not be fixed properly through current literature models. This study proposes a new RES model coupled with the original perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) for the thermal conductivities of 11 typical refrigerants (including hydrocarbons, hydrofluorocarbons, hydrofluoroolefins and carbon dioxide) and their multicomponent mixtures. Model parameters are adjusted to saturated vapor and liquid data obtained from REFPROP. The combination of a modified reference thermal conductivity term as an extension of the original work by Rosenfeld et al. and the new formulation of the reduced term could provide reasonable predictive results for the investigated pure and mixture systems with AAD% of 3.95% and 5.76%, respectively. As expected, quasi-linear behaviors in both dilute and dense fluid regions could be observed for the refrigerant systems under study. The extended corresponding states (ECS) and the other two RES approaches reported in the literatures are used as reliable tools for assessing the predictive capability of the proposed model. However, the critical enhancement of thermal conductivity and the correction of the reference EoS especially near the critical point will be the priority in the future.

Numerical study of real gas effects during bubble collapse using a disequilibrium multiphase model

An explicit density-based solver of the Euler equations for inviscid and immiscible gas–liquid flow media is coupled with real-fluid thermodynamic equations of state supporting mild dissociation and calibrated with shock tube data up to 5000 K and 28 GPa. The present work expands the original 6-equation disequilibrium method by generalising the numerical approach required for estimating the equilibrium pressure in computational cells where both gas and liquid phases co-exist while enforcing energy conservation for all media. An iterative numerical procedure is suggested for taking into account the properties of the gas content as derived from highly non-linear real gas equations of state and implemented in a tabulated form during the numerical solution. The developed method is subsequently used to investigate gaseous bubble collapse cases considering both spherical and 2D asymmetric arrangements as induced by the presence of a rigid wall. It is demonstrated that the predicted maximum temperatures are strongly influenced by the equations of state used; the real gas model predicts a temperature reduction in the bubble interior up to 41% space-averaged and 50% locally during the collapse phase compared to the predictions obtained with the aid of the widely used ideal gas approximation.

Liquid-Phase Speed of Sound and Vapor-Phase Density of Difluoromethane.

Difluoromethane (HFC-32, DFM), with a global warming potential (GWP) of 677, is of interest as a pure refrigerant and as a component in low-GWP refrigerant mixtures. Additionally, difluoromethane has recently been identified as a safe, liquefied-gas electrolyte material in batteries. Using state-of-the-art instruments for measurements, this paper presents new liquid-phase speed of sound and vapor-phase density data for difluoromethane. Two hundred and nine liquid-phase speed of sound values were measured using a dual-path pulse-echo instrument at temperatures from 230 to 345 K and pressures from 2.1 to 70 MPa. Accounting for all sources of uncertainty, the relative expanded combined uncertainty (k = 2) in the speed of sound ranged from 0.035 to 0.17%. One hundred and thirty-eight vapour-phase density values were measured using a two-sinker densimeter at temperatures from 240 to 340 K and pressures from 0.1 to 1.61 MPa with an uncertainty of 0.011 to 0.12%. These experimental data will be valuable in the ongoing development of a new fundamental thermodynamic equation of state for difluoromethane.

The Hawking-Page-like phase transition from FRW spacetime to McVittie black hole

In this paper, we investigate the thermodynamics especially the Hawking-Page-like phase transition of the McVittie space-time. We formulate the first law of thermodynamics for the McVittie black hole, and find that the work density W of the perfect fluid plays the role of the thermodynamic pressure, i.e. P:=-W. We also construct the thermodynamic equation of state for the McVittie black hole. Most importantly, by analysing the Gibbs free energy, we find that the Hawking-Page-like phase transition from FRW spacetime to McVittie black hole is possible in the case P > 0.

Development of Helmholtz Equation of State for Thermodynamic Properties of R-1233zd(E)

The application of refrigerants from the Hydro Chloro Fluoro Olefins (HCFO) group, namely R-1233zd(E), in addition to the Hydro Fluoro Olefins (HFO), is a solution to environmental problems. A thermodynamic equation of state for R-1233zd(E), which can predict all thermodynamic properties over a wide-range fluid phase, is proposed in a function of the Helmholtz free energy. Three ancillary equations support the development and application. A genetic algorithm method combined with a weighted least squares regression was applied. The Helmholtz equation of state has an average absolute deviation of 0.12% for the liquid density, 2.0% for the vapor density, 1.4% for the vapor pressure, 0.16% for the saturated liquid density, 4.1% for the saturated vapor density, 3.6% for the isobaric specific heat, and 0.15% for the speed of sound in the liquid phase, and 0.18% for the speed of sound in the vapor phase. The deviation of the isochoric and saturated specific heats was not evaluated yet due to unavailable experimental data. The reasonable behavior of its extrapolation and ideal characteristic curves confirm its reliability. The Helmholtz equation of state can predict thermodynamic properties from the triple point to 1000 K and 100 MPa.

Effect of nonfactorizable background geometry on the thermodynamics of galaxy clusters

In this paper, we study the effect of nonfactorizable background geometry on the thermodynamics of the clustering of galaxies. A canonical partition function is derived for the gravitating system of galaxies treated as point particles contained in cells of appropriate dimensions. Various thermodynamic equations of state, like Helmholtz free energy and entropy, among others, are also obtained. We also estimate the effect of the corrected Newton’s law on the distribution function of galaxies. Remarkably, the effect of the modified Newton’s law is seen only in the clustering parameter while the standard structure of the equations is preserved. A comparison of the modified clustering parameter [Formula: see text] with that of the original clustering parameter is made to visualize the effect of the correction on the time scale of clustering. The possibility of system symmetry breaking is also analyzed by investigating the behavior of the specific heat with increasing system temperature.

Velocity modeling of supercritical pore fluids through porous media under reservoir conditions with applications for petroleum secondary migration and carbon sequestration plumes

Computational methods to characterize secondary migration in porous media traditionally rely on fluid transport equations with assumptions of time invariance, such as flow-path modeling of buoyancy vectors, statistical percolation algorithms, capillary pressure curves, or a form of Darcy’s law, which presumes instantaneous fluid transport. However, in petroleum systems modeling, the timeframe of secondary migration from source to reservoir is important to quantify in relation to other geologic factors, such as the timing of petroleum generation, fault movement, and seal formation. In addition, quantifying migration velocities enables an estimation of the distance a plume of geologically sequestered carbon dioxide travels over time, as well as the identification of low-permeability strata appropriate for long-term containment. This study introduces a method to quantify transport velocities of supercritical fluids in low-permeability lithologies for a broad range of rock and fluid properties likely encountered in the sedimentary sequence. A time-dependent form of Darcy’s law for pressure-driven viscous flow through homogeneous isotropic porous media was used to model flow velocities within a carrier bed. Thermodynamic equations of state were used to determine thermophysical properties of supercritical pore fluids under reservoir pressures ranging from 0 MPa to 200 MPa (0–29,000 psi) to constrain the momentum equations. Three case studies were examined that (1) estimated fluid flow velocities of methane within the low-permeability Upper Jurassic Haynesville Formation, (2) defined permeability-based flow units to evaluate saline formations for long-term geologic carbon sequestration, and (3) calculated the migration distance of carbon dioxide plumes at the Decatur, Illinois injection and sequestration project.

Extensions to the Navier–Stokes equations

Historically, the mass conservation and the classical Navier–Stokes equations were derived in the co-moving reference frame. It is shown that the mass conservation and Navier–Stokes equations are Galilean invariant—they are valid in any arbitrary inertial reference frame. From the mass conservation and Navier–Stokes equations, we can derive a wave equation, which contains the speed of pressure wave as its parameter. This parameter is independent of the speed of the source—the fluid element velocity. The speed of pressure wave is determined from the thermodynamic equation of state of the fluid, which is reference frame independent. It is well known that Lorentz transformation ensures wave speed invariant in all inertial frames, and the Lorentz invariance holds for different inertial observers. Based on these arguments, general Navier–Stokes equations (conservation law for the energy–momentum) can be written in any arbitrary inertial reference frame, they are transformed from one reference frame into another with the help of the Lorentz transformation. The key issue is that the Lorentz factor is parametrized by the local Mach number. In the instantaneous co-moving reference frame, these equations will degrade to the classical Navier–Stokes equations—the limit of the non-relativistic ones. These extended equations contain a square of the Lorentz factor. When the local Mach number is equal to one (the Lorentz factor approaches infinity), the extended Navier–Stokes equations will embody an intrinsic singularity, meaning that the transitions from the subsonic flow to the supersonic flow will happen. For the subsonic flow, the square of the Lorentz factor is positive, while for the supersonic flow, the square of the Lorentz factor becomes a negative number, which represents that the speed of sound cannot travel upstream faster than the flow velocity.

Studying the strengthening mechanism and thickness effect of elastomer coating on the ballistic-resistance of the polyurea-coated steel plate

Experiments have shown that the ballistic performance of the steel plate improves significantly after coating polyurea at the impact face. However, the strengthening mechanism of the elastomer coating on the ballistic-resistance is not understood well until now. In this paper, the strengthening mechanism and thickness effect of the coating under fragment simulating projectiles (FSP) impact are studied through numerical and experimental analysis. Firstly, a viscoplastic constitutive model of polyurea is developed based on experimental characterization and thermodynamic equation of state. It can capture strongly nonlinear volumetric behavior and impact-induced shearing stiffening and strengthening considering the coupling of high pressure, high temperature, and high strain rate under ballistic impact. Then, by applying this newly developed viscoplastic model of polyurea, the underlying ballistic-resistant mechanism of the polyurea-coated steel plate is numerically studied. The analysis shows that the dynamic behavior of polyurea under ballistic impact and propagating regulation of impact wave jointly play the key role. One side, the impact-induced stiffening and strengthening of the polyurea change the failure modes of the steel plate and considerably improve the ballistic-resistance of the structure. The other side, increasing coating thickness can only provide the limited benefit for improving the ballistic-resistance of the structure, as obtained in experiments. For the impact-face-coated steel plate, the low-thickness polyurea coating suffering from FSP impact can increase the frequency of the compressive wave reflection-superposition on the high-impedance polyurea/steel interface before the energy transfer from the impacted location. This helps to improve the pressure level and enhance the instantaneous energy storage and dissipation of the polyurea per volume significantly. However, increasing coating thickness can make the pressure level and the specific energy density of polyurea decrease gradually during penetration. It is also found that the energy dissipation of FSP during penetration is reduced after increasing the coating thickness because of the erosion mitigation from the structure. The new insight on the strengthening mechanism of elastomer coating will be helpful for the light armor design.

A simple approach to estimate the melting temperature of alkali halides

In the present study, a thermodynamical formulation is developed to determine the melting temperature of alkali halides with increasing pressure. The model is used to estimate the melting temperature of alkali halides at different pressures qualitatively. The formulation is obtained using the Goyal and Gupta thermodynamic equation of state. The model calculations require the computed values of volume compression, bulk modulus and its first pressure derivative at varying pressures. It is noted from model calculations that melting temperature increases with pressure but not in linear manner. The present computed results for pressure dependent melting temperature are compared with the available simulated results. The approach is found to be valid as good agreement is observed between previous and present results. Graphs are plotted to depict the variation of melting temperature with isothermal pressure, thermal pressure and total pressure acting on the solid. The present study helps to understand the impact of pressure on melting temperature of alkali halides qualitatively. The model used can also extrapolate the results upto higher pressures.

Quotient of the Euler system on one class of curves

We consider the Euler system describing a one-dimensional inviscid flows in space along curves of a certain class. Using differential invariants for the Euler system, we obtain its quotient equation. The solutions of the quotient equation that are constant along characteristic vector field provide some solutions of the Euler system. We discuss solving the quotient using asymptotic expansions of unknown functions and virial expansion of thermodynamic state equations. Thus the quotient is reduced to a series of ODE systems.

Intercomparison of different state-of-the-art formulations of the mass density of humid air

The differences between one classical and three state-of-the-art formulations of the mass density of humid air were quantified. Here, we present both the calculi for direct determination of the humid-air mass density employing the virial form of the thermodynamic equation of state, and a sufficiently accurate look-up-table for the quick-look determination of the humid-air mass density, which is based on the advanced Thermodynamic Equation of Seawater 2010.

Deterministic symbolic regression with derivative information: General methodology and application to equations of state

AbstractSymbolic regression methods simultaneously determine the model functional form and the regression parameter values by generating expression trees. Symbolic regression can capture the complexity of real‐world phenomena but the use of deterministic optimization for symbolic regression has been limited due to the complexity of the search space of existing formulations. We present a novel deterministic mixed‐integer nonlinear programming formulation for symbolic regression that incorporates derivative constraints through auxiliary expression trees. By applying the chain rule to mathematical operations, binary expression trees are capable of representing the calculation of first and second derivatives. We apply this formulation to illustrative examples using derivative information to show increased model discrimination capability. In addition, we perform a case study of a thermodynamic equation of state to gain insight on valid functional forms with thermodynamics‐based constraints on the first and second derivatives.

Experimental Determination of Coefficients for the Renner Model of the Thermodynamic Equation of State of the Poly(butylene succinate) and Wheat Bran Biocomposites.

This paper presents the assumptions of a thermodynamic equation of state for polymers according to the Renner model. The experiments involved extruding a biocomposite based on poly(butylene succinate) that was filled with ground wheat bran with its size not exceeding 200 μm. The biocomposite was produced in pellet form with three different contents by weight of wheat bran, i.e., 10%, 30% and 50%. All specimens were examined for their thermodynamic p-v-T characteristics. Taking advantage of the SimFit module of Cadmould 3D-F, experimental results were used to determine the coefficients of thermodynamic equation of state for the tested biocomposite according to the Renner model. The coefficients were then used to calculate transition temperature and to create diagrams illustrating the relationship between pressure, temperature and specific volume for the tested biocomposite. The obtained results can serve as a basis for assessing the suitability of the biocomposite for injection molding, selecting technological parameters of this process, as well as for analyzing shrinkage and defects of injection-molded parts.

Effective field theory for hydrodynamics without boosts

We formulate the Schwinger-Keldysh effective field theory of hydrodynamics without boost symmetry. This includes a spacetime covariant formulation of classical hydrodynamics without boosts with an additional conserved particle/charge current coupled to Aristotelian background sources. We find that, up to first order in derivatives, the theory is characterised by the thermodynamic equation of state and a total of 29 independent transport coefficients, in particular, 3 hydrostatic, 9 non-hydrostatic non-dissipative, and 17 dissipative. Furthermore, we study the spectrum of linearised fluctuations around anisotropic equilibrium states with non-vanishing fluid velocity. This analysis reveals a pair of sound modes that propagate at different speeds along and opposite to the fluid flow, one charge diffusion mode, and two distinct shear modes along and perpendicular to the fluid velocity. We present these results in a new hydrodynamic frame that is linearly stable irrespective of the boost symmetry in place. This provides a unified covariant stable approach for simultaneously treating Lorentzian, Galilean, and Lifshitz fluids within an effective field theory framework and sets the stage for future studies of non-relativistic intertwined patterns of symmetry breaking.

Zitterbewegung particles tunneling from Reissner-Nordström AdS black hole surrounded by quintessence

We study quantum gravity corrections to the thermodynamic quantities of Reissner-Nordström anti-de Sitter black hole surrounded by quintessence by evaluating the Hawking radiation of zitterbewegung particles under the Generalized Uncertainty Principle (GUP) effect. By using the modified Hawking temperature of the black hole, we derive the thermodynamic equation of state and the Helmholtz free energy. We discuss global stability conditions and phase transitions by plotting the and planes. We also assume that the black hole is a heat engine where the Carnot cycle is performed. We observe that the efficiency under the GUP effect is higher than the standard one when attractive Coulomb interaction is considered with the influence of zitterbewegung particles.

Local Shape of the Vapor–Liquid Critical Point on the Thermodynamic Surface and the van der Waals Equation of State

Differential geometry is a powerful tool to analyze the vapor–liquid critical point on the surface of the thermodynamic equation of state. The existence of usual condition of the critical point (∂p/∂V)T=0 requires the isothermal process, but the universality of the critical point is its independence of whatever process is taken, and so we can assume (∂p/∂T)V=0. The distinction between the critical point and other points on the surface leads us to further assume that the critical point is geometrically represented by zero Gaussian curvature. A slight extension of the van der Waals equation of state is to letting the two parameters a and b in it vary with temperature, which then satisfies both assumptions and reproduces its usual form when the temperature is approximately the critical one.

Возможности измерения локальной плотности в зондирующих приборах

In modern expeditionary research carried out on research vessels, the local density of seawater is not measured, but calculated using the TEOS-10 thermodynamic equation of state based on cumulative measurements of pressure, temperature and electrical conductivity or speed of sound. The equation of state with satisfactory accuracy is valid only for oceanological waters, and is used with corrections for the waters of the marginal and inland seas. These corrections are permanently refined and will be refined in the future. It is desirable to have direct measurements of the local density of any waters directly in the environment with sufficient accuracy, which is now 4·10-6. Laboratory measurements made by various methods and with different devices, not always automated, give accuracy ~ . Attempts to automate these measurements and introduce them into sounding devices have been made several times. Experimental samples of vibration, refractometric and hydrostatic devices have been created and tested, none of which have been introduced into the practice of expeditionary work for various reasons. However, scientific and technological progress makes possible technical solutions previously difficult to implement. This also applies to modifications of the hydrostatic method for measuring the local density, which have recently attracted the interest of developers. The paper analyzes the possibilities of implementing the hydrostatic method using multi-element resistor distributed pressure and temperature sensors, resistance profilers of these sensors and determining the spatial pressure drop on a given base by measuring and subtracting sensor resistances, taking into account temperature correction. The use of three such sensors, mounted on three rods, oriented downward along the axes of a rectangular coordinate system, provides the possibility of measuring the local density with an arbitrary orientation of the probe relative to the vertical. The analysis shows the possibility of measuring the local density of seawater by the hydrostatic method with the required accuracy in probe instruments.

Analysis of p, T, ρ-data for all fluid states via comparison with a transcribed Van der Waals equation

The old concepts of the Van der Waals equation (vdW EOS) and of the virial equation, truncated to the second virial coefficient, can be rewritten with a function defined as cc=(p−R·T/M)/ρ2. This function arises from introducing the ideal gas law into the thermodynamic equation of state, which describes the volume dependence of the internal energy. Thus both concepts can be superimposed and then complement one another. This allows an almost complete and correct description of the p/T/ρ-behavior of real fluids in the gaseous, liquid, critical and supercritical state up to the Joule inversion curve. The function cc, used as structure parameter with constant values, reveals a hidden feature of the vdW EOS, which describes a linear function between density and temperature for iso-cc-lines. This is empirically correct for 16 examined regular pure fluids. Non-regular fluids like He, H2, water and acetonitrile, which show more or less slightly curved iso-cc-lines, can be linearized using a simple factor, applied to the density, and then be described in a similar manner like regular fluids.

Equation of State Measurements on Iron Near the Melting Curve at Planetary Core Conditions by Shock and Ramp Compressions

AbstractThe outer core of the Earth is composed primarily of liquid iron, and the inner core boundary is governed by the intersection of the melt line and the geotherm. While there are many studies on the thermodynamic equation of state for solid iron, the equation of state of liquid iron is relatively unexplored. We use dynamic compression to diagnose the high‐pressure liquid equation of state of iron by utilizing the shock‐ramp capability at Sandia National Laboratories’ Z‐Machine. This technique enables measurements of material states off the Hugoniot by initially shocking samples and subsequently driving a further, shockless compression. Planetary studies benefit greatly from isentropic, off‐Hugoniot experiments since they can cover pressure‐temperature (P‐T) conditions that are close to adiabatic profiles found in planetary interiors. We used this method to drive iron to P‐T conditions similar to those of the Earth’s outer‐inner core boundary, along an elevated‐temperature isentrope in the liquid from 275 GPa to 400 GPa. We derive the equation of state using a hybrid backward integration – forward Lagrangian technique on particle velocity traces to determine the pressure‐density history of the sample. Our results are in excellent agreement with SESAME 92141, a previously published equation of state table. With our data and previous experimental data on liquid iron we provide new information on the iron melting line and derive new parameters for a Vinet‐based equation of state. The table and our parameterized equation of state are applied to provide an updated means of modeling the pressure, mass, and density of liquid iron cores in exoplanetary interiors.