Simple Equation Of State Research Articles

Near-absolute equations of state of diamond, Ag, Al, Au, Cu, Mo, Nb, Pt, Ta, and W for quasi-hydrostatic conditions

Using the modified formalism of [Dorogokupets, Oganov, 2005, 2007], equations of state are developed for diamond, Ag, Al, Au, Cu, Mo, Nb, Pt, Ta, and W by simultaneous optimization of shock-wave data, ultrasonic, X-ray, dilatometric and thermochemical measurements in the temperature range from ~100 K to the melting temperature and pressures up to several Mbar, depending on the substance. The room-temperature isotherm is given in two forms: (1) the equation from [Holzapfel, 2001, 2010] which is the interpolation between the low pressure (x≥1) and the pressure at infinite compression (x=0); it corresponds to the Thomas-Fermi model, and (2) the equation from [Vinet et al., 1987]. The volume dependence of the Grüneisen parameter is calculated according to equations from [Zharkov, Kalinin, 1971; Burakovsky, Preston, 2004] with adjustable parameters, t and δ. The room-temperature isotherm and the pressure on the Hugoniot adiabat are determined by three parameters, K', t and δ, and K0 is calculated from ultrasonic measurements. In our study, reasonably accurate descriptions of all of the basic thermodynamic functions of metals are derived from a simple equation of state with a minimal set of adjustable parameters. The pressure calculated from room-temperature isotherms can be correlated with a shift of the ruby R1 line. Simultaneous measurements of the shift and unit cell parameters of metals are conducted in mediums containing helium [Dewaele et al., 2004b; 2008; Takemura, Dewaele, 2008; Takemura, Singh, 2006], hydrogen [Chijioke et al., 2005] and argon [Tang et al., 2010]. According to [Takemura, 2001], the helium medium in diamond anvil cells provides for quasi-hydrostatic conditions; therefore, the ruby pressure scale, that is calibrated for the ten substances, can be considered close to equilibrium or almost absolute. The ruby pressure scale is given as P(GPa)=1870⋅Δλ/λ0⋅(1+6⋅Δλ/λ0). The room-temperature isotherms corrected with regard to the ruby scale can also be considered close to equilibrium or almost absolute. Therefore, the equations of state of the nine metals and diamond, which are developed in our study, can be viewed as almost absolute equations of state for the quasi-hydrostatic conditions. In other words, these equations agree with each other, with the ruby pressure scale, and they are close to equilibrium in terms of thermodynamics. The PVT relations derived from these equations can be used as mutually agreed pressure scales for diamond anvil cells in studies of PVT properties of minerals in a wide range of temperatures and pressures. The error of the recommended equations of the state of substances and the ruby pressure scale is about 2 or 3 per cent. Calculated PVT relations and thermodynamics data are available at http://labpet.crust.irk.ru.

Applicability of equations of state for modeling helium systems

Proper design of helium systems with large number of components and involved configurations such as helium liquefiers/refrigerators requires the use of tools like process simulators. The accuracy of the simulation results, to a great extent, depends on the accuracy of property data. For computation of thermodynamic properties of helium, the 32-parameter MBWR equation of state proposed by McCarty and Arp [1] is widely used. However, it is computationally involved, makes the simulation process more time-consuming and sometimes leads to computational difficulties such as numerical oscillations, divergence in solution especially, when the process operates over a wide thermodynamic region and is constituted of many components. Substituting MBWR EOS by simpler equations of state (EOS(s)) at selected thermodynamic planes, where the simpler EOS(s) have the similar accuracy as that of MBWR EOS may enhance ease of computation. In the present paper, the methodology to implement this concept has been elucidated with examples of steady state and dynamic simulation of helium liquefier/refrigerator based on Collins cycle. The above concept can be applied to thermodynamic analysis of other process cycles where computation of fluid property is involved.

Shape of a Ponytail and the Statistical Physics of Hair Fiber Bundles

A general continuum theory for the distribution of hairs in a bundle is developed, treating individual fibers as elastic filaments with random intrinsic curvatures. Applying this formalism to the iconic problem of the ponytail, the combined effects of bending elasticity, gravity, and orientational disorder are recast as a differential equation for the envelope of the bundle, in which the compressibility enters through an "equation of state." From this, we identify the balance of forces in various regions of the ponytail, extract a remarkably simple equation of state from laboratory measurements of human ponytails, and relate the pressure to the measured random curvatures of individual hairs.

A comprehensive approach to an equation of state for hard spheres and Lennard Jones fluids

We present a simple method of obtaining various equations of state for hard sphere fluid in a simple unifying way. We will guess equations of state by using suitable axiomatic functional forms (n = 1, 2, 3, 4, 5) for surface tension Smn(r), r ≥ d/2 with intermolecular separation r as a variable, where m is an arbitrary real number (pole). Among the equations of state obtained in this way are Percus—Yevick, scaled particle theory and Carnahan—Starling equations of state. In addition, we have found a simple equation of state for the hard sphere fluid in the region that represents the simulation data accurately. It is found that for both hard sphere fluids as well as Lennard—Jones fluids, with m = 3/4 the derived equation of state (EOS) gives results which are in good agreement with computer simulation results. Furthermore, this equation of state gives the Percus—Yevick (pressure) EOS for the m = 0, the Carnahan—Starling EOS for m = 4/5, while for the value of m = 1 it corresponds to a scaled particle theory EOS.

Perturbation approach for equation of state for hard-sphere and Lennard–Jones pure fluids

In this paper we have established the equation of state (EOS) for liquids. The EOS was established for hard-sphere (HS) fluid along with Lennard–Jones (LJ) fluid incorporating perturbation techniques. The calculations are based on suitable axiomatic functional forms for surface tension S m (r), r ≥ d/2 with intermolecular separation r, as a variable, and m is an arbitrary real number (pole). The results for βP/ρ from the present EOS thus obtained are compared with Percus-Yevick (PY), scaled particle theory (SPT), and Carnahan–Starling (CS). In addition, we have found a simple EOS for the HS fluid in the region which represents the simulation data accurately.

Density calculation of liquid organic compounds using a simple equation of state up to high pressures

Experimental densities of three groups of liquid organic substances (acids, esters, alcohols) have been correlated using Goharshadi–Morsali–Abbaspour (GMA) equation of state and then the values calculated from the equation of state have been compared with the experimental data. The paper reports new correlation for the density of 20 organic liquids (7 acids, 7 esters and 6 alcohols) at temperatures between 293.15 K and 393.15 K and pressures between 0.1 MPa and 35 MPa. A comparison with experimental data in the specified range of temperature from low to high pressures has been made. Some generalized correlations are also used for comparison with GMA equation of state and experimental data. The results show that the equation of state reproduces the experimental PρT data of liquid organic compounds with good accuracy. The excellent agreement with experimental data indicates that this equation of state can be used to calculate the density of liquid organic compounds with a high degree of certainty. The comparison with other correlations shows that the GMA equation of state is better to some extent and reliable in the given temperature and pressure range.

Pressure–volume–temperature measurements of phosphonium-based ionic liquids and analysis with simple equations of state

In spite of the great importance of the ( P, V, T) data of phosphonium–based ionic liquids, only limited information on these data seems to be available in the open literature. In this work, we present the results for the density measurements of the trihexyltetradecylphosphonium chloride, [(C 6H 13) 3P(C 14H 29)][Cl] and trihexyltetradecylphosphonium dicyanamide, [(C 6H 13) 3P(C 14H 29)][N(CN 2)] with an estimated uncertainty of ±0.5 kg · m −3. The ranges of temperature and pressure are T = (273.15 to 318.15) K and p = (0.1 to 25) MPa for [(C 6H 13) 3P(C 14H 29)][Cl] and T = (273.15 to 318.15) K and p = (0.1 to 35) MPa for [(C 6H 13) 3P(C 14H 29)][N(CN 2)]. The high consistency of our data for [(C 6H 13) 3P(C 14H 29)][Cl] compared with those measured by other authors allowed all the experimental data for this IL to be combined and correlated using the Goharshadi–Morsali–Abbaspour equation of state over a wide range of temperature and pressure. From this equation, thermomechanical coefficients as the isothermal compressibility, thermal expansivity, thermal pressure, and internal pressure were calculated for the two ILs. The Sanchez–Lacombe equation of state was used also for ( P, V, T) correlation and the estimation of the free volume in these phosphonium ionic liquids. Finally ionic volumes for trihexyltetradecylphosphonium cation and several anions available in the literature made possible the calculation of the free (hole) volume.

Linking Trehalose Self-Association with Binary Aqueous Solution Equation of State

In aqueous solutions, trehalose possesses a high propensity to form hydrogen bonds with water as well as other trehalose molecules. This hydrogen bonding not only affects water structure but also promotes extensive concentration dependent aggregation of trehalose molecules, which may impact trehalose's role as a protective cosolute to biomacromolecules. To study the association of trehalose in aqueous solutions over a wide concentration range, we used molecular dynamics simulations based on two different force fields, as well as vapor pressure osmometry. By analyzing trehalose cluster size and fractal dimension in simulations, we estimate the cluster percolation threshold at 1.5-2.2 m. Experimentally, trehalose solutions showed positive deviations from ideal van't Hoff's law that grew with concentration. These variations in osmotic pressure can be explained using a simple equation of state that accounts for the repulsive excluded volume interactions between trehalose molecules as well as attractions reflected in sugar clustering. We find that simulations with both applied force fields result in reasonable representations of the solution equation of state. However, in contrast to experiments, the balance between the repulsive and attractive trehalose-trehalose interactions in simulations results in a slightly negative deviation from ideality, probably due to the moderately overaggregative nature of the force fields used.

Numerical study of unsteady turbulent cavitating flows

The simulation of cavitating flows is a challenging problem both in terms of modelling the physics and developing robust numerical methodologies. Such flows are characterized by important variations of the local Mach number, compressibility effects on turbulence and involve thermodynamic phase transition. To simulate these flows by applying homogeneous models and Reynolds averaged codes, the turbulence modelling plays a major role in the capture of unsteady behaviours. This paper presents a one-fluid compressible Reynolds-Averaged Navier–Stokes (RANS) solver with a simple equation of state (EOS) for the mixture. A special focus is devoted to the turbulence model influence. Unsteady numerical results are given for Venturi geometries and comparisons are made with experimental data.

Phenomenological Method for Construction of the Liquid and Gas Equation of State

The semiempirical method was suggested for construction of a simple equation of state for liquid and gas. With this purpose, some new conditions and restrictions, which should be satisfied by the “right” equation of state, were formulated. In particular, the preferable structure of the equation of state was determined. The equations for description of thermodynamic properties of real gases and liquid phase were derived with involvement of additional restrictions. The second virial coefficient calculated on the basis of the modified step potential of intermolecular interactions was used as the temperature functions in the derived equations, and the density functions were determined from analysis of experimental data. A new equation of state for liquid and gas with six individual adjustable parameters was suggested. This new equation describes the thermodynamic properties of substances much better than the known Benedict−Webb−Rubin equation of state. Some caloric properties of substances were calculated.

Nonlinear waves in a quark gluon plasma

Recent measurements at RHIC suggest that a nearly perfect fluid of quarks and gluons is produced in AA collisions. Moreover the passage of supersonic partons through this medium seems to produce waves. These waves might pile up and form Mach cones, which would manifest themselves in the so called away-side jets, forming a broad structure in the angular distribution of the particles recoiling against a trigger jet of moderate energy. In most of the theoretical descriptions of these phenomena, the hydrodynamic equations are linearized for simplicity. We propose an alternative explanation for the observed broadening of the away-side peak. It is based on hydrodynamics but it is a consequence of the non-linearities of the equations, which instead of simple waves may lead to localized waves or even solitons. We investigate in detail the consequences of including the non-linear terms. We use a simple equation of state for the QGP and expand the hydrodynamic equations around equilibrium configurations. The resulting differential equations describe the propagation of perturbations in the energy density. We solve them numerically and find that localized perturbations can propagate for long distances in the plasma. Under certain conditions our solutions mimick the propagation of Korteweg - de Vries solitons.

An equation of state for condensed matter; application to solid chemical elements

A particularly simple equation of state derived from the definitions of the compressibility k and the volume thermal expansion β of homogeneous condensed phases is applied to 100 solid elemental species, which are found to show two types of colligative characteristics: typical elements whose compressibility and expansivity depend on the packing coefficient of the crystal structures and isovalent elements whose values of k and β depend on the valence of the element in the solid state.

Breaking waves in a quark gluon plasma

Using a simple equation of state for the quark gluon plasma (QGP) we expand the hydrodynamic equations around equilibrium configurations. The resulting differential equations describe the propagation of perturbations in the energy density. We solve them numerically and find that localized perturbations can propagate for long distances in the plasma. Under certain conditions our solutions mimick the propagation of Korteweg–de Vries solitons.

Experimental and theoretical study of penetration of spheroconincal bodies into a sand target

The results of direct and reverse normal impact laboratory experiments performed to study spheroconical bodies with barriers of sand filled in metal cylindrical containers are given. The data are compared with the numerical simulation results obtained based on the modified Godunov difference method. S.S. Grigoryan's model with relatively simple equations of state is used as a model of the dynamic behavior of sandy media. The model parameters are based on the results of the plane-wave experiments and the experiments with a modified Kolsky method. The behavior of forces and coefficients of resistance to penetration at different phases of impact interaction is analyzed.

Numerical simulations of interfaces in relativistic hydrodynamics

We consider models of relativistic matter containing sharp interfaces across which the matter model changes. These models will be relevant for neutron stars with crusts, phase transitions or for viscous boundaries where the length scale is too short to be modelled smoothly. In particular we look at the numerical techniques that allow us to evolve stable interfaces, for the interfaces to merge, and for strong waves and shocks to interact with the interfaces. We test these techniques for ideal hydrodynamics in special and general relativity for simple equations of state, finding that simple level set-based methods extend well to relativistic hydrodynamics.

Erratum to "On the computation of the fundamental derivative of gas dynamics using equations of state"

The value of the fundamental derivative of gas dynamics, Γ, is a quantitative measure of the variation of the speed of sound with respect to density in isentropic transformations, such as those occurring, for example, in gas-dynamic nozzles. The accurate computation of its value, which is a constant for a perfect gas, is key to the understanding of real-gas flows occurring in a thermodynamic region where the polytropic ideal gas law does not hold. The fundamental derivative of gas dynamics is a secondary thermodynamic property and so far, no experiments have been conducted with the aim of measuring its value. Several studies document the estimation of Γ for fluids composed of complex molecules using mainly simple thermodynamic equations of state, e.g., that of Van der Waals. A review of these studies has revealed that the calculated values of Γ are affected by large uncertainties; these uncertainties are due to the functional form of the adopted equations and because of uncertainties in the available fluid property data on which these equations were fitted. In this work, the fundamental derivative of gas dynamics of molecularly simple fluids is computed with the aid of, among other models, modern reference equations of state. The accuracy of these computations has been assessed. Reference thermodynamic models however, are not available for molecularly complex fluids; some of these molecularly complex fluids are the substances of interest in studies on the so-called nonclassical gas dynamics. Therefore, results of the computation of Γ for few, molecularly simple hydrocarbons, like methane, ethane, etc., are used as a benchmark against which the performance of simpler equations of state, can be assessed. For the selected substances, the Peng-Robinson, Stryjek-Vera modified, cubic equation of state yields good results for Γ-predictions, while the modern multiparameter technical equations of state, e.g., the one in the Span-Wagner functional form, are preferable, provided that enough accurate thermodynamic data are available. Another notable result of this study, is that Γ for a fluid composed of complex molecules is less affected by the inaccuracy of C 1 ,-information (C 1 , is the isochoric heat capacity), if compared to the estimation of Γ for simple molecules. Inspection of the results of the calculation of Γ in the proximity of the critical point confirms that analytical equations of state fail to predict the correct physical behavior, even if they include terms which allow for the correct estimation of thermodynamic properties.

Phase diagram of hard colloidal platelets: a theoretical account

We construct the complete liquid crystal phase diagram of hard plate-like cylinders for variable aspect ratio using Onsager's second virial theory and employing the Parsons–Lee decoupling approximation to account for higher-body interactions in the isotropic and nematic fluid phases. The stability of the solid (columnar) state at high packing fraction is included by invoking a simple equation of state based on a Lennard–Jones–Devonshire cell model which has proven to be quantitatively reliable over a large range of packing fractions. By employing an asymptotic analysis based on the Gaussian approximation we are able to show that the nematic–columnar transition is universal and independent of particle shape. The predicted phase diagram is in qualitative agreement with simulation results.

On measuring the strength of metals at ultrahigh strain rates

The strain rate sensitivity of materials is normally measured through a combination of quasistatic, Hopkinson bar, and pressure-shear experiments. Recent advances in uniaxial strain ramp loading provide a new means to reach strain rates significantly higher than achievable in pressure-shear experiments. One way to determine strength in ramp loading is by comparing the uniaxial stress-strain response to an appropriate pressure-density response obtained from an equation of state for the material. Using this approach, strengths for aluminum are obtained for strain rates of 105–108 s−1. Two issues arise in this calculation: heating due to plastic work and the effect of the superimposed hydrostatic stress on the strength. Heating due to plastic work is calculated and accounted for within the context of the equation of state for the material in a straightforward manner, but neglecting this heating can lead to significant errors in the calculated strength at higher compression levels. A simple scaling of strength with the pressure-dependent shear modulus is utilized to estimate the strength at zero pressure for ramp loading and pressure-shear experiments. When examined in this manner, the strain rate dependence of aluminum is found to be less than previously reported, with little increase in strength below strain rates of about 107s−1. The effects on ramp loading strength measurements of heating due to plastic work and of hydrostatic pressure are also examined for copper and tantalum using simple equation of state and strength models. The effect of plastic heating is similar for the three materials for a given strain level but quite different for a constant stress, with aluminum showing greater effects than the other materials. The effect of hydrostatic pressure in ramp loading experiments is similar for all three materials, but the effect is likely to be much greater in pressure-shear experiments for aluminum than the other materials.

Thermodynamic properties of the mixtures of some ionic liquids with alcohols using a simple equation of state

In the present work, we have used a simple equation of state called the GMA EoS to calculate the density of three ionic liquid mixtures including 1-butyl-3-methylimidazolum hexafluorophosphate, [BMIM] [PF 6] + methanol, 1-butyl-3-methylimidazolum tetrafluoroborate, [BMIM] [BF 4] + methanol, and [BMIM] [BF 4] + ethanol at different temperatures, pressures, and compositions. The isothermal compressibility, excess molar volumes, and excess Gibbs molar energy of these mixtures have been computed using this equation of state. The values of statistical parameters show that the GMA EoS can predict these thermodynamic properties very well within the experimental errors. The results show that isothermal compressibility of ionic liquids is lower than alcohols and the effect of temperature and pressure on the isothermal compressibility of ionic liquids is lower than alcohols. The excess molar volumes and excess molar Gibbs energy for these ionic liquid mixtures with alcohols are all negative at various temperatures and pressures over the whole composition range. The results have been interpreted in terms of intermolecular interactions and structural factors of the ionic liquids and alcohols.

Simple equation of state for hard disks on the hyperbolic plane

A simple equation of state for hard disks on the hyperbolic plane is proposed. It yields the exact second virial coefficient and contains a pole at the highest possible packing. A comparison with another very recent theoretical proposal and simulation data is presented.