Der Waals Equation Of State Research Articles

A notable difference between ideal gas and infinite molar volume limit of van der Waals gas

The van der Waals equation of state does not sufficiently represent a gas unless a thermodynamic potential with two proper and independent variables is simultaneously determined. The limiting procedures under which the behaviour of the van der Waals gas approaches that of an ideal gas are letting two van der Waals coefficients be zero rather than letting the molar volume become infinitely large; otherwise, the partial derivative of internal energy with respect to pressure at a fixed temperature does not vanish.

Daniel Berthelot. Part I. Contribution to thermodynamics

Daniel Berthelot (1865-1927) did fundamental research on the conductivity of electrolyte solutions, thermodynamics, pyrometry, and photochemistry. In thermodynamics he proved that the limiting density of gases can be used to determine their atomic and molecular weights; he provided the first proof that the principle of corresponding states requires a third parameter and discovered several thermodynamic loci that could be used for this purpose. He improved the van der Waals equation of state by making the term a/v2 a function of the temperature and developed a new equation of state that may be considered the basis of the Redlich-Kwong equation of state.

A diffuse-interface approach to two-phase isothermal flow of a Van der Waals fluid near the critical point

A novel numerical method for simulations of isothermal, compressible two-phase flows of one fluid component near the critical point is presented on the basis of a diffuse-interface model and a Van der Waals equation of state. Because of the non-convexity of the latter, the nature of the set of governing equations is mixed hyperbolic–elliptic. This prevents the application of standard numerical methods for compressible flow. Moreover, the Korteweg capillary stress tensor, characteristic for the diffuse-interface approach, introduces third-order spatial derivatives of mass density in the Navier–Stokes equation, resulting in a dispersive behavior of the solution. Our computational method relies on a transformation of the conserved variables, which controls dispersion, stabilizes the numerical simulation and enables the use of coarser grids. A one-dimensional simulation shows that this method provides better stability and accuracy than without transformation of variables. Two- and three-dimensional simulations for isothermal liquid–vapor flows, in particular the retraction of a liquid non-spherical drop in vapor and the binary droplet collision in vapor, show the applicability of the method. The surface tension calculated from the numerical results is in good agreement with its theoretical value if the computational grid is sufficiently fine.

Generic van der Waals equation of state for polymers, modified free volume theory, and the self-diffusion coefficient of polymeric liquids

In this paper, a molecular theory of self-diffusion coefficient is developed for polymeric liquids (melts) on the basis of the integral equation theory for site–site pair correlation functions, the generic van der Waals equation of state, and the modified free volume theory of diffusion. The integral equations supply the pair correlation functions necessary for the generic van der Waals equation of state, which in turn makes it possible to calculate the self-diffusion coefficient on the basis of the modified free volume theory of diffusion. A random distribution is assumed for minimum free volumes for monomers along the chain in the melt. More specifically, a stretched exponential is taken for the distribution function. If the exponents of the distribution function for minimum free volumes for monomers are chosen suitably for linear polymer melts of N monomers, the N dependence of the self-diffusion coefficient is N − 1 for the small values of N , an exponent predicted by the Rouse theory, whereas in the range of 2.3 ≲ ln N ≲ 4.5 the N dependence smoothly crosses over to N − 2 , which is reminiscent of the exponent by the reptation theory. However, for ln N ≳ 4.5 the N dependence of the self-diffusion coefficient differs from N − 2 , but gives an N dependence, N − 2 − δ ( 0 < δ < 1 ) , consistent with experiment on polymer melts in the range. For polyethylene δ ≈ 0.48 for the parameters chosen for the stretched exponential. Because the stretched exponential function contains undetermined parameters, the N dependence of diffusion becomes semiempirical, but once the parameters are chosen such that the N dependence of D can be successfully given for a polymer melt, the temperature dependence of the self-diffusion coefficient can be well predicted in comparison with experiment. The theory is satisfactorily tested against experimental and simulation data on the temperature dependence of D for polyethylene and polystyrene melts.

1-D Van der Waals foams heated by ion beam energy deposition

One dimensional simulations of various initial average density aluminum foams (modeled as slabs of solid metal separated by low density regions) heated by volumetric energy deposition are conducted with a Lagrangian hydrodynamics code using a Van der Waals equation of state (EOS). The resulting behavior is studied to facilitate the design of future warm dense matter (WDM) experiments at LBNL. In the simulations the energy deposition ranges from 10 to 30 kJ/g and from 0.075 to 4.0 ns total pulse length, resulting in temperatures from approximately 1–4 eV. We study peak pressures and temperatures in the foams, expansion velocity, and the phase evolution. Five relevant time scales in the problem are identified. Additionally, we present a method for characterizing the level of inhomogeneity in a foam target as it is heated and the time it takes for a foam to homogenize.

Finite Size Effects in Equations of State under Nontrivial Boundary Conditions

We study free particles in a one-dimensional box with combinations of two types of boundary conditions: the Dirichlet condition and a one-parameter family of quasi-Neumann conditions at the two walls. We calculate energy spectra approximately and obtain equations of state having the same (one-dimensional) volume dependence as van der Waals equations of state. The dependence of the equations of state is examined for particles obeying Maxwell-Boltzmann, Bose-Einstein, or Fermi-Dirac statistics. Our results suggest that the deviation from ideal gas may also be realized as finite size effects due to the interaction between the particles and the walls.

Cálculo do volume na equação de van der Waals pelo método de cardano

Analytical solutions of a cubic equation with real coefficients are established using the Cardano method. The method is first applied to simple third order equation. Calculation of volume in the van der Waals equation of state is afterwards established. These results are exemplified to calculate the volumes below and above critical temperatures. Analytical and numerical values for the compressibility factor are presented as a function of the pressure. As a final example, coexistence volumes in the liquid-vapor equilibrium are calculated. The Cardano approach is very simple to apply, requiring only elementary operations, indicating an attractive method to be used in teaching elementary thermodynamics.

ASYMPTOTICAL SOLUTIONS FOR A VIBRATIONALLY RELAXING GAS

Using the weakly non‐linear geometrical acoustics theory, we obtain the small amplitude high frequency asymptotic solution to the basic equations governing one dimensional unsteady planar, spherically and cylindrically symmetric flow in a vibrationally relaxing gas with Van der Waals equation of state. The transport equations for the amplitudes of resonantly interacting waves are derived. The evolutionary behavior of non‐resonant wave modes culminating into shock waves is also studied.

A study on the correlations of supercritical temperature of fluid alkali metals based on the generalised van der Waals equation of state

The supercritical temperature, the inversion temperature and the critical compressibility factor of caesium, rubidium and potassium are correlated through the generalised van der Waals equation of state. This three-parameter equation differs from the known van der Waals equation of state by the introduction of a third parameter in the expression for molecular pressure. For caesium, rubidium and potassium, the ratios of the supercritical temperature to the inversion temperature are estimated. The results of estimation are in agreement with the results from other equations which are based on experimental data. It has been established that caesium, rubidium and potassium obey the single-parameter law of corresponding states with the ratio of the supercritical temperature to the inversion temperature as the thermodynamic similarity parameter.

Nonlinear diffusion in two-dimensional ordered porous media based on a free volume theory

A continuum nonlinear diffusion model is developed to describe molecular transport in ordered porous media. An existing generic van der Waals equation of state based free volume theory of binary diffusion coefficients is modified and introduced into the two-dimensional diffusion equation. The resulting diffusion equation is solved numerically with the alternating-direction fully implicit method under Neumann boundary conditions. Two types of pore structure symmetries are considered, hexagonal and cubic. The former is modeled as parallel channels while in case of the latter equal-sized channels are placed perpendicularly thus creating an interconnected network. First, general features of transport in both systems are explored, followed by the analysis of the impact of molecular properties on diffusion inside and out of the porous matrix. The influence of pore size on the diffusion-controlled release kinetics is assessed and the findings used to comment recent experimental studies of drug release profiles from ordered mesoporous silicates.

The special features of heat transfer in a supercritical fluid: Mathematical and physical modeling results

The special features of heat transfer in a supercritical fluid were considered for the example of two problems, those of Rayleigh-Benard convection in a horizontal layer heated from below and nonstationary heat transfer in a closed volume with heated boundaries. Isentropic equilibrium of a compressible medium that obeyed the van der Waals equation of state was studied. The calculation results were generalized, and the special features of convective heat transfer in a supercritical fluid beyond the stability threshold of hydrostatic equilibrium were discussed. The results of numerical and experimental studies of the relaxation of density and temperature nonuniformities as the temperature of volume walls changed were presented. The calculations were performed using the complete system of Navier-Stokes equations and the equation of state of an ideal or van der Waals gas.

Excluded volume effect on entropy and clustering behaviors in supercooled liquids and glasses

We study the significant entropy changes in supercooling and glass transition by developing a two (H and L) state cluster model, where the H-state represents the clusters with the coordination number z being larger than the reference size z R and the L-state those with z < z R , respectively. We exploit the excluded volume effect on the clustering behaviors of various glass-formers and the temperature dependence (at fixed pressure of 1 atmosphere) of the entropy difference Δ s o between the two cluster states on the basis of z R = 1 + z R o [ exp ( − Δ s o / k B ) − 1 ] , where z R o = 1.8 as taken universally for all the glass-formers investigated. The temperature dependence of the average size 〈 z c ∗ 〉 of clusters with z ≥ z R is then determined for some representative glass-formers, such as GeO 2, 1-propanol, glycerol, propylene carbonate, o-terphenyl, and so forth, covering the entire fragility spectrum studied experimentally. As a result, the temperature dependence of 〈 z c ∗ 〉 is found to highly depend on the fragility index ϑ associated with the temperature dependence of the excluded volume B ∗ ( ϕ , T ) of which statistical mechanical representation is given by the generic van der Waals equation of state. With decreasing T toward the glass transition temperature T g , 〈 z c ∗ 〉 increases to attain universally about 13 at T g regardless of the fragility and molecular details of the glass-formers. We specifically analyze in depth the theoretical predictions for the cluster sizes and the experimental estimations for the length scales associated with temporary clusters of slow segments in supercooled glycerol as about 1.4±0.5 nm for glycerol at 10 K above T g . Given the large uncertainties associated with the experiment, the present theory appears to be in good agreement with the experiment.

Evolution, Propagation, Reflection, and Interactions of 1D-normal Shocks in Air and FC70 Using hpk Finite Element Computational Framework

In this paper, the mathematical models based on Navier-Stokes equations incorporating physical mechanism of viscosity and conductivity are utilized in h, p, k finite element computational framework to investigate 1-D normal shocks in air and FC70. In the case of air, shock evolution, propagation, repeated reflection and interactions are simulated using ideal gas law with constant viscosity and thermal conductivity. For FC70, numerical studies are presented for initial conditions inside the BZT zone as well as for those away from the BZT zone to investigate the possibility of compression as well as rarefaction shocks. The Van der Waals equation of state with constant transport properties is used for FC70. In all cases, “the rate of production of entropy per unit volume” is used to establish: (i) evolution of shock; (ii) sustained propagation; (iii) repeated reflections and subsequent sustained propagation; and (iv) interactions and sustained propagation of reflected shocks. In all numerical studies care is taken in choosing h, p, k computational parameters to ensure that the computational processes are free of measurable numerical dispersion and that the non-discretized form of the governing differential equations are satisfied accurately in the point wise sense and hence, ensuring time accuracy of the evolutions. 1-D Riemann shock tube is used as a model problem.

Lagrangian particle model for multiphase flows

A Lagrangian particle model for multiphase multicomponent fluid flow, based on smoothed particle hydrodynamics (SPH), was developed and used to simulate the flow of an emulsion consisting of bubbles of a non-wetting liquid surrounded by a wetting liquid. In SPH simulations, fluids are represented by sets of particles that are used as discretization points to solve the Navier–Stokes fluid dynamics equations. In the multiphase multicomponent SPH model, a modified van der Waals equation of state is used to close the system of flow equations. The combination of the momentum conservation equation with the van der Waals equation of state results in a particle equation of motion in which the total force acting on each particle consists of many-body repulsive and viscous forces, two-body (particle–particle) attractive forces, and body forces such as gravitational forces. Similar to molecular dynamics, for a given fluid component the combination of repulsive and attractive forces causes phase separation. The surface tension at liquid–liquid interfaces is imposed through component dependent attractive forces. The wetting behavior of the fluids is controlled by phase dependent attractive interactions between the fluid particles and stationary particles that represent the solid phase. The dynamics of fluids away from the interface is governed by purely hydrodynamic forces. Comparison with analytical solutions for static conditions and relatively simple flows demonstrates the accuracy of the SPH model.

Van der Waals model for phase transitions in thermoresponsive surface films

Phase transitions in polymeric surface films are studied with a simple model based on the van der Waals equation of state. Each chain is modeled by a single bead attached to the surface by an entropic-Hooke's law spring. The surface coverage is controlled by adjusting the chemical potential, and the equilibrium density profile is calculated with density functional theory. The interesting feature of this model is the multivalued nature of the density profile seen at low temperature. This van der Waals loop behavior is resolved with a Maxwell construction between a high-density phase near the wall and a low-density phase in a "vertical" phase transition. Signatures of the phase transition in experimentally measurable quantities are then found. Numerical calculations are presented for isotherms of surface pressure, for the Poisson ratio, and for the swelling ratio.

Adiabatic Heating and Convection in a Porous Medium Filled with a Near‐Critical Fluid

Dynamics and heat transfer in a porous medium filled with a fluid phase at parameters near the gas-liquid critical point are studied. A two-dimensional numerical solver based on the hydrodynamic model for a porous medium with a high compressible fluid phase including the van der Waals equation of state is used. In weightlessness, adiabatic heating of fluid phase under the step-temperature heat supply is investigated analytically and numerically. In terrestrial conditions, gravity-driven convection in vertical rectangular cells generated by lateral heating in unsteady and steady-state regimes is simulated. The effects of high compressibility of near-critical fluid phase on convection are studied. Convective motions and heat transfer in horizontal rectangular cells consisting of two porous layers at different porosity and permeability heated from below are simulated as well. Adiabatic heating subjected to hydrostatic compressibility effects, the onset and development of convection, and convective structures in a steady-state regime are analyzed.

Improved Method for the Self-Diffusion Coefficient in the Modified Free Volume Theory: Simple Fluids

In this paper, we further examine the modified free volume (MFV) theory of diffusion in an effort to improve the accuracy of self-diffusion coefficients calculated with the help of the generic van der Waals equation of state. It is shown that the self-diffusion coefficient of the Lennard-Jones (LJ) fluid can be improved significantly over the results obtained by the previously employed method. In the previous method, we have used the hard sphere self-diffusion coefficient for the pre-exponential factor in the MFV theory formula for the self-diffusion coefficient. We show in this paper that if the hard sphere self-diffusion coefficient used for the pre-exponential factor is replaced with the Chapman-Enskog formula for the LJ fluid the accuracy of the self-diffusion coefficient improves impressively in the liquid density for a wide range of temperature.

GPU-based meshless fluids interactive rendering

An approach based on smoothed particle hydrodynamics to simulate fluids with free surfaces was proposed, while surface tension was modeled by using Van der Waals equation of state with a combination of short-range repulsive and longer-range attractive interactions between fluid particles, finally a GPU-based particle splatting algorithm was designed for rendering. This approach completely avoids the temporal discretization artifacts, which results in the interactive rendering effects with high quality. In contrast to traditional Lagrangian approaches, this method uses the simplified surface tension model with fast rendering algorithm, reduces the complexity of the simulation and efficiently advances the run-time speed.

Package FLUIDS. Part 3: correlations between equations of state, thermodynamics and fluid inclusions

AbstractThe computer package FLUIDS (Bakker 2003) has been revised to calculate fluid properties in pores and inclusions. The programs are provided with a Graphical User Interface (GUI) and are adapted to new operating systems, including MacOS, Windows and Linux. The van der Waals equation of state has been added to the group Loners and is used to illustrate a large variety of calculation procedures for many thermodynamic parameters and properties. The mathematical transformations can be applied to any equation of state with the form p(V,T,n), i.e. the pressure of a multi‐component fluid is expressed as a function of volume, temperature and amount of substance. The fluid properties that are usually observed in microthermometric analyses of fluid inclusions, such as phase separation, phase coexistence and stability, can be predicted with these equations of state by using its spinodal and critical point calculations, in addition to fugacity calculations of liquid and vapour phases. The computer program LonerW can be freely downloaded from the website: http://fluids.unileoben.ac.at/Computer.html.

Turbulent Rayleigh–Bénard convection in a near-critical fluid by three-dimensional direct numerical simulation

This paper presents state of the art three-dimensional numerical simulations of the Rayleigh–Bénard convection in a supercritical fluid. We consider a fluid slightly above its critical point in a cube-shaped cell heated from below with insulated sidewalls; the thermodynamic equilibrium of the fluid is described by the van der Waals equation of state. The acoustic filtering of the Navier–Stokes equations is revisited to account for the strong stratification of the fluid induced by its high compressibility under the effect of its own weight. The hydrodynamic stability of the fluid is briefly reviewed and we then focus on the convective regime and the transition to turbulence. Direct numerical simulations are carried out using a finite volume method for Rayleigh numbers varying from 106 up to 108. A spatiotemporal description of the flow is presented from the convection onset until the attainment of a statistically steady state of heat transfer. This description concerns mainly the identification of the vortical structures in the flow, the distribution of the Nusselt numbers on the horizontal isothermal walls, the structure of the temperature field and the global thermal balance of the cavity. We focus on the influence of the strong stratification of the fluid on the penetrability of the convective structures in the core of the cavity and on its global thermal balance. Finally, a comparison with the case of a perfect gas, at the same Rayleigh number, is presented.