Equation of State for Nonattracting Rigid Spheres
TL;DR: In this paper, a new equation of state for rigid spheres has been developed from an analysis of the reduced virial series, which possesses superior ability to describe rigid-sphere behavior compared with existing equations.
Abstract: A new equation of state for rigid spheres has been developed from an analysis of the reduced virial series. Comparisons with existing equations show that the new formula possesses superior ability to describe rigid‐sphere behavior.
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TL;DR: This paper presents a meta-modelling procedure called "Continuum Methods within MD and MC Simulations 3072", which automates the very labor-intensive and therefore time-heavy and expensive process of integrating discrete and continuous components into a discrete-time model.
Abstract: 6.2.2. Definition of Effective Properties 3064 6.3. Response Properties to Magnetic Fields 3066 6.3.1. Nuclear Shielding 3066 6.3.2. Indirect Spin−Spin Coupling 3067 6.3.3. EPR Parameters 3068 6.4. Properties of Chiral Systems 3069 6.4.1. Electronic Circular Dichroism (ECD) 3069 6.4.2. Optical Rotation (OR) 3069 6.4.3. VCD and VROA 3070 7. Continuum and Discrete Models 3071 7.1. Continuum Methods within MD and MC Simulations 3072
13,286 citations
TL;DR: In this paper, the authors studied the flow of an idealized granular material consisting of uniform smooth, but nelastic, spherical particles using statistical methods analogous to those used in the kinetic theory of gases.
Abstract: The flow of an idealized granular material consisting of uniform smooth, but nelastic, spherical particles is studied using statistical methods analogous to those used in the kinetic theory of gases. Two theories are developed: one for the Couette flow of particles having arbitrary coefficients of restitution (inelastic particles) and a second for the general flow of particles with coefficients of restitution near 1 (slightly inelastic particles). The study of inelastic particles in Couette flow follows the method of Savage & Jeffrey (1981) and uses an ad hoc distribution function to describe the collisions between particles. The results of this first analysis are compared with other theories of granular flow, with the Chapman-Enskog dense-gas theory, and with experiments. The theory agrees moderately well with experimental data and it is found that the asymptotic analysis of Jenkins & Savage (1983), which was developed for slightly inelastic particles, surprisingly gives results similar to the first theory even for highly inelastic particles. Therefore the ‘nearly elastic’ approximation is pursued as a second theory using an approach that is closer to the established methods of Chapman-Enskog gas theory. The new approach which determines the collisional distribution functions by a rational approximation scheme, is applicable to general flowfields, not just simple shear. It incorporates kinetic as well as collisional contributions to the constitutive equations for stress and energy flux and is thus appropriate for dilute as well as dense concentrations of solids. When the collisional contributions are dominant, it predicts stresses similar to the first analysis for the simple shear case.
2,631 citations
TL;DR: In this article, an equation of state is proposed for the mixture of hard spheres based on an averaging process over the two results of the solution of the Percus-Yevick integral equation.
Abstract: An equation of state is proposed for the mixture of hard spheres based on an averaging process over the two results of the solution of the Percus–Yevick integral equation for the mixture of hard spheres. Compressibility and other equilibrium properties of the binary mixtures of hard spheres are calculated and they are compared with the related machine‐calculated (Monte Carlo and molecular dynamics) data. The comparison shows excellent agreement between the proposed equation of state and the machine‐calculated data.
1,894 citations
TL;DR: An equation of state for associating liquids is presented as a sum of three Helmholtz energy terms: Lennard-Lones (LJ) segment (temperature-dependent hard sphere + dispersion), chain (increment due to chain formation), and association as mentioned in this paper.
Abstract: An equation of state for associating liquids is presented as a sum of three Helmholtz energy terms: Lennard-Lones (LJ) segment (temperature-dependent hard sphere + dispersion), chain (increment due to chain formation), and association (increment due to association). This equation of state has been developed by extending Wertheim’s theory obtained from a resummed cluster expansion. Pure component molecules are characterized by segment diameter, segment-segment interaction energy, for example, Lennard-Jones u and E, and chain length expressed as the number of segments. There are also two association parameters, the association energy and volume, characteristic of each site-site pair. The agreement with molecular simulation data is shown to be excellent at all the stages of development for associating spheres, mixtures of associating spheres, and nonassociating chains. The model has been shown to reproduce experimental phase equilibrium data for a few selected real pure compounds.
1,844 citations
Cites methods from "Equation of State for Nonattracting..."
...Since we approximate our segments as hard spheres, we approximate g(d)w as the hard sphere radial distribution function (Carnahan and Starling, 1969): (19) 2 -7 2 0 -d3 g(d)seg = g(d)h " = where 7 is the reduced density defined as...
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TL;DR: In this paper, a predictive two-phase flow model was derived starting with the Boltzman equation for velocity distribution of particles, which is a generalization of the Navier-Stokes equations of the type proposed by R. Jackson, except that the solids viscosities and stresses are computed by simultaneously solving a fluctuating energy equation for the particulate phase.
Abstract: Detailed knowledge of solids circulation, bubble motion, and frequencies of porosity oscillations is needed for a better understanding of tube erosion in fluidized bed combustors. A predictive two-phase flow model was derived starting with the Boltzman equation for velocity distribution of particles. The model is a generalization of the Navier-Stokes equations of the type proposed by R. Jackson, except that the solids viscosities and stresses are computed by simultaneously solving a fluctuating energy equation for the particulate phase. The model predictions agree with time-averaged and instantaneous porosities measured in two-dimensional fluidized beds. Observed flow patterns and bubbles were also predicted.
1,583 citations
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TL;DR: In this paper, the three-dimensional classical many-body system is approximated by the use of collective coordinates, through the assumed knowledge of two-body correlation functions, and a self-consistent formulation is available for determining the correlation function.
Abstract: The three-dimensional classical many-body system is approximated by the use of collective coordinates, through the assumed knowledge of two-body correlation functions. The resulting approximate statistical state is used to obtain the two-body correlation function. Thus, a self-consistent formulation is available for determining the correlation function. Then, the self-consistent integral equation is solved in virial expansion, and the thermodynamic quantities of the system thereby ascertained. The first three virial coefficients are exactly reproduced, while the fourth is nearly correct, as evidenced by numerical results for the case of hard spheres.
2,358 citations
TL;DR: In this article, an equilibrium theory of rigid sphere fluids is developed based on the properties of a new distribution function G(r) which measures the density of rigid spheres molecules in contact with a rigid sphere solute of arbitrary size.
Abstract: An equilibrium theory of rigid sphere fluids is developed based on the properties of a new distribution function G(r) which measures the density of rigid sphere molecules in contact with a rigid sphere solute of arbitrary size. A number of exact relations which describe rather fully the functional form of G(r) are derived. These are based on both geometrical considerations and the virial theorem. A knowledge of G(a) where a is the diameter of a rigid sphere enables one to arrive at the equation of state. The resulting analytical expression which is exact up to the third virial coefficient gives the fourth virial coefficient within 3% and the fifth, insofar as it is known, within 5%. Furthermore over the entire range of fluid density, the equation of state derived from theory agrees with that computed using machine methods. Theory also gives an expression for the surface tension of a hard sphere fluid in contact with a perfectly repelling wall. The dependence of surface tension on curvature is also given. ...
1,237 citations
TL;DR: In this paper, simple and exact expressions for the compressibility and pressure equations of state predicted by the Percus-Yevick equation for hard spheres were found for Wainwright and Alder.
Abstract: Simple and exact expressions have been found for the compressibility and pressure equations of state predicted by the Percus—Yevick equation for hard spheres. The equations of state are in good agreement with the machine calculations of Wainwright and Alder, and Wood, Parker, and Jacobson.
987 citations
TL;DR: In this article, the equation of state and the collision rate for systems ranging in size from four to 500 particles are described, and the dependence of the results on the number of particles is qualitatively discussed and insight is gained as to what is required of more accurate analytical theories.
Abstract: The equation of state and the collision rate for systems ranging in size from four to 500 particles are described. The dependence of the results on the number of particles is qualitatively discussed and in this way insight is gained as to what is required of more accurate analytical theories. By comparing the results to various analytical theories now available their region of validity is established. The number of particles necessary at various densities to obtain a quantitative description of the equilibrium properties is delineated. Whether a first‐order phase transition exists for hard spheres remains open until larger systems are investigated.
657 citations
TL;DR: In this paper, the modified stars contain both Mayer f functions and f functions (f≡f+1) and it is shown that the number of topologically distinguishable graphs occurring in the new expressions is about half the number required in previous expressions.
Abstract: New expressions for the fourth, fifth, and sixth virial coefficients are obtained as sums of modified star integrals. The modified stars contain both Mayer f functions and f functions (f≡f+1). It is shown that the number of topologically distinguishable graphs occurring in the new expressions is about half the number required in previous expressions. This reduction in the number of integrals makes numerical calculation of virial coefficients simpler and more nearly accurate. For particles interacting with a hard‐core potential, values of the modified star integrals are shown to depend strongly on dimension; for example, several modified star integrals are identically zero for hard disks (two dimensions), but give nonzero values for hard spheres (three dimensions). Of all the modified star integrals contributing to the fourth, fifth, and sixth virial coefficients, the complete star integrals are shown to be the largest. Mayer's expressions for these coefficients made the complete star integrals the small...
450 citations