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

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Quantum mechanical continuum solvation models.

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.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/0BD9AC7975884A5D5F360188D647F2B6/S0022112084000586a.pdf/div-class-title-kinetic-theories-for-granular-flow-inelastic-particles-in-couette-flow-and-slightly-inelastic-particles-in-a-general-flowfield-div.pdf

Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield

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.

Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres

Hansen solubility parameters (HSPs) are used to predict molecular affinities, solubility, and solubility-related phenomena. Revised and updated throughout, Hansen Solubility Parameters: A User's Handbook, Second Edition features the three Hansen solubility parameters for over 1200 chemicals and correlations for over 400 materials including p

Hansen Solubility Parameters : A User's Handbook, Second Edition

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.

New reference equation of state for associating liquids

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.

Equation of State for Nonattracting Rigid Spheres

Generalized multiparameter correlation for nonpolar and polar fluid transport properties

Kinetic gas theories have been applied for the development of a correlation of gas viscosity and thermal conductivity. Employing the acentric factor (..omega..), the dipole moment (..mu..), and the association parameter (kappa) to characterize the effects of molecular shape and anisotropic intermolecular forces, the resultant multiparameter correlations are self-consistent for viscosity and thermal conductivity and generalized for polar and nonpolar gases. The results for pure gases are outstanding, not only in accuracy but also in applicability for such wide classes of fluids which include polar and hydrogen-bonding compounds.

Applications of kinetic gas theories and multiparameter correlation for prediction of dilute gas viscosity and thermal conductivity

This investigation demonstrates the advantages of an improved expression for intermolecular repulsions in equations of state. A highly accurate equation of state for the rigid-sphere fluid is used as the model of repulsion behavior in real fluids. A method for general usage of this new rigid-sphere equation in real fluid models is presented via reformulation of van der Waal's equation using the accurate new rigid-sphere equation to describe repulsion effects. To test this reformulation, properties of methane, ethane, propane, n-butane, isobutane, hydrogen sulfide, nitrogen, ethylene, acetylene, methyl chloride, cyclohexane, pentane and octane have been calculated. The advantage of using the accurate rigid-sphere equation was demonstrated by the general improvement in calculated properties relative to the original van der Waals and Redlich-Kwong equations. Calculations of enthalpy departures for mixtures of nitrogen and methane, propane and methane, pentane and cyclohexane, and pentane and octane also indicated the desirability of using the accurate rigid-sphere equation to describe repulsion effects in mixtures.

Intermolecular repulsions and the equation of state for fluids

An equation of state for carbon dioxide is developed here that yields high accuracy in P-V-T calculations over wide ranges of temperatures (216-423 K) and pressures (to 310.3 MPa). It is particularly accurate around the critical region due to the use of "nonanalytical" terms to model the critical isotherm. Thus it is suited to applications in supercritical states. The density calculation is reliable to within 0.1-0.2 % outside the critical region, and to within 1% near the critical point. The equation has also been tested for vapor pressure and enthalpy calculations (with deviations less than 0.06 %, and 2-5 J/g, respectively). Comparison with a number of existing equations of state shows that the present equation is more dependable.

Kenneth E. Starling

Papers

Equation of State for Nonattracting Rigid Spheres

Generalized multiparameter correlation for nonpolar and polar fluid transport properties

Applications of kinetic gas theories and multiparameter correlation for prediction of dilute gas viscosity and thermal conductivity

Intermolecular repulsions and the equation of state for fluids

An accurate equation of state for carbon dioxide