Local Kinetic Energy Models

With the quasi-probability distribution function as an intermediary, the local kinetic energy can be defined as  

Because the quasi-probability distribution is not uniquely defined, neither is the local kinetic energy. ° 3 If one can model the local kinetic energy using the electron density, then the kinetic energy functional can be written as 

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Spademan [96] demonstrated that the local kinetic energy density at the BCP, as given by Eq. 2, is rather insensitive to local density rearrangements due to HB formation. Reinterpretation of the above HB data reveals that the observed behavior of Gbcp can be well reproduced by a two-atom ED model (spherical... 

The functionals considered in Section 1.3 are all semilocal the local kinetic energy at the point r depends only on the electron density and its derivatives at the point r. Improved models for the kinetic energy require considering how the electron density at other points r affects the local kinetic energy at the point r. Without including these effects, the oscillations in electron density that are essential for modeling the shell structure of atoms and differentiating between core and valence electrons in molecules cannot be recovered. 

This suggests, for this very simple kinetic energy model, that the local erosion rate within the cone section is inversely proportional to the 3 power of the local cone diameter. In deriving Eq. (12.1.8), the local particle velocity near the wall (shown in Eq. 12.1.7) was assumed proportional to the local gas velocity. This local gas velocity near the wall is, in turn, assumed to be that of a free vortex and, thus, inversely proportional to the cone radius, as in Eq. (12.1.7). 

In one of the earliest DFT models, the Thomas-Fermi theory, the kinetic energy of an atom or a molecule is approximated using the above type of treatment on a local level. That is, for each volume element in r space, one... 

The model is able to predict the influence of mixing on particle properties and kinetic rates on different scales for a continuously operated reactor and a semibatch reactor with different types of impellers and under a wide range of operational conditions. From laboratory-scale experiments, the precipitation kinetics for nucleation, growth, agglomeration and disruption have to be determined (Zauner and Jones, 2000a). The fluid dynamic parameters, i.e. the local specific energy dissipation around the feed point, can be obtained either from CFD or from FDA measurements. In the compartmental SFM, the population balance is solved and the particle properties of the final product are predicted. As the model contains only physical and no phenomenological parameters, it can be used for scale-up. 

Several methods have been used for analyzing the electron density in more detail than we have done in this paper. These methods are based on different functions of the electron density and also the kinetic energy of the electrons but they are beyond the scope of this article. They include the Laplacian of the electron density ( L = - V2p) (Bader, 1990 Popelier, 2000), the electron localization function ELF (Becke Edgecombe, 1990), and the localized orbital locator LOL (Schinder Becke, 2000). These methods could usefully be presented in advanced undergraduate quantum chemistry courses and at the graduate level. They provide further understanding of the physical basis of the VSEPR model, and give a more quantitative picture of electron pair domains. 

As mentioned before in Eq. (3), the most common source of SGS phenomena is turbulence due to the Reynolds number of the flow. It is thus important to understand what the principal length and time scales in turbulent flow are, and how they depend on Reynolds number. In a CFD code, a turbulence model will provide the local values of the turbulent kinetic energy k and the turbulent dissipation rate s. These quantities, combined with the kinematic viscosity of the fluid v, define the length and time scales given in Table I. Moreover, they define the local turbulent Reynolds number ReL also given in the table. 

The standard wall function is of limited applicability, being restricted to cases of near-wall turbulence in local equilibrium. Especially the constant shear stress and the local equilibrium assumptions restrict the universality of the standard wall functions. The local equilibrium assumption states that the turbulence kinetic energy production and dissipation are equal in the wall-bounded control volumes. In cases where there is a strong pressure gradient near the wall (increased shear stress) or the flow does not satisfy the local equilibrium condition an alternate model, the nonequilibrium model, is recommended (Kim and Choudhury, 1995). In the nonequilibrium wall function the heat transfer procedure remains exactly the same, but the mean velocity is made more sensitive to pressure gradient effects. 

Within this local-density approximation one can obtain exact numerical solutions for the electronic density profile [5], but they require a major computational effort. Therefore the variational method is an attractive alternative. For this purpose one needs a local approximation for the kinetic energy. For a one-dimensional model the first two terms of a gradient expansion are ... 

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