Discrete distribution function, general

General Dynamic Equation for the Discrete Distribution Function 3Q7... 

GENERAL DYNAMIC EQUATION FOR THE DISCRETE DISTRIBUTION FUNCTION... 

The change in the discrete distribution function with time and position is obtained by generalizing the equation of convective diffusion (Chapter 3) to include terms for particle growth and coagulation ... 

Steinhauer and Gasteiger [30] developed a new 3D descriptor based on the idea of radial distribution functions (RDFs), which is well known in physics and physico-chemistry in general and in X-ray diffraction in particular [31], The radial distribution function code (RDF code) is closely related to the 3D-MoRSE code. The RDF code is calculated by Eq. (25), where/is a scaling factor, N is the number of atoms in the molecule, p/ and pj are properties of the atoms i and/ B is a smoothing parameter, and Tij is the distance between the atoms i and j g(r) is usually calculated at a number of discrete points within defined intervals [32, 33]. 

The probability distribution of a randoni variable concerns tlie distribution of probability over tlie range of tlie random variable. The distribution of probability is specified by the pdf (probability distribution function). This section is devoted to general properties of tlie pdf in tlie case of discrete and continuous nmdoiii variables. Special pdfs finding e.xtensive application in liazard and risk analysis are considered in Chapter 20. 

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

In solution theory the specialized distribution functions of this kind should appear in the theory of ion pairs in ionic solutions, and a form of the Bjerrum-Fuoss ionic association theory adapted to a discrete lattice is generally used for the treatment of the complexes in ionic crystals mentioned above. In fact, the above equation is not used in this treatment. Comparison of the two procedures is made in Section VI-D. 

Kiparissides, et al. (8) developed mathematical models of two levels of sophistication for the vinyl acetate system a comprehensive model that solved for the age distribution function of polymer particles and a simplified model which solved a series of differential equations assuming discrete periods of particle nucleation. In practice, the simplified model adequately describes the physical process in that particle generation generally occurs in discrete intervals of time and these generation periods are short in duration when compared with operation time of the system. The simplified model is expanded here for a series of m reactors. The total property balances for number of particles, polymer volume, conversion, and area of particles, are written as ... 

The discrete set of the nuclear point distribution is generalized to more general three-dimensional objects, for example, to the continuous molecular charge density functions p(r). 

A test of the PMH which is quite different, and more general, has recently been given.It follows earlier work by Gyftopoulos and Hatsopoulos, who used a grand canonical ensemble with a limited number of discrete energy levels, so that the distribution function was known.Those authors then calculated the electronic chemical potential, /z, which was found to be /z = (/-(-y4)/2. The ensemble was a collection of systems containing the three species M , M+, M and with energy levels E°, E° + /) and E° — A). 

An aerosol distribution can be described by the number concentrations of particles of various sizes as a function of time. Let us define Nk(t) as the number concentration (cm-3) of particles containing k monomers, where a monomer can be considered as a single molecule of the species representing the particle. Physically, the discrete distribution is appealing since it is based on the fundamental nature of the particles. However, a particle of size 1 pm contains on the order of 1010 monomers, and description of the submicrometer aerosol distribution requires a vector (N2, N-j,..., N10io) containing 1010 numbers. This makes the use of the discrete distribution impractical for most atmospheric aerosol applications. We will use it in the subsequent sections for instructional purposes and as an intermediate step toward development of the continuous general dynamic equation. 

A generalization of these population balance methods to reactions with arbitrary RTD was given by Rattan and Adler [126]. They expanded the phase space of the distribution functions to include the life expectation as well as concentration of the individual fluid elements i/ (C, A, 0- The population balance then reduces to all of the previous developments for the various special cases of segregated or micromixed flow, the perfect macromixing coalescence-redispersion model, and can be solved as continuous functions or by discrete Monte Carlo techniques. Goto and Matsubara [127] have combined the coalescence and two-environment models into a general, but very complex, approach that incorporates much of the earlier work. 

In this appendix, we present the generalized Euler theorem for homogeneous functions of order one. We first write the Euler theorem for a discrete quasi-component distribution function QCDF) and then generalize by analogy for a continuous QCDF. A more detailed proof is available. ... 

The development in this section arises from the generalization of an idea due to Kendall (1950) for a simple birth-and-death process. This generalization, accomplished by Shah et al (1977) was significant in that it provides a route to the simulation of a particulate system of arbitrary complexity and is limited only by the amount of computational power available. It is statistically exact in that the random numbers to be generated satisfy exactly calculated distribution functions from the model for particle behavior, thus allowing not only the calculation of the average system behavior but also the fluctuations about it. Furthermore, we shall see that it is free from arbitrary discretizations of time (or any other governing evolutionary coordinate) that were characteristic of the simulations of Section 4.6.1. 

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