Characteristics functionalized particles

We now turn our attention to functions that are of more direct interest the probability densities that describe the momenta of specified sets of particles. It will prove advantageous to work with characteristic functions rather than probability densities. We consider a specified set of r particles labelled (q, i2,..., ir) in which the particles are not necessarily related as nearest neighbors. Let Pr(j>ti,pi2,. .., pir t) be the probability density for the set of momenta (ph,..., pir) at time t, and let the characteristic function (p cOij, co(-2,. .., coiV t) be defined by... 

In similar fashion, the relaxation of the two-particle characteristic function can be described in terms of a function A2(cor, cos ) ... 

Figure 3 summarizes the rates of deposition calculated for hydrosols depositing onto a rotating disk. The four curves correspond to the four pairs of surfaces whose interactions are characterized in Fig. 1. Surface characteristics of particle and collector are interchangeable in the calculation of the rates. Values of other parameters include a = 0.1 p,m, cx = I08 cm-3, to = 6 rev/sec, and v = 0.01 cm2/sec. Rates are presented as a function of Hamaker s constant, which characterizes the van der Waals attraction, because this parameter is most difficult to experimentally determine, and because the rate is sensitive to its value.

Fig. 7. The characteristic functions of rj vs.

The measured responses were fitted using the characteristic function of micropore diffusion in isotropic spherical particles of uniform size (see the symbols for the measured responses and the best fit curves in Figure 5), The larger the deviation of the data from the best fit the wider is the particle-size distribution (cf Figures 2 and 5). 

The computation of a cyclone fractional or grade efficiency depends on cyclone parameters and flow characteristics of particle-laden gases. The procedure involves a series of equations containing exponential and logarithmic functions. Koch and Licht [ 12] described a cyclone using seven geometric ratios in terms of its diameter as ... 

As a second example, it is instructive to derive the Kramers stationary flux function which serves as a basis for practical application in the Rayleigh quotient variational method (34,35). In principle there are an infinity of stationary flux functions, as any function in phase space which is constant along a classical trajectory will be stationary. Kramers imposed in addition the boundary condition that the flux is associated with particles that were initiated in the infinite past in the reactant region. Following Pechukas (69), one defines (68) the characteristic function of phase points in phase space Xr, which is unity on all phase space points of a trajectory which was initiated in the infinite past at reactants and is zero otherwise. By definition x, is stationary. The distribution function associated with the characteristic function Xr projected onto the physical phase space is then... 

The filtration layer is characterized by the main functional parameters, i.e. pressure losses (including their changes with time, depending on the dust content) and total or fractional separation efficiency. The efficiency depends on the characteristics of particles to be retained, on the parameters of the gas to be purified, on the physical and chemical characteristics of the filtration layer and on its structure. It is represented by complicated functions of several variables. For the fractional separation efficiency of the filtration layer of a fibrous structure it is possible to write... 

Polarimetry is a powerful method for studying solar-system bodies. It has allowed the determination of such parameters as the complex refractive index of particles in planetary atmospheres, the size distribution functions of these particles, the methane concentrations, the atmospheric pressure values above the cloud layers, etc. Independent spectral analyses of linear P) and circular (V) polarization observational data also may facilitate the determination of physical characteristics of particles at different heights in a planetary atmosphere. Polarimetiy enables us to make qualitative conclusions about... 

In-phase and out-of-phase characteristic functions for macro-particle shape factor sm... 

The benefit of the analytical treatment presented thus far for the calculation of the characteristic functions of the single-file system is only limited by the increasing complexity of the joint probabilities and the related master equations. This treatment, however, has suggested a most informative access to the treatment of systems subjected to particle exchange with the surroundings and to internal transport and reaction mechanisms [74,75]. Summing over all values (Ji = 0 and 1 and, subsequently, over all sites i, Eq. 31 may be transferred to the relation Eq. 34... 

Characteristic Function and Transport Equation for the Particle Density... 

Our goal is to find an equation for the density of particles that follow the random walk (3.156). First, let us find the characteristic function... 

As was just mentioned, quantum theory limits the accuracy with which the classical variables of position and velocity can be specified. On the other hand, it introduces a new characteristic for particles their spin. Properly speaking, this should also be one of the variables of the flux O, or, more concretely, two flux functions are actually needed to specify the neutron distribution completely. One of these, 4>r, would describe the flux due to neutrons of right helicity (spin parallel to velocity), the other, Oz, would describe the flux due to neutrons of left helicity (spin antiparallel to velocity). There are transitions in which the helicity of neutrons changes so... 

The problem of estimation of the kinetic parameters from linear FR characteristic functions, has been solved long ago, for simple isothermal kinetic models [15]. The process time constant can be estimated from the extremum of the so-called out-of-phase function [15], which is identical to the negative imaginary part of the first-order particle FRF Fi,p(o)) [28]. 

Tables of computed characteristic functions of scattering on dielectric, absorbing, and polydisperse particles have been compiled (van de Hulst, 1957 Shifrin and Telmanovich,...

The main premises of this technique are given in the monograph (Klenin et al., 1977a) with calibrations of all the characteristic functions for inonodisperse, spherical particles within wide ranges of a Euid m computed with the use of Mie s theory. Since then, new results have been obtained concerning the problem of phase sepsu ation in polymer systems. 

The characteristic functions of a monodisperse particle system are of complex oscillating nature. However, real systems are always polydisperse, and intuition prompts to adapt the characteristic functions of monodisperse systems to polydisperse ones by smoothing the oscillations. This has been done using simple symmetry. 

Klenin et al. (1977a), Ramazanov et al. (1983a) have established that the characteristic functions of polydisperse systems fit the smoothed functions of monodisperse systems in the best way if the A average value of particle size is used (Shchyogolcv and Klenin, 1971a Klenin el al.. 1977a)... 

The other characteristic functions of anisodiametric particles are calculated through the efficiency factor (Equation 115) similarly to Equations 99-105. 

Figures 2.21-2.25 report some of the obtained results showing that random arrangement of the anisodiametric particles smooths oscillations of the characteristic functions as well.

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