Particle spreading parameter

In this section, we present results found with the scalar-conditioned velocity algorithm introduced in the previous section using the NDF representation in Eq. (8.108). All of these calculations have been done with beta EQMOM using n = 2 nodes for the first quadrature and M = 50 nodes for the second. The initial NDF, where f e [0,1] is the particle volume, has one of two forms. In the first case, referred to as small spread, the NDF is represented by two beta functions, one near = 1/3 and one near f = 2/3, with a small spread parameter cr. The initial moments for this case are shown in Figure 8.28. As can be observed from this figure, the moments are nonzero only within the spatial domain x (-0.8, -0.2). Some statistics computed from the initial NDF are given in Figure 8.29. Note that the initial conditional velocity u(0,x,f) = 1 is independent of the particle size, and the mean and standard deviation of the volume do not depend on x. 

In the case of styrene as monomer and hexadecane as model oil, the cohesion energy density of the polymer phase is closer to that of the oil, and therefore the structure of the final particles depends much more on those parameters which critically influence the interfacial tensions. A variety of different morphologies in the styrene-hexadecane system can be obtained by changing the spreading parameter. This was done by changing the monomer concentration and the type and amount of surfactant, as well as the initiator and the functional comonomer. 

We investigated the mechanistic memory of the PE 12,000 atoms to reform a sphere-like shape for the Si, C, and Al surfaces by calculating the asymmetric parameter in Eq. (20). As shown in Fig. 32, the particle with the initial velocity 5 A/ps tended to self-organize into the spherical shape during equilibration. Since the particles spread (coats) on the surfaces on impact, the asymmetric parameter had the largest value at the impact (oblate top). It was interesting to note... 

The coal particles can be tracked as parcels in an Eulerian-Lagrangian framework. Discrete phase model (DPM) are used to define the injected particles that enter the reactor. In the case of INCI, simulation values for axial velocity of -1.732 m/s and a radial velocity of -l.Om/s of the particles must be provided. If agglomeration is neglected, a maximum particle diameter of 0.1 mm, a mean diameter of 0.09 mm, and a minimum diameter of 0.001 mm are assumed according to a Rosin-Rammler-Sperling-Bennett distribution with a spread parameter of = 0.688 in 10 individual groups for fluid-bed coal (see also Section 3.12.3.3). Particles can be treated as nonspherical with a shape factor of 0.85. 

Methods of estimating gaseous effluent concentrations have undergone many revisions. For a number of years, estimates of concentrations were calculated from the equations of Sutton, with the atmospheric dispersion parameters C, C, and n, or from the equations of Bosanquet with the dispersion parameters p and Q. More common approaches are based on experimental observation that the vertical distribution of spreading particles from an elevated point is... 

Since the parameter y is non-vanishing, the wave packet will disperse with time as indicated by equation (1.28). For a gaussian profile, the absolute value of the wave packet is given by equation (1.31) with y given by (1.43). We note that y is proportional to m, so that as m becomes larger, y becomes smaller. Thus, for heavy particles the wave packet spreads slowly with time. 

In addition to the positional and thermal parameters of the atoms, least-squares procedures are used to determine the scale of the data, and parameters such as mosaic spread or particle size, which influence the intensities through multiple-beam effects (Becker and Coppens 1974a, b, 1975). It is not an exaggeration to say that modern crystallography is, to a large extent, made possible by the use of least-squares methods. Similarly, least-squares techniques play a central role in the charge density analysis with the scattering formalisms described in the previous chapter. 

The behaviour of the correlation function X% (r, t) is defined by the auto-catalytic reaction stage the probability to find some particle B near another B is rather high if they are reproduced by a division. For the short relative distances r the function X% (r,t) has a singularity which is, however, weakly pronounced, i.e., particles B are quasi-randomly distributed in space. For a chosen parameter k = 0.02 the relative diffusion coefficients is large Db =2(1 - k), Db Dp,. The aggregates emerging under reproduction of B s are spread out rapidly due to the diffusion. 

In both methods the least squares parameters consisted of 40 histogram step heights spread over a two decade particle size range. The maximum number of parameters allowed in Provencher s program is 50. 

The solution of the above equation in order to obtain W(y) requires an appropriate form of the spreading function and the numerical values of its parameters. Furthermore, to convert W(y) into a size distribution requires a relationship between the mean retention volume y and the particle diameter D (i.e., a calibration curve). 

The equations used to describe the combustion wave propagation for microstructural models are similar to those in Section IV,A [see Eq. (6)]. However, the kinetics of heat release, 4>h may be controlled by phenomena other than reaction kinetics, such as diffusion through a product layer or melting and spreading of reactants. Since these phenomena often have Arrhenius-type dependences [e.g., for diffusion, 2)=9)o exp(— d// T)], microstructural models have similar temperature dependences as those obtained in Section IV,A. Let us consider, for example, the dependence of velocity, U, on the reactant particle size, d, a parameter of medium heterogeneity ... 

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