Size distribution function moments

The theory of particle clouds proceeds from consideration of the dynamics of the particle size distribution function or its integral moments. This distribution can take two forms. The first is a discrete function in which particle... 

The moments of the size distribution function are useful parameters. These have the form ... 

The shape of the size distribution function for aerosol particles is often broad enough that distinct parts of the function make dominant contributions to various moments. This concept is useful for certain kinds of practical approximations. In the case of atomospheric aerosols the number distribution is heavily influenced by the radius range of 0.005-0.1 /xm, but the surface area and volume fraction, respectively, are dominated by the range 0.1-1.0 fxm and larger. The shape of the size distribution is often fit to a logarithmic-normal form. Other common forms are exponential or power law decrease with increasing size. 

Several parameters can be used to express the mean size of the milk fat globules. These parameters are derived from the so-called moments of the size distribution function the th moment of the distribution function (Sn) is equal to ... 

Related Calculations. This procedure can be used to calculate average sizes, moments, surface area, and mass of solids per volume of slurry for any known particle size distribution. The method can also be used for dry-solids distributions, say, from grinding operations. See Example 10.7 for an example of a situation in which the size distribution is based on an experimental sample rather than on a known size-distribution function. 

In the quadrature method of moments (QMOM) developed by McGraw [131], for the description of sulfuric acid-water aerosol dynamics (growth), a certain type of quadrature function approximations are introduced to approximate the evolution of the integrals determining the moments. Marchisio et al [122, 123] extended the QMOM for the application to aggregation-breakage processes. For the solution of crystallization and precipitation kernels the size distribution function is expressed using an expansion in delta functions [122, 123] ... 

The general moment of the particle size distribution function can be defined by the expression... 

Different parts of the particle size distribution function make controlling contributions to the various moments. In a polluted urban atmosphere, the number concentration or zeroth moment is often dominated by the 0.01- to 0, l- um size range, and the surface area is often dominated by the 0.1- to 1.0-/zm range contributions to the volumetric concentration come from both the 0.1 to 1.0- and 1.0- to 10-/xm size ranges (Chapter 13). Muincnts of fractional order appear in the theory of aerosol convective diffusion (Chapter 3). 

In this section we discu.ss briefly several size distribution functions that can be used to fit experimental data for aerosols or to estimate average particle size or the effects of aerosols on light scattering. The examples discussed are nornud.power-law, and self-sirniUir distributions. Selecting a distribution function depends on the specific application. In some cases, fragmentary information may be available on certain moments or on sections of the... 

To evaluate a it is necessary to solve for the SPD, which depends on Df. Values of a for the free molecule regime vary little with Df in the range 2 to 3 as shown in Table 8.1. along with the 1/0/ moment of the size distribution function. Mi/Of For Df 3, ctsa function of the size of the primary particle, ctpo. 

At the beginning of the chapter it is shown that the usual models for coagulation and nucleation presented in Chapters 7 and 10 arc special cases of a more general theory for very small particles. An approximate criterion is given for determining whether nucleation or coagulation is rate controlling at the molecular level. The continuous form of the GDE is then used to derive balance equations for several moments of the size distribution function. 

Figure 11.4 Evolution of the moments of the size distribution function for the aerosols shown in Fig. 11.3, The peak in the number distribution probably results when formation by homogeneous nudeaiiun is balanced by coagulation. Total aerosol volume increases with time as gas-tO partide conversion takes place. Total. surface area, A, increases at first and then approaches an approximately constant value, due probably toa balance between growth and coagulation (Husarand Whitby, 1973). The results should be compared with Pig. 11.2.

FIGURE 5.11 Sketch of the droplet size distribution function,/(i ,t) vs. the droplet radius i at a given moment t. 8 is the length of the mesh used when solving the problem by discretization. 

Another important characteristic of the late stages of phase separation kinetics, for asymmetric mixtures, is the cluster size distribution function of the minority phase clusters n(R,T)dR is the number of clusters of minority phase per unit volume with radii between R and R + dR. Its zeroth moment gives the mean number of clusters at time x and the first moment is proportional to the mean cluster size. 

However, the pore-size distribution function /(r) is not normalized, so the zeroth moment must be included in the expression for the average pore radins ... 

Important quantities to characterize the particle size distribution function / are the moments Ma of order a, defined by... 

The whole curve fitting method has attracted a lot of attention in the literature and continues to do so today. The latest effort in its use for reducing the amount of data handled by computer models of particulate processes is to find the necessary parameters from the moments of the particle size distribution function instead of finding the best fit from the closeness of the measured points to the fitted function, as is more usual in practice. 

Mroczka, J., Method of Moments in Light Scattering Data Inversion in the Particle Size Distribution Function, Opfics Comm., 1993,99, 147-151. 

Because the inequalities among the various mean diameters are usually strengthened when the drop sizes are widely dispersed, the ratio of some higher-order moment to a lower-order moment is often useful as a measure of the dispersion of the drop sizes. For example, the coefficient of variation for the surface-weighted size distribution is a function of the ratio of the weight-weighted mean drop size to the volume-surface mean drop size. The variance of the drop-size distribution may also be expressed in terms of the moments of the unweighted size distribution. 

Let the system contain at the initial time a certain number N of nuclei of exactly the same size. The distribution function Z is equal to zero everywhere except at one specific point (curve 1, Fig. 3). In the macroscopic theory, each nucleus changes with time in a quite definite way, depending on its size and external conditions N nuclei which were identical at the initial moment will remain identical even after a certain time interval t, and curve 1 will be shifted as a whole to another place (curve 2) that corresponds to the change in the size of the nucleus, in accordance with the kinetics equation of the form... 

The formalism of nonlocal functional density theory provides an attractive way to describe the physical adsorption process at the fluid - solid interface.65 In particular, the ability to model adsorption in a pore of slit - like or cylindrical geometry has led to useful methods for extracting pore size distribution information from experimental adsorption isotherms. At the moment the model has only been tested for microporous carbons and slit - shaped materials.66,67 It is expected that the model will soon be implemented for silica surfaces. 

Thus, given gparticle size distribution. For narrow size distributions, the autocorrelation function is satisfactorily analyzed by the method of cumulants to give the moments of the particle size distribution.(7) However, the analysis of QELS data for samples with polydisperse or multimodal distributions remains an area of active research.(8)... 

A number of distribution functions have been identified experimentally for a variety of systems ( 3) and, in particular, the Log-Normal distribution is extensively used for the calculation of the integral and for the evaluation of the moments of the particle size distribution (8—12). The problem with this approach is that, in general, the shape of the particle size distribution is unknown and thus, the average particle diameters obtained are conditional upon... 

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