Particle-size distribution moments

Figure 3. The total particle size distribution moments...

Equation 2.101 enables calculation of local average quantities such as moments of the particle size distribution. Baldyaga and Orciuch (2001) review expressions for local instantaneous values of particle velocity and diffusivity of particles, Z)pT, required for its solution and recover the distribution using the method of Pope (1979). 

The second moment of the particle size distribution used in the mass balances is obtained from... 

The moment equations of the size distribution should be used to characterize bubble populations by evaluating such quantities as cumulative number density, cumulative interfacial area, cumulative volume, interrelationships among the various mean sizes of the population, and the effects of size distribution on the various transfer fluxes involved. If one now assumes that the particle-size distribution depends on only one internal coordinate a, the typical size of a population of spherical particles, the analytical solution is considerably simplified. One can define the th moment // of the particle-size distribution by... 

The other state variables are the fugacity of dissolved methane in the bulk of the liquid water phase (fb) and the zero, first and second moment of the particle size distribution (p0, Pi, l )- The initial value for the fugacity, fb° is equal to the three phase equilibrium fugacity feq. The initial number of particles, p , or nuclei initially formed was calculated from a mass balance of the amount of gas consumed at the turbidity point. The explanation of the other variables and parameters as well as the initial conditions are described in detail in the reference. The equations are given to illustrate the nature of this parameter estimation problem with five ODEs, one kinetic parameter (K ) and only one measured state variable. 

Fig. 9. Snapshots of particle sizes and their spatial distribution in a vertical midway baffle plane at two moments in time, along with the pertinent respective overall particle size distributions. The diameter of the particles is enlarged by a factor of 10 for reasons of clarity. Grey colors represent particle size with respect to the original particle size. Reproduced with permission from Hartmann (2005).

Hollander (2002) and Hollander et al. (2001 a,b, 2003) studied agglomeration in a stirred vessel by adding a single transport equation for the particle number concentration mQ (actually, the first moment of the particle size distribution)... 

X-ray line broadening provides a quick but not always reliable estimate of the particle size. As Cohen [9] points out, the size thus determined is merely a ratio of two moments in the particle size distribution, equal to /. Both averages are weighted by the volume of the particles, and not by number or by surface area, as would be more meaningful for a surface phenomenon such as catalysis. Also, internal strain and instrumental factors contribute to broadening. 

This expression, which only depends on the particle size distribution through the generalized mean size m, is our desired moment free energy. 

Fine-grained dolomite flour was used in the experiments. Raw material for testing was prepared on the basis of five commercially available size fractions of this material classified as <10, <15, <60, <100 and <250 pm. The particle size distribution of each fraction was determined using a laser particle size analyser Analysette 22 and described by the statistical moments. 

Studies on the application of the theory of statistical moments in the description of grinding in ball mills have been carried out in the Department of Process Equipment, Lodz Technical University [1-3]. The research was carried out in a laboratory scale for selected mineral materials. Results obtained confirmed applicability of the theory of statistical moments in the description of particle size distribution during grinding. 

Here, z0 10-10 -10 13s is the inverse attempt frequency that depends on the damping of the magnetic moments by the phonons. The superparamagnetic blocking occurs when r equals the measuring time of each experimental point, te, therefore TB = all / kB ln(/(, / r0 ), where a is a constant that depends on the width of the particle size distribution. 

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... 

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... 

For polydisperse systems, replacement of equation 13 into equation 2 for every particle diameter, yields an approximation to the turbidity in terms of ratios of moments of the particle size distribution without having to make assumptions regarding the shape of the distribution ... 

Note that the formulation stated by equation 18 is different from that of Dobbins et al (Equation 11) in that we are seeking a solution to the integral equation 18 in terms of one or several of the moments of the particle size distribution and in that, simultaneously, we are inquiring as to the information content of the scattering function with regards to the particle size distribution. 

By first replacing equation 23 into 10, it can be readily shown, that the resulting integral can be expresed as a weighted sum of the moments of the particle size distribution... 

In the small particle size regime, two equivalent formulations lead to the interpretation of the data in terms of ratios of moments of the particle size distribution or in terms of powers of the D32 average (equations 15 and 20). It is clear that in either case a sufficient number of terms in the series has to be included in order to account for the behavior of the extinction as function of a. The number of terms required cannot be decided a priori, rather the data itself has to dictate how many terms in the power series approximation the measurements can detect. 

This appendix reports the weights for the moments of the particle size distribution obtained from an eight order Taylor Series approximation to the scattering efficiency for the anomalous diffraction case... 

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. 

Two main techniques have been used to determine the particle size distribution of colloidal systems PCS and electron microscopy including both SEM and TEM. The QELS technique for Brownian moment measurement, offers an accurate procedure for measuring the size distribution of nanoparticles. The PCS technique does not require any particular preparation for analysis and is excellent due to its efficiency and accuracy. However, its dependency on the Brownian movement of particles in a suspended medium may affect the particle size determination. 

Very often one does not require as much detail as presented in Figure 2 and the model can be simplified considerably. For example, one may only be interested in the first few moments of the latex particle size distribution, F(V,t) so as to get a mean and variance of the distribution. This can be readily calculated from the definition of the jth moment ... 

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