Moments of particle size

Fig.2.3. Moments of particle size distribution for nucleation and growth of sulfuric acid aerosol from sulfuric acid vapor at low humidity. Number concentration, cm , x 2E5, 0 mass concentration, g/ccxlE-13, + surface, cm xlE-7, 1/Number, cm ...

Unzueta and Forcada [93] developed a mechanistic model for the emulsion copolymerization of methyl methacrylate and n-butyl acrylate stabilized by mixed anionic and nonionic surfactants, which was verified by the experimental data. This model is based on the mass and population balances of precursor particles and the moments of particle size distribution. It is sensitive to such parameters as the composition of mixed surfactants and the total surfactant concentration. A competitive particle nucleation mechanism is incorporated into the model to successfully simulate the evolution of particle nuclei during polymerization. 

Thus, we plot M(x,t)IMi vs xls(t). As noted earlier, the cluster size distribution and the first moment of the size distribution are averaged over the entire journal bearing. As indicated by the behavior of P in Fig. 39b, the cluster size distribution becomes self-similar when the average size is about 10 particles per cluster. 

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).

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. 

International Standards Organization is preparing a Standard, ISO/FDIS 9276-2 Calculation of average particle sizes/diameters and moments from particle size distributions. For a discussion of this draft Standard see Alderliesten [5]. 

This is a Fredholm integral equation of the first kind. The regularized solution to this equation has been applied to the measurement both for the moments and the size distribution of a wide range of latices [46]. K has been given by van de Hulst [45] in terms of particle size/refractive index domain. Mie theory applies to the whole domain but in the boundary regions simpler equations have been derived. 

Moments Moments represent a PSD by a single value. With the help of moments, the average particle sizes, volume specific surfaces, and other mean values of the PSD can be calculated. The general definition of a moment is given by (ISO 9276, Part 2 Calculation of Average Particle Sizes/Diameters and Moments from Particle Size Distributions)... 

Example 2 The Sauter mean diameter and the volume weighted particle size and distribution given in Table 21-1 can be calculated by using FDIS-ISO 9276-2, Representation of Results of Particle Size Analysis—Part 2 Calculation of Average Particle Sizes/Diameters and Moments from Particle Size Distributions via Table 21-2. 

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. 

As mentioned above, macroscale models are written in terms of transport equations for the lower-order moments of the NDF. The different types of moments will be discussed in Chapters 2 and 4. However, the lower-order moments that usually appear in macroscale models for monodisperse particles are the disperse-phase volume fraction, the disperse-phase mean velocity, and the disperse-phase granular temperature. When the particles are polydisperse, a description of the PSD requires (at a minimum) the mean and standard deviation of the particle size, or in other words the first three moments of the PSD. However, a more complete description of the PSD will require a larger set of particle-size moments. 

As it is possible to see, the drift term has disappeared since the continuous growth of particle size does not change the total number concentration (if Gl > 0). However, N is influenced by the rate of formation of particles (e.g. nucleation), and the rates of aggregation and breakage, which cause appearance and disappearance of particles. These processes are all contained in the source term /tL.o The third-order moment mL,3 is related to the fraction of volume occupied by particles with respect to the suspending fluid and can be easily found fromEq. (2.18) ... 

Another approach being developed that shows great promise is to use the modes of the size distribution to account for the effect of particle size distribution on flow. Brown et al. (1995) have shown that this modal approach can be applied to polydisperse aerosol particles in air flow. Here, moments of the PSD are used to couple the evolving PSD with the IC model. The additional computational load imposed is relatively low. 

The moments of the size distribution, and can be used to determine the relative extent of Ag condensation and particle coagulation on the growth mechanism. By measuring the arithmetic mean radius, r =Y rJN > cube-mean... 

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