Particle size, statistics distribution, characterization

The concepts described in Section 4.1 apply equally well to particle size distributions and to other quantities commonly characterized by number statistics— for example, dollars, populations, or the dimensions of machined parts. What makes particle size statistics more complicated is that we frequently measure or need to know some quantity that is proportional to particle size raised to a power (moment), such as surface area, which is proportional to d, or mass, which is proportional to d. There is no counterpart of this in number statistics, for what is the meaning of or (people) The need to use moment averages for particle statistics arises because aerosol size is frequently measured indirectly. For example, if you have a basket of apples of different sizes, you could determine the average size by measuring each apple with calipers, summing the results, and dividing by the total mun-ber of apples. This procedure is direct measurement. If, however, each apple were... 

When a distribufion of particle sizes which must be collected is present, the aclual size distribution must be converted to a mass distribution by aerodynamic size. Frequently the distribution can be represented or approximated by a log-normal distribution (a straight line on a log-log plot of cumulative mass percent of particles versus diameter) wmich can be characterized by the mass median particle diameter dp5o and the standard statistical deviation of particles from the median [Pg.1428]

The extraction of more complex particle size distributions from PCS data (which is not part of the commonly performed particle size characterization of solid lipid nanoparticles) remains a challenging task, even though several corresponding mathematical models and software for commercial instruments are available. This type of analysis requires the user to have a high degree of experience and the data to have high statistical accuracy. In many cases, data obtained in routine measurements, as are often performed for particle size characterization, are not an adequate basis for a reliable particle size distribution analysis. 

This chapter has described the various techniques of ceramic powder characterization. These characteristics include particle shape, surface area, pore size distribution, powder density and size distribution. Statistical methods to evaluate sampling and analysis error were presented as well as statistical methods to compare particle size distributions. Chemical analytical characterization although veiy important was not discussed. Surface chemical characterization is discussed separately in a later chapter. With these powder characterization techniques discussed, we can now move to methods of powder preparation, each of which 3uelds different powder characteristics. 

Other statistical measurements used in geology for particle size distribution characterization (moment, quartile and others) have been defined [135,136]. 

Log normal distribution (logarithmic normal distribution). A statistical probabiUty-density function, characterized by two parameters, that can sometimes provide a faithful representation of a polymer s molecular-weight distribution or the distribution of particle sizes in ground, brittle materials. It is a variant of the familiar normal or Gaussian distribution in which the logarithm of the measured quantity replaces the quantity itself. Its mathematical for is... 

In most applications, the systems and processes contain large amounts of particles with size distribution each size may also possess a distinguished shape. To describe properly these systems and processes for design and analysis, they need to be adequately characterized to reflect their physical and chemical potentials. In the following sections, different average particle diameter definitions are introduced along with statistical descriptions of particle size distribution. Depending on applications, one definition may be more suitable than others. Thus care must be exercised to select the proper characterization for each process. 

Statistically, the particle size distribution can be characterized by three properties mode, median, and mean. The mode is the value that occurs most frequently. It is a value seldom used for describing particle size distribution. The average or arithmetic mean diameter, d, is affected by all values actually observed and thus is influenced greatly by extreme values. The median particle size, is the size that divides the frequency distribution into two equal areas. In practical application, the size distribution of a typical dust is typically skewed to the right, i.e., skewed to the larger particle size. The central tendency of a skewed frequency distribution is more adequately represented by the median rather than by the mean (see Fig. 9). Mathematically, the relationships among the mean, median, and mode diameter can be expressed as... 

Most physical properties of a particulate system are ensembles or statistical values of the properties from their individual constituents. Commonly evaluated particle geometrical properties are counts, dimension (size and distribution), shape (or conformation), and surface features (specific area, charge and distribution, porosity and distribution). Of these properties, characterization of particle size and surface features is of key interest. The behavior of a particulate system and many of its physical parameters are highly size-dependent. For example, the viscosity, flow characteristics, filterability of suspensions, reaction rate and chemical activity of a particulate system, the stability of emulsions and suspensions, abrasiveness of dry powders, color and finish of colloidal paints and paper coatings, strength of ceramics, are all dependent on particle size distribution. Out of necessity, there are many... 

The size of the particles of a monodisperse aerosol is conq)letely defined by a single parameter, the particle diameter. Most aerosols, however, are polydisperse and may have particle sizes that range over two or more orders of magnitude. Because of this wide size range and the fact that the physical properties of aerosols are strongly dependent on particle size, it is necessary to characterize diese size distributions by-statistical means. For the purposes of this chapter, we neglect the effect of particle shape and consider only spherical particles. 

In this section, we characterize size distributions and their properties by using examples based on a specific set of particle size data. The result of a careful size analysis might be a list of 1000 particle sizes. In some situations, keeping the data in this form may be desirable— for example, if the list is stored in a computer. In most situations, however, we would like to have a picture of how the particles are distributed among the various sizes and to be able to calculate several different kinds of statistics that describe the properties of the aerosol. For that purpose, a list of 1000 numbers is an awkward format, so it is necessary to resort to descriptive statistics to summarize the information. 

Figure 2.6 The figure shows the different types of analyses that can be performed on chemical imaging data. The types of analyses that are performed can be grouped into three categories component abundance estimation, statistical analysis of component distribution, and morphological analysis of discrete particles. All three analyses are used to make inter- and intrasample comparisons, generating abundance and content uniformity estimates, sample heterogeneity and blend uniformity characterization, as well as domain statistics and domain size uniformity data.

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