Properties of Size Distributions

It is often convenient to summarize the features of an aerosol distribution using one or two of its properties (mean particle size, spread of distribution) than by using the full function nN(Dp). Growth of particles corresponds to a shifting of parts of the distribution to larger sizes or simply an increase of the mean particle size. These properties are called the moments of the distribution, and the two most often used are the mean and the variance. 

Let us assume that we have a discrete distribution consisting of M groups of particles, with diameters Dk and number concentrations Nk, k = 1.2. M. The number concentration of particles is therefore 

The variance a2, a measure of the spread of the distribution around the mean diameter Dp, is defined by 

A value of a2 equal to zero would mean that every one of the particles in the distribution has precisely diameter Dp. An increasing a2 indicates that the spread of the distribution around the mean diameter Dp is increasing. 

We will usually deal with aerosol distributions in continuous form. Given the number distribution nN(Dp), (8.27) and (8.28) can be written in continuous form to define the mean particle diameter of the distribution by 

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Minimum Fluidizing Velocity U,nj, the minimum fluidizing velocity, is frequently used in fluid-bed calculations and in quantifying one of the particle properties. This parameter is best measured in small-scale equipment at ambient conditions. The correlation by Wen audYu [A.l.Ch.E.j., 610-612 (1966)] given below can then be used to back calculate d. This gives a particle size that takes into account effects of size distribution and sphericity. The correlation can then be used to estimate U, at process conditions, if U,nj cannot be determined experimentally, use the expression below directly. 

Although they are a relatively small volume product—approximately 75,000 tons produced in 1949 (126)—interest in asphalt emulsion has continued at a high level. Abraham (6) has reviewed the patent literature relative to the types of emulsifying agents used, while commercial practice has been discussed by Day (16). The most common emulsifiers are sodium or potassium soaps of tall oil, abietic acid, or Vinsol resin, or colloidal clays such as bentonite for adhesive base emulsions. Lyttleton and Traxler (53) studied the flow properties of asphalt emulsions, and Traxler (122) has investigated the effect of size distribution of the dispersed particles on emulsion viscosity. A decrease in particle size uniformity was found to be accompanied by a decrease in consistency because particles of various size assume a more loosely packed condition than do those of the same size. 

Size-resolved chemical information is much more difficult to obtain. The many applications of the differential mobility analyzer in measuring properties of size-classified particles are important tools for the characterization of aerosol systems, but the approaches demonstrated to date yield limited data. Vapor pressures, surface tension, and optical absorption have been measured on mobility-classified aerosols. Direct measurements of the distribution of chemical composition with particle size are needed. Elemental... 

Farah, Z., Riiegg, M. 1991. The creaming properties and size distribution of fat globules in camel milk. J. Dairy Sci. 74, 2901-2904. 

Mugnai a., Fiocco G. and Grams G., Effects of aerosol optical properties and size distributions on heating rates induced by stratospheric aerosols. Q. J. Roy. Met. Soc., 104, 783-796 (1978). 

Although surface phenomena determine the fundamental properties of emulsions in terms of size distributions and stability, the bulk properties or bulk compositions are the yardsticks by which plant operators and process personnel measure process efficiency. Accurate determination of the oil, water, and solids (if present) is therefore one of the most important aspects of emulsion characterization. 

Before discussing our method for determining particle size, it is necessary to briefly review the definition of size distribution. If all particles of a given system were spherical in shape, the only size parameter would be the diameter. In most real cases of irregular particles, however, the size is usually expressed in terms of a sphere equivalent to the particle with regard to some property. Particles of a dispersed system are never of either perfectly identical size or shape A spread around the mean distribution) is found. Such a spread is often described in terms of standard deviation. However, a frequency function, or its integrated (cumulative) distribution function, more properly defines not only the spread but also the shape of such a spread around the mean value. This is commonly referred to as the particle size distribution (PSD) profile of the dispersed sample. 

This average property of the distribution is defined through the number density itself and it represents the mean particle size with respect to the number of particles in the system. Of course, other definitions of the mean particle size are possible, as will become clear below. If we define the th moment of the length-based NDF as... 

Determination of size distributions can be done by several methods, each having its limitations and pitfalls. Accurate determination is notoriously difficult. Systematic and random errors are involved. Several methods are indirect ones, determining some macroscopic property, for instance, a light scattering spectrum. The conversion of these data to a size distribution is generally difficult and may lead to considerable error. 

The nature of size distributions in etched track lengths is further complicated by the difficulty in determining the dimensions of unetched, or latent tracks. While there were some assumptions about the shape of a latent fission track (see Carlson 1990), their geometry has proven to be notoriously difficult to determine (Kobetich and Katz 1968). In a classic case of the act of observation modifying the observed property—radiolytic annealing was observed in apatite when samples were exposed to the electron beams... 

Color, clarity, particulate matter, pH, formation of precipitate, osmolarity. Powder for Injection Solution color, water content, reconstitution time, and in use stabdity of solution Polymorphic conversion, pH, viscosity and other rheological properties, particle size distribution/morphology/habits, settUng/caking/redispersibility, content uniformity, dissolution profile... 

One step further to tune the electronic properties of a 2-D array of Ag nanoparticles with a size distribution of 7% was reported by Remade et al. [94]. These authors discussed the experimental and computational results of temperature-dependent conductivity measurements as a function of size distribution, compression of the array, and the applied gate voltage. From the temperature-dependent source-drain measurements they obtained sigmoidal-shaped and nonlinear curves (Figure 5.65). 

By using the distributions with varying and a, the variation in the properties of the distribution with time can be analyzed. For the distributions with a broad width, the standard deviation of the distribution decreases to the characteristic steady-state value, whereas for the distributions with a narrow width, the standard deviation increases to the steady-state value. We may interpret this behavior during the transient regime to mean that the scaling regime acts as a strong attractor for the evolution of the precipitate size distribution. 

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