The size distribution

Since the size of atmospheric particles covers several orders of magnitude (see Subsection 4.1.1) the concentration alone is not sufficent to characterize atmospheric particles. For more complete aerosol characterization the size 

It is obvious that a similar size distribution function can be given for the surface, volume and mass of aerosol particles. Thus, e.g. the volume concentration (aerosol volume per unit volume of air) distributes according to particle radius in the following way  

It goes without saying that the number size distribution may be converted to volume size distribution, given an assumed particle shape. Furthermore by using a constant particle density the size distribution of particle mass can be calculated from equation [4.9] by a simple calculation. 

It is customary, in the interconversion of these distribution functions, to assume that the particles are spherical this simplifies the mathematics, but is somewhat questionable physically. The method of measurement determines the nature of the reported radii of these hypothetical spheres e.g. in the case of microscopic sizing, the so-called surface radius is obtained, which is the radius of a circle having the same surface area as the orthogonal projection of the particle. 

On the basis of his atmospheric impactor measurements Junge (1963) proposed a power law to describe the size distribution of large and giant particles  

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The distribution curves may be regarded as histograms in which the class intervals (see p. 26) are indefinitely narrow and in which the size distribution follows the normal or log-normal law exactly. The distribution curves constructed from experimental data will deviate more or less widely from the ideal form, partly because the number of particles in the sample is necessarily severely limited, and partly because the postulated distribution... 

Now the relationship between v and A is given by the size distribution curve the value of A merely represents the lengths of the particles measured in terms of a particular, arbitrary, unit. Thus, if the size distribution curve remains of exactly the same shape during the grinding process, the values of... 

Most tests of the validity of the BET area have been carried out with finely divided solids, where independent evaluation of the surface area can be made from optical microscopic or, more often, electron microscopic observations of particle size, provided the size distribution is fairly narrow. As already explained (Section 1.10) the specific surface obtained in this way is related to the mean projected diameter through the equation... 

Everett concludes that in systems where pore blocking can occur, pore size distribution curves derived from the desorption branch of the isotherm are likely to give a misleading picture of the pore structure in particular the size distribution will appear to be much narrower than it actually is. Thus the adsorption branch is to be preferred unless network effects are known to be absent. 

Whereas at the lower end of its range mercury porosimetry overlaps with the gas adsorption method, at its upper end it overlaps with photomicrography. An instructive example is provided by the work of Dullien and his associates on samples of sandstone. By stereological measurements they were able to arrive at a curve of pore size distribution, which was extremely broad and extended to very coarse macropores the size distribution from mercury porosimetry on the other hand was quite narrow and showed a sharp peak at a much lower figure, 10nm (Fig. 3.31). The apparent contradiction is readily explained in terms of wide cavities which are revealed by photomicrography, and are entered through narrower constrictions which are shown up by mercury porosimetry. 

In a pore system composed of isolated pores of ink-bottle shape, the intrusion curve leads to the size distribution of the necks and the extrusion curve to the size distribution of the bodies of the pores. In the majority of solids, however, the pores are present as a network, and the interpretation of the mercury porosimetry results is complicated by pore blocking effects. 

Experimental exponents for cake thickness vary from 0.5 to as much as 3.0. The theoretical value of //2 may be approached only by incompressible cakes of a narrow range of sizes. The proper and characteristic value for the mean particle size, d, is difficult to ascertain. In practice, the most finely divided particles, eg, 10—15 wt % of soHds, almost whoUy determine the Hquid content of a cake, regardless of the rest of the size distribution. It seems reasonable to use a d closely related to Hquid content, eg, the 10% point on a cumulative weight-distribution curve. 

Particle Size Distribution. Almost every feed slurry is a mixture of fine and coarse particles. Performance depends on the frequency of distribution of particle size ia the feed. Figure 5 shows that whereas all of the coarse particles having a diameter greater than some are separated, fewer of the very fine particles are, at any given feed rate. The size distribution frequency of particles ia feed and centrate for a fine and coarse feed are quite different. More coarse particles separate out than fine ones. Classification of soHds by size is often done by centrifugal sedimentation. 

The characteristic separation curve can be deterrnined for any size separation device by sampling the feed, and coarse and fine streams during steady-state operation. A protocol for determining such selectivity functions has been pubHshed (4). This type of testing, when properly conducted, provides the relationships among d K, and a at operating conditions. These three parameters completely describe a size separation device and can be used to predict the size distribution of the fine and coarse streams. 

There are relationships between the independent size separation device parameters and the dependent size separation efficiencies. For example, the apparent bypass value does not affect the size distribution of the fine stream but does affect the circulation ratio, ie, the ratio of the coarse stream flow rate to the fine stream flow rate. The circulation ratio increases as the apparent bypass increases and the sharpness index decreases. Consequendy, the yield, the inverse of the circulating load (the ratio of the feed stream flow rate to the fine stream flow rate or the circulation ratio plus one), decreases hence the efficiencies decrease. For a device having a sharpness index of 1, the recovery efficiency is equal to (1 — a). 

A wide variety of particle size measurement methods have evolved to meet the almost endless variabiUty of iadustrial needs. For iastance, distinct technologies are requited if in situ analysis is requited, as opposed to sampling and performing the measurement at a later time and/or in a different location. In certain cases, it is necessary to perform the measurement in real time, such as in an on-line appHcation when size information is used for process control (qv), and in other cases, analysis following the completion of the finished product is satisfactory. Some methods rapidly count and measure particles individually other methods measure numerous particles simultaneously. Some methods have been developed or adapted to measure the size distribution of dry or airborne particles, or particles dispersed inhquids. 

Droplet Size Distribution. Most sprays comprise a wide range of droplet sizes. Some knowledge of the size distribution is usuaUy required, particularly when evaluating the overaU atomizer performance. The size distribution may be expressed in various ways. Several empirical functions, including the Rosin-Rammler (25) andNukiyama-Tanasawa (26) equations, have been commonly used. 

The larger the value of n, the more uniform is the size distribution. Other types of distribution functions can be found in Reference 1. Distribution functions based on two parameters sometimes do not accurately match the actual distributions. In these cases a high order polynomial fit, using multiple parameters, must be considered to obtain a better representation of the raw data. 

Precipitated Calcium Carbonate. Precipitated calcium carbonate can be produced by several methods but only the carbonation process is commercially used in the United States. Limestone is calcined in a kiln to obtain carbon dioxide and quicklime. The quicklime is mixed with water to produce a milk-of-lime. Dry hydrated lime can also be used as a feedstock. Carbon dioxide gas is bubbled through the milk-of-lime in a reactor known as a carbonator. Gassing continues until the calcium hydroxide has been converted to the carbonate. The end point can be monitored chemically or by pH measurements. Reaction conditions determine the type of crystal, the size of particles, and the size distribution produced. 

Size Distribution Relationships. Different models have been used to describe the size distribution of particles experiencing single and multiple fractures. A model based on fracture at the site of the weakest link and a distribution of weakest links in the system gave results that could be described as well by the Rosin-Rammler relation (56). The latter is based on the concept that fracture takes place at pre-existing flaws that are distributed randomly throughout the particle. 

Products. In all of the instances in which crystallization is used to carry out a specific function, product requirements are a central component in determining the ultimate success of the process. These requirements grow out of how the product is to be used and the processing steps between crystallization and recovery of the final product. Key determinants of product quaHty are the size distribution (including mean and spread), the morphology (including habit or shape and form), and purity. Of these, only the last is important with other separation processes. 

Identification of an initial condition is difficult because of the problem of specifying the size distribution at the instant nucleation occurs. The difficulty is mitigated through the use of seeding which would mean that the initial population density function would correspond to that of the seed crystals ... 

In industrial practice, the size-distribution cui ve usually is not actually construc ted. Instead, a mean value of the population density for any sieve fraction of interest (in essence, the population density of the particle of average dimension in that fraction) is determined directly as AN/AL, AN being the number of particles retained on the sieve and AL being the difference between the mesh sizes of the retaining sieve and its immediate predecessor. It is common to employ the units of (mm-L)" for n. 

Had only the growth rate been known, the size distribution of the solids could have been calculated from the equation... 

Size Recovery and Yield Centiifuges have been apphed to classify polydispersed fine particles. The size distribution of the paiticles is quantified by the cumulative weight fraction F less than a given particle size d for both the feed and the centrate streams. It is measured by a particle size counter which operates based on piinciples such as sedimentation or optical scatteiing. 

The breakage function AB gives the size distribution of product breakage of size u into all smaller sizes k. Since some fragments from size u are large enough to remain in the range of size u, the term AB is not zero, and... 

The initial distribution of binding fluid can have a pronounced influence on the size distribution of seed granules, or nuclei, which are formed from fine powder. Both the final extent of and the rate at... 

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