Size distribution particle

1 PARTICLE SIZE DISTRIBUTION General Discussion and Principle 

Particle size analysis separates the inorganic mineral portion of the soil into classified grades according to particle size and determines their relative proportion by weight. The determination involve three district stages viz. 

A particle falling in a vacuum will encounter no resistance, as it is accelerated by gravity and hence its velocity increases as it falls. A particle falling in a fluid on the other hand will encounter a fi ictional resistance proportional to the product of its radius and velocity and to the viscosity of the fluid. 

The resisting force or the viscous drag due to friction was shown by Stokes G.G. (1851) to 

T = viscosity of the fluid r = radius of the particle V = velocity of the particle 

However, the term particle size does not describe an unambiguous quantity, but rather a variety of measurands which are related to the outer particle dimensions. Indeed, particle size is always derived from a geometrical or physical property. If the property is not a length, it is usually converted to a diameter of a sphere being equivalent to the particle with regard to this property. The corresponding diameter is called equivalent diameter (e.g. equivalency in volume V leads to volume equivalent diameter xty. Table 2.1 lists some of these properties and the associated equivalent diameters. 

In general, particle properties vary within a particle system and the particle size X should be regarded as a distributed parameter. Size distributions can be depicted by the cumulative function Q, the density function qr, or the transformed density function q  

The details of size distributions are frequently summarised in a few parameters that reflect the average size or the polydispersity of the particle system  

AU three types of location parameters— modes, medians and means—are fre-quentiy used in practice yet their meaning and robustness is not identical. 

The modal values are characteristic values of the density functions q and q, which appear to be the immediate result of most sizing techniques (e.g. line-start disc centrifugation, spectroscopic techniques). Modal values represent the most 

Because the average particle size is often not adequate to characterize a sample, a PSD, either volume- (i.e. mass-) or number-weighted, must usually be determined. In the former case. 

Number, volume mean diameter c/nv = Length, surface mean diameter c/ls = 

Let us deal with the relationship between the PSD and probability distribution. 

The particle size distribution can be plotted in terms of the cumulative percent oversize or undersize in relation to the particle diameters. The weight, volume, number, and so on are used for percentage. By differentiating the cumulative distribution with respect to the diameter of the particle, the PSD can be obtained. 

Since the particle volume is proportional to x, the larger particles contribute much more to the volume distribution than to the number distribution. This can be seen in the shapes of the cmves in Fig. 2.3.3. The larger particles contribute negligibly to the number distribution, which appears to go 

If one of the distributions is known, the others can be calculated at least this is the case if one assumes the particles to be spherical (Allen, 1990). For instance, we can calculate the volume density distribution from the number density distribution. Since the number fraction of particles in the diameter interval x — 1/2 dx and x + 1/2 dx is /jv(x)dx, then  

We have to choose the proportionality constant so that the area under our volume density distribution becomes unity  

In addition to density distributions, a very widely used method of reporting a particle size distribution is through the use of its cumulative undersize distribution F x), defined as the fraction of particles with a diameter less than X. F x) is related to the density function /(x) by  

In Fig. 2.3.4 the cumulative undersize distributions corresponding to the density functions in Fig. 2.3.3 are shown. 

Relative viscosity for bimodal distributions of hard spheres as function of the proportion of small particles (0,). The parameter is the ratio of particle radii. Note that increasing total volume fraction from 0.6 (dashed line) to 0.65 has a large effect. After Chong et al. (1971). 

Very few process slurries contain particles of uniform size. A large proportion of slurries, processed by decanters, contain solids which have a particle size distribution which conforms closely to a logarithmic probability distribution. The logarithmic probability equation was derived by Hatch and Choate [3] in 1929  

Integrating this equation gives the formula for a cumulative number distribution  

It can be shown, by using equation (4.16), that the equation for the cumulative weight distribution is similar  

Since Hatch and Choate first published their equation, special graph paper has been developed and printed whereby plotting the cumulative percent of particles by number or weight, oversize or undersize, against particle size, results in a straight line. The mathematics of the distribution are such that one can readily transfer between weight and number distributions, and even area and diameter distributions [4], 

For example, the total surface area, Af, of the solids in the slurry can be calculated  

Packing materials are characterized by the average diameter of their particles and the distribution of the particle size around the average value. 

Particle size distribution itself does not affect chemical behavior of HPLC adsorbent, although it is known to influence the efficiency of packed column. Packings with wide particle size distribution contain a significant amount of 

Halasz and Naefe [9] and Majors [10] suggested that if the distribution is not wider than 40% of the mean, then acceptable flow resistance and column efficiency can be obtained. The narrower the particle size distribution, the better and the more reproducibly the columns could be packed. Generally accepted criteria is that 95% of all particles should be within 25% region around the mean particle diameter [11,12]. 

Surface area of HPLC adsorbents is probably the most important parameter, although it is almost never used or accounted for in everyday practical chromatographic work. As shown in the theory chapter (see Chapter 2), HPLC retention is proportional to the adsorbent surface area. The higher the surface area, the greater the analyte retention, although as we discuss later, depending on the surface geometry, analytes of a different molecular size could effectively see different surface areas on the same adsorbent. 

The region between 0.05 and 0.25 relative pressures is called the BET region, and it is used for the determination of the so-called monolayer capacity—the amount of nitrogen molecules adsorbed on the sample surface in a compact monolayer fashion. The BET equation represents the dependence of amount of adsorbed nitrogen as a function of the relative equilibrium pressure iplpf)  

The particle sizes relevant for inorganic pigments stretch between several tens of nanometers for transparent pigment types to approximately two micrometers. For practical applications it is very desirable to determine not only the mean particle size but also the whole distribution. These parameters must not be confused with the crystal size determined by X-ray diffraction, as pigment particles usually are not monocrystals. 

The determination of the particle size distribution is a complex issue and the subject of voluminous monographs [1.15] so only an introduction to the questions relevant for applications concerning inorganic pigments can be given. 

Two methods are mainly used for the determination of particle size of inorganic pigments Sedimentation methods (centrifuges) and Fraunhofer diffraction with additional correction due to Mie scattering. 

When evaluating the results of these measurements one has to remember that a property of the particles (light scattering or the velocity of sedimentation) is determined. With models relying on a number of assumptions (for example that all particles are spherical) and further input (for example the complex index of refraction or the density) the particle size distribution is calculated in the final step. Applying the results of the measurement this and other deviations from the model have to be taken into account. Different measurement techniques usually result in different results for the measurements of particle size distributions. 

While using a dilute suspension in a pump-through cell there is the possibility to determine even particles far from the main distribution, present only in minor amounts, with high precision. This makes it possible to catch the particles several times, resulting in good reproducibility of the measurement result. 

Any sample of parficulafe food solids, whefher nafurally occurring or the result of a manufacfuring process, will confain a disfribufion of particle sizes. However, despite the fact that the existence of a disfribu-tion of sizes can affecf fluidized bed behaviour very significanfly, predictive models, especially those which have been proposed for fhe prediction of minimum fluidizing velocify, usually require a single 

The characteristics of a particle size distribution can be defined as fhe fofal number, fofal lengfh, total surface area and fofal volume of fhe parficles. A disfribufion of particle sizes can be represented by a set of uniformly-sized parficles which retains two characteristics of fhe original disfribufion. The mean particle size of a disfribufion is fhen equal fo fhe size of fhe uniform particles with respect to the two characteristics. Thus, for fhe definifion of mean parficle size, Mugele and Evans (1951) proposed 

Substituting all the possible combinations of characteristics, i.e. values of p and q, info equation 1.10 gives rise to a number of differenf definitions of the mean size of a distribution. At minimum fluidization the drag force acting on a particle due to the flow of fluidizing gas over the particle is balanced by the net weight of fhe particle. The former is a function of surface area and the latter is proportional to particle volume. Consequently the surface-volume mean diameter, with p = 2 and = 3, is the most appropriate particle size to use in expressions for minimum fluidizing velocity. It is defined by equafion 1.11 

The definition of the surface-volume mean diamefer given by equation 1.11 must be modified for use wifh dafa from a sieve analysis. By assuming that the shape and density of fhe particles are constant for all size fractions, a number distribution can be transformed fo a mass distribution (Smith, 2003) and therefore the surface-volume diameter becomes 

For pine, salt cedar, and juniper, with aspect ratio 4.0, 3.2, and 4.4, respectively, image analysis using more than 500 particles per species showed the following particle size (in terms of area, in square microns) distribution [130]  

There are two components to the reflectance spectra as defined by Kubelka-Munk theory, namely absorption and scatter. RutQe pigment particles exhibit maximum scatter at approximately 0.2 pm. The scatter decreases slowly with increasing particle size, but drops off rapidly with finer particle size, the particles becoming transparent at sizes 50 nm. Therefore, the reflectance spectra will be a function of the particle size distribution of the pigment. 

Color values are a derived set of numbers calculated from the reflectance spectra. A commonly used system is CIE L a b with values calculated for Illuminant D65 and a 10° observer. The Lvalue, with a range from O(black) to lOO(white), correlates to lightness, -i-a =redness, -a =greenness, -i-b =yellowness, and -b =blue-ness. Texts can be consulted for greater detail [47]. 

For bright masstone appUcations, it is important to optimize the cleanness or color purity of a pigment The cleaimess can be roughly interpreted from the reflectance spectra by the difference between the maximum and rriinimum reflectances. A more accurate approach is to develop modified color values. Correlations can be developed from grind studies to account for changes in scatter due to differences in particle size distributions. For the NiSbTi yellows, a formula to calculate b values (e.g. b =b -3AL ) can be developed for a particular polymer system to account for the increased scatter and whiteness from finer-sized pigments. 

It must be remembered that pigments are controlled for color properties. Although the particle size affects the color properties, so do other factors. Lot-to-lot variations in mean particle size of 10% are not uncommon. 

Although a modern instrument may indicate 100% 3 pm, it is important to realize that there will still be a certain portion of particles 3 pm. In fact, most pig- 

It seems reasonable to conclude that experimentally determined specific surface areas can only qualitatively relate to the physical characteristics of organic pigments. Instead, their value emerges in combination with other physical or physico-chemical parameters or in the context of application properties such as oil absorption [16] or wettability (Sec. 1.6.5). 

The particles of a synthetic pigment, far from being uniform, cover a more or less wide range of sizes. Normally discontinuously produced pigment batches are usually combined so as to yield mixtures that meet the technical standards of certain target applications. This explains why it is possible for different batches of the same pigment to exhibit somewhat divergent particle size distributions. 

There are various methods for the determination of the size distribution of organic pigment particles, the most common are sedimentation techniques in ultracentrifuges and specialized disk centrifuges as well as electron microscopy. These methods require considerable experimental skill, since the results depend largely on sample preparation and especially on the quality of the dispersion. 

Size analyses are commonly carried out by mixing the pigment powder with an organic solvent or with water and adding appropriate surfactants to enhance the dispersibility of the powder. Aqueous dispersions frequently undergo size separation in ultracentrifuges, while organic solvents are more appropriate for electron microscopic techniques. 

The three talcs have the same composition (talc 40-41%, chlorite 57-59%). The differences in properties can be attributed to the way in which they were processed. 

A general conclusion from this is that industry can manufacture a variety of particle size distributions tailored to the requirements of the application. Particle 

Graphing does not always provide the best means of comparing particle size distribution unless the materials are very divergent (as the selected examples). A mathematical form of data presentation is sometimes more convenient. Granulometry in number and in weight is calculated from the following equations 

The results are either expressed as a ratio - L /Ln or a dispersity factor is calculated  

In a study of the synthesis of a monodisperse colloidal silica, it was possible to control the particle size distribution. A range of products was obtained with ratios Lw/Ln=l. 03-33. This again shows that it is possible to tailor particle size to the requirements. We now need to determine what the ratio should be and why. 

Modification and optimization of a slurry, whether amorphous or crystalline, in the laboratory can yield significant improvements in filtration rates. By modeling the process in the laboratory, one can model what is occurring in the plant. 

It is evident that attention paid in the laboratory to the factors affecting particle size distribution will save on capital investments made for separation equipment and downstream process equipment. Specific cake resistance (a) can be determined in the laboratory over the life of a batch, to evaluate if time in the vessel and surrounding piping system is degrading the product s particle size to the point it impedes filtration, washing and subsequent drying. 

In the plant, the type of pump and piping system used to feed the filter are often of great importance, as time spent on crystallization and improving crystal size and particle size distributions can be quickly undone through particle damage. Recirculation loops and pumps for slurry uniformity may not always be necessary. 

A review of the most commonly used process pumps are discussed below  

Diaphragm pumps. These offer very gentle handling of slurries and are inexpensive and mobile. However, the pulsating flow can cause feeding and distribution problems in some types of filtration systems, e.g., conventional basket centrifuges. They can also interfere with process instrumentation e.g., flowmeters and loadcells. 

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Rowell and co-workers [62-64] have developed an electrophoretic fingerprint to uniquely characterize the properties of charged colloidal particles. They present contour diagrams of the electrophoretic mobility as a function of the suspension pH and specific conductance, pX. These fingerprints illustrate anomalies and specific characteristics of the charged colloidal surface. A more sophisticated electroacoustic measurement provides the particle size distribution and potential in a polydisperse suspension. Not limited to dilute suspensions, in this experiment, one characterizes the sonic waves generated by the motion of particles in an alternating electric field. O Brien and co-workers have an excellent review of this technique [65]. 

Fig. 1.13 Gaussian particle size distributions. Curve I represents a more uniform size distribution than does Curve II.

Only one additional stipulation needs to be made before adapting the results that follow from Eq. (5.24) to addition polymers. The mode of termination must be specified to occur by disproportionation to use the results of Sec. 5.4 in this chapter, since termination by combination obviously changes the particle size distribution. We shall return to the case of termination by combination presently. 

A fundamental requirement in powder processing is characterization of the as-received powders (10—12). Many powder suppHers provide information on tap and pour densities, particle size distributions, specific surface areas, and chemical analyses. Characterization data provided by suppHers should be checked and further augmented where possible with in-house characterization. Uniaxial characterization compaction behavior, in particular, is easily measured and provides data on the nature of the agglomerates in a powder (13,14). 

The term essentially a drag coefficient for the dust cake particles, should be a function of the median particle size and particle size distribution, the particle shape, and the packing density. Experimental data are the only reflable source for predicting cake resistance to flow. Bag filters are often selected for some desired maximum pressure drop (500—1750 Pa = 3.75-13 mm Hg) and the cleaning interval is then set to limit pressure drop to a chosen maximum value. 

Scmbbers make use of a combination of the particulate coUection mechanisms Hsted in Table 5. It is difficult to classify scmbbers predominantly by any one mechanism but for some systems, inertial impaction and direct interception predominate. Semrau (153,262,268) proposed a contacting power principle for correlation of dust-scmbber efficiency the efficiency of coUection is proportional to power expended and more energy is required to capture finer particles. This principle is appHcable only when inertial impaction and direct interception are the mechanisms employed. Eurthermore, the correlation is not general because different parameters are obtained for differing emissions coUected by different devices. However, in many wet scmbber situations for constant particle-size distribution, Semrau s power law principle, roughly appHes ... 

The constants a and y depend on the physical and chemical properties of the system, the scmbbing device, and the particle-size distribution in the entering gas stream. 

The RDX particle size distribution must be carefully controlled to produce castable slurries of RDX and TNT having acceptable viscosity. Several classes of RDX are produced to satisfy requirements for the various pressed and cast RDX-based compositions. A continuous process for medium-scale production of RDX has been developed by Biazzi based on the Woolwich process (79,151—154). 

For large amounts of fillers, the maximum theoretical loading with known filler particle size distributions can be estimated. This method (8) assumes efficient packing, ie, the voids between particles are occupied by smaller particles and the voids between the smaller particles are occupied by stiH smaller particles. Thus a very wide filler psd results in a minimum void volume or maximum packing. To get from maximum packing to maximum loading, it is only necessary to express the maximum loading in terms of the minimum amount of binder that fills the interstitial voids and becomes adsorbed on the surface of the filler. 

Fluidized-bed design procedures requite an understanding of particle properties. The most important properties for fluidization are particle size distribution, particle density, and sphericity. 

Particle Size. The soHds in a fluidized bed are never identical in size and foUow a particle size distribution. An average particle diameter, is generally used for design. It is necessary to give relatively more emphasis to the low end of the particle size distribution (fines), which is done by using the surface mean diameter, to calculate an average particle size ... 

Particle size distribution is usually plotted on a log-probabiHty scale, which allows for quick evaluation of statistical parameters. Many naturally occurring and synthetic powders foUow a normal distribution, which gives a straight line when the log of the diameter is plotted against the percent occurrence. However, bimodal or other nonnormal distributions are also encountered in practice. 

Bubble size control is achieved by controlling particle size distribution or by increasing gas velocity. The data as to whether internal baffles also lower bubble size are contradictory. (Internals are commonly used in fluidized beds for heat exchange, control of soflds hackmixing, and other purposes.)... 

The heat-transfer coefficient depends on particle size distribution, bed voidage, tube size, etc. Thus a universal correlation to predict heat-transfer coefficients is not available. However, the correlation of Andeen and Ghcksman (22) is adequate for approximate predictions ... 

Figure 18 is an entrainment or gas-carryiag capacity chart (25). The operating conditions and particle properties determine the vertical axis the entrainment is read off the dimensionless horizontal axis. For entrainment purposes, the particle density effect is considered through the ratio of the particle density to the density of water. When the entrainable particle-size distribution is smaller than the particle-size distribution of the bed, the entrainment is reduced by the fraction entrainable, ie, the calculated entrainment rate from Figure 18 is multipfled by the weight fraction entrainable. 

Gopolymerization. The chemistry of the resin matrix, the type and degree of porosity, the particle size, and the particle size distribution are estabhshed in the copolymerization step. Formulations and operating procedures must be strictiy foHowed. Reaction vessels must be weH designed. Mistakes made during copolymerization are rarely corrected during functionalization. 

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