Size distribution function averaging

Particle Size Distribution Functions Averaged Over Measurements Made in Pasadena, August to September 1969 (Whitby, et al., 1972)... 

Fig. 11.6. Grain size distribution function / (/.) defined by TEM image of Fcx(Si()2)i thin films. Solid line is the Gaussian fit / (/.) oc exp[—(L — (L))2 /2a ] with the average value < ) = 3.8nm and the dispersion 07 = 1.9.

The size distribution of micellar aggregates Ng/F is plotted against the aggregation number, g, for an amphiphile with an octyl hydrocarbon tail and for a 2 X 1G4 cal A2t molsb (Figure 1). Equation 19 leads to wjt = 3.88. For < crit, the size distribution is a monotonic decreasing function of g. At = CT t, the size distribution function has ah inflection point. At > mt, the size distribution function has two extrema. It can be seen that if increases both the number and the average size of the micellar aggregates increase. 

Related Calculations. This procedure can be used to calculate average sizes, moments, surface area, and mass of solids per volume of slurry for any known particle size distribution. The method can also be used for dry-solids distributions, say, from grinding operations. See Example 10.7 for an example of a situation in which the size distribution is based on an experimental sample rather than on a known size-distribution function. 

The first three columns of Table 10.5 show sieve data for a 100-cc slurry sample containing 21.0 g of solids taken from a 20,000-gal (75-m3) mixed suspension-mixed product removal crystallizer (MSMPR) producing cubic ammonium sulfate crystals. Solids density is 1.77 g/cm3, and the density of the clear liquor leaving the crystallizer is 1.18 g/cm3. The hot feed flows to the crystallizer at 374,000 lb/h (47 kg/s). Calculate the residence time r, the crystal size distribution function n, the growth rate G, the nucleation density n°, the nucleation birth rate B°, and the area-weighted average crystal size L3 2 for the product crystals. 

Calculate the crystal size distribution function n. The crystal size distribution for the ith sieve tray is n, = 1012M3AVTj/(L3ALj), where AIT, is the weight fraction retained on the ith screen, /., is the average screen size of material retained on the ith screen (see Example 10.7, step 2), and A/., is the difference in particle sizes on the ith screen (see Example 10.7, step 3). For instance, for the Tyler mesh 100 screen, nio = 1012(0.119)(0.040)/(1613 x 28) = 40.7 crystals per cubic centimeter per micron. Table 10.5 shows the results for each sieve screen. 

The formulae given so far are correct only under the assumption that the foam is monodisperse. In order to describe the expansion ratio of a real polydisperse foam it is necessary to use the average values of a average linear ab, average surface as and average volume a. The relation between them depends on the type of bubble size distribution function... 

Furthermore, the explicit-water simulations do include the CDS terms to the extent that dispersion and hydrogen bonding are well represented by the force field. Finally, by virtue of the solvent being explicitly part of the system, it is possible to derive many useful non-entropy-based properties "" (radial distribution functions, average numbers of hydrogen bonds, size and stability of the first solvation shell, time-dependent correlation functions, etc.). Since many of these properties are experimentally observable, it is often possible to identify and correct at least some deficiencies in the simulation. Simulation is thus an extremely powerful tool for studying solvation, especially when focused on the response of the solvent to the solute. 

Table 5.1 Comparison of crystallite sizes calculated from the variance (s. ) and Scherrer (So) formula, with the area-weighted and volume-weighted < Ml > average sizes corresponding to different cases of simulated size distribution functions P n).

In this section we discu.ss briefly several size distribution functions that can be used to fit experimental data for aerosols or to estimate average particle size or the effects of aerosols on light scattering. The examples discussed are nornud.power-law, and self-sirniUir distributions. Selecting a distribution function depends on the specific application. In some cases, fragmentary information may be available on certain moments or on sections of the... 

Forpolydisperse aerosols, the simple expression (5.36) for the autocorrelation function must be averaged over the particle size distribution function. In the Rayleigh scattering range... 

Particle collision and coagulation lead to a reduction in the total number of particles and an increase in the average size. An expression for the time rate of change of the particle size distribution function can be derived as follows. 

As a result of time averaging, several new terms appear in the GDE. The fourth term on the left-hand side, the tiiictuating growth term, depends on the correlation between the fluctuating size distribution function n and the local concentrations of the gaseous species converted to aerosol. It result.s in a tendency for spread to occur in the particle size range—-a turbulent diffusion through v space (Levin and Sedunov, 1968). 

Hd = time-averaged size distribution function V — mean wind field K = eddy diffusion coefficient Cj = terminal settling velocity... 

The size distribution function dp) % AN/ dp for the Pasadena aerosol averaged over the measurements in August and September 1969 is shown in the table. 

The most complete information that can be obtained in dispersion analysis comes from the determination ofparticle size distributionfunction (in some cases one may be interested in obtaining particle shape distribution). Some methods yield only the information on the average particle size, which in some cases may be accompanied by some conditional distribution width. These terms require a more detailed discussion, as different methods may yield different size distribution functions and average sizes for the same disperse system. 

However, the pore-size distribution function /(r) is not normalized, so the zeroth moment must be included in the expression for the average pore radins ... 

Heat of adsorption, phase transitions of adsorbates Weight loss as a function of temperature Enthalpic (exothermic or endothermic) changes upon heating surface area, pore size distribution, and average pore diameter... 

With respect to the fact that coc s co/jo, ( >s(R) = oor (see Eq. 3.95) and substituting (3.97) into (3.96), one can find the integral intensity dependence on particle radius. The observed lineshape can be obtained by averaging of this intensity with particles size distribution function determined by (3.83), i.e. 

To describe the polymerization process we introduce a particle size distribution function / = /(t, v) depending on time t > 0 and a volume variable t > 0. The physical interpretation of / may be given in a heuristic way as follows The differential /(t, v)dv is the average number of particles whose volumes at time t belong to the infinitesimal volume interval (v, v + dv). That is, we make the usual assumption of statistical physics that the particle number has a density - which is justified by the large number of particles. 

In a subsequent analysis, Srolovitz et al. (22) showed that when the grain size is normalized to the average grain size, then the size distribution function becomes time invariant, as also predicted by the mean field theories. A comparison of the distribution function obtained from the computer simulations with the lognormal distribution, and the distribution functions derived by Hillert and that by Louat are shown in Fig. 9.13. As shown in Fig. 9.18, the distribution function can also provide an excellent fit to some experimental data. 

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