Cumulative number size distributions

Figure 3. Cumulative number size distributions of samples from fructose crystallization for Run 5.

It is out of the scope of this book to describe the AP mechanics, i. e. microphysics and dynamics (Friedlander 1977, Hinds 1882, Kouimtzis and Samara 1995, Harrison and van Grieken 1998, Meszaros 1999, Spumy 1999, 2000, Baron and Willeke 2001). Here, we only summarize the important topic of atmospheric aerosol size distribution (Jaenicke 1999). Fig. 4.15 shows that the size range covers several orders of magnitude. Therefore, the common logarithm of the radius is useful to describe the different distribution functions dN r)ld gr = f( gr) or dN r)ldr =/(Igr)/2.302 r. N r) cumulative number size distribution (or the integral of radii) having dimension cm , r radius of particle ... 

Fig. 7-8. Influence on cloud nuclei formation of the mass fraction e (water-soluble material/particle dry mass). Left Critical supersaturation of aerosol particles as a function of particle dry radius. Right Cloud nuclei spectra calculated for e = 0.1 and 1 on the basis of two size distributions each for continental and maritime aerosols (solid and dashed curves, respectively). [Adapted from Junge and McLaren (1971).] The curves for the maritime cloud nuclei spectra are displaced downward from the original data to normalize the total number density to 300 cm-3 instead of 600 cm-3 used originally. The curves for e = 1 give qualitatively the cumulative aerosol size distributions starting from larger toward smaller particles (sk = 10 4 corresponds to r0 0.26 p.m, sk = 3 x 10 3 to rs 0.025 Atn). Similar results were subsequently obtained by Fitzgerald (1973, 1974). The hatched areas indicate the ranges of cloud nuclei concentrations observed in cloud diffusion chambers with material sampled mainly by aircraft [see the summary of data by Junge and McLaren (1971)] the bar represents the maximum number density of cloud nuclei observed by Twomey (1963) in Australia.

One can distinguish between the differential and integral (or cumulative) particle size distribution functions. These two types of functions are related to each other by the differentiation and integration operations, respectively. The adequate description of distribution function must include two parameters the object of the distribution (i.e. what is distributed), and the parameter with respect to which the distribution is done. The first parameter may be represented by the number of particles, their net weight or volume10, their net surface area or contour lengthen some rear cases). The second parameter typically characterizes particle size. It can be represented as a particle radius, volume, weight, or, rarely, surface area. Consequently, the differential function of the particle number distribution with respect to their... 

Figure 16 Simulated granulation with increasing amounts of binder. Cumulative particle size distribution vs. relative agglomerate diameter. Stokes number St = 0.1 and capillary number Ca = 1.0.

Example 6.4.1 For the MSMPR crystal number density function (6.4.4), determine the expressions for Nt,F rp),fp, Ar and Mp. Develop expressions for the cumulative crystal size distribution fraction and the cumulative surface area distribution fraction. 

A number based cumulative crystal size distribution fraction may be defined by... 

At the conclusion of the calculation, a fragment size distribution as well as fragment number is provided. A cumulative number distribution is shown in Fig. 8.22 and compared with aluminum ring data acquired at = lO s (Grady and Benson, 1983). With the assumed fracture site nucleation law, the calculated distribution appears to agree reasonably well with the data. The calculation better predicts the tails of the distribution which have trends which deviate from strict exponential behavior as was noted in the previous section. 

The moment equations of the size distribution should be used to characterize bubble populations by evaluating such quantities as cumulative number density, cumulative interfacial area, cumulative volume, interrelationships among the various mean sizes of the population, and the effects of size distribution on the various transfer fluxes involved. If one now assumes that the particle-size distribution depends on only one internal coordinate a, the typical size of a population of spherical particles, the analytical solution is considerably simplified. One can define the th moment // of the particle-size distribution by... 

As AD is made smaller, a histogram becomes a frequency distribution curve (Fig. 4.1) that may be used to characterize droplet size distribution if samples are sufficiently large. In addition to the frequency plot, a cumulative distribution plot has also been used to represent droplet size distribution. In this graphical representation (Fig. 4.2), a percentage of the total number, total surface area, total volume, or total mass of droplets below a given size is plotted vs. droplet size. Therefore, it is essentially a plot of the integral of the frequency curve. 

Particles come in all shapes and sizes and in large numbers. Data are presented graphically using histograms, fractional plots, or cumulative plots. These graphs are primarily useful as pictures of the size distribution of the mixture. Table 15.4 gives a typical screen analysis for a 900-g sample. The measured experimental data are the mesh sizes, and the masses of the particles on each of the sieves are the masses of the residuals or fines. The other quantities are calculated. 

Figure 6 shows the size distributions for the samples taken from one of the runs, presented as the cumulative number oversize per ml of slurry. From the lateral shift of the size distributions, the growth rate can be determined. Figure 7 shows values of growth rate, G, supersaturation, s, and crystal content determined during the run. As a material balance check, the crystal contents were evaluated from direct measurements, from solution analyses and from the moments of the size distribution. The agreement was satisfactory. No evidence of size dependent growth or size dispersion was observed. 

Marsh (1988), Cashman and Marsh (1988), and Cashman and Ferry (1988) investigated the application of crystal size distribution (CSD) theory (Randolph and Larson, 1971) to extract crystal growth rate and nucleation density. The following summary is based on the work of Marsh (1988). In the CSD method, the crystal population density, n(L), is defined as the number of crystals of a given size L per unit volume of rock. The cumulative distribution function N(L) is defined as... 

A conveniendy expressed coordinate for plotting filtration performance is the drainage number, d(G )1//2 /V, where d is the mean particle diameter in micrometers, V is the kinematic viscosity of the mother liquor in m2/s (Stokes x 10-4) at the drainage temperature, and G is (02r/g r is the largest screen radius in a conical bowl. Because the final moisture content of a cake is closely related to the finest 10—15% fraction of the solids and is almost independent of the coarser material, it is suggested that d be used at the 15% cumulative weight level of the particle size distribution instead of the usual 50% point. 

FIGURE 8.4 Cumulative size distribution of gelatin (lime-cured type B, bloom number 225) microparticles prepared at high pH obtained using a Coulter Counter and plotted as a log-probit function. (From Lou, Y. and Groves, M.J. (1995). J. Pharm. Pharmacol., 47, 97-102. With permission.)... 

The principal quantities related by these equations are tpm, d4>m/dx, L, Lpl, t, n°, and B°. Fixing a certain number of these will fix the remaining one. Size distribution data from a CSTC are analyzed in Example 16.6. In Example 16.7, the values of the predominant length Lpr and the linear growth rate G are fixed. From these values, the residence time and the cumulative and differential mass distributions are found. The effect of some variation in residence time also is found. The values of n° and B° were found, but they are ends in themselves. Another kind of condition is analyzed in Example 16.4. 

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. 

The normal probability function table given in the appendix d this book can also be used for values of the log-normal distribution function, f, and the log-normal cumulative distribution function, F. In these tables Z = [ln(d/cy/(In o- )] is used. A plot of the cumulative log-normal distribution is linear on log-normal probability paper, like that shown in Figure 2.11. A size distribution that fits the log-normal distribution equation can be represented by two numbers, the geometric mean size, dg, and the geometric standard deviation,. The geometric mean size is the size at 50% of the distribution, d. The geometric standard deviation is easily obtained finm the following ratios ... 

The particle size distribution as a function of the number tanks, N, is given in Figure 6.26(a) in terms of the cumulative wei t distribution. From this figure, it is evident that the size distribution becomes narrower as the number of tanks in the cascade, N, increases. 

Figures 7 and 8 show typical particle size distributions for vinyl acetate emulsions produced in a single CSTR. A large number of particles,are quite small with 80 to 90% being less than 500 A in diameter. The large particles, though fewer in number, account for most of the polymer mass as shown by the cumulative volume distributions. Data are also presented on Figures 7 and 8 for the number of particles based on diameter measurements (N ), the average number of free radicals per particle, and the steady state conversion.

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