Size distribution function normal distributions

The size distribution function f(D) pertaining to the event as a whole has variously been taken as a log-normal or as a power law distri-... 

Packer and Rees [3] extended the work of Tanner and Stejskal by the development of a theoretical model using a log-normal size distribution function. Measurements made on two water-in-oil emulsions are used to obtain the self-diffusion coefficient, D, of the water in the droplets as well as the parameters a and D0 0. Since then, NMR has been widely used for studying the conformation and dynamics of molecules in a variety of systems, but NMR studies on emulsions are sparse. In first instance pulsed field gradient NMR was used to measure sdf-diffusion coefficients of water in plant cells (e.g. ref. [10]). In 1983 Callaghan... 

In the applications of gas-solid flows, there are three typical distributions in particle size, namely, Gaussian distribution or normal distribution, log-normal distribution, and Rosin-Rammler distribution. These three size distribution functions are mostly used in the curve fitting of experimental data. 

The shape of the size distribution function for aerosol particles is often broad enough that distinct parts of the function make dominant contributions to various moments. This concept is useful for certain kinds of practical approximations. In the case of atomospheric aerosols the number distribution is heavily influenced by the radius range of 0.005-0.1 /xm, but the surface area and volume fraction, respectively, are dominated by the range 0.1-1.0 fxm and larger. The shape of the size distribution is often fit to a logarithmic-normal form. Other common forms are exponential or power law decrease with increasing size. 

The normalization of the size distribution function P ( g) finally yields... 

Fig. 5 Pore size distribution function of microporous membranes, parameterization with logarithmic normal distribution, for the parameter sets specified in Table 4. (a) Differential psd,...

Examples of size distribution functions are shown ill Figs. 1.4 and 1.5. Figure 1.4 shows number distributions of commercially produced silica particles in terms of the fraction of particles in the,size range around dp, dN/N d dp) = na(,dp)fNxs where is the total particle concentration. The total particle surface area corresponding to each size distribution is shown. Commercial silica manufactured by the oxidation of SiCU is used as a filler (additive) in rubber. Both coordinate axes in Fig. 1.4 are linear, and the area under each curve should be normalized to unity. A bimodal volume distribution with a minimum near a particle size of 1 is shown in Fig. 1.5. Distributions of this type are often observed for atmospheric aerosols (Chapter 13) the volume of aerosol material per unit volume of gas above and below a micron is about the same as shown by the area under the curve. Bimodal distributions are also often observed in aerosols from industrial sources as discus.sed below. 

The growth law for a polydisperse aerosol can be determined by measuring the change in the size distribution function with lime. In experiments by Heksler and Friedlander (1977), small quantities of organic vapors that served as aerosol precursors were added to a sample of the normal atmospheric aerosol contained in an 80-m bag exposed to solar radiation. The bag was made of a polymer film almost transparent to. solar radiation in the UV range and relatively unreactive with ozone and other species. Chemical reaction led to the formation... 

We can define a normalized size distribution function tisiDp) by ri/ /(Dp) = ni (Dp)/N, such that... 

The units of Jis Dp) are /xm . The normalized size distribution function ns Dp) can also be viewed as the probability that a randomly selected particle has a diameter in the range Dp, Dp -E dDp) it is therefore equivalent to the normalized probability density of particle size. 

In equation (21-6) for the void fraction Sp, the pore-size distribution function is given by /(r), and fir) dr represents the fraction of the total volume of an isolated catalytic peUet with pore radii between r and r +dr. This is not a normalized distribution function because... 

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 ... 

From such measurements, surface areas (normalized cumulative and relative), pore radii (choice of three measuring units), pore volumes (raw, normalized, cumulative and relative) and pore-size distribution functions of samples can calculated. Figure 8 presents the graphs of mercury-penetrated volume versus pressure in pores of Na- and La-montmorillonite samples. Figure 9 shows pore-size distribution functions from porosimetry data. 

If the particle size distribution is normal or log normal, then the data can be linearized by plotting the particle frequency as a function of particle rize on arithmetic or logarithmic probability graph p r respectively. The 50% value of sudi plots yields the geometric median diameter and the geometric standard deviation is the ratio of the 84.1% m the 50% values. 

Figure 53 Typical size distributions for normal (a) and malignant (b) lymphocytes. The results of fitting by Gauss distribution function are shown by the solid line D is the mean diameter of cells. (From Ref. 72. With permission from Elsevier Science B.V.)...

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. 

Fig. 1. Time-independent normalized size-distribution function given by Lifshitz and Slezov(i2). Here r is the particle radius, f is the mean particle radius.

Lifshitz and Slezov on the other hand showed that in a dispersion in which grain growth was occurring according to Eq. (4) a time-independent normalized size distribution function would be approached in which the radius of the largest particles would be only 1.5 r (See Fig. 1.) From this steady-state distribution function, however, they derived an equation that predicts growth rates of the same order as Greenwood s equation, viz.. 

Another consequence of LSW theory is the prediction that the size distribution function g( ) for the normalized droplet radius u = r/rc adopts a time-independent form given by ... 

Most frequently, an aerosol is characterized by its particle size distribution. Usually this distribution is reasonably well approximated by a log-normal frequency function (Fig. 4A). If the distribution is based on the logarithm of the particle size, the skewed log-normal distribution is transferred into the bell-shaped, gaussian error curve (see Fig. 4B). Consequently, two parameters are required to describe the particle size distribution of an aerosol the median particle diameter (MD), and an index of dispersion, the geometric standard deviation (Og). The MD of the log-normal frequency distribution is equivalent to the logarithmic mean and represents the 50% size cut of the distribution. The geometric standard deviation is derived from the cumulative distribution (see Fig. 4C) by... 

Before we go back to Eq. (14.30), we shall evaluate the mass distribution function for the particles whose size distribution is of the form (14.33), i.e., normal probability size distribution. 

Here is the embryo diffusivity in the space of sizes a, while the factor M is the normalizing constant for the distribution function /(a,r) ... 

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