Log-normal law

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... 

Region 2 It was discovered that region 2 sieving followed a log normal law (Ref 19) ... 

Both the A1 and AP samples appeared to follow very closely the log normal law of particle distribution as evidenced by the straight line plots of the size distribution data obtained microscopically and Micromerographically, This... 

Particle size, like other variables in nature, tends to follow well-defined mathematical laws in its distribution. This is not only of theoretical interest since data manipulation is made much easier if the distribution can be described by a mathematical law. Experimental data tends to follow the Normal law or Gaussian frequency distribution in many areas of statistics and statistical physics. However, the log-normal law is more frequently found with particulate systems. These laws suffer the disadvantage that they do not permit a maximum or minimum size and so, whilst fitting real distributions in the middle of the distribution, fail at each of the tails. 

According to the normal law, differences of equal amounts in excess or deficit from a mean value are equally likely. With the log-normal law, it is ratios of equal amounts that are equally likely. In order to maintain a symmetrical bell-shaped curve, it is therefore necessary to plot the relative frequency against size in a geometric progression. 

For each analytical property, the proportion of individual pellets lying outside specification may be tabulated. This level should be minimal if the capability of the process and of the lyophilizer is consistent with the specification limits. If the distribution of pellet contents obeys the normal or log-normal law, it is further possible to predict for the whole batch the proportion of samples that can be expected to be outside the specifications. Alternately, process capability indices can be calculated [15. ... 

In Eq. (1.9), the adhesion number Ypid) allows us to weight the final distribution of particles with respect to the initial i.e., Tf( ) is the number of particles of various sizes that are removed under the influence of external forces. In order to calculate the adhesion number, in addition to 7f( ) we need to know the size distribution of the adherent poly disperse particles. This distribution most frequently follows a log-normal law [16]... 

Thus we can say that the relationship between ap and log Fad is characterized as a stationary process. This means that its mathematical expectation is close to the line OLp=a- b og Fad. This situation is in agreement with the experimental data shown in Fig. 1.4. The function ap =/(Fad) is continuous, and the distribution of particles with respect to adhesive force follows a log-normal law this will be used subsequently in characterizing adhesive interaction. 

Quantitative Changes in Force of Adhesion. On the basis of information obtained on the modification of glass surfaces, we have carried out certain studies on the distribution of particles with respect to force of adhesion in relation to the wetting angle for the original substrate [21]. The results of these studies, which are shown in Fig. II. 5, indicate that the distribution of monodisperse spherical particles with respect to the force of adhesion to modified glass surfaces follows a log-normal law, the same as the distribution of particles with respect to force of adhesion to ordinary unmodified surfaces (see Section 3, p. 13). Values are plotted on the horizontal axis of Fig. II.5 for the force of... 

The dependence of the adhesion number on the maximum double-layer charge density, in the case of adhesion of spherical glass particles to a painted steel surface, follows a log-normal law, and the standard deviation of this function o is the variable quantity [151-153]. 

In Fig. V. 13 we show integral curves of adhesion for spherical glass particles with a diameter of 20-40 [xm and for irregularly shaped particles of equivalent size. The distribution of irregularly shaped particles with respect to adhesive force, the same as the spherical particles, follows a log-normal law. 

Thus we see that the distribution of irregularly shaped particles follows a log-normal law. Those definitions and calculational formulas (see p. 141) that were proposed for use in finding the average force of adhesion of spherical particles can also be used for irregular particles for which the dimensions have been reduced to a single equivalent dimension. 

Features of Adhesion of Cylindrical Particles and Other Regularly Shaped Particles. As in air, particle adhesion in liquid media has a number of special features when the particles are not spherical. First, let us consider the adhesion of particles to a glass surface in an aqueous medium when the particles are cylinders with a diameter of 40 fim and a length within the range of 100-600 jLim. The distribution of adherent particles with respect to adhesive force in a liquid medium, the same as in air (see Fig. V.l 1), follows a log-normal law [194]. 

Thus, the adhesive-force distribution of cylindrical particles in a liquid medium follows a log-normal law. This opens up the possibility of determining the median and average forces of adhesion. The average force of adhesion of cylindrical particles in an aqueous medium is less than in the air because of the disjoining pressure of the thin layer of liquid. 

The adhesive-force distribution of adherent particles with irregular and spherical shapes is shown in Fig. VI. 16. With respect to equivalent diameter, the adhesive-force distribution of irregular particles also follows a log-normal law. The adhesion of irregular particles in the aqueous medium is greater than that of the equivalent spherical particles. This higher level of adhesion for the irregular particles is observed over the entire range of values of ap. 

Thus we see that, in a liquid medium, the adhesive-force distribution for irregularly shaped particles also follows a log-normal law. This enables us to determine the median force of adhesion and to calculate the average force of adhesion. The values found for the median force of adhesion of irregularly shaped particles are generally greater than those for the equivalent spherical particles. In a liquid medium, in all cases, for particles of different shapes, we find that the adhesive force varies directly with the particle size. 

The relationships found for the adhesion of particles to surfaces in air and in hquid media (see Chapters IV-Vl) are valid in the case of particle adhesion to painted surfaces. The distribution of particles of different shapes adhering to paint and varnish coatings likewise follows a log-normal law [194]. 

The adhesive-force distribution of adherent particles on oil-contaminated surfaces, the same as on oil-free surfaces, follows a log-normal law. This enables us to determine the median and average forces of adhesion (see Section 3) for particles of different sizes adhering to an oil layer that has a density of 0.1 mg/cm. ... 

Since there is a distribution of adherent particles with respect to adhesive force, the detachment velocity will depend on this distribution and on the sizes of the adherent particles. It has been shown experimentally that in the detachment of identical particles, the velocity required for detachment will vary. In Fig. X.3 we show as an example the fractional distribution of particles removed, as characterized by adhesion number, in relation to detaching velocity. A probability-logarithmic scale has been used for these plots. Similar distributions have been obtained for other particle-surface systems [277]. On a probability-logarithmic scale, the distribution of particles removed as a function of detaching velocity is approximated by a straight line. This means that the distribution of detaching velocities, as the distribution of adhesive forces (see Section 3), follows a log-normal law. 

We can see now that the detachment of adherent particles by an air stream can be characterized by the velocity of detachment. This velocity depends on the adhesive force, the particle size, and the properties of the contiguous bodies. The distribution of the detached particles with respect to adhesion number, in relation to the velocity of detachment, follows a log-normal law. If we know the parameters of this distribution, we can find the median and average velocities of detachment for the adherent particles the average velocity gives an unambiguous quantitative characterization of the effect of the air stream on the dusty surface across which it is blowing. 

As descried by Herdan (1960), if a powder is obtained by comminution processes, such as milling, grinding, or crushing, its distribuhon appears to be governed very often by the log-normal law. Furthermore, the log-normal function is the most useful one among the different types of functions (Beddow and Meloy, 1980). It can be given in this form ... 

Graph papers are available on which the integral function F(x) is plotted against In x and distributions that follow the log-normal law give straight lines. The evaluation of the two necessary parameters from the plots can be done in two somewhat different ways. 

Any suitable feed sohds material may be used for the testing but it has to be one with a mono-modal particle size distribution that approximately follows the log-normal law. This requirement is not too hmiting because the test material is a matter of choice. 

On the basis of the available experimental evidence it can be concluded that for a given cyclone design and low feed solids concentrations (say below 1% by volume) the shape of the reduced grade-efficiency curve is reasonably constant. At higher solids concentrations, it becomes dependent on the feed material and experimental measurement may be necessary. Alternatively, with smaller diameter cyclones, the log-normal law with a measured geometric standard deviation may also be used. Knowledge of the cut size is of course necessary in any of these cases to obtain the fiiU curve. 

Log-normal law of crystalline particles size distribution is the basis of Selivanov-Smyslov technique. This technique can be used in the investigation of nanomaterials with a spherical shape and a unimodal distribution and with the size of structural components from 5 to 150 nm [2]. 

The feed solids in a slurry to a hydrocyclone obey the following log-normal law ... 

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