Probability, logarithmic

Fig. 1.14 is a logarithmic probability system that shows bubble distribution in a foam produced from 1% solution of mixed sulphanol NP and trisodiumphosphate [10], It is clearly seen that the polydispersity of foam strongly increases with time. 

If the bubble distribution analysis does not take into account a certain fraction, for example R < Rn, then the linear character of the distribution curves in the logarithmic probability system is sharply disturbed close to the point corresponding to radius R the curves acquire a vertical asymptotic character. 

Fig. 1.14. Bubble size distribution in foam (in logarithmic probability co-ordinate system) lime from the...

Particle-Size Analysis Methods for particle-size analysis are shown in Fig. 17-34, and examples of size-analysis methods are given in Table 17-1. More detailed information may be found in Lapple, Chem. Eng, 75(11), 140 (1968) L ple, Particle-Size Analysis, in Encyclopedia of Science and Technology, 5th ed., McGraw-Hill, New York, 1982 Cadle, The Measurement of Airborne Particles, Wiley, New York, 1975 Lowell, Introduction to Powder Surface Area, 2d ed., Wiley, New York, 1993 and Allen, Particle Size Measurement, 4th ed. Chapman and Hall, London, 1990. Particle-size distribution may be presented on either a frequency or a cumulative basis the various methods are discussed in the references just cited. The most common method presents a plot of particle size versus the cumulative weight percent of material larger or smaller than the indicated size, on logarithmic-probability graph paper. 

The use of probability plots is of value when the arithmetic or geometric mean is required, since these values may be read directly from the 50% point on a logarithmic probability plot. By definition, the size corresponding to the 50% point on the probability scale is the geometric mean diameter. The geometric standard deviation is given (for % LTSS) by ... 

The linear normal distribution (bell curve Gaussian normal distribution ) is generally suitable for very narrow particle-size distributions. The standardized, dimensionless shape of the normal distribution produces a straight line on semi-logarithmic probability paper. 

If the results in Fig. 6.2 are plotted on double-logarithmic probability paper. Fig. 

Figure 8, on logarithmic-probability coordinates, shows the behavior of the J function. The concentration histories, as plotted against time on... 

Plot the cumulative distribution given in Table 28.2 on logarithmic-probability paper. Is the plot linear over any range of particle sizes How does the amount of fine material (smaller than 20-mesh) differ from what would be predicted from the size distribution of the coarser material ... 

The logarithmic probability plot has two linear branches one for sizes coarser than 14 mesh, the other for sizes smaller than 2Q mesh. There is much more fine material of any given size in the sample than would be predicted from the distribution of the +14-mesh material. 

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. 

The results of studies on the distribution of particles with respect to adhesive force are presented in the form of integral curves. These curves can also be represented on logarithmic probability coordinates (Fig. 1.3), with values of the... 

The parameters F and a characterize the distribution of particles of different fractions. The standard deviation a, on the logarithmic probability scale used... 

Figure 11 Log-normal distribution plotted on a logarithmic probability graph paper.

However, if we plot a graph of the particle distribution with respect to the forces of adhesion in logarithmic probability coordinates, then this distribution will be normally logarithmic, which will enable us to determine the geometric standard deviation (a), and the median (F) value of the force of adhesion. The parameters a and F give a fuller description of the particle distribution in terms of the forces of adhesion. Using the resultant parameters, we may write... 

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 ... 

The cut point size is the smallest particle size that has to be settled in the decanter. Technically 50% of particles of that size settle and 50% are lost in the centrate above that size the separational efficiency increases and below it vice versa. In consequence, the size distributions in both the cake and the centrate will also exhibit logarithmic probability distributions. 

Plot the Schulz distribution in cumulative form on logarithmic probability paper for xjx = 2. What parameter for the Wesslau model gives the best fit ... 

Compare the cumulative distribution of Problem 6.14 with the Wesslau model on logarithmic probability paper using the product ax as the measure of molecular weight. 

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