Cumulative undersize distribution

Beside the average particle size, the size distribution is also an important parameter of the product. Comparing the obtained cumulative undersize distribution of the precipitates it has been found that the ratio of the smallest particles (below 10 im) in the distribution was more than 50% when d E 10 im. In the other cases (d > 10 fxm) the ratio was less than 50% and showed decreasing tendency with increasing size. The Malvern 2600 size-analyzer calculates... 

Schematic drawing of a separator showing the flow rates Q, the cell concentrations X, and the cumulative undersize distributions y.

Table 8.12 Conversion of cumulative undersize distribution by surface into a cumulative undersize distribution by mass...

Figure 2.9. The cumulative undersize distribution of fineparticle size is an important way of displaying size distribution data. Shown above, plotted on log-log scales, are the size distributions of the fragments produced when two different amorphous materials were shattered by impact after being cooled to low temperatures [18]. From the perspective of chaos theory and applied fractal geometry explained in more detail in a later chapter, the slope of this type of data line is described as a fractal dimension in data space.

It follows from this equation that the cumulative undersize distribution function of a fineparticle system can be calculated by measuring the concentration within a setding, initially homogeneous, suspension at a series of specified times and/or depths and by plotting against d. ... 

In addition to density distributions, a very widely used method of reporting a particle size distribution is through the use of its cumulative undersize distribution F x), defined as the fraction of particles with a diameter less than X. F x) is related to the density function /(x) by ... 

The simple approximation is to assume the cyclone to have a sharp cut at X50, i.e. that all material below X50 is lost and all material above is collected. If the cumulative undersize distribution of the feed is Ff x), then (see Fig. 3.2.2) ... 

Let us say that we have measurements of the cumulative undersize distributions for the feed and the lost fractions. The overall fractional efficiency has been measured to be rj=0.SS. We wish to calculate the grade-efficiency curve. The cumulative size distributions are ... 

Figure 1.12, Cumulative and differential distributions. Dotted line Fractional (or differential) volume distribution with the right ordinate. A Cumulative-undersize distribution with the left ordinate. B Cumulative-oversize distribution with the left ordinate.

In histogram form, the cumulative distribution can be obtained by summing all the columns from the smallest x value up to the point i from the fractional (or density, or differential) distribution. This representation of the cumulative term is also called the cumulative-undersize distribution, in contrast to the cumulative-oversize distribution, wherein the summation is performed from Xmax to Xi. For percentage distributions, there exists the relation Q(x,)cumuiative-undersize = 1 Q(Xi)cumuiative-oversize- Because of the integration relation between the fractional distribution and the cumulative distribution, the absolute values of the maximum and minimum slopes in the cumulative distribution correspond to the... 

Most modem instrumental particle size analysers readily present data in a variety of forms, such as frequency, cumulative undersize or oversize, and interconvert between number, mass and other distributions. Acquisition of data in a suitable form is therefore not usually a problem. 

Cell size distribution of HeLa cells grown in serum-free culture medium (Luebberstedt, 2000), presented as (A) frequency and (B) cumulative undersize (y) and oversize (z) distributions. 

Table 8.11 Conversion of attenuation of homogeneous centrifuge into a cumulative surface undersize distribution assuming constant extinction and shape coefficients...

Table 5 shows the sieve data of two batches of materials prepared by milling. These materials with a wide distribution exhibit a linear plot when plotted on log probability paper. When the percent cumulative undersize is plotted on the log probability paper. Batch 0903 shows a linearized log normal distribution. Hence, the value for the geometric mean (/x = 115 pm) can be obtained directly from the graph in Figure 14. The geometric standard deviation can also be easily obtained from the graph by the following equation ... 

Incremental Methods. Based on the results of motion of a particle in a gravitational or drag field reported earlier, it can be shown that the solids concentration at depth h can be related to the cumulative undersize mass distribution as (8)... 

By measuring absorbance changes as a function of time, the cumulative undersize particle distribution can be obtained by mathematical manipulation. However, in converting Equation 34-18 to the size distribution, it is necessary to know how the absorption coefficient K varies with d,. Alternatively, an external standard can be used for calibration. If correction is not made for variations in K. results are valid only for comparison purposes. 

The graphical representation of a particle size distribution is usually plotted in a cumulative form. In a typical cumulative plot, points are entered showing the amount of particulate material contributed by particles below or above a specified size. Hence, the curve presents a continuously rising or decreasing character. These oversize and undersize distributions, as illustrated in Figure 2.26, are simply related by... 

Feed line specified by a flow rate, solids concentration and particle size distribution of the solids (this is in turn specified either by a table of cumulative % undersize or by three parameters of a bounded log-normal distribution)... 

A random mixture consists of two components A and B in proportions 60 and 40% by mass, respectively. The particles are spherical and A and B have particle densities 500 and 700 kg/m, respectively. The cumulative undersize mass distributions of the two components are shown in Table llW.l. 

Figure 3 Comparison Between the BI-XDC, Sedigraph 5100 and BCR 66 Standard Data Cumulative Undersize Mass Distribution...

The X value at which w% of particles has smaller x values. It is most commonly used in diameter distribution presentation, in which d % means the diameter at which w% of particles is smaller (cumulative undersize by volume.)... 

The normal distribution is commonly encountered in the cumulative form, that is, as the fraction of particles larger (oversized) or smaller (undersized) than a particular tt value. Since the total area under the normal curve equals unity, the area under one tail of the curve from t, to oo gives the fraction of the population having t values greater than the integration limit t . 

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. 

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